My special thanks go to the "less new" PIs, Jean and Sébastien, for open-handedly sharing their treasurable scientific expertise. I would also like to thank Florent Meyniel for his advice and helpful feedbacks.

Thank you to my very lucid mates, Lindsay, Alex, Ben, and Delphine again, for giving me the opportunity to participate in a brilliant project, and for your beautiful smiles and perpetual good vibes.

Merci à mes amis et à mes (littéralement innombrables) colocataires pour tous ces excellents moments à Paris ou en région, pendant lesquels il m'est même arrivé de penser à autre chose que ma thèse. Une spéciale dédicace à Mélanie, pour ces vingt précieuses années d'amitié qui ne nous rajeunissent pas, à Julio, pour sa détermination sans bornes et les conversations toujours passionnantes, et à Thomas, pour les bons plans, les tisanes, les parties de Catane dans un esprit toujours bon enfant, et les aventures passées et à venir.

Enfin, j'adresse un immense merci à ma famille, à mes frères Jérémy et Maxime, et tout particulièrement à mes parents, Pascal et Valérie. Et j'adresse un merci tout aussi immense à Florent. Ce travail n'aurait pu être accompli sans eux. Grâce à leur amour et à leur soutien sans faille, même les jours de gros temps, cette traversée ne s'est jamais faite en solitaire.

In the first study, we had participants anticipate the energetic cost of composite running routes whose later completion could entail more or less valuable monetary outcomes, either framed as obtained gains or avoided losses. We found that a given route was anticipated as more costly when it was paired with a larger monetary stake compared to a smaller one (either in the gain or loss frame). Besides, our behavioural data were most in line with a cognitive scenario according to which individuals' prospective effort judgment is contaminated by the output of a cost-benefit computation.

In the second study, we moved on to the investigation of interferences between value and probability, with a focus on a particular case of probability judgment: confidence, that is, the subjective probability of being correct or successful at a given task. Our participants were trained to perform a motor precision task; next, they underwent testing trials in which they were first presented with information about the upcoming motor challenge: its difficulty (adjusted via the size of the target to be hit), the magnitude of their monetary gain in case of success, and the magnitude of monetary loss in case of failure; then, they were prompted to estimate their prospective chances of success (i.e. to report their confidence); finally, they vi tried to hit the target but did not receive any feedback about their performance. Our main finding was that individuals tend to be more confident when faced with a larger prospective gain, but less confident when faced with a larger prospective loss. This pattern of distortions is evocative of an affect-as-information type of misattribution.

In our third study, we drew inspiration from earlier neuroimaging findings to predict that value-confidence misattributions may occur bidirectionally, and between unrelated items. To test our predictions, we had participants answer more or less difficult general-knowledge quiz questions while hearing more or less pleasant musical extracts, after which we asked them to make a confidence rating about their answer to the quiz question, or a likeability rating about the musical background. Partly in line with our predictions, higher confidence led to higher likeability ratings, whereas the opposite effect was more elusive.

Overall, we managed to isolate, at the behavioural level, various cases of interference between decisional variables in a well-controlled setting, and we gained some insight about their underlying cognitive mechanisms, which seem to differ from one type of interference to the other. viii effectuer correctement une tâche de précision motrice consistant à atteindre une cible en serrant un dynamomètre à main, nos participants étaient amenés, lors des essais de test proprement dits, à prendre connaissance de la difficulté du challenge moteur à venir (ajustée via la taille de la cible à atteindre) ainsi que de la taille du gain (ou de la perte) monétaire en cas de succès (ou d'échec), puis à estimer a priori leurs chances de réussir à atteindre la cible ; enfin, ils tentaient d'atteindre la cible mais ne recevaient aucun feedback quant à leur performance. Les participants se montraient plus confiants lorsque le challenge pouvait conduire à un gain plus important, et inversement moins confiants lorsqu'ils risquaient une perte plus importante. Ces distorsions de la confiance pourraient résulter d'une « misattribution » de l'état affectif induit par les pertes et gains prospectifs.

Dans notre troisième étude, nous nous sommes inspirés de récents résultats en neuroimagerie pour prédire que les misattributions entre jugements de valeur et de confiance pourraient avoir lieu dans les deux sens, y compris lorsque ces jugements portent sur des stimuli ou objets logiquement indépendants. Afin de tester nos prédictions, nous avons soumis des volontaires à des questions de culture générale plus ou moins difficiles, auxquelles ils répondaient tout en entendant des extraits musicaux plus ou moins plaisants, après quoi ils étaient invités à rapporter la confiance qu'ils avaient dans leur réponse à la question de culture générale, ou bien à noter à quel point ils avaient apprécié l'extrait musical. En accord partiel avec nos prédictions, une confiance plus élevée dans la réponse au quiz donnait lieu à une appréciation plus élevée de la musique, mais l'effet inverse s'est avéré moins robuste. Dans l'ensemble, nous sommes parvenus à isoler, au niveau comportemental, divers cas d'interférence entre variables décisionnelles dans un contexte expérimental bien contrôlé, et nous avons approfondi la compréhension des mécanismes cognitifs sous-jacents, lesquels semblent différer d'un type d'interférence à l'autre.

If you are a standard human being, you likely engage several times a day in courses of action which imply spending time, energy, and other resources, in the pursuit of certain outcomes that you think valuable, but that you are seldom guaranteed to obtain.

In this work, we explored how some fundamental cognitive processes that typically precede and orient the selection of a course of action might interfere together and generate inferential and behavioural biases.

The purpose of this introductory chapter is 1) to present these cognitive processes and their relation to observable behaviour, 2) to review current knowledge about their neural implementation, 3) to review the various cases of interferences that have been reported in the literature to this day, 4) to introduce the research questions that we have addressed during this thesis.

We will approach the topic with a prosaic (yet purely fictitious) scenario.

Let us consider Edouard, a former intern at the NCC1 lab. Despite timid first steps, Edouard has done undeniably well for himself in life, and he is now the proud tenant of a one-bedroom Parisian flat with a balcony from which one can almost see the Sacré-Coeur. With just a little more greenery around him, his happiness would be complete. Besides nature, Edouard also loves orange juice, so, as he strolls along the aisles of the garden center, he thinks he might as well grow an organic orange tree on his balcony. But will it survive the drizzly, chilly local weather? Also, how long will he have to wait before the tree bears fruit? At least one month, he assumes. And how much orange juice can he expect it to yield each year? Wouldn't it be more reasonable to grow an apple tree, which might be more robust and productive, although its fruit is less delicious? Besides, Edouard notices that these baby trees are quite pricey, and might need trimming and other forms of care that take time and skill, either of which Edouard is not entirely certain to have in large amounts... 2Of particular significance to us are the several estimation processes that seem to have been occurring in the protagonist's mind while he was pondering whether to try and grow an orange tree: he envisioned the possible outcomes of this endeavour and estimated how valuable, or enjoyable, these outcomes would be; he assessed the likelihood that these more or less valuable outcomes would occur, and in how long, taking into account his gardening skills; he evaluated the different costs entailed by the project: the monetary cost of buying the tree, the temporal and physical cost of watering it, trimming it and repotting it, etc.

We shall see now how these estimation processes have received interest from economists, psychologists, and neuroscientists over the last centuries, and how they have been incorporated into modern theories of action selection.

A. Decisional variables: the cognitive components of action selection

In this section, we will take a historical tour of the main economic and psychological theories of preference-based option or action selection, and in doing so, we will get acquainted with three recurring constructs: outcome value, outcome probability, and action cost.

1) Apples and oranges: the science of choice and value in its early days a. Value & price in nascent economics

If we were to pinpoint -somewhat arbitrarily -a triggering event in the ongoing quest of economists, psychologists and philosophers for an insightful and powerfully predictive theory of human choice, a good candidate would be the formulation of the "diamond-water paradox" by Adam Smith (1723Smith ( -1790) ) 1 . This paradox is in fact pretty unsophisticated: why are diamonds (which are practically useless) so expensive, while drinking water (which is vital) is almost free? Or, as Adam Smith puts it:

"Nothing is more useful than water: but it will purchase scarcely anything; scarcely anything can be had in exchange for it. A diamond, on the contrary, has scarcely any use-value; but a very great quantity of other goods may frequently be had in exchange for it." (Smith, 1776a) b

. The Marginal Revolution

What happened in the middle of the nineteenth century was that three European economists, William Stanley Jevons [START_REF] Jevons | The Theory of Political Economy[END_REF], Carl Menger [START_REF] Menger | Principles of economics[END_REF] and Léon Walras (Walras, 1874), almost simultaneously noticed something peculiar about prices, which sparked an insight about the psychological value of things. Their first step was to realize that the exchange price of a given good, such as a litre of water, depended on the quantity that the transactioners already possessed. Looking at the price curve of water as a function of the volume exchanged, one could see that the price at which each additional (or marginal) litre of water was sold and bought diminished with the quantity initially possessed (or just acquired) by the agents. This observation seemingly solved Smith's paradox, by accounting for the cheapness of a litre of water in societies such as theirs where water was abundant and owned in large quantities by most citizens, and for the expensiveness of diamonds, which were obviously much scarcer. But these economists, now called Marginalists, were not content with that feat, and sought to understand why marginal price declined as a function of the possessed quantity of goods. Indeed, why should it be that, under some circumstances, people are willing to expend large sums for a good, whereas under other circumstances, they would not give a penny for the exact same good? Why, in fact, should price depend on scarcity? They reasoned that an individual's willingness to pay for a good must reflect the degree of pleasure, or contentment, or happiness, all encompassed by the notion of utility, that she expects to derive from owning or consuming it. More specifically, they made the decisive assumption that people are striving to maximize their overall utility, and that their pricing behaviour must thus relate to the marginal utility that they perceive the traded good to carry. Looking at empirical price curves from contemporary markets, and noticing their recurring concave pattern, they concluded that individuals must experience less utility from each subsequent quantity of good that they came to own. Our modern understanding of value-based decision-making owes a lot to the Marginal Revolution: perhaps even more important than their establishing of a diminishing marginal utility, is their bridging of economics with psychology, through explicit incorporation of subjectivity aspects, and in particular hedonic feelings, in the scientific study of human choice.

c. The Ordinal and Revealed Preference revolutions

By the end of the nineteenth century, the Marginal Revolution had drawn a horde of economists in its wake, among whom many sought to shed more light on the determinants of individual utility. They attempted to quantify its relation to other economic constructs such as use-value and scarcity, and to capture it in all its functional details [START_REF] Caplin | Chapter 1-Basic Methods from Neoclassical Economics[END_REF]. Some of them, like Francis Edgeworth [START_REF] Edgeworth | Mathematical Psychics[END_REF], inspired by utilitarian thinkers such as Jeremy Bentham and John Stuart Mill, were hoping to apply their findings to a large-scale maximization of society's welfare [START_REF] Drakopoulos | The Historical Perspective of the Problem of Interpersonal Comparisons of Utility[END_REF].

However, as Vilfredo Pareto soon remarked (and even proved mathematically, [START_REF] Pareto | Manual of Political Economy[END_REF]), these enthusiastic scholars had missed a crucial point: any utility function could only be specified up to a monotonically increasing transformation, such that one could not simply resort to numerical values drawn from utility curves to predict which of two bundles of items an individual would like better, or to arbitrate which of two individuals liked an item better. Besides, he argued, there was no convincing evidence that utility had any grounding in reality, not to mention the assumption that agents were indeed acting to maximize it.

Choices and prices, he claimed, were the only things that economists could empirically observe (in those days, self-introspection as a tool for scientific inquiry was falling out of popularity, and the "hedonimeter" that some longed for in their cardinalist investigation of utility was still pure fantasy [START_REF] Hands | Economics, psychology and the history of consumer choice theory[END_REF]). As such, choice behaviours were the only sound basis on which to erect theories of decisionmaking. Utility could at best summarize a set of preferences already observed in reality; it could be a way to express which, of A or B, an agent prefers, but it could never tell us how much A was preferred to B. In short, utility was only about ordering, not about measuring. Pareto's dismissal of the cardinal conception of utility, instead of which he advocated a purely ordinal approach, initiated the socalled Ordinal Revolution, and found its way into the mind of the next revolutionary thinker in this simplified (his)tory of decision theory.

Despite its name, the Revealed Preference Revolution which came after the Ordinal Revolution did not represent a conceptual break from Pareto's view, but rather followed and built upon it. Yet, it marked a turning point regarding the general flavour of economic science. Until then, theories of consumer choice had displayed mostly descriptive ambitions, and the Revealed Preference Revolution brought about a shift towards a more normative, mathematicised approach. The aforementioned "next revolutionary thinker" was Paul Samuelson. His initiation of the Revealed Preference movement can be traced back to 1938, when he published an article [START_REF] Samuelson | A Note on the Pure Theory of Consumer's Behaviour[END_REF] in which he argued that economists should, once and for all, draw away from the concept of utility, and in particular, from the conception that utilities are causing choices. He advocated a radical focus on choice as the only acceptable way to reveal preference, and the only valid starting point from which to infer theoretical abstractions. In his view, not only was utility intrinsically unable to provide a cardinal metric of preference, as Pareto had pointed out, but the very assumption that agents are seeking to maximize utility, that is, the cornerstone of Marginalism, should be made questionable, or more precisely, falsifiable. In fact, Samuelson came forth with a formal way to challenge the validity of the utility maximization theory. He proposed a very simple, seemingly undemanding axiom about choice, which constituted a necessary condition that any decision-maker should verify for his set of choices to be seen as consistent with utility maximization. This axiom is known as the Weak Axiom of Revealed Preference (WARP), and can be written as follows: if a subject is observed to choose A over B, then he cannot strictly prefer B to A. Practically, in terms of choice, this latter proposition means that the subject cannot choose B over a bundle comprised of A and an infinitesimal amount of money.

Thus, what Samuelson demonstrated is that, if the choice behaviour of an agent was shown, in any situation, to violate the WARP, then it must be concluded that this agent's overall choice behaviour cannot be described as maximizing a single internal utility function. Some years later, Hendrik Houthakker [START_REF] Houthakker | Revealed Preference and the Utility Function[END_REF] went a crucial step further by providing the field with a cleverly augmented version of Samuelson's axiom: for an agent's choice behaviour, satisfying Houthakker's axiom was both a necessary and sufficient condition for being declared consistent with the maximization of some utility function. This extended axiom is known as the Generalized Axiom of Revealed Preference, or GARP. It can be expressed as follows: If a subject is observed to choose A over B, and also observed to choose B over C, then he cannot strictly prefer C to A.

If an agent is observed to consistently satisfy the GARP, then it does not mean that his choices are actually driven by the will to maximize a monotonically increasing utility function, but that they may be described as if it were the case, i.e., that there exists a utility function that could fully account for his choice behaviour.

By virtue of Pareto's demonstration, the exact functional form of that function can never be determined based on choices only, and seeking the functional form is intrinsically wrong. Discerning readers may have recognized this axiom as belonging to the class of transitivity axioms. In fact, transitivity now serves as the very standard defining preference consistency, and is part of the definition of economic rationality. To illustrate this, let us consider someone who strictly prefers apples over oranges, oranges over grapes, and grapes over apples. Because of the circularity (or, intransitivity) of his preferences, that person could get turned into a "money pump" [START_REF] Davidson | Outlines of a Formal Theory of Value, I[END_REF]: if he was given an apple, he would certainly be willing to pay an insignificant fee (say, 1 cent) to exchange his currently owned fruit for a preferred fruit (here, grapes), and find himself, after 3000 such exchanges, with the original apple in hand and 300€ less in his wallet. This is, indeed, the kind of behaviour that would be intuitively qualified as "irrational".

2) Will the tree bear fruit? Incorporating uncertainty in choice problems

a. Expected Value: Blaise Pascal giving gamblers a hand

It is often stated that we are living in an increasingly uncertain world. Yet, more than three centuries before people became aware of the threat posed by global warming and started debating whether they should keep bringing sentient beings into this world (or whether Liverpool FC or Real Madrid would win the next Champion's League), Louis XIV's subjects were already faced with pretty serious matters of worry or hope: will the harvests be good enough for us to make it through winter? Will the doctor (who also happens to be the village's shoemaker) be able to save my wife from pneumonia? Meanwhile, in more frivolous settings, aristocrats were pondering whether they should enter the latest game of chance, in which they might make or lose a small fortune. In 1654, one of these aristocrats, known as Chevalier de Méré, stumbled over one particular gambling problem: if a game, opposing two players across several rounds for a pot to which they had contributed equally, was interrupted before either player had achieved victory, then what would be a fair division of the pot? He turned to his friend -and mathematics genius -Blaise Pascal, asking for his assistance in solving this "problem of points". Pascal agreed to take the case, and started discussing it in his correspondence with Pierre de Fermat (another decent mathematician of the time).

Although it was intuitive that the division of the pot should somehow reflect the current relative score of each player, Pascal and Fermat realized that the focus should not be on how many rounds each player had already won, but how many additional rounds would have been needed by each player to achieve victory, had the game not been interrupted.

Soon enough, they had laid down the foundations for the theory of probability, and Pascal had come up with a new mathematical concept: expected value. Expected value formally quantifies the average value that a random variable (e.g., the number of rounds played until victory) would take, across a large (actually infinite) number of trials. If we imagine that on each trial, the random variable 𝑋 can take one of three possible values, {𝑥 1 , 𝑥 2 , 𝑥 3 }, with respective probabilities {𝑝 1 , 𝑝 2 , 𝑝 3 }, then its expected value is written:

𝐸𝑉[𝑋] = ∑ 𝑝 𝑖 • 𝑥 𝑖 3 𝑖=1
But what definitely earned Pascal a seat in the pantheon of decision theorists was his following step: he proposed a universal solution to any gambling problem involving uncertain outcomes whose probabilities are known. Imagine that you are offered to buy a 15€ ticket to a lottery in which 3 of the 150 sold tickets will yield a prize of 1000€. Should you buy it? -'You certainly should', would Pascal reply. Indeed, he would argue, -assuming you would rather own more money than less -you should always choose whatever option holds the highest expected value.

And in this specific case, not buying the lottery ticket has an expected value of 0€ (you don't spend nor earn anything), while buying it has an expected value of: Thus, Pascal formulated a simple yet powerful precept for optimal decisionmaking between options involving some form of uncertainty, or risk: maximize expected value. He seemed fairly confident in the universal validity of this rule, and, in an effort to instil some Christian faith in an (imaginary) atheist interlocutor, he went on and famously applied it to a rather significant wager: that of believing in the existence of God, which, he explained, had a higher expected value than not believing in God. [START_REF] Pascal | [END_REF] b

. Expected Utility: one Bernoulli may conceal another

As one can see, Pascal's approach to gambling problems was quite prescriptive, or equivalently, normative. It provided decision-makers with a policy that they should follow if they wanted to maximize their wealth -at least in the long run. A few decades after its formulation, Pascal's injunction was challenged by another mathematician, Nicolas Bernoulli [START_REF] Eves | An Introduction To The History of Mathematics[END_REF]; not because of its mathematical invalidity, but because it sometimes failed so completely to account for choices made by real-world agents, however sane or erudite they might be. To illustrate this, he depicted the following (hypothetical) situation: a casino offers a game of chance for a single player, in which a fair coin is tossed at each round. The initial money stake is $2, and gets doubled every time heads appears. The first time that tails appears, the game ends, and the player wins whatever is in the pot. Now, the question is: what would be a fair price to pay the casino to enter this game?

Most people, Bernoulli argues, would not be willing to pay more than $20 to enter the game; and yet, had they been able to turn to the late Pascal for advice, he would have urged them to pay whatever price was asked, until their last penny if need be, to get an opportunity to play this game. The reasoning behind this seemingly senseless instruction is in fact quite sensible: trusting that the coin is indeed fair, the player has a 1 2 probability of winning $2, a 1 4 probability of winning $4, etc., such that the expected value of the winning pot is:

𝐸𝑉[𝑃𝑜𝑡] = 1 2 • 2 + 1 4 • 4 + 1 8 • 8 + ⋯ = 1 + 1 + 1 + ⋯ = +∞
This game has an infinite positive expected value, therefore, regardless of the price paid to play it, the expected balance will be in favour of the player.

A first attempt at solving Bernoulli's paradox was made by Gabriel Cramer, who, considering that "the mathematicians estimate money in proportion to its quantity, and men of good sense in proportion to the usage that they may make of it" [START_REF] Pulskamp | Correspondence of Nicolas Bernoulli concerning the St. Petersburg Game[END_REF], demonstrated that, if men of good sense were indeed seeking to maximize their usage of money, and if that usage was a square root function of its face value, then the expected usage of the winning pot was actually finite, and so should be the price that these men would be willing to pay to enter the game.

The canonical resolution of Bernoulli's paradox, however, is attributed to another Bernoulli: Daniel, Nicolas' cousin, also a prominent mathematician. In 1738, he published his solution [START_REF] Bernoulli | Exposition of a New Theory on the Measurement of Risk[END_REF] in the journal of the Imperial Academy of Science of Saint Petersburg (which is why this paradox is now widely known as the "Saint Petersburg Paradox"). 𝑼(𝒘) = 𝒍𝒏(𝒘). From [START_REF] Caplin | Chapter 1-Basic Methods from Neoclassical Economics[END_REF] Bernoulli's argument was largely in line with Cramer's, but instead of proposing that usage, or utility, of the decision-maker was a square-root function of the face value of gambling gains, he suggested to quantify it a natural logarithmic function of the total wealth of the decision-maker:

"The determination of the value of an item must not be based on the price, but rather on the utility it yields… There is no doubt that a gain of one thousand ducats is more significant to the pauper than to a rich man though both gain the same amount." [START_REF] Bernoulli | Exposition of a New Theory on the Measurement of Risk[END_REF] Again, due to such concave transformation of the future wealth of the agent after winning the pot, the expected utility of playing the game was finite, irrespective of whether he was a pauper or a rich man, and thus he was right in balking at the idea of investing his whole current wealth in this game.

By trying to better account for real-world gambling decisions, Cramer and

Bernoulli pioneered the concept of decreasing marginal utility, as a descriptive addition to Pascal's prescriptive framework, more than a century before Marginalists introduced it to economics. Remarkably, they also pioneered the rule of expected utility maximization, two centuries before its first formalization in the economics literature.

c. Expected Utility Theory, by Von Neumann & Morgenstern

In 1944, mathematicians (and part-time economists) John von Neumann and Oskar Morgenstern came forward with a new theory [START_REF] Von Neumann | Theory of games and economic behavior[END_REF], which was built up from the revealed preference approach developed by Samuelson, and which aimed to address an entire class of decision problems that principles of revealed preference and ordinalism alone were unable to tackle. One such decision problem could be, for instance: given that Alice was observed to choose a lottery offering a 50% chance of winning an orange (or nothing otherwise) over the certain prospect of gaining an apple, what choice can we expect her to make when offered to pick one of two new lotteries: 25% chance of winning an orange, OR, 50% chance of winning an apple? Another decision problem that Von Neumann and Morgenstern aimed to address was: given that Bob was observed to prefer strawberries over bananas, and bananas over grapes, could we somehow devise a lottery between strawberries and grapes that would be as valuable to him as a banana?

In short, this new theory aimed at covering an entire part of choice behaviour that had been largely ignored by earlier economists: gambling, that is, decision-making between options involving risk, i.e., uncertain outcomes whose probabilities are known by the decision-maker.

Their contribution mainly consisted in identifying a set of four axioms, and in demonstrating that compliance with these axioms was a necessary and sufficient condition for someone's choice behaviour to be described as maximizing an expected utility function, where the expected utility of a lottery leading to outcomes 𝑥 𝑖 with probabilities 𝑝 𝑖 is written:

𝐸𝑈[𝐿𝑜𝑡𝑡𝑒𝑟𝑦] = ∑ 𝑝 𝑖 • 𝑢(𝑥 𝑖 )
𝑖 and the utility function 𝑢 is continuous and monotonically increasing.

The four axioms of Expected Utility Theory (EUT) are:

Completeness axiom: for any two lotteries A and B, the agent either strictly prefers A to B, or B to A, or is indifferent between A and B.

Transitivity axiom (similar to the GARP): for any three lotteries A, B and C, if the agent prefers A to B and B to C, then he prefers A to C.

Continuity axiom: for any three alternative lotteries A, B and C, if the agent prefers A to B and B to C, then there exists a probability 𝑝 such that the agent is indifferent between B and the following mixture between A and C: 𝑝𝐴 + (1 -𝑝)𝐶.

Independence axiom: for any three lotteries A, B and C, and for any probability 𝑝 ∈ [0; 1], if the agent prefers A to B, then he also prefers 𝑝𝐴 + (1 -𝑝)𝐶

to 𝑝𝐵 + (1 -𝑝)𝐶.

Thus, an individual whose choice behaviour constantly obeys these four axioms can be seen as maximizing a single internal expected utility function, i.e., as summing the subjective values of all the possible outcomes of a lottery, weighted by their respective probability, and then selecting the lottery with the highest result. (Besides, such an individual is regarded as a rational decision-maker by economists.)

In relation to Bernoulli's ad hoc assumption that people's choices maximize an expected utility function (where marginal utility decreases with total wealth),

Von Neumann & Morgenstern's theory of expected utility maximization is also normative, but it has an additional -and substantial -quality: that of being falsifiable (at the individual level). Furthermore, it leverages probability as a ruler to quantify relative preference between items (at least for agents complying with their axioms), thus allowing to determine the shape of an agent's utility function based on her choices, and taking cardinalists one step closer to their Grail (however, there still can be no unique utility function -considering the linear structure of EUT computation, all positive affine transformations of a given utility function make the same predictions).

Finally, another noteworthy benefit of EUT is that it provides a way to translate risk attitude (observed through choices) into value representation. For instance, an agent who chooses a sure gain of 8€ over a lottery with 50% chances of yielding 20€ (and 0€ otherwise) is said to be risk-averse: he is willing to sacrifice an expected value of 2€ to be granted the certain option (or, the sure thing, as some economists say) instead of the risky one. The remarkable thing here is that this agent's attitude towards risk is embedded in the shape of his utility function. A concave utility function in the value domain corresponds to (or "causes", or "describes" -depending on the relation that one assumes to exist between utility and choice) risk-aversion in the choice domain. Conversely, a convex utility function corresponds to risk-seeking.

Fig. 1.3: Graphical Representation of Risk Attitudes in Expected

Utility Theory. Here, the decision-maker is given a choice between receiving 5000€ for sure ("sure thing"), or entering a lottery in which he has 50% chances of winning 10000€ ("gamble"). Left: Risk-averse; due to the concavity of the utility function, the sure thing has a higher expected utility than the gamble. Ten years after the publication of Von Neumann & Morgenstern's theorem, and inspired by the works of Frank Ramsey [START_REF] Ramsey | The Foundations of Mathematics and Other Logical Essays[END_REF] and Bruno de Finetti [START_REF] De Finetti | La prévision: Ses lois logiques, ses sources subjectives[END_REF], Leonard Savage proposed an extension of EUT that finally filled that uncomfortable gap in decision theory [START_REF] Savage | The Foundations of Statistics[END_REF]. He had identified a set of seven axioms (some of which were equivalent to EUT's axioms) that constituted a necessary and sufficient set of conditions for an agent's behaviour to be described as maximizing expected utility computed from a utility function and a well-defined internal probability function (that is, a function that complies with the standard mathematical definition of a probability measure, and that assigned a subjective probability to each possible outcome following each possible act in each possible state.) This extension of EUT is called Subjective Expected Utility theory.

It is worth noting that one of Savage's axioms, known as the "qualitative probability" axiom, postulates that the probability ranking between any two events does not depend upon the utility of their consequences. As an illustration, let us picture that, to celebrate the kick-off of the Football World Cup, a generous liquor merchant is offering a bottle of exceptionally fine wine to anyone who will correctly predict the winning team. Savage postulates that, if in such circumstances an agent bets on a French rather than German victory, then he would not have bet on Germany had the merchant been less generous and had the prize been a can of warm beer. Thus, one of the cornerstones of SEU theory is the assumption that, regardless of whether subjective probabilities are in line or not with reality, they should be independent from the subjective value (that is, utility) of the corresponding outcomes.

3) The reconciliation of economics and psychology

a. Man versus savage axioms

For all of the elegance of the EU theorem, it was not long before it came under the fire of critiques pointing out that it made for a very unrealistic description of the behaviour of real-world agents. Among the first whistle-blowers was the economist Maurice Allais [START_REF] Allais | Le Comportement de l'Homme Rationnel devant le Risque: Critique des Postulats et Axiomes de l'Ecole Americaine[END_REF]. He had devised a clever set of pairwise choices that revealed systematic deviations of people's preferences from EU axioms, and even managed to trick Savage himself [START_REF] Glimcher | Introduction: A Brief History of Neuroeconomics[END_REF]. Allais' paradox involved two (hypothetical) choice situations: The interesting result is that most people prefer option A in situation 1 and option B in situation 2. The reason why it is interesting is that it violates EUT's independence axiom. Indeed, in situation 1, option A (resp. B) can be seen as a mixture between a lottery a (resp. b) occurring with a 11% chance (where a leads to a sure win of 1 M€, and b leads to a 5 M€ 10 out of 11 times) and a lottery L occurring with a 89% chance, which leads to a sure 1 M€ win. According to EUT's independence axiom, the fact that people strictly prefer A over B means that they strictly prefer a to b. However, the same reasoning can be applied in situation 2, where A (resp. B) is a mixture between a lottery a (resp. b) (which are exactly the same as in situation 1), again occurring with an 11% chance, and a lottery L' (89% chance) leading to a sure 0€. Given that people have already expressed their strict preference for a over b in situation 1, it follows from the independence axiom that they will again prefer A to B. Yet, this is not the case. Therefore, Allais concluded, the independence axiom is wrong, at least for every agent who made different choices in situations 1 versus 2.

Another such paradox was identified by Daniel Ellsberg a few years later [START_REF] Ellsberg | Risk, Ambiguity, and the Savage Axioms[END_REF]. He came up with two urn-and-marbles scenarios across which people consistently showed a preference for the urn with a known ratio of winning marbles (the risky urn) over the urn with an unknown ratio (the ambiguous urn), even when they implicitly believed (as inferred from their first urn choice) the ambiguous urn to contain a higher ratio of winning marbles. This is the first wellknown demonstration of an ambiguity aversion in the general population.

Incidentally, this is also a demonstration that individuals routinely violate some of Savage's axioms of decision-making.

b. The birth of behavioural economics

Allais' paradox certainly sparked intense questioning about the legitimacy of the Expected Utility theory as a descriptive model of human value-based decisionmaking under risk. Yet, the real turning point came in 1979, when two economists, Amos Tversky and Daniel Kahneman, published a seminal paper [START_REF] Kahneman | Prospect Theory: An Analysis of Decision under Risk[END_REF] in which they laid out a long list of empirically observed violations of EU axioms, and proposed a couple of adjustments to be made in economists' decision theoretic model to better account for (in the sense of describe) the choice behaviour of real-world agents.

First, having observed in field and laboratory experiments that people tend to be risk-averse when faced with gain prospects and risk-seeking when faced with loss prospects, they proposed that utility -or "(subjective) value" in their terminology -should not be a function of the total wealth of the decision-maker, but instead should be defined according to a reference point. This reference point sets a frontier between gains and losses along the outcome axis, as well as a frontier between pleasure and displeasure along the utility axis. It should be noted that in Kahneman and Tversky's model, the subjective gain/loss framing is not necessarily aligned with the objective framing of the outcomes. For example, if an individual enters a game which is supposed to yield a 10€ gain on average, then,

given the individual's overall expectations, an 8€ gain outcome might be internally framed as a 2€ loss. (Influences of expectations and aspirations on reference point setting have since been actively explored [START_REF] Heath | Goals as Reference Points[END_REF][START_REF] Kőszegi | A Model of Reference-Dependent Preferences[END_REF][START_REF] Post | Deal or No Deal? Decision-making under Risk in a Large-payoff Game Show[END_REF].) Furthermore, given the pattern of risk-attitude exhibited by their subjects, they proposed that the utility function accounting for generic human behaviour should be concave in the gain domain, and convex in the loss domain.

Second, they argued that some of the choices made by their subjects implied that the displeasure associated to a loss was larger than the pleasure associated to a gain of same numerical amplitude, i.e., that, at a psychological level, losses loom larger than gains. To account for that loss aversion -and assuming that an outcome of 0 (no gain nor loss) maps to a utility of 0 (no pleasure nor displeasure)

-, the generic utility curve should be, on average and around 0, steeper in the loss than in the gain domain.

Third and last, choice paradoxes such as Allais' -which they replicated experimentally across several variants: modest outcomes, non-monetary outcomes, etc. -seemed to imply that people overweighted, in their decision, the utility of low-probability outcomes, and underweighted the utility of highprobability ones. They proposed that individuals do not maximize their mathematical expectation of subjective value, but maximize instead a weighted sum of subjective values, where the weights, called "decision weights", are certainly related to objective probabilities but display consistent deviations from them, e.g.

due to the influence of ambiguity or cognitive biases (among which, the overestimation of low probabilities and underestimation of high ones).

Importantly, decision weights are not constrained to satisfy the mathematical properties of a probability measure (such as summing to 1 across the space of possible elementary outcomes -in fact, they typically sum to less than 1, a feature that they call subcertainty), and they should not be merely seen as subjective probabilities, or degrees of belief, but as deriving from them as well as other factors. Because much of their argument was based on the results of behavioural experiments which involved offering a choice between alternative prospects (conceptually equivalent to EUT's lotteries), their theory was named Prospect Theory.

It was a turning point in decision theory in that it marked the reconciliation of economists and psychologists, and the birth of the behavioural economy movement, which incorporated psychological considerations into economics models for a better description of choice behaviour, as well as experimental methodology from the psychology field for better controlled observations of human behaviour.

4) Seeing the bigger picture: from options to actions a. Introducing action: Markov Decision Processes & instrumental learning

As behavioural economists were prompt to point out, there is more to human behaviour than choosing which fruit to pick in a basket or which race horse to bet on. Neoclassical economists had tended to reason about idealised situations in which outcomes occur almost immediately (although possibly stochastically) upon the agent choosing between two or more options. By doing that, they had ignored the broader, real-life context in which decisions are made, which often implies that agents perform an action or sequence of actions beyond the elementary act of choosing, in order to transition from their current state to the outcome state.

One of the earliest mathematical frameworks used to model the behaviour of an agent as she transitions, through action and chance, from one state to the next, is that of Markov Decision Processes (MDP) [START_REF] Bellman | A Markovian decsion process[END_REF]. Specifically, it is assumed that, at each time step 𝑡, the agent is in one of a set of states {𝑆 1 , 𝑆 The thickness of action-to-state arrows represents the transition probability. In this situation, performing 𝒂 𝟏 in state 𝑺 𝟏 leads to a deterministic transition (decision-maker stays in 𝑺 𝟏 for sure) while every other transition is probabilistic.

Some transitions (e.g. from 𝑺 𝟐 to 𝑺 𝟏 via 𝒂 𝟏 ) are rewarded (in this example, +5), while other transitions are punished (e.g. staying in 𝑺 𝟐 after action 𝒂 𝟏 ).

MDPs have been applied to individual decision-making problems where outcomes depend partly on the agent's actions and partly on chance, and where, given the set of transition probabilities and rewards, an optimal policy 𝜋, which is optimal in that it maximizes the expectancy of future rewards, can be derived.

Formally, 𝜋 is a function that maps each possible state with an optimal action. The MDP framework has received extensions such as the Partially Observable Markovian Decision Process (POMDP) [START_REF] Kaelbling | Planning and acting in partially observable stochastic domains[END_REF] to account for decision problems in which an agent has incomplete information about her current state and the transition probabilities to other states (which she learns through observation), and in which she discounts rewards which are in a distant future (t+2, t+3, etc.) compared to the one that comes immediately (t+1).

When rewards and transition probabilities are initially unknown, MDP optimization is conducted through reinforcement learning, or Q-learning.

MDPs were thus particularly suitable to model the behaviour of ideal agents learning to maximize their rewards in a stochastic and initially unknown (or variable) environment. Such behaviour had been studied in the previous decades in (non-ideal) cats, rats and pigeons, first by Edward Thorndike, and later, by B.F.

Skinner.

Thorndike famously placed cats in custom-made "puzzle boxes", in which a specific action, such as pulling a lever or pushing a button, had to be performed by the animal in order to open the escape hatch and access a food reward. He observed that, at first, a large amount of time would elapse before the cat eventually pressed the escape button by chance, but that over repeated trials, the animal would perform the correct behaviour more and more quickly [START_REF] Thorndike | Animal intelligence: An experimental study of the associative processes in animals[END_REF]. He generalized this observation in his "law of effect", according to which "responses that produce a satisfying effect in a particular situation become more likely to occur again in that situation, and responses that produce a discomforting effect become less likely to occur again in that situation" [START_REF] Gray | Psychology[END_REF].

His work was followed on by Skinner, who conducted extensive operant conditioning (as opposed to Pavlov's classical conditioning) experiments in which he studied how modulations in the reward or punishment schedule ensuing a particular behaviour (typically, pressing a lever) impacted the propensity of the animal to perform the said behaviour [START_REF] Skinner | The behavior of organisms: An experimental analysis[END_REF]. Thus, decades before the advent of behavioural economics, Thorndike and Skinner pioneered the investigation of the link between outcome values, (simple) animal behaviours, and putative mental content, such as feelings, desire or satisfaction.

b. From pressing a lever to climbing a mountain: the costs of action

Would you rather enjoy the view from the top of Mount Everest, or the view from the top of the chair on which you are currently sitting?

In most cases, transitioning from one state (e.g., hungry), to another, potentially more valuable state (e.g., satiated), involves not just pressing a button, but performing an extensive sequence of actions (e.g., getting up, going down a flight of stairs, crossing a busy street, walking in the freezing cold to the nearest kebab house, ordering a sandwich, waiting for it to be ready, paying the cashier). In Markov Decision Processes, performing an action is cost-free and essentially delayfree (in that it involves a fixed transition from t to t+1, regardless of the specific action performed), but in the real world, actions are resource-consuming. They typically spread across time (delay), involve taking risks (i.e., incurring a non-null probability of being carried away to low-value states, such as getting hit by a car while crossing the street or having your wallet stolen), expending energy and other bodily resources, such as muscle integrity and contractility, through physiological processes (effort) or other resources such as money, as well as going through transitory states of relatively low hedonic value, e.g. in terms of physical or psychological well-being 1 : pain, distress, etc.. Among the various dimensions that might make an action aversive or costly, we will now focus on effort.

All other things equal, effort tends to be avoided. The "principle of least effort" was coined more than a century ago [START_REF] Ferrero | L'inertie Mentale Et la Loi du Moindre Effort[END_REF], then passed on to psychologists by Clark Hull as the "law of less work" [START_REF] Hull | Principles of behavior: An introduction to behavior theory[END_REF], and has since received mountains of supporting empirical and experimental evidence.

Perhaps the first scientific field to take an interest in effortful behaviour was ethology, or behavioural ecology. The kind of questions that ethologists were asking then was: "what motivates a wild animal to travel in search of a new patch of food instead of staying in its current patch?", "how does a predator chooses between pursuing a small, easy prey and a fleshier but faster one?" [START_REF] Charnov | Optimal foraging, the marginal value theorem[END_REF][START_REF] Krebs | Optimal foraging: Theory and experiment[END_REF][START_REF] Stephens | Foraging Theory[END_REF]. Some researchers, such as Eric Charnov,

1 This is, of course, not always true. A wide variety of resource-consuming actions, such as doing crosswords or playing tennis, are reported as entailing higher hedonic states, and as being enjoyable in themselves. This can be related to the extrinsic versus intrinsic motivation dichotomy [START_REF] Deci | Intrinsic Motivation[END_REF]: actions which are accompanied with a relatively low hedonic state (e.g. which are physically painful or emotionally distressful) are usually performed in order to reach a high-value future state, and as such are said to be "extrinsically motivated".

proposed optimal foraging models describing the behaviour of an (ideal) animal acting to maximize its metabolic resources, which are increased by food consumption and depleted by effort production. Still in the early days of effort research, as interest spread to the investigation (and formalization) of the economic and neurobiological determinants of effortful behaviour, researchers had animals face food dilemmas in the lab (i.e., a well-controlled environment), such as whether to take a left turn and go retrieve one food pellet without trouble, or take a right turn and climb a high wire-mesh barrier behind which four food pellets were waiting to be devoured [START_REF] Salamone | Anhedonia or anergia? Effects of haloperidol and nucleus accumbens dopamine depletion on instrumental response selection in a T-maze cost/benefit procedure[END_REF][START_REF] Walton | The Role of Rat Medial Frontal Cortex in Effort-Based Decision Making[END_REF][START_REF] Walton | Weighing up the benefits of work: Behavioral and neural analyses of effort-related decision making[END_REF][START_REF] Schweimer | Dopamine D1 receptors in the anterior cingulate cortex regulate effort-based decision making[END_REF][START_REF] Floresco | Amygdala-Prefrontal Cortical Circuitry Regulates Effort-Based Decision Making[END_REF][START_REF] Hillman | Neurons in the Rat Anterior Cingulate Cortex Dynamically Encode Cost-Benefit in a Spatial Decision-Making Task[END_REF].

Fig. 1.6: A rat in a T-maze experiment.

In the left arm of the maze, one food pellet can be retrieved effortlessly (low effort/low reward option). In the right arm is a 30 cm-high wire-mesh barrier, that the rat will have to climb in order to get four food pellets (high effort/high reward option).

In a similar paradigm, developed soon after the T-maze one, rodents would choose whether to press a lever three times in a row and get two food pellets, or press another lever ten times in a row and get 6 food pellets [START_REF] Kennerley | Optimal decision making and the anterior cingulate cortex[END_REF][START_REF] Walton | Weighing up the benefits of work: Behavioral and neural analyses of effort-related decision making[END_REF]. Regardless of the specifics of the paradigm, it was consistently observed that animals tended to select an option more often as the associated reward increased, and less often as the energetic cost of the action required to obtain the reward increased. Thus, animals seemed to maximize a trade-off between reward and effort. When animals were not given a choice but instead forced to perform a given effort for a cued reward [START_REF] Bouret | Ventromedial and Orbital Prefrontal Neurons Differentially Encode Internally and Externally Driven Motivational Values in Monkeys[END_REF][START_REF] Minamimoto | Is working more costly than waiting in monkeys?[END_REF][START_REF] Varazzani | Noradrenaline and Dopamine Neurons in the Reward/Effort Trade-Off: A Direct Electrophysiological Comparison in Behaving Monkeys[END_REF], their appetitive responses and error rates were modulated by effort and reward magnitudes in opposite directions. Overall, effort could be seen as discounting the subjective value of a reward.

These conclusions are unsurprising, even to the naïve reader, because they are in line with our subjective, internal experience of effort production.

Phenomenologically, -in a vast majority of cases -exerting effort is aversive, unpleasant, or even painful [START_REF] Kurzban | The sense of effort[END_REF]. From an evolutionary point of view, it is quite easy to conceive why physical effort feels bad: an organism for which usual rewards such as food and play feel good and effort also feels good (or neutral) would likely engage in much more effortful behaviours in order to obtain marginally larger rewards, and, when resources are not available (e.g. in winter),

would not be particularly inclined to stop playing or foraging, thus depleting its metabolic resources faster than it can restore them (and therefore would probably die of exhaustion before reaching reproductive maturity). In this scenario, we only have considered the case of physical effort; it is widely accepted, though, that effort can come in two distinct flavours: physical or cognitive (i.e., mental). Mental effort, such as the one needed to compute 34 x 56 in your head or to speak a foreign language, also happens to be experienced as aversive (or even "painful", thus placing it very close to the phenomenology of physical effort), but the reasons why it is the case are much less clear than for physical effort [START_REF] Kurzban | An opportunity cost model of subjective effort and task performance[END_REF][START_REF] Kurzban | The sense of effort[END_REF][START_REF] Shenhav | Toward a Rational and Mechanistic Account of Mental Effort[END_REF]. In what follows, we will keep our focus on physical effort.

In recent years, there have been multiple attempts at modeling both qualitatively and quantitatively the discounting of the subjective value of a course of action by the physical effort attached to it. In the first experimental implementations of reward-effort trade-offs, effort cost was confounded with delay in reward obtention (it takes more time to climb a wire-mesh barrier than to simply walk to the end of the maze arm, or to press a lever ten times instead of three). This confound, which is also present in several human studies [START_REF] Croxson | Effort-Based Cost-Benefit Valuation and the Human Brain[END_REF][START_REF] Treadway | Worth the 'EEfRT'? The Effort Expenditure for Rewards Task as an Objective Measure of Motivation and Anhedonia[END_REF][START_REF] Wardle | Amping Up Effort: Effects of d-Amphetamine on Human Effort-Based Decision-Making[END_REF], prevented a proper isolation of discounting effects due to effort only. More suitable paradigms have since been developed, in which effort is only modulated via its intensity (and not its duration)

dimension. One such paradigm, now a classic in the field, consists in having the (human or primate) subject squeeze a handgrip at a certain force level (usually expressed as a percentage of their maximum voluntary contraction, or MVC) for a fixed duration in order to receive a reward, whose expected magnitude is either fixed [START_REF] Kurniawan | Choosing to Make an Effort: The Role of Striatum in Signaling Physical Effort of a Chosen Action[END_REF][START_REF] Burke | Neural Integration of Risk and Effort Costs by the Frontal Pole: Only upon Request[END_REF][START_REF] Hartmann | Parabolic discounting of monetary rewards by physical effort[END_REF][START_REF] Skvortsova | Learning To Minimize Efforts versus Maximizing Rewards: Computational Principles and Neural Correlates[END_REF][START_REF] Klein-Flügge | Behavioral Modeling of Human Choices Reveals Dissociable Effects of Physical Effort and Temporal Delay on Reward Devaluation[END_REF][START_REF] Klein-Flügge | Neural Signatures of Value Comparison in Human Cingulate Cortex during Decisions Requiring an Effort-Reward Trade-off[END_REF][START_REF] Bouc | Computational Dissection of Dopamine Motor and Motivational Functions in Humans[END_REF][START_REF] Chong | Neurocomputational mechanisms underlying subjective valuation of effort costs[END_REF][START_REF] Hauser | Separate mesocortical and mesolimbic pathways encode effort and reward learning signals[END_REF][START_REF] Vinckier | Sour grapes and sweet victories: How actions shape preferences[END_REF], or proportional to the level of force exerted [START_REF] Pessiglione | How the Brain Translates Money into Force: A Neuroimaging Study of Subliminal Motivation[END_REF][START_REF] Bonnelle | Characterization of reward and effort mechanisms in apathy[END_REF][START_REF] Bouc | Computational Dissection of Dopamine Motor and Motivational Functions in Humans[END_REF][START_REF] Lopez-Persem | Choose, rate or squeeze: Comparison of economic value functions elicited by different behavioral tasks[END_REF]. Adapted from (Le [START_REF] Bouc | Computational Dissection of Dopamine Motor and Motivational Functions in Humans[END_REF] Several forms of discounting functions have been proposed to describe the devaluation of an outcome by the effort needed to obtain it. The simplest one comes from early animal research. Based on previous experimental observations, [START_REF] Phillips | Calculating utility: Preclinical evidence for cost-benefit analysis by mesolimbic dopamine[END_REF] proposed that the subjective value SV of a reward magnitude R was linearly discounted by the amount of exerted effort E:

𝑆𝑉(𝑅, 𝐸) = 𝑅 -𝜶 • 𝐸

where 𝜶 is a fixed parameter (linear coefficient).

Another functional form, known as hyperbolic discounting, accounts well for the discounting of outcomes by delay [START_REF] Mazur | Estimation of Indifference Points with an Adjusting-Delay Procedure[END_REF][START_REF] Laibson | Golden Eggs and Hyperbolic Discounting[END_REF][START_REF] Frederick | Time Discounting and Time Preference: A Critical Review[END_REF], and was also proposed to model effort discounting [START_REF] Mitchell | Discounting the Value of Commodities According to Different Types of Cost[END_REF][START_REF] Sugiwaka | Reformative self-control and discounting of reward value by delay or effort[END_REF][START_REF] Prevost | Separate Valuation Subsystems for Delay and Effort Decision Costs[END_REF]:

𝑆𝑉(𝑅, 𝐸) = 𝑅 1 + 𝒌 • 𝐸
where 𝒌 is a fixed parameter. At this point, it should be noted that discounted models predict subjective value, but what researchers observe is usually choices. In order to be able to invert their model based on observed data and to obtain an estimate for the parameter(s) of the model (i.e., to "fit the model"), they thus need an additional function (sometimes called "observation function" [START_REF] Daunizeau | VBA: A Probabilistic Treatment of Nonlinear Models for Neurobiological and Behavioural Data[END_REF]) which predicts (observable) choices based on (hidden) subjective value. Following on the utility maximization hypothesis which underlies most economic theories of value-based decision-making, such a function should be: 𝐶ℎ𝑜𝑠𝑒𝑛 𝑜𝑝𝑡𝑖𝑜𝑛 = 𝑎𝑟𝑔𝑚𝑎𝑥 𝑖 (𝐸𝑈 𝑖 ), where 𝐸𝑈 𝑖 are the expected utilities of alternative options 𝑖. In reality, however, decision-makers do not systematically opt for the option with the highest expected utility (or overall subjective value, regardless of the way it is computed): their choices are noisy. To account (descriptively) for this noise, choices are predicted probabilistically, by a softmax function in the vast majority of cases:

𝑃(𝐶ℎ𝑜𝑠𝑒𝑛 𝑜𝑝𝑡𝑖𝑜𝑛 = 𝑖) = 𝑒 𝜷•𝑆𝑉 𝑖 ∑ 𝑒 𝜷•𝑆𝑉 𝑗 𝑗
where 𝑆𝑉 𝑗 is the subjective value (or equivalently here, expected utility) of each option (or equivalently, course of action) 𝑗, and 𝜷 is a linear coefficient, usually called "inverse temperature", scaling the subjective values and reflecting how "precise" (in terms of subjective value maximization) the decision-maker is.

More recently yet, building on the old finding that perceived effort increases as a power function of the exerted force [START_REF] Stevens | On the Psychophysical Law[END_REF], a parabolic discounting model was added to the pool [START_REF] Burke | Neural Integration of Risk and Effort Costs by the Frontal Pole: Only upon Request[END_REF][START_REF] Hartmann | Parabolic discounting of monetary rewards by physical effort[END_REF][START_REF] Klein-Flügge | Behavioral Modeling of Human Choices Reveals Dissociable Effects of Physical Effort and Temporal Delay on Reward Devaluation[END_REF][START_REF] Białaszek | Physical and cognitive effort discounting across different reward magnitudes: Tests of discounting models[END_REF][START_REF] Chong | Neurocomputational mechanisms underlying subjective valuation of effort costs[END_REF]:

𝑆𝑉(𝑅, 𝐸) = 𝑅 -𝜶 • 𝐸 𝛾
where 𝜶 is a linear coefficient and 𝜸 is an exponential coefficient scaling the effort E.

In several studies quantitatively comparing parabolic models with other functional forms [START_REF] Hartmann | Parabolic discounting of monetary rewards by physical effort[END_REF][START_REF] Białaszek | Physical and cognitive effort discounting across different reward magnitudes: Tests of discounting models[END_REF][START_REF] Chong | Neurocomputational mechanisms underlying subjective valuation of effort costs[END_REF], parabolic discounting was found to account best for the reward-effort choices made by the participants. In another study, however [START_REF] Klein-Flügge | Behavioral Modeling of Human Choices Reveals Dissociable Effects of Physical Effort and Temporal Delay on Reward Devaluation[END_REF], an additional, sigmoidal model was inverted:

𝑆𝑉(𝑅, 𝐸) = 𝑅 • (1 -( 1 1 + 𝑒 -𝒌(𝐸-𝒑) - 1 1 + 𝑒 𝒌𝒑 ) (1 + 1 𝑒 𝒌𝒑 ))
where 𝒌 is a linear coefficient reflecting the steepness of the sigmoid, and 𝒑 sets its turning point (i.e. the effort level at which the subjective value of a reward magnitude R is discounted by half). In this study, the sigmoidal effort discounting model was found to provide the best account for empirical data. It also captured choice behaviour better than hyperbolic and parameter-free subtractive or divisive discounting models in [START_REF] Klein-Flügge | Neural Signatures of Value Comparison in Human Cingulate Cortex during Decisions Requiring an Effort-Reward Trade-off[END_REF] Although effort discounting models have only been compared based on observed binary choices, they have been applied to other behavioural outputs, such as exerted force itself (Le [START_REF] Bouc | Computational Dissection of Dopamine Motor and Motivational Functions in Humans[END_REF] or even complete movement trajectories encompassing continuous-time decision-making [START_REF] Rigoux | A Model of Reward-and Effort-Based Optimal Decision Making and Motor Control[END_REF].

c. What if you fail? Confidence as a special kind of probability

When pondering whether to engage in a given course of action or project, such as growing an orange tree to eventually produce orange juice, one aspect that individuals typically consider is the subjective value of the various possible outcomes (e.g., within the next five years, a healthy tree and litres of delicious orange juice, or, a healthy tree yet barely any orange, or, a dying tree and no orange in sight). As we have seen above, individuals also consider the overall cost entailed by this course of action (e.g. the money to buy the tree, the time and effort to take care of it). Another crucial judgment generated during the decision process is about the subjective probability that the course of action will lead to each of the different envisaged outcomes. In many instances, as in the orange tree scenario, these outcomes are at least partly contingent on the individual's specific behaviour (or, in this context, performance), and can be situated with respect to a "success" or "correctness" (more or less arbitrary) criterion. In the above example, one may be a seasoned gardener and estimate that he is very likely to succeed (where his success threshold would be, e.g., keep the tree healthy and harvest oranges within five years) or he might be inexperienced but overestimate his skills and have the same prediction. In any case, as we have seen earlier, expected utility theory postulates that the subjective value of each outcome is multiplied with its subjective probability (before being summed and discounted by the cost of action).

Thus, according to this framework, and since "successful" outcomes are in general higher valued than "unsuccessful" ones, the more confident a decision-maker is in his endeavour being successful, the higher the overall expected utility of the course of action, and the likelier he is to initiate it. The notion of "confidence" has just entered the conceptual picture painted in this introductory chapter, and before it is further employed, it might be appropriate to settle on an explicit definition.

What confidence is, and what it is not

Confidence may seem like a straightforward concept: we all know what being unconfident feels like. Yet (or maybe because of that), over the last decades, psychologists and neuroscientists have been using this term to refer to disparate constructs, thus hampering progress towards an understanding of its computational and neurobiological determinants, as well as its role in action valuation and selection.

We henceforth define confidence as the subjective probability of being correct, or successful. It is in essence a "local" judgment, since it applies to a particular action sequence or first-order judgment, which one sets against a particular validity (correctness or success) criterion.

Following [START_REF] Pouget | Confidence and certainty: Distinct probabilistic quantities for different goals[END_REF], we distinguish confidence from certainty. In their framework, inferentially relevant variables -be they concrete or abstract (How heavy is this bag? How honest is this person?), continuous or discrete (How far is the nearest village? How many cafes does it have?), external or internal (How much does this cost? How much do I like this?), settled in past, present or future states of the world (What colour was her dress? When will the bus arrive?) -are assumed to be internally represented as probability density functions across envisaged values of the variable. 1 In this context, they propose that certainty refers to the narrowness (or inverse variance) of such a probability distribution, while confidence is the integral of that distribution across a subset of its domain (corresponding to, e.g., the subjective probability that the bus will arrive between now and now + 10 minutes). Although we adopt the same definition of certainty, our own definition of confidence is therefore more restrictive than Pouget and colleagues', since, in distributional terms, we define confidence as the integral of the probability distribution of an action-dependent 2 variable of interest (e.g.

1 See also [START_REF] Pouget | Probabilistic brains: Knowns and unknowns[END_REF] for a discussion about their neural implementation. 2 Or, judgment-supporting, if confidence is not about a behaviour per se, but about a first-order judgment where the ball we served will hit the ground) across the specific subset of correct or successful values (e.g. inside the court). 1 This definition nonetheless calls for clarification about what it means, from a subjective standpoint, to be correct.

What it means to be correct

We can distinguish (at least) two ways in which one can be correct, or successful: 1) epistemically, 2) behaviourally. The first case applies to a judgment, belief or prediction one has internally committed to or will commit to (e.g., thinking to yourself: "he will be at least five minutes late"), and being correct means that this judgment is in line with some ground truth or observable state. The second case applies to a behaviour, or behavioural sequence, one has performed or will perform, and being correct or successful means that either the behaviour per se, or its outcome, belongs to a set of correct or success states. The criteria defining truth, or correctness or validity can be more or less arbitrary, either defined externally (e.g. by encyclopaedias, the rules of a sport) or internally (e.g., by your own appreciation of what a "good soufflé" is) and applying to objective variables (e.g.

where the tennis ball lands, how puffy a soufflé is) and/or subjective ones (e.g. the elegance of a move or the deliciousness of a soufflé). Thus, generating confidence judgments typically implies a discretization (and often even a binarization) of the variable(s) of interest. This is not as artificial as it may seem, because in many natural situations, the value distribution of possible outcomes tends to be multinomial, discrete, or even binary (to take a rather extreme example, if you try 1 In an earlier attempt to lay out a clear theoretical framework for the study of confidence [START_REF] Meyniel | Confidence as Bayesian Probability: From Neural Origins to Behavior[END_REF], it was argued that two types of "confidence" should be distinguished: distributional versus summary confidence. This should not be confused with Pouget et al.'s certainty versus confidence dichotomy. In Meyniel et al., information about the uncertainty corrupting a representation can be conveyed either via the probability density distribution itself (distributional confidence) or by a readout that extracts a scalar, summary statistic from the distribution (specifically, its second moment, i.e. variance: summary confidence, analogous to Pouget et al.'s certainty).

to jump across a deep rift, either you reach the other side and survive -although you may have twisted your ankle in the process -, or you don't, and die).

A typology of confidence

There can be many ways to classify confidence judgments into subtypes, but we will only propose two distinctions here, as (hopefully) useful landmarks in the confidence field which we can refer to later.

The first one, expounded in the previous paragraph, is the distinction between epistemic and behavioural confidence. It can be noted, however, that in many recent studies about confidence, these two notions are basically treated as one and the same. These studies use paradigms in which the subject is required to make a perceptual or mnemonic -often binary -decision (e.g.: Is this cloud of dots moving to the left or to the right? Has this word been presented to you earlier?) and to report it explicitly, for instance by pressing a key on a keyboard. Then, subjects are typically prompted to make a confidence ("second-order") judgment about their perceptual ("first-order") answer. Therefore, in such cases, confidence is both about the correctness of their behaviour (pressing key A rather than key B) and that of their internal perceptual judgment (i.e., belief, or epistemic state), since there is a systematic mapping between both. One could argue, though, that what this kind of experiments probes is, in essence, epistemic confidence. [START_REF] Bang | Distinct encoding of decision confidence in human medial prefrontal cortex[END_REF][START_REF] Boldt | The impact of evidence reliability on sensitivity and bias in decision confidence[END_REF][START_REF] Gardelle | Does Confidence Use a Common Currency Across Two Visual Tasks[END_REF][START_REF] Hainguerlot | Metacognitive ability predicts learning cuestimulus associations in the absence of external feedback[END_REF][START_REF] Lebreton | Two sides of the same coin: Monetary incentives concurrently improve and bias confidence judgments[END_REF][START_REF] Rutishauser | Single-Neuron Representation of Memory Strength and Recognition Confidence in Left Human Posterior Parietal Cortex[END_REF][START_REF] Wagner | Building Memories: Remembering and Forgetting of Verbal Experiences as Predicted by Brain Activity[END_REF] The second distinction, which has also been touched upon earlier, is that of prospective versus retrospective confidence. Quite simply, prospective confidence applies to judgments or behaviours not yet performed (e.g., just before taking a mathematics exam), whereas retrospective confidence applies to judgments or behaviours already performed, but for which some uncertainty remains regarding their validity or that of their outcomes (e.g. when leaving the examination room). These two types of confidence are usually studied separately, but see [START_REF] Fleming | Metacognition about the past and future: Quantifying common and distinct influences on prospective and retrospective judgments of selfperformance[END_REF][START_REF] Aguilar-Lleyda | Confidence as a priority signal[END_REF] for counter-examples.

Where does confidence come from?

Humans and other animals assess their probability of success in multiple ways, based on multiple cues. These depend, among other things, on the prospective versus retrospective nature of the confidence judgment.

In a "Bayesian brain" framework such as the one defended in [START_REF] Pouget | Probabilistic brains: Knowns and unknowns[END_REF][START_REF] Meyniel | Confidence as Bayesian Probability: From Neural Origins to Behavior[END_REF][START_REF] Pouget | Confidence and certainty: Distinct probabilistic quantities for different goals[END_REF][START_REF] Paz | Decision Confidence As A Mapping Of Bayesian Posterior Belief[END_REF], confidence as we define it here can be viewed as resulting from some integration, or summation, of an internal, neurally-encoded density function representing a variable of interest (and obeying Bayes' rule of conditional probability, hence the Bayesian qualifying term) across correct or successful values. Regardless of whether we adhere to this view, some questions remain: how is the distribution constructed? What external or internal evidence does it incorporate? This latter question, in particular, is applicable to confidence judgments irrespective of their specific neural implementation (which we will address in more detail in section B).

Several studies have shed some light on this issue. First, in experimental contexts where the accuracy of the subjects' behaviour could be objectively evaluated, and where confidence reports were given by the subjects, a consistent, positive correlation between actual accuracy and confidence reports was found. This is, in particular, a recurring finding in the context of perceptual discrimination or other cognitive, or "epistemic" tasks, such as memory retrieval [START_REF] Baranski | The calibration and resolution of confidence in perceptual judgments[END_REF][START_REF] Busey | Accounts of the confidence-accuracy relation in recognition memory[END_REF][START_REF] Barthelmé | Flexible mechanisms underlie the evaluation of visual confidence[END_REF][START_REF] Rutishauser | Representation of retrieval confidence by single neurons in the human medial temporal lobe[END_REF][START_REF] Sanders | Signatures of a Statistical Computation in the Human Sense of Confidence[END_REF][START_REF] Lebreton | Two sides of the same coin: Monetary incentives concurrently improve and bias confidence judgments[END_REF][START_REF] Rutishauser | Single-Neuron Representation of Memory Strength and Recognition Confidence in Left Human Posterior Parietal Cortex[END_REF]. In such contexts, the process of assessing one's own judgment validity is known as metacognition. Thus, these studies have demonstrated that humans and other animals do have significant metacognitive abilities. However, this confidenceperformance correlation is not perfect, that is, confidence reports are to some extent dissociated from actual performance. In the particular case of retrospective confidence, they have not only been found to be correlated with actual accuracy, but also with the time needed to make a response (response time, or "RT") [START_REF] Pleskac | A dynamic and stochastic theory of choice, response time, and confidence[END_REF][START_REF] Zylberberg | The construction of confidence in a perceptual decision[END_REF][START_REF] Kiani | Choice Certainty Is Informed by Both Evidence and Decision Time[END_REF]. The direction of the causality relation (if there is one) between response time and confidence is not entirely clear, but it has been proposed, and supported by recent evidence, that response time was incorporated as a heuristic cue in confidence judgments [START_REF] Audley | A stochastic model for individual choice behavior[END_REF][START_REF] Kiani | Choice Certainty Is Informed by Both Evidence and Decision Time[END_REF].

Overall, these observations suggest that confidence judgments are built from 1) performance-modulating variables or cues, such as the size of the target to hitor in a more general sense, the inclusiveness of the validity criterion respective to the range of possible epistemic states or behavioural outcomes -or the signal-tonoise ratio in the case of perceptual confidence [START_REF] Wei | Confidence estimation as a stochastic process in a neurodynamical system of decision making[END_REF][START_REF] Sanders | Signatures of a Statistical Computation in the Human Sense of Confidence[END_REF][START_REF] Boldt | The impact of evidence reliability on sensitivity and bias in decision confidence[END_REF][START_REF] Lebreton | Two sides of the same coin: Monetary incentives concurrently improve and bias confidence judgments[END_REF], 2) (in the case of retrospective confidence judgments) performance-modulated cues or variables, such as response time or even partial feedback (e.g. proprioceptive or early visual signals about the direction given to a tennis ball).

Post-decision confidence in a perceptual discrimination task was additionally found to be influenced by transcranial magnetic stimulation of the motor representation associated with the unchosen response, regardless of whether TMS was applied before or after a response was selected [START_REF] Fleming | Action-Specific Disruption of Perceptual Confidence[END_REF], which suggests that confidence judgments are not only based on e.g. external signal-to-noise ratio or cues about the difficulty of a task, but also incorporate late-stage, performance-unrelated, downstream information.

The accuracy, or predictive quality of confidence judgments themselves can be quantified in several ways. One simple approach, when actual success or correctness can be observed, is to decompose this quality assessment into a calibration and a sensitivity dimension.

Calibration refers to the difference between average confidence and the success rate at a given task. Individuals with positive deltas are said to be overconfident; conversely, individuals exhibiting negative deltas are underconfident; and individuals with deltas close to zero are well-calibrated.

Interestingly, individual calibration was found to be correlated across various tasks [START_REF] West | The domain specificity and generality of overconfidence: Individual differences in performance estimation bias[END_REF][START_REF] Bornstein | I know I know it, I know I saw it': The stability of the confidence-accuracy relationship across domains[END_REF], suggesting that confidence judgments are subject to stable trait influences over and above the specific task at hand, and, further, that confidence may be represented as a common currency across tasks and modalities. This latter hypothesis has received consistent empirical support over the last years. For example, it has been shown that individuals are able to make a comparison between their confidence in two responses they made during identical or different perceptual discrimination tasks, and that this confidence ordering is as sensitive in the across-task comparison condition as in the within-task condition [START_REF] Gardelle | Does Confidence Use a Common Currency Across Two Visual Tasks[END_REF][START_REF] Gardelle | Confidence as a Common Currency between Vision and Audition[END_REF]. This brings us to the second dimension of confidence accuracy: sensitivity.

Sensitivity refers to how well performance variations are reflected in confidence judgments. One straightforward way to quantify it is to compute the difference between the average confidence reported in correct trials and the average confidence reported in incorrect trials. In situations where performance is not necessarily binary, one can derive another measure of sensitivity, namely the correlation between objective performance and confidence judgments.

Other, more sophisticated metrics of sensitivity (or metacognitive efficiency) have been proposed and implemented in empirical work. One of them is the metad' value derived from Signal Detection Theory formalism, which has been shown to be less sensitive to calibration biases than correlational approaches like the one presented above. We refer the interested reader to [START_REF] Maniscalco | A signal detection theoretic approach for estimating metacognitive sensitivity from confidence ratings[END_REF] for an extensive introduction to meta-d'.

Confidence and action

We have reviewed some evidence that people (and other animals) are able to estimate their probability of being correct or successful, with some degree of accuracy. We will now survey some evidence that these confidence judgments are not only generated when experimenters ask for a rating, but are even spontaneously incorporated in action selection, initiation and continuation processes.

Modern confidence research has roots in the study of self-efficacy, a construct proposed by Albert Bandura a few decades ago, and defined as the individual's belief in her ability to perform a certain task. Bandura argued that selfefficacy was a major determinant of high expectations of performance, and of the propensity to work hard and persevere in the face of obstacles to meet these expectations:

"Judgments of self-efficacy also determine how much effort people will expend and how long they will persist in the face of obstacles or aversive experiences. When beset with difficulties people who entertain serious doubts about their capabilities slacken their efforts or give up altogether, whereas those who have a strong sense of efficacy exert greater effort to master the challenges." [START_REF] Bandura | Self-efficacy mechanism in human agency[END_REF] More recently, in a laboratory experiment involving a search task, as individuals became less confident about their searching abilities, they also became less likely to keep searching [START_REF] Falk | Self-Confidence and Search[END_REF]. Besides, it has been observed that individuals exhibiting trait overconfidence tend to engage more and be more perseverant in risky professional ventures such as launching a start-up [START_REF] Koellinger | I think I can, I think I can": Overconfidence and entrepreneurial behavior[END_REF][START_REF] Hayward | Beyond hubris: How highly confident entrepreneurs rebound to venture again[END_REF]. Interestingly, in another recent study, individuals who displayed higher confidence in their knowledge in a specific domain also tended to invest more effort in an unrelated task, suggesting that trait confidence modulates general effort investment [START_REF] Pikulina | Do confident individuals generally work harder[END_REF].

In fact, humans are not the only beings whose perseverance depends on confidence. Rodents were found to commit more to decisions about which they were more confident: they waited longer for a potential reward before proceeding to the next trial (in this study, confidence was actually inferred from this degree of commitment, but also had a reliable proxy in task difficulty) [START_REF] Lak | Orbitofrontal Cortex Is Required for Optimal Waiting Based on Decision Confidence[END_REF]).

Yet humans have also been shown to prioritize actions and decisions about which they were more confident [START_REF] Aguilar-Lleyda | Confidence as a priority signal[END_REF], and to use confidence as a teaching signal in the absence of feedback, thus reinforcing actions about which they felt confident [START_REF] Daniel | Striatal activations signal prediction errors on confidence in the absence of external feedback[END_REF][START_REF] Guggenmos | Mesolimbic confidence signals guide perceptual learning in the absence of external feedback[END_REF][START_REF] Hainguerlot | Metacognitive ability predicts learning cuestimulus associations in the absence of external feedback[END_REF].

d. Putting it all together: a current framework of value-based action selection

In the present work, we adhere to modern frameworks of value-based decision-making, which themselves draw on earlier formalizations by economists and psychologists. The currently dominant view on action selection [START_REF] Rangel | A framework for studying the neurobiology of valuebased decision making[END_REF] is that it is contingent on action valuation, which itself derives from an internal representation of internal (e.g. hunger) and external states (e.g. month of the year) of the agent, as well as (of course) the set of feasible actions, and previous experiences of action-outcome associations. From [START_REF] Rangel | A framework for studying the neurobiology of valuebased decision making[END_REF] Here, we focus on the action valuation step, and build on recent cost-benefits models, such as the Expected Value of Control model of cognitive effort [START_REF] Shenhav | The Expected Value of Control: An Integrative Theory of Anterior Cingulate Cortex Function[END_REF], according to which the overall value (OV) of an action in a given state 𝑆 0 corresponds to the expected subjective value (SV) of all envisaged outcome states 𝑆 𝑗 , minus the costs (pain, energy…) entailed by that action: 𝑂𝑉(𝐴𝑐𝑡𝑖𝑜𝑛 𝑖 , 𝑆 0 ) = ∑ 𝑝(𝑆 𝑗 |𝑆 0 , 𝐴𝑐𝑡𝑖𝑜𝑛 𝑖 ) * 𝑆𝑉(𝑆 𝑗 ) -𝐶𝑜𝑠𝑡𝑠(𝐴𝑐𝑡𝑖𝑜𝑛 𝑖 ) 𝑗

5) A methodological parenthesis: eliciting decisional variables

Over decades of experimental investigation, behavioural economists and psychologists have come up with several methods to elicit and measure the components entering the computation of action value, i.e., decisional variables.

Importantly, the measurement can be made before or after the outcome is owned or consumed, or the action is performed, thus prompting either an a priori or a posteriori evaluation.

The most straightforward way to probe a decisional variable -whether it is the subjective value (be it positive or negative) attached to a given outcome, its subjective probability, or the subjective cost attached to an action -is to ask explicitly: "How much do you like this cookie?", "How likely is it that you will draw a winning card?", "How costly do you find it to climb six flights of stairs?". Typically, in experimental settings, the question is displayed on a computer screen, and the answer is made by placing a cursor on a rating scale, which is bounded and features either numeric or verbal labels, such as "Not at all" and "Extremely". This is arguably the most frequent approach to the measurement of decisional variables in modern cognitive neuroscience research. Likeability or pleasantness ratings have been used in a large number of studies over the past fifteen years (O'Doherty, [START_REF] O'doherty | Temporal Difference Models and Reward-Related Learning in the Human Brain[END_REF][START_REF] Lebreton | An Automatic Valuation System in the Human Brain: Evidence from Functional Neuroimaging[END_REF][START_REF] Tobler | Risk-dependent reward value signal in human prefrontal cortex[END_REF][START_REF] Rushworth | Frontal Cortex and Reward-Guided Learning and Decision-Making[END_REF][START_REF] Lebreton | Automatic integration of confidence in the brain valuation signal[END_REF][START_REF] Genevsky | When Brain Beats Behavior: Neuroforecasting Crowdfunding Outcomes[END_REF][START_REF] Schmidt | How context alters value: The brain's valuation and affective regulation system link price cues to experienced taste pleasantness[END_REF][START_REF] Vinckier | Sour grapes and sweet victories: How actions shape preferences[END_REF], as have effort ratings [START_REF] Schmidt | Get Aroused and Be Stronger: Emotional Facilitation of Physical Effort in the Human Brain[END_REF][START_REF] Rollwage | I judge you by your profit: Judgments of effort exerted by self and others are influenced by task contingent rewards[END_REF], probability ratings (e.g. [START_REF] Genevsky | When Brain Beats Behavior: Neuroforecasting Crowdfunding Outcomes[END_REF][START_REF] Lebreton | Automatic integration of confidence in the brain valuation signal[END_REF][START_REF] Sharot | How unrealistic optimism is maintained in the face of reality[END_REF][START_REF] Szpunar | Get real: Effects of repeated simulation and emotion on the perceived plausibility of future experiences[END_REF], and confidence ratings [START_REF] Aridan | Neural correlates of effort-based valuation with prospective choices[END_REF][START_REF] Bang | Distinct encoding of decision confidence in human medial prefrontal cortex[END_REF][START_REF] Berg | A common mechanism underlies changes of mind about decisions and confidence[END_REF][START_REF] Gardelle | Does Confidence Use a Common Currency Across Two Visual Tasks[END_REF][START_REF] Sanders | Signatures of a Statistical Computation in the Human Sense of Confidence[END_REF].

The main benefits of the rating approach are its ease of implementation, in most cases its intuitiveness for the subjects, and the direct interpretability of the measure. Among its drawbacks are the idiosyncrasy in the way subjects use of the scale to report their internal estimate, its transparency as regards the variable of interest for the experimenter, the usual absence of any form of commitment to the rating (in other words, subjects have no built-in incentive to be honest or accurate in their ratings), and the impossibility to use it with non-human animals or infants. were asked to rate the subjective likeability and probability of hypothetical events, and to make a second-order confidence judgment about their ratings.

From (Lebreton et al., 2015)

Another important class of elicitation techniques is that of incentivecompatible auctions, which involve an incentivization (usually, through monetary prospects) of the accuracy of the rating, respective to the subject's true internal estimate. In the context of outcome value or action cost elicitation, it typically consists in having the subject report the amount of money that she is willing to pay (WTP) in order to acquire a good (Breidert et al., 2006;[START_REF] Lusk | Experimental Auctions: Methods and Applications in Economic and Marketing Research[END_REF], or willing to accept (WTA) in order to abandon a good or perform an effortful action [START_REF] Horowitz | Willingness to accept, willingness to pay and the income effect[END_REF]. Most WTP or WTA elicitations are based on bidding procedures such as the Becker-DeGroot-Marschak (BDM) one [START_REF] Becker | Measuring utility by a single-response sequential method[END_REF], which is still used in modern-day experiments [START_REF] Plassmann | Orbitofrontal Cortex Encodes Willingness to Pay in Everyday Economic Transactions[END_REF][START_REF] Burke | Neural Integration of Risk and Effort Costs by the Frontal Pole: Only upon Request[END_REF][START_REF] Voigt | Hard Decisions Shape the Neural Coding of Preferences[END_REF]. One simple BDM variant consists in having the subject report a WTP (a "bid") which is then compared to a randomly generated number. If that number (concurrent bid) is lower than the subject's bid, then the subject pays the randomly drawn price and receives the item. Otherwise, she does not pay nor receive anything.

Similar auction procedures exist for eliciting beliefs, that is, subjective probabilities [START_REF] Schlag | A penny for your thoughts: A survey of methods for eliciting beliefs[END_REF][START_REF] Burfurd | Experimental guidance for eliciting beliefs with the Stochastic Becker-DeGroot-Marschak mechanism[END_REF]. One of them, called "Matching Probability", consists in the following: 1) the subject gives a probability (or confidence) rating 𝑟, 2) a lottery is run, i.e., a number 𝑙 in [0;1] is randomly selected with uniform probability, 3) if 𝑟 > 𝑙, then a monetary bonus is granted to the subject if and only if her response is correct; if 𝑙 > 𝑟, then another lottery is immediately played, with a 𝑙 probability of leading to the bonus. This procedure has been shown to be robust to idiosyncrasy in risk-attitudes [START_REF] Hollard | In search of good probability assessors: An experimental comparison of elicitation rules for confidence judgments[END_REF], and has been implemented in a couple of recent studies about confidence judgments [START_REF] Lebreton | Two sides of the same coin: Monetary incentives concurrently improve and bias confidence judgments[END_REF][START_REF] Lebreton | Contextual influence on confidence judgments in human reinforcement learning[END_REF][START_REF] Ting | Decreased confidence in loss-avoidance contexts is a primary meta-cognitive bias of human reinforcement learning[END_REF]. Here, the incentive for rating accuracy is a 1€ bonus. From (Lebreton et al.,

2019)

The main advantage of this kind of techniques is that it reduces the reporting biases and idiosyncrasy that might occur in non-incentivized the ratings. On the con side, one may argue that such procedures are too artificial, explicit and complicated, and thus that they are ill-suited to elicit spontaneous judgments and may even be misunderstood by the subjects.

More implicit, indirect approaches exist besides ratings and auctions. As for auction procedures, they typically rest on a utility maximization assumption, but further assume a well-defined model for the computation of the subjective expected value of an action or option. In those cases, which we will call the revealed approaches 1 , several decisional variables (e.g. probability and energetic cost) are objectively known or measurable, which allows inferring an estimate for the remaining dimension (e.g. subjective value) from an observed behaviour such as choosing, producing effort to obtain a reward [START_REF] Pessiglione | How the Brain Translates Money into Force: A Neuroimaging Study of Subliminal Motivation[END_REF][START_REF] Cléry-Melin | Why Don't You Try Harder? An Investigation of Effort Production in Major Depression[END_REF][START_REF] Lopez-Persem | Choose, rate or squeeze: Comparison of economic value functions elicited by different behavioral tasks[END_REF], and waiting for a reward after performing a behaviour [START_REF] Kepecs | Neural correlates, computation and behavioural impact of decision confidence[END_REF][START_REF] Lak | Orbitofrontal Cortex Is Required for Optimal Waiting Based on Decision Confidence[END_REF].

The main merit of such approaches is their implicitness, which prevents any kind of reporting bias from altering the measurement. Yet, because they are indirect, they require a model and objective measurability of the other decisional variables at play in the observed behaviour, to allow for a conversion of this observed behaviour to the variable of interest. Besides, inferences made from binary choices are less precise than a rating, and usually require repeated choice, until the estimate can be confined to a narrow-enough credible range.

In a comparative study recently carried out [START_REF] Lopez-Persem | Choose, rate or squeeze: Comparison of economic value functions elicited by different behavioral tasks[END_REF], subjective value functions of personal gain and charitable donation were elicited either via explicit ratings of various combinations of personal gain and charitable donation, or via more indirect approaches: having the subject choose between two alternative {gain, donation} combinations, or squeeze a handgrip in order to increase the probability to receive each proposed {gain, donation} combination. All three elicitation methods led to similar value functions and were found to be equivalently accurate in the prediction of (out-of-sample) binary choices.

An additional class of measurements relies on physical markers such as salivation, lipping [START_REF] Rushworth | Frontal Cortex and Reward-Guided Learning and Decision-Making[END_REF][START_REF] Abitbol | Neural Mechanisms Underlying Contextual Dependency of Subjective Values: Converging Evidence from Monkeys and Humans[END_REF], or facial expressions of pleasure [START_REF] Berridge | Dissecting components of reward: 'Liking', 'wanting', and learning[END_REF] or displeasure, either in reaction to reward-or punishment-predicting cues, or upon consumption of the reward (or administration of the punishment). This approach is mostly confined to studies on non-human animals, which cannot perform ratings but which tend to display more 1 as in the "revealed preferences" approach in economics, where preferences are inferred from choices systematic and decipherable physical reactions than humans. Even more indirect measures may be exploited, e.g. in infants, such as their looking time after their witnessed someone choosing between options (here, looking time serves as a measure of surprise and thus as an indicator of implicit personal preference) [START_REF] Liu | Ten-month-old infants infer the value of goals from the costs of actions[END_REF].

B. Neural correlates of action selection

In this section, we will briefly review the current understanding of the neural correlates of action selection, which mainly builds upon findings from neuroimaging experiments conducted in the three last decades.

1) Anticipated & experienced value of outcomes

For now, we will focus on basic item valuation processes (such as the one which would be triggered by the following question: "How much would you like to receive an orange now (without any delay nor cost of any sort)?") and put aside downstream valuation processes incorporating e.g. delay, risk, or effort. Thanks to the countless fMRI studies (as well as intracranial measures of neural activity and lesion studies in humans and other animals) that were conducted over the last thirty years in the novel field of "neuroeconomics", it is now well established that, when people are making a judgment about the subjective value of an item, the activity in a stable set of brain regions consistently reflects (or "encodes") the subjective value of the considered item, reported via ratings or inferred from choices. This set of regions mainly includes: the ventromedial prefrontal cortex (vmPFC), the ventral striatum (VS), and the posterior cingulate cortex (PCC); the network they form is known as the Brain Valuation System (BVS) [START_REF] Lebreton | An Automatic Valuation System in the Human Brain: Evidence from Functional Neuroimaging[END_REF][START_REF] Peters | Neural representations of subjective reward value[END_REF][START_REF] Levy | The root of all value: A neural common currency for choice[END_REF][START_REF] Bartra | The valuation system: A coordinate-based meta-analysis of BOLD fMRI experiments examining neural correlates of subjective value[END_REF][START_REF] Clithero | Informatic parcellation of the network involved in the computation of subjective value[END_REF]. vmPFC and ventral striatum (VS). Adapted from [START_REF] Bartra | The valuation system: A coordinate-based meta-analysis of BOLD fMRI experiments examining neural correlates of subjective value[END_REF] BOLD signal in the voxels highlighted in red was positively correlated to subjective value in both anticipation ("decision") and outcome receipt stages, for both monetary and primary (e.g. food) rewards.

Over years of neuroimaging studies, several interesting functional properties of the BVS have been identified.

Genericity

The BVS has been found to signal the subjective value of a very wide range of stimuli or items, be they primary or secondary, concrete or abstract, prospective or retrospective, presented or self-generated: food [START_REF] Chib | Evidence for a Common Representation of Decision Values for Dissimilar Goods in Human Ventromedial Prefrontal Cortex[END_REF]Levy & Glimcher, 2011;[START_REF] Lebreton | Automatic integration of confidence in the brain valuation signal[END_REF], trinkets, tools, toys or other objects [START_REF] Chib | Evidence for a Common Representation of Decision Values for Dissimilar Goods in Human Ventromedial Prefrontal Cortex[END_REF][START_REF] Cunningham | Orbitofrontal cortex provides cross-modal valuation of self-generated stimuli[END_REF][START_REF] Lebreton | Automatic integration of confidence in the brain valuation signal[END_REF], money [START_REF] Izuma | Processing of Social and Monetary Rewards in the Human Striatum[END_REF][START_REF] Chib | Evidence for a Common Representation of Decision Values for Dissimilar Goods in Human Ventromedial Prefrontal Cortex[END_REF]Levy & Glimcher, 2011;[START_REF] Lin | Social and monetary reward learning engage overlapping neural substrates[END_REF], paintings or sculptures [START_REF] Kawabata | Neural Correlates of Beauty[END_REF][START_REF] Dio | The Golden Beauty: Brain Response to Classical and Renaissance Sculptures[END_REF][START_REF] Lebreton | An Automatic Valuation System in the Human Brain: Evidence from Functional Neuroimaging[END_REF][START_REF] Abitbol | Neural Mechanisms Underlying Contextual Dependency of Subjective Values: Converging Evidence from Monkeys and Humans[END_REF], musical pieces [START_REF] Blood | Emotional responses to pleasant and unpleasant music correlate with activity in paralimbic brain regions[END_REF][START_REF] Abitbol | Neural Mechanisms Underlying Contextual Dependency of Subjective Values: Converging Evidence from Monkeys and Humans[END_REF], faces and social stimuli [START_REF] O'doherty | Beauty in a smile: The role of medial orbitofrontal cortex in facial attractiveness[END_REF][START_REF] Izuma | Processing of Social and Monetary Rewards in the Human Striatum[END_REF][START_REF] Lebreton | An Automatic Valuation System in the Human Brain: Evidence from Functional Neuroimaging[END_REF][START_REF] Lin | Social and monetary reward learning engage overlapping neural substrates[END_REF], odours [START_REF] Rolls | Different representations of pleasant and unpleasant odours in the human brain[END_REF][START_REF] De Araujo | Cognitive Modulation of Olfactory Processing[END_REF][START_REF] Grabenhorst | Different representations of relative and absolute subjective value in the human brain[END_REF][START_REF] Howard | Identity-specific coding of future rewards in the human orbitofrontal cortex[END_REF], tastes [START_REF] Plassmann | Marketing actions can modulate neural representations of experienced pleasantness[END_REF], hypothetical events [START_REF] Lebreton | Automatic integration of confidence in the brain valuation signal[END_REF], and so forth.

Therefore, it would seem that activity in the BVS serves as a "neural common currency" for the subjective value of otherwise incommensurable items, and supports our ability to compare and choose, e.g., between a new computer and a vacation trip (Levy & Glimcher, 2011;[START_REF] Levy | The root of all value: A neural common currency for choice[END_REF].

Automaticity

In most studies, neural correlates of subjective valuation have been identified by looking for voxels whose activity was higher when subjects were actively evaluating an item (specifically, a highly-valued one, based on the subject's behaviour) in order to comply with the instructions of a computer to make a value rating or a choice about it. However, the BVS has also been shown to signal the value of attended items or stimuli even when no evaluation or choice was asked of the subject. For instance, in [START_REF] Lebreton | An Automatic Valuation System in the Human Brain: Evidence from Functional Neuroimaging[END_REF], it was more active when subjects were assessing the age of preferred faces, paintings or houses compared to nonpreferred ones (see Fig. 1.14).

The finding that BVS activity during non-valuation oriented viewing of a stimulus predicts later preference manifestations (i.e., choices or likeability ratings) has been replicated in a number of studies [START_REF] Harvey | Monetary Favors and Their Influence on Neural Responses and Revealed Preference[END_REF][START_REF] Tusche | Neural Responses to Unattended Products Predict Later Consumer Choices[END_REF]Levy et al., 2011;[START_REF] Abitbol | Neural Mechanisms Underlying Contextual Dependency of Subjective Values: Converging Evidence from Monkeys and Humans[END_REF]. However, a few other studies have obtained results contradicting the view of a fully automatic BVS. [START_REF] Grabenhorst | Selective attention to affective value alters how the brain processes taste stimuli[END_REF], have reported that the strength of value encoding by the vmPFC was higher when participants focused on the hedonic quality of the taste stimulus (its pleasantness), compared to when they focused on a dimension less related to subjective valuation (its intensity). Along a similar line, [START_REF] Grueschow | Automatic versus Choice-Dependent Value Representations in the Human Brain[END_REF] found a dissociation between the PCC, which encoded subjective values regardless of their relevance to the choice at hand, and the vmPFC, whose value encoding was modulated by the relevance to current the decision-making situation.

Contextual dependency

Two instances of contextual dependency of BVS signals will be discussed here. The first one is due to temporal inertia, such that the offset corresponding to baseline BVS activity preceding the valuation of a given stimulus is conserved during the BVS response to the stimulus, and interestingly, impacts subjective value reports. Abitbol et al., 2015, observed this baseline dependency both in humans and in monkeys. In humans, the baseline activity of the vmPFC was modulated via a manipulation of (external) contextual pleasantness (exposure to more or less pleasant musical background), after which subjects were shown a painting and prompted to make a likeability rating about that painting. As would be predicted by a baseline dependency in the BVS, the experimenters found that participants gave higher likeability ratings to paintings that had been preceded with a pleasant musical background. And indeed, paintings ratings were correlated to vmPFC's pre-stimulus (here, the painting) activity. (See Fig. 1.15.) In monkeys, similar results were obtained, with the exception that baseline activity in vmPFC homolog was not directly modulated by a contextual manipulation, but affected by trial number and thus weariness, which can arguably be classified as an internal contextual influence (as opposed to the external contextual manipulation performed in humans). Another instance of BVS contextual dependency is the modulation of BVS response and, congruently, of behaviourally reported valuation by prior expectations about value. Plassmann et al., 2008, had participants taste Cabernet Sauvignons in an fMRI scanner, and "incidentally" announced the price of each wine before it was sampled. Participants' mOFC (here, the ventral part of the vmPFC) responded very differently to the exact same wine, depending on whether it was labelled as cheap or expensive beforehand. (See Fig. 1.16.) This modulation of neural response by prior value expectations was reflected in their actual liking ratings, which were higher for 'expensive' wine A than for 'cheap' wine A.

These observations were recently replicated in [START_REF] Schmidt | How context alters value: The brain's valuation and affective regulation system link price cues to experienced taste pleasantness[END_REF], and, in fact, extended to both the vmPFC/mOFC and the ventral striatum. Further, this second study demonstrated that the impact of price cues on 'experienced pleasantness' ratings was actually mediated by BVS activity. From [START_REF] Plassmann | Marketing actions can modulate neural representations of experienced pleasantness[END_REF]).

Although we have not distinguished anticipated, decision, and experienced

values until now, it is worth mentioning that these three kinds of (subjective) value have been claimed to refer to distinct psychological processes, and to be underpinned by distinct brain networks. Some scholars like Daniel Kahneman [START_REF] Kahneman | Back to Bentham? Explorations of Experienced Utility[END_REF] or Kent Berridge [START_REF] Berridge | Dissecting components of reward: 'Liking', 'wanting', and learning[END_REF] have proposed an extensive typology of subjective values, or utilities (both terms can be used interchangeably here). Largely borrowing from [START_REF] Berridge | Neuroeconomics: Chapter 18. From Experienced Utility to Decision Utility[END_REF], we will superficially review one possible division of utility into sub-types, how these sub-types are related, and what neural substrates for each of them have been identified to this date. The most intuitive type of utility is perhaps experienced utility. It refers to the extent of pleasure or satisfaction felt by the agent during or immediately after exposure to, or consumption of, an outcome. A network of small, specific brain regions has been causally implicated in signalling experienced value, via the opioid neuromodulatory system. These regions are called "hedonic hotspots" and include a fraction of the ventral striatum, as well as a fraction of the ventral pallidum [START_REF] Peciña | Hedonic Hot Spot in Nucleus Accumbens Shell: Where Do μ-Opioids Cause Increased Hedonic Impact of Sweetness[END_REF][START_REF] Smith | The Ventral Pallidum and Hedonic Reward: Neurochemical Maps of Sucrose "Liking" and Food Intake[END_REF][START_REF] Smith | Opioid Limbic Circuit for Reward: Interaction between Hedonic Hotspots of Nucleus Accumbens and Ventral Pallidum[END_REF]. As seen above, based on fMRI experiments in humans, it seems that experienced utility is also signalled, or encoded, in the mOFC, although the causal (rather than correlational) implication of this region is not clear [START_REF] Kringelbach | The human orbitofrontal cortex: Linking reward to hedonic experience[END_REF]. Moreover, reward prediction error signals have been observed upon delivery (or omission) of rewards (variable in size) in the dopaminergic system (specifically, in the ventral tegmental area (VTA) and its main output, the nucleus accumbens -which is more or less equivalent to the VS) [START_REF] Schultz | A Neural Substrate of Prediction and Reward[END_REF]. This suggests that experienced utility is an input to RPE signals involved in reinforcement (or "habitual") learning. Another utility type in Berridge's framework is remembered utility. It likely derives from (memories of) experienced utility, which may involve distortions both at the encoding and retrieval stages. Remembered utility may involve the hippocampus, which, beyond its well-established role in episodic memory, has been shown to be implicated in subjective valuation of displayed stimuli or imagined outcomes [START_REF] Lebreton | An Automatic Valuation System in the Human Brain: Evidence from Functional Neuroimaging[END_REF][START_REF] Lebreton | A Critical Role for the Hippocampus in the Valuation of Imagined Outcomes[END_REF]. Remembered utility is assumed to inform another type of utility judgment: anticipated, or predicted, utility. This refers to the pleasure or satisfaction that one expects to derive from a potential future outcome. Correlates have been mainly found in the ventral striatum as well as in the vmPFC [START_REF] Diekhof | The role of the human ventral striatum and the medial orbitofrontal cortex in the representation of reward magnitude -An activation likelihood estimation meta-analysis of neuroimaging studies of passive reward expectancy and outcome processing[END_REF][START_REF] Lebreton | Automatic integration of confidence in the brain valuation signal[END_REF][START_REF] Weber | Frontostriatal pathways gate processing of behaviorally relevant reward dimensions[END_REF]. Since humans are able to perform anticipated utility judgments about outcomes they have never experienced before (e.g., "How much do you think you will enjoy your vacation in this country you have never been to?"), it is likely that anticipated utility judgments can be derived through generalization of and comparison to similar past experiences, as well as more abstract reasoning. Finally, a fourth type of utility is decision utility: it is the judgment that directly guides option selection (assuming that agents act such as to maximize their decision utility). Berridge has proposed that decision utility might not only be based on anticipated utility, but might also be influenced by more 'Pavlovian' wanting signals (of "incentive salience") conveyed by the dopaminergic system, which could lead an agent to select an option not because he expects to derive utility from it (as would be the case in goaldirected decision-making [START_REF] Rangel | A framework for studying the neurobiology of valuebased decision making[END_REF]) but out of craving (as is the case, e.g., in drug addiction, and which is supposedly signalled by a Pavlovian system [START_REF] Rangel | A framework for studying the neurobiology of valuebased decision making[END_REF]).

Fig. 1.17: A schematic typology of utility.

Inspired from [START_REF] Berridge | Neuroeconomics: Chapter 18. From Experienced Utility to Decision Utility[END_REF]. Several types of utilities can be distinguished: anticipated utility (left), whose correlates have been found in the VS and vmPFC; decision utility (center); experienced utility (right), whose correlates have been found in so-called "hedonic hotspots" (HH), including a part of the nucleus accumbens (in VS), as well as in mOFC. Experienced utility seems to be signalled by the opioid neuromodulatory system, as well as conveyed in a prediction error signal supported by the dopaminergic (DA) system.

Remembered utility (bottom), may involve the hippocampus.

2) Probability & confidence

Outcome probability

Neural correlates of prospective reward probability have been found, either at a single neuron level or at a population level (BOLD), in a number of different regions across the brain. These include: the dopaminergic midbrain, whose phasic activity during the display of reward-predictive stimuli was found to correlate positively with reward probability [START_REF] Tobler | Adaptive Coding of Reward Value by Dopamine Neurons[END_REF], the basal ganglia, in which the correlation was either positive [START_REF] Oyama | Reward Prediction Error Coding in Dorsal Striatal Neurons[END_REF][START_REF] Berns | Striatal topography of probability and magnitude information for decisions under uncertainty[END_REF] or negative [START_REF] Knutson | Distributed Neural Representation of Expected Value[END_REF][START_REF] Apicella | Tonically active neurons in the striatum differentiate between delivery and omission of expected reward in a probabilistic task context[END_REF], and in particular, the NAcc/ventral striatum whose activity correlates positively with probability [START_REF] Tobler | Reward Value Coding Distinct From Risk Attitude-Related Uncertainty Coding in Human Reward Systems[END_REF][START_REF] Yacubian | Subregions of the ventral striatum show preferential coding of reward magnitude and probability[END_REF], as well as prefrontal cortical regions such as the mPFC [START_REF] Knutson | Distributed Neural Representation of Expected Value[END_REF][START_REF] Wu | The Neural Correlates of Subjective Utility of Monetary Outcome and Probability Weight in Economic and in Motor Decision under Risk[END_REF][START_REF] Studer | What are the Odds? The Neural Correlates of Active Choice during Gambling[END_REF], the lPFC [START_REF] Tobler | Reward Value Coding Distinct From Risk Attitude-Related Uncertainty Coding in Human Reward Systems[END_REF][START_REF] Kennerley | Double dissociation of value computations in orbitofrontal and anterior cingulate neurons[END_REF], and the ACC [START_REF] Wu | The Neural Correlates of Subjective Utility of Monetary Outcome and Probability Weight in Economic and in Motor Decision under Risk[END_REF][START_REF] Kennerley | Double dissociation of value computations in orbitofrontal and anterior cingulate neurons[END_REF], which all encode probability positively.

Despite the extensive spatial overlap between regions encoding reward magnitude (or, not equivalently, subjective value) and reward probability, some studies have highlighted an intra-regional dissociation between areas encoding these two variables, e.g. in the striatum, although the specifics of the dissociation differ from one study [START_REF] Berns | Striatal topography of probability and magnitude information for decisions under uncertainty[END_REF] to the others [START_REF] Yacubian | Subregions of the ventral striatum show preferential coding of reward magnitude and probability[END_REF][START_REF] Tobler | Reward Value Coding Distinct From Risk Attitude-Related Uncertainty Coding in Human Reward Systems[END_REF] (see Fig. 1.18). Adapted from [START_REF] Tobler | Reward Value Coding Distinct From Risk Attitude-Related Uncertainty Coding in Human Reward Systems[END_REF] Finally, it is worth noting that the mPFC in particular has been found to encode reward probability positively, regardless of whether that probability was announced numerically (classical lottery), or modulated via the difficulty of a motor precision task (motor lottery) [START_REF] Wu | The Neural Correlates of Subjective Utility of Monetary Outcome and Probability Weight in Economic and in Motor Decision under Risk[END_REF]. Besides, when that variance increases, it is on average more optimal (in the sense that it minimizes expected error) to give a rating closer to the center of the rating scale. (b) From the rating optimization model, it follows that confidence should vary quadratically with the punctual rating given by the subject on a bounded scale. This is what was observed behaviourally. (c) Traditional view, according to which activity in the vmPFC correlates linearly with subjective value ratings.

(d) In this study, the vmPFC was found to encode both subjective value and confidence positively (and presumably linearly), which, given the quadratic relationship between ratings and confidence, corresponds to an overall hookshaped activity as a function of subjective value ratings, and a parabolic-shaped activity as a function of value-unrelated ratings. Figure from [START_REF] Barron | Reassessing VMPFC: Full of confidence?[END_REF].

3) Effort cost

Although not the focus of this PhD work, we will now briefly review recent neuroimaging-based findings about the neural correlates of physical effort cost judgments and/or effort-discounted action valuation in humans.

The most recurring brain region in studies investigating the neural substrates of effort cost in action valuation is arguably the dorsal anterior cingulate cortex. However, the reported sign of effort encoding varied from one study to the other: it was found to positively encode prospective effort-discounted rewards in the absence of choice (i.e. when a given effort-reward combination was imposed to the subject) [START_REF] Croxson | Effort-Based Cost-Benefit Valuation and the Human Brain[END_REF] or the effort-discounted value of the chosen option when a choice was given between two effort-reward combinations [START_REF] Klein-Flügge | Neural Signatures of Value Comparison in Human Cingulate Cortex during Decisions Requiring an Effort-Reward Trade-off[END_REF]). Yet, it was also found to positively encode anticipated effort or, correspondingly, to negatively encode upcoming effort-discounted value in other studies, where subjects were either imposed a unique effort-reward combination on each trial [START_REF] Kurniawan | Effort and Valuation in the Brain: The Effects of Anticipation and Execution[END_REF], or given the choice to reject it [START_REF] Prevost | Separate Valuation Subsystems for Delay and Effort Decision Costs[END_REF] or given a choice between two (hidden, to be learned) effort-reward combinations [START_REF] Skvortsova | Learning To Minimize Efforts versus Maximizing Rewards: Computational Principles and Neural Correlates[END_REF]. Also based on functional magnetic resonance imaging in humans, other regions seem to be involved, such as: the supplementary motor area, which is adjacent to the dACC and, similarly, was reported to encode anticipated effort positively in [START_REF] Kurniawan | Effort and Valuation in the Brain: The Effects of Anticipation and Execution[END_REF][START_REF] Burke | Neural Integration of Risk and Effort Costs by the Frontal Pole: Only upon Request[END_REF], but effort-discounted outcome value positively in [START_REF] Croxson | Effort-Based Cost-Benefit Valuation and the Human Brain[END_REF], and the difference between chosen and unchosen effort negatively in [START_REF] Klein-Flügge | Neural Signatures of Value Comparison in Human Cingulate Cortex during Decisions Requiring an Effort-Reward Trade-off[END_REF]; the ventral striatum, which was found to positively encode effort-discounted outcome value posterior to effort exertion [START_REF] Kurniawan | Effort and Valuation in the Brain: The Effects of Anticipation and Execution[END_REF]; the dorsal striatum, which was both found to negatively encode high effort versus low effort choices [START_REF] Kurniawan | Choosing to Make an Effort: The Role of Striatum in Signaling Physical Effort of a Chosen Action[END_REF] and to positively encode anticipated effort in other studies [START_REF] Kurniawan | Effort and Valuation in the Brain: The Effects of Anticipation and Execution[END_REF][START_REF] Burke | Neural Integration of Risk and Effort Costs by the Frontal Pole: Only upon Request[END_REF]; the anterior insula, whose activity was positively correlated with anticipated effort in [START_REF] Prevost | Separate Valuation Subsystems for Delay and Effort Decision Costs[END_REF][START_REF] Skvortsova | Learning To Minimize Efforts versus Maximizing Rewards: Computational Principles and Neural Correlates[END_REF].

Finally, some researchers have argued that, when effort-involving valuation is temporally separated from effort exertion, the main regions involved in value discounting by effort cost are the vmPFC and primary motor areas, whose activity correlated negatively with prospective level of effort exertion in [START_REF] Aridan | Neural correlates of effort-based valuation with prospective choices[END_REF].

Overall, the neural network underlying effort cost judgments and effortdiscounting of outcomes seems to include a variety of meso-cortical and striatal regions (dACC, SMA, vmPFC, nucleus accumbens, putamen), whose exact function remains to be clarified. Besides, although substantial evidence from behavioural observations following lesions and intracranial recordings in animal experiments point to a central role of the dopaminergic system in favouring high-effort highreward actions over low-effort low-reward ones [START_REF] Walton | Weighing up the benefits of work: Behavioral and neural analyses of effort-related decision making[END_REF][START_REF] Salamone | The Mysterious Motivational Functions of Mesolimbic Dopamine[END_REF], the emerging consensus is that the dopaminergic system is primarily sensitive to prospective reward, rather than effort costs, and thus promotes reward-directed action [START_REF] Walton | What Is the Relationship between Dopamine and Effort[END_REF].

C. Interferences within and between decisional variables

In this section, we will review a collection of behavioural observations, made either in the field or in laboratory settings, which document intra-or interdimensional interferences between judgments of value, probability, and effort.

1) Intra-dimensional interferences a. Anchors & expectations

Out of the long list of biases that have been reported to affect human judgment, one of the most widely known is arguably the anchoring bias. The anchoring effect has been popularized -and partly theorized -by Kahneman & Tversky [START_REF] Tversky | Judgment under Uncertainty: Heuristics and Biases[END_REF], who observed that participants in their experiments tended to make higher numerical estimates (e.g. about the percentage of African countries in the United Nations, or the length of the Mississippi river) when first provided with a high anchor, that is, an arbitrary initial value from which they had to adjust their estimate (typically, they would first be asked whether the actual value was above or below that arbitrary number drawn by a wheel of fortune, and then to give a specific, point estimate of the actual value). Similarly, they gave lower estimates when provided with a low anchor.

Even before Tversky & Kahneman's seminal paper, anchoring effects had been reported in the domain of physical perception. In particular, judgments about the weight of a "target" object were found to be affected by the weight of an "anchor" object that subjects had lifted just before [START_REF] Harvey | Judgments of weight as affected by adaptation range, adaptation duration, magnitude of unlabeled anchor, and judgmental language[END_REF].

However, in this study, the effect was not one of assimilation as in Tversky & Kahneman's later observations, but instead a contrast effect, where heavy anchors led to lower weight estimates. A more recent and comprehensive study on anchoring of physical perception (including other dimensions than weight, and involving other perceptual modalities than proprioception, such as length (vision) or loudness (audition)) found the opposite pattern of results, that is, an assimilation of estimates to the preceding anchors [START_REF] Leboeuf | The long and short of it: Physical anchoring effects[END_REF]. This divergence with earlier results was attributed by the authors to differences in the way estimates were elicited (anchors and targets were interspersed and all rated sequentially in earlier studies, while later studies followed on Tversky & Kahneman's work and focused on anchoring-and-adjustment, wherein the anchor is presented as a potential estimate that the subject has to adjust subsequently).

Besides affecting judgments of weight (and by extension, effort), anchoring effects have been observed in probability assessments about either personally relevant or irrelevant events, e.g. the likelihood of a nuclear world war breaking out in the next 10 years [START_REF] Plous | Thinking the Unthinkable: The Effects of Anchoring on Likelihood Estimates of Nuclear War1[END_REF][START_REF] Chapman | Anchoring, Activation, and the Construction of Values[END_REF][START_REF] Mussweiler | Overcoming the Inevitable Anchoring Effect: Considering the Opposite Compensates for Selective Accessibility[END_REF], or the likelihood of a stranger being a biology or chemistry major [START_REF] Rottenstreich | Unpacking, repacking, and anchoring: Advances in support theory[END_REF]). In the line of the anchoring-and-adjustment scenario which predicts an assimilation effect, probability judgments were biased upwards by high anchors and downwards by low anchors.

Further, it would seem that anchoring is not restrained to judgments about external variables on which the estimator has no influence, but can also distort beliefs about one's ability to solve upcoming cognitive puzzles, that is, self-efficacy judgments (which are, in our terminology, a type of prospective confidence judgments) [START_REF] Cervone | Anchoring, efficacy, and action: The influence of judgmental heuristics on self-efficacy judgments and behavior[END_REF]. In this study, anchors modulated not only self-efficacy predictions but also actual persistence levels displayed by participants during task execution.

So far, all reported anchoring effects were about objectively measurable variables, but it has been shown that subjective evaluations of, e.g., familiar products, belongings, or hedonic experiences, were also prone to be biased by arbitrary anchors [START_REF] Mussweiler | Overcoming the Inevitable Anchoring Effect: Considering the Opposite Compensates for Selective Accessibility[END_REF][START_REF] Ariely | Coherent Arbitrariness": Stable Demand Curves Without Stable Preferences[END_REF].

Besides incidental anchors, prior expectations have been found to distort judgments, and in particular, judgments of subjective value or likeability. Oddly enough, many of the studies demonstrating such an effect involved the tasting and rating of alcoholic beverages. On each trial, prior expectations about the gustatory qualities of the upcoming sample were set by disclosing some (fake) information about its brand or price. Thus-induced high (resp. low) value expectations led to significantly higher (resp. lower) experienced pleasantness ratings of the exact same beverage [START_REF] Allison | Influence of Beer Brand Identification on Taste Perception[END_REF][START_REF] Lee | Try It, You'll Like It: The Influence of Expectation, Consumption, and Revelation on Preferences for Beer[END_REF][START_REF] Plassmann | Marketing actions can modulate neural representations of experienced pleasantness[END_REF][START_REF] Schmidt | How context alters value: The brain's valuation and affective regulation system link price cues to experienced taste pleasantness[END_REF]. The same effect was observed when subjects were asked to report their enjoyment of vacations or movies instead of beers or wines [START_REF] Klaaren | The role of affective expectations in subjective experience and decision-making[END_REF]. Overall, the distortion of valuation by expectations could be interpreted as the result of an internal, non-numerical anchor (the expected value) which is not sufficiently adjusted for upon experiencing the stimulus. It should be noted that, apart from insufficient adjustment, other accounts have been proposed to explain anchoring effects, see [START_REF] Furnham | A literature review of the anchoring effect[END_REF] for a review.

b. Affect as information about value

In parallel with the stream of research investigating anchoring effects on judgments, another, related stream emerged. Its focus was on how feelings impacted judgments, and in particular, subjective value judgments. It led to the development of the influential feeling-as-information theory [START_REF] Schwarz | Feelings and phenomenal experiences[END_REF].

In this literature, three types of feelings are usually distinguished [START_REF] Clore | Cognitive Phenomenology: Feelings and the Construction of Judgment[END_REF]: affective (e.g. positive mood, anger), bodily (e.g. pain, hunger), and cognitive (e.g. feeling of knowing, ease of processing). Because we are interested in cases of value-value interferences, and because the mapping from value to affect is much more straightforward than to other types of feelings, we will focus on the subconstruct of affect-as-information.

Affects can be decomposed, following Russell's circumplex model, into a valence and an arousal component [START_REF] Russell | A circumplex model of affect[END_REF], where valence refers to the hedonic quality of an affect, ranging from negative to positive, and arousal refers to the level of activation or excitement, ranging from low to high. Another classic distinction regarding affects consists in differentiating emotions from moods.

Although both belong to the class of affective feelings, emotions (such as anger, disgust, or relief) are conceptualised as acute, brief, and associated to an identifiable referent (they are about something), whereas moods are typically more prolonged, less intense, lack a clear referent and mostly unidimensional (moods differ from one another in terms of valence) [START_REF] Schwarz | Feelings-as-Information Theory[END_REF]. A wealth of behavioural studies have manipulated participants' affect, mainly along the valence dimension, mainly in the mood mode, and found repercussions on their pleasantness or likeability judgments about various stimuli or items. In one of the most influential of these studies [START_REF] Schwarz | Mood, misattribution, and judgments of well-being: Informative and directive functions of affective states[END_REF], healthy individuals were asked how satisfied they were with their life, while in a positive or negative mood which was induced either by an episodic recall procedure (Experiment 1) or simply by the current weather (Experiment 2). Quite interestingly, respondents who were in a good mood (e.g. on a sunny day) reported higher levels of life satisfaction than those in a neutral mood, who themselves reported a higher satisfaction than those in a bad mood (e.g. on a rainy day). Yet more interestingly, the effect of bad mood on life satisfaction disappeared when subjects were made aware of the extraneous origin of their mood (in Experiment 2, the interviewer on the phone pretended to be out of town, and inquired about the weather before conducting the life satisfaction survey). These observations implied two things: first, that the influence of affect could be incidental, i.e., occur in a context where affect was modulated by a factor unrelated to the target of the value judgment; second, that this incidental influence, called misattribution of affect, could be moderated by increasing awareness about the (unrelated) source of this affective state.

On top of this empirical work, Schwarz and Clore made an important contribution to the field in the form a theoretical proposal: when an individual is making an evaluation, her current affective state is incorporated as information about the target of the evaluation. As the authors have pointed out, this incorporation is not necessarily a bad thing for the accuracy of the evaluation, especially when the evaluation is inherently subjective ("How much do you like this song?") and affect is mainly modulated by the target itself. Of course, it can be more problematic when the affective state is modulated by a factor irrelevant to the judgment at hand: for instance, academics might evaluate a PhD applicant less positively if they have just received a rejection notice from a journal, or if the room in which they conduct the interview has a stale smell.

Following on Schwarz & Clore's seminal paper, several studies reported similar effects of affect misattribution in value judgments, about e.g. stereo speakers, or medical school candidates [START_REF] Schwarz | Soccer, rooms, and the quality of your life: Mood effects on judgments of satisfaction with life in general and with specific domains[END_REF][START_REF] Gorn | Mood, awareness, and product evaluation[END_REF][START_REF] Redelmeier | Rainy weather and medical school admission interviews[END_REF].

The factors moderating the extent to which affects are (mis)attributed to a target during a value judgment have been reviewed recently, based on decades of empirical work [START_REF] Greifeneder | When Do People Rely on Affective and Cognitive Feelings in Judgment? A Review[END_REF]. Four main factors have received extensive consensus:

1) Salience of the affect: on average, more salient feelings are more incorporated into judgments. However, misattribution effects have also been found when feelings were induced subliminally [START_REF] Winkielman | Unconscious Affective Reactions to Masked Happy Versus Angry Faces Influence Consumption Behavior and Judgments of Value[END_REF], and on the contrary, it has been consistently shown that making the source of the feeling cognitively salient typically decreases misattribution effects [START_REF] Schwarz | Feelings-as-Information Theory[END_REF].

2) Representativeness of the affect for the target being judged: the more an affect is being viewed as caused by or about the target, the stronger the (mis)attribution.

3) Relevance of the affect to the judgment: the more relevant an affect is perceived, the stronger the (mis)attribution. For instance, when evaluating the professional competence of a co-worker, feelings of sympathy towards this co-worker, although representative in the sense introduced above, may be considered irrelevant and thus put aside from the evaluation process. 4) Malleability of the judgment: if, aside from affect, cues to construct a judgment are scarce or ambiguous, then this judgment is said to be more malleable (e.g., when evaluating the trustworthiness of politicians versus their gender, or how good a movie is versus how much it costs to buy it), and affect is predicted to have a larger impact upon it.

2) Inter-dimensional interferences a. Value -probability

Judgment interdependences are not restricted to intra-dimensional cases. In fact, the literature documenting influences of value judgments on probability judgments might be even more abundant. One of the most famous instances of such (supposedly unwarranted) influence is the so-called desirability bias.

Over the last seventy years, it has been repeatedly observed that healthy people tend to perceive relatively more desirable (future or hidden) states of the world as relatively more likely. In a seminal study [START_REF] Marks | The effect of probability, desirability, and 'privilege' on the stated expectations of children[END_REF], Marks presented children with decks of cards, a proportion of which were picture cards, while the others were blank. After orally stating the proportion of picture cards in a given deck, she asked the children to predict if they were going to draw a picture card.

Considerably more children predicted a picture card draw when it meant winning one point versus losing one point. This finding was generalized to adults two years later [START_REF] Irwin | Stated expectations as functions of probability and desirability of outcomes[END_REF].

Since then, positive distortions of the perceived probability of an event by its desirability have been consistently observed in a wide range of paradigms, mainly involving games of chance (e.g., cards drawn from decks, beads drawn from jars), in which a subset of outcomes were incentivized monetarily or via other types of reward (e.g. food or raffle tickets), and where the dependent variable was either a binary prediction, a probability assessment, or a bet from the participant (Bar-Hillel & Budescu, 1995;[START_REF] Budescu | The relationship between the illusion of control and the desirability bias[END_REF][START_REF] Lench | Automatic optimism: Biased use of base rate information for positive and negative events[END_REF][START_REF] Lench | Motivated perception of probabilistic information[END_REF][START_REF] Lench | Moving toward a brighter future: The effects of desire on judgments about the likelihood of future events[END_REF]; see [START_REF] Krizan | The influence of outcome desirability on optimism[END_REF] for a comprehensive review of empirical evidence about the desirability bias).

It has been argued [START_REF] Chambers | Biases in social comparative judgments: The role of nonmotivated factors in above-average and comparative-optimism effects[END_REF]) that the desirability bias may underlie the related optimism bias, which also pervades in the general population. Overoptimism consists in overestimating the probability of positive events, and underestimating that of negative ones. In a now classic study [START_REF] Weinstein | Unrealistic optimism about future life events[END_REF], Weinstein asked hundreds of college students to judge how much more or less likely than their classmates they were to experience various positively or negatively valenced events in the future, such as living past 80 or divorcing a few years after getting married. He found that on average, participants thought themselves more likely than others to experience positive events, and less likely to experience negative ones. This observation of a propensity for comparative optimism has since been replicated in various settings [START_REF] Weinstein | Egocentrism as a Source of Unrealistic Optimism: Personality and Social Psychology Bulletin[END_REF][START_REF] Burger | The Illusion of Unique Invulnerability and the Use of Effective Contraception[END_REF][START_REF] Baker | When every relationship is above average[END_REF][START_REF] Helweg-Larsen | Do Moderators of the Optimistic Bias Affect Personal or Target Risk Estimates? A Review of the Literature: Personality and Social Psychology Review[END_REF].

Given the recurring observation that healthy individuals are unrealistically optimistic about future events, some researchers asked the following question: how can individuals stay overly optimistic in the face of the reality, which, by definition of unrealistic optimism, regularly brings disconfirmation and disappointment?

Sharot and her team have proposed that individuals do not cognitively absorb bad news as extensively as they absorb good news [START_REF] Sharot | How unrealistic optimism is maintained in the face of reality[END_REF][START_REF] Sharot | How Dopamine Enhances an Optimism Bias in Humans[END_REF]. In their experimental paradigm, participants sequentially estimated their likelihood of experiencing various adverse life events (e.g., robbing, Alzheimer's disease). After each prior estimate, the actual rate of occurrence was disclosed to them, and later in the testing session, they were again prompted to estimate their risk, a posteriori. The researchers found that the update of personal risk estimates following information about the actual base rate was larger when this information had come as good news (the statistical risk being lower than their prior estimate) and it implied decreasing one's risk estimate, than when it had come as bad news, and implied increasing one's risk estimate. This valence-dependent belief updating upon information receipt is evocative of a more recent finding from the same team, regarding information-seeking [START_REF] Charpentier | Valuation of knowledge and ignorance in mesolimbic reward circuitry[END_REF]. This study was motivated by the observation that, although people tend to value knowledge positively and to seek information actively, they sometimes reject opportunities to receive bad but useful news, e.g. about their current health status [START_REF] Thornton | The Demand for, and Impact of, Learning HIV Status[END_REF][START_REF] Persoskie | Association of cancer worry and perceived risk with doctor avoidance: An analysis of information avoidance in a nationally representative US sample[END_REF][START_REF] Dwyer | Predicting avoidance of skin damage feedback among college students[END_REF]. And indeed, participants were willing to pay to avoid negatively-valenced information about a potential monetary loss they may have incurred. This result lends some additional support to earlier proposals -empirically well-grounded already [START_REF] Karlsson | The ostrich effect: Selective attention to information[END_REF][START_REF] Ganguly | Fantasy and Dread: The Demand for Information and the Consumption Utility of the Future[END_REF] -that belief states carry subjective value in and of themselves, in the sense that, all else being equal (and in particular, when one cannot inflect a state of affairs upon knowing about it), holding the belief that you are healthy and well-liked by all is more valuable than holding the belief that you have cancer and that your relatives secretly despise you [START_REF] Caplin | Psychological Expected Utility Theory and Anticipatory Feelings[END_REF][START_REF] Kőszegi | Utility from anticipation and personal equilibrium[END_REF][START_REF] Bénabou | The Economics of Motivated Beliefs[END_REF][START_REF] Golman | Information gaps: A theory of preferences regarding the presence and absence of information[END_REF].

Furthering this view, a positive correlation has been found in the general population between trait optimism and psychological (or even physical) well-being [START_REF] Taylor | Illusion and well-being: A social psychological perspective on mental health[END_REF][START_REF] Scheier | Dispositional optimism and recovery from coronary artery bypass surgery: The beneficial effects on physical and psychological well-being[END_REF][START_REF] Taylor | Psychological resources, positive illusions, and health[END_REF]. Besides, clinical or sub-clinical depressive states have been linked to an absence of optimism bias (cf.

"depressive realism", [START_REF] Moore | Depressive realism: A meta-analytic review[END_REF]).

As we have seen in section A., a type of probability judgment that is particularly relevant for action selection is confidence. Since confidence refers to the subjective probability of a desirable state (being correct or successful), one could expect, from the aforementioned evidence about desirability bias and overoptimism, that healthy individuals would also tend to be overconfident. This is indeed what has been found in numerous instances, in the field as well as in the laboratory. In one of the earliest and most famous studies about overconfidence [START_REF] Svenson | Are we all less risky and more skillful than our fellow drivers?[END_REF], Svenson asked participants to situate their own driving skills respective to the other subjects in the room (all college students at the same university). In each of the several experiments that were conducted, half of the testing group placed themselves in the best 25%, and more than 80% of the respondents regarded themselves as above the median driver, both in terms of competence and safety, which means that a large proportion of the interviewees were probably overconfident about their skills. Beyond this better-than-average effect [START_REF] Alicke | The Self in Social Judgment[END_REF][START_REF] Sedikides | Self-Enhancement: Food for Thought: Perspectives on Psychological Science[END_REF] which is a manifestation of relative overconfidence, heathy individuals tend to exhibit absolute overconfidence, that is, they tend to overestimate their skills or their past or future performance in a wide array of laboratory or real-life tasks [START_REF] Baranski | The calibration and resolution of confidence in perceptual judgments[END_REF][START_REF] West | The domain specificity and generality of overconfidence: Individual differences in performance estimation bias[END_REF][START_REF] Camerer | Overconfidence and Excess Entry: An Experimental Approach[END_REF][START_REF] Dunning | Flawed Self-Assessment: Implications for Health, Education, and the Workplace[END_REF][START_REF] Berner | Overconfidence as a cause of diagnostic error in medicine[END_REF]. As for overoptimism, overconfidence has been linked to better mood and well-being [START_REF] Taylor | Illusion and well-being: A social psychological perspective on mental health[END_REF][START_REF] Taylor | Psychological resources, positive illusions, and health[END_REF], higher achievement in competitive domains, from academia to war by way of entrepreneurial innovation [START_REF] Koellinger | I think I can, I think I can": Overconfidence and entrepreneurial behavior[END_REF]Johnson et al., 2011;[START_REF] Hirshleifer | Are Overconfident CEOs Better Innovators[END_REF][START_REF] Stankov | Confidence: A better predictor of academic achievement than self-efficacy, self-concept and anxiety?[END_REF][START_REF] Chen | Looking at the Bright Side: The Motivation Value of Overconfidence[END_REF], and has been demonstrated to provide an evolutionary advantage to its carriers, under certain environmental circumstances [START_REF] Johnson | The evolution of overconfidence[END_REF]. Even in the clinical domain, high mood and overconfidence seem to co-occur, since excessive confidence is a diagnostic criterion of pathological mania, which, contrary to many psychiatric conditions, is often accompanied with hyperthymia (DSM -5, American Psychiatric Association,

2013).

Aside from the fact that it feels good to feel confident, it has been suggested that fluctuations in affective valence can induce fluctuations in confidence: in the lab, subjects whose mood was modulated upwards through an induction procedure reported higher levels of confidence (or self-efficacy, in the terminology of the paper) about various activities [START_REF] Kavanagh | Mood and self-efficacy: Impact of joy and sadness on perceived capabilities[END_REF]; in real stock markets, traders were found to take more risks when the sun shines [START_REF] Saunders | Stock Prices and Wall Street Weather[END_REF][START_REF] Hirshleifer | Good Day Sunshine: Stock Returns and the Weather[END_REF][START_REF] Bassi | O Sole Mio: An Experimental Analysis of Weather and Risk Attitudes in Financial Decisions[END_REF], and less risks in gloomy winter days [START_REF] Kamstra | Winter Blues: A SAD Stock Market Cycle[END_REF], or when their favourite sports team just lost an important match [START_REF] Edmans | Sports Sentiment and Stock Returns[END_REF]. However, the modulation of risk-aversion by mood may not be imputable to a change in confidence, or even optimism (given that stock trading outcomes are not entirely dependent on the trader's behaviour), but instead to a change in valuation: as we have seen above, moods can distort the perceived value of obtained outcomes, and might also distort that of anticipated ones.

On top of the incidental effects of mood, confidence is susceptible to influences from the overall value of the context in which this judgment is made, or from the expectancy of incurring gains or losses. In [START_REF] Maddux | Self-efficacy expectancy and outcome expectancy: Their relationship and their effects on behavioral intentions[END_REF], participants' "self-efficacy expectancy", that is, prior confidence about being able to perform a behaviour correctly, increased with increasing outcome expectancy (in terms of the expected gain resulting from a correct behaviour -no losses were involved). More recently [START_REF] Lebreton | Two sides of the same coin: Monetary incentives concurrently improve and bias confidence judgments[END_REF], also in a controlled experimental setting, it was found that participants were more confident a posteriori about their perceptual judgment when confidence ratings were incentivized by a gain prospect (via a "matching probability" procedure, see section A.5)) compared to the neutral condition (no incentivization), and less confident when ratings were incentivized by a loss prospect. The same team later found that, in a reinforcement learning paradigm, participants were less confident in choosing the best option (in terms of expected outcome) when they were learning to minimize losses, compared to when they were learning to maximize gains [START_REF] Lebreton | Contextual influence on confidence judgments in human reinforcement learning[END_REF].

b. Value -effort

Unlike interdependences between value and probability judgments, which are usually irrational, those linking outcome value or probability judgments with effort judgments are often sensible: all else being equal, in most real-life situations, exerting more effort in a task yields larger rewards or smaller losses with higher probability, and larger prospective rewards motivate higher effort exertion. Due to this general correlation, when direct cues on which to build a prospective or retrospective effort judgment (e.g. the weight of a bag to carry, the slope of a running trail) are scarce, information about expected or obtained outcome value can be rightfully incorporated into this effort cost estimation. And indeed, healthy individuals were found to adjust their retrospective effort judgments about a laboratory task (Pooresmaeili et al., 2015, see Fig. 1.23) as a function of the reward received upon task completion, whose magnitude was, on average and by design, contingent on task difficulty (and thus effort cost). In line with a Bayesian cue integration mechanism, the incorporation of reward as a source of information about effort became more extensive as the effort-reward actual contingency increased. This finding was recently extended to vicarious effort judgments: participants judged the effort exerted by other participants as higher when their effort exertion was followed by a larger reward [START_REF] Rollwage | I judge you by your profit: Judgments of effort exerted by self and others are influenced by task contingent rewards[END_REF].

Interestingly, reward information was integrated to a larger extent in effort judgments about others than about themselves, in accordance with the predictions of Bayesian optimal integration (the scarcer other information (e.g., proprioceptive feedback) is about effort produced, the stronger the influence of the reward source of information), and in contradiction with what would be entailed by an attribution bias, whereby individuals attribute their own success to effort and talent, but others' to luck [START_REF] Weiner | Attribution Theory, Achievement Motivation, and the Educational Process[END_REF]). alternately pressing two keys on the keyboard), subjects received a reward of variable size, and subsequently estimated the effort they had just expended. To make the purpose of the experiment less transparent, subjects were instructed to watch for a possible colour change of the ball while they were pushing it up the ramp, and to report it after their effort rating.

However, aside from supporting rational Bayesian inferences, the value of effort outcomes (or incentives) has been shown to bias the retrospective perception of exerted effort in the opposite direction to what structural effortoutcome causalities should entail. In [START_REF] Schmidt | Get Aroused and Be Stronger: Emotional Facilitation of Physical Effort in the Human Brain[END_REF], participants reported having exerted less effort, or rather, retrospectively perceiving an exerted effort as less costly when being more rewarded for it.

Conversely, it has been convincingly demonstrated that before the age of one, humans are able to infer other people's preferences towards various items from the effort they deploy to reach them [START_REF] Liu | Ten-month-old infants infer the value of goals from the costs of actions[END_REF].

Nevertheless, it would seem that sometimes, people also infer their own preferences, i.e., the subjective value of various outcomes, from their own or other people's efforts exerted towards those outcomes. This has been observed in numerous instances.

In one of them (Kruger et al., 2004), subjects attributed a higher value and reported a higher liking of poems, paintings, or suits of armour that allegedly took more time and effort to be created (by others). The authors conjecture that this effect reflects a cognitive "effort heuristic". As they further note, participants were more influenced by information about effort when the quality of the item was more ambiguous and difficult to evaluate, consistent with most views about cue integration, from affect-as-information wherein malleability is proposed to moderate distortion by affect, to Bayesian optimal integration theory.

Another, related instance where value judgments about an item are influenced by the effort invested into the production of that item is the so-called "IKEA effect", referring to the propensity of (healthy) individuals to ascribe a higher value (expressed, e.g., through WTA auctions) to objects that they have effortfully but successfully assembled themselves compared to identical but preassembled objects [START_REF] Norton | The IKEA effect: When labor leads to love[END_REF].

Very similar is the effort justification effect, according to which the fact that individuals assign a higher value to outcomes obtained after more effort exertion [START_REF] Aronson | The effect of severity of initiation on liking for a group[END_REF]) results from a need to reduce cognitive dissonance, and to justify effort expenditure by the value of its outcome.

In line with those behavioural effects, recent neuroimaging studies have found higher neural responses to rewards following higher effort exertion [START_REF] Hernandez Lallement | Effort increases sensitivity to reward and loss magnitude in the human brain[END_REF][START_REF] Ma | I endeavor to make it: Effort increases valuation of subsequent monetary reward[END_REF].

As we have seen so far, most distortions exerted by effort on value are observed in retrospective settings, that is, when effort is already expended. However, some cases of prospective distortions of value by effort have been highlighted in a recent review [START_REF] Inzlicht | The Effort Paradox: Effort Is Both Costly and Valued[END_REF], which further suggested that, at least in some circumstances, effort might intrinsically carry positive value. If this is the case, effort could influence cost-benefit computations through other, more subtle (and opposite) ways than it does in classical effort discounting (where effort bears negative value). This positive influence could take two distinct computational forms. In traditional cost-benefit models describing effort allocation, the positive value of effort could either be expressed as an enhancement of outcome value (in line with the IKEA and effort justification effects, but assuming that this distortion also applies prospectively), or as an additional value term (which would be, e.g., proportional to the amount of effort deployed), or both. Despite these amendments, an implicit assumption has remained enclosed in these models until today, that of an independence of judgment between the various decisional variables entering the cost-benefit computation: subjective value of outcomes, outcome (subjective) probability, and (subjective) costs entailed by the considered course of action (e.g. due to effort). In recent years, the neural substrates of these decisional variables have begun to be uncovered, while still reasoning under that assumption of mutual independence.

Yet, over the last five decades, psychologists have pointed out a number of instances in which value, probability, and effort judgments can influence, or "contaminate" each other. However, the computational and neural mechanisms underpinning these interferences have scarcely been explored.

In the work presented below, we sought to elucidate these mechanisms.

We conducted three studies on healthy human volunteers, with the aim of answering the following questions:

1) How does the prospective value of the endpoint of a sequence of actions (i.e., the goal) distort the amount of effort we think that action sequence will entail, at a behavioural and computational level? (See Chapter II: Article 1.)

2) When the outcome of an action is contingent on our own behaviour (or "performance"), how does the value of the possible success/failure outcomes distort our confidence that we will perform this action correctly, at a behavioural and neural level? (See Chapter III: Article 2.)

3) Can our confidence in a given behaviour influence how much we appreciate an unrelated but concurrent stimulus, and vice-versa? If so, are we able to prevent such interferences from arising when motivated to provide accurate judgments? (See Chapter IV: Article 3.)

Chapter II: Article 1

No sweat, no sweet? A motivational account of how expected benefits inflate anticipated effort cost. of choice, about whether or not to trade an effort for a reward, has been investigated and modelled in neuro-economic studies. The general idea of decision-making theory is that the subjective value of the goal (defined as a desired state) is discounted by some function of the cost [START_REF] Hull | Principles of behavior: An introduction to behavior theory[END_REF][START_REF] Mitchell | Discounting the Value of Commodities According to Different Types of Cost[END_REF][START_REF] Rangel | A framework for studying the neurobiology of valuebased decision making[END_REF][START_REF] Kable | The Neurobiology of Decision: Consensus and Controversy[END_REF][START_REF] Schmidt | Neural Mechanisms Underlying Motivation of Mental Versus Physical Effort[END_REF][START_REF] Klein-Flügge | Behavioral Modeling of Human Choices Reveals Dissociable Effects of Physical Effort and Temporal Delay on Reward Devaluation[END_REF][START_REF] Chong | Neurocomputational mechanisms underlying subjective valuation of effort costs[END_REF] (related to the energetic expense). Crucially, these theories make the implicit assumption that individual judgments about goal value and effort cost are made independently from each other, before being integrated into a discounted value on which decision is based. More generally, most choice models assume that the different costs integrated in the decision value computation (not only effort, but also delay, risk etc.) are estimated separately, a principle known as the independence axiom.

Yet, it has been several decades since the first reports of an interplay between goal value and effort cost judgments have appeared in the psychology literature [START_REF] Lewis | Psychological effect of effort[END_REF]. One line of research has described numerous instances in which exerted effort inflates the perceived value of the associated outcome [START_REF] Aronson | The effect of severity of initiation on liking for a group[END_REF][START_REF] Arkes | The Psychology of Windfall Gains[END_REF][START_REF] Inzlicht | The Effort Paradox: Effort Is Both Costly and Valued[END_REF]. Some authors have considered this phenomenon as an attempt to justify effort expenditure, following on a need to reduce cognitive dissonance [START_REF] Inzlicht | The Effort Paradox: Effort Is Both Costly and Valued[END_REF]. For example, yielding to the socalled "IKEA effect" [START_REF] Norton | The IKEA effect: When labor leads to love[END_REF], individuals tend to ascribe greater value to an object that they assembled themselves compared to one that was assembled by an expert. A distinct line of research has documented cases in which the value of a pursued goal modulates one's retrospective judgment about the amount of effort invested to reach it. For example, people tend to incorporate the monetary reward into their retrospective judgment of effort intensity, using the positive correlation between reward and effort in a Bayes-optimal fashion [START_REF] Pooresmaeili | Receipt of reward leads to altered estimation of effort[END_REF][START_REF] Rollwage | I judge you by your profit: Judgments of effort exerted by self and others are influenced by task contingent rewards[END_REF]. Thus, the foundational assumption of mutual independence between judgments of effort cost and goal value appears at odds with a large body of empirical observations. However, all cases of mutual influence between effort and value reported in the literature were only observed once effort was already expended. The question of whether effort cost judgments might be modulated by goal value prospectively has so far received little attention. Yet this seems an important question, given that decisions are obviously based on prospective, not retrospective, effort cost estimates.

The most widespread instance of systematic cost distortion is perhaps the "planning fallacy", that is, the tendency observed in healthy agents to underestimate the time necessary to complete a task or a project (Kahneman & Tversky, 1979a;[START_REF] Buehler | Exploring the 'Planning Fallacy': Why People Underestimate Their Task Completion Times[END_REF][START_REF] Brunnermeier | An Economic Model of the Planning Fallacy[END_REF], even when they have extensive experience about it . While this fallacy has mainly been reported in the context of delay estimation, it also occurs for other types of cost, such as money [START_REF] Peetz | Is There a Budget Fallacy? The Role of Savings Goals in the Prediction of Personal Spending: Personality and Social Psychology Bulletin[END_REF][START_REF] Peetz | Different goals, different predictions: Accuracy and bias in financial planning for events and time periods[END_REF] and effort [START_REF] Morgenshtern | Factors affecting duration and effort estimation errors in software development projects[END_REF]. The planning fallacy has been proposed to result from an excessive focus on the future goal, to the detriment of information accumulation about both the intermediate steps to be taken, and declarative or experiential knowledge about the task at hand [START_REF] Buehler | Planning, personality, and prediction: The role of future focus in optimistic time predictions[END_REF][START_REF] Kruger | If you don't want to be late, enumerate: Unpacking reduces the planning fallacy[END_REF]. This attentional bias could have been selected to favour engagement in strenuous actions, by partially ignoring their costs. If more attractive goals capture more attention, then effort costs should be even more underestimated when the goal is assigned a higher value.

Therefore, our research aimed to 1) assess the potential impact of goal value on prospective effort cost judgments, and if present, 2) elucidate the cognitive mechanisms underlying this impact. To this end, we set up a novel experimental paradigm in which healthy participants prospectively appraised the energetic cost of various action sequences leading to a more or less valuable outcome. The action sequence was a run on a treadmill, composed of multiple segments with varying speed and slope, which was rewarded with a financial payoff of varying magnitude. Specifically, participants first viewed animations depicting a cartoon character running along a more or less effortful multi-segment route, and subsequently estimated the energetic cost of the route. Critically, each route was explicitly paired with a monetary incentive, whose magnitude was varied orthogonally to energetic cost fluctuations. In order to trigger decision-oriented cost/benefit computations, participants were offered a choice between a high-effort high-value new route and fixed low-effort low-value reference route. We ensured that cost judgments and incentives had personal significance to the participants by having a randomly selected fraction of their choices implemented in reality, meaning that they were requested to perform the chosen action sequence on the treadmill, and were given the associated payoff. In line with our planning fallacy hypothesis, we predicted that participants would anticipate a lower energetic cost for routes paired with a higher-value monetary outcome.

Results -Experiment 1

With this experiment, we intended to measure the impact of goal value onto the anticipated effort cost of an action sequence. We used composite running routes, each consisting of 4 segments of varying slope and speed, as the action sequences about which participants would make a priori energetic cost judgments. We manipulated the value attached to the completion of each route by offering monetary rewards of varying magnitude (from 0.01€ to 5€). Specifically, in each trial, participants were first presented with the picture of a monetary amount (the incentive, or potential reward) on a computer screen, followed by a short animation depicting a cartoon character running along a computer-generated route (see Fig. 1). Next, subjects were required to choose between accepting this High-Effort High-Gain (or HEHG) offer (namely, running this route on a treadmill in order to earn the displayed incentive) or discarding it in favour of a fixed Low-Effort 0-Gain (or LE0G) option (namely, running a default 'comfort' route for no reward). Last, they were prompted to estimate the energetic cost of this route, which was our main behavioural measure of interest, by placing a cursor along an analog scale.

Of note, trials were grouped in blocks of 18, and once a block had been completed, one trial was randomly drawn from this block and executed: participants were asked to complete the selected running route (or the comfort route if they had declined the HEHG offer) on a treadmill, and the attached monetary reward was then added to their final payment. This procedure was intended to reinforce the personal relevance of the displayed reward magnitudes and of the energetic cost estimates made by our participants. Middle row: Structure of a session. Each session includes a training phase, 3 blocks of 18 trials, a recalibration of AEC ratings, and 3 additional blocks.

Bottom row:

Random draw performed at the end of each block among its 18 just-completed trials. Choice made by the participant during the randomly selected trial is reminded and then executed: if the participant accepted the HEHG offer, she must get onto the treadmill and complete the effortful running route, after which the concurrently displayed reward is added to her total earnings. If she declined the HEHG offer, she must complete a default low-effort running route, and her total earnings remain unchanged.

Critically, the reward magnitude and the real energetic cost (REC) of the route were orthogonal by design, so the incentive cue was in no way informative about the rating that participants should provide. Before the experiment, participants were trained to estimate the REC of various running routes, whose objective value was computed using standard equations in the physical exercise literature. Participants were trained again with objective feedbacks between the two halves of the experiments. Also, it was in their interest to make sound judgments during the experiment, as a monetary bonus could be earned on the basis of rating accuracy. Thus, any impact of incentive level we might observe on the anticipated energetic cost (AEC) would qualify as a bias (i.e., a manifestation detrimental to local optimality) rather than as the result of an appropriate inference.

Twenty-two healthy adults gave their consent to participate in this experiment and were tested one after the other.

Choice

As illustrated in Fig. 2a, participants tended to accept the HEHG option more often as the attached incentive level increased, and as the REC level decreased (with a sigmoid-like nonlinearity in the acceptance rate curve, characteristic of floor and ceiling effects for low and high HEHG values, respectively). For a more quantitative yet straightforward assessment of these effects, we inverted a sigmoidal model of choices between HEHG and LE0G options, including 𝑅𝐸𝐶 𝐻𝐸𝐻𝐺 and 𝐼𝑛𝑐𝑒𝑛𝑡𝑖𝑣𝑒 𝐻𝐸𝐻𝐺 as explanatory variables. Choice data (but not rating data) from 4 participants were excluded before model inversion, due to a total rate of "No" (resp. "Yes" for 1 participant) choices ≤ 5/108, i.e. a percentage of "Yes" (resp. "No") choices > 95%. Unsurprisingly, we found a negative effect of 𝑅𝐸𝐶 𝐻𝐸𝐻𝐺 (𝛽 ̅ = -1.1, 𝐶𝐼 = [-1.5; -.62], 𝑡(17) = -5.0, 𝑝 = 1.2 • 10 -4 ), and a strong positive effect of 𝐼𝑛𝑐𝑒𝑛𝑡𝑖𝑣𝑒 𝐻𝐸𝐻𝐺 (Fig. 2a

Anticipated Energetic Cost (AEC)

As expected, AEC ratings about routes were strongly influenced by the Real Energetic Cost (REC) of these routes (𝛽 ̅ = 15, 𝐶𝐼 = [14; 17], 𝑡(21) = 20, 𝑝 < 10 -14 ), whose effect was tested in a generalized linear model (GLM) that included both REC and Incentive Level as factors of interest.

This effect was significant despite a 'regression-to-the-mean' bias (i.e. overestimation for low levels and underestimation for high levels, apparent on Fig. 2b, top). Thus, after training subjects were able to estimate energetic cost of a running route with good accuracy (82 ± 3 %), defined as the complement to distance from target (REC-AEC, see methods).

Most interestingly, we found a robust positive effect of incentive level (i.e., ordinal value) on AEC ratings (Fig. 2b bottom, 𝛽 ̅ = 2.0, 𝐶𝐼 = [1.2; 2.7], 𝑡(21) = 5.3, 𝑝 = 2.8 • 10 -5 ). That is, all other things being equal, subjects tended to anticipate a higher energetic cost for better-rewarded running sequences. Because the relation between incentive level and AEC ratings seemed in fact slightly supralinear (see Fig. 2b), and in order to test more directly for the effect of the incentive magnitude (i.e., cardinal value, which is the feature primarily relevant to participants) we inverted the following complementary model:

𝐴𝐸𝐶 = 𝛽 0 + 𝛽 1 • 𝑅𝐸𝐶 + 𝛽 2 • 𝐼 𝑚 𝛼
where 𝐼 𝑚 is the incentive magnitude and 𝛼 a nonlinearity parameter which we expected to be below 1, in line with the typical observation of a concave magnitude-utility mapping [START_REF] Schoemaker | The Expected Utility Model: Its Variants, Purposes, Evidence and Limitations[END_REF][START_REF] Glimcher | Neuroeconomics: Decision Making and the Brain[END_REF].

We found a significant positive impact of the transformed incentive magnitude onto AEC (𝛽 2 ̅̅̅ =

.32, 𝐶𝐼 = [. 18; .46], 𝑡(21) = 4.7, 𝑝 = 1.1 • 10 -4 ), with an average nonlinearity parameter 𝛼 ̅ = 0.74.

Given the strong positive relation between incentive magnitude and acceptance rate of the HEHG option, one could hypothesize that the impact of Incentive on AEC was fully mediated by choices, with 'Yes' choices leading to higher cost estimates than 'No' choices. However, the incentive effect on AEC remained strongly significant even when the 𝐼𝑛𝑐𝑒𝑛𝑡𝑖𝑣𝑒 predictor was Another mechanism could have accounted for this phenomenon: despite the orthogonality of incentive and REC in our design, subjects may have assumed that these two variables were correlated in the task, just like effort and reward are generally correlated in real-life situations, and they may have used the incentive cue to inform their AEC estimate. Therefore, we sought to appraise the persistence of this incentive effect across session blocks, expecting that, if such a confound were true, the effect would decrease or even disappear as subjects progressed through the session and sampled more {incentive, REC} couples.

In a GLM of AEC ratings including REC, incentive level, block index, as well as the three Thus, the overall effect was not driven by the first blocks of trials, but was instead maintained across the whole session. This observation casts doubt on a scenario in which the inflation of AEC by prospective reward would be solely due to a strong prior belief in a rewardeffort correlation in our task. Yet further investigation is needed before we can confidently rule this scenario out, which we pursued in Experiment 2.

Interim Discussion

In Experiment 1, we observed an unexpected positive impact of potential reward onto anticipated effort cost. Because it was opposite to our predictions, a replication study was necessary to better establish this finding. We took this opportunity to specify the underlying cognitive mechanisms. Below are described the three types of cognitive mechanisms that we considered, hereafter denoted as the 'belief', 'affect' and 'motivation' scenarios.

First, given that most healthy individuals hold the belief that obtaining more reward typically requires investing more effort [START_REF] Nalebuff | Prizes and Incentives: Towards a General Theory of Compensation and Competition[END_REF][START_REF] Bénabou | Belief in a Just World and Redistributive Politics[END_REF][START_REF] Van-Yperen | Goal orientations, beliefs about success, and performance improvement among young elite Dutch soccer players[END_REF][START_REF] Frieze | Children's Beliefs About the Causes of Success and Failure in School Settings[END_REF], our participants might have used the incentive cue as implicit evidence informing their effort cost estimation. One slightly different possibility is that participants may have explicitly assumed that reward and effort cost were correlated in this specific task. We henceforth refer to this account as the "Belief" scenario, which can be rephrased as individuals carrying a non-flat p(Cost|Incentive) representation, that they integrate as a likelihood signal into a posterior cost estimate. This Belief scenario is the basis of ideal Bayesian observer models that capture the integration of reward magnitude into retrospective effort estimates, in paradigms where effort and reward levels are actually correlated by design [START_REF] Pooresmaeili | Receipt of reward leads to altered estimation of effort[END_REF][START_REF] Rollwage | I judge you by your profit: Judgments of effort exerted by self and others are influenced by task contingent rewards[END_REF].

Second, because of its personal relevance, information about reward is likely to trigger an affective reaction from the viewer, as previously demonstrated in similar contexts [START_REF] Knutson | Anticipatory affect: Neural correlates and consequences for choice[END_REF][START_REF] Wu | Affective traits link to reliable neural markers of incentive anticipation[END_REF]. In line with the affect-as-information theory, and its associated observations that internal affective states are incorporated as evidence into various kinds of judgments [START_REF] Clore | Affect as information about liking, efficacy, and importance[END_REF][START_REF] Clore | How emotions inform judgment and regulate thought[END_REF][START_REF] Storbeck | Affective arousal as information: How affective arousal influences judgments, learning, and memory[END_REF][START_REF] Schwarz | Feelings-as-Information Theory[END_REF], affect may have mediated the impact of incentives on cost anticipation. In the psychology literature, affective states are classically parametrized in a two-dimensional space: arousal by valence [START_REF] Russell | A circumplex model of affect[END_REF][START_REF] Posner | The circumplex model of affect: An integrative approach to affective neuroscience, cognitive development, and psychopathology[END_REF]. The mediating role of affect could therefore be specific to one of these dimensions, or rely on both. In particular, since both prospective and exerted effort has been shown to increase arousal [START_REF] Gellatly | The effects of goal difficulty on physiological arousal, cognition, and task performance[END_REF][START_REF] Schmidt | Get Aroused and Be Stronger: Emotional Facilitation of Physical Effort in the Human Brain[END_REF][START_REF] Vassena | Overlapping Neural Systems Represent Cognitive Effort and Reward Anticipation[END_REF][START_REF] Varazzani | Noradrenaline and Dopamine Neurons in the Reward/Effort Trade-Off: A Direct Electrophysiological Comparison in Behaving Monkeys[END_REF], it is possible that the cognitive system responsible for effort anticipation may use arousal state as one of its inputs. Moreover, because prospective rewards also increase arousal [START_REF] Knutson | Anticipatory affect: Neural correlates and consequences for choice[END_REF][START_REF] Wu | Affective traits link to reliable neural markers of incentive anticipation[END_REF], individuals might misinterpret incentive-triggered arousal as a cue about the upcoming effort, such that anticipated energetic cost would be higher when a higher reward is at stake. A similar reasoning could apply in a valence-based scenario, since it is well established that prospective reward positively modulates affective valence [START_REF] Knutson | Anticipatory affect: Neural correlates and consequences for choice[END_REF]). Yet this scenario would imply that prospective effort is assigned a positive valence in our group of participants, which is not implausible given that they all practised sports regularly (this was one of our inclusion criteria). We henceforth refer to this as the "Affect" scenario, which comprises an "Arousal" and a "Valence" variant.

Third, at a motivational level, information about reward is a crucial input to the cost-benefit computations that adjust both the direction and intensity of behaviour, according to standard decision theory [START_REF] Rangel | A framework for studying the neurobiology of valuebased decision making[END_REF][START_REF] Croxson | Effort-Based Cost-Benefit Valuation and the Human Brain[END_REF][START_REF] Glimcher | Neuroeconomics: Decision Making and the Brain[END_REF]. Since our participants were systematically required to decide whether to accept a new reward-effort combination, it is likely that they injected the reward information into some expected value computation which would later guide their choice. From there, two specific mechanisms can account for a positive influence of prospective reward on anticipated effort cost. In any case, the most parsimonious formalization is that some output of the cost-benefit computation (dependent on prospective reward) would be integrated as evidence about the energetic cost. Our two motivation models detailed below differ as to the nature of the computational output that is integrated into cost estimation.

According to what we hereafter call the "Maximal Budget" hypothesis, as soon as individuals

receive complete information about the potential reward, they compute the maximal energetic cost they would be willing to invest, that is, the cost above which they would rather choose the Low-Effort 0-Reward option. This preliminary computation may entail easier subsequent decision-making, since choice could then result from a direct comparison between the maximal budget and the estimated energetic cost.

Here, we assume a subtractive parabolic effort-cost discounting of subjective incentive values [START_REF] Hartmann | Parabolic discounting of monetary rewards by physical effort[END_REF][START_REF] Białaszek | Physical and cognitive effort discounting across different reward magnitudes: Tests of discounting models[END_REF], such that the decision variable, i.e. the difference in utility U between the HEHG and the LE0G option, can be written as:

𝐷𝑉 = 𝑈 𝐻𝐸𝐻𝐺 -𝑈 𝐿𝐸0𝐺 = (𝑅 𝜌 -𝐶 𝐸 • 𝐸𝐶 𝛼 ) -(0 -𝐶 𝐸 • 𝐸𝐶 0 𝛼 ) ( 1
)
Where 𝑅 is the reward obtained upon completion of the HE route, 𝜌 is the nonlinearity parameter controlling the curvature of magnitude-to-utility mapping (typically, 𝜌 < 1 [START_REF] Schoemaker | The Expected Utility Model: Its Variants, Purposes, Evidence and Limitations[END_REF][START_REF] Glimcher | Neuroeconomics: Decision Making and the Brain[END_REF]), 𝐶 𝐸 is the scaling coefficient of the effort cost discounting variable (EC and EC0 corresponding to the HEHG and LE0G option, respectively), 𝛼 is the nonlinearity parameter controlling the curvature of effort discounting.

Thus:

𝐷𝑉 = 𝑅 𝜌 -𝐶 𝐸 • (𝐸𝐶 𝛼 -𝐸𝐶 0 𝛼 ) (2)
therefore, the maximal energetic cost such that DV = 0 is:

𝐸𝐶 𝑚𝑎𝑥 = (𝐸𝐶 0 𝛼 + 𝑅 𝜌 𝐶 𝐸 ) 1 𝛼 (3)
The main proposal of this "Maximal Budget" model is that subjects' anticipated energetic cost derives from a mixture between the relevant visual information (slope and speed, underlying REC), and the maximal effort cost they are willing to expend:

𝑨𝑬𝑪 = 𝑩 𝟎 + 𝑩 𝟏 • (𝜸 • 𝑹𝑬𝑪 + (𝟏 -𝜸) • 𝑬𝑪 𝐦𝐚𝐱 ) (4)
As this "contaminated" cost judgment likely reflects the cost which subjects are planning to invest, it may in turn be incorporated in an updated decision variable:

𝐷𝑉 ′ = 𝑅 𝜌 -𝐶 𝐸 • (𝐴𝐸𝐶 𝛼 -𝐸𝐶 0 𝛼 ) (5)
This updated decision variable then leads to a choice via a softmax rule:

𝑷(𝑪𝒉𝒐𝒊𝒄𝒆 = 𝒀𝒆𝒔) = 𝒔𝒐𝒇𝒕𝒎𝒂𝒙(𝜷 𝟎 + 𝜷 𝟏 • 𝑫𝑽 ′ ) (6)
As evidenced by equations ( 3) and ( 4), this model indeed predicts a positive impact of monetary incentives on anticipated energetic cost.

In an alternative scenario, which we call the "Optimal Expense" hypothesis, we propose that it is the output of an optimal expense computation, instead of a maximal budget one, that is subsumed as evidence about the actual cost. Here, we build upon the intuition that subjects might not fully internalize the mapping rule between expended energy and the probability of reaching their goal (obtaining the reward). In our experiment, the effort-reward contingency is deterministic: the participant should be sure to win the full reward if running the high-effort route, and nothing otherwise. Thus, the mapping from effort to reward obtainment probability is a step function with a 0 to 1 discontinuity at EC = REC. Indeed, as this discontinuity is due to the fact that energetic expense is strongly constrained in our experimental setting (running on a treadmill at a defined slope and speed), it might not reflect most ecological situations, in which increasing effort would naturally increase the probability of reaching one's goal in a continuous fashion. Therefore, we propose that individuals hold a sigmoidal (rather than step-like)

𝑃(𝑆𝑢𝑐𝑐𝑒𝑠𝑠| 𝐸𝑛𝑒𝑟𝑔𝑒𝑡𝑖𝑐 𝑒𝑥𝑝𝑒𝑛𝑠𝑒) = 𝑓(𝐸𝑛𝑒𝑟𝑔𝑒𝑡𝑖𝑐 𝑒𝑥𝑝𝑒𝑛𝑠𝑒) internal mapping function, whose inflection point abscissa and slope may be adjusted depending on the specifics of the considered task. Here, to make the sigmoid parameter values easier to interpret, we express the energetic expense variable as relative to REC and normalised by 𝐸𝐶 max :

𝑃(𝑆𝑢𝑐𝑐𝑒𝑠𝑠|𝐸𝐶) = 𝑠𝑜𝑓𝑡𝑚𝑎𝑥 (𝜅 0 + 𝜅 1 • 𝐸𝐶 -𝑅𝐸𝐶 𝐸𝐶 max ) (7)
In line with classical expected utility theory [START_REF] Friedman | The Expected-Utility Hypothesis and the Measurability of Utility[END_REF], this subjective probability of success enters the computation of the decision variable DV by scaling the utility of prospective outcomes:

𝐸[𝑈 𝐻𝐸 (𝐸𝐶)] = 𝑃(𝑆𝑢𝑐𝑐𝑒𝑠𝑠|𝐸𝐶) • 𝑅 𝜌 -𝐶 𝐸 • 𝐸𝐶 𝛼 (8)
By definition, the optimal energetic cost to expend is the one maximizing 𝐸[𝑈 𝐻𝐸 (𝐸𝐶)]:

𝐸𝐶 * = 𝑎𝑟𝑔𝑚𝑎𝑥 𝐸𝐶 (𝐸[𝑈 𝐻𝐸 (𝐸𝐶)]) (9)
As with the "Maximal Budget" hypothesis, we postulate that subjects incorporate this optimal energetic expense (which they intend to invest) as a piece of evidence about the actual (or 'typical') energetic cost of the route:

𝑨𝑬𝑪 = 𝑩 𝟎 + 𝑩 𝟏 • (𝜸 • 𝑹𝑬𝑪 + (𝟏 -𝜸) • 𝑬𝑪 * ) ( 10
)
The decision variable being the difference between 𝐸[𝑈 𝐻𝐸 (𝐴𝐸𝐶)] and 𝑈 𝐿𝐸 , it is written:

𝐷𝑉′ = 𝑃(𝑆𝑢𝑐𝑐𝑒𝑠𝑠|𝐴𝐸𝐶) • 𝑅 𝜌 -𝐶 𝐸 • (𝐴𝐸𝐶 𝛼 -𝐸𝐶 0 𝛼 ) (11)
Finally, this decision variable is incorporated into a softmax rule to determine choice probability:

𝑷(𝑪𝒉𝒐𝒊𝒄𝒆 = 𝒀𝒆𝒔) = 𝒔𝒐𝒇𝒕𝒎𝒂𝒙(𝜷 𝟎 + 𝜷 𝟏 • 𝑫𝑽′) (12)
We henceforth refer to this couple of models as the "Motivation" scenario, which comprises a "Maximal Budget" and an "Optimal Expense" variant. (See Fig. 3 for an illustration of this scenario.)

Fig. 3 -Schematic illustration of the motivation scenario

According to this hypothetical mechanism, information about the monetary incentive attached to the High-Effort (HE) option enters a cost-benefit computation. According to the Maximal Budget variant (left panel), participants compute the maximal energetic expense above which the Low-Effort (LE) option would bear a higher Expected Utility (EU). According to the Optimal Expense variant, they compute the energetic expense optimizing the EU attached to the HE option. In both panels, the computation is illustrated for two different reward magnitudes, R1 (small, light colour) and R2 (large, dark colour). Both variants predict that larger incentives would result in a higher 'energetic expense' (whether maximal or optimal) computational output.

This output is next incorporated into an Anticipated Energetic Cost computation, which also integrates a preestimate that is exclusively based on the running animation depicting the multi-segment route, and is thus a noisy reflection of the Real Energetic Cost (REC). Finally, the resulting (hidden) AEC estimate affects (observable) AEC rating and HE versus LE choice.

Practical implementation & predictions

In order to assess the validity of these three alternative -albeit not mutually exclusivescenarios, we formulated critical modifications of our task design, in preparation for a second experiment. These changes were devised in such a way that each of our hypotheses would make a specific set of predictions about measured behavioural outputs.

Under the "belief" scenario, we reasoned that if we emphasized the orthogonality between monetary incentive and effort cost, we should observe a decreased impact of prospective reward on anticipated energetic cost. We made this orthogonality intuitive and salient to the subjects by introducing a lottery animation at the beginning of each trial. This virtual lottery depicted the roll of a slot machine, where a running route was selected in the left slot and a monetary amount was selected (independently) in the right slot. Additionally, the "belief" hypothesis predicted that the extent to which prospective reward impacted anticipated energetic cost should correlate, across individuals, with the strength of the belief in a Reward -Effort positive correlation. We measured the strength of this belief by appending a computerized question after arousal ratings, which was phrased as follows: "Do you think that, in life, the less effort one makes, the less reward one receives?", with four possible answers, ranging from "Absolutely yes" to "Not at all".

Under the "affect" scenario, incidental fluctuations in affective state should have repercussions on anticipated energetic cost. Thus, we introduced two novel features to the task in order to manipulate affective arousal and valence independently. First, to address the 'valence' variant, we complemented prospective gains with prospective losses (in a distinct set of trials), which, we assumed, would trigger negative affect (increasingly with increasing loss amplitude), which should in turn decrease anticipated energetic cost. Second, in order to address the 'arousal'

variant, we added a musical background to two-thirds of the running animations. Based on previous reports of music-based arousal manipulation [START_REF] Gomez | Relationships between musical structure and psychophysiological measures of emotion[END_REF][START_REF] Gingras | The Eye is Listening: Music-Induced Arousal and Individual Differences Predict Pupillary Responses[END_REF], we expected our musical extracts to modulate subjects' arousal, depending on the varying low-or high-level features of each extract (slow versus fast tempo, high versus low predictability, soothing versus epic impression, etc.). As arousal induction may vary a lot across individuals, we played again each music extract at the end of the experiment, and had every participant reporting subjective arousal on a rating scale. Using eye-tracking to measure pupil diameter during the rating task, we checked that subjective report of arousal level was following, across music extracts, arousal level in a physiological sense (activation of the autonomic nervous system).

Additionally, regardless of the characteristics of our musical extracts, we expected the presence of music to increase average arousal, compared to the testing blocks in which music would be absent [START_REF] Gomez | Affective and physiological responses to environmental noises and music[END_REF]. Therefore, this manipulation provided us with two ways of testing for a positive impact of incidental arousal fluctuations on anticipated energetic cost, which was predicted by the arousal variant of the affect scenario. Top part, bottom row: Random draw performed at the end of each block among its 18 just-completed trials.

Choice made by the participant is reminded then executed: in the Gain condition, if the participant accepted the High-Effort High-Gain (HEHG) offer, she must get onto the treadmill and complete the effortful running route, after which the concurrently displayed reward is added to her total earnings. If she declined the HEHG offer, she must complete a default low-effort running route, and her total earnings remain unchanged.

Middle part: Structure of a session. Each session includes a training phase, 3 blocks of 18 trials, a recalibration, 3 additional trial blocks, a rating of arousal induced by all 72 musical extracts, and eventually a question about their belief in a general effort-reward correlation. Here, Gain and Loss conditions are alternated across blocks (green and red, respectively), and all blocks except B3 and B4 include musical extracts during display of route & incentive.

Bottom part: Same as Top part, but in Loss condition. See details above.

Results -Experiment 2

Twenty-four healthy adults gave their consent to participate in this experiment and were tested one after the other.

Choice

Here, we replicated the observations made in Experiment 1, with an extension into the Loss domain: participants tended to accept the HEHG (resp. HE0L) option more often as gain (resp.

loss) level increased, and as REC level decreased. We inverted a sigmoidal model of choices between HE and LE options, including gain and loss magnitudes 𝐺𝑎𝑖𝑛 𝐻𝐸𝐻𝐺 and 𝐿𝑜𝑠𝑠 𝐿𝐸𝐻𝐿 , as well as 𝑅𝐸𝐶 𝐻𝐸 , as explanatory variables. Choice data (but not rating data) from 10 participants were excluded before model inversion, due to a total rate of "No" choices ≤ 5/108, thus a percentage of "Yes" choices > 95%. As expected, we found a a strong positive effect of 𝐺𝑎𝑖𝑛 𝐻𝐸𝐻𝐺 (Fig. 5a bottom, 𝛽 ̅ = 3.

Anticipated Energetic Cost

In order to evaluate the impact of experimental factors on AEC ratings, and similarly to

Pupil dilation & arousal manipulation

Next, we assessed the impact of our arousal manipulation procedure on AEC ratings. First, we found that arousal ratings made about each musical extract during the Arousal Rating phase were a significant positive predictor of pupil diameter around 2s after music onset (time window:

[1.67 -2.55] s post music onset, 𝑝 = 3 • 10 -4 , corrected for multiple comparisons; see Suppl.

Material, Figure S2), suggesting that ratings faithfully reflected the arousal state actually felt by participants while listening to the musical extracts. Second, we incorporated these individual arousal ratings as an additional linear regressor in our GLM of AEC ratings (see above). We found no significant effect of this predictor (𝛽 ̅ = .19, 𝐶𝐼 = [-.35 ; .74], 𝑡(23) = .73, 𝑝 = .47), nor did we find an effect of a binary "Music/NoMusic" regressor indicating whether music was played during a given trial (𝛽 ̅ = -.16, 𝐶𝐼 = [-.93 ; .61], 𝑡(23) = -.43, 𝑝 = .67). These observations suggest that incidental arousal fluctuations do not affect anticipated energetic cost, and therefore challenge the main prediction of the "Affect -Arousal" scenario.

Belief in an Effort => Reward implication

We measured the individual strength of belief in an Effort -Reward correlation by asking the following question (via the computer), once all trials were completed: "Do you believe that in life, the less effort one makes, the less reward one gets?". Participants answered on a 4-point Likert scale ranging from "Not at all" to "Absolutely". We found no correlation between the strength of their belief and the magnitude of the effect of Gain or Loss, or both Gain and Loss combined, on AEC (resp. 𝑟 = -.18, 𝑝 = .39; 𝑟 = -.049, 𝑝 = .82; 𝑟 = -.12, 𝑝 = .59). This finding is in contradiction with one of the predictions of the "Belief" scenario.

Exploring the motivation scenario: computational modeling

As introduced in the Interim Discussion, we propose two computational models which could explain the positive effect of goal value on anticipated energetic cost from high-level, decision-oriented cognitive processes. Specifically, the 'Maximum Budget' hypothesis proposes that, as soon as the goal value is known, individuals compute the corresponding maximal effort they are willing to invest to attain it. Subsequently, they (partly) intermingle what they are willing to invest with what they must invest. As an alternative, the 'Optimal Expense' scenario proposes that individuals compute the optimal amount of effort they should invest to maximize the expected utility of their action, i.e. to secure the outcome, assuming the effort-outcome contingency is uncertain. This 'optimal effort' variable is then (partly) confused with the objective effort which is demanded from them.

In order to assess the significance of these hypotheses, and to arbitrate between them based on the relevant data (AEC ratings and Yes/No choices), we inverted each of these two models (both predicting ratings and choices simultaneously) based on the datasets from Experiments 1 and 2. Within each model family, two variants were inverted. First, was a 'Standard'

variant described in the Interim Discussion (with an additional Loss term when fitting data from Experiment 2, see Methods), in which the probability of choosing the high-effort option is a sigmoidal function of the AEC estimate (also derived from the model). Second was an 'Null Choice' variant in which choices did not depend on AEC estimates, but directly on the Real Energetic Cost (i.e., the subjective distortions of effort cost were not incorporated into decision-making).

Additionally, we inverted purely descriptive models featuring linear terms which accounted for the influence of prospective Gain or Loss on AEC ratings (the first variant of these descriptive models -hereafter labelled 'Linear' -only included main effects, the second one, 'LinearInterac', further comprised terms of interaction with REC).

We first inverted each of our models on data from Experiment 1 (with agnostic priors on parameters' density functions), which allowed us to derive empirically-informed prior parameter distributions, before inverting the models again on data from Experiment 2. Corresponding model fits for standard 'Maximal Budget' and 'Optimal Expense' models are illustrated in Fig. 6. Both models do predict the qualitative pattern observed in high-effort choice rate and AEC ratings. Cross markers indicate data averaged across all REC levels and across all subjects in Experiment 1 or 2 (top or bottom line, respectively) for a given incentive level.

Triangles and circles indicate averaged predictions of the 'Maximal Budget' and 'Optimal Expense' models respectively. Predictions were computed from models fitted to each subject's data (both ratings and choices in both experiments).

Solid lines represent a linear interpolation between averaged data points or model predictions, for illustrative purposes only. Shaded area around each curve represents the standard error to the mean (s.e.m), computed across subjects.

In order to arbitrate between model families from a quantitative standpoint, we pooled free-energy approximations of model evidence across the two groups of participants (Exp. 1 and Exp. 2), and performed several group-level random-effects Bayesian model family comparisons.

First, we compared 'Standard' models, wherein choice probability is a function of the model's AEC estimate, to 'Null Choice' models, wherein choice probability is a function of REC (see Fig. 7, right part, top row). This resulted in a protected exceedance probability 𝑃𝐸𝑃 > 0.99 strongly in favour of the 'Standard' family. This provided support for the fact that AEC estimates, which incorporated several distortions of REC, including those relating to Gain or Loss influences, were a better predictor of choices than the Real Energetic Cost itself. Second, we compared a 'Cost-benefit' family (comprising both variants of 'Maximal Budget' or 'Optimal Expense' models) to a 'Linear' family (comprising both variants of 'Linear' and 'LinearInterac' models). This revealed a very likely dominance of the 'Cost-benefit' family over the purely descriptive 'Linear' family (again, 𝑃𝐸𝑃 > 0.99), suggesting that most healthy subjects are indeed incorporating the output of a cost-benefit computation into their prospective cost judgment. However, a third comparison between 'Maximal Budget' and 'Optimal Expense' families did not reveal any likely dominance of the 'Maximal Budget' mechanism over the 'Optimal Expense' one (𝑃𝐸𝑃 = 0.52) or vice-versa. PEP is the probability that a given model family is more frequent than any other one in the population, corrected for chance fluctuations of observed individual model evidences.

Here, 𝑃(𝑓𝑟𝑒𝑞 𝐶ℎ𝑜𝑖𝑐𝑒=𝑓(𝐴𝐸𝐶) > 𝑓𝑟𝑒𝑞 𝐶ℎ𝑜𝑖𝑐𝑒=𝑓(𝑅𝐸𝐶) ) > 0.99, 𝑃(𝑓𝑟𝑒𝑞 𝐶𝑜𝑠𝑡-𝑏𝑒𝑛𝑒𝑓𝑖𝑡 > 𝑓𝑟𝑒𝑞 𝐿𝑖𝑛𝑒𝑎𝑟 ) > 0.99 and 𝑃(𝑓𝑟𝑒𝑞 𝑀𝑎𝑥𝐵𝑢𝑑𝑔𝑒𝑡 > 𝑓𝑟𝑒𝑞 𝑂𝑝𝑡𝑖𝑚𝐸𝑥𝑝𝑒𝑛𝑠𝑒 ) = 0.52.

Discussion

In this study, we investigated the putative influence of goal value onto the effort cost individuals anticipate they will have to expend in order to achieve this goal. To this end, we devised a novel experimental paradigm in which healthy participants made prospective energetic cost estimates about composite running routes paired with a monetary outcome, whose magnitude was varied orthogonally to the actual energetic cost of the routes. In our first experiment, despite a financial incentive for accurate cost estimates, and contrary to our expectations, we found a positive impact of incentive magnitude on prospective effort cost estimates. To our knowledge, this is the first time such an effect has been isolated in a controlled laboratory setting. It questions a fundamental axiom of rational decision theory, according to which effort cost and goal value should be estimated independently of each other.

We next set out to elucidate the cognitive processes underlying the value dependency of cost estimates, which we hereafter call the motivation bias. Three main cognitive scenarios could be put forth: 1) a 'Belief' scenario, in which individuals incorporate information about goal value as a cue about effort cost, due to an implicit or explicit belief in a positive correlation between these two dimensions, 2) an 'Affect' scenario in which affective states are modulated (along their valence or arousal dimension) by prospective gains or losses and in turn inform prospective effort cost judgments, 3) a 'Motivation' scenario in which prospective effort cost judgments are partly contaminated by the output of cost-benefit evaluations related to effort investment (either the maximal effort one is willing to invest in the pursuit of the goal, or the optimal effort one should invest to maximize their expected utility). We extended our task design to improve the discriminability between these three scenarios based on behavioural data, and we performed a second experiment. Our new findings contradicted the predictions of the 'Belief' and 'Affect' scenarios, but were in line with those of the 'Motivation' scenario.

Indeed, there was no significant link between the motivation bias and the reported belief in a general (in real life) correlation between effort and reward. Moreover, demonstrating to participants the absence of correlation between effort and reward in the experiment, by playing lottery animations, did not change the motivation bias. This finding extends previous reports that reward is used as a cue for retrospective effort judgment [START_REF] Pooresmaeili | Receipt of reward leads to altered estimation of effort[END_REF][START_REF] Rollwage | I judge you by your profit: Judgments of effort exerted by self and others are influenced by task contingent rewards[END_REF], which was observed in experiments where reward and effort were actually correlated, and therefore interpreted as a rational Bayesian inference. Although such a Bayesian inference might be implemented in situations where it does produce valid conclusions, our results show that reward also impacts prospective effort estimates that drive decision-making even when this influence is not instrumental. This impact should therefore be seen as an irrational bias, imposed by a cognitive process that is automatically triggered despite its irrelevance in the context of the experiment. It is impossible with our design to know whether the putative cognitive process inducing a bias on effort cost estimates was triggered when receiving the information about incentive levels and running routes, when making the decision about whether to accept the offer, or when reporting subjective cost on the rating scale. By automatic, we just mean that the bias was not (fully) controlled, since it occurred despite the monetary bonus offered for accurate judgments.

This automatic process could have involved affective reactions to incentive presentation, following on affect-as-information theory [START_REF] Clore | Affect as information about liking, efficacy, and importance[END_REF]. The idea is that the brain inverses a general implication from information to affective state. For instance, anticipating an important effort goes with activating the autonomic nervous system, which is perceived as enhanced arousal level. Therefore, perceiving enhanced arousal level could signal that an important effort is to be exerted soon. We discarded this possibility by manipulating arousal level through music extracts played when participants viewed the running routes and the associated incentives. Although music did increase autonomic measures (pupil size) in proportion to arousal ratings, it did not influence effort cost estimates. Thus, the impact of incentives is unlikely to have been mediated by arousal state. However, we must acknowledge that arousal ratings were not recorded during the experiment, so it remains possible that the music was actually ineffective for modulating arousal level when participants performed the main task. A variant of this affect scenario would be that incentives modulate the affective state along a valence dimension.

Participants would overestimate effort costs not because they feel more excited with higher incentives but because they feel more excited in a positive way. This mediation by valence is also unlikely, because negative incentives (prospective losses) had the same impact as positive incentives (prospective gains) on effort cost estimates. Furthermore, it would imply that participants feel better when they have to exert more effort, which may be plausible in sporty people but not generalizable to the average person.

Finally, our findings are in line with the predictions of the motivation scenario, which assumes that participants automatically translate reward into effort. This ability would arise from the cost-benefit trade-off machinery responsible for the motivational control of behaviour.

Bayesian family model comparison provided strong evidence in favour of cost-benefit models, relative to purely descriptive, non-principled models (generalized linear regression). However, further comparison between the two variants of this hypothesis ('Maximal Budget' and 'Optimal Expense') did not suggest that one was more prevalent than the other in the sampled population of participants. This could mean that these two underlying mechanisms are equally prevalent in the population, or simply that statistical power was not sufficient to detect a difference in prevalence. Besides, it is possible that both mechanisms are at play within a given individual, either at the scale of a single effort estimate, or across several estimates, such that none of these mechanisms clearly dominates the other. In any case, we must assume that these computations are implicitly performed by the brain, because participants did not spontaneously report them during post-experiment debriefing interview.

The general conclusion remains that the positive distortion of effort cost ratings by goal incentive value, observed in model-free analyses, is likely due to a contamination of prospective effort cost judgments by the product of a prescriptive, decision-oriented cost-benefit computation.

Indeed, we found higher evidence for models in which the probability of choosing the high-effort option was a function of anticipated effort cost (including distortions of the real effort cost by monetary incentives) rather than a direct function of real effort cost. This suggests that subjective distortions induced by goal value affect not only ratings of effort cost but also decisions about whether to exert more effort for a higher benefit. Since choices were made before ratings, it is unlikely that decision making was biased because of explicit cost judgments, although one cannot exclude that participants prepared their ratings beforehand. In any case, this motivation bias makes ambivalent the role of goal value in expected utility computation: on the one hand, it increases the expected benefit of the action, but on the other hand, it also increases the perceived cost of the action. The net effect of the motivation bias is to make high-value goals less attractive than they should be, if participants had not anticipated that they would come at higher costs.

Although we think this work is an important step towards a mechanistic comprehension of how effort anticipation and other action-related judgments can interfere, further investigation will be needed to assess the robustness and generalizability of these findings, to circumscribe the scope of this phenomenon, and to gain a fine-grained understanding of its computational and neural bases. Specifically, one remaining question to be addressed is whether other types of prospective cost judgments, about e.g. delay or risk, are subject to the same distortions from goalvalue as effort cost judgments. Another crucial issue regards the impact of this effect on action execution. In the present study, the actual effort expended during running was controlled by the treadmill speed and slope. However, earlier studies have reported a positive effect of goal value on force production, even when it is non-instrumental or in fact detrimental to performance [START_REF] Dam | People toss coins with more vigor when the stakes are higher[END_REF][START_REF] Oudiette | A Pavlovian account for paradoxical effects of motivation on controlling response vigour[END_REF], which could indeed originate from a higher anticipated effort cost.

This observation leads us to interrogate the potential adaptiveness of such an influence of goal value on effort cost anticipation. We can speculate that, in environments where the invested effort and the value of the attained goal are strongly correlated, information about this goal value is on average informative about the effort that will be required. Thus, using such a heuristic might help prepare the action, in situations where value is known with precision whereas information about the cost is more uncertain. A better preparation, for an overestimated difficulty, may later result in higher persistence towards high-value goals in the face of fatigue, pain, or other forms of adversity that might be encountered during action execution. However, overestimating the cost may lead people to abandon the pursuit of high-value goals that they actually have the ability to reach. These missed opportunities may be particularly frequent in patients suffering from depression, who might be trapped in a paradoxical behaviour: ignoring attractive goals just because their high values are seemingly signalling high costs.

We acknowledge that, even if it provides a well-controlled proof of concept, the size of the effect was rather small in our data. Yet, this critically depends on the specifics of the experimental situation, notably the fact that participants were heavily trained (and encouraged) to make accurate effort cost judgments. Further research is needed to investigate how preponderant the effect is in more naturalistic situations where people have only vague ideas about the actual costs.

Another critical feature of our design is that the incentive level was clearly indicated when ratings and choices were made, while the effort cost had to be estimated and memorized while viewing the running animation. A question for future investigation is whether the phenomenon of contamination might manifest in the opposite direction when information about effort cost is available for a precise subjective estimate, while information about goal value remains vague. This opposite contamination would result in overestimated goal value when effort cost is obviously high, and would lead people to another paradox: engaging in strenuous actions just because they seem costly enough (to bring valuable reward, presumably).

Experiment 1

The general idea of our experimental paradigm was as follows: each subject was shown short on-screen animations depicting a cartoon character running various routes, each associated to a varying amount of reward, and was asked to estimate a priori the energetic cost of each given route. Subject was additionally asked whether she would agree to run each given route in reality (on a treadmill) in order to earn the associated reward.

More specifically, after giving their consent, each subject completed a single 3-hour session comprising these steps:

1) Training Subjects went through an extensive training in order to familiarize themselves with the task.

i. Comfort speed First, they performed a 3-minute physical warm-up run on the treadmill, which was pre-set at a 7 km.h -1 speed and a 4% slope. Then, while they kept running, they were instructed to adjust the treadmill speed to their "comfort" level, which was described to them as the speed at which they would be able to run for 30 minutes without interruption nor excessive fatigue. They tuned the speed by 0.5 km.h -1 increments via "plus" and "minus" buttons and received immediate motor feedback (but no numeric feedback about the current speed was given, in order to encourage them to rely on a proprioceptive rather than a strictly numeric benchmark), until they reported having reached such a "comfort" speed and had spent at least 2 minutes in the speed adjustment phase (in order to ensure a minimal warm-up duration of 5 minutes).

ii. Familiarization with speed and slope levels Running routes which were to be displayed during the testing phase would be comprised of 4 segments, each a combination of 1 of 3 speed levels (110%, 125%, and 140% of comfort speed, expressed via the on-screen scrolling speed of dashes symbolizing the road) and 1 of 3 slope levels (0%, 4%, 8%, represented by 0, 12 and 24° slopes, respectively, to make levels more distinguishable visually).

In order to allow participants to map visual, on-screen features characterizing each route segment with proprioceptive sensations (and in particular, an effort cost internal signal), we instructed them to run 6 "elementary" routes successively, each corresponding to a single 1-minute segment {𝑆𝑝𝑒𝑒𝑑 𝑖 ; 𝑆𝑙𝑜𝑝𝑒 = 0} then {𝑆𝑝𝑒𝑒𝑑 = 𝑆 𝑐𝑜𝑚𝑓𝑜𝑟𝑡 ; 𝑆𝑙𝑜𝑝𝑒 𝑖 } with 𝑖 ∈ ⟦1; 3⟧. Differences in speed or slope were thus experienced with the other dimension being held constant.

iii. Familiarization with global route display and energetic cost rating

This training phase was critical in that it allowed participants to map together, for a given route 1) its corresponding on-screen animation, 2) the physical sensations perceived while and after running this route (and associated internal effort signal), and 3) a cursor position on the energetic cost rating scale. First, subjects were instructed that their accuracy at the energetic cost estimation task would condition their final payment, and were advised to picture themselves in the shoes of the running character during the animation display in order to produce accurate estimates. This was to encourage them to rely on internal simulation rather than visual cues to build their cost estimate, as they would do in real-life situations. Then, they completed 2 trials in which they sequentially watched a running animation, ran the corresponding 2-minute route on the treadmill, and were finally shown the corresponding cursor position on the energetic cost scale. Next, they completed 2 trials similar to the two previous ones, except that subjects now tried to estimate each route's energetic cost by placing the cursor on the rating scale after having run the route and before receiving a feedback from the computer. This was followed by 12 additional trials, in which subjects estimated the energetic cost directly after seeing the running animation, i.e. without actually running the route, and were then given feedback about the actual energetic cost (still as a position on the rating scale), as well as numerical feedback about their estimation accuracy (e.g. "Accuracy: 83%"), computed as the complement of the actual distanceto-target relative to the maximal possible distance-to-target:

𝐴𝑐𝑐𝑢𝑟𝑎𝑐𝑦 = 100 (1 - |𝜔-𝜔 ̂| max{1-ω ;ω} )
, where 𝜔 is the actual energetic cost (on a normalized [0;1] scale) and 𝜔 ̂ is the estimated energetic cost.

iv. Training block

Before beginning the actual testing blocks, subjects performed a training block of trials, which was identical to later testing blocks except that it did not entail any real financial stake (as made clear to the subjects before the beginning of the training block).

2) Testing

The testing phase was composed of 6 blocks, each comprising 18 trials. Each trial went as follows:

i. Display of prospects First, the static picture of a monetary reward was displayed on the right side of the screen for 2s. This reward was one of 6 possible amounts: {0.01, 0.10, 0.50, 1, 2, 5} € and corresponded to a bonus that would be added to the subject's final payment if she actually ran the associated route. Next, a running animation was displayed on the left side of the screen for 10 s. The animation depicted a character running along a route composed of 4 segments, each corresponding to a (𝑠𝑝𝑒𝑒𝑑 * 𝑠𝑙𝑜𝑝𝑒) couple, pseudo-randomly drawn from the levels described above, and lasting a pseudorandom duration (between 25 and 45s in reality, i.e. between 2.08 and 3.75s onscreen). An in-house algorithm ensured that the total duration of the animation was 10s (corresponding to a real duration of 120s), and that the actual energetic cost of the corresponding route equalled one of 6 pre-determined energetic cost levels: 𝜔 ∈ {.20, .32, .44, .56, .68, .80} such that 𝐸𝐶 𝜔 = (1 -𝜔)𝐸 min + 𝜔𝐸 max , with 𝐸 min the energetic cost of a 2-minute run at the easiest speed and slope levels: {110% * 𝑆 𝑐𝑜𝑚𝑓𝑜𝑟𝑡 ; 0%}, and 𝐸 max the energetic cost of a 2-minute run at the hardest speed and slope levels: {140% * 𝑆 𝑐𝑜𝑚𝑓𝑜𝑟𝑡 ; 8%}. Subjects were not aware that energetic costs systematically belonged to one of 6 possible levels. The instantaneous metabolic cost (power) of a given {Speed; Slope} couple was computed following American College of Sports Medicine's (ACSM) metabolic equation for running, which has been shown to predict actual energetic expense with good accuracy [START_REF] Hall | Energy Expenditure of Walking and Running: Comparison with Prediction Equations[END_REF].

ii. Choice

Once the animation stopped playing, subjects were presented with a choice: "Do you prefer to run this route?". Selecting the "YES" answer meant that they preferred running the route just displayed in order to win the monetary reward simultaneously displayed, selecting the "NO" answer meant that they preferred running a "comfort" route (2 minutes at comfort speed with 0% slope) for no monetary reward. Choice was self-paced, and was not executed immediately afterwards.

iii. AEC rating Once they had validated their choice, subjects were prompted to estimate the energetic cost of the running route depicted by the animation they had just watched. This self-paced rating was made by placing a cursor along a horizontal scale (implicitly bounded by 𝐸 𝑚𝑖𝑛 and 𝐸 𝑚𝑎𝑥 ). Subjects were incentivized to produce accurate ratings via the following rule: if, by the end of each block, their average accuracy exceeded 80%, they would receive a 1€ bonus for this block. (Participants' average accuracy was 82.3 ± 3.8 % in Experiment 1.)

At the end of each block, one trial among the 18 just completed was randomly drawn by the computer; the corresponding route and the associated monetary reward were displayed, along with a reminder of the participant's choice ("You accepted/refused to run this route."). The participant was then instructed to step onto the treadmill and to run the effortful (resp. comfort) route if she had accepted (resp. refused) the offer. Upon completion of the effortful route, the associated monetary reward was added to her final payment. (Notably, all participants completed all running routes -except for one participant who chose to stop before the end of the most effortful run in the training session -so the risk dimension was negligible.)

Upon completion of the running sequence, participants were given a numerical feedback about the accuracy of their energetic cost estimates (averaged across the last 18 trials) along with a notification about their accuracy bonus ("You (do not) win a 1€ accuracy bonus for this block"). Their updated cumulative payment was also displayed ("Your current payment is 33.4€").

Before subjects began the 4 th block, they went through a short recalibration procedure, aimed at helping them maintaining accurate cost estimates, and consisting in two trials identical in structure to the last 12 trials at step iii. Familiarization with global route display and energetic cost rating of the training phase.

Since our experimental design involved 6 energetic cost levels * 6 reward levels, the prospect matrix comprised 36 possible cost-reward combinations, each repeated 3 times across the 108 trials, in a random order.

Finally, they completed a written survey and a subsequent oral debriefing with the experimenter. This allowed us to gather their general thoughts about the experiment, enquire about the heuristics or strategies they may have used to perform the task, or the influences they may have felt from the various experimental variables, and last, make sure that they had not guessed the real purpose of the experiment or explicitly assumed a correlation by-design between the Incentive and Effort Cost factors of the task.

Experiment 2

We adapted the design of Experiment 2 from that of Experiment 1, such that the various mechanisms we proposed to explain the effect of interest observed in Experiment 1 (see Interim Discussion) would each make a distinct set of predictions about the behavioural outputs measured in Experiment 2. We therefore introduced the following modifications to the design:

1) Each trial in the Testing phase would start with a lottery animation, akin to a slot machine roll and lasting 0.8s, in which a running route and a monetary incentive amount were shown to be randomly and independently selected on either side of the screen. The goal of this manipulation was to make the orthogonality of energetic cost and monetary incentive variables intuitive and salient to the subjects. In fact, the order in which {cost, incentive} couples would be displayed was still determined at the beginning of the session via a pseudo-random permutation performed by our computer program.

2) Additionally, we probed subjects' belief in a Reward => Effort general implication by asking (via the computer) the following question: "Do you think that, in life, the less effort you make, the less reward you receive?", with four possible answers, ranging from "Absolutely yes" to "Not at all".

3) Testing blocks would alternate between a GAIN (as in Exp. 1) and a LOSS condition. In the GAIN condition, a 'Yes' answer at the Choice stage meant that the participant preferred running the route depicted by the animation in order to win the monetary amount displayed within the green 'GAIN' frame (subsequently referred to as the High-Effort High-Gain, or HEHG, offer), rather than running the 'comfort' route (see Exp. 1 description for details) for no reward (Low-Effort 0-Gain, or LE0G, offer). In the LOSS condition, a 'Yes' answer meant that the participant preferred running the route depicted by the animation in order not to lose the monetary amount displayed within the red 'LOSS' frame (High-Effort 0-Loss, or HE0L), rather than running the 'comfort' route and having this monetary amount deducted from her final payment (Low-Effort High-Loss, or LEHL).

The GAIN/LOSS condition was displayed before each testing block began, and reminded in each trial by the coloured and labelled frame surrounding the monetary incentive picture.

In order to maintain a number of 3 repetitions for each {energetic cost level x incentive level x GAIN/LOSS condition} combination, while keeping a session duration that would avoid excessive fatigue effects, only 3 incentive levels were used: {0.01, 0.5, 10} €.

During the Testing phase, participants' arousal state was manipulated independently from energetic cost and incentive fluctuations via music. On each trial, once a running route and an incentive level had been selected by the lottery, and concomitantly with the running animation, a musical extract was played in the headset participants were wearing. For each {energetic cost level x incentive level x GAIN/LOSS condition} combination, 3 levels of arousal were tested: strongly arousing music, weakly arousing music, or no music (in blocks 3 and 4). Strongly and weakly arousing musical extracts were pseudo-randomly interleaved within each musical block. We pre-selected 128 extracts (each 17-second long) from moderately familiar to little-known classical music pieces, and had a small panel (N = 4, including EB and CT) rate the emotional arousal evoked by each extract. Next, we retained the 36 most and 36 least arousing excerpts based on this score, which allowed us to classify our music stimuli into a priori high vs. low arousal categories. Note that this procedure was only meant to ensure a sufficiently wide and approximately symmetrical distribution of arousal level, and that the external validity of this classification (relative to e.g. the physiological or reported arousal measured from a large sample of the population) was not crucial to test the "affect -arousal" hypothesis. Indeed, we additionally acquired a direct measure of the subjective arousal felt by participants in reaction to our musical extracts. Upon completion of the 6 th testing block, participants listened again to each of the musical extracts they had previously heard (without any concurrent visual stimulation, in a randomized order) and were asked to report the amount of arousal they felt "due to the musical extract", on an analog rating scale ranging from "None" to "Extreme".

Pupil dilation was recorded during each Testing and Arousal Rating trial. In order to mitigate motion-related noise in the data, participants laid their head on a chin-rest during all computerized phases of the experiment, from the training to the end of arousal ratings. The luminance of all visual stimuli was roughly equalized before data acquisition, and was subsequently measured with a Dr Meter LX1010B digital luxmeter, which we placed at the same position as participant's eyes in front of the computer screen, and exposed to all stimuli combinations.

Statistical analysis

All statistical analyses were performed with Matlab and its Statistics and Machine Learning Toolbox, as well as the Variational Bayesian Analysis (VBA) toolbox [START_REF] Daunizeau | VBA: A Probabilistic Treatment of Nonlinear Models for Neurobiological and Behavioural Data[END_REF].

Generalized Linear Models (GLM) were inverted with Matlab's built-in glmfit function. All regressors were z-scored prior to GLM inversion so as to enable direct comparison of the amount of explained variance between regressors.

We subsequently tested for group-level random effects of the regressors in our models by performing one-sample two-tailed Student t-tests on each vector of beta posterior estimates.

Non-linear models were inverted using a variational Bayes approach under the Laplace approximation, implemented in the VBA toolbox. The algorithm used in VBA not only inverts nonlinear models but also estimates their evidence, which represents a trade-off between accuracy (goodness of fit) and complexity (degrees of freedom).

Experiment 1

Choice

To assess the impact of our main experimental factors on preference for the HEHG option, we fitted a simple logistic model to binary choices made by each participant, formalized as follows:

𝑃(𝐶ℎ𝑜𝑖𝑐𝑒𝑌𝑒𝑠) = 1 1 + 𝑒 -(𝐵 0 +𝑘 𝑅𝐸𝐶 •𝑅𝐸𝐶+𝑘 𝐼 •𝐼) (1)
where 𝐵 0 is a bias parameter, 𝑅𝐸𝐶 is the real energetic cost of the HEHG route, 𝐼 is the incentive (or potential reward) level of the HEHG option, and 𝑘 𝑅𝐸𝐶 and 𝑘 𝐼 are the linear coefficients scaling the energetic cost and the incentive level, respectively.

Anticipated Energetic Cost

We fitted several GLMs to the AEC ratings made by each participant.

In order to evaluate the impact of our two main experimental factors, we started with a very simple model:

𝐴𝐸𝐶 = 𝛽 0 + 𝛽 𝑅𝐸𝐶 • 𝑅𝐸𝐶 + 𝛽 𝐼 • 𝐼 (2)
where 𝑅𝐸𝐶 is the real energetic cost of the route, 𝐼 is the incentive level, and 𝛽s are the linear coefficients of our model.

Next, we tested whether the incentive level factor had an impact on AEC over and above choices made by the participant. For this, we orthogonalized our factor of interest to the potentially confounding factor:

𝐴𝐸𝐶 = 𝛽 0 + 𝛽 𝑅𝐸𝐶 • 𝑅𝐸𝐶 + 𝛽 𝐶 • 𝐶ℎ𝑜𝑖𝑐𝑒 * + 𝛽 𝐼 • 𝐼 * (3)
where 𝐶ℎ𝑜𝑖𝑐𝑒 * is the choice made by the subject (encoded as 1 for Yes, 0 for No, and subsequently orthogonalized to the REC vector), and 𝐼 * is the incentive level, orthogonalized to 𝑅𝐸𝐶 and 𝐶ℎ𝑜𝑖𝑐𝑒 * .

Finally, we looked into a possible decrease in the effect of reward across experimental blocks. To that end, we added an 𝑖 𝐵𝑙𝑜𝑐𝑘 factor indicating the index of the current block, as well as all corresponding interaction terms, including 𝐼 * 𝑖 𝐵𝑙𝑜𝑐𝑘 which was our regressor of interest here:

𝐴𝐸𝐶 = 𝛽 0 + 𝛽 𝑅𝐸𝐶 • 𝑅𝐸𝐶 + 𝛽 𝐼 • 𝐼 + 𝛽 𝐵 • 𝑖 𝐵𝑙𝑜𝑐𝑘 + ⋯ 𝛽 𝑅𝐸𝐶 * 𝐼 • 𝑅𝐸𝐶 * 𝐼 + 𝛽 𝑅𝐸𝐶 * 𝐵 • 𝑅𝐸𝐶 * 𝑖 𝐵𝑙𝑜𝑐𝑘 + 𝛽 𝐼 * 𝐵 • 𝐼 * 𝑖 𝐵𝑙𝑜𝑐𝑘 (4)

Pupil dilation

We acquired two pupil diameter timeseries per subject (one for the main task, and one for the arousal rating task), that we pre-processed as follows:

1) Invalid data points (lost signal, mostly due to a blink or fixation outside the screen) were discarded, as well as all data points pertaining to a [-100;200] ms window around these timepoints 2) Outlier samples (outside 3 median absolute deviations from the median) were discarded 3) A shape-preserving piecewise cubic interpolation was performed on the remaining timeseries 4) An 11 th order median filtering was applied to the interpolated signal (for smoothing) 5) A 1 128 Hz high-pass filter was applied to correct for low-frequency drift 6) Timeseries were split into trial sequences 7) Trial timeseries were baseline-corrected (by subtracting the pupil diameter averaged across the [-200;0] ms window preceding trial onset)

Next, in order to assess the effect of music-related arousal on pupil diameter, we fitted the following minimal GLM to each timepoint in arousal-rating timeseries:

𝑃𝑢𝑝𝑖𝑙𝐷𝑖𝑎𝑚 = 𝛽 0 + 𝛽 𝐴 • 𝐴 (9)
where 𝐴 is the arousal rating made by the participant about the musical extract being played.

Finally, we smoothed the t-statistic timeseries derived from fitting this GLM by convolving them with a gaussian kernel (𝜎 = 50 𝑚𝑠) in order to ensure that the assumptions of the Random Field theory were met, and we applied the RFT_GLM_contrast function (from the VBA toolbox) to our smoothed timeseries. This allowed us to test the significance (in a random-effects approach) of each regressor at each timepoint while correcting for multiple comparison at cluster level.

Inversion of non-linear cost-benefit models of Choice and AEC

We inverted non-linear models predicting both AEC ratings and Choices as a function of the real energetic cost (REC), the nominal gain magnitude (G) and the nominal loss magnitude (L). (See Interim discussion for details about the models applied to Exp. 1 data.)

In Exp. 2, the 'Maximal Budget' model was extended from that presented in the Interim discussion, to account for the presence of a Loss condition. The decision variable was now written:

𝐷𝑉 = 𝑈 𝐻𝐸 -𝑈 𝐿𝐸 = (𝐺 𝜌 -𝐶 𝐸 • 𝐸𝐶 𝛼 ) -(-𝜆 • 𝐿 𝜌 -𝐶 𝐸 • 𝐸𝐶 0 𝛼 ) ( 10
)
Where 𝐺 and 𝐿 are the prospective Gain and Loss magnitudes attached to the HE and LE options respectively, 𝜌 is the nonlinearity parameter controlling the magnitude-to-utility mapping (typically, 𝜌 < 1 [START_REF] Schoemaker | The Expected Utility Model: Its Variants, Purposes, Evidence and Limitations[END_REF][START_REF] Glimcher | Neuroeconomics: Decision Making and the Brain[END_REF])), 𝜆 is a loss aversion parameter (typically, 𝜆 > 1 [START_REF] Kahneman | Choices, values, and frames[END_REF][START_REF] Tom | The Neural Basis of Loss Aversion in Decision-Making Under Risk[END_REF]), 𝐶 𝐸 is the scaling coefficient of the effort cost discounting variable, and 𝛼 is the nonlinearity parameter controlling the curvature of effort discounting.

Thus:

𝐷𝑉 = 𝐺 𝜌 + 𝜆 • 𝐿 𝜌 -𝐶 𝐸 • (𝐸𝐶 𝛼 -𝐸𝐶 0 𝛼 ) (11)
therefore, the maximal energetic cost such that DV = 0 is:

𝐸𝐶 𝑚𝑎𝑥 = (𝐸𝐶 0 𝛼 + 𝐺 𝜌 + 𝜆 • 𝐿 𝜌 𝐶 𝐸 ) 1 𝛼 (11)
And the updated decision variable was written:

𝐷𝑉 ′ = 𝐺 𝜌 + 𝜆 • 𝐿 𝜌 -𝐶 𝐸 • (𝐴𝐸𝐶 𝛼 -𝐸𝐶 0 𝛼 ) (12)
The 'Optimal Expense' model was extended in a similar fashion. The expected utility from the high-and low-effort option were now written (respectively):

𝐸[𝑈 𝐻𝐸 (𝐸𝐶)] = 𝑃(𝑆𝑢𝑐𝑐𝑒𝑠𝑠|𝐸𝐶) • 𝐺 𝜌 -(1 -𝑃(𝑆𝑢𝑐𝑐𝑒𝑠𝑠|𝐸𝐶)) • 𝜆 • 𝐿 𝜌 -𝐶 𝐸 • 𝐸𝐶 𝛼 𝐸[𝑈 𝐿𝐸 (𝐸𝐶)] = -𝜆 • 𝐿 𝜌 -𝐶 𝐸 • 𝐸𝐶 0 𝛼 (13)
Such that the updated decision variable was:

𝐷𝑉 * = 𝑃(𝑆𝑢𝑐𝑐𝑒𝑠𝑠|𝐸𝐶 * ) • (𝐺 𝜌 + 𝜆 • 𝐿 𝜌 ) -𝐶 𝐸 • ((𝐸𝐶 * ) 𝛼 -𝐸𝐶 0 𝛼 )
As befits a Bayesian approach, we set non-trivial prior probability distributions over model parameters. We set agnostic distributions (centered on 'null' values, e.g. 𝜌 and 𝛼 were centered on 1) before inverting models on our first dataset (Exp. 1), and used the posterior distributions derived from this first inversion stage as the prior distribution in our second inversion (Exp. 2, except for the 𝜆 parameter, which was absent from the models in Exp. 1 and was attributed an agnostic prior distribution centered on 1).

Model comparison

The log-evidences, estimated for each participant and each model via the VBA scheme, were submitted to a group-level random-effect analysis [START_REF] Rigoux | Bayesian model selection for group studies-Revisited[END_REF]. This analysis was used to generate family-level protected exceedance probabilities, which quantify the likelihood that a given family of models is more frequently implemented by the population than any other family in the comparison set. This allowed us to compare the evidence for 'Standard' versus 'Null Choice', 'Cost-benefit' versus 'Linear', and 'Maximal Budget' versus 'Optimal Expense' families, over and above the variants tested within each family. Over the course of their life, individuals are presented with a number of important or mundane challenges to be taken or avoided. One of the most robust principles of human behaviour is that one tends to select the course of action which maximizes one's expected utility. By definition, the expected utility of an action is computed by multiplying the subjective value, or utility, of each possible outcome of that action, with its perceived probability. In the context of an upcoming challenge, modern decision theory therefore stipulates that agents can generate and combine two types of judgments: a judgment about the utility (or disutility) that a success (or failure) would bring about, that is, a value judgment about the stakes of the challenge, and a judgment about the probability to succeed at the task, that is, a confidence judgment.

Aside from its cardinal principle of expected utility maximization, modern decision theory rests on an implicit assumption about how the very building blocks of expected utility are created before being combined together: that of a mutual independence between value and probability

judgments. Yet, just as humans have been reported to violate the expected utility maximization principle in various circumstances [START_REF] Allais | Le Comportement de l'Homme Rationnel devant le Risque: Critique des Postulats et Axiomes de l'Ecole Americaine[END_REF][START_REF] Kahneman | Prospect Theory: An Analysis of Decision under Risk[END_REF][START_REF] Maccrimmon | Utility Theory: Axioms Versus 'Paradoxes[END_REF], they have been reported to exhibit puzzling inter-dependencies between value and probability -and sometimes, more specifically, confidence -judgments. Among the observations that defy the aforementioned independence assumption, is that of a so-called "desirability bias" [START_REF] Krizan | The influence of outcome desirability on optimism[END_REF][START_REF] Giardini | Overconfidence in Predictions as an Effect of Desirability Bias[END_REF][START_REF] Tappin | The Heart Trumps the Head: Desirability Bias in Political Belief Revision[END_REF], which refers to the propensity of healthy individuals to attribute a higher probability to more desirable (i.e., prospectively valuable) states, all other things being equal. In one of these studies [START_REF] Giardini | Overconfidence in Predictions as an Effect of Desirability Bias[END_REF], when success at a task was made more desirable by modulating the performancecontingent reward, participants displayed higher levels of retrospective confidence in their performance. Another instance of value-confidence interplay is the observed impact of affective states (or, at longer timescales, moods) induced by more or less pleasant stimuli (e.g., nice versus gloomy weather) on self-confidence [START_REF] Kavanagh | Mood and self-efficacy: Impact of joy and sadness on perceived capabilities[END_REF][START_REF] Briñol | Happiness versus sadness as a determinant of thought confidence in persuasion: A self-validation analysis[END_REF][START_REF] Koellinger | Joy Leads to Overconfidence, and a Simple Countermeasure[END_REF]. In such cases, the stimuli eliciting the value judgment are largely unrelated to the task eliciting the confidence judgment, which suggests that the mechanisms underlying valueconfidence interferences are not restricted to integral judgments and can also operate incidentally.

In the present study, we aimed to explore how the stakes of an upcoming enterprise (i.e., the value of success-contingent outcomes) might distort associated prospective confidence judgments, in a context where actual success probability is not contingent on the degree of motivation of the agent. To our knowledge, this question had never been addressed before in a controlled laboratory setting, despite the substantial repercussions that such confidence distortions may have on action selection, initiation and persistence, and, at larger scales, on individual and collective risk-taking, social status, evolutionary fitness, and psychological wellbeing [START_REF] Taylor | Illusion and well-being: A social psychological perspective on mental health[END_REF][START_REF] Koellinger | I think I can, I think I can": Overconfidence and entrepreneurial behavior[END_REF][START_REF] Johnson | The evolution of overconfidence[END_REF][START_REF] Anderson | A status-enhancement account of overconfidence[END_REF]. For this investigation, we first trained healthy participants on a novel motor task requiring a high degree of precision, which constituted the challenge that they would subsequently be faced with. Critically, we chose this particular task such that participants' actual performance would not depend on their level of motivation, which mostly depends on stakes, to perform well. This independence was a prerequisite to a clear-cut demonstration of any putative value-confidence interplay above and beyond any value-performance dependency: indeed, if the chance of success objectively increases with expected value, then it is rational to feel more confident. If it is independent, then a boost in confidence can be considered as a bias. On each trial of the testing session, we presented them with information about the difficulty and the stakes of the upcoming challenge. The difficulty was adjusted via the width of a virtual target to hit, which determined the objective probability of success (whose exact value was unknown to the participant). The stakes consisted in a variable monetary gain in case of success, and a variable monetary loss in case of failure. Importantly, the difficulty, gain, and loss factors were manipulated orthogonally across trials. Next, participants were prompted to estimate their probability of succeeding at the motor challenge, that is, to express their prospective confidence, which was our main measure of interest.

Interestingly, the several pre-existing lines of investigation on value and confidence (and their interaction) yielded conflicting predictions about what pattern of results we should obtain.

If participants were subject to a desirability bias, they should display a higher level of confidence when success was made more desirable by either high prospective gains, or high prospective losses, or both. The same should be observed in the case of a misattribution of arousal, or "stakelikelihood effect", which has been put forth as an intriguing alternative to the desirability bias account of overoptimism [START_REF] Vosgerau | How prevalent is wishful thinking? Misattribution of arousal causes optimism and pessimism in subjective probabilities[END_REF]: according to this theory, higher stakes, irrespective of their valence, should elicit higher arousal, which in turn should increase the perceived probability of the considered event -in this case, a success at the motor challenge, and thus, confidence ratings. However, a different scenario, inspired from the long-standing "affect-asinformation" construct [START_REF] Schwarz | Mood as Information: 20 Years Later[END_REF][START_REF] Schwarz | Feelings-as-Information Theory[END_REF], as well as recent behavioural and neuroimaging observations, predicted that participants should not be more confident when faced with a larger potential loss, but in fact less so. Indeed, the ventromedial prefrontal cortex (vmPFC), which is considered a key node of the brain valuation system (BVS) since it was repeatedly found to linearly encode the subjective value of considered items or even background stimuli [START_REF] Lebreton | An Automatic Valuation System in the Human Brain: Evidence from Functional Neuroimaging[END_REF][START_REF] Bartra | The valuation system: A coordinate-based meta-analysis of BOLD fMRI experiments examining neural correlates of subjective value[END_REF][START_REF] Abitbol | Neural Mechanisms Underlying Contextual Dependency of Subjective Values: Converging Evidence from Monkeys and Humans[END_REF], has been recently shown to also correlate positively with the degree of confidence in one's judgment [START_REF] De Martino | Confidence in value-based choice[END_REF]Lebreton et al., 2015). This regional overlap could reflect the fact that the codes for value and confidence are not independent, such that a change in prospective value could be misread as a change in success probability. In short, this scenario predicts that confidence judgments should be commensurate, at least to some extent, to value-related affective states. Very recently, this line of reasoning has received some support from three behavioural studies in which participants displayed higher retrospective confidence in the correctness of their choice when success lead to a reward, and lower confidence when failure led to a loss [START_REF] Lebreton | Two sides of the same coin: Monetary incentives concurrently improve and bias confidence judgments[END_REF][START_REF] Lebreton | Contextual influence on confidence judgments in human reinforcement learning[END_REF][START_REF] Ting | Decreased confidence in loss-avoidance contexts is a primary meta-cognitive bias of human reinforcement learning[END_REF]. Our primary goal was thus to study the impact of stakes -and in particular, potential loss -on prospective confidence judgments; our secondary goal was to elucidate the brain regions and cognitive mechanisms involved in the distortions we expected to observe.

Results

Experiment 1

We designed our first experiment so as to probe the impact of the stakes of an action onto three variables of interest: an individual's prior confidence about her performance (before performing the action), the choice of whether to perform the action, and her posterior confidence (after performing the action). In this paradigm, the action to perform was a brief hand movement requiring a high degree of motor precision, consisting in briefly squeezing a handgrip, so that the peak of exerted force would reach a given, pre-defined level. We chose this action because it was likely new to every participant at the beginning of the experiment (no associated history of performance), it could be quickly learned, and, in previous pilot and published studies [START_REF] Vinckier | Neuro-computational account of how mood fluctuations arise and affect decision making[END_REF], we found subjects' accuracy at this task to be largely independent of their motivation to perform correctly. We manipulated the difficulty of this action, or "motor challenge", by varying the width of the force range that the force peak should belong to (or, more figuratively, the "target" to hit) in order for the movement to be successful. We also manipulated the magnitude of monetary gains or losses contingent on success at the challenge, which ranged from 1€ to 50€. All three task factors (difficulty, gain, loss) were varied orthogonally across trials.

As illustrated in Fig. 1, we structured experimental sessions as follows: first, subjects went through an extensive calibration phase, during which they learned to perform the required movement and were given some feedback about their motor variability, which would later help them appraise their probability to succeed at the motor challenge (i.e., form a confidence judgment). Next, they started the testing trials, which were of three types: Choice, Prior

Confidence rating, and Posterior Confidence rating. In all types of trials, subjects were first presented with information about the upcoming motor challenge: its difficulty (set by the size of the green target to hit), the monetary gain in case of success, and the monetary loss in case of failure. Then, in Choice trials, subjects could accept or decline the challenge, then squeezed the handgrip without receiving any feedback about the exerted force (without any monetary consequence if they had declined the challenge); in Prior Confidence trials, subjects first estimated their a priori chances of succeeding at the challenge (i.e., of hitting the target), then they squeezed the handgrip without any feedback; in Posterior Confidence trials, subjects first squeezed the handgrip without any feedback, then they estimated their a posteriori chances of having succeeded at the challenge.

Importantly, irrespective of the trial type, subjects were not given any feedback about their performance during or after their movement. Nevertheless, once all Choice (resp. all

Confidence rating) trials were completed, a subset (5%) of these trials was randomly drawn by the computer and the corresponding gains or losses (or 0 for Choice trials in which the subject had declined the motor challenge) were averaged together, to form a bonus (or malus) on the subject's initial monetary endowment.

Twenty-four healthy adults gave their consent to participate in this experiment and were included in our analyses (except for two of them, where mentioned). The focus of this article being the impact of the stakes of an action on the associated a priori confidence, we restrict the analyses below to Prior Confidence trials.

Fig. 1: Design of Experiment 1

a. Session structure. Each experimental session began with a calibration phase, in which participants learned to perform the required movement with decent precision and were given feedback about their motor accuracy (see Methods for details). This was followed by two testing phases: a CHOICE phase and a CONFIDENCE phase, each comprising 100 and 200 trials, respectively. Participants went through a recalibration procedure every 100 trials. At the end of each testing phase, 5 or 10 trials were randomly drawn, and the corresponding gains or losses were averaged, which constituted a bonus/malus to the compensation eventually paid to each participant.

The order of CHOICE and CONFIDENCE phases was pseudo-randomized across subjects.

b. Motor challenge. The required movement consisted in briefly squeezing a handgrip such that the force peak would fall within a target range around 40% of the participant's maximal force, which was represented as a green window on a virtual thermometer. Four example force timecourses are illustrated. The green line represents a successful trial. Participants could fail because they squeezed too much (upper red line), not enough (lower red line) or for too long (more than 400 ms, grey line).

c. Trial structure. At the beginning of each trial, a fixation cross was displayed at the center of the screen, followed by a representation of the motor difficulty and the financial stakes of the trial: the green target window on the thermometer varied in size across trials, as did the monetary gains (resp. losses) ensuing a motor success (resp. failure). Then, three different series of tasks could occur: 1) in CHOICE trials (n = 100), participants were asked whether they wished to accept the bet, i.e., take up the motor challenge and its potential monetary outcomes. Next, upon appearance of a 'Squeeze!' cue, participants would squeeze the handgrip, with potential monetary consequences depending on their precision (in case of a YES answer) or no monetary consequences (in case of a NO answer). 2) in PRIOR CONFIDENCE RATING trials (n = 100), a scale ranging from 0% to 100% was displayed, prompting participants to report their a priori chance of success at the challenge (which they could not refuse). Then, upon appearance of the 'Squeeze!' cue, they would squeeze the handgrip and try to hit the target. 3) in POSTERIOR CONFIDENCE RATING trials (n = 100), participants performed the motor challenge first, and then were prompted to estimate their a posteriori chance of success. All PRIOR and POSTERIOR confidence rating trials were randomly interleaved within the CONFIDENCE phase. Regardless of the trial type, after completing the series of tasks (which always included squeezing the handgrip), participants would proceed to the next trial without receiving any feedback about their motor performance.

Success rate

We first performed a logistic regression of individual motor success across trials (1 if target hit, 0 otherwise), which comprised the following regressors: the (objective) expected probability of success based on the size of the target (unknown to the participant, see Methods for details), the Gain level associated to a success, the Loss level associated to a failure. Reassuringly, we found a strong positive effect of the expected probability of success ("pSuccess") predictor (𝛽 ̅ = .66, 𝐶𝐼 = Two subjects were excluded from this random-effects analysis because their motor accuracy was extremely low (success rate = 1%) on this set of trials, although they had performed the training correctly, and although their confidence ratings were sensitive to the difficulty level and did not reflect much awareness of their poor performance.

Prior confidence

We next inverted a Generalized Linear Model (GLM) of prior confidence ratings made by each subject across trials, comprising the same predictors as the logistic model of motor success. We found a strong positive effect of the expected probability of success (𝛽 ̅ = .15, 𝐶𝐼 = [ .11 ; .18], 𝑡(23) = 9.5, 𝑝 < 10 -8 ) on confidence ratings, which confirms that subjects were able to extract information about the difficulty of the upcoming motor challenge from the visual display (green target of varying size in a virtual thermometer), and were careful enough in their confidence ratings for this information to be reflected. However, we did not observe any .10) on prior confidence ratings.

Experiment 2

In Experiment 1, we failed to detect any significant impact of prospective gain or loss on prior confidence judgments in spite of the predictions derived from two well-established psychological phenomena (desirability bias and affect-as-information). We sought to elucidate whether this absence of finding was due to an absence of true psychological effect, or rather due to a lack of statistical power (or some other design flaw) in our first attempt. Indeed, each individual set of 𝛽 estimates was derived from 100 trials, which might be too few to yield a reliable estimate.

Therefore, we performed a second experiment in which we increased the focus on prior confidence judgments, as well as the number of corresponding ratings made by each participant.

In order to reduce the information load resulting from the display of the prospect panel (difficulty, gain and loss), and thus to increase awareness of the monetary stake, we introduced trials in which no loss or no gain could be incurred ("Gain-only" and "Loss-only" types, respectively). One more minor change was the introduction of more frequent feedbacks (once every 10 trials) about the gains and losses incurred over the last trials, so as to preserve the subject's interest and engagement in the task throughout the testing session (see Fig. 2). Another minor change was the introduction of a lottery choice task at the end of the testing session. This task consisted in making binary choices between risky monetary lotteries and sure (but modest) monetary gains. The aim of this lottery task was to derive a loss aversion parameter from those binary choices, for each participant.

Twenty-one healthy subjects consented to take part in this experiment and were included in our analyses, which follow on from those performed on our Experiment 1 dataset.

Success rate

Again, and as intended, the probability of hitting the target was strongly modulated by the a. Session structure. Each experimental session began with a calibration phase, in which participants learned to perform the required movement with decent precision and were given feedback about their motor accuracy (see Methods for details). This was followed by a testing phase comprising 270 prior confidence rating trials, split into 3 blocks each including 9 games of 10 trials. At the end of each game, a feedback was given about the averaged monetary outcomes of the last 10 trials. Participants went through a recalibration procedure every 90 trials. After all 27 games were completed, one of them was randomly drawn by the computer and its averaged outcomes were added (or subtracted) to the participant's payment. This was followed by a lottery task, comprising 90 trials in which participants chose between a 50/50 gain/loss lottery and a certain smaller gain. (See Methods for details.)

b. Trial structure. This was similar to the structure of PRIOR CONFIDENCE trials in Experiment 1, with three variants regarding the format of the financial stakes attached to the motor challenge. In « gain-only » trials (n = 60), participants would incur no loss in case of failure, whereas in « loss-only » trials (n = 60), they would obtain no gain in case of success. The remaining 150 « gain & loss » trials involved non-null outcomes for both success and failure, as in Experiment 1. Trials of the three types were randomly interleaved across the experiment.

Experiment 3

In this follow-up experiment, we pursued two main goals: first, to assess the robustness of the previous two-fold finding of a positive impact of prospective gain, and a negative impact of prospective loss, on prior confidence judgments; second, to identify the brain regions and mechanisms involved in such a value-to-confidence spillover, with the help of functional magnetic resonance imaging (fMRI). Regarding the latter issue, our working hypothesis was that the ventromedial prefrontal cortex (vmPFC) was a major site of interference between value-encoding and confidence-encoding signals, and that individuals exhibiting a larger spillover effect at the behavioural level would also exhibit a higher degree of similitude between patterns of valueencoding and confidence-encoding respectively, at the neural level.

For this, we made a few minor changes in our experimental design, with the aim of increasing its efficiency and compatibility with fMRI constraints (Fig. 3). We also added post-block feedbacks about the subject's success rate over the past 50 trials, so as to help their confidence judgment stay calibrated on their true performance, without providing too much information about it (which would be less in line with real-life situations, where individuals rarely have the opportunity to refine their confidence judgment over hundreds of successive trials and feedbacks).

We recruited ten healthy participants for a pilot experiment, in which only behavioural data was acquired, then recruited twenty-four additional participants, who performed the same tasks in an fMRI scanner.

Fig. 3: Design of Experiment 3

a. Session structure. The calibration & training phase (see Methods for details) was followed by a testing phase comprising 250 prior confidence rating trials, split into 5 blocks each including 5 games of 10 trials. At the end of each game, a feedback was given about the averaged monetary outcomes of the last 10 trials. Participants went through a recalibration procedure every 50 trials. At the end of each block, they were informed of their success rate over the last 50 trials. After all 25 games were completed, one of them was randomly drawn by the computer and its averaged outcomes were added (or subtracted) to the participant's payment (this random draw procedure was repeated 5 times in the fMRI group). This was followed by a lottery task, identical to that in Experiment 2.

b. Trial structure. All testing trials involved a prior confidence rating similar to that of Experiments 1 and 2. See Methods for details.

Behaviour

We repeated our previous analyses on this third behavioural dataset. For both the success rate and prior confidence ratings, we replicated the findings of Experiment 2.

Success rate

As before, the probability of hitting the target was strongly modulated by the expected probability

fMRI

Univariate approach

First, we sought to identify the brain regions respectively involved in processing the monetary stakes of the motor challenge, and in encoding the confidence judgment subsequently expressed in the rating. Hoping to replicate an earlier finding that the vmPFC encodes both value (monetary stakes) and confidence judgments [START_REF] Lebreton | Automatic integration of confidence in the brain valuation signal[END_REF], we adopted a similar -conventionalmassively univariate approach. (See Methods for details about BOLD response models.) Although contrasts of non-interest such as "Squeeze Onset > Prospects Onset" did reveal an involvement of the expected regions (in the contrast mentioned above, we found a cluster in the primary motor cortex, or M1: cluster peak coordinates [-34 -24 54], 𝑇 = 15.7, 𝑍 = 7.5, 𝑝 𝑐𝑙𝑢𝑠𝑡𝑒𝑟 𝐹𝑊𝐸𝑐𝑜𝑟𝑟 < 10 -15 ) , no voxel crossed the significance threshold in a Confidence contrast, nor in a Gain or even an Expected Value contrast (whether expressed as Gain -Loss or as Confidence*Gain -(1 -Confidence)*Loss).

Multivariate Pattern Analysis (MVPA)

Despite this failure to identify brain regions involved in value and/or confidence processing via a univariate approach, we moved on to an MVPA analysis, which has been shown to yield more statistical power [START_REF] Norman | Beyond mind-reading: Multi-voxel pattern analysis of fMRI data[END_REF][START_REF] Allefeld | Valid population inference for information-based imaging: From the second-level t-test to prevalence inference[END_REF], and which would allow us to quantify, in each individual, the similarity (here, as a simple inter-voxel correlation) in the patterns of encoding of value versus confidence in pre-defined regions of interest (here, the vmPFC). This purpose followed from the intuition that a higher degree of behavioural interference between value and confidence would correspond to a higher degree of similarity or "overlap" in neural codes. We therefore derived a vector of 𝛽 weights for each subject and each variable of interest (various combinations of gain and loss, confidence) across the voxels of the vmPFC, then computed the correlation between the value and confidence weight vectors. However, contrary to our expectations, we found no significant relationship between the degree of similitude in valueand confidence-encoding patterns on the one hand, and the magnitude of behavioural effect of interest (positive effect of prospective gain, 𝑟 = -.16, 𝑝 = .49, negative effect of prospective loss on prior confidence judgments, 𝑟 = -.18, 𝑝 = .44, or overall effect of Gain -Loss, 𝑟 = -.26, 𝑝 = .26).

Experiments 1 to 3, pooled.

In order to increase statistical power and to achieve a more accurate estimate of the size of our effects of interest, and because that size estimate did not vary between our three experiments (one-way ANOVA on individual estimates of 𝛽 𝐶𝑜𝑛𝑓 𝐺𝑎𝑖𝑛: 𝐹 2 = .32, 𝑝 = .73; on 𝛽 𝐶𝑜𝑛𝑓 𝐿𝑜𝑠𝑠: 𝐹 2 = .21, 𝑝 = .81), we pooled the three datasets and reiterated the same statistical analyses. main predictors was found to have any significant impact, all p > .6). After excluding subjects whose success rate was sensitive to stakes, these results were all maintained: expected probability of success was still a significant predictor of confidence ratings (𝛽 ̅ = .094, 𝐶𝐼 = [. 077 ; .11], 𝑡(63) = 11, 𝑝 < 10 -15 ), as were prospective gain level (𝛽 ̅ = .0084, 𝐶𝐼 = [. 0030; .014], 𝑡(63) = 3.1, 𝑝 = .0029) and prospective loss level (𝛽 ̅ = -.0087, 𝐶𝐼 = [-.014 ; -.0031], 𝑡(63) = -3.1, 𝑝 = .0028) (Fig. 4b). Again, all interaction terms were nonsignificant, p > .5.

Remarkably, we found individual estimates of 𝛽 𝐶𝑜𝑛𝑓 𝐺𝑎𝑖𝑛 and 𝛽 𝐶𝑜𝑛𝑓 𝐿𝑜𝑠𝑠 to be negatively correlated across participants ( 𝑟 = -.50, 𝑝 < 10 -4 , 𝐅𝐢𝐠. 𝟓), with individuals exhibiting a larger positive distortion of confidence by prospective gain also exhibiting a larger negative distortion by loss. Besides, we did not find any evidence for an interindividual correlation between 𝛽 𝐶𝑜𝑛𝑓 𝐺𝑎𝑖𝑛 and 𝛽 𝑆𝑢𝑐𝑐𝑒𝑠𝑠 𝐺𝑎𝑖𝑛 (𝑟 = .024, 𝑝 = .83; 𝑟 = -.046, 𝑝 = .72 after excluding subjects whose performance was incentive-dependent), nor between 𝛽 𝐶𝑜𝑛𝑓 𝐿𝑜𝑠𝑠 and 𝛽 𝑆𝑢𝑐𝑐𝑒𝑠𝑠 𝐿𝑜𝑠𝑠 (𝑟 = .057, 𝑝 = .62; 𝑟 = .16, 𝑝 = .21 after excluding subjects whose performance was incentivedependent). Therefore, the distortion of an individual's confidence judgments by the stakes of the challenge was dissociated from the impact that these stakes could have on her actual probability of success.

Besides, it would seem that subjects were not aware of this impact of stakes on their confidence, as the vast majority of them denied, in the post-session debriefing, having been influenced in their confidence ratings by anything else than the size of the virtual target and psychomotor factors such as physical fatigue, perceived history of performance over the past trials, etc.

Finally, we did not find any inter-individual correlation between the loss aversion parameter estimates 𝜆 derived from binary choices made during the lottery task on the one hand, and various aggregates of beta estimates quantifying the relative impact of prospective gains and losses on confidence ratings on the other hand (we tested correlations between 𝜆 and 𝛽 𝐺𝑎𝑖𝑛 -

𝛽 𝐿𝑜𝑠𝑠 , 𝛽 𝐿𝑜𝑠𝑠 𝛽 𝐺𝑎𝑖𝑛 𝑜𝑟 |𝛽 𝐿𝑜𝑠𝑠 | |𝛽 𝐺𝑎𝑖𝑛 |
, all 𝑝 > .33). We found a significant negative correlation (𝑟 = -.50, 𝑝 < 10 -4 ) between 𝛽 𝐶𝑜𝑛𝑓 Gain and 𝛽 𝐶𝑜𝑛𝑓 Loss across subjects.

Middle panel: Individual 𝛽 estimates of Gain level predictor in a GLM model of Prior Confidence ratings, against individual 𝛽 estimates of Gain level predictor in a logistic model of motor Success. Green and grey dots represent included (N = 62) and excluded subjects (N = 17), respectively. (Inclusion criteria were: as above, no marginal impact of prospective gain and/or loss on motor success probability, and success rate > 5% so as to allow the inversion of the logistic model.)

We found no significant correlation between 𝛽 𝑆𝑢𝑐𝑐𝑒𝑠𝑠 Gain and 𝛽 𝐶𝑜𝑛𝑓 Gain.

Right panel: Individual 𝛽 estimates of Loss level predictor in a GLM model of Prior Confidence ratings, against individual 𝛽 estimates of Loss level predictor in a logistic model of motor Success. Red and grey dots represent included and excluded subjects (following the same criteria as above), respectively. We found no significant correlation between 𝛽 𝑆𝑢𝑐𝑐𝑒𝑠𝑠 Loss and 𝛽 𝐶𝑜𝑛𝑓 Loss.

In all panels, the red line illustrates a simple linear regression of the y-axis variable by the x-axis variable, computed only for the subset of subjects included in the correlation computation (N = 64).

Discussion

Here, we examined how the stakes of an upcoming challenge may affect one's perceived chances of success, that is, one's prospective confidence judgment. For this, in a series of three experiments, we presented healthy participants with a succession of motor challenges varying in difficulty, but also in the magnitudes of gain and loss that could be incurred depending on their success or failure at the challenge, and we had them estimate their a priori chances of hitting the virtual target. Overall, we found confidence judgments to be sensitive to both prospective gain and prospective loss, to a similar extent but in opposite directions: confidence was increased by larger prospective gain, and diminished by larger prospective loss. Importantly, the effect of prospective gain was negatively correlated, across participants, to the effect of prospective loss, but the influence of monetary stakes on confidence ratings was dissociated from their impact on actual probability of success. Besides, based on their reports during the written and oral postsession debriefing, it would seem that participants were largely unaware of the stakes-induced distortion of their confidence judgments.

Among the various mechanisms or psychological constructs that have previously been put forward to account for observed interferences between value and probability -and more specifically, confidence -judgments, several fail to explain our pattern of findings. First, if participants had been subject to a desirability bias, i.e., a tendency to perceive more desirable events as more likely to occur [START_REF] Krizan | The influence of outcome desirability on optimism[END_REF], then their confidence should have increased in the face of higher prospective losses (and thus more desirable success at the motor challenge). Similarly, if the positive distortion of confidence by prospective gains was underpinned by a misattribution of the arousal [START_REF] Vosgerau | How prevalent is wishful thinking? Misattribution of arousal causes optimism and pessimism in subjective probabilities[END_REF] elicited by large prospective gains, then, large prospective losses being more arousing than small ones [START_REF] Knutson | Distributed Neural Representation of Expected Value[END_REF][START_REF] Knutson | Anticipatory affect: Neural correlates and consequences for choice[END_REF], we should have observed an effect of prospective losses in the same direction as gains. Neither could our findings be explained by some strategic confidence adjustment, which would consist for individuals in artificially boosting their confidence reports, especially so in high-stakes situations, with the hope -presumably derived from some selffulfilment conception of performance prediction -that it would boost their chances of success.

Finally, we can rule out another scenario that might have accounted for a positive effect of reward incentives on confidence. According to this scenario, healthy subjects hold an internal "theory of motivated control", that is, they reason that, when they are more motivated to succeed (because of a large incentive), they will invest more effort into performing the task correctly, and therefore are more likely to succeed. Again, such a scenario predicts an increase in confidence in the face of larger prospective losses, and is thus in contradiction with our findings.

Instead, the incentive-confidence interplay isolated here was evocative of an "affect-asinformation" (Schwarz, 2012) mechanism. Indeed, confidence covaried with the utility (or subjective value) of prospective stakes; this covariation could have been mediated by the affective states elicited by those stakes, which could in turn have informed confidence estimation, in the sense that a region or network representing the current affective state might send a positive (or in neural terms, excitatory) input to the cognitive network in charge of building a confidence signal.

Given that high prospective gains or losses not only modulate affective state but also attract attention [START_REF] Anderson | A value-driven mechanism of attentional selection[END_REF][START_REF] Schmidt | Attentional capture by signals of threat[END_REF][START_REF] Hickey | Neural Mechanisms of Incentive Salience in Naturalistic Human Vision[END_REF][START_REF] Vuilleumier | Affective and motivational control of vision[END_REF][START_REF] Müller | Relevance drives attention: Attentional bias for gainand loss-related stimuli is driven by delayed disengagement[END_REF][START_REF] Schomaker | Attention in natural scenes: Affectivemotivational factors guide gaze independently of visual salience[END_REF], an alternative explanation for their opposing impact on confidence is that particularly large gains (resp. losses) captured participants' focus during the display of challenge difficulty and stakes, and that their attentional dwelling on the "GAIN" (resp.

"LOSS") potential outcome increased the perceived probability of such an outcome occurring, in line with the "repeated simulation" mechanism which has been proposed as an account for preference-related likelihood distortions [START_REF] Krizan | The influence of outcome desirability on optimism[END_REF] and has since received some empirical support [START_REF] Szpunar | Get real: Effects of repeated simulation and emotion on the perceived plausibility of future experiences[END_REF]. Finally, beyond their effect on attentional focus or affective state, the stakes of an action are obviously a major determinant of initiation of that action [START_REF] Vinckier | Neuro-computational account of how mood fluctuations arise and affect decision making[END_REF]. Thus, another possibility is that, even when not given the choice to engage or not, subjects had generated some internal signal of their determination to engage (which would be higher for larger prospective gains, lower for larger prospective losses), and that this signal would have influenced confidence judgments.

Unfortunately, we could not assess any of the candidate neural mechanisms, because we failed to identify the neural representations of value and confidence in our experimental settings.

Irrespective of its underlying mechanisms, it would be worth investigating the repercussions of this value-related distortion of confidence judgments on the propensity for individuals to engage and persevere in risky courses of action. One reason is that it might allow us

to gain insight about the potential adaptiveness of what would at first seem prejudicial to forecasting and decision-making processes, as has been undertaken for the related overconfidence bias [START_REF] Johnson | The evolution of overconfidence[END_REF]. One can speculate that this interplay would be advantageous in relatively safe and resource-abundant environments, where individuals might benefit from a supplement of confidence in daily enterprises involving no risk of substantial loss, whereas behaviours susceptible to lead to massive or even immeasurable losses such as major injury or death would be discouraged. Besides, if, following on Bayesian conceptualizations of confidence, we assume that little-informed (that is, uncertain, or distributionally wide, or entropic) confidence judgments are more sensitive to influences from action stakes, then this distortion might modulate the weight of outcome value as a decision variable in action selection, and especially amplify it in contexts where evidence is too scarce to yield a reliable confidence judgment as a basis for expected utility computation. This uncertainty-dependent modulation strength hypothesis could be easily tested by appraising the magnitude of confidence distortions in more or less ambiguous contexts (regarding probability of success). A second, more pragmatic and present-oriented reason to take a closer look at the consequences of stakes-induced confidence distortions is that it might motivate a leveraging of this phenomenon at a public-policy level, in order to modulate confidence in under-or over-confident populations and eventually promote or refrain audacious, or risky, behaviours.

24 participants were recruited for the fMRI task (13 M, aged 24.8 ± 3.6 years). Although they were told that they were endowed an initial 50€ and could win or lose up to that same amount depending on their performance at the precision task (with an additional bonus of up to 5€ for their accuracy at predicting their performance, and another bonus from the random draws of a subsequent lottery task), they were eventually paid a fixed amount of 75€ regardless of their performance or the outcome of the draws.

Apparatus

We programmed all tasks and data analyses with MATLAB (MathWorks ®) and used the Psychophysics Toolbox 3.0 [START_REF] Kleiner | What's new in psychtoolbox-3[END_REF] for stimuli presentation. Choices and ratings were made using a standard keyboard (or an MRI-compatible response device in Experiment 3 -fMRI task). Grip force was measured with a hand dynamometer (Software and Technology, Beaverton, Oregon, USA) interfaced with Matlab so as to readout force in Newtons (or a custommade MRI-compatible pneumatic dynamometer in Experiment 3 -fMRI task).

fMRI data acquisition

Functional echo-planar images (EPIs) were acquired with a T2*-weighted contrast on a 3 T magnetic resonance scanner (Siemens Trio). Interleaved 2 mm slices with a 2.5 mm interslice spacing and oriented along a 60° tilted plane were acquired in a multi-band fashion (factor 3) to cover the whole brain with a repetition time of 1100 ms. The first three scans were discarded to allow for equilibration effects. All analyses were performed using the Statistical Parametric

Mapping toolbox (SPM12, version 7219, Wellcome Trust Center for NeuroImaging, London, UK).

Structural T1-weighted images were coregistered with the mean EPI, segmented, and normalized to the standard Montreal Neurological Institute (MNI) T1 template. During preprocessing, EPIs were spatially realigned, normalized (using the same transformation as for structural images), and smoothed with an 8 mm full-width at half-maximum Gaussian kernel.

Experimental design

Experiment1

After giving their informed consent, participants read detailed written instructions about the tasks to be completed. It was explained to them that they were endowed 20€ at the beginning of the session, and could win or lose money depending on their motor precision, the choices they would make, and their luck during the experiment. Next, they went through an extensive calibration phase during which they gradually learned to perform the required movement and received feedback about their motor accuracy. Specifically, the calibration phase comprised the following stages:

1) Measurement of individual maximal voluntary contraction, or 'fMax'; 3 trials.

Participants were instructed to squeeze the dynamometer, or 'handgrip', with their right hand, as hard as possible. They were given online visual feedback about the exerted force, in the form of a fluid level moving up and down in a virtual thermometer depicted on the computer screen. The maximal force value recorded over their three attempts was set as their reference maximal force, or 'fMax', for the rest of the session (unless they overshot that value later on, in which case that new maximal value was set as the new fMax).

2) First stage of motor calibration, or 'Grip 1'; 15 trials. Participants were instructed to briefly squeeze the handgrip so as to produce an impulsion lasting less than 400 ms, the peak of which should land within a force range displayed as a green area within the thermometer. Although the size of the target could vary, it was always centered on 40%

of their maximal force. They were given online feedback about their exerted force in the form of a fluctuating fluid level, as well as offline feedback as a Force against Time plot displayed to the right of the thermometer (similar to Fig. 1b plots), which was accompanied with explicit written feedback about their performance ('Success', 'Too strong', 'Too weak', or 'Too slow').

3) Second stage of motor calibration, or 'Grip 2'; 15 trials. This stage was similar to the previous one, except that this time, no online visual feedback was delivered (a static green and red thermometer was displayed during the movement). The offline feedback plot was still displayed after their movement was completed.

4) Third stage of motor calibration, or 'Grip 3'; 15 trials. This stage was similar to the previous one, except that no target was displayed during movement execution. Upon presentation of a 'Squeeze!' cue (the same as in later testing trials), participants would squeeze the handgrip and try to be as close as possible to 40% fMax (which was referred to them as "the same force level as previously"). They were still given offline feedback.

5) Fourth stage of motor calibration, or 'Grip 4'; 30 trials. This stage was similar to the previous one, except that no offline feedback was given to the participants about their movement. Unbeknownst to them, the aim of this stage was to sample their peak force distribution so as to compute the target size corresponding to a 50% success rate. 6) Confidence calibration; 30 trials. This stage was similar to the previous one, except that a thermometer with a green target (whose size was fixed and corresponded to a 50% probability of success, cf. 5)) was displayed instead of the 'Squeeze!' cue, and participants were instructed to aim at this target. After completing all trials, they were prompted to estimate their success rate, by positioning a cursor along a pseudo-analog scale ranging from 0% to 100% (with an additional 50% reference tick). Upon validation of their estimate, their actual success rate was disclosed to them, along with a Peak force vs Trial number plot, which depicted their motor variability and drift over 30 trials without feedback.

Once they were calibrated, participants began the testing trials, which were split into a CHOICE phase (100 trials) and a CONFIDENCE phase (200 trials). Before each phase, they received additional written instructions and performed 4 training trials identical to the subsequent testing trials, in order to familiarize themselves with the series of tasks. Even-numbered subjects started with the CHOICE phase whereas odd-numbered subjects did the opposite.

CHOICE phase

Each trial comprised four steps:

1) A fixation cross was displayed for 3 s at the center of the screen

2) The difficulty of the proposed motor challenge was depicted in the form of a thermometer with a green target (which represented, as in the calibration phase, the range of force around 40% fMax to aim at) varying in size across trials. Unbeknownst to the participants, the size of the target was such that the corresponding probability of success was one of the following: {0.2, 0.4, 0.6, 0.8}, and was pseudo-randomly sampled such that each Probability x Gain x Loss level combination was presented once over the CHOICE phase, and Probability, Gain and Loss factors were varied orthogonally. The matching target sizes were re-adjusted every 20 trials so as to be in line with the participant's recent history of motor precision. On either side of the thermometer were displayed the pictures of the monetary gain in case of success at the motor challenge (in a green frame) and loss in case of failure (in a red frame). Gains and losses were pseudorandomly sampled among the following values: {1, 5, 10, 20, 50} €. The side of gain versus loss was randomized across trials. This screen was displayed for 4 s.

3) Participants accepted or declined the motor challenge (and the associated monetary stakes) by selecting 'YES' or 'NO' with standard keyboard arrows. There was no constraint on their response time.

4) Upon presentation of the 'Squeeze!' cue, they were required to squeeze the handgrip. If they had accepted the motor challenge, they attempted to execute the movement as required and to hit the target. In any case, they did not receive any feedback about their motor output. At this stage, regardless of their choice or their motor performance, their cumulative payment remained unchanged.

Random draw

As announced to the participants, after all CHOICE trials were completed, 5 of them were randomly drawn by the computer, and the corresponding monetary outcomes (gains or losses if they had accepted the motor challenge, 0 if they had declined it) were revealed to the participant, then averaged, and that average outcome was added -or subtracted -to their cumulative payment.

CONFIDENCE phase

Two types of trials occurred (in equal proportion but in random order) in this phase: PRIOR confidence rating and POSTERIOR confidence rating trials. In either case, each trial comprised 4 steps. The first 2 steps were identical to CHOICE trials, and the order of steps 3) and 4) differed between PRIOR and POSTERIOR trials. For PRIOR trials:

3) After they were presented with the difficulty of the motor challenge and the magnitude of the attached monetary stakes (step 2), they were prompted to estimate their a priori chances of success, i.e., their subjective probability of hitting the target in their upcoming attempt. This rating was made by positioning a cursor along a pseudo-analog horizontal scale ranging from 0% to 100% (with a middle 50% tick). There was no constraint on their response time.

4) Upon appearance of the 'Squeeze!' cue, they squeezed the handgrip and attempted to hit the target previously shown. They did not receive any feedback about their motor output. At this stage, regardless of their motor performance, their cumulative payment remained unchanged.

For POSTERIOR trials, step 4) consisted in rating their a posteriori chances of success, i.e., their subjective probability of having just hit the target, in a similar fashion to step 3) of PRIOR trials.

Random draw

As announced to the participants, after all CONFIDENCE trials were completed, 10 of them were randomly drawn by the computer, and the corresponding monetary outcomes (gains or losses, depending on whether they had hit the target) were revealed, then averaged, and that average outcome was added -or subtracted -to their cumulative payment.

Recalibration

Every 100 trials across CHOICE and CONFIDENCE phases, participants went through a short recalibration procedure designed to help them stay in line with the average force peak required of them (40% fMax), and to give them feedback on their actual success rate as well as their motor variability and drift across trials. This recalibration comprised 10 trials identical to 'Grip 3' trials of the calibration phase, followed by a stage identical to the 'Confidence' calibration stage (see above).

Experiment 2

Experiment 2 was similar in design to Experiment 1. Participants gave their informed consent, received similar instructions and went through the same calibration procedure before beginning the testing phase. The differing features were:

-A focus on prior confidence estimates: the testing phase comprised 270 trials, all of PRIOR CONFIDENCE type (see nomenclature above).

-Some of these trials were gain-only (failure at the motor challenge would result in no loss) or loss-only (success would result in no gain). All three types of trials were randomly interleaved across the testing phase.

-Gain/Loss magnitudes proposed across trials were: {5c, 50c, 2€, 10€, 50€}. The probability levels according to which the size of the target was set were: {0.35, 0.45, 0.55, 0.65, 0.75, 0.85}. The average probability of success was thus raised to .6 in order to ensure a positive expected value of the motor challenge, and to preserve participants' general motivation over the experiment. Each Gain x Loss x Probability combination was proposed once over the testing phase (150 'gain & loss' trials), and each Gain x Probability or Loss x Probability combination was proposed twice (60 'gain-only' and 60

'loss-only' trials).

-The total 270 testing trials were split into 3 blocks of 90 trials interleaved with 3-minute breaks followed by a recalibration (identical to Experiment 1). Each block comprised 9

"games" of 10 trials. After completing each game, the average monetary outcome over its 10 trials was disclosed to the subject.

-After all 270 trials were completed, one of the 27 games was randomly drawn by the computer, the average monetary outcome over its 10 trials was added (or subtracted) to the participant's initial endowment (20€).

-The main testing phase was followed by a 'Lottery' phase, which aimed to measure idiosyncratic loss aversion. This phase consisted of 90 trials in which participants were presented with a binary choice between a risky lottery with 50% chances of gain (varying in magnitude across trials) and 50% of loss (variable too), and a certain option with 100% chances of a smaller gain. The magnitude of gain and loss outcomes attached to the lottery was adjusted across trials following a staircase procedure, so as to ensure proximity to each subject's indifference curve and to optimize the subsequent estimation of individual loss aversion parameters. To make monetary outcomes more incentivizing, 10% of trials were covertly selected in advance and the corresponding choice was executed immediately after being made, i.e., a "Selected trial" warning appeared on screen upon choice validation, then, if the risky lottery had been chosen, a virtual coin was flipped and the outcome was kept hidden but was stored by the computer. At the end of the Lottery phase, the outcomes of the 9 selected trials were averaged and added (or subtracted) to the participant's final payment.

Experiment 3

In this experiment, we tested two groups of healthy participants. The Pilot group was tested first, and allowed us to confirm the viability of our experimental design for subsequent fMRI data acquisition. Both groups went through the same series of task, but the Pilot group remained in the behavioural testing room (thus, we only acquired behavioural data), whereas the fMRI group performed all testing trials in an fMRI scanner. Overall, the design of Experiment 3 was similar to that of Experiment 2, except for the following features:

-For design efficiency, all testing trials involved two-sided monetary stakes, that is, a nonzero gain in case of success, and a non-zero loss in case of failure (i.e. all trials were of "gain & loss" type).

-The probability levels according to which the size of the target was set were: {0.36, 0.48, 0.60, 0.72, 0.84}. Each Gain x Loss x Probability combination was proposed twice over the testing phase, in a pseudo-random order.

-The duration of the fixation cross preceding the display of the target and monetary stakes was randomly sampled from a uniform [0.5; 4.5] s distribution. We additionally inserted a waiting screen of variable duration (sampled from a uniform [0; 4] s distribution) between the validation of the confidence rating and the onset of the 'Squeeze!' cue. This was 1) to make the trial events' onset less predictable (and the overall session less monotonous), 2) to minimize correlation (as well as autocorrelation) between the timecourses of our several independent variables of interest (e.g. the onset/offset of monetary stakes, and the onset of the movement) as well as inadvertent correlation of these timecourses with periodic signal fluctuations of noninterest (e.g. due to physiological functions), which would be detrimental in subsequent analyses of fMRI data.

-The design of the 'Squeeze!' cue was simplified (reduced to its text component), and its onset was preceded (t -500 ms) with the onset of a grey square, which cued participants to get ready for the motor execution.

-Due to time constraints on fMRI acquisition time, a time limit was set on confidence ratings (< 5 s between scale onset and rating confirmation) and on motor execution (< 5

s between the onset of the 'Squeeze!' cue and movement initiation).

-For half the testing trials in the Pilot group, and all testing trials in the fMRI group, the motor target was centered on 25% fMax. Participants in the pilot group were notified of this shift at the middle of the session and re-calibrated on the new force level before proceeding with the trials.

-In the fMRI group, participants went through a preliminary recalibration phase in the scanner, even before the beginning of testing trials, in order to re-sample their maximal force and to help them stay aligned with the desired force level after they had changed environment and body position.

-The 250 testing trials were split into 5 blocks (or 'runs' in fMRI terminology) of 5 'games'

of 10 trials each. At the end of each block, participants were informed of their success rate over the 50 trials they had just completed.

Behavioural analysis

For each dataset, we performed a logistic regression of the binary Success variable. The inverted model was:

𝑃(𝑆𝑢𝑐𝑐𝑒𝑠𝑠) = 1 1 + exp (-(𝛽 0 + 𝛽 1 * 𝑃 𝐸𝑥𝑝𝑒𝑐𝑡𝑒𝑑 (𝑆𝑢𝑐𝑐𝑒𝑠𝑠) + 𝛽 2 * 𝐺𝑎𝑖𝑛 + 𝛽 3 * 𝐿𝑜𝑠𝑠))
with 𝑃 𝐸𝑥𝑝𝑒𝑐𝑡𝑒𝑑 (𝑆𝑢𝑐𝑐𝑒𝑠𝑠) the expected probability of success (computed based on the target size and individual motor output in recent trials), 𝐺𝑎𝑖𝑛 the level of prospective gain to be obtained in case of success (coded from 1 to 5), and 𝐿𝑜𝑠𝑠 the level of prospective loss to be incurred in case of failure. The 𝛽 𝑖 parameters are the linear coefficients of each regressor.

We first obtained an estimate of each 𝛽 parameter for each subject, using Matlab's glmfit.

We next performed a group-level random-effects analysis by applying a Student t-test to each group vector of 𝛽 parameters.

Additionally, for each subject, we inverted a simple Generalized Linear Model (GLM) of prior confidence ratings:

𝐶𝑜𝑛𝑓𝑖𝑑𝑒𝑛𝑐𝑒 𝑅𝑎𝑡𝑖𝑛𝑔 = 𝛽 0 + 𝛽 1 * 𝑃 𝐸𝑥𝑝𝑒𝑐𝑡𝑒𝑑 (𝑆𝑢𝑐𝑐𝑒𝑠𝑠) + 𝛽 2 * 𝐺𝑎𝑖𝑛 + 𝛽 3 * 𝐿𝑜𝑠𝑠
We also inverted a similar model augmented with the corresponding interaction terms, which were orthogonalized to the main terms. We next performed a group-level random-effects analysis by applying a Student t-test to each group vector of 𝛽 parameters. where 𝑃(𝑊𝑖𝑛) (resp. 𝑃(𝐿𝑜𝑠𝑒)) is the probability to win (resp. lose) at the lottery (fixed at 0.5), 𝐺 𝐿𝑜𝑡𝑡𝑒𝑟𝑦 (resp. 𝐿 𝐿𝑜𝑡𝑡𝑒𝑟𝑦 ) is the amount of money that can be won (resp. lost) at the lottery, 𝜆 is the loss aversion parameter, quantifying the relative weight of potential losses compared to potential gains in expected utility (EU) computations (as in [START_REF] Tom | The Neural Basis of Loss Aversion in Decision-Making Under Risk[END_REF]), and 𝐺 𝑆𝑎𝑓𝑒 𝑂𝑝𝑡𝑖𝑜𝑛 is the amount of reward guaranteed by the alternative, safe option.

fMRI analysis

Massively univariate approach

Preprocessed individual time series were regressed for each voxel against the following GLMs. All GLMs included three separate categorical regressors for the main events of the task: a 3.5 s boxcar regressor for the display of challenge difficulty and stakes, a variable-duration boxcar regressor from onset to validation of the confidence rating, and a stick regressor (null duration) for the onset of the squeezing movement. In the first GLM, the prospect-display regressor was parametrically modulated by the expected probability of success (as computed by our algorithm), the prospective gain level, and the prospective loss level. In the second GLM, that same regressor was instead modulated by the expected value of the challenge, that is: 𝐶𝑜𝑛𝑓𝑖𝑑𝑒𝑛𝑐𝑒 * 𝐺𝑎𝑖𝑛 + (1 -𝐶𝑜𝑛𝑓𝑖𝑑𝑒𝑛𝑐𝑒) * 𝐿𝑜𝑠𝑠, with 𝐺𝑎𝑖𝑛 and 𝐿𝑜𝑠𝑠 cardinally encoded. In all GLMs, the confidence regressor was parametrically modulated by the given confidence rating, and the motor onset regressor was modulated by the peak of the produced force curve.

All regressors were convolved with the canonical hemodynamic response function of SPM12. To correct for motion artefacts, participant-specific realignment parameters were modelled as covariates of no interest. Regression coefficients were estimated at the individual level using the restricted maximum-likelihood estimation. Linear contrasts of regression coefficients were computed at the participant level and then taken to group-level random effect analyses, using one-sample t-tests.

Statistical maps were family-wise corrected for multiple comparisons at the cluster level.

Multivariate approach

With the help of The Decoding Toolbox [START_REF] Hebart | The Decoding Toolbox (TDT): A versatile software package for multivariate analyses of functional imaging data[END_REF], we performed a region-of-interest (ROI) multivariate pattern analysis (MVPA) with a leave-one-run-out cross-validation approach, to decode high versus low confidence ratings (median-split for each participant) and high versus low stake value (gain -loss, median-split) based on individual spatially MNI-normalized but non-smoothed BOLD activity of voxels included in the ROI, using a linear support vector machine algorithm. The ROI was an 8 mm-radius sphere centered on the vmPFC peak coordinates ([0 40 -12]) from a metaanalysis of value representation [START_REF] Bartra | The valuation system: A coordinate-based meta-analysis of BOLD fMRI experiments examining neural correlates of subjective value[END_REF]. For each subject, we thus obtained a vector of weights (for each voxel in the ROI) for high versus low confidence decoding, and another vector for high versus low stake value decoding. Finally, we computed inter-voxel correlation between these two vectors, which provided us with an individual metric of pattern similarity between value and confidence encoding.

Chapter IV: Article 3

Are you positive? Misattribution of value and confidence judgments A classical formalization of decision-making that has dominated economic studies for several decades is expected utility theory [START_REF] Von Neumann | Theory of games and economic behavior[END_REF][START_REF] Friedman | The Expected-Utility Hypothesis and the Measurability of Utility[END_REF]. At the core of this theory is the principle, originally suggested by Blaise Pascal [START_REF] Pascal | [END_REF], that the optimal course of action is the one that maximizes the product of value and probability, summed over all potential outcomes. Thus, the maximization principle assumes that when considering an action, the agent must envisage all potential outcomes, and estimate for each outcome its value (how pleasurable) and its probability (how plausible, given the action). A

key assumption here is that value and probability are estimated independently from each other.

The maximization principle, along with the independence axiom, are still at the heart of modern decision theories, even those which include both stochasticity and systematic distortions in the estimation of value and probability [START_REF] Cane | Individual Choice Behavior: A Theoretical Analysis[END_REF][START_REF] Becker | Stochastic models of choice behavior[END_REF][START_REF] Kahneman | Prospect Theory: An Analysis of Decision under Risk[END_REF].

Here, we question the independence axiom in the case where the outcome depends upon the agent's behaviour. In this case, the (subjective) probability of obtaining the outcome is the (subjective) probability that the behaviour is correct, which is a standard definition of confidence [START_REF] Fleming | How to measure metacognition[END_REF]. What we question is therefore the independence of value and confidence estimates. The motivation for this question comes from recent neuroimaging studies in decision neuroscience. Meta-analyses of functional MRI studies have identified the ventromedial prefrontal cortex (vmPFC) as a central node of the so-called "brain valuation system", which signals values expressed as likeability judgments or as preferences in binary choices [START_REF] Peters | Neural representations of subjective reward value[END_REF][START_REF] Bartra | The valuation system: A coordinate-based meta-analysis of BOLD fMRI experiments examining neural correlates of subjective value[END_REF][START_REF] Clithero | Informatic parcellation of the network involved in the computation of subjective value[END_REF]. More recently, it has been demonstrated that the vmPFC also signals confidence in the behavioural response, whether this response is a choice or a judgment [START_REF] De Martino | Confidence in value-based choice[END_REF][START_REF] Lebreton | Automatic integration of confidence in the brain valuation signal[END_REF][START_REF] Morales | Domain-General and Domain-Specific Patterns of Activity Supporting Metacognition in Human Prefrontal Cortex[END_REF]. These findings raise the intriguing possibility that a brain system reading out the vmPFC output may mistake value for confidence and vice-versa.

Other functional properties of value neural signals may shape our expectations about a putative interplay between value and confidence. A first property is anticipation, meaning that baseline vmPFC activity has an impact on subsequent value estimates [START_REF] Abitbol | Neural Mechanisms Underlying Contextual Dependency of Subjective Values: Converging Evidence from Monkeys and Humans[END_REF][START_REF] Vinckier | Neuro-computational account of how mood fluctuations arise and affect decision making[END_REF]. A second property is generality, meaning that the vmPFC represents the value of items pertaining to different categories, such as money, food or music [START_REF] Chib | Evidence for a Common Representation of Decision Values for Dissimilar Goods in Human Ventromedial Prefrontal Cortex[END_REF][START_REF] Abitbol | Neural Mechanisms Underlying Contextual Dependency of Subjective Values: Converging Evidence from Monkeys and Humans[END_REF], in accordance with the notion of a common neural currency used to arbitrate between seemingly incommensurable options [START_REF] Levy | The root of all value: A neural common currency for choice[END_REF]. A third property is automaticity, meaning that the vmPFC signals the value of attended items even when they are irrelevant to the task at hand [START_REF] Lebreton | An Automatic Valuation System in the Human Brain: Evidence from Functional Neuroimaging[END_REF]Levy & Glimcher, 2011). All of these properties have been replicated in the case of value representation, but they might also apply to confidence, as suggested by recent reports [START_REF] De Martino | Confidence in value-based choice[END_REF][START_REF] Lebreton | Automatic integration of confidence in the brain valuation signal[END_REF][START_REF] Morales | Domain-General and Domain-Specific Patterns of Activity Supporting Metacognition in Human Prefrontal Cortex[END_REF].

Taken together, these properties make the prediction that a value judgment may distort a subsequent confidence judgment (or vice-versa), even when it relates to an incidental item from a different category and irrelevant to the ongoing task. In other words, we should expect not only some correlation between value and confidence estimates, but also some misattribution, the value of a particular contextual feature affecting the confidence in an unrelated behavioural response (or vice-versa).

The present study aims at exploring the possibility of an interplay between value and confidence judgments. To our knowledge, this specific type of misattribution has not been investigated so far. Nevertheless, several cases of contamination between logically unrelated judgments have been well documented in psychology studies. Many cases have been subsumed

under the term "halo effect", coined by Edward Thorndike [START_REF] Thorndike | A constant error in psychological ratings[END_REF], to describe the correlations between judgments about personality traits of the same valence. The idea is that a positive trait seems to confer the person a magical aura that calls for other positive judgments.

For instance, experiments have shown that attractiveness and trustworthiness judgments assigned to series of face pictures are typically correlated [START_REF] Langlois | Maxims or myths of beauty? A meta-analytic and theoretical review[END_REF][START_REF] Todorov | Understanding evaluation of faces on social dimensions[END_REF][START_REF] Palmer | Halo Effects and the Attractiveness Premium in Perceptions of Political Expertise[END_REF]). Yet the notion of trustworthiness, attached to the face stimulus, is different from our notion of confidence, which qualifies participants' perception of their own behaviour. Other cases of contamination, closer to an interplay between value and confidence, have been subsumed under the term "desirability bias". This term designates the propensity to inflate probability estimates of future events in proportion to their desirability [START_REF] Krizan | The influence of outcome desirability on optimism[END_REF][START_REF] Giardini | Overconfidence in Predictions as an Effect of Desirability Bias[END_REF][START_REF] Scherer | Hoping for more: The influence of outcome desirability on information seeking and predictions about relative quantities[END_REF][START_REF] Lench | Motivated perception of probabilistic information[END_REF][START_REF] Tappin | The Heart Trumps the Head: Desirability Bias in Political Belief Revision[END_REF].

When these future events are behavioural outcomes (success or failure), then the desirability bias is evidence for a contamination between value and confidence.

Yet the effects mentioned above concern value and confidence estimates assigned to a same entity: a person for the halo effect, or an event for the desirability bias. However, we intend to test whether the misattribution phenomenon generalizes to value and confidence estimates attached to different and logically-independent items. In addition, the desirability bias has been observed when manipulating the expected benefit of behavioural performance and collecting selfreports of confidence. This corresponds to an influence of expected value on confidence ratings (subjective probability of success). Nothing in our working hypothesis, based on vmPFC properties, indicates that causality should only go this way, so we also intend to test the influence of confidence on value judgment.

Another line of research explored the covariations of value and confidence at longer time scales, from days to months. At such time scales, likeability or desirability are not attached to any event in particular, but rather are assimilated to mood. According to a psychological phenomenon sometimes called the "rosy outlook", good mood can bias economic decisions (e.g., people gamble more in periods of nicer weather) [START_REF] Saunders | Stock Prices and Wall Street Weather[END_REF][START_REF] Hirshleifer | Good Day Sunshine: Stock Returns and the Weather[END_REF][START_REF] Kamstra | Winter Blues: A SAD Stock Market Cycle[END_REF][START_REF] Otto | Unexpected but Incidental Positive Outcomes Predict Real-World Gambling: Psychological Science[END_REF]. Consistently, sad mood has been associated with lower confidence, and joyful mood with overconfidence [START_REF] Briñol | Happiness versus sadness as a determinant of thought confidence in persuasion: A self-validation analysis[END_REF][START_REF] Moore | Depressive realism: A meta-analytic review[END_REF][START_REF] Koellinger | Joy Leads to Overconfidence, and a Simple Countermeasure[END_REF][START_REF] Rouault | Psychiatric Symptom Dimensions Are Associated With Dissociable Shifts in Metacognition but Not Task Performance[END_REF]. These correlations are also observed in clinical conditions, confidence being lower during depressive episodes and higher in manic episodes [START_REF] Johnson | Mania and dysregulation in goal pursuit: A review[END_REF]. Here, we intend to assess whether the putative contamination between value and confidence estimates can occur at a much shorter time scale, compatible with the formation of value and confidence representations observed with fMRI in the vmPFC. This corresponds to the duration of a typical trial in fMRI studies (a few seconds), from the onset of a stimulus with variable value to the response given with variable confidence.

We therefore opted for experimental situations that are known to vary value and confidence representations in the vmPFC on a trial-by-trial basis. We manipulated value representations by playing more or less pleasant background music, and confidence representations by presenting more or less difficult multiple-choice questions. The musical extracts and quiz questions had been employed in previous fMRI studies and shown to modulate vmPFC activity [START_REF] Abitbol | Neural Mechanisms Underlying Contextual Dependency of Subjective Values: Converging Evidence from Monkeys and Humans[END_REF][START_REF] Vinckier | Neuro-computational account of how mood fluctuations arise and affect decision making[END_REF]. To assess causal influence in both ways, we implemented two conditions: one in which we recorded value (likeability) judgments about the background music, one in which we recorded confidence judgments about the responses given to quiz questions. Thus, the value and confidence (independent) factors were varied in both cases, but the recorded (dependent) variables differed, allowing us to assess the two causal pathways.

To recapitulate, we exploited the functional properties of the vmPFC: anticipation, because value and confidence factors are expected to affect subsequent value or confidence judgments; generality, because background music and quiz questions are related to distinct domains; automaticity, because there was no request of explicit value or confidence judgment for the incidental factor. Our neuroscience-based perspective on misattribution between value and confidence makes the following predictions: 1) it should occur even when value and confidence judgments concern unrelated stimulus and response; 2) it should go both ways, from valuation to confidence and vice-versa; 3) it should be observed at a short timescale, corresponding to one task trial (a few seconds).

Results

Experiment 1

In Experiment 1, we had participants complete a control session, followed by an interference session one month later (see Fig. 1 for an overview of the experimental design). In the control session, participants completed a likeability rating task ('Like') in which they reported how much they liked various musical extracts that were played in their headset (see Methods for details about task design), and a confidence rating task ('Conf') in which they answered various general-knowledge quiz questions and reported how sure they were about their answer. This control session was meant to verify that group-averaged judgments of likeability and confidence were good predictors of individual judgments on the same items. If verified, then individual judgments on the incidental factor are unnecessary, so further experiments can skip the control session, and test the interference directly. In the interference session, participants completed two tasks which both involved answering to quiz questions while hearing musical extracts. In this session, since participants were simultaneously exposed to a context of varying pleasantness and to questions of varying difficulty, likeability and confidence judgments would be concurrently generated and thus susceptible to interfering with each other. In one of these tasks ('LIKE'), participants additionally made likeability ratings about the musical extracts they had heard, while in the other task ('CONF'), they additionally made confidence ratings about their answers to the quiz question. Experiment 1 comprised two sessions: a control session (tasks a. and b.), followed by an interference session one month later (tasks c. and d.). All subjects performed the four tasks. Two different sets of music extracts and quiz questions were used in top and bottom tasks (a.c. and b.d.).

a. 'Like' task. On each trial of this control task, subjects listened to a musical extract and reported how much they liked it. For each subject, the same set of 128 musical extracts played in this task would be played again in task c.

b. 'Conf' task. On each trial of this control task, subjects answered a quiz question by choosing one out of 4 possible answers, then reported their confidence in the correctness of their answer, and were finally given a feedback about this answer. For each subject, the same set of 128 quiz questions asked in this task would be asked again in task d.

c. 'CONF' task. On each trial of this interference task, subjects answered a quiz question while a musical extract (from task a. set) played in their headset, after which they reported their confidence in the correctness of their answer.

d. 'LIKE' task. On each trial of this interference task, subjects answered a quiz question (from task b. set) while a musical extract played in their headset, after which they reported how much they liked the musical extract and received a feedback about their answer to the quiz question.

The horizontal dashed-line arrows represent the pairing of musical extracts or quiz questions set between tasks a. and c. and between tasks b. and d.

For each task, we fitted individual ratings with a linear model, using sigmoidal transformation for a projection to the bounded scale. This model included likely modulators of the ratings, as well as potentially confounding factors, all regressors being z-scored. In the 'Conf' task .39], 𝑡(17) = 8.1, 𝑝 < 10 -6 ) with the confidence ratings averaged across all other subjects ('avgCR(t)'), which suggests that average confidence ratings were a good proxy for individual confidence judgments.

In the 'Like' task (Fig. 2b) , we found no significant effect (𝜷 ̅ =. 𝟎𝟒, 𝑪𝑰 = [-. 𝟎𝟒 ; . 𝟏𝟏],

𝒕(𝟏𝟕) = 𝟏. 𝟏, 𝒑 =. 𝟐𝟗) of the initial position of the rating cursor ('cursorInit(t)') on individual likeability ratings ('LR(t)'), and a small but significantly positive relationship (𝜷 ̅ =. 𝟎𝟕, 𝑪𝑰 = [. 𝟎𝟏 ; . 𝟏𝟐], 𝒕(𝟏𝟕) = 𝟐. 𝟓, 𝒑 =. 𝟎𝟐) with the rating made on previous trial ('LR(t-1)'). More importantly, we found a strong positive relationship (𝜷 ̅ =. 𝟔𝟔, 𝑪𝑰 = [. 𝟓𝟏 ; . 𝟖𝟐], 𝒕(𝟏𝟕) = 𝟗. 𝟏, 𝒑 < 𝟏𝟎 -𝟕 ) between individual likeability ratings and likeability ratings averaged across all other subjects ('avgLR(t)'), which demonstrates that subjects tended to have similar taste for music, and that average likeability ratings were a good proxy for individual likeability judgments. Thus, analysis of the control session indicated that some potential artifacts, such as the initial cursor position, should be regressed out. It also showed some remanence of the ratings made during previous trials, and some expected effects such as quiz performance (correctness and response time) affecting confidence. Finally, it confirmed that average ratings of a given item were good proxies for individual ratings. We next proceeded to the analysis of the interference session, using the same regression model, augmented with the factors related to the concurrent manipulation (quiz difficulty for likeability ratings, and music pleasantness for confidence ratings).

In the 'CONF' task (Fig. 3a), we observed no impact of the initial position of the cursor on the rating scale on individual confidence ratings (𝛽 ̅ = .06, 𝐶𝐼 = [-.02 ; .14], 𝑡( 17 Having found evidence for the predicted contamination between value and confidence in both directions, we tested for a possible carry-over effect across tasks. We found no influence of the order in which subjects had performed the two interference tasks on the size of the contamination in any direction: i.e., subjects who first performed the 'LIKE' task and then the 'CONF' task did not show a stronger influence of contextual likeability onto confidence ratings ((𝛽 𝐿𝐼𝐾𝐸 𝑓𝑖𝑟𝑠𝑡 ̅̅̅̅̅̅̅̅̅̅̅̅ -𝛽 𝐶𝑂𝑁𝐹 𝑓𝑖𝑟𝑠𝑡 ̅̅̅̅̅̅̅̅̅̅̅̅ ) = -0.076, 𝑡(16) = -1.7, 𝑝 = 0.94, in a one-tailed t-test), nor did subjects who performed 'CONF' before 'LIKE' show a stronger influence of confidence onto likeability judgments ((𝛽 𝐶𝑂𝑁𝐹 𝑓𝑖𝑟𝑠𝑡 ̅̅̅̅̅̅̅̅̅̅̅̅ -𝛽 𝐿𝐼𝐾𝐸 𝑓𝑖𝑟𝑠𝑡 ̅̅̅̅̅̅̅̅̅̅̅̅ ) = 0.032, 𝑡(16) = 0.81, 𝑝 =0.21, one-tailed ttest). This precludes the possibility that the misattribution effect observed at group-level would be driven by some behavioural perseveration (subjects reporting the first-rated dimension in their second task ratings). In Experiment 2, to further discard the possibility of carry-over effects across tasks, we had two separate groups of participants perform 'LIKE' and 'CONF' tasks, similar to those from the interference session in Experiment 1 (see Fig. 4 for an overview of the experimental design).

In both tasks, participants answered quiz questions of varying difficulty while hearing musical extracts of varying pleasantness, so we again expected them to generate both judgments (likeability and confidence) within the same trial, one being explicit and the other implicit. In one of these tasks ('CONF'), participants made explicit confidence ratings about the accuracy of their answers to the quiz questions, while in the other task ('LIKE'), participants made explicit likeability ratings about the musical extracts they had heard.

A further aim of this new experiment was to assess the robustness of the misattribution bias to some motivation for judgment accuracy. Therefore, a critical manipulation was that for some participants (half of each group) confidence and likeability ratings were now incentivized: confidence ratings determined whether subjects would get a financial bonus, and likeability ratings which music they would listen to at the end of the experiment (see Methods for details about the procedure). A more minor change was the suppression of feedbacks in the quiz tasks, as we realized they could only complicate the isolation of the predicted effect. Indeed, we aimed at testing the impact of an internal confidence feeling related to the current quiz question, and not the impact of external feedbacks about previous quiz questions. Experiment 2 comprised two distinct tasks each comprising two variants, performed by four separate groups of subjects.

a. 'CONF': Confidence rating task. Top: 'No incentive' variant. On each trial, subjects answered a quiz question while a musical extract played in their headset, after which they reported their confidence in the correctness of their answer. The whole session comprised 4 blocks (only one block is represented here), each comprising: 8 mini-blocks (themselves composed of 8 trials at a given music pleasantness level), and a break (light grey bar). Bottom: 'Financial incentive' variant. Trials were identical to those in the 'no incentive' variant. After all 64 trials in the block had been completed, the computer randomly drew 2 of those trials, then selected that with the highest confidence rating. If the subject had correctly answered the selected quiz question, a bonus of 2€ would be added to her cumulative reward (no gain otherwise).

b. 'LIKE': Likeability rating task. Top: 'No incentive' variant. On each trial, subjects answered a quiz question while a musical extract played in their headset, after which they reported how much they liked the musical extract. The whole session comprised 4 blocks (only one block is represented here), each comprising 8 miniblocks (themselves composed of 8 trials at a given difficulty level) followed by a break (light grey bar). Bottom: 'Hedonic incentive' variant. Trials were identical to those in the 'no incentive' variant. After all 64 trials in the block had been completed, the computer randomly drew one of these trials, and a 2€ bonus was delivered if the participant had correctly answered that quiz question. Next, the computer randomly drew 2 of those trials, selected that with the highest music likeability rating, and the subject would then have to listen to a 2-minute version of the selected musical extract.

In the 'CONF' task (Fig. 5a This set of findings suggests that the misattribution of contextual value to confidence judgments is either non-existent, or weak or elusive, whereas confidence-to-value misattribution stands across experimental replications. Furthermore, the incentivization procedure did not impact the extent of misattribution in either direction, not did it modulate the weight of predictors of non-interest in the logistic model of neither confidence nor likeability ratings (see tables S1 and S2 in supplementary material).

In all analyses of interference sessions, the level of the incidental factor (pleasantness or difficulty) in the previous trial had no significant influence on the current explicit rating (confidence or likeability). This suggests that the interplay between value and confidence occurs within a trial, as opposed to a progressive buildup of positive or negative feelings across trials. We tested more directly for a possible buildup of the misattribution effect (when statistically significant: Experiment 1, 'CONF' & 'LIKE'; Experiment 2, 'LIKE') within each block of given difficulty or pleasantness level. We performed logistic regressions against the same models as above, except that the incidental variables in the last two regressors were replaced with the following three regressors: 𝑖𝑇𝑟𝑖𝑎𝑙_𝑖𝑛_𝐵𝑙𝑜𝑐𝑘, indicating the trial index within its block, 𝐼𝑛𝑐𝑖𝑑𝑒𝑛𝑡𝑎𝑙𝐿𝑒𝑣𝑒𝑙_𝑖𝑛_𝐵𝑙𝑜𝑐𝑘, indicating the pleasantness or difficulty level of the current block (in the 'CONF' or 'LIKE' tasks respectively), and an interaction term 𝑖𝑇𝑟𝑖𝑎𝑙 * 𝐼𝑛𝑐𝑖𝑑𝑒𝑛𝑡𝑎𝑙𝐿𝑒𝑣𝑒𝑙_𝑖𝑛_𝐵𝑙𝑜𝑐𝑘. We found no evidence for a within-block buildup of the misattribution effect, as the interaction regressor was not a significant predictor of confidence or likeability 𝑡(34) = .30, 𝑝 = .76). In other words, the misattribution effects, when present, were not driven by the latest trials of particularly easy or difficult blocks for likeability ratings, and respectively pleasant or unpleasant blocks for confidence ratings. This suggests that blocking trials by difficulty or pleasantness levels brings no advantage to inducing a strong misattribution effect.

Discussion

Partly in line with a neuroscience-based perspective inspired by the encoding properties of the vmPFC, our results provide consistent evidence for a contamination from confidence to value representations, even when these representations apply to unrelated stimuli and responses, and even when they fluctuate on a short timescale. However, the opposite misattribution, that is an influence of contextual pleasantness onto confidence judgments, seemed to be less robust.

In a first experiment, subjects were instructed to answer trivia questions of varying difficulty, while listening to musical extracts of varying pleasantness. Averaged likeability ratings from other participants were used as a proxy for music pleasantness, and averaged confidence ratings as a proxy for quiz difficulty. We observed a positive influence of average music likeability on confidence judgment about the answer to the quiz, as well as a positive influence of average quiz confidence on likeability judgments about the music.

Since we define confidence as the subjective probability of being correct, it could be considered a particular case of probability judgment. The psychology literature already includes several reports about the impact of subjective value onto probability judgment, a phenomenon that has been coined desirability bias [START_REF] Kunda | The case for motivated reasoning[END_REF][START_REF] Giardini | Overconfidence in Predictions as an Effect of Desirability Bias[END_REF][START_REF] Sharot | Forming Beliefs: Why Valence Matters[END_REF][START_REF] Lebreton | Two sides of the same coin: Monetary incentives concurrently improve and bias confidence judgments[END_REF]. The opposite influence, that of probability estimate onto desirability judgment, has scarcely been explored [START_REF] Kay | Sour Grapes, Sweet Lemons, and the Anticipatory Rationalization of the Status Quo[END_REF]. Moreover, the desirability bias refers to situations where value and probability judgments relate to the same future events. In short, people tend to believe that events they desire more are more likely to happen. Here, likeability and confidence judgments were connected to logically independent events, which is the reason why we describe the correlation between the two judgments as a misattribution bias. Since the vmPFC encodes desirability but not probability estimates about future events, the desirability bias might have different underpinnings than the anatomical overlap observed between value and confidence representations [START_REF] Lebreton | Automatic integration of confidence in the brain valuation signal[END_REF]. Also, the absence of carry-over across trials distinguishes the misattribution bias from mood fluctuations, which could drive correlations between value and confidence [START_REF] Kamstra | Winter Blues: A SAD Stock Market Cycle[END_REF][START_REF] Briñol | Happiness versus sadness as a determinant of thought confidence in persuasion: A self-validation analysis[END_REF][START_REF] Moore | Depressive realism: A meta-analytic review[END_REF][START_REF] Koellinger | Joy Leads to Overconfidence, and a Simple Countermeasure[END_REF][START_REF] Otto | Unexpected but Incidental Positive Outcomes Predict Real-World Gambling: Psychological Science[END_REF]. Indeed, mood fluctuations, including those observed in fMRI studies and reflected in vmPFC activity, are best modeled as an accumulation of positive and negative events over several trials [START_REF] Rutledge | A computational and neural model of momentary subjective well-being[END_REF][START_REF] Vinckier | Neuro-computational account of how mood fluctuations arise and affect decision making[END_REF].

Another phenomenon that could explain the correlation between value and confidence ratings is the anchoring bias, which can be exerted by incidental numerical stimuli onto any kind of judgment [START_REF] Ungemach | How Incidental Values From the Environment Affect Decisions About Money, Risk, and Delay[END_REF]. The fact that the (value or confidence) judgment about the incidental factor was always kept implicit in our tasks makes such a numerical anchoring effect an unlikely explanation for the misattribution bias. Besides, the order in which subjects performed the two rating tasks had no influence on the size of the misattribution effect in any direction, ruling out the possibility that the bias arose from an internal representation of the incidental dimension in a numerical format (elicited by the first rating task).

To avoid one judgment priming the other, we had separate groups of participants performing likeability and confidence ratings in the second experiment. Additionally, we aimed to assess the robustness of the observed misattribution effects, and their suppressibility when subjects were financially or hedonically motivated to provide accurate judgments. The influence of quiz-related confidence on music likeability ratings was maintained, providing further evidence against priming or anchoring explaining the misattribution bias. However, we found no significant evidence for the reciprocal misattribution effect, from incidental likeability to explicit confidence judgments.

Although our neuroscience-based perspective does not predict, in general, such an asymmetry in value-confidence interferences, we acknowledge that one effect being stronger than the other may depend on the specifics of the experimental manipulation. In the present case, several features of our paradigm may account for the larger robustness of the confidence-to-value misattribution. First, in the context of a quiz task followed by a confidence rating task, music is instrumentally irrelevant and thus might be relatively easy to filter out. In contrast, in a task involving both answering to the quiz and rating the musical extracts, since it is likely that confidence about the quiz question is the metric used to select the answer (in the sense that people pick the answer about which they feel most confident), confidence judgments are relevant to performing the task correctly and are therefore used by the participants, rather than ignored. In other words, metacognition about task performance would be more intrusive than valuation of the context, which could be ignored. If music likeability was indeed filtered out, it must have been done in an implicit fashion, since no subject reported using such a strategy in the debriefing interview. Moreover, most subjects stated that their confidence on the quiz had not affected their appreciation of the music (and vice-versa), so they had little awareness of the misattribution bias, and hence no explicit intention to fight against it. A second, complementary explanation is that value construction may be more susceptible to incidental influences than confidence estimation.

Value is an expression of subjective preference, for which there is no obvious notion of accuracy.

Confidence can be based on more objective information, such as the time taken to select the response, or the presence of memories that can help with answering the quiz question. In a Bayesian or distributional framework, these additional cues informing the confidence judgment might lead to a prior internal estimate which is relatively more precise and thus less liable to being shifted with the incorporation of an additional (incidental) cue. Such a view would imply that in another context where external, non-affective cues for building a confidence estimate are scarce or noisy, confidence judgments would be more vulnerable to incidental affective signals.

Besides the scarcity of the cues informing the target judgment, another factor that we thought could modulate the extent of misattribution, in one direction or the other, is the motivation to form accurate judgments. However, we did not observe any impact of our incentivization procedure on misattribution effect sizes. This implies that either our manipulation of incentives was not successful in affecting participants' motivation to form accurate judgments, or that the change in motivation did not translate into an actual change in their rating behaviour (e.g., they were unable to suppress the incidental influence of confidence judgments onto value judgments).

The interplay between value and confidence judgments was detrimental in our experimental context, in the sense that judgments about the relevant dimension were blurred by the incidental dimension, which was orthogonally manipulated. However, one can envisage at least two reasons why such a contamination would have survived the pressure of natural selection. First, value and confidence might be correlated in natural situations, such that mistaking one dimension for the other would not be so devastating. The intuition here is that positive feedback would bring both valuable reward and information that the behaviour is correct.

Symmetrically, negative feedbacks would be both punishing and informative about the behaviour being incorrect. Second, value and confidence signals might have a similar influence on behavioural regulation, since alternative actions should be looked for, in case of either a poor (expected) outcome or a poor (expected) performance. This idea is evidently captured by the multiplication of probability and value estimates in the calculation of expected utility. Thus, any potential gain in adaptive fitness might not have been enough to overcome the burden of developing a brain machinery that would properly disambiguate value and confidence judgments.

An obvious limitation of the present study is the absence of neural recording. As a consequence, we cannot conclude that the vmPFC is indeed implicated in the interference between value and confidence representations observed here. However, the vmPFC remains the most likely candidate for mediating this interference, since we were careful to use stimuli, tasks and timing that yielded specific vmPFC activation with both value and confidence in previous studies [START_REF] Lebreton | Automatic integration of confidence in the brain valuation signal[END_REF][START_REF] Abitbol | Neural Mechanisms Underlying Contextual Dependency of Subjective Values: Converging Evidence from Monkeys and Humans[END_REF][START_REF] Vinckier | Neuro-computational account of how mood fluctuations arise and affect decision making[END_REF]. Moreover, the vmPFC possesses the functional properties of being both a generic and automatic valuation device [START_REF] Chib | Evidence for a Common Representation of Decision Values for Dissimilar Goods in Human Ventromedial Prefrontal Cortex[END_REF][START_REF] Lebreton | An Automatic Valuation System in the Human Brain: Evidence from Functional Neuroimaging[END_REF]Levy & Glimcher, 2011;Levy et al., 2011), which are necessary properties for an interference from incidental stimuli of a distinct category. Our finding of an asymmetry between confidence-to-value and value-to-confidence misattribution suggests, however, that value judgments about incidental features may be less automatic than confidence judgments, or at least that it is easier for an individual to ignore a feature of her environment than the correctness of what she is doing. Overall, the demonstration of a misattribution bias can be considered as complementing the preceding fMRI studies, which alone could not establish that the anatomical overlap between value and confidence representations had (detrimental) consequences on the behaviour. Indeed, given the spatial resolution of functional MRI, overlapping activations do not necessarily imply that at the neuron level, the brain uses the same code for representing value and confidence.

However, it remains possible that other brain regions contribute to the contamination, possibly components of the brain valuation system such as the ventral striatum [START_REF] Peters | Neural representations of subjective reward value[END_REF][START_REF] Bartra | The valuation system: A coordinate-based meta-analysis of BOLD fMRI experiments examining neural correlates of subjective value[END_REF][START_REF] Clithero | Informatic parcellation of the network involved in the computation of subjective value[END_REF]. Other brain regions, notably in the dorsal and lateral prefrontal cortex, have also been implicated in metacognitive accuracy and confidence judgment [START_REF] Vaccaro | Thinking about thinking: A coordinate-based meta-analysis of neuroimaging studies of metacognitive judgements[END_REF]. The recruitment of these brain regions could explain the elimination of the absence of value-to-confidence misattribution. In this scenario, the vmPFC would underpin the affective component of confidence, which is susceptible to be confounded with the affective valuation of the context. The dorsal metacognition region would be responsible for a more explicit estimation of confidence, based on available information. Such a view would accord well with the affect-as-information theory [START_REF] Clore | How emotions inform judgment and regulate thought[END_REF][START_REF] Schwarz | Feelings-as-Information Theory[END_REF], according to which a generic affect representation (probably supported by the vmPFC) could be not only an output of information about context and behaviour, but also an input to value or confidence representation, together with other, more specific input. Further studies are needed to test this global scenario and to specify the mechanisms of contamination between value and confidence representations at the neural level.

In this study, we were able to isolate for the first time a manifestation of confidence-tovalue misattribution in healthy individuals. This phenomenon may be pervasive yet elusive, as relevant judgments were contaminated by incidental, unrelated events that were implicitly processed by the brain. While manipulations of state value had previously been found to affect probability estimates, the confidence-to-value direction of contamination had never before been demonstrated, to our knowledge. While a tendency to misattribution might seem strictly prejudicial to its carriers, there is room for optimism. Indeed, we speculate that this very misattribution bias could be exploited towards the alleviation of various societal issues. For example, in educational contexts, instilling confidence through positive feedback could drive pupils to like school better and attend classes on a more regular basis.

Experiment 1, the difficulty of the questions was estimated by averaging the accuracy of participants in a separate pilot (N = 66). In Experiment 2, every participant answered the full set of 256 questions during their single testing session. The difficulty of the questions was estimated by averaging the accuracy of participants in Experiment 1.

We used 256 musical extracts of 17 seconds each, covering a large variety of styles, tonalities and instruments (from Gregorian polyphonies to hardcore grind music). 60 of those extracts had already been used in a previous fMRI study [START_REF] Abitbol | Neural Mechanisms Underlying Contextual Dependency of Subjective Values: Converging Evidence from Monkeys and Humans[END_REF]. All extracts were strictly instrumental. We equalized sound volume across all extracts (via root mean square, or RMS, normalization). In Experiment 1, 10 participants listened to the uneven items in tasks a. and c. and to the even items in task d., while 8 participants did the opposite. Before Experiment 1, the pleasantness of each musical extract was estimated by averaging the likeability ratings of participants from a previous study [START_REF] Abitbol | Neural Mechanisms Underlying Contextual Dependency of Subjective Values: Converging Evidence from Monkeys and Humans[END_REF] (N = 26), and from a pilot experiment (N = 10). In Experiment 2, every participant listened to the full set of 256 musical extracts during their single testing session. We estimated each extract's likeability by averaging the ratings made in Experiment 1.

Tasks were programmed with MATLAB (MathWorks) and the Cogent 2000 toolbox (Wellcome Department of Imaging Neuroscience, London, UK) (Experiment 1) or the Psychophysics Toolbox 3.0 [START_REF] Kleiner | What's new in psychtoolbox-3[END_REF] (Experiment 2) for stimuli presentation.

For all ratings in Experiments 1 and 2, the initial position of the scale cursor was drawn from a uniform distribution spanning the central half of the rating scale.

Experiment 1

Subjects took part in two testing sessions: first a control session, in which they made likeability and confidence judgments in separate tasks, and then an interference session about one month later, designed such that both likeability and confidence judgments were now likely to be generated within the same task and were thus susceptible to interfere together.

Both sessions comprised a likeability rating task and a confidence rating task.

The likeability rating task in the control session, hereafter referred to as 'Like' task (see Fig. 1) was coupled with the confidence rating task in the interference session, 'CONF', in the sense that the musical extracts rated on Like were subsequently played in the background during CONF. This Like--CONF task couple aimed to measure the impact of context pleasantness (measured in Like) on confidence judgments (measured in CONF). Similarly, the quiz questions about which subjects made confidence ratings in Conf (control session, confidence rating task) were subsequently asked during LIKE (interference session, likeability rating task). Thus, Conf--LIKE aimed to measure the impact of quiz difficulty on context likeability judgments.

• Like task (128 trials): On each trial, a musical extract was first played in the headset during 15 seconds while a fixation cross was displayed at the centre of the screen, then music stopped and subjects rated the likeability of the extract they had just heard ("How much do you like this extract?") by moving a cursor along a discrete scale ranging from -10 to 10 (step = 1). The order of the musical extracts was randomized across subjects.

• Conf task (128 trials): This task was divided into 4 blocks, each comprising 32 quiz questions corresponding to a given difficulty range: questions were first ordered according to their difficulty and split into 4 blocks of increasing difficulty. The order of the questions inside each block and the order of the blocks were then randomized. Each trial went as follows: 1) a fixation cross appeared for 1 s; 2) a quiz question was presented during 4 s; 3) four possible answers appeared, and subjects then had 5 s to select the answer they thought was correct by moving a selection frame (whose initial position was randomized) using 'up' and 'down' arrows on the keyboard. 4) Once subjects had confirmed their selection by pressing 'Space', they reported their confidence in the correctness of their answer ("How sure are you about your answer?") by moving a cursor on an analog scale ranging from "Not sure at all" to "Absolutely sure". 5) Once participants had validated their rating, they were given a binary feedback about the correctness of their answer. Unbeknownst to the participants, we manipulated feedbacks: errors were given a positive feedback 60% of the time in the easy session, 40% of the time in the medium-easy session, 20% of the time in the medium-difficult session, and never in the difficult session. The purpose of the manipulation was to increase the effect of question difficulty on participants' confidence judgment.

In the interference session occurring one month later, subjects again performed two tasks.

In both of them, they answered quiz questions while musical extracts were played in their headset (Fig. 1.). In one of the tasks ('CONF' task), participants additionally gave a confidence rating about the correctness of their answer to each quiz question, whereas in the other task ('LIKE' task), participants gave a likeability rating about each musical extract.

• CONF task: This task included 128 trials, divided into 4 blocks of 32 trials of similar music pleasantness. The order of the extracts within each block and the order of the blocks were randomized. Each block was divided into 4 sub-blocks of 8 trials of similar question difficulty. Each trial went as follows: 1) a musical extract started playing, while a fixation cross was displayed at the centre of the screen for a duration varying from 1 to 10 seconds, adjusted so that trials would last 17 s on average (average fixation cross duration = 5.9 s, average s.d. of duration = 1.4 s); 2) a quiz question was presented during 4 s; 3) four possible answers appeared, and subjects then had 5 s to select the answer they thought was correct by moving a selection frame (whose initial position was randomized) using 'up' and 'down' arrows on the keyboard. 4) Once subjects had confirmed their selection by pressing 'Space', music stopped, and they reported their confidence ("How sure are you about your answer?") by moving a cursor on an analog scale ranging from "Not sure at all" to "Absolutely sure".

• LIKE task: This task included 128 trials, divided into 4 blocks of 32 questions of similar difficulty (see 'Conf' task for details). The order of the questions within each block and the order of the blocks were randomized. Each block was divided into 4 sub-blocks of 8 trials of similar music pleasantness. Each trial went as follows: 1) a musical extract started, and a fixation cross was displayed at the centre of the screen for a duration varying from 2 to 10 seconds, adjusted so that trials would last 15 s on average (average fixation cross duration = 6.6 s, average s.d. of duration = 1.6 s); 2) a quiz question was presented during 4 s; 3) four possible answers appeared, and subjects then had 5 s to select the answer they thought was correct by moving a selection frame (whose initial position was randomized) using 'up' and 'down' arrows on the keyboard. 4) Once subjects had confirmed their selection by pressing 'Space', music stopped, and they rated the likeability of the musical extract they had just heard ("How much do you like this extract?") by moving a cursor on a discrete scale ranging from -10 to 10 (step = 1). 5) Once participants had confirmed their rating, they were given a feedback about the correctness of their answer (see 'Conf' task for details).

Experiment 2

In order to rule out one potential confound present in Experiment 1, that is the fact that the same subjects performed both the likeability rating and the confidence rating tasks on the same day, which could have led to confusion as to which dimension they should pay attention to and report in their ratings, we had separate groups of subjects perform each of the rating tasks in Experiment 2. Therefore, each subject took part in a single testing session, in which they answered quiz questions while musical extracts were played in their headsets. Additionally, they either performed confidence ratings (Group 1), or music likeability ratings (Group 2). The other critical difference in task design with Experiment 1 was that for a subset of each group of participants (N = 17/35), answers to the quiz questions, as well as the confidence or likeability ratings, were incentivized in order to assess the robustness of misattribution biases to financial or hedonic motivation.

• CONF task: This task included 256 trials, divided into 8 blocks of 32 trials of similar music pleasantness. Each trial went as follows: 1) a musical extract started, and a fixation cross appeared at the centre of the screen for a fixed duration of 1 s; 2) a quiz question was presented during 4 s; 3) four possible answers appeared on diagonal positions, which were mapped with horizontally adjacent 'A', 'B', 'C', 'D' response buttons. Subjects had 7 s to answer by pressing one of these buttons, upon which a selection frame appeared to highlight the selected answer. The four possible answers and the selection frame remained on-screen until maximal response time (7 s) was elapsed. This was to equalize the time of exposure to the musical extracts across trials, and to ensure its independence from quiz response time. 4) Music stopped, and subjects reported their confidence in their answer ("How sure are you about your answer?") by moving a cursor along an analog scale ranging from 'Not at all' to 'Absolutely sure'.

• LIKE task: This task included 256 trials, divided into 8 blocks of 32 trials of similar quiz question difficulty. Each trial went as follows: 1) a musical extract started playing, and a fixation cross appeared at the centre of the screen for a fixed duration of 1s; 2) a quiz question was presented during 4 s; 3) four possible answers appeared on diagonal positions, which were mapped with 'A', 'B', 'C', 'D' response buttons. Subjects had 7 s to answer by pressing one of these buttons, upon which a selection frame appeared to highlight the selected answer. The four possible answers and the selection frame remained on-screen until maximal response time (7 s) was elapsed. 5) Music stopped, and subjects rated the likeability of the musical extract they had just heard ("How much did you like this extract?") by moving a cursor along a discrete scale ranging from -10 to 10 (step = 1).

Incentivization

In Experiment 2, for a subset of each group of participants (N = 17/35), both the accuracy in the general-knowledge quiz task and the confidence/likeability ratings were financially or hedonically incentivized.

Specifically, 1) in the confidence rating task: at the end of each block of 64 trials, 2 of these trials were randomly drawn, and among these two, that with the highest confidence rating was selected and executed. That is, if the participant had chosen the correct answer on this trial, he would receive a 2€ bonus. Otherwise, he would not receive any bonus on this block; 2) in the likeability rating task: at the end of each block of 64 trials, one of these trials was randomly drawn and executed. If the participant had chosen the correct answer on this trial, he would receive a 2€ bonus. (Otherwise, he would not receive any bonus on this block.) Then, 2 of the 64 musical extracts played during the block were randomly drawn, and among these two, that with the highest likeability rating was selected. The participant would subsequently have to listen to a 2-minute extract of this musical piece.

In the instructions preceding both the confidence and the likeability rating tasks, an example random draw was simulated and commented, and it was explicitly stated that due to this selection procedure, it was in the interest of the participants to give ratings which reflected as closely as possible their subjective probability of being correct (in the confidence rating task) or the pleasure derived from listening to the musical extract (in the likeability rating task).

Data analysis

Statistical analyses were performed with MATLAB, and all models were inverted using a variational Bayes approach under the Laplace approximation, implemented in the Variational Bayesian Analysis (VBA) toolbox [START_REF] Daunizeau | VBA: A Probabilistic Treatment of Nonlinear Models for Neurobiological and Behavioural Data[END_REF]. The algorithm used in VBA not only inverts nonlinear models but also estimates their evidence, which represents a trade-off between accuracy (goodness of fit) and complexity (degrees of freedom). The log-evidences, estimated for each participant and model, were submitted to a group-level random-effect analysis [START_REF] Rigoux | Bayesian model selection for group studies-Revisited[END_REF]. This analysis was used to generate exceedance probability, which measures the likelihood that a given model is more frequently implemented by participants than any other model in the comparison set. We performed logistic rather than linear regressions of confidence and likeability ratings because of the boundedness of the rating scale, which makes sigmoidal transformations naturally more suitable to modeling subjects' ratings, while introducing no additional complexity. A random-effects model comparison between this logistic model and its linear equivalent confirmed that the logistic variant was much more likely, as it had an exceedance probability XP > 0.999 for all model comparisons (task-wise), except for the likeability rating task in Experiment 2, where the superiority of the logistic model was less clear-cut (𝑋𝑃 𝑙𝑜𝑔𝑖𝑠𝑡𝑖𝑐 = 0.65; 𝑋𝑃 𝑙𝑖𝑛𝑒𝑎𝑟 = 0.35). It must be noted however that all significant effects of interest remain significant (and all non-significant effects remain non-significant) when linear regressions are performed instead of logistic ones.

All the variables included in our models were z-scored across the whole task, so as to enable direct comparison of the amount of explained variance between regressors. All prior probability density functions over beta parameters were 'agnostic' gaussians centred on 0, with a variance of ln²(3), set such that parameter values within one standard deviation from the centre of the distribution would entail a [-.25;

.25] change in rating (expressed between 0 and 1) around the mean (0.5) when the associated regressor would vary within one standard deviation from its mean, which corresponds to a reasonably large effect. We subsequently tested for group-level random effects of the parameters in our models by performing one-sample two-tailed Student ttests on each vector of beta posterior estimates. with 𝐶𝑅 𝑡 the subject's confidence rating on current trial (t), 𝐼𝑛𝑖𝑡𝐶𝑢𝑟𝑠𝑜𝑟 𝑡 the initial position of the cursor on the rating scale, 𝐶𝑅 𝑡-1 the confidence rating made on previous trial, 𝑆𝑢𝑐𝑐𝑒𝑠𝑠 𝑡 a binary variable indicating whether the subject's answer on trial t was correct or not, 𝑅𝑇 𝑡 the time elapsed between the onset of the four possible answers and the selection of an answer by the subject, 𝐴𝑣𝑔𝐶𝑅 𝑡 the confidence rating made about the current quiz question, averaged across all other subjects.

c) CONF task

For each subject, we inverted a logistic model of individual confidence ratings: with 𝐶𝑅 𝑡 the subject's confidence rating on current trial (t), 𝐼𝑛𝑖𝑡𝐶𝑢𝑟𝑠𝑜𝑟 𝑡 the initial position of the cursor on the rating scale, 𝐶𝑅 𝑡-1 the confidence rating made on previous trial, 𝑆𝑢𝑐𝑐𝑒𝑠𝑠 𝑡 a binary variable indicating whether the subject's answer on trial t was correct or not, 𝑅𝑇 𝑡 the time elapsed between the onset of the four possible answers and the selection of an answer by the subject, 𝐴𝑣𝑔𝐶𝑅 𝑡 the confidence rating made about the same quiz question, averaged across all other subjects, 𝐴𝑣𝑔𝐿𝑅 𝑡-1 the likeability rating made about the musical extract played on previous trial, averaged across all other subjects, and 𝐴𝑣𝑔𝐿𝑅 𝑡 the likeability rating made about the musical extract played on current trial, averaged across all other subjects.

Only the last two regressors in this model directly address our effect of interest, while the other regressors represent factors of non-interest which were nonetheless likely to influence ratings, and which may incidentally have covaried with average likeability fluctuations, such that we removed their potential influence from the estimation of 𝛽 6 and 𝛽 7 . Specifically, 𝛽 1 • 𝐼𝑛𝑖𝑡𝐶𝑢𝑟𝑠𝑜𝑟 accounts for a possible anchoring effect of the initial position of the rating cursor, 𝛽 2 • 𝐶𝑅 𝑡-1 accounts for the likely temporal auto-correlation in the ratings, due to the blocking of quiz questions by difficulty level, as well as to some perseveration bias from the subjects.

d) LIKE task

For each subject, we inverted a logistic model of individual likeability ratings: with 𝐿𝑅 𝑡 the subject's likeability rating on current trial (t), 𝐼𝑛𝑖𝑡𝐶𝑢𝑟𝑠𝑜𝑟 𝑡 the initial position of the cursor on the rating scale, 𝐿𝑅 𝑡-1 the likeability rating made on previous trial, 𝐹𝑒𝑒𝑑𝑏𝑎𝑐𝑘 𝑡-1 a binary variable indicating whether the subject received a 'correct' feedback on previous trial, 𝐴𝑣𝑔𝐿𝑅 𝑡 the likeability rating made about the same musical extract, averaged across all other subjects, 𝐴𝑣𝑔𝐶𝑅 𝑡-1 the confidence rating made about the quiz question answered on previous trial, averaged across all other subjects, and 𝐴𝑣𝑔𝐶𝑅 𝑡 the confidence rating made about the quiz question answered on current trial, averaged across all other subjects.

As in the CONF logistic model, only the last two regressors in this model directly address our effect of interest, while the other regressors represent factors of non-interest which were nonetheless likely to influence ratings, and which may incidentally have covaried with average confidence fluctuations, such that we removed their potential influence from the estimation of 𝛽 4 and 𝛽 5 . Specifically, 𝛽 1 • 𝐼𝑛𝑖𝑡𝐶𝑢𝑟𝑠𝑜𝑟 accounts for a possible anchoring effect of the initial position of the rating cursor, 𝛽 2 • 𝐿𝑅 𝑡-1 accounts for the likely temporal auto-correlation in the ratings, due to the blocking of musical extracts by pleasantness level, as well as to some perseveration bias from the subjects, and 𝛽 3 • 𝐹𝑒𝑒𝑑𝑏𝑎𝑐𝑘 𝑡-1 accounts for possible effects of feedback on mood and thus on likeability ratings. While the feedback received on previous trial is also likely to have modulated subjects' feeling of confidence, our last two regressors are a proxy for confidence fluctuations over and above the impact of feedbacks. The only difference is that, here, 𝐴𝑣𝑔𝐿𝑅 𝑡 was obtained by averaging the likeability ratings made about the current musical extract by the other group of subjects.

b) Likeability rating task

As for LIKE task in Experiment 1, we inverted a logistic model of likeability ratings:

𝐿𝑅 𝑡 = 𝑠𝑖𝑔𝑚(𝛽 0 + 𝛽 1 • 𝐼𝑛𝑖𝑡𝐶𝑢𝑟𝑠𝑜𝑟 𝑡 + 𝛽 2 • 𝐿𝑅 𝑡-1 + 𝛽 3 • 𝐴𝑣𝑔𝐿𝑅 𝑡 + 𝛽 4 • 𝐴𝑣𝑔𝐶𝑅 𝑡-1 + 𝛽 5 • 𝐴𝑣𝑔𝐶𝑅 𝑡 )

The only difference here is that, since no feedback was given to the subjects about the correctness of their answer, we did not include any feedback regressor in our model.

1) Overview

In our modern understanding, three major types of judgment are at play during action valuation: the subjective value of each action's possible outcomes, the perceived probability of these various outcomes, and the costs (e.g., the energetic expense) entailed by each action. In this PhD work, we set out to delineate some circumstances under which interferences can occur between these decisional variables, and we tried to elucidate their underlying computational and neural mechanisms.

In the first of the three studies presented above, we focused on the interaction between the value of a pursued goal and the anticipated effort cost of the sequence of actions needed to reach that goal. We had participants prospectively estimate the energetic cost of composite running routes, whose completion could entail a monetary gain (or prevent a monetary loss) of variable magnitude. We found that routes associated to larger stakes (gains to be obtained or losses to be avoided) were anticipated as more costly by our participants, although the real energetic cost and monetary stakes were orthogonal by design.

We replicated this finding in a follow-up experiment, in which we additionally investigated its computational underpinnings. Of all the cognitive scenarios we had considered to account for the positive distortion of anticipated energetic cost by goal value, the one whose predictions best aligned with our set of behavioural findings involves a contamination of the prospective cost judgment by the output of a cost-benefit computation. This computation could either be that of the optimal energetic expense to invest in order to maximize a cost-benefit trade-off under the assumption that effort investment continuously improves the chances of reaching the high-value outcome (either a gain or no loss, depending on the framing), or that of the maximal energetic expense above which it is preferable to opt out of the highvalue high-effort course of action. Our model comparison did not reveal any difference in frequency between these two scenarios within our sample of subjects, either because they are indeed similarly prevalent in the general population, or because we lacked statistical power to detect a difference in prevalence. Besides, we did not find consistent evidence for other putative interference mechanisms: that of an affect-as-information misattribution, and that of an incorporation of a belief in a general correlation between goal value and action cost; their predictions were not supported by our results.

In the second study, we shifted our focus to the influence of performancecontingent outcome value on perceived probability of success, that is, confidence judgments. For this, we first trained participants to perform a specific motor task, which consisted in briefly squeezing a handgrip at a given force level, with more or less error margin, without any visual feedback on the exerted force. Then, on each trial, three pieces of information were disclosed about the upcoming motor challenge: its difficulty (adjusted via the size of the force target to hit), the amount of money that could be won in case of success, and the (generally different) amount of money that could be lost in case of failure. In the following stage of each trial, participants were prompted to make an a priori estimate of their chances of success at the motor challenge, that is, to give a prospective confidence rating. Finally, they squeezed the handgrip and tried to reach the target without receiving any feedback about their exerted force, either during or after their motor execution. Our main finding is that subjects tend to be more confident in their upcoming performance when success is associated with a large monetary gain, and less confident when failure is associated with a large loss. Interestingly, this pattern of confidence distortions is in contradiction with the predictions of several popular psychological constructs, such as the desirability bias, and instead points to a misattribution-ofaffect kind of mechanism. We sought to clarify the neural underpinnings of this apparent affect-to-confidence spillover via functional neuroimaging. Based on earlier reports that the vmPFC encodes both value and confidence judgments in an automatic and generic way (with some carry-over effects also mentioned between distinct value judgments), our hypothesis was that the vmPFC might be the main site in which value-confidence misattributions occur. However, we failed to identify any region whose activity covaried with our variables of interest (the value of prospective outcomes and confidence judgments), or to observe an inter-subject correlation between the multi-voxel patterns in vmPFC encoding prospective value and prospective confidence, respectively.

In our third study, we tested the hypothesis -derived from the aforementioned findings about the vmPFC encoding properties -that valueconfidence interferences might occur even between unrelated objects or stimuli, and bidirectionally, that is, from confidence to value judgments as well as from value to confidence. We ran two behavioural experiments in which subjects answered more or less difficult general-knowledge quiz questions while hearing more or less pleasant musical extracts. After they had answered each quiz question, in a subset of trials or participants, they were asked to report their confidence in having selected the correct quiz answer; in the complementary subset, they were asked to report how much they liked the musical extract they had just heard. We found a consistent positive impact of quiz-related confidence (proxied by confidence ratings averaged across participants about the presented quiz question) on music pleasantness judgments, even when participants were hedonically incentivized to provide accurate ratings; however, the opposite effect was more elusive. Thus, our neurally-derived predictions were only partly verified. At a more cognitive level, our findings suggest that the relative salience, precision, and taskrelevance of value and confidence judgments might determine the direction and strength of the misattribution.

2) What have we learned?

First, we add to the existing literature about interferences between decisional variables, by isolating such effects in well-controlled laboratory settings, which notably include loss frames that had scarcely been investigated before. Our work thus casts additional doubt on the validity, or viability, of the classical assumption of an independence between the various judgments implicated in action valuation.

In all three studies, we isolated interferences between judgments of value on the one hand and judgments of either confidence or effort cost on the other hand. Specifically, we extended previous knowledge by showing that 1) larger prospective gains entail higher confidence and larger anticipated effort cost, while larger prospective losses entail lower confidence as well as larger anticipated effort cost, 2) interferences between value and confidence judgments can be observed in both directions 3) they can be observed even when these judgments bear on unrelated objects, such that in some circumstances, incidental confidence can modulate target value judgments, 4) different instances of interference are likely underpinned by different cognitive mechanisms: value and confidence judgments tend to covary positively, and their mutual influence is evocative of a misattribution of affective states, while value and effort cost judgments display a more complex relationship, which is best explained by another cognitive mechanism involving cost-benefit computations by the agent.

Besides, it can be noted that our behavioural data about the force exerted by participants in our second study contributed to the identification of a "reward effect" on action vigour, which was primarily investigated by another researcher in our team (Oudiette et al., 2019, see complete publication in Appendix). Across several experiments conducted by D. Oudiette and others (including E. Bioud), it was found that healthy individuals tend to produce more force in actions incentivized with larger rewards, even when this increase in vigour is actually detrimental to the obtention of the reward. Posterior force reports by the participants reflected that reward effect, suggesting that they are aware of their bias but unable to suppress it. This automatic invigoration of response by potential reward (which was observed to a much lesser extent for potential loss) might result from a Pavlovian stimulus-response shortcut, which seems to occur independently from the specific task context (and thus can deteriorate performance in precision tasks, as opposed to force tasks such as weightlifting). Interestingly, this pattern of response to potential gain versus loss departs from the pattern of distortion of anticipated energetic cost by gain versus loss in our first study, which in both cases inflated to a similar extent with stake magnitude. This dissociation could be due to the different nature of the running task versus the precision grip or putting tasks in which the "reward effect" was observed: in the former, all the cost laid in the energetic expense, while the latter also involved a motor control cost. Besides, explicit effort anticipation and actual effort production are probably underpinned by non-identical systems, the second one being potentially more sensitive to Pavlovian influences (which usually impact action more than cognition).

3) Limitations & Perspectives

We will now review the main limitations of this work, a number of which actually open perspectives for future research.

First, it can be pointed out that the interference effects isolated here are quite small, in the sense that the independent variable of interest does not account for a large fraction of the variance of the dependent variable. However, this does not necessarily mean that these interferences are always minor in real-life situations, and that they could be considered as anecdotal overall. One explanation for the relatively modest effect size we observed could be that the fluctuations of the independent variable (either a value judgment in studies 1 to 3, or a confidence judgment in study 3) were themselves relatively small compared to that of other factors informing the dependent judgment (such as the size of the target, or the genre of the music). We speculate that massively increasing the range of prospective monetary gains or losses, to an extent that may be difficult to implement in laboratory settings, would allow to observe more substantial distortions of concurrent decisional variables. Along the same line, and as evoked in the Discussion sections of the above articles, if we adopt a Bayesian view of cue integration in inferential judgments [START_REF] Pouget | Confidence and certainty: Distinct probabilistic quantities for different goals[END_REF], then it seems likely that, had non-interferential evidence for building the dependent judgment been scarcer or noisier than it actually was in our experimental paradigms, this judgment would have been more liable to interferential influences from concurrent value or confidence judgments. More generally, it would be interesting to examine how interferences between two decisional variables vary in direction and strength with the relative salience, the relative relevance, and the relative uncertainty of these two judgments. Finally, another reason for our small effect sizes could be that our data were acquired in the lab rather than in the field, such that subjects might have been highly motivated to provide unbiased judgments, even in the absence of any incentivization procedure. This latter scenario implies that individuals might be able to suppress or at least moderate these interferences when motivated to do so.

Although we did not find evidence for such a moderation upon incentivization in our third study, it might simply be that our hedonic incentivization was not effective enough in increasing their motivation for rating accuracy. This point would be worth investigating via more extensive, or better-established [START_REF] Hollard | In search of good probability assessors: An experimental comparison of elicitation rules for confidence judgments[END_REF], incentivization procedures.

Another blind spot in our characterization of interferences between decisional variables is that, beyond outcome or state value, confidence, and effort cost judgments, we ignored other types of decisional variables such as delay, pain, or other types of cost or probability judgments. This is mainly due to a lack of time, combined with the fact that we prioritized decision variables that could be easily manipulated or measured, as well as couples of variables for which interactions had been previously reported in the literature, for we eventually aimed to investigate their underlying mechanisms; however we believe that such variables also deserve investigation in the future.

Perhaps the major limitation of this work is that we failed to elucidate the neural mechanisms responsible for these interferences. In our second study, we could not identify any 'non-trivial' region or set of regions (that is, not the primary visual areas, whose apparent implication in encoding confidence or value was very likely due to luminance confounds) whose BOLD signal would covary with any of our two variables of interest, namely, prospective value and confidence. This absence of finding, especially in the vmPFC, contrasts with the hundreds of studies that have found this region to be more active when subjects were considering valuable stimuli or prospects, or were highly confident in their behaviour. From a pragmatic point of view, our null result might be due to a combination of insufficient statistical power (in particular, regarding group sample size), insufficient intertrial interval (and thus, given slow hemodynamics, too much overlap of value and/or confidence signals from one trial to the next), and excessive information load during prospects presentation (the first stage of each trial, during which we displayed the size of the motor target, as well as potential gains and losses simultaneously). From a more theoretical standpoint, over and above the fact that we were unable to isolate clusters of above-threshold correlation with value or confidence, we acknowledge that the regional overlap of value-and confidenceencoding voxels in the vmPFC does not imply that these variables partly share the same neural code in the vmPFC. A network overlap is not even a necessary condition to observe value-confidence misattribution; it would be enough, e.g., that one downstream output of the network in charge of building a confidence judgment be fed in to a node of the network in charge of building a value judgment; in fact, this interference could occur in such a distributed way, and/or at such a microscopic scale, that fMRI might not be a suitable imaging technique to detect it.

Another important limitation of this work is that we did not assess the impact of judgment interferences on actual action selection. Beyond the fact that our small effect sizes would lead us to expect a rather minor impact on decision-making, this assessment is made difficult by the fact that decisional variables have (by definition) a main impact on action valuation, on top of which they might have a second congruent or incongruent impact through their distortion of another decisional variable. This difficulty may be overcome, for example, by finding a way to modulate the extent of judgment interferences (e.g. via incentivization, or by manipulating the scarcity of non-interferential cues informing the dependent judgment), and measuring the ensuing modulation in action selection. Until this investigation is conducted, we can speculate that: 1) the inflation of anticipated effort cost by goal value (whether as an obtained gain or an avoided loss) may, all other things equal, deter engagement in the considered course of action, but in return, if the course of action is still taken on, it may incite more resource accumulation prior to action engagement (and thus may compensate the planning fallacy that healthy individuals typically exhibit, and eventually entail higher chances of reaching the goal); 2) the positive influence of prospective outcome value on confidence may accentuate agents' engagement in courses of actions attached to an overall positive balance of outcomes, as well as their avoidance of courses of action with an overall negative balance of outcomes. One can speculate that several judgment distortions may occur during a given action valuation; for instance, prospective outcome value might both distort confidence and effort cost judgments. In such a case, we would expect that highly valuable prospective outcomes, by boosting confidence but concurrently increasing the anticipated energetic cost of the sequence of action, may not, overall, have a strong effect on action selection per se, but may improve resource storage prior to action if it is indeed taken on (although it may also, due to an inflation of performance confidence, deteriorate other forms of action preparation).

An additional avenue for future research regards the impact of these judgment interferences across time: is a distortion of experienced pleasantness, or posterior confidence judgment, just local or could it be fed to a learning process, or transferred to episodic memory storage, such that these distortions might actually accumulate over time, and/or bias learning processes? At a larger scale, following on the findings that overoptimism bias brings about higher well-being and better health status [START_REF] Taylor | Psychological resources, positive illusions, and health[END_REF], and that overconfidence is linked to higher risk-taking and perseverance in professional ventures [START_REF] Koellinger | I think I can, I think I can": Overconfidence and entrepreneurial behavior[END_REF], it would be interesting to systematically assess the consequences of these interferences at the level of individual traits, e.g. by looking for correlations between statistical parameters quantifying the extent of interference between two decisional variables (ideally, across different contexts) and various psychometric scores.

Fig. 1 . 1 :

 11 Fig. 1.1: Illustrative curve depicting the exchange-value of water as a function of the volume already possessed. From (Caplin & Glimcher, 2014).

Fig. 1 . 2 :

 12 Fig. 1.2: Daniel Bernoulli's proposal of a decreasing marginal utility as a function of total wealth. This function is a natural logarithmic one:

Fig. 1 . 4 :

 14 Fig. 1.4: The two functions describing generic human decisionmaking, according to Prospect Theory. Left: Utility (or "value") U(x) as a function of outcome x. Right: Decision weight 𝜋(𝑝) as a function of the stated probability 𝑝 of an outcome.

Fig. 1

 1 Fig. 1.5: Example Markov Decision Process Here, at each time-step, the decision-maker can be in one of two states, 𝑺 𝟏 or 𝑺 𝟐 . In each state, she can perform an action 𝒂 𝒊 , which takes her to state 𝑺 𝟏 or 𝑺 𝟐 with a given probability.

Fig. 1 . 7 :

 17 Fig. 1.7: Force production task. (A) Final reward is proportional to the maximal exerted force. (B) Choice between two fixed reward-effort bundles.

Fig. 1

 1 Fig. 1.8: Hyperbolic model of effort discounting. Subjective value as a function of effort cost level. Each curve represents a different value of parameter k. Adapted from (Klein-Flügge et al., 2015)

Fig. 1 . 9 :

 19 Fig. 1.9: Sigmoidal model of effort discounting. Left: Subjective value as a function of effort cost level. Each curve represents a different set of parameters (k,p). Right, individual fits (and average in yellow) of experimental data. Adapted from (Klein-Flügge et al., 2015)

Fig. 1 .

 1 Fig. 1.10: A modern multi-step framework of value-based action selection.

Fig. 1 .

 1 Fig. 1.11: Example of rating paradigm. In this experiment, participants

Fig. 1 .

 1 Fig. 1.12: A Probability Matching procedure for confidence ratings.

Fig. 1 .

 1 Fig. 1.13: The two main components of the Brain Valuation System:

Fig. 1 .

 1 Fig. 1.14: Extract of task design and BVS BOLD Preferred versus Nonpreferred response, in (Lebreton et al., 2009).

Fig. 1 .

 1 Fig. 1.15: Testing paradigm, theoretical predictions and results of a study investigating baseline dependency effects in the vmPFC. (Abitbol et al., 2015)

Fig. 1 .

 1 Fig. 1.16: mOFC response to identical but differently priced wines.

Fig. 1 .

 1 Fig. 1.18: Spatial dissociations of reward magnitude and probability in the striatum. (A) Adapted from[START_REF] Berns | Striatal topography of probability and magnitude information for decisions under uncertainty[END_REF]. (B) Adapted from[START_REF] Yacubian | Subregions of the ventral striatum show preferential coding of reward magnitude and probability[END_REF]

Fig. 1 .

 1 Fig. 1.20: Neural correlates of probability weights in classical andmotor lotteries. From[START_REF] Wu | The Neural Correlates of Subjective Utility of Monetary Outcome and Probability Weight in Economic and in Motor Decision under Risk[END_REF]

Fig. 1 .

 1 Fig. 1.22: Schematic of possible mood dysfunctions. Euphoria or mania vs. dysphoria or depression, from (Eldar et al., 2016). (A) Mood fluctuations over time, depending on relative learning rates from positive versus negative prediction errors and outcome history. (B) Positive feedback loops reinforcing high versus low mood, and possible transition between them.

Fig. 1 .

 1 Fig. 1.23: Effort exertion and estimation task, from Pooresmaeili et al., 2015. After each effort exertion episode (pushing a virtual ball up a ramp by

Fig. 1 .

 1 Fig. 1.24: Schematic consequences of effort-enhanced value in a costbenefit trade-off. From (Inzlicht et al., 2018) Left: Scenario A, effort enhances product value. Right: Scenario B, effort as an intrinsic reward.Irrespective of the specific scenario, the value-enhancing property of effort predicts an increased effort exertion.

Fig. 1 -

 1 Fig. 1 -Experiment 1 design

  bottom, 𝛽 ̅ = 4.1, 𝐶𝐼 = [3.1 ; 5.1], 𝑡(17) = 8.8, 𝑝 = 9.4 • 10 -8 ), on the probability to accept the HEHG offer.

  orthogonalized to a binary 𝐶ℎ𝑜𝑖𝑐𝑒 predictor(𝛽 ̅ = 2.4, 𝐶𝐼 = [1.3; 3.5], 𝑡(21) = 4.7, 𝑝 = 1.3 • 10 -4 ). Moreover, alternative regressors 𝐼𝑛𝑐𝑒𝑛𝑡𝑖𝑣𝑒 𝐻𝐸𝐻𝐺 and 𝐼𝑛𝑐𝑒𝑛𝑡𝑖𝑣𝑒 𝐿𝐸0𝐺 conditional on the choice made (respectively the HEHG option or the LE0G one) were both significant positive predictors of AEC ratings (𝐻𝐸𝐻𝐺: 𝛽 ̅ = 2.4, 𝐶𝐼 = [1.4; 3.4], 𝑡(21) = 5.0, 𝑝 = 6.9 • 10 -5 ; 𝐿𝐸0𝐺: 𝛽 ̅ = 2.0 𝐶𝐼 = [. 70; 3.4], 𝑡(21) = 3.2, 𝑝 = 5.0 • 10 -3 ), meaning that the effect of Incentive on AEC was present within both 'Yes' and 'No' trials. Therefore, prospective reward inflated anticipated energetic cost over and above choices made by the participant.

  corresponding interaction terms (orthogonalized to the main factors), we found no evidence for a 𝐼𝑛𝑐𝑒𝑛𝑡𝑖𝑣𝑒 * 𝑖𝐵𝑙𝑜𝑐𝑘 interaction (𝛽 ̅ = -.47, 𝐶𝐼 = [-2.5; 1.5], 𝑡(21) = -.49, 𝑝 = .63), nor did we find evidence for a main effect of block index (𝛽 ̅ = .71, 𝐶𝐼 = [-.54; 2.0], 𝑡(21) = 1.18, 𝑝 = .25), or other kinds of interaction effects (𝑅𝐸𝐶 * 𝐼𝑛𝑐𝑒𝑛𝑡𝑖𝑣𝑒: 𝛽 ̅ = 2.6, 𝐶𝐼 = [-3.5; 8.8], 𝑡(21) = .88, 𝑝 = .39; 𝑅𝐸𝐶 * 𝑖𝐵𝑙𝑜𝑐𝑘: 𝛽 ̅ = -.039, 𝐶𝐼 = [-6.6; 6.6], 𝑡(21) = -.012, 𝑝 = .99). Additionally, the effect of incentive on AEC ratings was significant within each block of the experimental session (e.g. block 6: 𝛽 ̅ = 1.4, 𝐶𝐼 = [. 57; 2.3], 𝑡(21) = 3.5, 𝑝 = .0023).

Fig. 2 -

 2 Fig. 2 -Choices and AEC ratings in Experiment 1 a. Choices: Rates of acceptance of the High-Effort High-Gain offer, plotted against incentive levels. Top: Cross markers indicate Acceptance rate averaged across all subjects for a given incentive level (x-axis) and a given Real Energetic Cost (REC) level (marker colour). Bottom: Purple cross markers indicate Acceptance rate, averaged across all REC levels and across all subjects for a given incentive level. Black dots indicate averaged predictions of a logistic regression model (see main text).

Finally

  , under the motivation scenario, and given the previously introduced experimental changes, we expected a distinct set of observations: a positive effect of incentive on anticipated energetic cost in both Gain and Loss conditions (and despite the introduction of a lottery animation), no inter-individual correlation between the magnitude of this effect and the strength of the belief in a Reward -Effort correlation, and finally, no effect of our music-based arousal manipulation on anticipated energetic cost. To further arbitrate between the 'Maximal Budget' and 'Optimal Expense' variants, we planned to perform group-level Bayesian model selection based on choice and rating data.

Fig. 4 -

 4 Fig. 4 -Experiment 2 designTop part, top row: Example trial in the Gain x With_Music condition. Each trial includes 4 phases: 1) lottery animation, during which a running route and an incentive level are independently and (seemingly) randomly selected, 2) display of the running route animation and the associated incentive (reward here) while a musical extract plays in subject's headset, 3) choice between this High-Effort High-Gain (or 0-Loss) offer and an implicit Low-Effort 0-Gain (or High-Loss) option (not displayed), 4) rating of Anticipated Energetic Cost (AEC).

  7, 𝐶𝐼 = [2.8 ; 4.6], 𝑡(13) = 8.5, 𝑝 = 1.1 • 10 -6 ) and 𝐿𝑜𝑠𝑠 𝐿𝐸𝐻𝐿 (𝛽 ̅ = 3.7, 𝐶𝐼 = [2.7 ; 4.8], 𝑡(13) = 7.8, 𝑝 = 2.9 • 10 -6 ), and a negative effect of 𝑅𝐸𝐶 𝐻𝐸 (𝛽 ̅ = -1.2, 𝐶𝐼 = [-1.7; -.76], 𝑡(13) = -5.6, 𝑝 = 8.5 • 10 -5 ) on the probability to prefer the High-Effort offer over the Low-Effort one.

Fig. 5 -

 5 Fig. 5 -Choices and AEC ratings in Experiment 2 a. Choices: Rates of acceptance of the High-Effort offer, plotted against incentive levels. Top: Cross markers indicate Acceptance rate averaged across all subjects for a given incentive level (x-axis) and a given Real Energetic Cost (REC) level (marker colour). Bottom: Cross markers indicate Acceptance rate, averaged across all REC levels and across all subjects for a given incentive level. Black dots indicate averaged predictions of a logistic regression model (see main text).

Experiment 1 ,

 1 we inverted a generalized linear model comprising the following regressors: the real energetic cost REC, the prospective Gain level (0 in Loss condition), and the prospective Loss level (0 in Gain condition) (Fig.5b). Beyond the unsurprising positive effect of REC on AEC (𝛽 ̅ = 15, 𝐶𝐼 = [14 ; 16], 𝑡(23) = 24, 𝑝 < 10 -14 ), we replicated the positive effect of prospective Gain level (𝛽 ̅ = 2.8, 𝐶𝐼 = [1.4 ; 4.3], 𝑡(23) = 4.0, 𝑝 = 6.3 • 10 -4 ). We found no significant difference in effect size with that observed in Experiment 1 (𝛽 ̅ = -.89, 𝐶𝐼 = [-2.6 ; .78], 𝑡(44) = -1.1, 𝑝 = .29), thus contradicting one of the predictions of the "Belief" scenario. Importantly, we also found a significant positive effect of prospective Loss level on AEC (𝛽 ̅ = 2.8, 𝐶𝐼 = [1.2 ; 4.4], 𝑡(23) = 3.6, 𝑝 = .0016), which contradicts the main prediction from the Valence variant of the "Affect" scenario. These effects were also significant in a model of AEC depending on Gain and Loss magnitudes rather than levels (Gain: 𝛽 ̅ = 1.6, 𝐶𝐼 = [. 72 ; 2.5], 𝑡(23) = 3.7, 𝑝 = .0012; Loss: 𝛽 ̅ = 1.7, 𝐶𝐼 = [. 66 ; 2.8], 𝑡(23) = 3.4, 𝑝 = .0027).

Fig. 6 -

 6 Fig. 6 -Comparison of modelled and observed data in Experiments 1 and 2

Fig. 7 -

 7 Fig. 7 -Bayesian model comparison across Experiments 1 & 2

Fig. S2 -

 S2 Fig. S2 -Pupil diameter during arousal rating task in Experiment 2

[

  .51 ; .81], 𝑡(21) = 9.1, 𝑝 < 10 -8 ) on the actual probability of success. As expected based on former observations, we found no effect of neither prospective gain level (𝛽 ̅ = -.0043, 𝐶𝐼 = [-.14; .14], 𝑡(21) = -.065, 𝑝 = .95) nor prospective loss level (𝛽 ̅ = -.074, 𝐶𝐼 = [-.21 ; .06], 𝑡(21) = -1.1, 𝑝 = .28).

  significant impact of neither prospective gain level (𝛽 ̅ = .0073, 𝐶𝐼 = [-.0050; .020], 𝑡(23) = 1.2, 𝑝 = .23) nor prospective loss level (𝛽 ̅ = -.0094, 𝐶𝐼 = [-.021 ; .0021], 𝑡(23) = -1.7, 𝑝 =

Fig. 2 :

 2 Fig. 2: Design of Experiment 2

  of success (𝛽 ̅ = .64, 𝐶𝐼 = [ .57; .71], 𝑡(33) = 18, 𝑝 < 10 -15 ), but did not show any influence from either the prospective gain level (𝛽 ̅ = -.031, 𝐶𝐼 = [-.086; .024], 𝑡(33) = -1.2, 𝑝 = .26) or the prospective loss level (𝛽 ̅ = -.030, 𝐶𝐼 = [-.088 ; .028], 𝑡(33) = -1.1, 𝑝 = .29). Prior confidence Again, expected probability of success (𝛽 ̅ = .086, 𝐶𝐼 = [. 069 ; .10], 𝑡(33) = 10, 𝑝 < 10 -11 ) and, more interestingly, prospective gain level (𝛽 ̅ = .0073, 𝐶𝐼 = [. 0011; .013], 𝑡(33) = 2.4, 𝑝 = .022) had a positive impact on prior confidence ratings, whereas prospective loss level had a negative impact (𝛽 ̅ = -.011, 𝐶𝐼 = [-.018 ; -.0042], 𝑡(33) = -3.3, 𝑝 = .0024). There was no evidence for a difference in absolute magnitude of the gain versus loss effects (|𝛽 𝐺𝑎𝑖𝑛 | -|𝛽 𝐿𝑜𝑠𝑠 | ̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅ = -.0012, 𝐶𝐼 = [-.0076 ; .0052], 𝑡(32) = -.38, 𝑝 = .71, in a paired t-test).

Fig. 4 :

 4 Fig. 4: Confidence ratings and motor success, pooled across experiments a. Prior confidence ratings. Left: Individual 𝛽 estimates of target p(Success), Gain level, and Loss level regressors in a GLM model of prior confidence ratings. Subjects were pooled across Experiments 1 to 3, and those exhibiting an impact of prospective gain and/or loss on their motor success were excluded from this analysis (N = 64 remaining). In each whisker plot, horizontal line is the median, and subjects are drawn as outliers (circles) if their parameter estimate is larger than 𝑄3 + 1.5 * (𝑄3 -𝑄1) or smaller than 𝑄1 -1.5 * (𝑄3 -𝑄1), where Q1 and Q3 are the 25th and 75th percentiles, respectively. Right: Prior confidence ratings, averaged across subjects, plotted against stake level (gain or loss). Dots are inter-subject averages; error bars represent s.e.m between individual averages (on each stake level) corrected for individual average confidence overall. b. Motor success rate. Left: Individual 𝛽 estimates of target p(Success), Gain level, and Loss level regressors in a logistic model of motor success. Right: Success rate, averaged across subjects, plotted against stake level (gain or loss). * : 𝑝 < .05 * * : 𝑝 < .01 * * * : 𝑝 < .001

Fig. 5 :

 5 Fig. 5: Inter-individual correlations between 𝜷 estimates in Confidence and Success modelsLeft panel: Individual 𝛽 estimates of Loss level predictor in a GLM model of Prior Confidence ratings, against individual 𝛽 estimates of Gain level predictor in the same GLM. Yellow dots represent subjects included in the correlation computation (N = 64), grey dots represent subjects excluded from it (N = 15) because their motor success probability was significantly (or marginally so, p < .1) impacted by prospective gain and/or loss. We found a significant negative correlation (𝑟 = -.50, 𝑝 < 10 -4 ) between 𝛽 𝐶𝑜𝑛𝑓 Gain and 𝛽 𝐶𝑜𝑛𝑓 Loss across subjects.

Finally, in

 in Experiments 2 & 3, we estimated individual loss aversion by inverting, with the help of the Variational Bayesian Algorithm (VBA) toolbox(Daunizeau et al., 2014), the following model of choices made during the lottery task: 𝑃(𝐴𝑐𝑐𝑒𝑝𝑡𝐿𝑜𝑡𝑡𝑒𝑟𝑦) = 1 1 + exp (-(𝐵 0 + 𝐵 1 * (𝐸𝑈 𝐿𝑜𝑡𝑡𝑒𝑟𝑦 -𝐸𝑈 𝑆𝑎𝑓𝑒 𝑂𝑝𝑡𝑖𝑜𝑛 ))) with 𝐸𝑈 𝐿𝑜𝑡𝑡𝑒𝑟𝑦 = 𝑃(𝑊𝑖𝑛) * 𝐺 𝐿𝑜𝑡𝑡𝑒𝑟𝑦 -𝑃(𝐿𝑜𝑠𝑒) * 𝜆 * 𝐿 𝐿𝑜𝑡𝑡𝑒𝑟𝑦 and 𝐸𝑈 𝑆𝑎𝑓𝑒 𝑂𝑝𝑡𝑖𝑜𝑛 = 𝐺 𝑆𝑎𝑓𝑒 𝑂𝑝𝑡𝑖𝑜𝑛

Fig. 1 :

 1 Fig. 1: Task design in Experiment 1.

(

  Fig.2a), we observed a positive effect (𝛽 ̅ = .13, 𝐶𝐼 = [. 06 ; .20], 𝑡(17) = 3.8, 𝑝 = .0015) of the initial position of the rating cursor on individual confidence ratings ('CR(t)'), a close-to-significant positive relationship (𝛽 ̅ = .06, 𝐶𝐼 = [-.05 ; .13], 𝑡(17) = 1.9, 𝑝 = .067) with the rating made on the previous trial ('CR(t-1)'), and no relationship (𝛽 ̅ = -.02, 𝐶𝐼 = [-.07 ; .03], 𝑡(17) = -.85, 𝑝 = .41) with the feedback given on the previous trial ('Feedback(t-1)'). As could be expected, we also observed a strong positive relationship (𝛽 ̅ = .28, 𝐶𝐼 = [. 19 ; .37], 𝑡(17) = 6.6, 𝑝 < 10 -5 ) between confidence ratings and the correctness of the participant's answer ('Success(t)'), and a strong negative relationship (𝛽 ̅ = -.40, 𝐶𝐼 = [-.54 ; -.26], 𝑡(17) = -5.9, 𝑝 = 1.6 • 10 -5 ) with the response time ('RT(t)'). Crucially, we found a strong positive relationship (𝛽 ̅ = .31, 𝐶𝐼 = [. 23 ;

Fig. 2 :

 2 Fig. 2: Logistic regression of individual ratings made during Control phase of Experiment 1 a. Left: box plot illustrates the group distribution of beta coefficients in the logistic regression of individual confidence ratings made during 'Conf' task. See Methods for details about regressors. Right: individual confidence ratings (subjective probability of being correct, between 0 and 1) are plotted against confidence ratings averaged across all other subjects (made during 'Conf' and 'CONF' tasks). For each subject, trials were binned according to the average confidence rating about the given quiz question. Each dot represents the confidence ratings averaged within each bin and across all subjects. The vertical error bars represent the standard error to the mean (SEM) of confidence ratings. The horizontal error bars (hidden, smaller than the central dots) represent the SEM of average likeability ratings. The continuous curve represents the model fit, averaged across participants, and the shaded area corresponds to the SEM of this model fit. 𝛽 ̅ = 0.31, 𝑝 < 10 -6 . b. Left: box plot illustrates the group distribution of beta coefficients in the logistic regression of individual likeability ratings made during 'Like' task. Right: individual likeability ratings (normalized to an analog scale between -1 and +1) are plotted against likeability ratings averaged across all other subjects (made during 'Like' and 'LIKE' tasks). 𝛽 ̅ = 0.66, 𝑝 < 10 -7 . Binning and fitting were done using the same procedures as in panel a. * : p < 0.05; ** : p < 0.01; *** : p < 0.001

  ) = 1.6, 𝑝 = .12), and no relationship (𝛽 ̅ = .03, 𝐶𝐼 = [-.04 ; .11], 𝑡(17) = .84, 𝑝 = .41) with the rating made on the previous trial. As expected, we again observed a strong positive relationship (𝛽 ̅ = .33, 𝐶𝐼 = [. 23 ; .43], 𝑡(17) = 6.9, 𝑝 < 10 -5 ) with the correctness of the participant's answer, a strong negative relationship (𝛽 ̅ = -.34, 𝐶𝐼 = [-.45 ; -.23], 𝑡(17) = -6.6, 𝑝 < 10 -5 ) with the response time, and a strong positive relationship (𝛽 ̅ = .32, 𝐶𝐼 = [. 24 ; .41], 𝑡(17) = 8.5, 𝑝 < 10 -6 ) with the confidence ratings averaged across all other subjects. Critically, we found a significant positive effect (𝛽 ̅ = .07, 𝐶𝐼 = [. 02 ; .11], 𝑡(17) = 2.8, 𝑝 = .012) of the average likeability of the background musical extracts on confidence ratings, with no effect (𝛽 ̅ = .02, 𝐶𝐼 = [-.04 ; .07], 𝑡(17) = .67, 𝑝 = .51) of the average likeability of the previous musical extract. Thus, the value of an unrelated contextual feature (background music) seemed to affect confidence in the reponse to the ongoing forced-choice trial.In the 'LIKE' task (Fig.3b), the initial position of the cursor on the rating scale (𝛽 ̅ = .10, 𝐶𝐼 = [. 05 ; .14], 𝑡(17) = 4.5, 𝑝 = 3.0 • 10 -4 ), as well as the likeability rating averaged across all other subjects (𝛽 ̅ = .55, 𝐶𝐼 = [. 40 ; .69], 𝑡(17) = 8.0, 𝑝 < 10 -6 ), were positively related to the individual likeability ratings. In addition, we also observed a positive relationship with the rating made (𝛽 ̅ = .08, 𝐶𝐼 = [. 03 ; .14], 𝑡(17) = 3.5, 𝑝 = .0027) and the feedback received (𝛽 ̅ = .06, 𝐶𝐼 = [. 01 ; .12], 𝑡(17) = 2.7, 𝑝 = .015) on the previous trial. Critically, we found a significant positive effect (𝛽 ̅ = .05, 𝐶𝐼 = [. 01 ; .09], 𝑡(17) = 2.7, 𝑝 = .017) of the average confidence triggered by answering a given quiz question on the contiguous likeability rating, while we found no effect (𝛽 ̅ = -.004, 𝐶𝐼 = [-.05 ; .04], 𝑡(17) = -.19, 𝑝 = .85) of the average confidence about the previous quiz question. Thus, the confidence in unrelated task performance did seem to affect the value assigned to the background context of the ongoing trial.

Fig. 3 :

 3 Fig. 3: Logistic regression of confidence and likeability ratings made during Interference session of Experiment 1

Fig. 4 :

 4 Fig. 4: Block design in Experiment 2.

  ), confidence judgments were sensitive to similar influences from factors of non-interest as in Experiment 1 (positive relationship with the initial position of the cursor: 𝛽 ̅ = .05, 𝐶𝐼 = [. 01 ; .09], 𝑡(34) = 2.8, 𝑝 = .0088; with the confidence rating made on previous trial: 𝛽 ̅ = .11, 𝐶𝐼 = [. 06 ; .16], 𝑡(34) = 4.7, 𝑝 < 10 -4 ; with the correctness of the subject's answer: 𝛽 ̅ = .35, 𝐶𝐼 = [. 27 ; .44], 𝑡(34) = 8.5, 𝑝 < 10 -9 ; and with confidence averaged across all other subjects regarding the current quiz question: 𝛽 ̅ = .60, 𝐶𝐼 = [. 52 ; .68], 𝑡(34) = 15, 𝑝 < 10 -15 ; negative relationship with response time: 𝛽 ̅ = -.43, 𝐶𝐼 = [-.53 ; -.33], 𝑡(34) = -8.7, 𝑝 < 10 -9 ; no effect of average likeability of the previous musical extract: 𝛽 ̅ = .02, 𝐶𝐼 = [-.02 ; .06], 𝑡(34) = .88, 𝑝 = .39). Remarkably, the positive effect of average contextual likeability previously observed in Experiment 1 was no longer present (𝛽 ̅ = .01, 𝐶𝐼 = [-.03 ; .05], 𝑡(34) = .61, 𝑝 = .55), regardless of the incentivization condition (𝛽 𝑁𝑜𝐼𝑛𝑐𝑒𝑛𝑡𝑖𝑣𝑒 ̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅ -𝛽 𝐹𝑖𝑛𝑎𝑛𝑐𝑖𝑎𝑙𝐼𝑛𝑐𝑒𝑛𝑡𝑖𝑣𝑒 ̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅ = .005, 𝐶𝐼 = [-.07 ; .08], 𝑡(33) = .14, 𝑝 = .89). In the 'LIKE' task (Fig. 5b), likeability judgments were again positively predicted by several factors of non-interest (initial position of the rating cursor: 𝛽 ̅ = .08, 𝐶𝐼 = [. 06 ; .10], 𝑡(34) = 6.8, 𝑝 < 10 -7 ; previous likeability rating: 𝛽 ̅ = .08, 𝐶𝐼 = [. 05 ; .10], 𝑡(34) = 6.1, 𝑝 < 10 -6 ; average likeability rating: 𝛽 ̅ = .64, 𝐶𝐼 = [. 56 ; .72], 𝑡(34) = 16, 𝑝 < 10 -15 ), and again no relationship was found with the average confidence about the previous quiz question (𝛽 ̅ = -.007, 𝐶𝐼 = [-.03 ; .02], 𝑡(34) = -.59, 𝑝 = .56). Most interestingly, the positive effect of average confidence about the quiz question previously observed in Experiment 1 was maintained (𝛽 ̅ = .03, 𝐶𝐼 = [. 01 ; .06], 𝑡(34) = 3.1, 𝑝 = .004), again, regardless of the incentivization condition (𝛽 𝑁𝑜𝐼𝑛𝑐𝑒𝑛𝑡𝑖𝑣𝑒 ̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅ -𝛽 𝐻𝑒𝑑𝑜𝑛𝑖𝑐𝐼𝑛𝑐𝑒𝑛𝑡𝑖𝑣𝑒 ̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅ = -.02, 𝐶𝐼 = [-.07 ; .02], 𝑡(33) = -.98, 𝑝 = .33).

  ratings (Experiment 1, 'CONF': 𝛽 ̅ = -.02, 𝐶𝐼 = [-.10; .06], 𝑡(17) = -.60, 𝑝 = .56; 'LIKE': 𝛽 ̅ = .01, 𝐶𝐼 = [-.04; .06], 𝑡(17) = .53, 𝑝 = .60. Experiment 2, 'LIKE': 𝛽 ̅ = .004, 𝐶𝐼 = [-.02; .03],

Fig. 5 :

 5 Fig. 5: Logistic regression of confidence and likeability ratings made during Experiment 2 a. 'CONF' task: The top left box plot illustrates the distribution of posterior averages of beta coefficients following the inversion of a logistic model of confidence ratings made by the first group. See Methods for details about regressors. Importantly, we do not find significant evidence (p = 0.55) for an impact of contextual likeability (averaged across ratings made by the second group) on confidence ratings. The top right graph illustrates this non-significant relationship. For each subject, trials were binned according to the average likeability of their musical extract, and each dot represents the confidence ratings averaged within each bin and across all subjects. The vertical error bars represent the standard error to the mean (SEM) of confidence ratings. The horizontal error bars (smaller than the central dots) represent the SEM of average likeability ratings. The continuous curve represents the model fit, averaged across participants, and the shaded area corresponds to the SEM of this model fit. b. 'LIKE' task: The bottom left box plot illustrates the distribution of posterior averages of beta coefficients following the inversion of a logistic model of likeability ratings made by the second group. We observe a significant positive impact (p = 0.0041) of confidence (averaged across ratings made by the first group) on likeability ratings. The bottom right graph illustrates this relationship. * : p < 0.05; ** : p < 0.01; *** : p < 0.001

  taskFor each subject, we inverted a logistic model of individual likeability ratings:𝐿𝑅 𝑡 = 𝑠𝑖𝑔𝑚(𝛽 0 + 𝛽 1 • 𝐼𝑛𝑖𝑡𝐶𝑢𝑟𝑠𝑜𝑟 𝑡 + 𝛽 2 • 𝐿𝑅 𝑡-1 + 𝛽 3 • 𝐴𝑣𝑔𝐿𝑅 𝑡 )with 𝐿𝑅 𝑡 the subject's likeability rating on current trial (t), 𝑠𝑖𝑔𝑚 the sigmoidal function such that 𝑠𝑖𝑔𝑚(𝑋) = 1 1+𝑒 -𝑋 , 𝐼𝑛𝑖𝑡𝐶𝑢𝑟𝑠𝑜𝑟 𝑡 the initial (random) position of the cursor on the rating scale, 𝐿𝑅 𝑡-1 the likeability rating made in previous trial, and 𝐴𝑣𝑔𝐿𝑅 𝑡 the likeability rating made about the current musical extract, averaged across all other subjects.b) Conf taskFor each subject, we inverted a logistic model of individual confidence ratings:𝐶𝑅 𝑡 = 𝑠𝑖𝑔𝑚(𝛽 0 + 𝛽 1 • 𝐼𝑛𝑖𝑡𝐶𝑢𝑟𝑠𝑜𝑟 𝑡 + 𝛽 2 • 𝐶𝑅 𝑡-1 + 𝛽 3 • 𝐹𝑒𝑒𝑑𝑏𝑎𝑐𝑘 𝑡-1 + 𝛽 4 • 𝑆𝑢𝑐𝑐𝑒𝑠𝑠 𝑡 + 𝛽 5 • 𝑅𝑇 𝑡 + 𝛽 6 • 𝐴𝑣𝑔𝐶𝑅 𝑡 )

  𝐶𝑅 𝑡 = 𝑠𝑖𝑔𝑚(𝛽 0 + 𝛽 1 • 𝐼𝑛𝑖𝑡𝐶𝑢𝑟𝑠𝑜𝑟 𝑡 + 𝛽 2 • 𝐶𝑅 𝑡-1 + 𝛽 3 • 𝑆𝑢𝑐𝑐𝑒𝑠𝑠 𝑡 + 𝛽 4 • 𝑅𝑇 𝑡 + 𝛽 5 • 𝐴𝑣𝑔𝐶𝑅 𝑡 + 𝛽 6 • 𝐴𝑣𝑔𝐿𝑅 𝑡-1 + 𝛽 7 • 𝐴𝑣𝑔𝐿𝑅 𝑡 )

  𝐿𝑅 𝑡 = 𝑠𝑖𝑔𝑚(𝛽 0 + 𝛽 1 • 𝐼𝑛𝑖𝑡𝐶𝑢𝑟𝑠𝑜𝑟 𝑡 + 𝛽 2 • 𝐿𝑅 𝑡-1 + 𝛽 3 • 𝐹𝑒𝑒𝑑𝑏𝑎𝑐𝑘 𝑡-1 + 𝛽 4 • 𝐴𝑣𝑔𝐿𝑅 𝑡 + 𝛽 5 • 𝐴𝑣𝑔𝐶𝑅 𝑡-1 + 𝛽 6 • 𝐴𝑣𝑔𝐶𝑅 𝑡 )

  rating task As for CONF task in Experiment 1, we inverted a logistic model of confidence ratings:𝐶𝑅 𝑡 = 𝑠𝑖𝑔𝑚(𝛽 0 + 𝛽 1 • 𝐼𝑛𝑖𝑡𝐶𝑢𝑟𝑠𝑜𝑟 𝑡 + 𝛽 2 • 𝐶𝑅 𝑡-1 + 𝛽 3 • 𝑆𝑢𝑐𝑐𝑒𝑠𝑠 𝑡 + 𝛽 4 • 𝑅𝑇 𝑡 + 𝛽 5 • 𝐴𝑣𝑔𝐶𝑅 𝑡 + 𝛽 6 • 𝐴𝑣𝑔𝐿𝑅 𝑡-1 + 𝛽 7 • 𝐴𝑣𝑔𝐿𝑅 𝑡 )

No sweat, no sweet? A motivational account of how expected benefits inflate anticipated effort cost.

  Should you walk to the nearest bakery, or cycle to your favourite one? Should you climb a little higher to get a better view or just stay below and enjoy what you can see already? This kind

	Introduction
	Emmanuelle Bioud, Corentin Tasu, Mathias Pessiglione
	Motivation, Brain and Behavior lab, Institut du Cerveau et de la Moelle épinière (ICM) ;
	Inserm U1127 ; CNRS U7225 ; Sorbonne Universités, Paris, France

  Cross markers indicate AEC ratings averaged across all subjects for a given incentive level (x-axis) and a given Real Energetic Cost (REC) level (marker colour). Dotted lines indicate the target rating for each REC level. Bottom: Purple cross markers indicate AEC ratings, averaged across all REC levels and across all subjects for a given incentive level. Black dots indicate averaged predictions of a generalized linear model (GLM, see main text).

). Bottom: Purple cross markers indicate Acceptance rate, averaged across all REC levels and across all subjects for a given incentive level. Black dots indicate averaged predictions of a logistic regression model (see main text). ***: Positive effect of incentive level on Acceptance rate, 𝑝 < 0.001 b. Anticipated Energetic Cost (AEC) ratings, plotted against incentive levels. Top: ***: Positive effect of incentive level on AEC ratings, 𝑝 < 0.001 Solid lines represent a linear interpolation between averaged data points (or model predictions), for illustrative purposes only. Shaded area around curves or error bars across dots represent the standard error to the mean (s.e.m), computed across subjects.

  Positive effect of Loss and Gain level on Acceptance rate, both 𝑝 < 0.001 b. Anticipated Energetic Cost (AEC) ratings, plotted against incentive levels. Top: Cross markers indicate AEC ratings averaged across all subjects for a given incentive level (x-axis) and a given Real Energetic Cost (REC) level (marker colour). Dotted lines indicate the target rating for each REC level. Bottom: Cross markers indicate AEC ratings, averaged across all REC levels and across all subjects for a given incentive level. Black dots indicate averaged predictions of a generalized linear model (GLM, see main text). **: Positive effect of Loss level on AEC ratings, 𝑝 < 0.005

). Bottom: Cross markers indicate Acceptance rate, averaged across all REC levels and across all subjects for a given incentive level. Black dots indicate averaged predictions of a logistic regression model (see main text). ***: ***: Positive effect of Gain level on AEC ratings, 𝑝 < 0.001 Solid lines represent a linear interpolation between averaged data points, for illustrative purposes only. Shaded area around each curve represents the standard error to the mean (s.e.m), computed across subjects.

Neuron, Cognition & Computation

For the record, Edouard was seen leaving the garden center about an hour later with three guppy fish in a plastic bag.

This paradox had already appeared in much earlier works, such as Plato's Euthydemus, Nicolaus Copernicus' and John Law's, and had likely been submitted by numerous Antiquity and Middle Ages children to their parents, but Adam Smith is often considered to be its most influential presenter.

Besides, in some of the works cited above, probability predictors were not orthogonal by design (nor orthogonalized a posteriori) to expected value, or expected utility (regardless of the specific theory, EU, SEU, PT, that we use to compute it), such that some of the identified correlates of reward probability might actually encode expected value. A few studies, however, have explicitly shown spatial dissociations between magnitude, probability, and expected value encoding(Tobler et al., 2007, see Fig. 1.19) which strongly suggest that some subpart of the striatum encodes reward probability specifically.

Over the last few years, an interesting proposal has emerged about the interactions between mood and the perceived value of outcomes or stimuli[START_REF] Eldar | Mood as Representation of Momentum[END_REF]. First, good (resp. bad) mood was hypothesized to result from positive (resp. negative) prediction errors between the expected and experienced value of an outcome, drawing on previous modeling and empirical findings about reward prediction errors and reported well-being[START_REF] Rutledge | A computational and neural model of momentary subjective well-being[END_REF]. But more importantly, Eldar and colleagues proposed that mood would in turn influence the perception of obtained outcomes, with more positive (resp. negative) mood inducing higher (resp. lower) valuation, thus creating a positive reinforcement loop between mood and valuation. Under certain conditions, this positive feedback loop could elicit some cases of mood dysfunctions, such as mania or depression. This theoretical proposal was accompanied with some converging empirical evidence, suggesting that affect and valuation could interact bi-directionally.

Acknowledgements

First, I would like to thank the researchers who kindly agreed to be part of my jury and to do me the honour of evaluating my work: Vincent de Gardelle, Nathalie George, Emmanuel Guigon, with a special thank you to Miriam Klein-Flügge and Philippe Tobler, who accepted to review this dissertation.

Acknowledgements

The study was funded by a Starting Grant from the European Research Council (ERC-BioMotiv).

This work also benefited from the program "Investissement d'avenir" ANR-10-IAIHU-06. E.B. received funding from the French Ministry of Higher Education and Research as well as a grant from the French Fondation pour la Recherche Médicale (FRM). R.A. was financially supported by the French Direction Générale de l'Armement (DGA).

F.V. received a private foundation grant from the Fondation d'entreprise Groupe Pasteur Mutualité.

Code availability.

All computer codes used during the current study are available from the corresponding author on reasonable request.

Data availability.

The datasets analysed during the current study are available from the corresponding author on reasonable request.

Code availability

All computer codes used during the current study are available from the corresponding author upon reasonable request.

Data availability

The datasets analysed during the current study are available from the corresponding author upon reasonable request.

Confidence

The neural substrates of confidence, as a special kind of probability judgment (see section A. 4) c.), have also been investigated over the past decade. In memory tasks, activity in the ventral striatum, the vmPFC, and the hippocampus has been found to correlate positively with confidence in recognition judgments [START_REF] Chua | Understanding metamemory: Neural correlates of the cognitive process and subjective level of confidence in recognition memory[END_REF][START_REF] Moritz | Neural correlates of memory confidence[END_REF][START_REF] Schwarze | Ventral Striatal Activity Correlates with Memory Confidence for Old-and New-Responses in a Difficult Recognition Test[END_REF].

In binary perceptual discrimination tasks, a wide network of brain areas seems implicated as well : the OFC in rodents [START_REF] Kepecs | Neural correlates, computation and behavioural impact of decision confidence[END_REF][START_REF] Lak | Orbitofrontal Cortex Is Required for Optimal Waiting Based on Decision Confidence[END_REF], the lateral intra-parietal sulcus [START_REF] Kiani | Representation of Confidence Associated with a Decision by Neurons in the Parietal Cortex[END_REF] and the midbrain dopaminergic system [START_REF] Lak | Midbrain Dopamine Neurons Signal Belief in Choice Accuracy during a Perceptual Decision[END_REF] in monkeys, the ventral striatum [START_REF] Hebart | The Relationship between Perceptual Decision Variables and Confidence in the Human Brain[END_REF], the ACC (negative correlation, Fleming & Dolan, 2012;[START_REF] Hebart | The Relationship between Perceptual Decision Variables and Confidence in the Human Brain[END_REF] and the rostro-lateral prefrontal cortex (negative correlation, Fleming & Dolan, 2012) in humans. In [START_REF] Bang | Distinct encoding of decision confidence in human medial prefrontal cortex[END_REF], BOLD signal in dorso-medial PFC, which the authors refer to as the perigenual ACC, correlated positively with subjects' confidence in their binary judgments about the average direction of a continuous dot-motion stimulus.

In judgment tasks about non-perceptual variables, the mPFC was also found to encode self-reported confidence as well as confidence inferred from first-order judgments, although in a more ventral area than the aforementioned pgACC. Specifically, in Lebreton et al., 2015, BOLD signal in the vmPFC correlated with second-order confidence judgments (explicitly prompted or not) about first-order age, probability or value judgments. Additionally, the vmPFC had been found, in an earlier study [START_REF] De Martino | Confidence in value-based choice[END_REF], to encode confidence in value-based choices.

Abstract

In order to decide whether an alternative is better than the default course of action, individuals typically weigh the expected costs and benefits. Optimal decision-making relies upon accurate effort cost anticipation, which is generally assumed to be performed independently from goal valuation. Challenging this independence axiom of standard decision theory, we investigated whether, and how, goal value may distort anticipated effort cost. For this, we asked healthy volunteers to rate the energetic cost of running routes paired with an orthogonally varied monetary outcome, after deciding whether they would be willing to take these routes programmed on a treadmill. In two experiments (n=46), we observed a positive impact of incentive magnitude on prospective effort cost estimates, contrary to the prediction derived from an extended planning fallacy perspective. To better understand this effect, which affected both ratings and choices, we introduced various experimental manipulations and computational models. Far from ideal observer models exploiting a putative correlation between effort and reward, our findings support a motivational account, according to which prospective effort cost judgments are biased by the output of a decision-oriented cost-benefit computation. This motivation bias might lead human agents to abandon the pursuit of valuable goals that are in fact not so costly to reach.

Methods

The study was approved by the Ethics Committee for Biomedical Research of the Pitié-Salpêtrière Hospital, where both experiments were conducted.

Participants

Participants were recruited via the Relais d'Information en Sciences Cognitives (RISC) website and screened for exclusion criteria: age below 18 or above 39; less than 1-hour physical exercise (and/or less than 30 minutes of cardio-respiratory training) per week; regular use of drugs or medications; history of psychiatric, neurological, cardiac, or respiratory disorders.

All participants gave informed consent prior to partaking in the study.

Experiment 1 22 participants (10 F, 12 M; mean age = 25.6 ± 4.7 y) were recruited for Experiment 1. One participant could not complete the last testing block due to a technical failure, but data from the first 5 blocks were nevertheless included in the analyses.

Subjects received an initial endowment of 18€, and earned financial bonuses during the session, leading to final payment of 30.7€ on average (s.d. = 4.5€).

Experiment 2 24 participants (12 F, 12 M; mean age = 26.0 ± 5.0 y) were recruited for Experiment 2 and were all included in the analyses.

Because subjects incurred a risk of monetary loss during this experiment, they received an initial endowment of 30€, which ensured a non-negative final payment. This final payment was 41.8€ on average (s.d. = 4.4€).

Apparatus

We programmed all tasks and data analyses with MATLAB (MathWorks ®) and used the Psychophysics Toolbox 3.0 [START_REF] Kleiner | What's new in psychtoolbox-3[END_REF] for stimuli presentation. Participants ran on a Tech med Cardio 270 treadmill, whose speed and inclination were controlled by the computer. In Experiment 2, pupil dilation was measured with a The Eye Tribe ET1000 eye-tracker (60 Hz sampling frequency) attached below the computer screen, and participants' head position was stabilized with a chin-rest placed 70 cm away from the 23" screen. Choices and ratings were made using a standard keyboard.

Task design

In this study, we designed a novel behavioural task aimed at measuring the influence of goal value on the anticipated energetic cost of reaching this goal.

We also measured the reward effect for each block separately by fitting the following model:

where 𝐼 𝐵𝑙𝑜𝑐𝑘 𝑖 indicates the incentive level within Block i (and equals 0 elsewhere).

Experiment 2

Choice

We performed a similar analysis to that in Experiment 1, by inverting the following logistic model (adapted from equation 1 to include a Loss factor):

where 𝑠𝑜𝑓𝑡𝑚𝑎𝑥: 𝑥 → 1 1+𝑒 -𝑥 , 𝐵 0 is a bias parameter, 𝑅𝐸𝐶 is the real energetic cost of the High Effort route, 𝐺 is the gain level of the High Effort option (0 in Loss blocks), 𝐿 is the Loss level of the Low Effort option (0 in Gain blocks), and 𝑘 𝑅𝐸𝐶 , 𝑘 𝐺 and 𝑘 𝐿 are the linear coefficients scaling the energetic cost, reward level, and loss level, respectively.

Anticipated Energetic Cost

Again, we fitted several GLMs to the AEC ratings made by each participant.

We started with a very simple model, extended from eq. 2 to include a Loss factor:

where 𝑅𝐸𝐶 is the real energetic cost of the route, 𝐺 is the gain level, 𝐿 is the loss level, and 𝛽s are the linear coefficients of our model.

Next, in order to assess the effect of the arousal evoked by the musical extracts onto AEC ratings, we fitted the following GLM:

where 𝐴 is the arousal rating made by the participant about the same musical extract during the arousal rating phase, and 𝛽 𝐴 is the corresponding coefficient. Whisker plots, right y-axis: All ratings and summary statistics of each subject. Each dot represents a rating, main vertical grey bar is the median, left and right ends of boxplot are first and third quartile of the distribution, respectively.

Supplementary figures

Chapter III: Article 2

The battle for confidence: a tug-of-war between prospective gain and loss

Abstract

Before taking on a challenge, healthy individuals contemplate two crucial notions: the outcomes in case of success versus failure, i.e., the stakes of the challenge, and their own chances to succeed versus to fail, i.e., their confidence. Here, we question an implicit assumption made by modern decision theory, that confidence judgments are independent from, and therefore insensitive to, the outcomes at stake. In a series of three experiments, we asked participants to express their prospective confidence in succeeding at an upcoming motor challenge, whose stakes were gains and losses of varying magnitude. Contrary to the predictions that could be derived from either an assumption of stake-confidence independence or various existing psychological constructs such as the "desirability bias", we found that larger prospective gains boosted individuals' confidence in their future success, while larger prospective losses hindered it. Our behavioural findings are evocative of another well-established psychological construct, that of "affect-as-information", in the sense that positive/negative affect (related to the perspective of gain/loss), is interpreted as signaling a higher/lower chance of success.

Success rate

When pooling all subjects from Experiments 1-3 (N = 77 after exclusion of 2 subjects from Experiment 1 for the analyses of success rate, see Exp. .043). This deterioration was likely related to the fact that participants, who on average overshoot the target (i.e., squeezed the handgrip too hard), tended to produce more force in high-stakes trials, in line with some "choking-under-pressure" effects reported in the literature [START_REF] Chib | The Effects of Incentive Framing on Performance Decrements for Large Monetary Outcomes: Behavioral and Neural Mechanisms[END_REF][START_REF] Oudiette | A Pavlovian account for paradoxical effects of motivation on controlling response vigour[END_REF]. Because we were interested in whether stakes impacted prior confidence judgments all other things being equal, and especially, actual success probability being held constant, we performed a second run of the same statistical analyses after having excluded all subjects whose success rate was even marginally impacted by stakes (either gain or loss) (N = 15 excluded, with a 𝑝 < .1 threshold on individual 𝛽 𝐺𝑎𝑖𝑛 and 𝛽 𝐿𝑜𝑠𝑠 estimates). Expected .0086], 𝑡(61) = -1.6, 𝑝 = .12) had an overall impact on success rate (Fig. 4a).

Prior confidence

Across all subjects from Experiments 1-3 (N = 79), prior confidence ratings were strongly influenced by expected probability of success (𝛽 ̅ = .099, 𝐶𝐼 = [. 083 ; .11], 𝑡(78) = 13, 𝑝 < 10 -15 ). Critically, they were also positively predicted by prospective gain level (𝛽 ̅ = .0084, 𝐶𝐼 = [. 0037; .013], 𝑡(78) = 3.6, 𝑝 = .00060) and negatively predicted by prospective loss level (𝛽 ̅ =

Methods

The study was approved by the Ethics Committee for Biomedical Research of the Pitié-Salpêtrière Hospital and carried out in accordance with French guidelines and regulations.

Participants

Participants were recruited from the Relais d'Information en Sciences Cognitives (RISC)

volunteer database and screened for exclusion criteria: age below 18 or above 39; lefthandedness; regular use of drugs or medications; history of psychiatric or neurological disorders.

A total of 81 healthy participants were recruited. All participants gave informed consent prior to partaking in the study.

Experiment 1 24 participants were recruited (12 M, aged 23.6 ± 3.2 years). They were endowed 20€ at the beginning of the experimental session, and could obtain monetary gains or incur losses across the experiment, depending on their choices, on their performance at the task (based on motor precision) and on the pseudo-random draws of the computer among completed trials (which, unbeknownst to the participants, were biased such that they could not incur large losses overall). If participants incurred more losses than gains overall, their final payment was raised back to 20€ (this was not disclosed to the participants before the end of the session), or unchanged otherwise. Their final compensation was 22.4€ on average (range [20.0; 33.9] €).

Experiment 2 23 participants were recruited, two of which were excluded before data analysis [one failed to perform the task correctly, one fell asleep during the experiment] (10 M, aged 24.9 ± 3.9 years in the remaining sample, N = 21). The payment procedure was similar to that in Experiment 1, the main difference here being that participants received an additional bonus from random draws in a subsequent lottery task (see details below). Participants' final compensation was 26.4€ on average (range [21.5; 30.1] €).

Experiment 3 10 participants were recruited for a behavioural pilot (5 M, aged 24.5 ± 3.6 years). The payment procedure was similar to that in Experiment 2 (see details below). Participants' final compensation was 27.9€ on average (range [25.0; 30.9] €).

Are you positive? Misattribution of value and confidence judgments

Emmanuelle Bioud 1 , Raphaëlle Abitbol

Abstract

When making a decision about whether or not to accept a challenge, agents must estimate both their chance of success and the benefits associated with success. According to prevailing decision theories, these confidence and value estimates should be constructed independently. Here, we question this independence axiom, based on functional neuroimaging studies reporting that value and confidence representations overlap in the same brain system. We predicted that the human brain would misattribute value judgments about the context with confidence judgments about the behaviour (or vice-versa). In our experiment, healthy volunteers answered more or less difficult quiz questions while listening to more or less pleasant musical extracts. In addition to this baseline task, they performed either confidence ratings about their answers or likeability ratings about the music. As predicted, explicit ratings were reciprocally influenced by incidental factors manipulating value and confidence, violating the independence axiom of decision theory.

However, a follow-up experiment indicated that the confidence-to-value influence was more robust than the value-to-confidence influence, denoting as more intrusive the judgment about the behaviour being correct, and/or more malleable the judgment about the environment being pleasant.

Methods

The study was approved by the Ethics Committee for Biomedical Research of the Pitié-Salpêtrière Hospital, where both experiments were conducted. All research was performed in accordance with the French guidelines and regulations.

Participants

Participants were recruited via the Relais d'Information en Sciences Cognitives (RISC) website and screened for exclusion criteria: age below 18 or above 39, regular use of drugs or medications and history of psychiatric or neurological disorders.

All participants gave informed consent prior to partaking in the study.

In Experiment 1, 20 participants (11F, 9M; mean age = 24) first performed a control session, followed by an interference session one month later. Each session included a confidence rating task and a likeability rating task. They were paid 20€ for their time (regardless of their performance at the tasks). Two participants (1M, 1F) were excluded for guessing the aim of the experiment after the control session. The remaining 18 participants were included in the analyses.

In Experiment 2, a group of 35 participants (21F, 14M; mean age = 23) performed a confidence rating task, and a separate group of 35 participants (19F, 16M; mean age = 25) performed a likeability rating task. Each group was further split into two subgroups, one performing a 'No incentive' paradigm variant (N = 18x2) and one performing a 'Financial/Hedonic incentive' variant (N = 17x2).

'No incentive' variant: Participants were paid 30€ for their participation regardless of their quiz answers or ratings.

'Financial/Hedonic incentive' variant: Participants received an initial endowment of 18€. As explained in the instructions they read prior to the task onset, they could additionally earn up to 8€ depending both on luck and on their performance at the quiz (for details, see Experiment 2. Incentivization paragraph below).

At the end of the session, participants were required to complete a debriefing questionnaire, followed by an oral interview with the experimenter, which confirmed that none of them had guessed the aim of the experiment. All 70 participants were thus included in the analyses.

Materials

We used 256 general-knowledge questions gathered from the French edition of Trivial Pursuit Master ©. Questions were chosen from all game categories to cover a large variety of subjects. The game's open questions were transformed into multiple-answer questions by adding three false answers to the true answer. Questions and answers were already used in a previous fMRI study [START_REF] Vinckier | Neuro-computational account of how mood fluctuations arise and affect decision making[END_REF]. In Experiment 1, 10 participants answered the uneven items in task c. (see Fig. 1) and the even items in tasks b. and d., and 8 participants did the opposite. Before

Additional Information

Author contributions

Supplementary material