Niels Boissonnet
email: niels.boissonnet@gmail.com.

Alexis Ghersengorin

Roland Bénabou

Yves Breitmoser

Franz Dietrich

Jean-Marc Tallon

Reactance: a Freedom-Based Theory of Choice *

Keywords: Niels Boissonnet, Alexis revealed preferences, freedom, reactance

published or not. The documents may come

"Prohibitions create the desire they were intended to cure" Lawrence Durell

Freedom is arguably an important reason for action that potentially conflicts with material welfare maximization. For instance, people that were reluctant to accept Covid 19 vaccination policy also suffered significant welfare costs, including the renunciation of previous occupations or leisure activities. Psychologists have long observed this propensity of freedom to counterbalance wellbeing motives-by means of natural and lab experiments-and developed the theory of psychological reactance to explain their findings. Despite the wide range of its applications, reactance has received little attention in economics.

A possible reason is that the existing theory neither yields a formal criterion to distinguish reactance from other motivations to deviate from welfare maximization, nor clarifies whether reactance lends itself to rigorous modeling. The main contribution of this article is an axiomatic foundation of a choice procedure that captures reactance and highlights trade-offs between welfare and freedom motives.

Reactance occurs when the decision makers (DM) reverse their choice to protest against the loss of what they perceive as part of their freedom. In a field experiment, [START_REF] Mazis | Elimination of phosphate detergents and psychological reactance[END_REF] consider how consumers reacted when Miami-Dade county decided to forbid phosphate use for laundry. Despite its strong environmental rationales, this decision raised significant protests as well as unexpected reactions. For the sake of the "American freedom", some consumers, that were not using phosphate-based detergent prior to the law, started buying it in neighbouring counties, smuggling it at extra cost and stockpiling the (now) precious product for the 20 years to come. 1Such a scenario violates the standard requirement of rationality, namely, the Weak Axiom of Revealed Preferences (WARP), and is incompatible with a straightforward utility representation. Indeed, denoting by x the phosphate detergent in a neighbouring county, y the same product in Miami, and z a phosphatefree detergent in Miami, the following choice is observed: z is chosen when the three options are available, i.e in the menu {x, y, z}, while x is chosen over z once y is removed, i.e in the menu {x, z}. When this occurs, we say that the choice of x protests against the loss of y. We interpret this as revealing that the DM perceives the loss of y as a threat to her freedom that can be voiced with the choice of x (Section 1.2). 2 This kind of WARP violation, however, is not exclusive to reactance and need not be interpreted this way. For instance, the analysis of the attraction effect by [START_REF] Ok | Revealed (p) reference theory[END_REF] is also based on similar choice patterns. The key challenge, then, is to provide choice-based axioms ensuring that an ascription of a choice reversal to reactance is valid and that distinguishes reactance from alternative psychological phenomena (Section 1.3).

Psychologists emphasize that reactance reflects an attempt to protest against the loss of concrete freedoms, that is freedoms to choose diverse types of option. 3 Reactance is therefore observed between options embodying similar freedoms: if the choice of x protests against the loss of y, then x and y are relevant to the same type of freedom. Importantly, types are not objectively observed but subjectively perceived by the DM. Hence they cannot be postulated a priori by an observer and must be revealed through the analysis. For instance, some DMs may perceive that buying environmentally harmful vehicles is relevant to protest against car environmental regulations, 4 but not against the phosphate ban studied by [START_REF] Mazis | Elimination of phosphate detergents and psychological reactance[END_REF], because cars and detergents are different types of product-their consumption relates to different freedoms. In contrast, some DMs may perceive it as relevant because they take the right to pollute as a relevant type of freedom. Hence, we do not presume which type of options are relevant freedom-wise for the DM. A contribution of the paper is to reveal which of them are, under the requirement that they partition the set of options.

We show that any DM making choices consistent with our reactance axioms can be represented by the following choice procedure that we call a reactance choice rule (Section 2.1). First, the DM sorts the options into types (forming a partition), where objects of each type share features that the DM subjectively perceives as relevant for her freedom. Implicitly, the DM will choose an option x to protest against the absence of y, only if x and y belong to the same DM: "For the theory [of reactance], freedoms are the creatures of subjective reality. If a person thinks he or she has the freedom to do X, and the ability to exercise this freedom, and perceives the conditions of the freedom's existence to be met, then this freedom exists for this person. It is theoretically irrelevant whether this freedom exists according to some more objective criteria." (Brehm and Brehm, 2013, p. 22 ) 3 "Contrary to some interpretations (e.g., Dowd, 1975), the freedoms addressed by the theory are not "abstract considerations," but concrete behavioral realities. If a person knows that he or she can do X (or think X, or believe X, or feel X), then X is a specific, behavioral freedom for that person." (Brehm and Brehm, 2013, p.12) 4 Here we are referring to the "rolling coal": in reaction to President Obama's environmental laws, some drivers modified their engine at significant costs in order to pollute more. See for instance https://www.nytimes.com/2016/09/05/business/energy-environment/rollingcoal-in-diesel-trucks-to-rebel-and-provoke.html type. Second, each type forms a set that is well ordered (i.e. WARP is satisfied) by a welfare criterion represented by a utility function u. Third, the DM determines a set that we interpret as a her freedom requirement. These options have in common that she would protest against their removal from her opportunity set, which can be interpreted as the set of options that she considers vital to satisfy her freedom demands. These options share an intuitive threshold property for each type (according to u), and thus the threshold for each type represents the DM's minimum welfare level to satisfy her freedom request. Choice is then made sequentially. First, the DM determines the best available options, according to u, for each type. Second, from these best options, she chooses the one with the highest valuation according to a criterion v that we interpret as reactance function. For, v(.) equates with u(.) for all options in the freedom requirement set, but v(.) > u(.) otherwise. We interpret this as v combining welfare and freedom motives; that is, the DM assigns a strictly positive freedom value only for options not in the freedom requirement set, which captures her propensity to protest against her lost freedom by choosing these options. We further study the uniqueness of our representation (Section 2.2). We show that the type partition is unique and that there exists a maximal freedom requirement set that includes all the others. We also find necessary and sufficient conditions for increasing transformations of u and v to preserve the reactance choice rule.

We study two social phenomena that, though often documented by the psychology literature, are not readily explained using existing (economic) models of choice (Section 3). First, reactance is often introduced as a possible determinant of the formation of conspiracy theories. To accommodate this phenomenon, we study how reactance impacts the DM's belief when she has to choose a biased source of information. By removing an unchosen moderately biased source, the DM might reverse his choice and choose a more biased source in the opposite direction. This can represent why, if a DM feels that some information is not accessible or hidden, he might end-up holding extreme belief or adhere to conspiracy theories. Our second application is related to integration policy. It explains why repressive policies towards immigrant minorities may generate backlash, as suggested by the empirical evidence. Additionally, it provides an argument for the evolutionary efficiency of reactance and its persistence in the long run.

In our final section 4, we ask the following question: how does a DM, whose choices can be explained by a reactance choice rule, would evaluate the freedom of choice offered by the distinct opportunity sets she is facing? Building on the series of papers by Pattanaik and Xu, we axiomatize a criterion to rank menus, which simply counts the number of types from which sufficiently good options (i.e in the freedom requirement set) are feasible (Section 4). We argue that this ordering integrates considerations about similarities between options (see [START_REF] Pattanaik | On diversity and freedom of choice[END_REF][START_REF] Nehring | A theory of diversity[END_REF] and the role of the preferences of the agent (see [START_REF] Pattanaik | On preference and freedom[END_REF]. Hence, it helps reconcile these two aspects, which have been studied separately, and it formally connects our choice rule to the existing literature on freedom of choice.

Related literature. The theory of reactance originated in [START_REF] Brehm | A theory of psychological reactance[END_REF]. Since then, a huge and vivid literature has spurred in psychology (see [START_REF] Brehm | Psychological reactance: A theory of freedom and control[END_REF][START_REF] Reynolds-Tylus | Psychological Reactance and Persuasive Health Communication: A Review of the Literature[END_REF]. In economics, few papers mention reactance, among which it is worth citing [START_REF] Arad | The People's Perspective on Libertarian-Paternalistic Policies[END_REF], who use this concept to explain results of their experiment about people's perspective on libertarian-paternalistic policies (the so-called nudges). Once aware of the implementation of the policy, some subjects backfire by willingly choosing the reverse of what they are nudged to, although they would have chosen it without the policy. Nonetheless, to our knowledge, no theoretical model of reactance has been proposed so far.

Our work contributes to investigating mechanisms that can cause menudependence. The first to argue that menu-independence can be theoretically counterintuitive is Sen in many works (see [START_REF] Sen | The formulation of rational choice[END_REF][START_REF] Sen | Maximization and the Act of Choice[END_REF] among others). Since then, many contributions have been proposed in the literature about conflicting motivations, among which [START_REF] Kalai | Rationalizing choice functions by multiple rationales[END_REF], [START_REF] Manzini | Sequentially rationalizable choice[END_REF], [START_REF] Cherepanov | Rationalization[END_REF], [START_REF] Bernheim | Choice-theoretic foundations for behavioral welfare economics[END_REF], [START_REF] Borie | Description-dependent Preferences[END_REF], [START_REF] Dietrich | Reason-based choice and context-dependence: An explanatory framework[END_REF][START_REF] Dietrich | Reason-based choice and context-dependence: An explanatory framework[END_REF][START_REF] Ridout | Choosing for the Right Reasons[END_REF] consider choices that can be explained (or justified) by different rationales that may depend on menus. [START_REF] Masatlioglu | Revealed attention[END_REF] and [START_REF] Lleras | When more is less: Limited consideration[END_REF] focus on the role of attention which can differ across menus. The contribution of [START_REF] Ok | Revealed (p) reference theory[END_REF]-henceforth OOR-is worth emphasizing, for they propose a theory of revealed referencedependent preferences that relies on similar choice irregularities as ours. The latter are however interpreted in the opposite direction: while they stress that choice reversals result from the addition of options (which then play the role of a reference point), we push forward the opposite interpretation that choice reversals result from the feeling of freedom threat induced by the deprivation of some options. Consequently, the axioms and the obtained representation are very different (see Section 1 for a more detailed comparison).

Finally, our model offers a choice theoretic counterpart to the normative literature on freedom of choice-see, among others, [START_REF] Sen | Freedom of Choice: Concept and Content[END_REF], [START_REF] Pattanaik | On Ranking Opportunity Sets in Terms of Freedom of Choice[END_REF][START_REF] Pattanaik | On preference and freedom[END_REF][START_REF] Pattanaik | On diversity and freedom of choice[END_REF], and [START_REF] Baujard | Conceptions of freedom and ranking opportunity sets. A typology[END_REF] for a survey. These works mainly studied the objective value of freedom, focusing on the opportunity aspect of freedom. While we similarly endorse this latter view, we rather study how the subjective perception of freedom may have an impact on agents' decisions. Following [START_REF] Sen | Rationality and Freedom[END_REF], we think that a theory of rational choice should encompass freedom of choice, "not merely because without [it], the idea of rational choice would be quite vacuous, but also because the concept of rationality must accommodate the diversity of reasons that may sensibly motivate choice" (p. 5).

REACTANCE-INDUCED CHOICES

Preliminaries

We work with a finite set of options X and denote by X = 2 X \∅ the collection of non-empty subsets of X. Elements of X stand for the menus faced by the DM, i.e., the set of options available to her, and will typically be denoted A, B, C, . . . A choice function 5 c : X -→ X associates to each menu the option chosen by the DM in this menu. Namely, for any menu A, c(A) ∈ A. 6 Let us stress that options are defined by objective features that can incorporate contextual properties-for instance, in our introductory example, we differentiated the phosphate laundry in a supermarket in Miami from the same product in a supermarket in a neighbouring county. Yet, we need not formalize these objective features, we only require the observer to be able to distinguish the different options. Some of these features may matter for the DM's subjective perception of freedom and thus will be revealed through the representation.

Revealed Reactance

Following [START_REF] Brehm | A theory of psychological reactance[END_REF], reactance is meaningful only when freedom conflicts with another motive (e.g welfare maximization). 7 We argue that reactance is 5 We focus on choice functions for the sake of simplicity: dealing with choice correspondences would add another layer of complexity that, we think, is not necessarily relevant in the present context. Nonetheless, we are confident that, with an additional axiom, our results would extend to choice correspondences. Such axiom is a weakening of Sen's α property :

If A ⊂ B and x, y ∈ C(B) ∩ A, then x ∈ C(A) ⇐⇒ y ∈ C(B).
6 For simplicity, if we enumerate a set {x 1 , . . . , x k }, we write c{x 1 , . . . , x k } instead of c({x 1 , . . . , x k }).

7 "Reactance is conceived to be a counterforce motivating the person to reassert or restore the threatened or eliminated freedom. It exists only in the context of other forces motivating the person to give up the freedom and comply with the threat or elimination." (Brehm and Brehm, 2013, p.37).

to be revealed through choice reversals that are inconsistent with the maximization of a single ordering, i.e that violate WARP. As was illustrated by our introductory phosphate example, such choice reversals are induced by the removal of an option that, when feasible, was not chosen. Namely, consider a triplet of options x, y, z and suppose that z is chosen from {x, y, z} but a reversal happens once y is no more available, i.e x = c{x, z}. When reactance is at work, this choice reversal results from the DM being concerned by the freedom loss he suffers when being denied the access to y. The choice of x over z is an effective way to protest against the loss of y.

Definition 1. Let c be a choice function on X and x, y ∈ X. We say that the choice of x protests against the loss of y, relative to c, if there exists z such that, z = c{x, y, z}, and x = c{x, z}. We denote it xR c y.

Such behaviors could well be explained by different motives. In particular, OOR base their definition of revealed reference on similar choice patterns. Their interpretation is however significantly different: they argue that z beats x only with the "help" of y. Hence, while they interpret these reversals as revealing a relationship between y and z, we interpret it as revealing a relationship between x and y. 8In addition, a definition of potential reactance is required. Assume we observe that (i) y is preferred to x, and (ii) for each t such that yR c t, we also observe that xR c t. In such a situation we might suspect x to be an effective choice to protest against the loss of y. Indeed, (i) states that y is preferred to x, while (ii) suggests that the choice of x is as efficient as the choice of y to protest against the DM's threatened freedom. Yet, it need not be the case that xR c y, for there might be no third option z that allows for revealing a reversal as stated by definition 1-i.e. no z is chosen in {x, y, z} while x is chosen in {x, z}. When this happens, we posit that the choice of x potentially protests against the loss of y.

Definition 2. Let c be a choice function on X and x, y ∈ X. We say that the choice of x potentially protests against the loss of y, relative to c, if y = c{x, y}, there exists t such that yR c t, and for any such option t, xR c t. We denote it xP c y.

The next section is dedicated to providing conditions based on these relations in order to make our interpretation meaningful and to restrict violations of WARP to reactance.

Reactance Choice Properties

Our first axiom relaxes WARP by observing that the DM should feel threatened in her freedom only when some options are made unavailable. Conversely, when her menu expands, the threats should disappear. The axiom states that if an option x is chosen in two menus A and B, then no freedom concern could justify a choice reversal when these menus are gathered, i.e in A∪B. Otherwise, the reversal from A ∪ B to A would be triggered by the loss of an option from B \ A, which would prevent x from being chosen in B. This leads to our first axiom.9

EXPANSION (Exp).

For any

x ∈ X, A, B ∈ X , if x = c(A) = c(B), then x = c(A ∪ B).
Note that if the choice of x protests against the loss of y, then Exp implies that y = c{x, y}. That is, for reactance to be meaningful, it must trigger a choice of an even "'worse" option than the one that is no more available, or equivalently, a sacrifice of welfare, as otherwise, reactance would hardly express a discontent. Furthermore, if z plays the same role as in definition 1, then Exp also implies that z = c{y, z}. Therefore, a typical pattern of reactance is revealed through a binary choice cycle-this is forbidden by OOR's axiom No Cycle.

We now posit axioms on the binary relations R c , P c . Consider our introductory example. Assume that, when only an expensive phosphate-free detergent is available in her county, the DM reacts to the prohibition by going to the neighbouring county to get some phosphate detergent, while she stays in her own county when a cheap phosphate-free detergent is available. This reveals that "buying phosphate detergent in the neighbouring county" protests against the loss of "buying phosphate detergent in Miami supermaket", though it is revealed only when the price of the phosphate-free detergent is high. Assume also that, while she prefers not to transgress the law when she can buy phosphate in the neighbouring county, the DM finally decides to go on the black market when the latter is forbidden, and that she does so whatever the price of the available phosphate-free detergent may be. This reveals that "buying phosphate detergent on the black market" protests against the deprivation of "buying phosphate detergent in the neighbouring county". We would like to conclude from these two assumptions that the DM goes to the black market because she feels even more threatened in her freedom now that the detergent is prohibited in both counties. This necessarily implies that "buying phosphate detergent on the black market" also protests against the loss of "buying phosphate detergent in Miami supermakets". Hence our first axiom requires R c to be transitive-note that P c is transitive by definition. Because R c and P c are typically incomplete, we also require their negative transitivity.

REACTANCE TRANSITIVITY (R-Tran).

For any x, y, z ∈ X, (i) if xR c y and yR c z, then xR c z,

(ii) let y = c{x, y}, z = c{y, z} = c{x, z}: if ¬[xR c y] and ¬[yR c z], then ¬[xR c z]; if ¬[xP c y] and ¬[yP c z], then ¬[xP c z].
Remark: let us stress that R-Tran imposes also transitivity in the similarity between options, that is, it prevents the following situation: x is sufficiently close to y and xR c y, y is sufficiently close to z and yR c z, but x and z are too different to consider the possibility of protesting against the loss of z by choosing x.

Consider an option y that is never chosen to protest against any lost opportunity, but whose absence triggers the DM to protest by choosing some option x-i.e xR c y. This means that as long as the DM has access to y, she never reacts to some limitation of her freedom of choice by choosing y to protest against it. At the same time, when y is no more available, she is ready to protest by choosing x. Our interpretation of such an option is that it satisfies the DM's freedom requirement, or alternatively, the choice of y is never motivated by freedom. Consider a third option z that is chosen over y-i.e z = c{y, z}-and such that also xR c z, then z should satisfy the DM's freedom requirement at least as well as y does. Our third axiom imposes two conditions in that direction. First, any option whose choice protests against the loss of z must be effective in protesting against the loss of y. Conversely, any option whose choice protests against the loss of y might not be good enough to protest against the loss of z, but at least z must be chosen over this option.

REACTANCE CONSISTENCY (R-Con).

For any x, y, z ∈ X, if xR c y, xR c z, z = c{y, z}, and there exists no t such that yR c t, then for any u ∈ X:

(i) uR c z =⇒ uR c y; (ii) uR c y =⇒ z = c{u, z}.
Remark: point (ii) can alternatively be seen as requiring that uR c y cannot be revealed through the choice with z, hence z = c{u, z}, which is consistent with the interpretation that y and z satisfy the same freedom requirements.

To motivate our last axiom, we extend the phosphate example. Suppose that "buying phosphate on the black market" (x) and "buying phosphate in a neighbouring county" (z) both protests against the loss of "buying phosphate in Miami supermakets" (t). Add the third option "buying phosphate in a further county" (y): quite naturally, z is chosen over y, and assume further that y is chosen over x. Suppose that the DM considers going on the black market as a means to protest against the prohibition in a further county, that is, xR c y. Said differently, the DM's concern for freedom is greater when x is the only phosphate detergent available than when y is. Because the choice of both x and z protests against the loss of a common option t, then one would expect that similarly the DM's concerns for freedom be reinforced when only y is available as compared to the same situation with z available. Hence our third axiom requires that the choice of y potentially protests against the loss of z. The second point says that if in addition xP c z, that is, whenever the DM considers going in a neighbouring county as a way to protest against a threat to her freedom, she would also consider the choice of going on the black market as a means to protest if necessary, the same conclusion, that is, yP c z, should follow even if we only observe xP c y and not necessarily xR c y.

REACTANCE MONOTONICITY (R-Mon).

For any x, y, z ∈ X, such that z = c{y, z}, y = c{x, y}:

(i) if xR c t and zR c t for some t ∈ X, then [xR c y =⇒ yP c z]; (ii) if xP c z, then [xP c y =⇒ yP c z].

REPRESENTATION

Reactance Choice Rule

We now state and discuss our main result: a representation for choice functions that satisfy the four axioms presented above. To state it formally, let us denote

T c 0 = {x ∈ X | t, xR c t ∨ tR c
x} the set of options for which reactance never arises. Omitted proofs can be found in the appendices.

Theorem 1. A choice function c satisfies Exp, R-Tran, R-Con and R-Mon if and only if there exist a freedom requirement set F ⊆ X, a partition T of the options into types such that T c 0 ∈ T , a utility function u : X -→ R, and a reactance function v : X -→ R such that:

(i) for any T ∈ T and any x, y ∈ T , if x ∈ F and u(x) < u(y), then y ∈ F ;

(ii) • v(x) > u(x) for all x / ∈ F ; • v(x) = u(x) for all x ∈ F ; (iii) for any T ∈ T , v • u -1 is single-peaked on u(T \ F );
(iv) for any menu A,10

c(A) = arg max x∈d(A) v(x), (1)
where:

d(A) = T ∈T arg max x∈T ∩A u(x).
We name a choice function that satisfy our four axioms a reactance choice rule (RCR). When the conditions (i)-(iv) of Theorem 1 are satisfied, we say that the ordered tuple < T , F, u, v > is a reactance structure that represents c.

Our interpretation of Theorem 1 is the following. The DM evaluates options either through an instrumental (or 'welfarist') preference, represented by the utility function u, or through the lens of a reactance function v, that combines welfare and freedom motives and thus exceeds (weakly) the former. Options are partitioned into classes of similar types. Each type comprises options that provide comparable consumption experience but with different levels of welfare, or satisfaction-i.e they are ranked by u. Hence, it reveals a specific freedom. One clear example of this is when these are actually the same good but obtained or consumed through different channels. For instance, buying phosphate laundry in a supermarket is less costly than getting it from the black market, but both goods are of similar type. Within each type, welfare maximization is the only motive for choice. Therefore, the binary comparisons are transitive-and thus represented by u-and freedom concerns cannot yield reactance cycles.

The freedom requirement set consists of options that meet the DM's freedom requirements; that is, for any T , F ∩ T captures her demand regarding this specific freedom. Point (i) and (ii) express how these requirements depend on welfare. Point (i) says that if an option x meets a freedom requirement of the DM (i.e x ∈ T ∩ F ), so do options of the same type that increase the DM's welfare (i.e any y ∈ T such that u(y) > u(x)). Hence, when all options in F ∩ T are unavailable, the DM deems that she does not have access to a sufficiently good option regarding the freedom associated to T . Since, in this case, the DM feels threatened in her freedom, point (ii) says that she must be even more willing to choose options of this type; that is, v(x) > u(x) for x / ∈ F . Conversely, when an option in F ∩ T is available, the DM does not feel threatened regarding the freedom embodied by T . Hence, freedom is not a reason for choosing these options, which is captured by the fact that u(.) = v(.). In this sense, x T,F ≡ arg min x∈T ∩F u(x) provides the minimal welfare requirement of the type of freedom associated to T .

Choices represented by (1) happen sequentially: within a menu, only the best options (according to u) from each type are retained-this is the set d(A)and are then compared according to v.11 The reasoning process can actually be decomposed into two maximization stages separated by an interim stage in which the DM evaluates for each type the threat to her freedom. More precisely, at the interim stage, if the best available option from a type T is worse than x T,F , then freedom becomes an additional reason to choose this option. Hence, while welfare alone is relevant to describe the first maximization stage, the second one is also driven by the freedom concerns introduced at the interim stage.

Finally, point (iii) imposes a specific shape of the reactance function v with respect to the utility function u. Specifically, if x, y, z are in type T but not in F and u(x) > u(y) > u(z), it prevents that v(y) = min t∈{x,y,z} v(t). It reflects the DM's increasing willingness to react-up to a certain point-as the limitations on their liberty is tightened. In our representation, this indicates that the less welfare the DM can obtain from a type of options, the more she is willing to restore her freedom and, thus, the more reactance is at work. Of course, this is true up to a certain point where welfare motives might weigh more in the trade-off between welfare and freedom-i.e you reach a point where you are no more willing to sacrifice welfare in the name of your liberty.

Figure 1 In the light of the representation, it is worth noting that the absence of preference reversal (or binary cycles) is to be interpreted as a lack of traceable reactance. This does not necessarily mean that the DM has no concern for freedom related to her opportunity, i.e. to the extent of her menu. Rather, this means that her freedom concerns (if any) are either too weak or, equivalently, too aligned with her welfare to be identified as a force counterbalancing welfare.

Uniqueness of Reactance Choice Rules

This section elaborates on the extent to which the ingredients of a reactance structure may be uniquely identified. We first provide a uniqueness result regarding the types and the freedom requirement set. It states that the collection of types is unique and that the union of two freedom requirement sets can also rationalize the DM's choices.

Proposition 1. If c is an RCR represented both by < T , F, u, v > and < T , F , ũ, ṽ >, then T = T and there exist û, v such that < T , F ∪ F , û, v > also represents c. 12We say that a reactance structure < T , F, u, v > is maximal if the freedom requirement set F is maximal according to the inclusion relation ⊆, that is, if for any other reactance structure < T , F , ũ, ṽ > that represents the same choice function, F ⊆ F . Proposition 1 ensures that for any RCR c, there exists an unique maximal reactance structure that represents it. The next proposition shows that the main characteristic of this structure is that the relation R c depends, for options in type T , on whether they provide more welfare than x T,F . Indeed, suppose that the option just below x T,F in T according to u-denote it x-does not protest against the loss of x T,F . Then, the choice of x does not protest against the loss of any option in T better than x T,F , and hence any option at all. 13 This means that the presence of x in T comes from the fact that there exists y such that yR c x. Namely, while x is never an effective way to protest against a lost freedom, the DM feels threatened when he is denied the feasibility of x. Hence it would seem natural that x is included in the freedom requirement set. This turns out to be necessarily true for the maximal freedom requirement set.

Proposition 2. Let < T , F, u, v > be a reactance structure that represents an RCR c and define for any

T = T c 0 , x T,F ≡ arg min x∈T ∩F u(x). < T , F, u, v > is maximal if and only if T c 0 ⊂ F , and for any T = T c 0 , x ∈ T : (i) if u(x) < u(x T,F ), then xR c x T,F ;
(ii) if u(x) > u(x T,F ), then there exists y such that yR c x and yR c x T,F .

The maximal freedom requirement set is therefore the relevant one to evaluate the DM's freedom demands and the thresholds x T,F s can then appropriately be interpreted as the minimal welfare requirements to satisfy these demands.

We now turn to a discussion about the uniqueness of the functions u and v. Let c be an RCR represented by the reactance structure < T , F, u, v >. What are the joint conditions on functions ũ, ṽ that ensures that < T , F, ũ, ṽ > also represents c? One obvious sufficient condition is if there exists an increasing function f : R -→ R such that ũ = f • u and ṽ = f • v. Now let suppose that there exist two functions f, g : R -→ R such that ũ = f • u and ṽ = g • v.

One clear necessary condition is that f |u(F ) = g |u(F ) and that both functions be increasing on u(F ). Because u is not directly used as a choice rule on options that are not in F , it is not necessary that f be increasing on u(X). Yet, it represents choices within types, which implies that f is increasing on u(T ) for every T . Because v is ultimately the function through which choices are made, one might be tempted to say that g must be increasing on v(X). This is however not exact because within types, the function v is never used to make choices. This problem does not arise as long as we impose one additional condition regarding the reactance function on certain pairs of options of similar types. This is captured by the following definition. Definition 3. A reactance structure < T , F, u, v > is a reactance structure if for any T ∈ T and any x, y ∈ T , such that {x, y} ⊆ F , c{x, y} = x and for all z ∈ T , c{x, z} = z ⇐⇒ c{y, z} = y, we have v(y) > v(x).

The next proposition states first that reactance structure exists and second that if we restrict ourselves to reactance structure , the conditions regarding the utility and the reactance functions stated above are not only sufficient, but also necessary.

Proposition 3. Let c be an RCR.

(i) There exists a reactance structure < T , F, u, v > that represents it.

(ii) Furthermore, let f, g : R -→ R be two real-mappings, < T , F, f • u, g • v > also represents c if and only if f is increasing on u(T ) for every T ∈ T , g is increasing on v(X) and f |u(F ) = g |u(F ) .

Behavioral Properties of Reactance Choice Rules

The relation between our behavioral definition of reactance and our representation is a legitimate concern. According to the sequential reasoning of an RCR, options are first compared according to u within types, and then according to v across types. Because u and v do not rank options in X similarly, choice reversals may happen when removing options in a type that are better ranked according to u but worse according to v. Point (i) of the next proposition shows that these are are necessary conditions. Conversely, they might not be sufficient. Hence point (ii) states conditions under which these are sufficient conditions for the choice of x to potentially protest against the loss of y.

Proposition 4. Let c be an RCR represented by the reactance structure S =< T , F, u, v >.

For any x, y ∈ X:

(i) if xR c y, then there exists T ∈ T such that x, y ∈ T , x / ∈ F , u(x) < u(y) and v(x) > v(y);

(ii) if S is maximal and there exists T ∈ T such that x, y ∈ T \ F , u(x) < u(y) and v(x) > v(y), then xP c y.

Proof. (i) Let c be an RCR. Consider x, y, z such that z = c{x, y, z} and x = c{x, z}, so xR c y. One can easily check that it is not possible that x, y, z are either all in the same type, or all in different types. Hence exactly two among them must be of the same type: denote it T . To allow such a choice behavior, this means that either the valuation of z changes from {x, y, z} to {x, z}, or the valuation of x changes. But given that z is chosen in the bigger menu, this means that it was the best, according to u, among the options of the same type in this menu, which is necessarily true also in a smaller menu. Hence it is the valuation of x that increases from {x, y, z} to {x, z}, meaning that it is evaluated according to u in the former and according to v in the latter. This is only possible x, y ∈ T and u(x) < u(y). Furthermore, the choices imply that v(x) > v(z) > v(y) ≥ u(y) > u(x), from which we conclude that x / ∈ F and v(x) > v(y).

(ii) Because S is maximal, x, y ∈ T \F implies that both xR c x T,F and yR c x T,F . Because u(x) < u(y), y = c{x, y}, and at the same time for any z ∈ T such that yR c z, u(z) > u(y) > u(x) and there exists t such that v(y) > v(t) > v(z), hence v(x) > v(t) and xR c z. Therefore, xP c y.

APPLICATIONS

Conspiracy Theories

As [START_REF] Sensenig | Attitude change from an implied threat to attitudinal freedom[END_REF] suggest, reactance has its counterpart in the realm of beliefs, namely the boomerang effect for psychologists [START_REF] Hovland | Communication and persuasion[END_REF] or the backfire effect for political scientists [START_REF] Nyhan | When corrections fail: The persistence of political misperceptions[END_REF][START_REF] Wood | The elusive backfire effect: Mass attitudes' steadfast factual adherence[END_REF]. 14 In the wake of Covid 19 pandemics, scholars argued such effects to be closely related to the formation of conspiracy theories and extreme beliefs [START_REF] Adiwena | Psychological Reactance and Beliefs in Conspiracy Theories During the Covid-19 Pandemic: Overview of the Extended Parallel Process Model (EPPM)[END_REF]. 15 We propose to accommodate this mechanism by adapting Che and Mierendorff (2019)'s single period model of attention allocation with reactance.

A DM must choose from two actions, l or r, whose payoffs depend on an unknown state i ∈ {L, R}. His prior belief that the state is R is denoted p and we assume that p ∈ (0, 1/2]. Before choosing his action, the DM acquires information. To that purpose, he can allocate his attention across four sources of information (e.g newspapers). Two of them are L-biased and the two others are R-biased.

The sources are represented by statistical experiments, or signals. The Lbiased ones, denoted σ LL and σ L , can only reveal the state R. Symmetrically, the R-biased ones, denoted σ RR and σ R , can only reveal the state L. For i = L, R, σ ii is an extreme source, whereas σ i is a moderate one, i.e the former is more biased than the latter. Formally, σ i sends signal s i with probability 1 in state i and with probability 1 -λ in state -i, and σ ii sends signal s i with probability 1 in state i and with probability 1 -δ in state -i. We assume that 3/4 > λ > δ = 1/2. The experiments induced by the moderate sources σ L and σ R are described in table 1. The signals σ LL and σ RR are obtained by replacing λ with δ.

σ L State/signal s L s R L 1 0 R 1 -λ λ σ R State/signal s L s R L λ 1 -λ R 0 1
Table 1: Experiments induced by the moderate sources.

Initially the DM faces the complete menu M = {σ LL , σ L , σ R , σ RR }. In terms of our representation, the set of L-biased sources and the one of R-biased sources each represent a type of options. For i = L, R, σ i is strictly more Blackwell informative than σ ii , therefore the DM will never choose any of the extreme sources when his opportunity set is M , that is: d(M ) = {σ L , σ R }. The DM's demands from freedom are satisfied when the moderate sources are available, that is, his freedom requirement set is F = {σ L , σ R }. When facing the menu M , the DM foresees that his payoff from choosing action a ∈ {l, r} in state i ∈ {L, R} is u i a where :

u R r = u L l = 1, u R l = u L r = -1.
Hence the DM will prefer action r if and only if his posterior belief is greater than 1/2. One can show that the DM's optimal allocation of attention is to choose the "own-biased news source", namely the signal biased toward one's prior: in our case σ L given that p ≤ 1/2. The rationale for this is the following. The prior indicates the action l as the optimal one. Hence, a breakthrough signal s R from σ L is more valuable than a breakthrough signal s L from σ R . And the biased signal s L from σ L is more aligned with the DM's prior belief than s R from σ R . Hence, he is better off allocating his attention to σ L (see [START_REF] Che | Optimal Dynamic Allocation of Attention[END_REF]Mierendorff, 2019, pp. 2999-3000, for the complete argument).

In the next period, the moderate R-biased source σ R is no more available, either because the government actually banned this newspaper or simply be-cause the DM perceives that this source is no more existing: only L-biased or extremely R-biased ones are present. The DM now faces the menu N = {σ LL , σ L , σ RR }. He interprets this removal as revealing that the disutility from making a mistake in state L-i.e choosing action r-is lower than expected: he now foresees it to be v L r = 0. σ RR is no more removed from consideration by σ R , hence d(N ) = {σ L , σ RR }. His anticipated utility from choosing σ L is unchanged while the one attached to σ RR is v(σ RR ) = p + (1 -p)δ (for p sufficiently close to 1/2 such that after signal s R from σ RR , the DM chooses action r).

As a consequence, some DMs with prior beliefs sufficiently close to 1/2, who would have chosen news source σ L in menu M , will choose the extreme source σ RR in menu N and their default option becomes r.

Proposition 5. There exists p < 1/2 such that if p ∈ [p , 1/2]:

(i) The DM prefers σ RR to σ L in menu N ;

(ii) After a realisation of signal s R from σ RR , the DM chooses action r. This is in strong opposition as what would be obtained without reactance. Indeed, if the DM does not modify his anticipated utility when the menu shrinks, by removing σ R , some DMs with prior belief strictly higher than 1/2 would now choose the source σ L instead and action l after a signal s L .

Integration Policy Backlash

Can forced assimilation policy foster the integration of immigrants communities? While [START_REF] Alesina | Nation building[END_REF]'s theory of nation building assumes that repressing the cultural practices of minorities spurs homogeneity, [START_REF] Bisin | The economics of cultural transmission and the dynamics of preferences[END_REF] suggest that the success of such policy may be mitigated by an increasing effort of parents to influence their children's cultural trait. In this application we show that, with reactance, one can even predict this policy to yield a backlash effect: the repressed immigrants react to repression by becoming more prompt to self-isolation. This, in addition, provides a rational to the persistence of reactance as an evolutionary efficient behavior.

Such a backslash effect has been recently documented by several papers. Some evidence suggests that the "burkha ban" in France in 2004 has strengthened the religious identity of French-Muslims [START_REF] Abdelgadir | Political Secularism and Muslim Integration in the West: Assessing the Effects of the French Headscarf Ban[END_REF]. [START_REF] Fouka | Backlash: The unintended effects of language prohibition in US schools after World War I[END_REF] shows that, in states which prohibited German Schools in the aftermath of World War I, German-Americans "were less likely to volunteer in World War II and more likely to marry within their ethnic group and to choose decidedly German names for their offspring".

To show how this backlash operates, we complement [START_REF] Bisin | The economics of cultural transmission and the dynamics of preferences[END_REF]'s account of cultural transmission with a reactance mechanism: as the repression increases, parents' educational freedom decreases and, reacting to this repression, they may endeavour to influence their children even more. 16 There are two cultural traits {m, M }-for minority and Majority. The proportion of the minority q is assumed to be positive but lower than 1/2. Each generation is composed of parents who have only one child. Intergenerational transmission results from two socialization mechanisms. First, by vertical socialization the parents may directly transmit their cultural trait i with probability d i . If, with probability 1 -d i , vertical socialization fails, then horizontal transmission occurs and the child adopts the traits of a random individual in society. Hence, the probability that a child from the minority be socialized by her parent's trait is:

P (d i ) ≡ d i + (1 -d i )q. (2)
As [START_REF] Bisin | The economics of cultural transmission and the dynamics of preferences[END_REF], we argue that parents endeavour to influence their child. They have a unit of time to allocate between their effort to fix d iwhich costs (d i ) β unit of time, with β > 1-and a leisure activity t i ∈ [0, 1], whose cost and utility are t i . In addition, the government can implement a repressive policy g i ≥ 1 that may increase the parents' cost of influencing their child: a pair (t i , d i ) costs t i + (d i ) β g i units of time for the parents. We posit that parents get a utility of 0 when their child is socialized to the other trait, while they get a utility V (g i ) when she is socialized to their own trait. Hence, their expected utility of their child's socialization is P (d i )V (g i ). This means that, given a repressive policy, parents choose options (t i , d i ) ∈ [0, 1] 2 from the menu

K g i ≡ {(t i , d i ) : t i + (d i ) β g i ≤ 1}, to maximize t i + P (d i )V (g i ), (3)
In what follows, we also assume that V has the following shape:

V (g) = V if ĝ ≥ g V g λ ĝ ĝ < g
For some ĝ > 1 with λ > 1 and V > 1. Hence, after a threshold ĝ, the more repressive is the policy g, the greater is V (g). The interpretation is that parents react to the repressive policy when they feel that their freedom to educate their child is threatened. In other words, more repression may create incentives to dedicate more resources to transmit their traits to their children. Note that λ represents some kind of reactance rate since as it increases, parents' willingness to influence their child also increases.

From the first order condition, we obtain that the unique equilibrium educational effort-the program 3 being concave-must satisfy:

d i (g i , q) = 1 -q β V (g i ) g i 1 β-1 (4)
Given the shape of V , d strictly decreases with g on (1, ĝ) and strictly increases with g on (ĝ, +∞). In other words, when the repressive policy exceeds ĝ, the more repression, the more parents invest in having their child socialized to their own trait. This suggests that reactance is at work in this model. In the following lemma, we establish the precise connection between this model and our reactance framework.

Lemma 1. The function C defined on {K g } g≥1 , such that for all g

C(K g ) = {(t, d) ∈ K g : (t, d) solves (3)}.
is a well-defined choice function and there exists an RCR C defined on all compact subsets of [0, 1] 2 such that C(K g ) = C (K g ) for all g ≥ 1. 17

Assuming the repressive policy to solely concern the minority (i.e. g M = 1), what does reactance imply for the population dynamics in this model? Let time τ ∈ [0, +∞) be continuous and q τ be the share of the population with the minority cultural trait at time τ . Then, we have 18

q = q(1 -q) d m g m , q -d M 1, 1 -q .
Given (4), d satisfies the cultural substitution property. 19 This implies that q 17 For convenience, we construct a reactance structure on this infinite collection of compact sets. Obviously, analogous results could be obtained by making the set of possible policies g and the menus K g finite. 18 See Bisin and Verdier (2001, equation (3), footnote 9) for discussions about this differential equation. 19 In Bisin and Verdier (2001, Definition 1), this property states that d is continuous, decreasing with q, and d = 0 when q = 1. converges to some q ∈ (0, 1), which satisfies d m g m , q = d M 1, 1 -q (see [START_REF] Bisin | The economics of cultural transmission and the dynamics of preferences[END_REF], Proposition 1). Hence,

q (g m ) = V (g m )/g m V (1) + V (g m )/g m (5)
Given that V (g)/g increases with g when g ∈ (ĝ, +∞) this means that repressive policy increases the size of the minority. This prediction contrasts with Alesina and Reich (2015)'s suggestions.

Noting that reactance is presumably a characteristic cultural trait [START_REF] Jonas | Culture, self, and the emergence of reactance: Is there a "universal" freedom?[END_REF], this model also provides a rational for why reactance can be evolutionary efficient. Minorities which are more prompt to exhibit reactance are more likely survive to repressive attempts to hinder their cultural practices.

To make precise this comparative statics statement, consider two minorities: one with a high reactance rate λ H and one with a low reactance rate λ L < λ H . Denoting by q L (.) and q H (.) the equilibrium population share for these two minorities, the following proposition establishes that q is always higher for the high-reactance minority. Proposition 6. For all g > ĝ, q H (g) > q L (g).

RANKING OF OPPORTUNITY SETS

The freedom of choice literature has mainly tackled the issue of valuing freedom through the ranking of opportunity sets. Starting with [START_REF] Jones | Evaluating choice[END_REF] and [START_REF] Pattanaik | On Ranking Opportunity Sets in Terms of Freedom of Choice[END_REF], a wide diversity of measures have been characterized (see [START_REF] Baujard | Conceptions of freedom and ranking opportunity sets. A typology[END_REF] for a survey of this literature). Two dimensions have been pointed out as relevant to the agents' valuation of their freedom: their (potential) preferences over options (see Pattanaik andXu (1998)henceforth PX98-and[START_REF] Sen | Markets and Freedoms: Achievements and Limitations of the Market Mechanism in Promoting Individual Freedoms[END_REF]) and the similarity between different options (see [START_REF] Pattanaik | On diversity and freedom of choice[END_REF]-henceforth PX00-and [START_REF] Nehring | A theory of diversity[END_REF]). We propose a ranking of opportunity sets based on reactance structures that arguably offers novel insights and helps reconcile these two aspects.

According to a (maximal) reactance structure, this is through the interaction between the types and the freedom requirement set that freedom concerns impact choices. Hence it suggests that these two channels should impact the DM's assessment of freedom offered by a given menu. The types represent classes of similar options, 20 suggesting that adding options of a similar type should not increase the DM's freedom of choice. At the same time, the maximal freedom requirement set represents the DM's freedom demands. Hence, it seems natural that adding options increases the DM's valuation of freedom only if it gives access to items of a type from which no options in the freedom requirement set were available.

We characterize with two axioms a rule that reflects these arguments. As before, we denote X a finite set of options and X := 2 X \ ∅ the collection of menus of options in X. Let < T , F, u, v > be a maximal reactance structure defined on X and is a complete and transitive binary relation defined on X .

To state our two axioms, we need to introduce the following definition. A menu A is richer than a menu B if for any T ∈ T , if T ∩ F ∩ A = ∅, then T ∩F ∩B = ∅. So A is richer than B means that any type whose no element from F is available in A must also have no feasible options from F in B. Furthermore we say that A is strictly richer than B if A is richer than B but the reverse is not true.

Our first axiom, R-DOMINANCE says that (strictly) richer sets are always (strictly) preferred and imposes that it is an equivalence for singletons.

R-DOMINANCE.

(i) For any A, B ∈ X : A richer than B =⇒ A B, with a strict preference if A strictly richer;

(ii) For any x, y ∈ X: {x} {y} =⇒ {x} strictly richer than {y}.

Note that part (i) of the axiom implies monotonicity in the sense of [START_REF] Kreps | A representation theorem for "preference for flexibility[END_REF], i.e: for any A, B ∈ X , A ⊇ B =⇒ A B. Indeed, in this case, A is richer than B. Part (ii) is an adaptation of [START_REF] Pattanaik | On Ranking Opportunity Sets in Terms of Freedom of Choice[END_REF]'s Indifference Between no Choice Situations, which simply imposes an indifference between every singleton. They argue that singletons do not offer any freedom of choice, hence they cannot be strictly ranked. This is still true in our case, except if only one the two options is in F , which is exactly what is implied by (ii).

Our second axiom, R-COMPOSITION, is an adaptation of the composition axioms used in Pattanaik and Xu's series of papers. Combining menus that do not overlap should preserve the ranking. This is however true only if combining really provides additional freedom, which is captured by the requirement that A is not richer that C (see PX00 for a complete discussion of their axiom).

R-COMPOSITION.

For any menu A, we define Φ(A) = {T (x) ∩ F ∩ A|x ∈ A}, the collection of subsets of A containing every option of one type from F that is available in A.

Theorem 2. satisfies R-DOMINANCE and R-COMPOSITION iff for any menu A and B:

A B ⇐⇒ #Φ(A) ≥ #Φ(B). ( 6
)
The interpretation is the following: what matters for the DM is to have access to more options, but only dissimilar objects-as captured by the distinct types-are valued. On top of that, within a certain class of similar options, the DM demands a minimal level of satisfaction to meet her freedom requirements, which is captured by the freedom requirement set F . This measure is close to PX00's one. In addition to their representation, there is a role for preferences in this evaluation that is captured through the set F . Although PX98 also incorporate preferences, let us stress the key difference. Their starting point is a collection of possible preferences (i.e complete and transitive orderings over the options) that a reasonable person may have. The resulting measure simply counts in a menu the number of options that are a maximiser of at least one of these preferences over the given menu. This approach integrates preferences relatively to a menu, simply attributing values to options that could be chosen in this menu. In contrast, in our approach, preferences are integrated in a more absolute way, in the following sense: below a certain level of satisfaction, even though the DM will have to choose an option, he does not attribute any freedom value to these potentially chosen items. 21Even more, keeping the reactance choice rule in mind, some options that might be chosen later on, simply because of reactance, will not matter in the assessment of freedom, while some unchosen ones will matter.

APPENDICES A Proof of Theorem 1

Proof of the necessity. Let < T , F, u, v > be a reactance structure that represents the RCR c. We denote T (x) for the type of option x. First, we need to state the following lemma, that simply says that WARP is satisfied for each collection of menus that contains options of the same type.

Lemma 2. For any

T ∈ T , A ⊂ B ⊆ T , x ∈ X, if x = c(B) and x ∈ A, then x = c(A). Proof. Given that B ⊆ T , x = c(B) implies that c(B) = d(B), that is, u(x) > u(y)
for any y = x, y ∈ B. Because A ⊂ B, it means that x = d(A), and therefore x = c(A).

To show Exp., let x ∈ X and A, B ∈ X such that x = c(A) = c(B). This means that x ∈ d(A) ∩ d(B). Hence, u(x) > u(y) for all y = x such that y ∈ (A ∪ B) ∩ T (x), which implies that x ∈ d(A ∪ B). Moreover, x = c(A) = c(B) implies that v(x) > v(z) for all z = x such that z ∈ d(A) ∪ d(B). Besides, d(A ∪ B) ⊆ d(A) ∪ d(B), hence v(x) > v(z) for all z = x, z ∈ d(A ∪ B). Hence, x = c(A ∪ B).
To show R-Tran. Let x, y, z ∈ X such that xR c y and yR c z. By definition, xR c y =⇒ ∃t ∈ X, t = c{x, y, t} and x = c{x, t}, yR c z =⇒ ∃t ∈ X, t = c{y, z, t } and y = c{y, t }.

Proposition 4 implies that T (x) = T (y) = T (z). Coupled with lemma 2, this shows that T (t ) = T (x) = T (y) = T (z) = T (t). Given that Exp is satisfied, we also know that z = c{z, y} and y = c{y, x}. Hence, u(z) > u(y) > u(z) and

v(x) > v(t) > v(y) > v(t ) > v(z). Therefore, t = c{x,
(x) = T (z) = T (t). Moreover, we have that v(x) > v(t) > v(z) so that v(x) > v(z). Hence, if T (y) = T (x), then z = c{y, z} = c{x, z} imply that v(y) > y=c{x,y} v(x) > v(z) > z=c{y,z} v(y) (7)
A contradiction. Now if T (y) = T (x), then u(z) > u(y) > u(x). But then either v(y) > v(t), and then yR c z, or v(t) > v(y) and then xR c y. In both cases we have a contradiction.

Finally, let x, y, z ∈ X such that y = c{x, y}, z = c{y, z} = c{x, z}, ¬[xP c y] and ¬[yP c z]. Assume by contradiction that xP c z. Hence, there exists t such that zR c t and for any such t, xR c t. By proposition 4, T (z) = T (t) = T (x) ≡ T , u(t) > u(z) > u(x), v(z) > v(t), and v(x) > v(t), which implies that x, z / ∈ F . Suppose that y / ∈ T , then by proposition 4, ¬[xR c y] and ¬[yR c z]. We have already proved that this implies ¬[xR c z]. Therefore, for any t / ∈ T , z = c{z, t} =⇒ x = c{x, t}. But then, given that z = cz, t and y = cx, y, it must be that y ∈ T . A contradiction. Hence, y ∈ T , which means that u(z) > u(y) > u(x). Because z / ∈ F , then y / ∈ F . Given that v • u -1 is single peaked on u(T \F ) we have that v(y) > min{v(x), v(z)}. If v(y) > v(z), then proposition 4 implies that yP c z, a contradiction. Hence v(z) > v(y) > v(x). Let t be such that, zR c t, then xR c t, which means that v(x) > v(t), and therefore v(y) > v(t), and hence yR c t. This proves that yP c z, again a contradiction.

To show R-Con., let x, y, z ∈ X such that xR c y, xR c z, z = c{y, z}, and such that there exists no t with yR c t. Proposition 4 implies that T (z) = T (x) = T (y) ≡ T and u(z) > u(y). Furthermore, by proposition 1, it is without loss of generality to assume that F is maximal, and by proposition 2, yR c t for no t implies that y ∈ F , and hence z ∈ F . Let u ∈ X such that uR c z, so there exists t / ∈ T such that v(u) > v(t) > v(z) = u(z) > u(u). Hence, u / ∈ F which, together with y ∈ F and u ∈ T (since uR c z ∈ T ) , implies that u(y) > u(u). This means that v(u) > v(t) > v(y) = u(y) > u(u). Hence, uR c y. This completes the proof of (i) in R-Con. Now assume that uR c y. This means that u ∈ T and u(z) > u(y) > u(u), which proves (ii) in R-Con.

To show R-Mon., let x, y, z ∈ X such that z = c{y, z}, y = c{x, y}, xR c t, and zR c t for some t. Assume that xR c y. By proposition 4, this means that

T (x) = T (y) = T (t) = T (z) ≡ T , u(z) > u(y) > u(x), v(y) < v(x) and x, y, z / ∈ F . The single peakedness of v • u -1 on u(T \F ) implies that v(y) > v(z)
, from which we can conclude that yP c z. This proves (i). Assume now that xP c z and xP c y. By proposition 4,

T (x) = T (y) = T (z) ≡ T , u(z) > u(y) > u(x) and x, y, z / ∈ F . The fact that v • u -1 is single-peaked on u(T \F ) implies that v(y) > min{v(x), v(z)}. If v(y) > v(x)
, then for any t such that zR c t, given that xR c t, also yR c t and hence yP c z. Similarly, if v(y) > v(z), yP c z, which ends the proof of (ii).

Proof of the sufficiency. Let define the binary relation ⊂ X 2 by x y if and only if x = c{x, y} or x = y. It is clear that is complete and antisymmetric. For any transitive and complete binary relation > defined on a set A, we write max(A, >) ≡ {x ∈ A | x > y, ∀ y ∈ A}. When > is a linear order, with a slight abuse of notation, when no confusion can be made, we indifferently write max(A, >) for the singleton or for the element of the singleton.

Lemma 3. Let K a subset of X such that, (x, y) ∈ K 2 ⇐⇒ ¬[xR c y] and ¬[yR c x] , (8)
then restricted to K 2 is linear order and for all K ⊆ K, c(K) y for all y ∈ K .

Proof. Let K satisfying (8), x, y, z ∈ K and x y z. Suppose by contradiction that z

x. If x = c{x, y, z}, then zR c y, which contradicts the fact that (y, z) ∈ K 2 and K satisfies (8). The same reasoning applies if either y = c{x, y, z} or z = c{x, y, z}. Hence, we conclude that x z.

Moreover, let K ⊆ K, the transitivity of on K, implies that there exists x ∈ K such that x y for any y ∈ K . By Exp, x = c(K ).

Define now

X ↓ = y∈X {x ∈ X : xR c y} X ↑ = y∈X t∈X {x ∈ X : yR c x, ¬[xR c t]} Let X = X ↑ ∪ X ↓ and for all x ∈ X ↓ , R ↑ (x) = {y ∈ X ↑ : xR c y}. Lemma 4. If x ∈ X ↓ , then R ↑ (x) = ∅.
Proof. Let x ∈ X ↓ , i.e. xR c y for some y ∈ X. If y ∈ X ↑ , this terminates the proof. Suppose that y / ∈ X ↑ , then there exists z 1 such that yR c z 1 , which by R-Tran implies that xR c z 1 . Either z 1 ∈ X ↑ , which ends the proofs, or there exists z 2 such that z 1 R c z 2 , which again by R-Tran implies that xR c z 2 . At each step k, we replicate the same reasoning. Because X is finite, there must exist n such that for all t ∈ X,

¬[z n R c t], i.e, z n ∈ X ↑ . Yet, R-Tran also implies that xR c z n . Hence, R ↑ (x) = ∅.
Note that lemma 3 implies that is transitive on X ↑ . Hence, lemma 4 implies the existence, for all x ∈ X ↓ , of m(x), defined by:

{m(x)} ≡ min(R ↑ (x), ) = {y ∈ R ↑ (x) | z y, ∀ z ∈ R ↑ (x)}. (9) Lemma 5. For all x, y ∈ X ↓ , if R ↑ (x) ∩ R ↑ (y) = ∅, then m(x) = m(y); Proof. Let x, y ∈ X ↓ . Assume there exists t ∈ R ↑ (x) ∩ R ↑ (y) and let t = m(x).
We show that t ∈ R ↑ (y). If t = t there is nothing to prove. If t = t , then by definition of t and since t ∈ R ↑ (x), we have t t . Given that t ∈ X ↑ we have that ¬[t R c z] for any z ∈ X. Since xR c t, xR c t and t t , by R-Con(i) yR c t, implies that yR c t , i.e t ∈ R ↑ (y).

We prove symmetrically that m(y) belongs to R ↑ (x). Hence, if m(x) = m(y) it would be, by definition, that m(x) m(y) and m(y) m(x). A contradiction. Hence, m(x) = m(y).

Since X is finite there exists n * such that we can index the set {m(x) : x ∈ X ↓ } of every minimal option by a sequence (m(i)) 1≤i≤n * such that i = j ⇐⇒ m(i) = m(j). Define now for all 1 ≤ i ≤ n * :

T ↓ (i) = {x ∈ X ↓ : xR c m(i)}, T ↑ (i) = {x ∈ X ↑ : ∃y ∈ X ↓ , yR c m(i), yR c x}, and
T (i) = T ↑ (i) ∪ T ↓ (i).
Define finally:

T (0) ≡ T c 0 = X\ X = y∈X t∈X {x ∈ X : ¬[xR c y], ¬[tR c x]}
These will define the types. We denote T = {T (i) : 0 ≤ i ≤ n * } the collection of types.

Lemma 6. T forms a partition of X.

Proof. Given the definition of T (0), it is sufficient to show that the collection

{T (i) : 1 ≤ i ≤ n * } partitions X.
We first show that X = 1≤i≤n * T (i). Note that for all 1 ≤ i ≤ n * , if x ∈ T (i), then there exists y such that xR c y or yR c x, so that x ∈ X. Hence, i≤n T

(i) ⊆ X. Similarly, if x ∈ X, then either x ∈ X ↓ or x ∈ X ↑ . If x ∈ X ↓ , then xR c y for some y ∈ X and by (9), xR c m(x), i.e. x ∈ T ↓ (i) for some 1 ≤ i ≤ n * . If x ∈ X ↑ , then yR c x for some y ∈ X and ¬[xR c z] for all z ∈ X. But then x ∈ R ↑ (y) and (9) implies that yR c m(y) = m(i) for some 1 ≤ i ≤ n * . Therefore x ∈ T ↑ (i). Hence, in either cases, x ∈ 1≤i≤n * T (i).
We now assume that for some 1 ≤ i, j ≤ n * , x ∈ T (i) ∩ T (j), and show that this implies i = j.

Case 1: Assume x ∈ T ↓ (i). Then, because X ↑ and X ↓ are disjoint, x necessarily belongs to T ↓ (j). x ∈ T ↓ (i) means that xR c m(i). By definition of the m(i)'s, there exists y such that m(y) = m(i). Applying lemma 5, we conclude that m(x) = m(y) = m(i). Similarly, we prove that m(x) = m(j). Hence m(i) = m(j) and therefore i = j.

Case 2: Assume x ∈ T ↑ (i). Then, because X ↑ and X ↓ are disjoint, x ∈ T ↑ (j). Hence, there exists y i , y j ∈ X ↓ such that y i R c m(i), y j R c m(j), y i R c x, and y j R c x. Hence, x ∈ R ↑ (y i ) ∩ R ↑ (y j ), which by lemma 5, implies that m(y i ) = m(y j ). Using the same argument as in case 1, we conclude that m(i) = m(y i ) = m(y j ) = m(j) which means that i = j.

Note that, given lemma 6, T (x) is well defined as the type of the option x ∈ X, i.e. T (x) = T (i) ⇐⇒ x ∈ T (i). We now prove that is transitive on every type T (i).

Lemma 7. For all 0 ≤ i ≤ n * , the relation is transitive on T (i).

Proof. That is transitive on T (0) is a direct consequence of lemma 3. We now focus on 1 ≤ i ≤ n * .

First, we show that for all x ∈ T ↓ (i) and

y ∈ T ↑ (i), x ≺ C y. If y = m(i) this follows directly. If y = m(i), there exists z ∈ X such that zR c y and zR c m(i). Hence, y, m(i) ∈ R ↑ (z) and y m(i). Moreover, y ∈ X ↑ so that ¬[m(i)R c y]. Since xR c m(i), R-Con(ii) implies that y x.
Second, we show that is transitive on T ↓ (i). Let x, y, z ∈ T ↓ (i) such that x y z. Assume by contradiction that z x. Suppose (w.l.o.g) that x = c{x, y, z}. In this case, zR c y. Given that xR c m(i), zR c m(i), and x y z, R-Mon (i) entails yP c x. But since y, z ∈ X ↓ , zR c y implies zP c y. Hence, R-Tran (ii) yields zP c x which contradicts z x.

Finally, we prove that is transitive on each type. Let i and x, y, z ∈ T (i) such that x y z. If x ∈ T ↓ (i) then, according to the first part of the proof, y ∈ T ↓ (i) and therefore similarly z ∈ T ↓ (i). Similarly, if z ∈ T ↑ (i), the first part of the proof implies that y ∈ T ↑ (i), which in turn also triggers that x ∈ T ↑ (i). In both cases, we already proved that is transitive on T ↓ (i) (second part of the proof) and on T ↑ (i) (a consequence of lemma 3). The last case are if x ∈ T ↑ (i) and z ∈ T ↓ (i), but then from the first part of the proof we obtain x z.

For any menu A we define:

d(A) ≡ {x | x = max(T (x) ∩ A, )}.
A direct implication of lemma 3 is that is transitive on d(A). Hence we can state the following lemma:

Proof. Note that 0 = ∩T (0) 2 ; hence, when i = 0 the conclusion follows from lemma 3.

Let 1 ≤ i ≤ n * . Showing that i is antisymmetric and reflexive is straightforward and thus left to the reader. Regarding the connectedness, for any x, y ∈ T (i), x = y, if x y, then it is not possible that xQ c y, so either yQ c x, and therefore y i x, or not, and therefore x i y. The proof is symmetric if y x. Given that is connected, this terminates the proof of the connectedness of i .

We finally show the transitivity. Let x, y, z ∈ T (i) such that x i y i z. We have to deal with several cases separately.

(1) If z y x, then xQ c y and zQ c x. Thus, by lemma 9, xQ c z and x i z.

(2) If x y z, then ¬[yQ c x] and ¬[zQ c y]. Given that lemma 7 implies x z, we need to show that ¬[zQ c y]. This directly follows from R-tran(ii).

(3) If x z y, then x y (lemma 7) and thus ¬[yQ c x] and yQ c z. Assume by contradiction that zQ c x. Then lemma 9 implies that yQ c x, a contradiction. So ¬[zQ c x] and x i z.

( This completes that proof that i is transitive.

Denote ˜ = i i . Let be the relation on X defined by:

∀x, y ∈ X, x y ⇐⇒ x ˜ y if x ∈ T (y) x y if x / ∈ T (y)
Lemma 11. The relation is a linear order.

Proof. Given that and ˜ are both antisymmetric and complete, so is . We now prove the transitivity. Let us consider x, y, z such that x y z. If there exists i such that x, y, z ∈ T (i), then this follows from lemma 10. Suppose finally that we are in the case T (x) = T (z) = T (y). Note that this implies that x y and y z. Assume by contradiction that z x. If x z, then zQ c x. But given that ¬[zQ c y], ¬[yQ c x], x y, and y z, R-Tran(ii) implies that ¬[zQ c x], a contradiction. Hence, z x is possible only if z

x and ¬[xQ c y]. But, given that ¬[xQ c z], this would imply either yR c x or zR c y. Given that T (x) = T (z) = T (y), both cases lead to a contradiction. Now let F = 1≤i≤n * T ↑ (i) ∪ T (0). Given that is transitive on F (lemma 3), there exists a function w : F -→ R that represents on F . Furthermore, for every i = 0, . . . , n * , is transitive on every T (i) (lemma 7), hence there exists a function u i : T (i) -→ R representing on T (i), and such that u T (x) = w(x) for every x ∈ F . We now define the function u : X -→ R such that for every i, x ∈ T (i), u(x) = u i (x). We clearly have, for any menu A,

d(A) = T ∈T arg max x∈T ∩A u(x)
Lemma 12. There exists a function v representing such that for any x ∈ F , v(x) = u(x) and for any x / ∈ F , v(x) > u(x).

Proof. Note first that, restricted to F 2 , = . Indeed let x, y ∈ F such that x y. If x / ∈ T (y), this directly follows from definition of . If x ∈ T (y), then if, by contradiction, y x, then we would have xQ c y, in contradiction with x ∈ F . Now given that is transitive and complete on X, there exists (up to a monotone transformation) a function v representing . Given that = on F , we can set this function such that u(x) = v(x) for all x ∈ F .

Next, if x / ∈ F , then xR c m(x), which implies v(x) > v(m(x)) and m(x) x, which means that u(m(x)) > u(x). Given that m(x) ∈ F , this means that v(x) > v(m(x)) = u(m(x)) > u(x) Lemma 13. For any menu A, c(A) = To complete the proof of theorem 1, we show that we can construct a function v such that v • u -1 is single-peaked on u(T (i) \ F ) for any 1 ≤ i ≤ n . For any 1 ≤ i ≤ n , let define M (i) ≡ max(T (i), i ). Note that for any z ∈ T ↑ (i), there exists t such that tR c z, i.e t i z, hence necessarily M (i) ∈ T ↓ (i).

The following lemma shows that v as constructed above satisfies singlepeakedness on the "right" of M (i), that is:

Lemma 14. For all 1 ≤ i ≤ n , x, y ∈ T ↓ (i), if u(y) > u(x) > u M (i) , then v(x) > v(y).
Proof. u(y) > u(x) > u M (i) means that y x M (i) and y M (i). We know that yR c m(i), M (i)R c m(i). Given that R c ∩ T ↓ (i) 2 ⊆ P c ∩ T ↓ (i) 2 and the definition of M (i), it must be that M (i)P c y, M (i)P c x. Therefore, R-Mon (ii) implies that xP c y, which implies v(x) > v(y).

For any 1 ≤ i ≤ n , we define the set

T ↓ (i) ≡ {x ∈ T ↓ (i) | u(x) < u(M (i))}. We order this set T ↓ (i) = {x i 1 , . . . , x i K i } such that for any 1 ≤ k ≤ K i -1, u(x i k ) > u(x i k+1
). Lemma 15. For all 1 ≤ i ≤ n , if x, y ∈ T ↓ (i), u(x) > u(y), and v(x) < v(y), then for any z / ∈ T (i), either v(z) < v(x) or v(y) < v(z).

Proof. Let 1 ≤ i ≤ n such that there exists x, y ∈ T ↓ (i) with u(x) > u(y), and v(x) < v(y). Assume by contradiction that there exists z / ∈ T such that v(x) < v(z) < v(y). Hence, yR c x. But then, by R-Mon (i), because yR c m(i) and M (i)R c m(i), xP c M (i), which contradicts the definition of M (i). Now let us define for all i ≤ n and all x ∈ T ↓ (i),

a i + (x) = min y: / ∈T (x):v(y)>v(x) v(y) a i -(x) = max y: / ∈T (x):v(y)<v(x) v(y) Lemma 16. for all i ≤ n x, y, z ∈ T ↓ (i), 1. a i + (x) = a i + (y) if and only if a i -(x) = a -(y); 2. if u(x) > u(y) > u(z) and v(z) ∈]a i -(x), a i + (x)[, then v(y) ∈]a i -(x), a i + (x)[= ]a i -(y), a i + (y)[.
Proof. The first claim stems from the fact that a + (x) > a + (y) implies that v(x) > a + (y) so that a -(x) ≥ a + (y) > a -(y). Hence, a -(x) = a -(y) implies a + (x) = a + (y). The reverse implication is similar. The second claim is a direct consequence of lemma 15.

We now define a function v which satisfies the conclusion of lemma 13 and is single-peaked as required by our representation. First, v(x) = v(x) for any

x ∈ 1≤i≤n [T (i) \ T ↓ (i)] ∪ T (0). Second, for each 1 ≤ i ≤ n , set vi on T ↓ (i) such that: vi (x) ∈]a i -(x), a i + (x)[ (11) vi (x) ∈ [ min y∈T ↓ (i):v(y)∈]a i -(x),a i + (x)[ v(y), max y∈T ↓ (i):v(y)∈]a i -(x),a i + (x)[ v(y)] (12) vi (x) > vi (y) ⇐⇒ u(x) > u(y) (13)
Finally, let v|T ↓ (i) = vi for each i.

By construction v is single-peaked. It remains to check that for all A, c(A) = arg max x∈d(A) v(x). To see why this is the case assume that arg max x∈d(A) v(x) = arg max x∈d(A) v(x) for some A. This means that there exist x, y ∈ A with x / ∈ T (y) such that v(x) > v(y) and v(x) < v(y). Given lemma 13, this is possible only if

x ∈ T ↓ (i) or y ∈ T ↓ (i) for some i. Assume that x ∈ T ↓ (i). Then, this would imply that v(x) > a i -(x) ≥ v(y). If y / ∈ T ↓ (i) for all i ≤ n , then v(x) > a i -(x) ≥ v(y) = v(y), while v(x) < v(y) ≤ a i -(x)
, in contradiction with (11). If y ∈ T ↓ (j) for some j ≤ n , then v(x) ≥ a j + (y) > v(y) and v(y) ≤ a -(x) < v(x). Note that a i -(x) ≥ a j + (y) would be in contradiction with (11), hence a i -(x) < a j + (y). This is possible only if a j + (y) = v(x ) and a i -(x) = v(y ) for some x ∈ T ↓ (i) and some y ∈ T ↓ (j). Note also that v

(x ) ∈]a i -(x), a i + (x)[ and v(y ) ∈]a j -(y), a j + (y)[. Moreover, it must be that v(x ) = min z∈T ↓ (i):v(z)∈]a i -(x),a i + (x)[ v(z),
as otherwise this would contradict the definition of a + (y). Similarly, it must be that v(y ) = max

y∈T ↓ (i):v(z)∈]a i -(y),a i + (y)[ v(z).
Hence, by (12) we have v(x) ≥ v(x ) > v(y ) ≥ v(y). A contradiction.

B Proof of Proposition 1

We first prove the following corollary of proposition 4.

Corollary 1. Let c be an RCR represented by a reactance structure < T , F, u, v >.

For any T ∈ T : T ⊂ F =⇒ T = T c 0 .

Proof. T ⊂ F implies that ∀x ∈ T , u(x) = v(x). Given proposition 4 (i), this implies that for any y ∈ X ¬[xR c y]. If, by contradiction, there exists z such that zR c x, then proposition 4 (i) implies again that there exists T such that x, z ∈ T , z / ∈ F . Given that T forms a partition, this means that T = T , and thus T \F = T \F = ∅. A contradiction. Hence, T = T 0 .

Proof of proposition 1. Assume that c is represented both by < T , F, u, v > and < T , F , ũ, ṽ >. We first prove that T = T . For that purpose, we define the binary relation E c ⊂ X 2 as the reflexive extension of R c , that is: for any x, y ∈ X, xE c y ⇐⇒ [xR c y ∨ yR c x]. We simply show that for any x, y ∈ X, y ∈ T (x) = T c 0 if and only if there exists a sequence (x k ) n+1 k=0 such that x 0 = x, x n+1 = y, and for any i ∈ {0, . . . , n}, x i E c x i+1 . Or equivalently, we show that the collection of types on X \ T c 0 is defined by the collection of the components of the graph generated by the binary relation E c on the set X \ T c 0 . Only if. Consider x, y ∈ T ∈ T with T = T c 0 . We show there exists a sequence (x k ) n+1 k=0 such that x 0 = x, x n+1 = y, and for any i ∈ {0, . . . , n}, x i E c x i+1 . By corollary 1, T \ F = ∅. First, define x ≡ arg max v(T \ F ). Given that v • u -1 is single-peaked on u(T \ F ), for any s, t ∈ T \ F , if u(s) < u(t) ≤ u(x ), then v(s) < v(t) ≤ v(x ). Hence there exists no z such that zR c s (or equivalently zR c t). Hence, T = T c 0 implies that sR c z for some z ∈ T with u(z) > u(x ). Second, define x = arg min v {x ∈ T | u(x) ≥ u(x )} . Given the definition of the freedom requirement set F and the fact that v

• u -1 is single-peaked on u(T \ F ), for any s, t ∈ T , if u(s) > u(t) ≥ u(x ), then v(s) > v(t) ≥ v(x ).
Hence there exists no z such that sR c z (or equivalently tR c z). Hence, T = T c 0 implies that zR c s for some z ∈ T with u(z) < u(x ).

Hence, for any s ∈ T , if sR c z for some z ∈ T , then u(s) < u(x ). Therefore, given the definition of x , and the single-peakedness of v • u -1 on u(T \ F ), for any such s, this is also the case that sR c x . Indeed, this means that u(s) < u(z), u(z) > u(x ) (from our first point made above) and there exists t / ∈ T such that v(s) > v(t) > v(z). But note then that u(x ) > u(s) and v(s) > v(t) > v(z) > v(x ), hence sR c x .

Finally, there are three cases to consider for x, y. (1) If both u(x) ≥ u(x ) and u(y) ≥ u(x ), then there exist s, t such that sR c x and tR c y. Both sR c x and tR c x . Hence xE c sE c x E c tE c y. (2) If u(x) ≥ u(x ) > u(y), then there exists s such that sR c x and sR c x . Furthermore, either yR c z, in which case yR c x , or zR c y, in which case zR c x , for some z. In both cases, we can conclude similarly as in (1). (3) If both u(x) < u(x ) and u(y) < u(x ), then either αR c z, in which case αR c x , or zR c α, in which case zR c x , for α = x, y, for some z. In any of the four possible combinations, we can similarly conclude.

If. Consider x, y ∈ X such that there exists a sequence (x k ) n+1 k=0 such that x 0 = x, x n+1 = y, and for any i ∈ {0, . . . , n}, x i E c x i+1 . By proposition 4, for any i ∈ {0, . . . , n}, x i E c x i+1 implies that there exists T such that x i , x i+1 ∈ T , from which we conclude that there exists T such that x, y ∈ T .

We now prove that there exists û, v such that < T , F ∪ F , û, v > also represents c. First note that given the definition of a freedom requirement set, for any type T ∈ T , either

F ∩ T ⊆ F ∩ T or F ∩ T ⊆ F ∩ T . Let us suppose w.l.o.g that F ∩ T c 0 ⊆ F ∩ T c 0 .
Then, for any T such that F ∩ T ⊆ F ∩ T , we simply define û|T = u |T and v|T = v |T . Now, let T be such that F ∩ T F ∩ T and define when they exist x (T ) = arg min u(F ∩ T ) and x (T ) = arg max u(T \ F ). Suppose first that x (T ) does not exist, this means that T ⊂ F , in which case, by corollary 1, T = T c 0 . But we assumed that F ∩ T c 0 ⊆ F ∩ T c 0 . Hence, x (T ) necessarily exists. Suppose now that x (T ) exists. There exists a sequence of options (x k ) n(T ) k=1 such that T ∩ ( F \ F ) = {x 1 , . . . , x n(T ) } and u(x (T )) > u(x 1 ) > • • • > u(x n(T ) ) > u(x (T )). Note that this must be that v(x (T )) > v(x n(T ) ) > • • • > v(x 1 ) > v(x (T )). Indeed, suppose by contradiction that v(x (T )) < v(x n(T ) ), then point (iii) of theorem 1 implies that for any x ∈ T such that u(x) < u(x (T )), v(x) < v(x n(T ) ). If there exists y such that xR c y, then this would imply that x n(T ) R c y, which is not possible given proposition 4 and the fact that x n(T ) ∈ F . This implies that there exists no y such that xR c y, which in turn triggers that for any y ∈ T there exists no z such that yR c z, from which we obtain that T = T c 0 , a contradiction. Hence we conclude that v(x (T )) > v(x n(T ) ) > • • • > v(x 1 ) > v(x (T )). At the same time, ṽ(x (T )) < ṽ(x n(T ) < • • • < ṽ(x 1 ) < ṽ(x (T )). Given that < T , F, u, v > and < T , F , ũ, ṽ > both represent c, this implies that for any k ∈ {1, . . . , n(T )} and any z / ∈ T , c{x (T ), z} = x (T ) ⇐⇒ c{x k , z} = x k . Let ν T > 0 be sufficiently small and define û(x k ) = v(x k ) = u(x (T )) -kν T for every k ∈ {1, . . . , n}.

Let us now consider the case where x (T ) does not exist. This means that T ∩ F = ∅. There exists similarly a sequence of options (x k ) n(T ) k=1 such that T ∩ ( F \ F ) = {x 1 , . . . , x n(T ) } and u(x 1 ) > • • • > u(x n(T ) ) > u(x (T )). Following a similar reasoning, one can show that v(x (T )) > v(x n(T ) ) > • • • > v(x 1 ) while ṽ(x (T )) < ṽ(x n(T ) < • • • < ṽ(x 1 ). Hence for any k ∈ {2, . . . , n(T )} and any z / ∈ T , c{x 1 , z} = x 1 ⇐⇒ c{x k , z} = x k . Let ν T > 0 be sufficiently small and define û(x 1 ) = v(x 1 ) = v(x 1 ) and û(x k ) = v(x k ) = v(x 1 ) -(k -1)ν T for every k ∈ {2, . . . , n}.

In both cases, finally define û|T\( F \F ) = u |T \( F \F ) and v|T\( F \F ) = v |T \( F \F ) . Repeat these operations for any T such that F ∩ T F ∩ T . One can easily check that by appropriately choosing the ν T s, û, v so defined are such that the reactance structure < T , F ∪ F , û, v > represents c.

C Proof of Proposition 2

Proof of the necessity. Let < T , F, u, v > be a a maximal reactance structure that represents c. The fact that T c 0 ⊂ F directly follows from our proof of theorem 1. Let T = T c 0 , x ∈ T and define x T,F as in the proposition. Note that T ⊂ F , as otherwise, by proposition 4, it would be the case that T = T c 0 . Hence there exist a sequence of options (x k ) n k=1 such that T \ F = {x 1 , . . . , x n } and u(x

T,F ) > u(x 1 ) > • • • > u(x n ).
First, this must be that v(x 1 ) > u(x T,F ), as otherwise, one could redefine û by û(x 1 ) = v(x 1 ), û = u elsewhere, and F = F ∪ {x 1 } such that < T , F , û, v > also represents c, which contradicts that < T , F, u, v > is maximal.

Second, there exists k ∈ {1, . . . , n} such that v(x k ) = max v(T \ F ). Given that v • u -1 is single-peaked on u(T \ F ), for any k < k < k , v(x k ) < v(x k ) < v(x k ). Hence, it is sufficient to show that x 1 R c x T,F . Suppose by contradiction that it is not the case. This means that for any z / ∈ T , c{x T,F , z} = x T,F =⇒ c{x 1 , z} = x 1 . But v(x 1 ) > u(x T,F ) implies that c{x 1 , z} = x 1 =⇒ c{x T,F , z} = x T,F . A contradiction.

Third, the single-peakness of v • u -1 on u(T \ F ) implies that for any k > k > k , v(x k ) < v(x k ) < v(x k ). Hence, let k ≥ k , by proposition 4, there exists no x such that xR c x k . This implies that there exists x ∈ T , with u(x) > u(x k ), such that x k R c x, as otherwise, x ∈ T c 0 a contradiction. In any case we have v(x k ) > v(x) > v(x T,F ). Indeed, if x = x k for some k < k , then v(x k) > v(x 1 ) > v(x T,F ) by the previous step. If x ∈ F , then v(x) = u(x) > u(x T,F ) = v(x T,F ), by definition of x T,F . But applying the same reasoning as for x 1 , it can be shown that x k R c x T,F . This concludes the proof of (i). Now let us consider x ∈ T such that u(x) > u(x T,F ). Hence, x ∈ F , therefore, there exists no y such that xR c y (by proposition 4). Given that T = T c 0 , there must exist y ∈ T such that yR c x. Again by proposition 4, this can be possible only if u(y) < u(x) and v(y) > v(x), which is possible only if u(y) < u(x T,F ). Point (i) implies that yR c x T,F , which ends the proof of (ii).

Proof of the sufficiency. Let < T , F, u, v > be a reactance structure that represents the choice function c and that satisfies the conditions stated in the proposition. Suppose by contradiction that it is not maximal. Therefore, there exists T = T c 0 such that T \ F = ∅, and there exists < T , F , ũ, ṽ > that represents c with F ⊃ F and F ∩ T F ∩ T . But this contradicts the fact that for any x with u(x) < u(x T,F ), i.e x ∈ T \ F , xR c x T,F .

D Proof of Proposition 3

Proof. The proof of (i) directly follows from the proof of theorem 1. The proof of the if part of (ii) is not complicated and thus left to the readers. We only prove the only if part.

The fact that f must be increasing on u(T ) for every T ∈ T simply follows from the fact that the function u represents the binary choices within each type. f |u(F ) = g |u(F ) is a direct consequence of the requirement that utility and the reactance functions be equal on the freedom requirement set.

We now prove that g must be increasing on v(X). Suppose by contradiction that there exists x, y ∈ X such that v(x) > v(y) but g • v(x) ≤ g • v(y). Note that there must exist a type, denote it T , such that x, y ∈ T , as otherwise v(x) > v(y) =⇒ x = c{x, y}, which cannot be accommodated by < T , F, f • u, g • v > if g • v(x) ≤ g • v(y). Define x = arg max v(T \ F ) and x T,F as in proposition 2.

(1) Consider the case where u(x) > u(y). If u(y) ≥ u(x T,F ), this would mean that x, y ∈ F , in which case, given that u |F = v |F , f |u(F ) = g |u(F ) and f is increasing on u(T ), it is impossible that g • v(x) ≤ g • v(y). If u(y) ≤ u(x T,F ), it means that y / ∈ F . The fact that v(x) > v(y) implies that there exists z / ∈ T such that x = c{x, z} while z = c{y, z}, that is v(x) > v(z) > v(y). Hence, g • v(x) ≤ g • v(y) cannot accommodate these choices.

(2) Consider the case where u(x) < u(y). Then necessarily u(x) < u(x T,F ), that is x / ∈ F . Given that v(x) > v(y), there exists no z / ∈ T such that z = c{x, z} while y = c{y, z}. Conversely, if there exists z / ∈ T such that x = c{x, z} while z = c{y, z}, that is v(y) > v(z) > v(x), then again g • v(x) ≤ g • v(y) cannot accommodate these choices. If it is note the case, that is for every z / ∈ T such that x = c{x, z} ⇐⇒ y = c{y, z}, then g • v(x) ≤ g • v(y) does not satisfy the requirement of reactance structure .

E Proof of Theorem 2

Proof. The necessity part of the theorem is left to the readers. We only prove the sufficiency.

Figure 1 :

 1 Figure 1: Reactance structure in one type.

  For any A, B, C, D ∈ X , such that A∩C = B ∩D = ∅, C ⊆ T and D ⊆ T for some T, T ∈ T , and A is not richer than C: if A B and C D, then A ∪ C B ∪ D.

  z, t} and x = c{x, t}, which means that xR c z. Now let x, y, z ∈ X such that y = c{x, y}, z = c{y, z} = c{x, z}, ¬[xR c y] and ¬[yR c z]. Assume by contradiction that xR c z. Then there exists t such that t = c{x, z, t} and x = c{x, t} and, by proposition 4 and lemma 2, T

  ) If y x z, then y z and thus xQ c y and ¬[zQ c y]. The similar reasoning as in (3) gives the expected conclusion. (5) If z x y, then z y and thus ¬[yQ c x] and yQ c z. Suppose that ¬[xQ c z], then R-Tran(ii) implies that ¬[yQ c z], a contradiction. Hence, xQ c z and x i z. (6) If y z x, then y x and thus xQ c y and ¬[zQ c y]. The similar reasoning as in (5) gives the expected conclusion.

  If T (x) ∩ T (y) = T (x) ∩ T (z) = T (y) ∩ T (z) = ∅, then this follows from lemma 3. Suppose we are in the case T (x) = T (y) = T (z). Note that this implies that y z. Assume first that xQ c y and suppose by contradiction that z x. Given that T (x) = T (y) = T (z), we have ¬[zQ c y] and ¬[xQ c z]. Moreover, by assumption, y z x, so that R-Tran(ii) implies ¬[xQ c y]. A contradiction. Assume now that x y and ¬[yQ c x]. Suppose by contradiction that z x, then, since ¬[yQ c x], we have either xR c z or zR c y. In any case a contradiction arises since T (x) = T (y) = T (z). We deal with the case T (y) = T (z) = T (x) similarly.

  Let x, y ∈ d(A), T (x) = T (y) and suppose (w.l.o.g) that x y. Given that v represents and T (x) = T (y), this means that x y =⇒ x y =⇒ v(x) > v(y). Hence, we just established that max d(A), ⊆ arg max x∈d(A) v(x). Given that max d(A), = ∅ and arg max x∈d(A) v(x) is a singleton (since is antisymetric), this proves that max d(A), = arg max x∈d(A) v(x).

[START_REF] Mazis | Elimination of phosphate detergents and psychological reactance[END_REF] showed, this astonishing effect on behavior was consistent with consumers' beliefs reversal: Miami consumers were, on average, more prone to praise phosphate detergent for its efficiency than their Tampa county neighbors.

2 As emphasised by psychologists, reactance is about freedom as subjectively perceived by the

More precisely, the application of OOR's definition to these choices would identify y as a revealed reference of z.

It was already present in[START_REF] Sen | Choice Functions and Revealed Preference[END_REF], named property γ, and was later used by[START_REF] Manzini | Sequentially rationalizable choice[END_REF] under the name Expansion.

With a slight abuse of notation, if x is the unique maximizer of a function f on the set E, we simply write x = arg max y∈E f (y), instead of {x} = arg max y∈E f (y).

Note that an RCR is a specific case of a rational shorlist method (RSM) à la[START_REF] Manzini | Sequentially rationalizable choice[END_REF]. Indeed, one can define the two orders 1 and 2 in the following way: x 1 y ⇐⇒ y ∈ T (x) ∧ x = c{x, y}; x 2 y ⇐⇒ x = c{x, y}. In that case, if c is an RCR, then for any menu A, c(A) = max(max(A, 1 ), 2 ). That is, the DM chooses as if she first keeps only options that are the best in each available type, and second, she chooses the best remaining one according to the binary comparisons. As a consequence, although we do not use it in our characterization, an RCR also satisfies the second property of RSMs, namely Weak WARP.

Note that the uniqueness of the collection of types relies on the requirement in our theorem 1 that T c 0 is one type. Hence, proposition 1 alternatively says that the collection of types is unique on the set of options on which reactance phenomena are observed.

See proposition 4 below (page 15) for a more precise statement of why this is true.

The boomerang effect is "a situation in which a persuasive message produces attitude change in the direction opposite to that intended". The backfire effect is a concept from political science that refers to a situation in which evidence contradicting the subjects' prior belief may reinforce their belief in the opposite direction.

 15 The fact that mass media did not give any credit to conspiracy theories has been pointed out as playing a role in reinforcing such theories through reactance.

For clarity, we adopt a continuous setting, while our own framework is discrete. The ideas would be exactly the same with a discrete setting.

It is actually a specific case of PX00's analysis where the equivalence classes induced by the similarity relation form a partition.

To illustrate this, our measure can be equal to 0 on some non-empty menus, which is impossible either in PX98 or in PX00.

Lemma 8. For any A ∈ X , c(A) = max(d(A), ) (10) Proof. For any menu A, denote i(A) = #{i | T (i) ∩ A = ∅}. We prove that for any 1 ≤ n ≤ n * + 1, for any A such that i(A) = n, (10) holds.

If i(A) = 1, the conclusion follows from lemma 7. Assume now that i(A) = 2. Let x, y ∈ A be such that T (x) ∩ T (y) = ∅, x = max(T (x) ∩ A, ), y = max(T (y) ∩ A, ), and y

x. Assume by contradiction that y = c(A). First note that by definition of y, Exp, and lemma 7, y z for any z ∈ T (y) ∩ A. Hence, there must exist z ∈ T (x) such that z y and y = c{x, y, z}; otherwise Exp would imply that y = c(A). This implies that x z y x. Since y = c{x, y, z}, this is only possible if either yR c z or xR c y, which in any case contradicts that x, z / ∈ T (y) (given that, according to lemma 6, types partition X). Hence we conclude that y = c(A).

Then fix 3 ≤ n ≤ n * + 1 and let A a menu such that i(A) = n. We denote y = max(d(A), ). Given the preceding proof for any menu A such that i(A ) = 2, for any z ∈ A, y = c T (y)∪T (z) ∩A . This implies by Exp that y = C(A).

Lemma 9. Q c is asymmetric and transitive.

Proof. By definition for no x, y ∈ X, xR c y and yP c x; otherwise x y and y x. Hence, given that both R c and P c are asymmetric, so is Q c .

To show that Q c is transitive, let x, y, z ∈ X such that xQ c y and yQ c z.

If yR c z, then, when xR c y, the conclusion directly follows from R-Tran. When xP c y, for any t such that yR c t, we have xR c t. Hence, yR c z implies xR c z, i.e. xQ c z.

If yP c z, then there exists t such that zR c t and for any of such t, yR c t. If xP c y, then the conclusion follows from the transitivity of P c . If xR c y, R-Tran implies that xR c t for any t such that yR c t, and therefore for any t such that zR c t, i.e. xP c z and xQ c z.

For each 0 ≤ i ≤ n * define the relation i on T (i) such that for all x, y ∈ T (i), x i y if and only if either xQ c y or (x y and ¬[yQ c x]).

Lemma 10. For all 0 ≤ i ≤ n * , x i y is a linear order.

(a) We first show that for any A, B such that A ⊆ T and B ⊆ T for some Assume now that B = ∅. By a similar reasoning as for A and B , one can easily show that

By reiterating the same argument, we obtain that i A i ∼ i B i . Finally, note that A is richer that i A i and conversely i A i is richer than A, hence, by RD, A ∼ i A i ; similarly B ∼ i B i . Therefore, by transitivity, we obtain that A ∼ B.

(c) We finally prove that for any A, B, if

A similar argument as before shows that A ∼ n i=1 A i and B ∼ k i=1 B i . Hence by transitivity A B.

F Proof of Proposition 5

Proof. (i) Denote u(σ L ) and v(σ RR ) the DM's anticipated utility from choosing respectively σ L and σ RR in the menu N :

We define p := 1/2 5/2-2λ and verify that p < 1/2:

which is true by assumption.

(ii). We first compute the value q of the posterior such that for any q ≥ q , action r is preferred. q solves (1 -q) -q = q, hence q = 1/3.

Then we simply compare the posterior obtained after the realisation of a signal s r from the news source σ RR with 1/3. The posterior is, p p+(1-p)1/2 , which is greater or equal than p. We are in the case where p ≥ p , hence it is sufficient to show that p ≥ 1/3: p ≥ 1 3 ⇐⇒ λ ≥ δ which is true by assumption.

G Proofs of Section 3.2

Proof of lemma 1. The maximand of the program ( 3) is strictly concave and the set K g is compact. Hence, C is well-defined (Weierstrass theorem) and is a choice function. Now let us build the reactance structure < T , F, u, v > that represents C. Given (4), d strictly increases with g if and only if g > ĝ. Let g(t, d) be the g such that t + gd β = 1.

Let us introduce the three following sets:

From these sets we can define the set of types and the freedom set

Given the uniqueness of g(t, d) for each (t, d), v is well-defined. It can easily be shown that < T , F, u, v > is a reactance structure. Consider the choice function C which is the RCR defined on the compact subsets of [0, 1] 2 and represented by < T , F, u, v >. We claim that for all g, C(K g ) = C (K g ). To check this claim let (t, d) and g such that (t, d) = C(K g ) and (t , d ) such that (t , d ) = C (K g ). Note that (t, d) = C(K g ) implies g = g(t, d). Similarly, (t , d ) = C (K g ) implies that u(t , d ) ≥ u(t , d ) for all t such that (t , d ) ∈ K g , that is, for all t ≤ t . Hence, g = g(t , d ).

Assume first that g ≤ ĝ. Then, note that (t , d ) ∈ F . Suppose that there exists (t , d ) ∈ K g \ F , this means that g(t , d ) > ĝ, and hence there exists t > t , such that g(t , d ) = g, which implies (t , d ) ∈ K g and u(t , d ) > u(t , d ). Therefore, only elements in F can be considered for choices in K g according to the choice procedure (1). Hence, both (t, d) and (t , d ) are elements of arg max u(K g ). Because the latter is a singleton, (t, d) = (t , d ).

Assume next that g(t, d) > ĝ. Suppose that (t , d ) ∈ F , then this implies that d / ∈ D ↑ , that is d = d. Because, g(t, d) = g(t , d ), this in turn implies that t = t. Furthermore, by definition, (t, d) / ∈ F , and from (t , d ) = C (K g ), we deduce that u(t , d ) > v(t, d). We also know that v(t , d ) ≥ u(t , d ), so v(t , d ) > v(t, d), which contradicts that (t, d) = C(K g ). Therefore, (t , d ) / ∈ F . Because (t, d) / ∈ F , (t , d ) = C (K g ) and (t, d) = C(K g ) imply that v(t , d ) ≥ v(t, d) = max v(K g ). Therefore, (t , d ) ∈ arg max v(K g ), and given that this set is a singleton, this implies that (t , d ) = (t, d).

Proof of proposition 6. This is a straightforward consequence of (5).