Quentin Cavalan
email: quentin.cavalan@hotmail.fr

Vincent De Gardelle

Jean-Christophe Vergnaud

I did most of the work! Three sources of bias in bargaining with joint production

Keywords: overcondence, bargaining, joint production, belief updating JEL C91, D03, D74, D81

Although conicts in bargaining have attracted a lot of attention in the literature, situations in which bargainers have to share the product of their performance have been less commonly investigated empirically. Here, we show that overplacement leads to conict in these situations: individuals overestimate their contribution to the joint production and consequently make unreasonable claims. We further decompose overplacement into three types of cognitive biases: overestimation of one's own production (i.e. overcondence bias), underestimation of others' production (i.e. superiority bias) and biases in information processing. We show that they all contribute to overplacement. To quantify these biases, we develop a novel experimental setting using a psychophysically controlled production task within a bargaining game, where we elicit participants' subjective estimation of their performance, both before and after they receive information about the joint production. In addition, we test several interventions to mitigate these biases, and successfully decrease disagreements and overplacement through one of them. Our approach illustrates how combining psychophysical methods and economic analyses could prove helpful to identify the impact of cognitive biases on individuals' behavior.

Introduction

Many studies in behavioral economics and psychology have established that agents often overestimate their skills relative to others, which can lead to economic failures. For example, entrepreneurs overestimate their chances of success when entering a market, leading to excess entry [START_REF] Camerer | Overcondence and excess entry: An experimental approach[END_REF]. Individuals who believe themselves to be better than average underweight the advice they get [START_REF] Gino | Eects of task diculty on use of advice[END_REF] and exhibit more aggressive behaviors in experimental wargames [START_REF] Johnson | Overcondence in wargames: experimental evidence on expectations, aggression, gender and testosterone[END_REF]. However, it is not always clear where this overplacement bias comes from.

In this paper, we focus in particular on situations of bargaining over a joint production where there is uncertainty surrounding the contributions of both parties. These situations are in fact ubiquitous in day-today life. For instance, in a household, the contributions of each member are uncertain, since they include not only wages but also time, sacriced career opportunities, participation in chores, etc. As such, both members may overestimate their own contribution to the household's wealth and/or they might fail to fully acknowledge the contribution of the other member, leading to disagreements over the merit of each member.

Similarly, in collective bargaining between labor unions and management, parties may have incompatible beliefs about their respective merits and consequently about the distribution of wealth between employees and share-holders. Such overestimation on both sides would lead to disagreements and bargaining ineciencies.

Starting with [START_REF] Gantner | Equitable choices in bargaining games with joint production[END_REF] and [START_REF] Cherry | Hardnose the dictator[END_REF], an important literature has recently ourished on bargaining problems over joint production [START_REF] Cappelen | The pluralism of fairness ideals: An experimental approach[END_REF][START_REF] Konow | Morals and mores: Experimental evidence on equity and equality[END_REF][START_REF] Cappelen | Responsibility for what? Fairness and individual responsibility[END_REF][START_REF] Karagözo §lu | Performance information, production uncertainty, and subjective entitlements in bargaining[END_REF][START_REF] Fischbacher | Non-additivity and the salience of marginal productivities: Experimental evidence on distributive fairness[END_REF][START_REF] Soldà | Absolute versus relative success: Why overcondence creates an inecient equilibrium[END_REF][START_REF] Santos Pinto | Does overcondence lead to bargaining failures? Working Paper[END_REF]. This literature has mostly focused on fairness issues, using games such as the dictator or ultimatum game, with a very limited strategic component [START_REF] Cappelen | Responsibility for what? Fairness and individual responsibility[END_REF]. Moreover, most studies in this literature have used experimental designs where participants have perfect knowledge about their individual production, either because they receive feedback about it [START_REF] Konow | Morals and mores: Experimental evidence on equity and equality[END_REF][START_REF] Fischbacher | Non-additivity and the salience of marginal productivities: Experimental evidence on distributive fairness[END_REF] or because production is chosen explicitly [START_REF] Gantner | Equitable choices in bargaining games with joint production[END_REF][START_REF] Cappelen | The pluralism of fairness ideals: An experimental approach[END_REF]. How individuals bargain when they are uncertain about their own production, by contrast, has been mostly unexplored. Critically though, in these situations uncertainty leaves room for each party to overestimate their relative contribution.

Furthermore, the dierent mechanisms that can generate this overplacement bias have not been clearly identied. Overcondence in one's own skills might lead individuals to overestimate their contribution but other cognitive biases may also play a role, such as biases in how individuals estimate the production of others, or biases in how they update their beliefs when receiving information about the production. These dierent biases may all contribute to overplacement and ultimately aect bargaining outcomes.

To the best of our knowledge, three experimental studies have investigated bargaining over a joint production with real eort made under uncertainty. [START_REF] Karagözo §lu | Performance information, production uncertainty, and subjective entitlements in bargaining[END_REF] have shown that providing participants with information about their relative performance leads them to increase their entitlements, which impact rst proposals, bargaining duration and settlements. [START_REF] Soldà | Absolute versus relative success: Why overcondence creates an inecient equilibrium[END_REF] have shown that overplacement generates costly delays and disagreements in bargaining. Similarly, Santos Pinto and Colzani (2021) have found higher rates of bargaining failure when participants exhibit overplacement.

Whereas these previous studies have mainly focused on the impact of overplacement on bargaining outcomes, here we further seek to disentangle the various biases which may lead to overplacement and ultimately bargaining ineciencies. More precisely, in the present study we shall decompose overplacement in three cognitive biases: 1) an overcondence bias by which individuals overestimate their production, 2) a superiority bias by which they estimate the production of others as lower than their own, 3) updating biases by which they update their beliefs in a non-Bayesian manner following feedback about the joint production.

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In our experiment, participants produced joint wealth by performing a task, and then shared this wealth through a particular version of a Nash demand game. In case of a disagreement, payos depended on the true performance of individuals, which was not known with certainty by participants beforehand. If however claims added up to an amount less than the joint production, the remainder was not lost but shared in proportion to the claims of the two players. 1 After the production task (but before making a claim in the bargaining game), participants also provided an initial estimate of their own performance by means of a condence judgment, then received feedback about the joint production, after which they could revise their condence. The production and bargaining game were repeated several times, and in the middle of the experiment an intervention was conducted to provide information to participants about their behavior.

Our rst goal was to identify dierent routes to overplacement: overestimation of self, underestimation of others, and belief updating biases. To this end, we built a model in which beliefs about one's own performance are aected by overcondence, beliefs about the performance of others are aected by a superiority bias and ultimately, updating biases translate those beliefs into a nal evaluation of performance. To estimate this model, we directly measured overcondence by comparing the initial condence of each participant to their actual performance in the production task. In addition, taking advantage of the multiple rounds in our design, we estimated a superiority bias and updating biases by examining how participants revised their condence. Our second goal was to evaluate how these biases may change after the intervention, when participants are told about their overall contribution or about their overplacement bias. We evaluated how such interventions aect overplacement, and ultimately bargaining outcomes.

The primary outcome of the negotiation is whether an agreement was reached or not, and we dened the disagreement index as the probability that the negotiation would fail. Quite obviously, a social planner might want to limit disagreements because they are costly: when the two parties have to resort to court, dead-weight losses are incurred by society. However, settlements might not always be desirable if they do not reect the respective contributions of individuals: a social planner might want individuals to claim what they have produced so that they get what they deserve in case of an agreement. Thus, to check how individuals' claims correspond to their true contribution, we also considered the distance between claims and contributions, hereafter labelled mismatch index.

With respect to previous studies on bargaining, our paradigm also introduced two original methodological features. Firstly, our production task was based on a perceptual task and we used psychophysical methods to obtain comparable conditions of performance for all participants. Given that overcondence heavily depends on the diculty of the task [START_REF] Kruger | Unskilled and unaware of it: How diculties in recognizing one's own incompetence lead to inated self-assessments[END_REF][START_REF] Grieco | Overcondence in absolute and relative performance: The regression hypothesis and bayesian updating[END_REF][START_REF] Santos Pinto | Does overcondence lead to bargaining failures? Working Paper[END_REF], this ensured that heterogeneity in performance was not an important confound in our setup when assessing cognitive biases. Secondly, our participants interacted with a computer-simulated human. Indeed, because multiple Nash equilibria often coexist in bargaining games, the process through which an equilibrium is selected might be an important confound of participants' behavior. To mitigate this issue, we rst performed a pilot study, in which participants interacted with other humans. Then, we performed the main study in which we used the behavior of participants in the pilot to program the computer's behavior. This allowed us to better isolate the impact of biases on bargaining behavior by reducing strategic uncertainty in the bargaining game.

The rest of the paper is organized as follows. Section 2 presents the design, methods and measures of the experiment. Section 3 describes our experimental results. First, we document overplacement and its eect 1 We introduced this feature to be closer to a real legal proceeding in which what is not claimed is certainly not lost. Moreover, this might reduce the appeal of always opting for fty-fty splits which reduce statistical power if they occur too much.

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The real-eort production task

In each round, the production task consisted of 5 trials of a simple visual task. On each trial, two stimuli (Gabor patches) with dierent orientations (θ 1 and θ 2 ) appeared sequentially and for 500 ms on the screen. Participants had to judge whether the second stimulus was oriented clockwise (θ 2θ 1 = ∆θ > 0) or counterclockwise (∆θ < 0) with respect to the rst stimulus. They responded using the computer's keyboard. After 5 such trials, the number of correct answers denoted X 1 corresponds to the participant's production in this round. Importantly, participants received no feedback about X 1 at this stage.

The diculty of the task was determined by |∆θ|: the larger the dierence between the two angles, the easier. We calibrated this quantity for each participant to aim for 75% of correct responses, based on individual performance in the 100 practice rounds. 3 Participants were not made aware that diculty was calibrated.

Condences elicitation

In each round, after the production phase, participants had to give their condence regarding their answers, on a continuous scale from 0% to 100%. This condence judgment was incentivized with a canonical BDM mechanism [START_REF] Becker | Measuring utility by a single-response sequential method[END_REF] applied to one randomly selected perceptual decision in the round. Under expected utility, this mechanism incentivized participants to report the probability that a randomly chosen trial in the round was correct, or equivalently, the average probability of being correct across trials in the round. Additionally, it incentivized them to perform the task as accurately as possible.

4 Participants were trained with this incentive mechanism before starting the main experiment.

Then, participants learnt the joint production of the dyad X for that round, which is the sum of the number of correct answers for the participant, noted X 1 , and for the computer , noted X 2 , that is: X = X 1 + X 2 .

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Journal Pre-proof Following this information, participants had to give again their condence in their performance, now knowing X. This revised condence, was incentivized in the same way as the initial condence judgment. 5The bargaining game Finally, participants bargained with the computer over the joint production X. The bargaining game was a slightly modied version of a one-shot Nash Demand Game, where payos were proportional to individual production (minus a cost) if claims couldn't be satised. More precisely, the participant and the computer made simultaneous claims (Claim 1 and Claim 2 ) which represent the share of the joint production they wanted for themselves. If claims were compatible i.e. Claim 1 + Claim 2 ≤ 100%, an agreement was reached and joint production was shared in proportion to the two claims. Otherwise, if Claim 1 + Claim 2 > 100%, players settled in court, where they received their own contribution minus a cost of 50.

In both cases, participants were informed about the claim of the computer, and whether an agreement was reached. In case of an agreement, participants also learnt the number of points they received. In case of a disagreement, they were simply reminded about the rule that converted their (unknown) performance into points. Overall, participants received the same amount of information regarding their individual performance, irrespective of whether they agreed or disagreed. Then, once the bargaining game was over, the next round started.

Computer's behavior

In each round, the number of correct answers for the computer X 2 was drawn from a binomial distribution B(5, 0.75). Computer's accuracy was thus similar to participants'. For the bargaining phase, the computer's claim Claim 2 was drawn from participants' claims in the pilot experiment in similar conditions, that is, in rounds where X 2 and X 1 had the same values as in the current round. The inset in Figure 1 illustrates claims as a function of true contribution for participants in the pilot experiment (and thus for computers in the main experiment), showing a clear increase in claims when contributions exceed 55% of the total. The mean and standard deviations of the computer's claims conditional on (X 1 , X 2 ) are presented in Online Appendix 4.

The intervention

In all experimental sessions, after 26 rounds of the bargaining game, participants were asked questions and received feedback about one or several of the variables of the game. There were 5 dierent interventions (4 treatments and a baseline condition) which varied the nature and the number of questions asked.

• In the baseline condition, T 0 , we asked Q 0 : In your opinion, what was your average claim in the previous rounds of bargaining?

• In treatment T 1 , we asked Q 1 : In your opinion, what was your average contribution to the joint production in the previous rounds of bargaining?

• In treatment T 2 , we asked Q 2 : In your opinion, how many times was your claim higher than your contribution in the previous rounds of bargaining?

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• In treatment T 3 , we asked Q 3 : In your opinion, how many times was your initial condence higher than your performance in the previous rounds of bargaining?

• In treatment T 4 , we asked Q 1 , Q 2 and Q 3 altogether.

In each session, the experimenter explained in detail the question(s) asked to participants and the feedback they would obtain. Participants then answered the question(s), and obtained two dierent types of feedback: one screen showed the correct answer along with their actual answer, and the subsequent screen showed the entire distribution of the variable at stake (e.g. in T 1 the distribution of the participant's past contributions) with a visual animation. The duration of all interventions was the same (approximately 10 minutes).

The goal of these interventions was to impact participants' beliefs in order to aect their bargaining behavior. In T 1 , we targeted participants' tendency to overestimate their contribution to the joint output, to make participants aware that, on average, they were contributing less than what they thought. In T 2 we also targeted participants' estimation of their contribution, but now with an explicit link with their claims, to make them realize that they might have made unreasonable claims with respect to their contribution. Finally, T 3 was designed to assess more directly how participants' knowledge of their individual performance, independently of the other player, impacted their bargaining behavior.

Finally, our baseline condition involves a question about past claims, to control for potential eects of time or increased motivation after the intervention, as we do not expect participants to make systematic mistakes on this question or to change their behavior based on this intervention. 6

Main measures

We used two measures for beliefs about performance: overcondence and overplacement. In addition, we dened two measures regarding the outcomes of the bargaining game: a disagreement index and a mismatch index. Finally, we also dened a measure of how claims are sensitive to participants' beliefs about their contribution to the joint production.

Overcondence and overplacement biases

For each participant, we dened overcondence bias as the mean dierence between the initial judgment of condence and the actual performance.

Overcondence bias = condenceactual performance Overplacement bias was dened as the mean dierence between revised condence once the joint production is known and actual performance, a positive overplacement bias indicating that participants overestimated the relative share they produced.

Overplacement bias = revised condenceactual performance 6 This can be checked on the data of the pilot experiment where similarly to this baseline condition, we ask this question in the middle of the experiment. We nd no systematic mistake: participants slightly underestimate it but this is not signicant (ErrorQ 0 = -2.18%, t(32) = -1.50, p = 0.143). Moreover, we do not nd any signicant correlation between mistakes made at this question and participants' change in claims after the intervention (Cor = 0.09, t(31) = 0.48, p = 0.633).

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Disagreement index

The main outcome of bargaining was whether players reached an agreement or not. Here, to smooth out the eect of the randomly drawn claim C 2 of the computer, we considered the expected disagreement corresponding to the claim of the participant given X 1 and X 2 the production of each player. 7 The disagreement index is the average of the expected disagreement over bargaining rounds. It lies between 0% and 100%. From a social planner perspective, the closer to 0 the better.

Disagreement index = E C2 1 {C1+C2>100} |C 1 , X 1 , X 2

Mismatch index

The goal of our interventions was to decrease the disagreement index. However, another measure of interest was the distance between participants' claims and their actual contributions. We thus dened a mismatch index as the average squared dierence between participant's claims and actual contribution, divided by 25, such that on average it lied between 0 and 100. 8 Here again, from a social planner perspective, the closer to 0, the better.

Mismatch index = 1 25 actual contribution -claim 2
Sensitivity to perceived contribution Finally, we measured how participants' beliefs related to their claims in the bargaining game. First, given participants' revised condence, the perceived contribution is dened as the estimated production (5 times their revised condence) divided by the joint production.

Then, participant's sensitivity to their perceived contribution was dened as the eect of these perceived contributions on claims, in a linear regression (least-square). The higher this measure, the more participants were sensitive to their perceived contributions when making claims in the bargaining game. The objective of this measure was to check whether the interventions changed participants' claims, regardless of their beliefs.

Claim sensitivity to perceived contribution = Cov(claim, perceived contribution) V ar(perceived contribution) 7 Our results were similar when using the average number of disagreements by participants instead. 8 Indeed, since participants were calibrated, actual contribution was equal to 50% on average. Thus, when claiming 100% (or 0%) of the joint production, the squared distance between claim and actual contribution would be 2500.

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In this section, we rst describe the eect of overplacement bias on disagreement and mismatch indices, using the pre-intervention data.

9

We then introduce a model for revised condence in which the role of several cognitive biases (overcondence bias, superiority bias and biases in belief updating) can be quantied. We then study the intervention eect on biases and bargaining outcomes. Finally, we provide some striking elements about gender dierences in the reaction to interventions.

3.1 Overcondence, overplacement and bargaining outcomes (before intervention)

Figure 2 illustrates the distributions of overcondence, overplacement and bargaining outcomes (disagreement and mismatch indices) across participants, before the intervention. Before receiving feedback about the joint production, participants exhibited an overcondence bias as their condence (M = 79.79, SD = 8.31) was signicantly higher (t(217) = 9.394, p < .001) than their actual performance (M = 74.24, SD = 3.11). After receiving feedback, there was still a clear overplacement bias, as participants' revised condence (M = 79.90, SD = 7.43) was signicantly higher (t(217) = 10.342, p < .001) than their actual performance (M = 74.24, SD = 3.11). In the bargaining game, we found 51.21% of disagreement and a mismatch of 9.54 on average.

As a comparison, for virtual agents who would maximize their expected earnings, best respond to the feedback about joint production and perfectly estimate their own production, disagreement and mismatch would still be present (even in the absence of overplacement, due to strategic uncertainty) but they would be significantly lower than in our data (disagreement: t(217) = 14.685, p < .001; mismatch: t(217) = 9.764, p < .001). We then investigated the relation between overplacement and bargaining outcomes. Under the assumption that claims should reect revised condence, we expected i) that disagreement would increase with overplacement and ii) that mismatch, would be minimal in the absence of overplacement. These two predictions were conrmed in panel regressions (Table 1), both across and within participants. Figure 3 illustrates these relations. 9 Out of the 234 participants, we removed 16 of them from the analysis due to the failure of the calibration procedure: their average performance was more than two standard deviations away from the average performance in our sample equal to 74%.

Updating

For a Bayesian agent, the revised condence in a given round t (noted rc * t ) can be calculated from Bayes' rule once the joint production of the pair X t is revealed, under the assumption that the number of correct answers (the production) was generated by a binomial distribution with success probability c t for the participant and c other t for the other player (equation 1).

rc * t = 5 k=0 k × P(k|X t , c t , c other t ) 5 with P(k|X t , c t , c other t ) = P(X t ∩ k|c t , c other t ) P(X t |c t , c other t ) (1)
In our model, belief updating could depart from the Bayesian benchmark: updating could be under- 10 We thank an anonymous reviewer who suggested incorporating non-Bayesian updating into our model. Initially, we had planned to recover beliefs about others by comparing condence and revised condence which would have merged biases about others with updating biases.

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Journal Pre-proof sensitive or over-sensitive to feedback and this could depend on whether this feedback represented good or bad news compared to the expected joint production. Indeed, an important literature shows that in comparison to Bayesian agents, individuals often exhibit conservatism [START_REF] Edwards | Conservatism in human information processing[END_REF][START_REF] Fischho | Hypothesis evaluation from a Bayesian perspective[END_REF][START_REF] Huck | Do players correctly estimate what others do? Evidence of conservatism in beliefs[END_REF] and asymmetry [START_REF] Ertac | Does self-relevance aect information processing? Experimental evidence on the response to performance and non-performance feedback[END_REF][START_REF] Eil | The good news-bad news eect: Asymmetric processing of objective information about yourself[END_REF][START_REF] Charness | Conrmation bias with motivated beliefs[END_REF][START_REF] Coutts | Good news and bad news are still news: experimental evidence on belief updating[END_REF][START_REF] Coutts | What to blame? Self-serving attribution bias withmultidimensional uncertainty[END_REF][START_REF] Drobner | Motivated belief updating and rationalization of information[END_REF][START_REF] Möbius | Managing self-condence: Theory and experimental evidence[END_REF] in their updating.

11 As seen in equation 2, we captured these biases with two parameters, β which is the responsiveness to feedback following bad news (when Bayesian revised condence is below the initial condence) and γ which is the dierence in responsiveness between good and bad news. For the Bayesian agent, β is equal to 1, and γ is equal to 0.

rc t -c t = β × (rc * t -c t ) + γ × Good News × (rc * t -c t ) with rc *
t the ideal Bayesian revision and Good News = 1 when rc * t > c t and 0 otherwise

(2)

Figure 5A illustrates belief updating biases in our data, under the hypothesis that other-condence is constant during the experiment and equal to .75 (the actual performance of the other player). The gure presents participants' updating rc tc t against the Bayesian benchmark rc * tc t . Overall, observations form a atter line than the Bayesian benchmark and particularly so in the case of bad news. In other words, even without estimating our full model (which involves two additional parameters described below, jointly estimated with β and γ) we could obtain evidence for conservative and asymmetric updating in our data. cother t is dened under the assumption that the updating process is Bayesian, that is β = 1 and δ = 0 (see main text for details). The dotted line represents the average correlation across participants. 11 We note that the evidence for asymmetric updating is actually mixed in the literature. While most studies nd oversensitivity to good news [START_REF] Eil | The good news-bad news eect: Asymmetric processing of objective information about yourself[END_REF][START_REF] Charness | Conrmation bias with motivated beliefs[END_REF][START_REF] Coutts | What to blame? Self-serving attribution bias withmultidimensional uncertainty[END_REF][START_REF] Drobner | Motivated belief updating and rationalization of information[END_REF][START_REF] Möbius | Managing self-condence: Theory and experimental evidence[END_REF], other studies nd the opposite asymmetry [START_REF] Ertac | Does self-relevance aect information processing? Experimental evidence on the response to performance and non-performance feedback[END_REF][START_REF] Coutts | Good news and bad news are still news: experimental evidence on belief updating[END_REF], or no asymmetry [START_REF] Grossman | An unlucky feeling: Overcondence and noisy feedback[END_REF][START_REF] Buser | Responsiveness to feedback as a personal trait[END_REF][START_REF] Barron | Belief updating: does the `good-news, bad-news' asymmetry extend to purely nancial domains[END_REF]. [START_REF] Barron | Belief updating: does the `good-news, bad-news' asymmetry extend to purely nancial domains[END_REF] recently concluded that dierences in information structure, priors, domains and stake sizes between the dierent studies could not account for those mixed results. A detailed investigation of the mechanims by which deviations may (or may not) arise, however, is outside of the scope of the present paper.

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Other-condence

Belief updating requires individuals to entertain a belief c other t about the performance of the other player.

We modelled this other-condence by means of two parameters. One parameter b captured the superiority bias, which is the extent to which participants estimated that the other player's performance was lower than their own. The other parameter δ captured the degree to which participants relied on their condence in each round to form their other-condence in that round. Putting together these two elements, participant's other-condence in round t could be written as

c other t = (1 -δ) × c + δ × c t -b (3)
with δ ∈ [0, 1] and c is participant's average condence during the experiment.

Notice that averaging across rounds equation 3, b corresponds to how much participant's average estimation of the performance of the other player (c other ) is lower than his own (c). The parameter δ by contrast does not relate to overplacement. We introduced δ to account for the possibility that participants' othercondence may not have been constant during the experiment and instead depended on their own condence, as appeared to be the case in our data. For instance, when setting β and γ in equation 1 to xed values, we could compute the other-condence cother t that would be consistent with the observables X t , c t and rc t , and we found that cother t was positively correlated with c t across trials. This gives some indication that participants use their belief about their own performance to form their estimates about the performance of the other. Figure 5C illustrates this phenomenon for β = 1 and δ = 0 (i.e. Bayesian benchmark for belief updating). In this case, the correlation between cother t and c t , estimated for each participant, was signicantly positive across participants (C or = .32, t(217) = 13.764, p < .001).

Estimation of the full model

In the full model, revised condence was thus determined for each participant on each round by participant's condence c t , the joint production X t , and 4 free parameters characterizing the formation of beliefs about the performance of the other (b, δ) and the updating of beliefs following feedback (β, γ). Assuming normally distributed noise on the predicted revised condence, we estimated our model using maximum likelihood, separately for each participant, using the 26 observations available for each participant before the intervention.

In Figure 6, we show the distribution of these 4 parameters across participants.

In addition, we plot the average revised condence predicted by our tted model against the actual average revised condence. On average, b was slightly higher than zero. Indeed, participants' estimated other-condence (M = 73.77, SD = 17.98) was on average lower than their condence (M = 79.79, SD = 8.31). Furthermore, the extent to which participants' condence inuenced their other-condence was highly heterogeneous: δ was equal to 1 for 34% of participants, to 0 for 13% of them and fell between these two values for 53% of participants. Finally, participants processed feedback about the joint production in a biased manner, exhibiting conservatism (β < 1) and asymmetry (γ > 0) in their belief updating.

Decomposition of overplacement

To evaluate the contribution of the dierent biases to overplacement, we examined how the revised condence predicted by the model would change if a particular bias was to be removed. Eliminating overcondence bias was done by setting condence in each round to the actual performance, that is setting for each round t c t = p t . Eliminating the superiority bias was done by setting b to the true dierence between the participant's performance and the performance of the other (such that on average, the dierence in beliefs would be equal to the actual dierence in performance between the participant and the other player). Eliminating biased belief updating corresponded to setting β = 1 and γ = 0. Starting from the full model, we eliminated one bias at a time such that after three steps the model would be fully unbiased. There were 6 possible orders for doing so, as illustrated on Figure 7. When removing a bias, we then evaluated how revised condence should change in the model. Finally, for each bias, we considered the average eect of removing this bias (across the 6 dierent paths) to quantify its contribution to overplacement.

For the full model, predicted revised condence was 79.71% on average (close to the 79.90% in the actual data). For the fully unbiased model, it dropped to 74.88%. On average, eliminating overcondence in our model reduced predicted revised condence by 1.40 points, eliminating superiority bias reduced it by 1.68 points, and eliminating biased updating by 1.75 points. In other words, each bias contributed about equally (by 29%, 35% and 36% respectively) to overplacement.

The interventions

We now turn to the analysis of the interventions conducted at the middle of the experiment. Recall that there were 5 experimental conditions: a baseline in which we asked participants about their average claim (T 0 ) and 4 treatments in which we focused on their average contributions (T 1 ), on how their claims may have exceeded their contributions (T 2 ), on how their condence may have exceeded performance (T 3 ), or on these last 3 variables at the same time (T 4 ).

Mistakes in the question asked during intervention

First, we present descriptive statistics about participants' answers during the interventions. In the baseline condition, we were not expecting any systematic error but participants slightly overestimated their past claims as they reported asking for 54.75% of the joint production while they were only asking for 53.03% on average (Error Q0 = 1.71, t(47) = 1.84, p = 0.072). This slight misconception is not a worry though, as it should not impact behavior. By contrast, when asked about their average contribution to the joint production (in T 1 and T 4 ), participants strongly overestimated it, declaring that they had produced 60.68% of the joint production while their actual contribution was 49.68% on average (Error Q1 = 11.06, t(82) = 9.80, p < 0.001).

Conrming the robustness of participants' answers, mistakes made on this question were correlated with overplacement bias (Cor = 0.51, p < 0.001). In T 2 and T 4 , participants largely underestimated the number of times their claim was above their contribution, with an average estimate of 7.84 times, compared to the true value of 13.55 times out of 26 (Error Q2 = 5.71, t(74) = 8.80, p < 0.001). In T 3 and T 4 , participants also consistently underestimated the number of times their condence was above their individual performance, with an estimate of 10.08 times instead of 13.18 times out of 26 (Error Q3 = 3.10, t(74) = 4.31, p < 0.001).

This last observation is consistent with participants being not only overcondent but also naive with respect to

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Journal Pre-proof this bias. This was conrmed by the correlation between the mistakes made on this question and participant's overcondence (C or = -0.50, p < 0.001). Finally, examining answers to all questions for participants in T 4 , we found that all estimation errors were signicantly correlated in the expected direction.

Intervention eect

The eect of the intervention on bargaining outcomes and overplacement bias are presented in Figure 8 and in Table 2 and3. Disagreement index signicantly fell by 5.57 points after the intervention in the baseline condition, and even more in the four treatments. However, only T 4 signicantly diered from baseline (p = .002) with an additional decrease of 8.16 points. Results were similar for the mismatch index. The distance between claims and contribution was reduced after the intervention in all treatments, but only for T 4 was the intervention eect signicant in comparison to baseline (p = .021), with a reduction of mismatch by 3.61 points. In sum, intervention T 4 successfully reduced disagreements and brought claims closer to actual contributions, thereby improving both bargaining outcomes. Overplacement bias showed a similar pattern, and was reduced in comparison to baseline only in T 4 , with a drop of 2.90 points (p = .027). As a robustness check, we conrmed that this intervention eect in T 4 (compared to baseline) was directly observable on the raw number disagreements (p = .016) as well as in claims (p = .004). For completeness, we further checked that our results were not driven by pre-intervention dierences between treatments: we found no pre-intervention dierences between baseline and T 4 in beliefs (overplacement, overcondence) and in bargaining behavior (claims, claim sensitivity, disagreement and mismatch indices).

We also found that the change in participants' claim sensitivity following the intervention was not significantly dierent in T 4 compared to baseline, suggesting that our intervention did not work solely because individuals changed their claim strategy with respect to their revised condence.

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Decomposition of the intervention eect on overplacement

To evaluate how the change in overcondence bias, superiority bias and updating biases may have contributed to the intervention eect on overplacement, we extended the methodology used above in our decomposition of overplacement. Specically, we rst estimated our model for revised condence as before but now on all observations, allowing the parameter b capturing the superiority bias, and the parameters β and γ capturing updating biases to change after the intervention. To evaluate how much each bias contributed to the intervention eect, in our model we then eliminated the intervention eect on each bias, one at a time, as if the bias was unaected by the intervention. For updating biases, we simply set β and δ after the intervention to their values estimated before the intervention. To eliminate the change in overcondence bias, we translated the reported condence after the intervention such that participants would exhibit the same overcondence J o u r n a l P r e -p r o o f

Journal Pre-proof bias as the one measured before the intervention. We proceeded in a similar way to eliminate the change in superiority bias, by setting b after the intervention to its value estimated before the intervention, with a correction for the potential change in the dierence in actual performances between the participant and the other player.

12 From this, we could evaluate how the intervention eect on revised condence would be aected if each bias was kept constant across the experiment and could not participate in this eect. As before, there were 6 possible dierent paths to consider (i.e. there were 6 possible orders in which biases could be removed) and we calculated the average eect of removing a given bias across these 6 paths. The results of this decomposition for each treatment are summarized in Figure 9A. In particular, in T 4 , where the intervention signicantly decreased overplacement and disagreements compared to baseline, predicted revised condence dropped by 4.22 after the intervention (4.43 in the actual data). We estimated that 51% (2.06 points) of this eect was due to a decrease in overcondence bias, 37% (1.49 points) to a decrease in superiority bias and 13% (0.52 points) to a decrease in updating biases.

Finally, we also carried out this procedure to the treatment eect, that is, the comparison of the intervention eect between T 4 and baseline treatments. The predicted treatment eect on revised condence was J o u r n a l P r e -p r o o f Journal Pre-proof 4.2 points in the full model (4.46 points in the data), and 0.15 points in the model where all changes in biases were eliminated. As can be seen in Figure 9B, eliminating overcondence reduced the predicted treatment eect by 1.42 points (35%), eliminating superiority bias reduced it by 0.80 points (20%), and eliminating biased updating reduced it by 1.82 points (45%). Notice that the relative contributions of overcondence and superiority biases were higher when we decomposed the intervention eect in T 4 alone than when we decomposed the treatment eect (T 4 vs. baseline): this was expected since in the baseline overcondence and superiority biases slightly decrease while updating biases increase (see Figure 9A).

Gender dierences

We conclude our results by presenting some elements about gender dierences.

Before the intervention, we found no dierence between men and women regarding the outcomes of the bargaining game (gender eect on disagreement index: F (216) = 0.966, p = 0.327; mismatch index F (216) = 0.361, p = 0.549), regarding the elicited performance beliefs (overplacement bias: F (216) = 0.197, p = 0.583; overcondence bias: F (216) = 0.113, p = 0.737) and regarding biases estimated in our model (superiority bias: F (216) = 0.187, p = 0.665; conservatism: F (216) = 2.607, p = 0.108; asymmetry F (216) = 1.955, p = 0.163).

This similarity in cognitive evaluation between men and women is not surprising. Indeed, although men are often more overcondent than women [START_REF] Lundeberg | Highly condent, but wrong: Gender dierences and similarities in condence judgments[END_REF][START_REF] Hügelschäfer | On condent men and rational women: It's all on your mind(set)[END_REF], gender dierences in condence were found to be highly task dependent [START_REF] Deaux | Attributing causes for one's own performance: The eects of sex, norms, and outcome[END_REF][START_REF] Lundeberg | Highly condent, but wrong: Gender dierences and similarities in condence judgments[END_REF][START_REF] Bordalo | Beliefs about gender[END_REF]. For instance, [START_REF] Bordalo | Beliefs about gender[END_REF] showed that on a quiz where men and women were expected to perform similarly, they were also similar in how they evaluated themselves or others. On the other hand, the fact that men and women reacted similarly to the joint production feedback contrasts with previous ndings that women tend to revise more pessimistically than men when receiving feedback about relative performance [START_REF] Berlin | Gender dierences in reactions to feedback and willingness to compete[END_REF].

13

We found one dierence between men and women regarding the sensitivity to perceived contribution: men were more sensitive to their perceived contribution than women (F (216) = 6.35, p = 0.012). This result is somewhat consistent with D'Exelle et al. ( 2017) who showed that in a Nash demand game the inuence of beliefs about what the other will claim is stronger for men than women and with [START_REF] Durante | Preferences for redistribution and perception of fairness: an experimental study[END_REF] who showed that the dierence in individuals' demand for redistribution when initial earnings are arbitrary vs earned is much more pronounced for men than women. One other possible interpretation is that our sensitivity measure might have captured risk or fairness principles (risk averse participants' should have made claims that are more disconnected from their condence and favoured safer claims at 50%) on which men and women may dier (e.g. women being more risk averse than men [START_REF] Eckel | Men, women and risk aversion: Experimental evidence[END_REF]).

We also considered gender dierences with respect to the eect of interventions on bargaining outcomes (see Table 4). Interestingly, it appeared that the eect of the intervention was mostly driven by women, as for men we did not nd any treatment eect on disagreement or mismatch indices. This pattern suggests that women reacted to the interventions while men did not. When focusing on the comparison between baseline and T 4 , we found that in T 4 the disagreement index improved signicantly more for women than for men (p = .002), with a large dierence of 16.38 points between genders. The mismatch index also improved more for women than for men in T 4 compared to baseline, but the dierence between genders was not signicant in this case. The dierence between men and women was also present in overplacement: when comparing T 4 to 13 This may be due to the fact that our perceptual task was more gender neutral than the mathematics task in [START_REF] Berlin | Gender dierences in reactions to feedback and willingness to compete[END_REF]. However, although we expect the gender gap on prior beliefs to depend on the type of task [START_REF] Deaux | Attributing causes for one's own performance: The eects of sex, norms, and outcome[END_REF][START_REF] Lundeberg | Highly condent, but wrong: Gender dierences and similarities in condence judgments[END_REF][START_REF] Bordalo | Beliefs about gender[END_REF], it is not clear why it would also impact the gender gap in information processing.

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Journal Pre-proof baseline, overplacement decreased after the intervention but this decrease was signicantly more pronounced for women than men (a dierence of 5.50 points, p = .035).

Discussion

In this paper, we show that overplacement bias prevents settlement when individuals bargain over a joint production, and we decompose this overplacement into several cognitive biases. In practice, by eliciting participants condence in their performance both before and after they receive feedback about the joint production, and repeating the bargaining game a large number of times, we decompose overplacement into three dierent biases: overestimation of one's own performance (overcondence bias), estimation of others' performance lower than one's own (superiority bias), and non-Bayesian information processing. We show that these three biases are present in our data, and contribute in roughly the same proportions to the overplacement bias.

Our paper thus contributes to the empirical literature that identies cognitive biases at the origin of bargaining failures when individuals have to share a jointly produced surplus under uncertainty (Karagözo §lu J o u r n a l P r e -p r o o f

Journal Pre-proof [START_REF] Karagözo §lu | Performance information, production uncertainty, and subjective entitlements in bargaining[END_REF][START_REF] Santos Pinto | Does overcondence lead to bargaining failures? Working Paper[END_REF][START_REF] Soldà | Absolute versus relative success: Why overcondence creates an inecient equilibrium[END_REF]). Among the previously cited studies, only [START_REF] Soldà | Absolute versus relative success: Why overcondence creates an inecient equilibrium[END_REF] measured beliefs twice as we do, before and after participants receive a noisy signal about their relative performance. They found no biases for prior beliefs (consistent with participants knowing that they were paired with equally performing players), but overplacement in posterior beliefs, due to biased updating.

14 Our results on biased updating are in line with these ndings, but in our case updating was based on exact information about the joint production, not on noisy feedback regarding relative performance.

Furthermore, our study goes beyond the updating issue, by examining the contribution of other biases as well.

To the best of our knowledge, our approach (i.e. evaluating the contribution of dierent cognitive biases to overplacement) has never been used in the past, and one could argue that our generative model of revised condence deserves further scrutiny. In this regard, note that our model was parsimonious in terms of hypotheses. We used a classic model of belief updating to capture non-Bayesian information processing and for other-condence our specication has the main advantage of directly incorporating the superiority bias with a single parameter. We agree that having other-condence vary across rounds depending on condence could be criticized as unnecessary a priori, but we provided evidence to support this specication. An interesting avenue for further research would be to measure this belief regarding the performance of the other player directly. This should allow assessing more precisely whether participants incorporate their own condence in their belief about the other player.

The other novelty of our study was to test interventions aimed at decreasing cognitive biases and improving bargaining outcomes. In practice, we tested 4 interventions and showed that the most comprehensive one successfully reduced disagreements, mismatch and overplacement. In this intervention, participants received information about their true contribution to the joint production, about whether they claimed more than what they contributed and about their overcondence, and after this intervention overplacement decreased signicantly, compared to the baseline treatment. This treatment eect was in fact driven not only by participants in the treatment group whose overcondence and superiority biases decreased, but also by participants in the baseline group who became increasingly biased in their updating (more conservative and asymmetric), which also raises an interesting question about the evolution of belief updating biases in competitive environments. Importantly, this decrease in overplacement coincided with a decrease in disagreement, so our intervention has produced the benets that were expected. In addition, these benets were not obtained at the cost of a greater mismatch between claims and contributions. In this regard, we note that the intervention in which participants only received feedback about their overcondence (T 3 ) seems to have reduced cognitive biases almost as much as our most successful intervention (T 4 ), but with no reduction in disagreements and mismatch indices.

Our work thus points at a possible way to improve bargaining between individuals from a social planner perspective. We are of course aware that there are gaps between the present experimental procedure and what can be done outside the laboratory. For instance, here participants received feedback about their true production, which is not always feasible in real bargaining situations. More realistically, one could provide the two parties information about their overcondence in tasks for which their true performance is measurable.

Assuming that overcondence is domain-general [START_REF] West | The domain specicity and generality of overcondence: Individual dierences in performance estimation bias[END_REF][START_REF] Kelemen | Individual dierences in metacognition: Evidence against a general metacognitive ability[END_REF][START_REF] Ais | Individual consistency in the accuracy and distribution of condence judgments[END_REF], de-biasing them for such tasks may correct their bias also in the bargaining situation.

Finally, one important but unexpected aspect of our results is the dierence between women and men in our study. Specically, we found that the intervention had a stronger impact on women than men. Although this eect was clear in our sample, we are aware that it would need to be conrmed in future studies with a larger sample size, as in the present study each treatment only involves 20 men and 20 women.

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Nonetheless, the fact that men and women may react dierently to de-biasing procedures would imply that gender-specic procedures have to be implemented in order to be successful. We can only speculate about the possible reasons underlying this gender eect. One possibility is that the strategic aspect of the game played a role, as suggested in other studies [START_REF] Niederle | Do Women Shy Away From Competition? Do Men Compete Too Much?[END_REF][START_REF] Dreber | Gender dierences in deception[END_REF][START_REF] Malik | Gender dierences in bargaining with asymmetric information[END_REF]. It is also possible that women reacted more to the intervention because they cared more about the social benets of reducing conicts. Indeed, it has been argued that men and women are socialized dierently in that boys are taught not to care too much about other people, while girls are encouraged to do so [START_REF] Gilligan | Why Does Patriarchy Persist?[END_REF]. In any case, understanding this gender dierence, and designing interventions that are eective for both men and women, constitute an interesting challenge for future research.

Figure 1 :

 1 Figure 1: Schematic representation of the experimental procedure for one round of the game and timeline of the experiment. See section 2.3 for details about computer's claim.

D

  Figure 2: (A) Distribution of overcondence for participants (n = 218). The line represents the absence of bias. (B) Same as A but for overplacement. (C) Distribution of disagreement index. The line represents the benchmark for unbiased agents making best-response claims. (D) Same as C) but for mismatch index. Dotted lines represent averages across participants.

  Figure 3: (A) Disagreement index regressed against overplacement bias. Each dot represents the average of observations over a percentile of overplacement. The black line represents a linear regression. Below the main plot are shown the distributions across participants of the regression coecients (slope and intercept), with dotted lines representing the mean across participants. (B) Same as A but for mismatch index, and with a quadratic regression, with intercept, slope, and coecient for the quadratic term.

  J o u r n a l P r e -p r o o fJournal Pre-proof3.2 A generative model of revised condenceTo investigate more precisely the determinants of overplacement, we considered a model in which revised condence is predicted from initial condence and joint production, while taking into account biases in the estimation of others' performance and biases in belief updating.10 By doing so, we could decompose over- placement into three specic components: one related to beliefs about own performance (i.e. condence) which are aected by an overcondence bias, one related to beliefs about the performance of the other player (thereafter other-condence) which are aected by a superiority bias and one related to belief updating which is aected by biases in information processing (conservatism and asymmetry). A schematic representation of this model is shown in Figure4. We present below evidence supporting the existence of the dierent components of the model before reporting our empirical estimation.

Figure 4 :

 4 Figure 4: Schematic representation of the generative model of revised condence. Orange boxes represent the empirically observed variables: performance, condence, feedback and revised condence. Blue boxes represent the internal variables of the model, namely the performance of the other player c other t

Figure 5 :

 5 Figure 5: (A) Participants' updating (revised condence minus condence) against Bayesian updating (Bayesian revised condence -condence) for bad news (left side of the plot, in red) and good news (right side of the plot, in green) before the intervention. Each dot corresponds to a given participant in a given round. For this illustration, Bayesian revised condence is dened under the assumption that c other t

F

  Figure 6: (A-E) Distribution across participants of the estimated parameters: b quanties the superiority bias, δ the extent to which participants' condence inuences their other-condence on a round-by-round basis, c other is the average other-condence across rounds as dened by parameters b and δ in equation 3 and β and γ characterize conservatism and asymmetry in belief updating. Dotted lines represent averages across participants. When relevant, blue solid lines represent the absence of bias as dened in the main text. (F) Actual and predicted value of the average revised condence for each participant in the full model (N = 218).

Figure 7 :

 7 Figure 7: Representation of each bias' contribution to overplacement. At the top is indicated the predicted revised condence in our full model (with the 3 biases) and at the bottom the predicted revised condence in the fully unbiased case (when all biases are removed from the model). Each line corresponds to the elimination of one type of bias (blue: superiority bias, red: belief updating bias, black: overcondence bias), and indicates the change in the predicted revised condence when doing so. See main text for details.

Figure 8 :

 8 Figure 8: Change in disagreement index (panel A), mismatch index (panel B) and overplacement bias (panel C) after (vs. before) the intervention, for each treatment. Positive values indicate an increase in the variable of interest after the intervention. Error bars represent mean and s.e.m. across participants.

Figure 9 :

 9 Figure 9: (A) Decomposition of the change in revised condence after (vs. before) the intervention by each bias, for each treatment. Error bars represent mean and s.e.m. across participants. (B) Decomposition of the treatment eect, that is, the change in T 4 relative to the change in T 0 . Asterisks represent the change in revised condence observed in our data. Circles, squares and triangles represent the estimated change in revised condence due to a change in overcondence bias, superiority bias and updating biases, respectively (see main text for details).

Table 1 :

 1 Panel data regressions of disagreement index and mismatch index against overplacement bias

		Disagreement BE FD	Mismatch BE FD
	Intercept	46.97 * * * (1.34)	-0.81 (0.54)	5.18 * * * (1.23)	-0.035 (0.301)
	Overplacement	0.75 * * * (0.14)	0.36 * * * (0.02)	-0.058 (0.093)	0.054 * * * (0.015)
	Overplacement 2	x	x	0.013 * * *	0.010 * * *
		x	x	(0.003)	(0.000)
	Observation used F Statistic	218 28.90 * * *	5450 226.28 * * *	218 8.20 * * *	5450 459.62 * * *
	Note:	* p<0.05; * * p<0.01; * * * p<0.001
		BE = between estimator
		FD = rst dierence estimator

Table 2 :

 2 Regression of disagreement index, average disagreement, average claim and mismatch index's change (After -Before) by treatment (reference=Baseline)

		∆ Disagreement	∆ Disagreement (raw)	∆ Claim	∆ Mismatch
		Estimate	Std	Estimate	Std	Estimate	Std	Estimate	Std
	T 1	-4.20	(2.67)		-5.71	(3.39)	-1.69	(0.96)	-2.88	(1.56)
	T 2	-4.13	(2.61)		-3.40	(3.31)	-1.93 *	(0.94)	-2.91	(1.53)
	T 3	-3.76	(2.81)		-2.22	(3.56)	-1.29	(1.01)	-0.65	(1.64)
	T 4	-8.16 * *	(2.66)		-8.15 *	(3.37)	-2.78 * *	(0.96)	-3.61 *	(1.65)
	Constant	-5.57 * *	(1.84)		-6.01 *	(2.33)	-0.26	(0.66)	0.35	(1.07)
	Observations	218			218		218		218
	F Statistic	2.37			1.71		2.28		2.01
	Note:		* p<0.05;	* * p<0.01; * * * p<0.001
					∆ OP bias	∆ Sensitivity
					Estimate	Std	Estimate	Std
		T1			0.09	(1.31)	-0.28	(0.18)
		T2			-0.53	(1.28)	-0.19	(0.18)
		T3			-1.17	(1.38)	-0.01	(0.19)
		T4			-2.90 *	(1.54)	-0.21	(0.19)
		Constant		0.71	(0.90)	0.13	(0.12)
		Observations		218		218
		F Statistic		1.71		0.92
		Note:		* p<0.05; * * p<0.01; * * * p<0.001

Table 3 :

 3 Regression of overplacement bias and sensitivity to perceived contribution's change (After -Before)

by treatment (reference=Baseline)

Table 4 :

 4 Regression of disagreement index, mismatch index, overplacement bias and sensitivity to perceived

		∆ Disagreement	∆ Mismatch	∆ OP bias	∆ Sensitivity
	T1	-0.83 (3.91)	-2.29 (2.31)	-0.07 (1.94)	-0.14 (0.26)
	T2	-0.67 (3.69)	-3.36 (2.11)	0.19 (1.77)	0.04 (0.24)
	T3	1.70 (3.75)	-0.51 (2.21)	-1.01 (1.86)	-0.08 (0.25)
	T4	-0.39 (3.62)	-1.54 (2.14)	-0.37 (1.79)	-0.18 (0.27)
	Women	7.10 (3.62)	-0.85 (2.14)	0.66 (1.79)	0.25 (0.25)
	Women × T1	-6.78 (5.31)	-0.89 (3.13)	0.18 (2.63)	-0.29 (0.36)
	Women × T2	-6.91 (5.16)	0.90 (3.04)	-1.50 (2.55	-0.48 (0.35)
	Women × T3	-11.67 * (5.58)	-0.51 (3.29)	-0.27 (2.76)	0.19 (0.38)
	Women × T4	-16.38 * * (5.25)	-6.65 (3.10)	-5.50 * (2.60)	-0.08 (0.38)
	Constant	-9.12 * * * (2.56)	0.77 (1.51)	-1.04 (1.27)	-0.00 (2.17)
	Observations F Statistic	218 2.30 *	218 1.68	218 1.57	218 0.92
	Note:		* p<0.05; * * p<0.01; * * * p<0.001

contribution's change (After -Before) by treatment (reference=baseline) interacted with gender

For instance, always claiming 100% of the joint production, or claiming a xed high or low amount depending on whether one's own production is estimated to exceed a certain threshold, are already simple Nash equilibria, but there are many more equilibria. When participants play against each other, the selection of one of these equilibria could take a long time and render the link between cognitive biases and behavior virtually unidentiable. The case is even more problematic when assessing the eect of a cognitive intervention, as any intervention eect could result from a change in how participants expect others to behave following the intervention, rather than an eect on cognitive biases. Although human players' beliefs and expectations can still lead to multiple equilibria in our main experiment, introducing a computer player, whose strategy is known to be constant, helps mitigate those issues.

See Online Appendix 2 for further details on the mechanism.

Because participants' earnings from condence and revised condence are drawn from the same round, one might worry that risk averse subjects hedge with their stated belief for one condence elicitation against the other. However, such hedging is unlikely in our experiment as it would only be expected for extreme levels of risk aversion (see Online Appendix 3 for further details).

That way, if a participant became relatively better (worse) than the other player after the intervention, we accounted for the fact that ideally the superiority bias should have increased (decreased) accordingly.

[START_REF] Soldà | Absolute versus relative success: Why overcondence creates an inecient equilibrium[END_REF] also measured absolute overestimation biases, but they did not analyse them in detail.

This work was supported by the Agence Nationale de la Recherche [ANR-16-CE28-0002] * Data are available and can be downloaded at https://osf.io/b8z4m/ 1

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domain: spi phys math info

text:
Weikun Deng
email: weikun.deng@enit.fr

Khanh T P Nguyen
email: tnguyen@enit.fr

Christian Gogu
email: christian.gogu@gmail.com

Jérôme Morio
email: jerome.morio@onera.fr

Kamal Medjaher
email: kamal.medjaher@enit.fr

Physics-informed lightweight Temporal Convolution Networks for fault prognostics associated to bearing stiffness degradation

Keywords: Physics-informed machine learning, Non-trending vibration, Prognostic, Bearing contact stiffness degradation

This paper proposes hybrid methods using physics-informed (PI) lightweight Temporal Convolution Neural Network (PI-TCN) for bearings' remaining useful life (RUL) prediction under stiffness degradation. It includes three PI hybrid models: a) PI Feature model (PIFM) -constructing physicsinformed health indicator (PIHI) to augment the feature space, b) PI Layer model (PILM) -encoding the physics governing equations in a hidden layer, and c) PI Layer Based Loss model (PILLM) -designing PI conflict loss, taking into account the difference before and after integration of the physics inputoutput relations involved module to the loss function. We simulated 200 different bearing stiffness degradations, using their discrete monitored vibration signals to verify the effectiveness of the proposed method. We also investigate their inference process through feature heat map analysis to interpret how the models melt physics knowledge to assist in capturing the degradation trend. The physics knowledge considered in this paper is the dynamic relationship between vibration amplitude and stiffness in a damped forced vibration model. The results show that all three PITCN models effectively capture degradation-related trend information and perform better than the vanilla lightweight TCN. Furthermore, the visualization of the feature channels highlights the important role of physics information in model training. Channels containing physics information demonstrate higher correlation with results as they significantly dominate the heat map compared to other channels.

INTRODUCTION

Bearings are critical components that are susceptible to fatigue damage and need to be monitored. Under certain degraded working conditions, its crack propagation process is not obviously characterized in the early stage monitoring. [START_REF] Li | A new damage-mechanics-based model for rolling contact fatigue analysis of cylindrical roller bearing[END_REF]. As a result, the monitoring signal and the corresponding periodical statistics tend to show "slight trend" before severe degradation, and the dramatic feature variation only appears in signal collected from the end of services life [START_REF] Porotsky | Remaining useful life estimation for systems with non-trendability behaviour[END_REF]. Due to the incomplete knowledge of the bearing degradation mechanism, high cost in dynamics failure modeling [START_REF] Massi | Degradation of high loaded oscillating bearings: Numerical analysis and comparison with experimental observations[END_REF] and the sparsity of the degradation information, the classic solutions, neither the traditional physics-based methods nor the data-driven machine learning (ML) methods [START_REF] Shi | A dual-lstm framework combining change point detection and remaining useful life prediction[END_REF] are applicable to capture the information about the nonlinear degradation from past data and working conditions (H. [START_REF] Liu | Generalized cauchy degradation model with long-range dependence and maximum lyapunov exponent for remaining useful life[END_REF]. As a result, researchers have turned to the quest to develop hybrid approaches. On the one hand, guiding machine learning to explore the embedding properties of the data by imposing additional constraints during training has been shown to learn better representations of degradation trends [START_REF] Liao | Enhanced restricted boltzmann machine with prognosability regularization for prognostics and health assessment[END_REF]. On the other hand, the physics-informed Machine Learning (PIML) method using incomplete physics model design constraints can maintain the physics consistency of ML training results and improve vanilla ML model performance [START_REF] Karniadakis | Physics-informed machine learning[END_REF]. For this purpose, the incorporation of physics knowledge in the main parts of the ML pipeline, including the augmented input space (Q. [START_REF] Wang | Integrating expert knowledge with domain adaptation for unsupervised fault diagnosis[END_REF][START_REF] Chao | Fusing physics-based and deep learning models for prognostics[END_REF], the algorithm architecture [START_REF] Yucesan | A physics-informed neural network for wind turbine main bearing fatigue[END_REF][START_REF] Viana | Estimating model inadequacy in ordinary differential equations with physics-informed neural networks[END_REF] and the objective function (J. [START_REF] Wang | Physics guided neural network for machining tool wear prediction[END_REF], has received much attention in recent years in PHM. These researches prove that we can integrate physics knowledge directly in ML, especially the Deep neural networks [START_REF] Nascimento | Hybrid physics-informed neural networks for lithium-ion battery modeling and prognosis[END_REF] and in turn, ML can compensate for incomplete physics models [START_REF] Yucesan | A hybrid physicsinformed neural network for main bearing fatigue prognosis under grease quality variation[END_REF], thus establishing a mapping between structural parameters (causal factors) and degradation states (phenomena). However, in the face of such limited data, the performance of the PIML model is yet unknown. Moreover, many PIML improvements are primarily based on over-parametric Neural Networks [START_REF] Caixian | Halving the error rate requires more than 500 times the computational power[END_REF] that often have more parameters than the data points available for a single training batch [START_REF] Deepmind | Alphastar: Mastering the real-time strategy game starcraft ii[END_REF]. The large number of parameters can assist in expressing the complexity of the association between the casual factor and data to some extent, but may lead to over-fitting issue and becomes infeasible for real-time applications. Meanwhile, we still lack an intuitive sense of the mechanism of physics information in ML. An interesting question will be investigated and explored in this paper is whether the vanilla non-over-parametric lightweight model has the flexibility to embed the same knowledge in different ways to achieve better performance gains. This paper is organized as follows.Section 2 aims to present problem statement while in Section 3, we describe the simulation procedure for the stiffness deterioration of a bearing. In Section 4, three different methods for integration of physics knowledge in lightweight TCN are detailed. The performance of three proposed PI-TCN models as well as the physics knowledge's role in training process of these models are investigated in Section 5. Finally, conclusions and perspectives of this work are discussed in Section 6.

PROBLEM STATEMENT

Bearing damages start from inside. Until the initial crack extends to its surface, there are no obvious signs of failure that can be observed because the geometry of the rollers is not altered. After that, the crack accelerates and the bearing fails rapidly [START_REF] Khan | Vibration and acoustic method for detection of cracks in bearings: A critical review[END_REF]. It is the root of the slight trend data. Hence, one can cite the following challenges for prediction of the bearing's RUL based on vibration signals, illustrated by Fig. 1: 1. Throughout continuous stiffness degradation of bearings, the vibration signal varies insignificantly in the early phases but changes dramatically only near the failure time. As a result, it is not trivial to capture trend information from vibration signals which reflect the ongoing degradation evolution. 2. As the stiffness degradation is hidden, when investigating historical run-to-failure data we only know the linear function of RUL in working time and do not know the duration of bearing health state and degradation state. Then, it is not trivial to match the linear RUL values with hidden non-linear stiffness degradation process.

To address the above challenge, we propose a lightweight TCN as a benchmark purely data-driven model, using timedomain statistics as the input features to predict the bearing RUL in stiffness degradation. The input space includes mean, variance, max, min-max, root mean square, skew, kurtosis, peak factor, waveform factor, impulse factor and margin values. The output of the TCN is a deterministic value of the RUL (days). Then, we improve the performance of this benchmark model by integrating incomplete physics knowledge about the analytical relationship between stiffness degradation and vibration signal in terms of the augmented input space, modified hidden layer, and conflict loss function. These integration form three PITCN models: PI Features model (PIFM), PI Layers model (PILM), and PI Layer Based Loss models (PILLM). The relationship between stiffness and vibration amplitude is shown in Eq.1 [START_REF] Blake | Basic vibration theory[END_REF], where V ib p is the peak value of the vibration signal and stiff represents the corresponding equivalent contact stiffness level. ε denotes the relevant imbalance in the system load. It is the extrinsic excitation of the bearing vibration. m represents the equivalent system mass. Ω is the rotation speed. In real conditions, the exact values of ε and m are unknown. Only the parameters Ω and V ib p are available in vibration based RUL prediction.

V ib p = εΩ 2 stif f m -Ω 2 (1)
The main objective of the proposed PI-TCN models is to approximate the mapping function g between the features extracted from vibration signals and the bearing's RUL values.

RU L = g Ω 2 V ib p , ϵ, m (2)

CASE STUDY DESCRIPTION

This case study aims to predict the RUL of a roller bearing, subject to stiffness degradation caused by the effect of crack expansion of its rollers. We assume that this bearing operates at a constant speed. And the bearing's state is monitored via vibration signals. In subsection 3.1, we describe how to simulate continuously degraded stiffness curves while subsection 3.2 presents how to generate vibration signals.

Continuous stiffness degradation simulation

As shown in Fig. 2, we generate a total of 200 bearing stiffness run to failure degradation trajectories. The mean value of the failure times of those 200 trajectories is 8.04 × 10 5 and its standard deviation is 1.47×10 4 . Among these trajectories, 50 sets are randomly selected as the test sets, and the remaining 150 sets are randomly divided into training and validation sets in 4:1 ratio. Each stiffness degradation trajectory is composed of health state, nonlinear degradation period and uncertain factors associated to operating environment effects.

Figure 2. Simulation process of contact stiffness degradation.

We use the Eq.3 from (J. [START_REF] Liu | A new dynamic model for vibration analysis of a ball bearing due to a localized surface defect considering edge topographies[END_REF] to calculate stiffness in different damage states:

stif f = 1 2 (cosγ) 5/2 ) nskp 2/3 + U stif f (3)
We set contact angle γ to 20°and k p stands for the Hertz elastic contact stiffness between the ball and the smooth surface. We compute k p according to Eq.5. The term U stif f represents the stiffness model uncertainty. It is due to the complex environmental effects of the actual process, the simplifying assumptions of the model, and other factors that make the real value not strictly adhering to the physics model. We assume U stif f satisfy a Skewed distribution in which the mean and the variance are equal to 10% and 5% of stif f respectively. If the bearing is healthy, we calculate the stiffness via Eq.4:

stif f 0 = 1 2 (cos20) 5/2 ) 180kp 2/3 (4)
where n s is the number of contact surfaces. Duration of health condition days generated by random seeding rand (2,20). Bearing degradation tends to be severe when n s decreases. In this simulation, the initial value of n s0 is set to 180 while the duration, in which the bearing state is healthy, is generated by a random seed as shown in Eq.4. The coeffi-cient k p is computed with

k p = 4R 1/2 6( (1-ν 2 ) E ) (5)
We set bearing roller radius R to 0.003 m, Poisson's ratio ν to 0.3, Young's modulus E to 2.1×10 11 in the different simulations. We assume that the expansion of the bearing defect will result in a continuous reduction in the number of contact surfaces between the roller and raceway, with more areas changing from a face-to-face contact to a defective edge line-to-face contact, as shown in Fig. 3. This process can be simulated by reducing n s in Eq.6.

Figure 3. Defective contact and the roller failure schematic.

We generate an in-homogeneous degradation by the Eq.6. We set steps to 0.0001. We use Eq.1 to get vibration amplitude, where ε is 20 g.cm, m is 5 kg.

Generation of discrete vibration monitoring signals

The stiffness values are assumed to be monitored by vibration sensors whose measurements are recorded every six hours.

In this case study, the vibration amplitudes are generated by Eq.1 and then substituted into Eq.7 (if the bearing is in healthy state) or into Eq.8 (if the bearing is in degradation state) to obtain the vibration sequences. Shocks caused by defects are commonly accompanied by both frequency and amplitude noise. We add frequency noise (n d1 ) according to the Skewed distribution of (3, 1).With 3 as the mean and 1 as the variance, this indicates an uncertainty shock due to a roller defect introducing a high multiple of the rotational frequency during the bearing rotation. We also add amplitude noise (n h ) according to the signal-to-noise ratio of 1/100 in the healthy state and 1/10 in (n d2 ).

x(t) = vib p × sin( 2πΩ 60 t) + n h (7)

x(t) = vib p ×sin( 2πΩ 60

DESCRIPTION OF THE PROPOSED METHODS

Informed learning is the seamless incorporation of constraints in an ML pipeline. PIML means transforming physics knowledge as potential or direct constraints of ML. In PHM, knowledge is parametrically representable perceptions of system behavior and failure mechanisms. In this section, we will present the purely data driven-model, i.e., a lightweight TCN model to predict bearing RUL, and then propose three PITCN models based on the same knowledge to improve the prediction performance. The first one is the PI Feature model (PIFM), which guides the extraction of data features through parameter relationships in the physics formulation. The second one is the PI Layer model (PILM), which embeds the physics input-output model in the computational function of the layer. And finally, the PI Layer Based Loss model (PILLM) adds regularization terms associated with the output of the physics model to the loss function of the vanilla TCN.

Purely data-driven model

CNN allows parallel computation of outputs and thus can achieve better performance than RNNs in sequence modeling [START_REF] Lea | Temporal convolutional networks for action segmentation and detection[END_REF]. TCN, consisting of dilated, 1D convolution layers with the same input and output lengths, avoid common pitfalls of recursive models, such as gradient explosion or disappearance problems or lack of retention. We build lightweight TCN as "Benchmark" based with causal separable Conv1D, in Fig. 5. The total number of parameters in the model is 3,411. Note that the subsequent implementation of PI-TCN will compute the physics features related to stiffness in the hidden layer, the values of which have a high probability of putting the neurons in the saturation zone. Therefore, the activation function is chosen for the nonlinear function h-swish with no upper bound, lower bound, smooth, and non-monotonic characteristics.

Physics-informed feature augmented input space

Eq.1 inspires us that the factor Ω 2 /V ib p can be used as a physics feature to predict the RUL of the bearing due to the RUL's dependence on the stiffness degradation level. Among the original 11 time-domain statistical features, we removed the Max feature while physics-informed health indicator (PHI) Although Ω 2 /V ib p is not an exact stiffness estimation, it has a clear physical meaning for being self-compiled quantities as stiffness. It contains trend information. PIFL is essentially an extension of the series combination structure to create a hybrid model. The output based on the physics model is part of the TCN input. The PI layer adds potentially physically consistent weak constraints to the input space through physics model parameter relationships based feature extraction.

Physics embedded layer

Fault dynamics models can be converted into an input-output module in ML, and in turn ML compensates for model incompleteness, as shown in Fig. 7. Based on this paradigm, we have developed the PILM. Particularly, Eq.1 is transformed into a neural network linear model in Fig. 8. The unknown function g(•) in Eq.2 can be approximated using a custom layer function h(•) of the neural network in the structure presented by Fig. 9. This model allows extracting the PHI Ω 2 /V ib p by embedding the transformation layer of Eq.1. Then, the PHI is used as the input of the hidden custom layer whose structure is defined as an approximate function of stiffness degradation in Compared to the PIFM, PILM provides induction bias for TCN, as shown in Fig. 7. It is able to ensure that the computational process based on physics knowledge is forced during the data processing of TCN, thus completely embeds the physics knowledge into the computational paradigm and overall derivation process of the TCN model.

Physics-informed layer based conflict loss

We also introduce the physics inconsistency by designing the loss function according to conflict between physics model's output and ML output. The whole PILLM consists of two parts: branch network and main network. An output layer is added after the physics informed layer to provide the PHI in the branch network. These features participate in the training process of the main network at one hand, and influence the hyper-parameter optimization of the main network through the loss function on the other hand, as shown in Fig. 10. The outputs of each are measured with two losses and assigned corresponding weights of 1.0 and 0.2. In the prediction process, only the prediction results of the main network are utilized.

Figure 10. Build PI-loss based on different branches conflict.

In contrast to the first two methods, PILLM aims to intervene in the adaptive search process of the TCN in the solution space. Firstly, due to the single finite feature, it can be conjectured that the prediction accuracy of the PI Branch network is limited. resulting the high level loss, so the feature of minimizing the loss function in the training process of the TCN is used to improve the prediction accuracy of the Main network part. Secondly, by allowing the PHI of the PI branch network to participate in the training process of TCN, the two parts are permitted to share optimization process, thus allowing the TCN to satisfy a certain degree of physics consistency.

RESULT DISCUSSION

We fix the training epochs to 1000, design an early stop mechanism with patience equal to 80 epochs. We initialize the network parameters with uniformly drawn weights. The batch size of 128. The input-shape of each batch is (30,11). All models are trained in the same conditions. Moreover, we use Adam Optimizer in training [START_REF] Kingma | Adam: A method for stochastic optimization[END_REF].

Investigation of the PITCN models' performance

Fig. 11 presents different model's prediction results through 10 randomly selected trajectories of the test set while Fig. 12 shows the box plots of the differences between the predicted and the truth RUL on the overall test set. The results highlight the performance of the PIML models compare to the one of the purely data-driven model: the same physics knowledge has different incorporation possibilities and potential for improving the benchmark. Particularly, we find that the PILLM model has the best performance. Its error range is only [17.97, 15.65] while the ones of the BENCHMARK, PIFM and PILLM are respectively [-95.51, 79.83], [-26.85, 36.66], and [-33.38, 26.11]. Table 1 presents the performance of the proposed models compared with the benchmark on the overall test sets. We get the following conclusions from Fig. 11, and Fig. 12: 1. PI-TCN models show more accurate prediction with the smaller prediction error limits compared to Benchmark.

2. The predicted results of PI-TCN converge with the trend in the real value, showing the possibility to effectively solve the "Challenge II" in Fig. 1.

3. Among the three different PI-TCN models, PILLM has the best prediction stability with the most compact upper and lower error limits and the minimum error mean, as shown in Fig. 12.

Investigation of physics knowledge's role in ML

To investigate the role of physics knowledge in the training process of the proposed models, we build the channel information model to generate the channel heat-map. After training, we extract the layers in which physics knowledge is integrated to investigate the correlation between physics information and results. More concretely, for the PIFM, we choose the input layer as the channel information model while for the PILM and PILLM, we choose the input layer as input and the "Add" layer as output. The brightness of the colors in the heat map reflects the correlation. The obtained results are presented in Fig. 13, 15, and 16.

Figure 13. Weight heat-map of the input layer in PIFM.

In the PI Feature model's channel information heat-map, we find that the model focuses more on the channels where the Ω 2 /V ib p and Kurtosis are located. Kurtosis as a higher order statistic with the ability to capture dramatic trends from flat data. However, the Ω 2 /V ib p feature are assigned a higher weight than Kurtosis. This result highlight the intuition that the physics-informed feature generated based on analytic relational formulations provide additional crucial information to improve the model performance. Figure 15. Heat-map of the PILM "Add" Layer.

For the heat-map of PILM, it can be seen in Fig. 16, the features flowing from the PI Layer receive higher attention relative to the other channels.

Figure 16. Heat-map of the PILLM "Add" Layer.

For the PILLM's heat-map, as shown in Fig. 16 the part incorporated into the feature from the PI Loss is dominant compared to the other channels. In contrast to the feature map of the benchmark on the same layer, the feature map of the model is dimensionless, with a size of 16 x 15, because the features of PI Loss are not incorporated. The overall value of this feature map is smaller than the corresponding value of PI Loss. So this result highlights that physical knowledge plays a significant role in improvement of the prediction results.

CONCLUSIONS

In this paper, the physics knowledge about the relationship between vibration signals and stiffness degradation is exploited to create the physics-informed TCN models in three ways: augmented input space, physics equation embeded layer, and physics-informed conflict loss. The simulation result demonstrates the flexibility of methods incorporating physics knowledge and also highlight the significant improvements they can bring to vanilla TCN when working with "slight trend" data.

In comparison with the benchmark model, PIFM, PILM, and PILLM reduce the mean prediction error by 69.39% (from 2.81 days to 0.8685 days), 90.35% (from 2.81 days to 0.2710 days) and 96.29% (from 2.81 days to 0.1045 days), respec-tively. By investigating channel weights in the related layer, we found that those improvements mainly stem from the focus of three PIML models placed on data streams containing physical information. Indeed, the underlying logic of the PIML models is to guide ML to capture features related to degradation by encoding physics knowledge and then to establish the underlying relationships between features and RULs thanks of ML's non-linear mapping capabilities. In our case, the stronger the physics constraints imposed by the encoding in TCN, the better the PITCN model performs. In future works, we will use sparse noise monitoring data and conduct in-depth research on the translatability of incomplete physics knowledge containing uncertainty to ML pipeline. Furthermore, PIML methods will be developed in real scenarios with complex systems and complex operating conditions.

Figure 1 .

 1 Figure 1. Challenges of bearing's RUL prediction under stiffness degradation.

  Figure 3. Defective contact and the roller failure schematic.We generate an in-homogeneous degradation by the Eq.6. We set steps to 0.0001. We use Eq.1 to get vibration amplitude, where ε is 20 g.cm, m is 5 kg. Ω takes the value 4200 rpm to indicate the wear at constant speed and external load. With U deg denoting the uncertainty to reflect the non-uniform expansion of defects. U deg conforms a Skewed distribution in which the mean and the variance values equal to 10% and 5% of the n si respectively. n si = n s0 ×steps×i×(180-steps×i)+U deg , i ∈ N (6)

  t)+vib p ×sin( 2πΩ 60 t(1+n d1 ))+n d2 (8) Each vibration sample is generated according to 4096 Hz sampling frequency of 3 s time length. More than 700 vi-bration samples were collected for each trajectory, as shown in Fig.4. It can be seen that the generative data satisfy the non-trending characteristics.

Figure 4 .

 4 Figure 4. Illustration of the generated vibration signal.

Figure 5 .

 5 Figure 5. Lightweight TCN architecture diagram.

Figure 6 .

 6 Figure 6. Create PIFM based on physics analytic relationships.

Figure 7 .

 7 Figure 7. Embedding physics equations in NN layer.

Fig. 8 .

 8 In the training process, the unknown parameters ε, m and U stif f reflected by the weights (ω) and biases (b) of the hidden layer are updated to optimize the prediction results.

Figure 8 .

 8 Figure 8. Embedding physics equations in NN layer.

Figure 9 .

 9 Figure 9. PI Layer outputs join the Resblocks' training.

Figure 11 .

 11 Figure 11. Prediction results of different models.

Figure 12 .

 12 Figure 12. Model evaluation results.

Figure 14 .

 14 Figure 14. Heat-map of Benchmark model.

Table 1 .

 1 Performance evaluation on the test dataset.

		Benchmark PIFM PILM PILLM
	RMSE 16.191	8.059	5.818	3.157
	MAE	10.878	5.939	3.942	1.965
	R2	0.862	0.959	0.982	0.994

BIOGRAPHIES