Niels Boissonnet
email: niels.boissonnet@gmail.com

Alexis Ghersengorin
email: alexis.ghersengorin@pse-mail.eu

Simon Gleyze
email: gleyze.simon@gmail.com

Douglas Bernheim

Franz Dietrich

Marco Mariotti

Pietro Ortoleva

Revealed Deliberate Preference Change *

We propose a model of rational preference change that is identifiable, empirically testable and founded on two normative principles. First, the decision maker (DM) must be able to justify her preference change by making attributes of the alternatives relevant or irrelevant. For instance, an employer who stops discriminating against candidates based on race or gender must make the attribute "race" or "gender" irrelevant for all her future choices. Second, DM must be consistent in her preference change. For instance, in the future the employer cannot make relevant again the attribute "race" or "gender", unless she makes (ir)relevant other complementary attributes. We show that two axioms on choice data which reflect these principles are necessary and sufficient to represent preference change by the maximization of a meta-preference. Finally, we apply our model to derive new insights on the polarization of political preferences.

INTRODUCTION

Understanding how individuals change their behavior is critical for social sciences. Economists traditionally argue that decision makers (DMs) are Bayesian; that is, they adapt their behavior by updating their beliefs about the environment. Although this mechanism has proved powerful and normatively appealing, a wide range of phenomena seem better described with preference change because they involve values such as fairness, conservatism, etc. For instance, [START_REF] Barrera | Facts, Alternative Facts, and Fact Checking in Times of Post-Truth Politics[END_REF] show experimentally that exposure to fake news about the European refugee crisis increases voting intentions toward far-right politicians, even though voters' beliefs may not change in case of fact-checking. Their explanation is that by raising voters' awareness of the migration issue, politicians may alter preferences. Another example is the expansion of abortion rights in western societies-along with its economical and political implications-that is more plausibly due to the diffusion of new values such as women's rights than to changing beliefs on some underlying state of the world.

Modeling preference changes raises two challenges: first, the lack of normative foundations compared to Bayesian updating; second the lack of testability of the model. To fill these gaps, we propose and axiomatize two testable normative principles: a principle of sufficient reason and a principle of deliberation. To express these normative principles, we use the attribute-based approach. Our primitive is the observation of successive preferences, as well as the attributes of each alternative. This allows us to define the attributes that are relevant to DM's choice at each period and, thereby, to reveal DM's reasoning behind preference changes. In doing so, we make progress toward a testable and normatively founded model of preference change.

The principle of sufficient reason states that DM changes her preferences if and only if it can be justified by an attribute of the alternative that is made relevant or irrelevant. For instance, if an employer becomes aware that her hiring decision is based on the attribute "gender", she might make this attribute irrelevant in the future to stop being discriminatory. 1 Formally, this translates into an identification axiom called Restricted Reversals, which guarantees that preference reversals can be explained by changes of relevant attributes alone (proposition 1).

The principle of deliberation states that DM should not make mistakes (from her perspective) when changing preferences; that is, she cannot change her mind twice regarding an attribute if no additional event occurred meanwhile. Otherwise, this would indicate that she fails to deliberate and lacks internal consistency. Formally, this translates into an Acyclicity axiom, which guarantees that if an attribute is made relevant and then irrelevant it must be explained by other attributes becoming (ir)relevant meanwhile.

Our main representation theorem states that Restricted Reversals and Acyclicity hold if and only if (i) preferences are represented by an ordering on the alternatives' attributes-we call this the attribute ordering-, and (ii) preference changes are explained by the maximization of an ordering on preferences themselves-we call this the meta-preference (theorem 1).

Preference changes take the following form: whenever DM becomes aware of an attribute-through education, social interactions, medias or introspectionshe can decide to make it relevant or irrelevant for the next period, inducing a preference change. The succession of such changes is consistent with the maximization of a meta-preference relation, capturing DM's moral values, motivated reasoning, social objectives, norms, etc. Therefore, the reasoning behind preference changes is revealed through the meta-preference relation and the sequence of awareness. Such a sequence represents DM's constraint regarding which preferences are reachable at each period. The existence of such a constraint follows from the principle of deliberation and the observation of multiple preference reversals. Indeed, should DM be unconstrained in the maximization of her meta-preference she would directly reach her most preferred set of relevant attributes and never change preferences again. Note that the attribute ordering remains stable and that only the set of relevant attributes changes; this implies that if DM deems relevant the same set of attributes from one period to another, she must make exactly the same choices.2 See Figure 1 for a representation of the model. Models of chosen preferences are receiving renewed attention since [START_REF] Bernheim | A Theory of Chosen Preferences[END_REF], and the present paper is the first to investigate its revealed preference implications.

The attribute ordering and the meta-preference are essentially unique.3 Furthermore, if two sequences of awareness represent DM's constraint on meta-choices, so does their intersection (proposition 2). We, however, stress that the sequence of relevant attributes is not uniquely identified in general. Hence, we investigate specific conditions that make this sequence set-identified or point-identified (proposition 4 and theorem 2).

We then investigate a particular type of meta-preference (i.e., a particular type of reasoning) in which DM chooses the preferences that maximize her underlying utility (theorem 3). This captures motivated preference change in which DM's evaluation of the attributes is guided by her own-interest alone. We show that motivated preference change provides new insights on the formation of political preferences. For instance, if two voters with identical preferences become aware of the same attributes in a different order, they can end up endorsing antagonistic views. Whether a voter becomes aware that a politician is corrupted before or after learning his political affiliation can lead to very different outcomes: in the latter case, the voter might ignore this attribute because it undermines the view of her preferred candidate. This type of path-dependent motivated reasoning is specific to our model and provides empirically testable implications.

Our contribution is threefold: first, we show that models incorporating preference changes can have empirical content and normative foundations. Second, our model suggests that choice reversals need not be irrational, and may reflect DM aligning her choice behavior with her values. Any deliberate preference change must break (or create) indifference with respect to other pairs of alternatives that share the same attribute, which indicates that this attribute becomes relevant (resp. irrelevant). This is a necessary condition for preference change to be induced by a coherent reasoning from DM. Finally, we illustrate the explanatory power of our model through an application. 4Related Literature. The idea of representing objects by their attributes goes back to [START_REF] Lancaster | A New Approach to Consumer Theory[END_REF]. Moreover, we draw on an important literature on reasonbased theories of choice, most notably [START_REF] Simonson | Choice Based on Reasons: The Case of Attraction and Compromise Effects[END_REF], [START_REF] Shafir | Reason-Based Choice[END_REF]Simonson (1993), and Dietrich andList (2013, 2016). [START_REF] Boissonnet | Rationalizing Preference Formation by Partial Deliberation[END_REF] and [START_REF] Dietrich | A model of non-informational preference change[END_REF] also use an attribute-based approach to model noninformational preference change. Our paper should be seen as the first counterpart of these models within the revealed preference theory.

There is an important literature on "changing tastes" understood as time inconsistency. [START_REF] Strotz | Myopia and Inconsistency in Dynamic Utility Maximization[END_REF] is the first to uncover the problem of consistent planning and to investigate how should individuals with non-exponential discounting make dynamically consistent choices. Gul andPesendorfer (2001, 2005) and [START_REF] Dekel | Temptation-Driven Preferences[END_REF] provide behavioral foundations of preferences for commitment, namely choosing a smaller choice set for one's future self to avoid temptation. The main differences with our paper is that they consider deviations between expected behavior and actual behavior which are typically not deliberate (inconsistent) from the point of view of past selves. Instead, we look at preference changes that are deliberate but completely myopic, meaning that DM is unaware that she may change preferences in the future. The closest paper in this literature to our own is [START_REF] Nehring | Self-Control Through Second-Order Preferences[END_REF] who studies the revealed preference implications of second-order preferences as a self-control mechanism. The main differences with our paper are that he considers preferences over menus whereas we deal with preferences over alternatives, he does not introduce attributes, and the second order preferences act exclusively as a self-control mechanism whereas our meta-preference relation is completely general.

Our work relates to the literature on conflicting motivations-or justifiable choices-as we also obtain a representation with several (more precisely two) orderings. See among other contributions [START_REF] Kalai | Rationalizing Choice Functions by Multiple Rationales[END_REF], [START_REF] Heller | Justifiable choice[END_REF], De Clippel and Eliaz (2012), [START_REF] Cherepanov | Rationalization[END_REF], [START_REF] Dietrich | Reason-Based Choice and Context-Dependence: An Explanatory Framework[END_REF] and [START_REF] Ridout | Choosing for the right reasons[END_REF]. Despite this similarity, these works focus on static choice data that violates the usual rational requirements-namely the Weak Axiom of Revealed Preferences (WARP) or the Independence of Irrelevant Alternatives Axiom (IIA)whereas in our work, the choice data consists in an ordered sequence of choices on the same collection of menus of options. We explore two distinct situations, one in which within-period choices are represented by not necessarily transitive bi-nary relations, one in which within-period choices satisfy WARP. We focus on the irregularities in choices that arise between periods, hence the reversals can happen on the same menus. Furthermore, the time structure is used to rationalize the successive changes as being guided by a meta-maximization.

In the applied theory literature, the closest paper is [START_REF] Bernheim | A Theory of Chosen Preferences[END_REF]. Their model and ours share two important ideas. First, they argue that DM can choose "worldviews" which determine her valuation of future consumption streams. This is related to our concept of relevant attributes. Second, in their model DM is constrained by her "mindset flexibility" when changing worldviews. This echoes our constraint on awareness. For the purpose of falsification, our model makes some simplifications: in their model DM anticipates her preference change, and they allow for convex combinations of worldviews. Despite the differences in modelling assumptions, their paper is complementary with ours as we focus on the identification and falsification of deliberate preference changes. Other models of chosen preferences include [START_REF] Becker | The Endogenous Determination of Time Preference[END_REF], [START_REF] Akerlof | Economics and Identity[END_REF], [START_REF] Palacios-Huerta | A Theory of Markets, Institutions, and Endogenous Preferences[END_REF].

DELIBERATE PREFERENCE CHANGE 2.1 Preliminaries

There is finite set X of alternatives, that are defined by their attributes. Formally, there are K attributes and an alternative is a vector x = (x 1 , . . . , x K ) in the vector space R K whose k th -coordinate describes the value x k of the attribute k. 5 For any subset M ⊆ {1, . . . , K}, denote x M = (x k ) k∈M and x -M = (x k ) k / ∈M . 6 We require that for any attribute k there exist x and y such that x k = y k , as otherwise this attribute could be removed.

The analyst observes (i) the value of each attribute for all alternatives, and (ii) choices over options for T periods of time. The latter are represented by a sequence of complete orders ( t ) t=1,...,T , where t and ∼ t denote the asymmetric and symmetric parts, respectively. For the first part of the analysis, we do not require each t to be transitive. We investigate the implications of transitivity within periods -that is, DM's choices satisfy WARP-in section 2.6.

Example 1: Labor Market Discrimination. An employer wants to hire a worker. Her decision is based on the resume of each candidate that provides information on three attributes: (1) "education", (2) "experience", and (3) "gender" (1 for female and 2 for male). Therefore, a female college-educated worker entering the labor market is represented by x = (4, 0, 1), while a male non-educated worker with ten years of experience is represented by y = (0, 10, 2).

Revealed Relevant Attributes

The attribute-based approach allows us to identify which attributes drive DM's choice behavior. These "relevant attributes" are easy to identify when the choice set X is sufficiently rich: the attribute k is revealed relevant at t if there is a pair of alternatives x and y that only differ on the k th -dimension (x -k = y -k ) and such that x ∼ t y. In this case, we are sure that DM uses attribute k in her decision making. This richness assumption-that we can always find two alternatives that differ only on one dimension-would be too restrictive, however. We illustrate the construction of the revealed relevant attributes using our running example and then provide a formal definition.

Example 1 (continued): Suppose that z t x for the two candidates x = (4, 0, 1), z = (4, 2, 2). The idea is to identify a set of attributes M ⊂ {1, 2, 3} that has to be relevant to explain this strict preference. From z t x, we can conclude that M = {2, 3} is revealed relevant because (i) the alternatives differ on M and are identical outside of M , and (ii) there is no pair of alternatives that differ on a strict subset of M and are ranked strictly. The second point captures conservatism in our definition of revealed relevant attributes: if we cannot disentangle which attributes drive DM's behavior exactly, we keep all attributes in M . The following definition formalizes points (i) and (ii).

Definition 1 (Revealed Relevant Attributes).

A set M of attributes is revealed relevant at period t if:

(i) there exists x, y ∈ X with x -M = y -M and x k = y k for every k ∈ M , such that x ∼ t y;

(ii) for every M M and every w, z ∈ X with w -M = z -M , w ∼ t z.

Remark: if two attributes are systematically revealed relevant together, they might be coded into a single attribute (for instance if colors have been coded into different binary attributes). Let P t denote the collection of sets of revealed relevant attributes at period t. We denote m t ∈ {0, 1} K the vector of revealed relevant attributes such that

m k t = 1 if k ∈ M ∈Pt M and m k t = 0 otherwise. 7
We emphasize that an attribute can be revealed relevant, yet DM might be unaware that it causes her behavior. For instance, it is well known that implicit discrimination can have a strong impact on job performance [START_REF] Bertrand | Implicit Discrimination[END_REF][START_REF] Glover | Discrimination as a Self-Fulfilling Prophecy: Evidence from French Grocery Stores[END_REF][START_REF] Bertrand | Field Experiments on Discrimination[END_REF].

Principle of Sufficient Reason

We impose the following principle of sufficient reason: DM changes preferences if and only if the revealed relevant attributes change. The interpretation is that DM does not "wake up" with different preferences but must be able to justify her new preferences by making some attributes relevant or irrelevant. We view this as a normative principle: unjustified changes would not be normatively compelling.

Formally, the axiom states that if two alternatives x and x have the same relevant attributes between periods t and t -namely, if x • m t = x • m t where • denotes the element-wise (Hadamard) product-DM should rank consistenly x against the other alternatives in period t and x against the other alternatives in period t . RESTRICTED REVERSALS. Preferences ( t ) t satisfy Restricted Reversals if for any t, t , and for any x, x , y, y ∈ X such that x

• m t = x • m t and y • m t = y • m t , x t y ⇐⇒ x t y .
Remark: although we do not impose restrictions on the values of the attributes, the value 0 has a specific role. If no attribute can take the value 0, this axiom can simply be stated as: if m t = m t , then t = t . This would imply that DM changes her evaluation towards every option each time an attribute is made (ir)relevant. We stress that, although it makes sense to remove 0 for some attributes (e.g a category attribute coding the color of an item), it is not necessarily the case for attributes measuring the intensity of a property (e.g years of experience) or binary attributes that indicate whether a property is possessed or not (e.g whether a job applicant is a foreigner or not). This axiom therefore imposes that some alternatives be ranked similarly in different periods, even though the revealed relevant attributes might change.

Example 1 (continued). Consider four candidates x = (6, 2, 1), x = (0, 2, 1), y = (5, 0, 2) and y = (0, 0, 1). Suppose that the only strict rankings of 1 are x 1 x 1 y whereas the only strict ranking of 2 is x 2 y . It is verified that the vectors of revealed relevant attributes are m 1 = (1, 1, 0) and m 2 = (0, 1, 0) respectively. Observe that x • m 1 = x • m 2 , hence x and x have the same relevant attributes at periods 1 and 2. Similarly, y • m 1 = y • m 2 . Therefore, this sequence of choices violate Restricted Reversals, given that x 1 y whereas x ∼ 2 y.

A consequence of this axiom is the existence of a bijection between vectors of revealed relevant attributes and preference relations. Namely, this axiom is necessary and sufficient to represent the sequence of preferences ( t ) t by the sequence of revealed relevant attributes (m t ) t together with a time-independent binary relation over vectors of attributes. Formally, for any period t, let X(m t ) = {x • m t : x ∈ X} be the set of alternatives "filtered" through the revealed relevant attributes m t , and denote X = t X(m t ).

Proposition 1. Preferences ( t ) t satisfy Restricted Reversals if and only if there exists a complete binary relation ⊆ X2 (called the attribute ordering), such that for any period t and any x, y ∈ X:

x t y ⇐⇒ x • m t y • m t . (1)
The interpretation is that DM has a fundamental preference-called the attribute ordering-that, unlike her choices ( t ) t , does not change over time. This attribute ordering ranks vectors of attributes and does not depend on the relevant attributes. 8 The main consequence of Proposition 1 is that preference change can only be induced by changes in relevant attributes. Observe that the attribute ordering need not be transitive. We derive necessary and sufficient conditions for a transitive attribute ordering in Section 2.6.

Principle of Deliberation

The second normative principle that guides our analysis is a principle of deliberation: DM must evaluate all possible preferences at time t and consistently choose the best feasible one according to some criterion. This translates into an acyclicity axiom, which states that if DM changes her preference once, every future change should be due to the discovery of some new attributes-i.e that were not involved in the first change-and towards which DM has changed her attitude meanwhile.

ACYCLICITY. Preferences ( t ) t satisfy Acyclicity if for any t and any

t > t + 1, if m t+1 = m t , then there exists k such that m k t = m k t+1 = m k t .
Note that, as soon as several choice reversals are observed, the principle of deliberation implies the existence of a constraint on preference change. Indeed, would preference change be unconstrained, DM would directly reach her most preferred preference once and for all. We interpret this constraint as DM's awareness: she can change only the attributes she is aware of, that is, the ones she is able to question.

Example 1 (continued). Suppose that m 1 = (0, 0, 1) and m 2 = (0, 1, 0), namely the recruiter makes gender relevant but experience irrelevant at the second period. This could be because on the market men are more experienced, implying an unfair discrimination. Therefore, she must have been able to modify her relevant attributes (at least) on these two attributes. Acyclicity implies that she could never choose the following relevant attributes in the future: (0, 0, 0) and (0, 1, 1) as they were accessible between period 1 and period 2. Since she did not change the relevance of the education attribute, we conclude that she was not aware of this attribute at this point. Assuming for instance that education provides a fair criterion to rank the candidates, she could later on decide to remove again gender only if education is made relevant jointly, reaching m 3 = (1, 0, 0).

The Representation

The constraint on preference change in the representation is formalized by a sequence of vectors (a t ) T -1 t=1 , which represents DM's awareness between each period t and t + 1. Namely, a t ⊆ {0, 1} K for any t and codes as 1 attributes that DM can modify and as 0 the ones that she cannot modify between t and t + 1. An awareness vector a ∈ {0, 1} K together with a vector of relevant attributes m ∈ {0, 1} K defines a set of reachable attributes for the next period R(m, a):

R(m, a) = m ∈ {0, 1} K : for all k, a k = 0 implies m k = m k .
To state our main result, define for any set A and any linear order

P ⊂ A 2 , max(A, P ) = {a ∈ A | aP b, ∀b ∈ A}.
Theorem 1 (Representation). Preferences ( t ) t satisfy Restricted Reversals and Acyclicity if and only if there exists a complete binary relation ⊆ X2 , a sequence of awareness (a t ) t (with a t ∈ {0, 1} K ), and a linear order ⊆ {0, 1} K × {0, 1} K ,9 such that, for any t and any x, y ∈ X,

x t y ⇐⇒ x • m t y • m t , (1)
{m t+1 } = max(R(m t , a t ), ). (2)
The principle of sufficient reason together with the principle of deliberation are necessary and sufficient for what we name a deliberate preference change model. If the tuple ( , , m t , a t ) satisfy the conditions in theorem 1, we say that it rationalizes ( t ) t . In this model, DM's behavior is represented by the maximization of two binary relations: a preference relation on alternatives that together with the relevant attributes determine choices in each period (1) and a meta-preference relation on vectors of relevant attributes that determine the change of preference between periods (2). The revealed preference implication of our model is that when we observe choice reversals between alternatives x and y, we should observe other choice reversals on alternatives that share attributes with x and y. For instance if an employer stops discriminating at work this should impact her preferences in other contexts, such as her political preferences.

The fact that attributes can only be made relevant or irrelevant-and that DM cannot change her "taste" (attribute ordering) towards an attribute due to the stability of the attribute ordering-might seem arbitrary at first. This is however important for the testability of the model, as otherwise almost any sequence of observed choices could be rationalized by changing DM's tastes. Furthermore, if the space of attributes is correctly specified from the beginning, there is no need to change DM's tastes. For instance, if the employer makes "gender" irrelevant to avoid discrimination, but makes it relevant again in the future due to an affirmative action policy, this policy should be thought of an attribute that is complementary with the attribute "gender". Therefore, it is not that DM changes her tastes toward the attribute "gender", but that the combination of "gender" and "affirmative action" is strictly preferred to "gender" alone. This suggests that the specification of the attributes is a crucial step that the researcher should discuss carefully, and commit to before observing choice data to avoid ex-post rationalization.

What can be inferred if one of the two axioms is violated? First, a violation of Restricted Reversal indicates that preference changes do not arise from changes in DM's revealed relevant attributes. Indeed, it is a necessary and sufficient condition for the existence of a time-independent attribute ordering that rationalizes each period's preference together with a set of relevant attributes (proposition 1). Therefore, the analyst's knowledge of what determines DM's preference is incomplete: we may not observe all attributes, or the attribute ordering may change because DM discovers new consequences of an attribute for instance. Second, a violation of Acyclicity suggests that DM does not change her preferences rationally, meaning that no linear order can rationalize the sequence of meta choices. Canonical examples of non-deliberate preference changes are nudges, conformism or random utility. Alternatively, a violation of these axioms may suggest that the revealed relevant attributes are not the "truly" relevant attributes for DM, and her behavior could be rationalized by our model with a different sequence (m t ) t . 10Finally, we emphasize that our model is complementary with Bayesianism to explain preference change. Even though evidence suggests that agents do not always follow Bayes' rule, we do not think that an exhaustive theory of social interactions could do without belief updating. Instead, we argue that preference change and belief updating can occur simultaneously. This thesis receives empirical support in experiments on fake news by [START_REF] Barrera | Facts, Alternative Facts, and Fact Checking in Times of Post-Truth Politics[END_REF] (cited in the introduction).

Regarding the uniqueness of the ingredients of a deliberate preference change model, without further restrictions, only the attribute ordering is uniquely re-vealed and the meta-preference identified up to an arbitrary completion. Furthermore, any intersection of two rationalizing sequences of awareness can also rationalize preference changes.

Proposition 2 (Uniqueness). Let ( , , m t , a t ) and ( , , m t , a t ) rationalize ( t ) t . Then, any completion of ∩ and ∩ , together with (m t , a t • a t ) t also rationalize ( t ) t .

In Section 2.7, we derive sufficient conditions for the identification of the relevant attributes, in the case where the the preferences at each period are transitive as well as the attribute ordering. Hence we first study in the next section the characterization of deliberate preference changes with transitive attribute ordering.

Transitive Attribute Ordering

Our main representation theorem does not guarantee that the attribute ordering is transitive and does not require that the observed preferences ( t ) t are transitive. Indeed Restricted Reversals constraints choices only between pairs of periods which is not enough to guarantee transitivity. For instance, suppose that x, y ∈ X(m t ), y, z ∈ X(m t ) and x, z ∈ X(m t ) but z / ∈ X(m t ), x / ∈ X(m t ) and y / ∈ X(m t ). It could be that x t y, y t z and z t x because Restricted Reversals does not constraint choices on triplets of periods. In fact, this problem is more general and may arise with any number of periods strictly greater than two.

Transitivity of preferences is sometimes viewed as a condition for rationality, hence it might be of interest to characterize transitivity of the attribute ordering. The following axiom extends Restricted Reversals to address this problem.

STRONG RESTRICTED REVERSALS.

For any {t 1 , . . . , t n } and any {x k ,

x k } k=1,...,n such that, for k = 1, . . . , n -1, x k • m t k = x k+1 • m t k+1 and x n • m tn = x 1 • m t 1 , preferences ( t ) t satisfy Strong Restricted Reversals if: x k t k x k , for every k = 1, . . . , n -1 =⇒ x n tn x n .
Proposition 3. Suppose that preferences ( t ) t are transitive. Preferences satisfy Strong Restricted Reversals and Acyclicity if and only if there exists a deliberate preference change model ( , , m t , a t ) that rationalizes them with being a complete preorder.

Identification of the Revealed Relevant Attributes

The relevant attributes are typically not identified without further restrictions on preferences. This is the case because when we observe an indifference, we cannot always identify whether this is due to an attribute being irrelevant, or whether DM is indifferent towards this attribute in the attribute ordering. Denote M( t ) = {m : ∃ a preorder s.t. (m, ) rationalizes t } the set of relevant attributes that rationalize preferences at t using a transitive attribute ordering. To explore the structure of M( t ) we make the following richness assumption.

RICHNESS ASSUMPTION.

For all x, y ∈ X that differ only on a subset M of n attributes, there is a sequence of alternatives z 1 , . . . , z n ∈ X such that z 1 = x, z n = y and

z -k i = z -k i+1 for some k ∈ M , for all i = 1, . . . , n -1. 11
We show that, under the richness assumption and the transitivity of the preferences t , the set of vectors of relevant attributes m that can be used to rationalize preferences in the baseline model has a lattice structure. The most parsimonious vector is the vector of revealed relevant attributes m t ,12 but in principle other vectors could be used to rationalize DM's preferences. Proposition 4. Assume richness and suppose that preferences t are transitive. If Restricted Reversal is satisfied, M( t ) is a lattice ordered by ≥. Its minimum is the vector of revealed relevant attributes m t and its maximum is (1, . . . , 1).

This indeterminacy problem between irrelevant attributes and indifference can be solved if we impose that indifference are only caused by an attribute being irrelevant. In this case, an indifference x ∼ t y has a clear interpretation in the sense that there is no attribute that motivates DM to choose x over y. This is the content of the following axiom. JUSTIFIED INDIFFERENCE. Preferences ( t ) t satisfy Justified Indifference if for any t and any alternatives x, y ∈ X,

x ∼ t y =⇒ |x -y| • m t = (0, . . . , 0).
When Justified Indifference is satisfied and if we restrict attention to strict attribute ordering, the relevant attributes are uniquely identified by the revealed relevant ones. Formally, let M ( t ) = {m : ∃ a partial order > s.t. (m, >) rationalizes t } be the set of relevant attributes that rationalize preferences at t using a strict attribute ordering. When Justified Indifference is satisfied, we have M ( t ) = {m t }.

Theorem 2. Assume richness and suppose that preferences ( t ) t are transitive. Preferences satisfy Strong Restricted Reversal, Acyclicity and Justified Indifference if and only if there a deliberate preference change model (>, , m t , a t ) that rationalizes ( t ) t with > being a partial order. Furhermore, for any period t, M ( t ) = {m t }.

MOTIVATED PREFERENCE CHANGE

Our main representation theorem shows that preference change can be represented by the maximization of a meta-preference. The representation, however, does not provide a straightforward interpretation of the meta-preference. It could be that DM is changing her behavior to make it more aligned with her values, or she may change preferences to serve her own-interests instead of purely disinterested motives-this is referred to as motivated preference change. In this section, we investigate the latter idea. We show that motivated preference change admits a tractable functional representation-this proves convenient for applications in the next section.

First, we construct an extension of the attribute ordering which allows us to keep track of (i) preferences over perceived alternatives at period t, and (ii) preferences over perceived alternatives at period t if she were to change her preferences to make good alternatives even better.

Definition 2. Let a, b ∈ R K . Denote a t b if x • m t = a for some x ∈ X and (i) y • m t = b for some y ∈ X and x t y; or (ii) y • m = b for some y ∈ X, m ∈ R(m t-1 , |m t -m t-1 |), and x t z for all z ∈ X.
The following axiom, which extends Strong Restricted Reversals, guarantees that DM makes attributes relevant if and only if these attributes are valued positivelythat is, making these attributes (ir)relevant increases DM's utility.

MOTIVATED RESTRICTED REVERSALS. Preferences ( t ) t satisfy Motivated Restricted

Reversals if for any {t 1 , . . . , t n } and any (a k ) k=1,...,n ∈ (R K ) n such that a k+1 t k a k for k = 1, . . . , n -1,

a 1 tn a n =⇒ a 1 tn a n .
The next axiom guarantees that there are no indifference between vectors of relevant attributes when changing preferences. Intuitively, the axiom states that if there is a tie between two vectors m and m that yield identical utility, DM breaks the tie in favor of one vector by virtually increasing her utility for some alternative x ∈ X so that m becomes strictly preferred to m . MOTIVATED TIE-BREAKING. Preferences ( t ) t satisfy Motivated Tie-Breaking if for all t, all x ∈ max(X, t ), and all y, y ∈ X such that there exists

m ∈ R(m t-1 , |m t - m t-1 |) with y • m t = y • m • m t , y ∈ max(X, t ) =⇒ m = m t .
These two axioms are necessary and sufficient for the motivated preference change representation.

Theorem 3 (Representation). Suppose that preferences ( t ) t are transitive. Preferences ( t ) t satisfy Motivated Restricted Reversals and Motivated Tie-Breaking if and only if there exists a sequence of awareness (a t ) t and a function u : R K -→ R such that for all t and all x, x ,

x t x ⇐⇒ u(x • m t ) ≥ u(x • m t ), {m t+1 } = argmax m∈R(mt,at) max x∈X u(x • m).
As in the previous representation, DM chooses alternatives to maximize her attribute ordering, which can be represented by a utility function here. The main difference is that preference change must maximize DM's utility. Therefore, all attributes that are "negatively valued" will be made irrelevant as soon as possible, and all attributes that are "positively valued" will be made relevant as soon as possible.

AN APPLICATION

An important feature of the model is path dependence-that is, the order in which DM becomes aware of certain attributes has a strong impact on the path of preference change. We illustrate this aspect in a voting context: ex-ante identical voters deliberately ignore what other voters think is relevant because this would undermine their view of their preferred candidate. 13 Therefore, we show that our model can account for polarization of political preferences among ex-ante identical voters in a simple and intuitive way.

Polarization refers to disagreement on policy issues or distrust of the other party members among politicians and citizens [START_REF] Iyengar | The Origins and Consequences of Affective Polarization in the United States[END_REF]. There is now widespread agreement concerning the growing importance of ideological divisions both among politicized and educated voters as well as non-politicized citizens [START_REF] Abramowitz | Is Polarization a Myth?[END_REF]. There is no agreement, however, on the causes of polarization. 14From a Bayesian perspective, it is surprising that polarization increases as rational agents whose posterior beliefs are common knowledge cannot agree to disagree, even if their posteriors are based on different observed information about the world [START_REF] Aumann | Agreeing to Disagree[END_REF]. Arguing that voters have different priors certainly explains polarization, but it only moves the goalpost: where do differences in prior come from? Instead, our model provides a foundation for the concept of "partisan social identity" introduced in the political science literature [START_REF] Iyengar | Fear and Loathing Across Party Lines: New Evidence on Group Polarization[END_REF]. This theory captures the tendency of voters to classify opposing partisans as members of an outgroup and copartisans as members of an ingroup. We show that our model can account for the construction of such opposing groups, and how partisan cues can reinforce division.

We consider a very stylized model with motivated preference change. There are two voters i and j and two candidates:

x D = (x 1 , x 2 , x 3 ) and x R = (x 1 , x2 , x3 ) with x1 < 0 < x 1 , x 2 < 0 < x2 , x 3 < 0 < x3 and x2 -x1 > x 1 -x 2 .
The first attribute captures the candidates' support for social policies (e.g. health care), the second attribute captures how conservative candidates are, and the third attribute represents corruption. Voters are ex-ante identical: they both value integrity and prefer candidates with strong convictions (represented by a high absolute value of the difference between the first and the second attributes). We can represent their preferences as follows:

u(x • m) = x 1 m 1 -x 2 m 2 2 -x 3 m 3 .
They both initially start with the vector of relevant attributes (0, 0, 0). Suppose that voter i attends a political debate with both candidates: a i 1 = (1, 1, 0). She will change her preferences and value more candidate x R who has stronger convictions: the meta-maximization writes

max x∈X u(x • (1, 1, 0)) = x2 -x1 2 > max x∈X u(x • (0, 1, 0)) = x2 ) 2 > max x∈X u(x • (1, 0, 0)) = x 1 2 > 0 = max x∈X u(x • (0, 0, 0)).
Later, voter i becomes aware that candidate x R is corrupted: a i 2 = (0, 0, 1). She decides to ignore this information and keep this attribute irrelevant if:

max x∈X u(x • (1, 1, 1)) = max x2 -x1 2 -x3 , x 1 -x 2 2 -x 3 < x2 -x1 2 = max x∈X u(x • (1, 1, 0)).
i.e. whenever x2 -x1 2 > x 1 -x 2 2 -x 3 . Namely, whenever candidate x R has strong convictions that counterbalance her corruption. The intuition is that making "corruption" relevant would undermine her view of candidate x R . In the end, voter i's most preferred candidate is x R .

Instead, voter j first becomes aware of a felony committed by candidate x R : a j 1 = (0, 0, 1). She will change her preferences to make it relevant: the metamaximization writes

max x∈X u(x • (0, 0, 1)) = -x 3 > 0 = max x∈X u(x • (0, 0, 0)).
At this point voter j prefers the upstanding candidate x D .

Later, voter j attends a political debate with both candidates: a j 2 = (1, 1, 0). She will lean toward the candidate x D even though he has less convictions than the candidate x R whenever (x 1 -x 2 ) 2 -x 3 > (x 2 -x1 ) 2 -x3 . Namely, whenever the convictions of x R does not make up for his felonies. In the end, voter j's most preferred candidate is x D .

It is quite striking that two identical voters who become aware of the same attributes can become polarized. This arises due to the path dependence of preference change: past justifications can conflict with new justifications leading to rich dynamics.

APPENDIX

Proof of Proposition 1. We say that a pair (m, ) rationalizes if for any x, y ∈ X, x y ⇐⇒ x • m y • m. The proof of the necessity is straightforward and therefore omitted.

(Sufficiency). Assume that ( t ) t satisfy Restricted Reversals. First, we fix a period t and show that we can indeed construct an ordering t ⊆ X 2 (m t ) such that (m t , t ) rationalizes t . We define the two following binary relations on X(m t ):

> t = {(a, b) ∈ X 2 (m t ) : ∃ x, y ∈ X, a = x • m t , b = y • m t , and x t y}, t = {(a, b) ∈ X 2 (m t ) : ∃ x, y ∈ X, a = x • m t , b = y • m t , and x ∼ t y}.
By definition, t is reflexive and symmetric. We show that > t is irreflexive, i.e. for any x and y such that x = y and x • m t = y • m t , x ∼ t y. Indeed, x • m t = y • m t implies that x and y do not differ on any attributes k such that m k t = 1. Hence, if by contradiction we had x t y, then there should be a subset of attributes on which x and y differ (i.e. with m k t = 0) and that are revealed relevant. This contradicts the definition of m t . Now let a, b ∈ X(m t ), with a = b, and x, x , y, y ∈ X such that x • m t = x • m t = a and y • m t = y • m t = b. Applying Restricted Reversal with t = t , we obtain x t y ⇐⇒ x t y . Given that > t is irreflexive, this establishes that it is asymmetric. It also proves that > t ∩ t = ∅.

Therefore, the relation t := t ∪ > t is complete on X(m t ) (by the completeness of t ) ; t and > t being respectively its symmetric and asymmetric parts. Furthermore, (m t , t ) rationalizes t .

Second, we show that for any two distinct periods t and t , t does not contradict t . Let x, x , y, y be such that x • m t = x • m t =: a and y • m t = y • m t =: b.

Then by Restricted Reversal we have,

a t b ⇐⇒ x • m t t y • m t ⇐⇒ x t y Restricted Reversal ⇐⇒ x t y ⇐⇒ x • m t t y • m t ⇐⇒ a t b
Finally, define := t t . By the previous argument, ∩X 2 (m t ) = t , so for any t, (m t , ) rationalizes t . Furthermore can be innocuously completed on X. We verify that is asymmetric. Suppose that m m and take t < t, such that m ∈ R(m t , a t ) and m = m t . First let us show that there cannot be t > t such that m = m t . Assume by contradiction that such a t exists. Then, we have

|m t -m t | = Def. m t |m -m t | ≤ m ∈R(m t ,|m t +1 -m t |) |m t +1 -m t |
where | • | is the element-wise absolute value: for any vector

b ∈ R K , |b| = (|b 1 |, . . . , |b K |). This means that for all k, if m k t = m k t +1 , then m k t = m k t = m k t +1
. Thus Acyclicity implies that m t +1 = m t = m . But, then we still have that m ∈ R(m t +1 , |m t +2 -m t +1 |) so that, applying the previous reasoning inductively, we obtain m = m t +2 = m t +3 = . . . m t = m = m . A contradiction. Second assume by contradiction that m = m t and m ∈ R(m t , |m t +1 -m t |) for some t , t such that t < t < t. By the same argument, Acyclicity would then imply that m = m t +2 = m t +3 = . . . m t = m = m. A contradiction.

We now verify that is transitive. Suppose that m m and m m . Then 20 there exist t, t with t > t , such that, m = m t and m = m t . Moreover,

m ∈ t :t <t R(m t , a t ) ⊆ t :t <t R(m t , a t )
where the inclusion follows from t > t . We conclude that m m , implying the transitivity of . Moreover, by the definition of , m t+1 = max(R(m t , a t ), ). By Szpilrajn's theorem, the meta-preference can be completed on {0, 1} K × {0, 1} K .

Proof of Proposition 2. For any t, any a, b ∈ X(m t ), a b if and only if there exist, x, y ∈ X such that x • m t = a, y • m t = b and x t y, which in turn is true if and only if a b. Which establishes that ∩X 2 (m t ) = ∩X 2 (m t ) for any t. Therefore, any completion of ∩ together with m t rationalizes t for any t. We next show that by considering ∩ and the sequence of awareness (a t • a t ) t , we can rationalize the meta-choices of each period t. Fix a period t, and suppose that being at m t , DM faces the meta-menu R(m t , a t • a t ). Note that R(m t , a t • a t ) = R(m t , a t ) ∩ R(m t , a t ). Hence it implies that (m t+1 , m) ∈ ∩ for any m ∈ R(m t , a t • a t ). This completes the proof than any completion of ∩ , together with ( ∩ , m t , a t • a t ) t rationalize ( t ) t .

Proof of Proposition 3. (Necessity.) Suppose there exists a complete preorder such that for every t, (m t , ) represents t . Take any {t 1 , . . . , t n } and any {x k , x k } k=1,...,n such that, for k = 1, . . . , n -1: x k • m t k = x k+1 • m t k+1 , x n • m tn = x 1 • m t 1 , and for every k ≤ n -1, x k t k x k . The latter implies that x k • m t k x k • m t k . Hence by the transitivity of , we can conclude that x n •m tn = x 1 •m t 1 x n-1 •m t n-1 = x n •m tn , i.e. x n tn x n . Hence Strong Restricted Reversal is satisfied.

(Sufficiency). We fix a period t and show that we can construct a complete preorder t ⊆ X 2 (m t ) such that (m t , t ) rationalizes t . We define t in the same way as in the proof of proposition 1. Given that Strong Restricted Reversal implies Restricted Reversal, the same arguments apply and we conclude that (m t , t ) rationalizes t . Furthermore, the transitivity of t is a direct consequence of the transitivity of t . We need now to construct a complete preorder on X that is time-independent.

From the proof of proposition 1, we know that for any two distinct periods t and t , t does not contradict t . We define 1;T := t t . We know therefore that for any t, 1;T ∩X 2 (m t ) = t .

We next show that the transitive closure of 1;T , denoted C 1;T , can rationalize the sequence ( t ) t together with the sequence (m t ) t . Namely, we show that for any period t, any a, b ∈ X(m t ), if b > t a, there cannot be a sequence (a k ) k=1,...,n and (t k ) 1≤k≤n-1 such that a 1 = a, a n = b and a k t k a k+1 . Let suppose by contradiction the existence of such a sequence. If t 1 = t, then complete the sequence with a 0 = a and t 0 = t; similarly, if t n-1 = t, then complete the sequence with a n+1 = b and t n = t. Therefore, w.l.o.g we consider the sequence (a k ) k=0,...,n+1 .

If t k = t k for some k = k , we show that we can restrict to a subsequence (a τ (k) ) k=1,...,n+1 with τ (0) = 0, τ (n+1) = n+1, such that τ (i) = τ (j) =⇒ t τ (i) = t τ (j) . Let suppose that t k = t k with k < k and that for any k ≤ i, j < k , if i = j then t i = t j . Let's consider the sequence (a i ) k≤i≤k +1 . There exists a sequence (x i , y i+1 ) k≤i≤k such that x k • m t k = a k , y k +1 • m t k = a k +1 , for any k ≤ i ≤ k -1, y i+1 • m t i = x i+1 • m t i+1 = a i+1 , and for any k ≤ i ≤ k , x i t i y i+1 . By applying Strong Restricted Reversal, this must be that x k t k y k +1 , i.e. a k t k a k +1 . Therefore, from the sequence (a k ) k=0,...,n+1 , we can construct a subsequence (a τ (k) ) k=0,...,n+1 , with τ (0) = 0, τ (n + 1) = n + 1, τ (i) = τ (j) =⇒ t τ (i) = t τ (j) , and such that for any k with τ (k) = τ (k + 1), a τ (k) τ (k) a τ (k+1) . From a similar reasoning, we conclude by Strong Restricted Reversal that a t b, a contradiction.

By an implication of Szpilrajn's theorem (see Corollary A.1 in [START_REF] Ok | Real Analysis with Economic Applications[END_REF]), there exists a complete, transitive and reflexive binary relation that extends C 1;T . We denote it . We proved that for any t, X 2 (m t )∩ = t , hence (m t , ) rationalizes t .

Figure 1 :

 1 Figure 1: The Dynamics of Deliberate Preference Change.

  Proof of Theorem 1. (Necessity). We only prove the necessity of Acyclicity. Let t and t such that t + 1 < t and assume m t = m t+1 . By equation (2), m t m t+1 and, thus, m t / ∈ R(m t , a t ). Hence, m k t = m k t and a k t = 0 for some attribute k. Yet, a k t = 0 and m t+1 ∈ R(m t , a t ) imply that m k t = m k t+1 , and thus m k t+1 = m k We know from proposition 1 that there exists an attribute ordering ⊆ X2 , such that for any period t, (m t , ) rationalizes t . Define the sequence of awareness as a t = |m t+1 -m t | for any t; and the revealed meta-preference relation as follows: m m if m = m and there exists t, such that m = m t and,

	m ∈

t (Sufficiency).

t : t <t R(m t , a t ).

Implicit discrimination would also imply that the attribute "gender" is relevant. Therefore, an attribute can be relevant even if DM does not consciously use this attribute.

We discuss why it would be problematic that DM changes her "taste" towards the attributes in section 2.5.

That is, if two distinct attribute orderings (resp. meta-preferences) rationalize the preference changes, any completion of their intersection do so.

All proofs until section 2.6 are in the Appendix. The remaining proofs can be found in the

Supplement[START_REF] Boissonnet | Supplement to "Revealed Deliberate Preference Change[END_REF].

Attributes can either code different categories (e.g colors, sex, etc.), indicate whether a property is possessed by the alternative (e.g whether a job applicant is a foreigner or not), or measure the intensity of a property (e.g years of experience of a candidate).

If M = {k} is a singleton, we simply write x -k instead of x -{k} .

Our definition of revealed relevant attributes is analogous to the definition of a non-null state in expected utility theory (taking the attributes as states and the alternatives as acts).

In a slightly different framework,[START_REF] Dietrich | A Reason-Based Theory of Rational Choice[END_REF] provide an equivalence result between this separability condition (their axiom 2) and the existence of an attribute ordering.

It is observationally equivalent to construct a linear order or a complete preorder together with a tie-breaking rule for the meta-choice such that if m t = m and m t = m for some t > t, then m τ = m for all τ > t .

Note that if one does not want to restrict attention to revealed relevant attributes, it is possible to write axioms on multiple "candidate" sequences of relevant attributes (details available upon request).

For this assumption to be satisfied, it might be necessary to regroup certain attributes. For instance, splitting a category attribute (e.g color) into binary attributes will violate this assumption.

If X is not rich, the vector of revealed relevant attributes m t need not be the minimum of the lattice.

Note that Bayesian updating cannot induce this type of path dependence because it is order invariant[START_REF] Cripps | Divisible Updating[END_REF].

Recent finding suggests that the emergence of the internet or rising economic inequality are less plausible causes than changes that are specific to the US-e.g., changing party composition, growing racial divisions, or the emergence of partisan cable news[START_REF] Boxell | Cross-Country Trends in Affective Polarization[END_REF].

A. Ghersengorin thanks ANR-17-CE26-0003 for its support. S. Gleyze acknowledges the support of the EUR grant ANR-17-EURE-0001.


Monique Tillard
email: mtillard@univ-montp2.fr

A phase width for CaGaSn. Crystal structure of mixed intermetallic Ca 4 Ga 4+x Sn 4-x and SmGa x Sn 3-x , stability, geometry and electronic structure

Keywords: Intermetallic compounds, Tin, Gallium, Crystal structure, DFT calculations 3

A phase width for CaGaSn. Crystal structure of mixed intermetallic Ca4Ga4+xSn4-x and SmGaxSn3-x, stability, geometry and electronic structure

Introduction

Numerous ternary combinations that involve tin and gallium are known and structurally characterized. Among the reports listed in the Pearson crystal database, a great number of compositions mainly includes Ba and adopts the famous clathrate type structures. In these ternary compounds, Sn and Ga atoms are 4-bonded within tridimensional networks composed by large cages that enclose Ba atoms. These arrangements are affected by some atomic disorder since Sn and Ga at cage vertices are statistically mixed at metal atoms positions. Various structures are described for the remaining compounds that contain p-block or transition elements. The M V -Sn-Ga compounds have been subject of interest for their superconducting behavior [1], [2], [3], [4] and [5] and the Sn/Ga atom distribution in structures would play a role in their properties.

Combinations that involve electropositive elements display more or less complicated intermetallic frameworks, as for example, layers in SrGaSn [6], double layers in Li5Ga4 [7], infinite chains of clusters in Na10Ga6Sn3 [8] or complex 3D network in Na3Ga8Sn3 in which coexist clusters and waved layers [9]. In most cases, Sn and Ga atoms are 4-bonded within networks and Sn/Ga atom disorder is often reported.

Only two compounds with very simple stoichiometries contrast with this general trend, SrGaSn and CaGaSn [6]. The structure of SrGaSn is described without atomic mixing and was solved from single-crystal diffraction while CaGaSn was stated isostructural on the basis of its similar powder pattern. Their hexagonal structure is related to the AlB2 structure-type in which alkaline-earth atoms occupy Al positions while Sn and Ga alternately fill the boron sites. Each metal atom on the hexagon network is then linked to three neighbors different in nature, it results in a waving of the layers which remain without any bonding interactions, as attested by shortest interlayer distances of 3.30 and 3.98 Å at Ga-Ga pairs in SrGaSn and CaGaSn, respectively. These structures fairly remind those of Li3Ga2 [10] and Li5Ga4 [7] Zintl compounds also deriving from the AlB2 structure. Layers therein are puckered and composed of 3-bonded Ga atoms (partially reduced by Li) at two independent crystallographic sites. Beyond the atom nature, the structural arrangements in Li3Ga2 and Li5Ga4 differ from that in SrGaSn by the staggered position of consecutive layers. Since layers are shifted against each other, interlinking is prevented in Li3Ga2. Instead, the shift two-by-two of the layers in Li5Ga4 allows association into double layers through Ga-Ga bonding of 2.68 Å.

The present work reports on two ternary intermetallic compounds of tin and gallium for which single crystal structures are determined. The compound SmGaSn2 displays an original network for a ternary composition and the hexagonal structure of Ca4Ga4.9Sn3.1 raises questions about the existence of a phase width for CaGaSn. Structural descriptions, network and bonding analyses are supported by DFT calculations.

Experimental section

The elements Ga (Rhone Poulenc, 6N), lumps of Sn and Sm (Alfa Aesar, 99.9%), Ca (Merck, 99%) were taken without further purification. Stoichiometric amounts were weighed to prepare alloys at compositions SmGa3Sn2 and Ca5Ga9Sn4, they were inserted in Ta tubes weld-sealed under argon atmosphere and protected from oxidation in Ar-filled sealed stainless steel jackets. Samples were heated to 900 °C for 16 h in a classical tubular furnace and then submitted to slow cooling at 10 °/h. The well crystallized products were handled under argon atmosphere, several crystals could be selected using a microscope placed in the glove box and then sealed into capillaries to be checked for crystallinity. The best diffracting single crystals that display the required quality were used for X-ray diffraction data collection and were further analyzed to establish their composition. EDX measurements were performed using an Oxford Instrument Environmental Scanning Electron Microscope, equipped with an X-Max large area SDD sensor that allows excellent sensitivity/precision/resolution. The EDX analysis led to Sm:Ga:Sn and Ca:Ga:Sn ratios of 26.4:24.1:49.5 and 33.5:37.9:29.1, respectively. Crystal structures were solved [START_REF] Sheldrick | SHELXS 97. A Program for Crystal Structures Solution[END_REF] and refined [START_REF] Sheldrick | SHELXL97: A Program for Refining Crystal Structures[END_REF] using SHELX programs. Details on single crystal data collection and structural refinements are given in table 1. Atom disorder Sn/Ga was considered for the two compounds and site occupation factors were refined together with atom positions and anisotropic displacement parameters. Within the standard deviation limits, site occupation at all positions does not deviate from full site occupancy. The final refined compositions from X-ray study are SmGaSn2 and Ca4Ga4.9Sn3.1, respectively. They quite well agree with atom proportions found from the EDX analysis of the single crystals previously used for diffracted intensity recording. It is noteworthy that Ga content is fairly lower in compounds than in the prepared alloys, this may underline the role of a Ga excess in the crystallization processes. The final refined atom positions and equivalent displacement parameters are given in Table 2. Further details on crystal structures may be obtained from FIZ Karlsruhe (crysdata@fiz-karlsruhe.de) on quoting the CSD deposition numbers 431519 (Ca4Ga4.9Sn3.1) and 431520 (SmGaSn2).

Calculation Method

Calculations at the DFT level have been performed with the code CASTEP [START_REF] Kresse | [END_REF] using the gradientcorrected GGA-PW91 exchange and correlation functional [14]. CASTEP uses plane-wave basis sets to treat valence electrons and pseudo potentials to approximate the potential field of ion cores.

Ultra-soft pseudo potentials (USPP) generated for each element according to the Vanderbilt [15] scheme were chosen. Tin was taken with four 5s 2 5p 2 valence states and calcium with ten 3s 2 3p 6 4s 2 valence states. The inner 3d and 4f levels were respectively considered for Ga 3d 10 4s 2 4p 1 and Sm 4f 6 5s 2 5p 6 6s 2 . Kinetic cut-off energies were set at ultrafine qualities (330 and 420 eV) and a Monkhorst-Pack grid of automatically generated k-points was used for numerical integration in the Brillouin zone [16].

All along this work, both atom positions and unit cell dimensions were fully relaxed in the geometry optimizations performed at the DFT level by minimizing the total energy. Calculations were carried out within the experimental cubic Pm3 m symmetry for Sm compounds, instead the P3m1 symmetry (which is common to all arrangements) was used for the hexagonal models in cells containing four layers.

Density of states DOS and crystal overlap populations COOP have also been calculated for CaGaSn using a rigid band model within the tight binding extended Hückel method provided in CAESAR2 package [START_REF] Ren | CAESAR2 Package for Windows. Crystal And Electronic Structure AnalyseR[END_REF].

Results and Discussion

The structural representation of compounds SmGaSn2 and Ca4Ga4.9Sn3.1 given in figure 1 highlights the differences in p-block sub-lattices yet characterized with a common feature, namely the occurrence of Sn/Ga atomic mixing.

SmGaSn2 structure

Compound SmGaSn2 crystallizes within cubic Pm 3 m symmetry with a = 4.5778(8) Å. Its structure is easily visualized as a 3D network of octahedral units sharing all vertices. Sn and Ga are statistically mixed on octahedra at the unique 3c (0,½,½) position and separated by 3.267 Å (octahedra edges) from alike neighbors. At origin of the cell, Sm atom is located at the center of a cuboctahedron formed by 12 Sn/Ga atoms. The Sn to Ga atomic proportion, freely refined in the structural refinement, converges to 2 and then the compound can be formulated Sm[GaSn2]. Its structure belongs to the AuCu3-type and is so far not observed for ternary tin and gallium combinations. Some isostructural ternary compounds including Sn have been already reported with atom mixing in their structure but it involves only the electropositive atom positions [START_REF] Wendorff | [END_REF]. This structural type is also adopted by SmSn3 and NpSn3, two binary compounds known as intermetallic heavy fermions whose interest is to display superconducting behavior at low temperature [19]. Note also that no structure has been reported in the literature for SmGa3, a stoichiometry yet included in the composition domain of ε-SmGax (2 < x < 4) with structures reminiscent of AlB2 type [20] and [21]. Even simply regarded as derived from SmSn3 by atomic substitution, compound SmGaSn2 deserves some attention since incorporation of gallium and occurrence of atom disorder in its structure may affect the physical properties. Replacement of Sn by Ga occurs at the unique crystallographic site 3c which is statistically occupied by one third of Ga atoms within the cubic symmetry (no additional diffraction spots). This atom substitution is responsible for the 6.5% contraction of the lattice volume compared to experimental SmSn3.

Geometries optimized at the DFT level (conditions described above) are in agreement with this lattice contraction and confirm that the cubic symmetry is retained for both SmSn3 and the ordered SmGaSn2 models. The density of states calculated for SmSn3 and for SmGaSn2 are very similar, particularly displaying a peak intensity at Fermi level.

Ca4Ga4.9Sn3.1 structure

The unit cell of Ca4Ga4.9Sn3.1 with dimensions a = 4.2233( 9) and c = 17.601(7) Å displays the hexagonal P63/mmc symmetry and belongs to the YPtAs-type as do the isostructural CaGaSn (a = 4.44, c = 17.53 Å [6]) and SrGaSn (a = 4.55, c = 18.71 Å) [6] and [22]. These structures derive from the hexagonal P6/mmm structure of AlB2 with Ca or Sr lying at the Al positions in interlayer space while Sn and Ga share the boron sites on hexagon layers. Subsequently layers in these ternary compounds are heteroatomic and no longer planar, thus the structural type YPtAs may be viewed as a superstructure of AlB2 by atomic differentiation and hexagon tilting. In the structure of Ca4Ga4.9Sn3.1, Ca fills 2a and 2b sites while heavier Sn and Ga are located at two cristallographically independent 4f positions. During the structural refinement, mixing of Sn and Ga atoms was considered at these 4f sites but it rapidly appeared that the substitution of Sn for Ga preferentially occurs at M1 site. At this position, the proportion of Sn reaches ~80 % while it remains close to zero at M2. This leads to a refined composition Ca4Ga4.7Sn3.3 which is fairly close to Ca4Ga4.5Sn3.5, the composition established by analysis for the studied single crystal. However, according to standard deviations, the very low Sn content at M2, refining to 0.001 (6), must be regarded as zero. Therefore, the final refinement has been performed with the M2 site entirely filled with Ga, leading to the slightly changed formulation Ca4Ga4.9Sn3.1, therefore used for the compound.

Layer packing in hexagonal structures

Before further analyzing the structural and geometrical parameters, let us have a look at the hexagon layers and their packing in various hexagonal structures. In the AlB2 parent [23], boron atoms are arranged within graphite-like planar layers stacked on top of each other and the Al atoms in the interlayer space are directly placed below and above hexagon centers. Boron atoms therein are 3-bonded and involved in B-B bonds of 1.79 Å within the layer while no bonding interaction occurs between layers separated by 3.26 Å. Note that the layer separation is reduced to 3.08 Å in isostructural MgB2 [24] known with special superconducting properties. Likewise, some ternary compounds are built with planar hexagon layers within the BaLiSi structure-type [25], as SrAlSi [26] or SrAlGe [27], or within the ZrBeSi structure-type [28], as LiBC [29]. One can note that superconductivity of some of these compounds, as for example CaAlSi and SrAlSi, has been investigated for their structural similarity with MgB2 [30].

Instead, layers may also deviate from flatness and stack in various ways as for instance with the same orientation in EuGe2 structure-type [31] or with opposite orientations in CaIn2 structure-type [32] (figure 2).

A great number of ternary compounds are found to display puckered layers of hexagons.

Nevertheless, the atomic composition, the ordering of atoms, the relative arrangement of layers and the level of distortion produce deviations to symmetry. Consequently the resulting structures belong to various space groups. A comprehensive study provides a description of structural relationships for structures originating from the AlB2-type and underlines the various distortions being the expression of flexibility and variety of the chemical bonding in these superstructures [33]. Among compounds that adopt the YPtAs structural type, one can cite the LnZnSn family of intermetallic compounds characterized with an ordered atom arrangement and a quasi-constant Zn-Sn bond length within the layers [34] and [35] whilst the Zn-Zn interlayer separation gradually increase along the series. For the ordered YbGaGe compound, a tolerance of a certain compound deviation has been evidenced [36] and [37] leading to occurrence of atom mixing in layers. Also interesting is the effect of atomic nature on the tendency of hexagon networks to pucker. Thereby Si and Ga are randomly distributed in plans of hexagons in EuGaSi structure of AlB2-type while Ga and Ge (or Sn) are arranged orderly in EuGaGe (or EuGaSn) structures of YPtAs-type [38].

Structures of Eu(Ga1-xTx)2 compounds supply a beautiful example of structural evolution with the composition, from the orthorhombic KHg2 to hexagonal AlB2 and further to tetragonal ThSi2 types, when T = Si. Instead for T = Ge, it changes step by step from KHg2 for hexagonal YPtAs and finally for trigonal EuGe2 via complex and disordered higher superstructures [39]. , BaLiSi [25], ZrBeSi [28], EuGe2 [31] and CaIn2 [32] structure-types

It is then convenient to define some useful parameters to compare the different structures and specially to quantify the geometry distortions. The flatness deviation of layers can be expressed by the angles αn and the relative position of successive layers by the interlayer distances dn (figure 3).

In the AlB2 parent structure, with planar equidistant layers, αn angles are 120° and all dn distances are equal to 3.26 Å. Starting from a planar sp 2 configuration with a 120° angle, the puckering of hexagon layers will reduce the value of the angle towards 109.45° which corresponds to the tetrahedral sp 3 configuration. A more substantial shift occurs in Li5Ga4 [7] for Sr4Ga4Sn4 from the available data [6] and [22]. The small lengthening in Sr4Ga4Sn4 could be attributed to the increase in radius of the electropositive element (1.34 for Ca 2+ and 1.44 for Sr 2+ ionic radii [41]). Likewise, the value of α angle of 112.1° is very close to that reported for Sr4Ga4Sn4 combination (111.1 or 111.6°). Noteworthy are these values of angle, higher than in Li combinations mentioned above, that indicate a weaker waving of the layers with the heavier electropositive elements.

It is important at this stage to point out the discrepancy in geometrical features for the two isostructural compositions of the Ca compound, however both consistent with the Ca4Ga4+xSn4-x formula. Curiously a much larger distance of 3.98 Å is reported for Ca4Ga4Sn4 [6] with also a very high angle of 118.8° meaning that layers are almost planar (only deviating by 1.2°). Seemingly, an increase by ~1 in the Ga-content (from Ca4Ga4Sn4 to Ca4Ga4.9Sn3.1) amplifies the waving of layers and reduces the distance between layers. At the same time, within layers, M-Ga bonds are elongated from 2.579 Å to 2.667 Å when Ga partially replaces Sn at M1 site. This does not appear really consistent with the covalent radii of the elements (1.26 Å for Ga and 1.40 Å for Sn) and might suggest a non-zero probability for the substitution of Ga for Sn at M2 site in Ca4Ga4.9Sn3.1.

Despite that, it is quite confusing to find larger differences in geometry between Ca4Ga4.9Sn3.1 and Ca4Ga4Sn4 differing by the layer content than between Ca4Ga4.9Sn3.1 and Sr4Ga4Sn4 contrasting by the nature of species in the interlayer space. These observations tend to indicate that size of the electropositive component would have a weak influence on the structural geometry, but comparison with the structure of BaGaSn does not confirm this statement. According to the literature, BaGaSn (Ba4Ga4Sn4) is indeed described in the hexagonal P63/mmc symmetry with weakly puckered layers (α = 117.7°) in which Sn and Ga are mixed in equal proportion at all positions [22] and [42]. Moreover, BaGaSn lattice is smaller, with a = 4.58, c = 10.33 Å and the orientation of layers does not agree with the YPtAs structure-type. At a first glance, one can imagine that a larger supercell might have been missed, but a detailed diffraction study unambiguously established that BaGaSn should be considered as a ternary disordered compound, isostructural with CaIn2 (figure 2).

Trying to understand these peculiarities, the geometry has been optimized for several ordered models (S1, D1, S3, D3 and D5, see in figure 5) built in a double cell that contains four layers.

Lattices do not deviate by more than 2% from the experimental, and planar geometry is never retained for the layers. The lowest total energy is calculated for model D3 associated to YPtAs structure-type but it only differs by 0.02 and 0.09 eV from the energies computed for models S3

and D5 associated with CaIn2 structure-type. One could then conclude that, only on an energetic point of view, an ordered Ba4Ga4Sn4 arrangement can exist under several geometries, a result which is likely to be modified by atomic disorder at all sites.

A composition domain for Ca4Ga4Sn4?

The characterization of a crystal whose composition Ca4Ga4.9Sn3.1 deviates from the already known Ca4Ga4Sn4 raises questions about coexistence in the ternary Ca-Sn-Ga system of two "line compounds" with close compositions and same structure. It is more reasonable to believe that these stoichiometries are included into a solid solution domain around CaGaSn, but the limits of such a domain remain unknown. Since no report was found on CaSn2 structure, the domain should be a priori limited on the tin-side. On the other hand, the existence of CaGa2 (P63/mmc, a = 4.4, c = 7.3

Å [43]) could let foresee an extension on the Ga-rich side. Nevertheless, in spite of its hexagonal structure, CaGa2 cannot be seen like a substructure of such a compound because of differences in the orientation of layers, which corresponds either to S3 arrangement in CaGa2 or to D3 arrangement in Ca4Ga4+xSn4-x (figure 5). Within a solid solution domain, the lattice dimensions are expected to follow a Vegard's law, especially in case of a progressive substitution phenomenon.

The c parameter of the hexagonal cell, directly related to the layer stacking, is then likely to vary significantly with the substitution. Remembering that substitution is only found at M1 site in Ca4Ga4.9Sn3.1, it is plausible that the domain would be limited to 0 ≤ x ≤1 for the Ca4Ga4+xSn4-x formulation. To evaluate the probability for CaGa2 to also exist within D3 configuration, its geometry was optimized starting from experimental S3 and hypothetical D3 models in 1×1×2 supercells. The total energy per CaGa2 formula unit differs by 0.13 eV in favor of the experimental S3 configuration, this corroborates the assumption of a limited phase width around Ca4Ga4Sn4.

Geometry and stability of Ca4Ga4+xSn4-x models

Considering the many possibilities to stack hexagon layers which provide structures being different by the atom arrangement and the orientation of layers, a theoretical approach is likely to improve our understanding of Ca4Ga4+xSn4-x. The study attempts to establish the relative stabilities of various models based on the experimental structure and to evaluate their dependence on composition. Models built from experimental lattice include four layers and correspond to one Ca4Ga4Sn4 formula unit. The repeat unit contains four atom pairs of A-type (Ga-Sn) or B-type (Sn-Ga) packed along the hexagonal c-axis and four Ca atoms used in calculations but not drawn in the schematic representations given in figure 5.

Packing alternately the layers in ABAB mode leads to S models while AABB mode gives D models. Note that orientation of the layers is identical in models S2, D2, D4 and in the Ca4Ga4+xSn4-

x experimental D3 configuration. While the planar model S1 corresponds to ZrBeSi structure-type, the puckered model S3 agrees with NdPtSb structure-type [44]. Instead, models S0, D1 and D5 can be seen as ternary derivatives of EuGe2, AlB2 and CaIn2, respectively (see in figure 2). Geometry has been optimized for nine models in the conditions described previously within the common P3m1 symmetry and in some cases additionally within the actual symmetry. The optimized lattice dimensions, the characteristic distances and angles are given together with total energies and formation enthalpies in supplementary material. The formation enthalpy which allows a good evaluation of the stability is defined as total energy of the compound minus total energies of the elements, calculated in their solid state structures.

The total energy and the formation enthalpy calculated for Ca4Ga4Sn4 with various arrangements are given in Figure 6. It clearly appears that models D1 and D3 distinguish from all other configurations by their high negative values of enthalpy/energy, a sign of stability. One must remark that starting arrangement (i.e. layer orientation) is maintained for all the optimized except planar models. The planar model S1 (ABAB packing mode) evolves towards the S0 "in-phase"

waving geometry and the model D1 (AABB packing mode) converts upon optimization to the D3 experimental geometry. Note that all the symmetry constrained calculations -P63/mmc, P3m1 and P1-for the D3 experimental arrangement led to optimized geometries that perfectly obey the P63/mmc symmetry as checked by using the "find symmetry" tool under the most strict (ultra-fine) tolerance conditions.

Structural accommodation of Ca4Ga4+xSn4-x

According to the present and the previous structural studies, Ca4Ga4+xSn4-x is found to exist at least for x = 0 and x ~ 1, with similar lattices and geometries. The question is how the structure can accommodate the composition and then the electron-content variations. A simple picture of the bonding is provided by a rigid band model calculation for the experimental arrangement in which Sn totally fills the M1 site and Ca is considered only as a potential electron donor.

The band structure computed for this network (figure 7) displays a small direct gap for an electron content of 36 that corresponds to the composition Ca4Ga4Sn4. A look at the crystal orbital overlap populations is more informative indicating the nature of interactions in a specific energy range.

The overall interactions are bonding up to 35.6 and clearly antibonding above 36 electrons, then filling the bands up to 36 electrons (Ca4Ga4Sn4) leads to an optimal situation in terms of bonding.

Nevertheless, the total overlap population has an extremely weak negative value between 35.6 and 36 and is naught for 35.6 electrons. An equilibrium is then reached also for an electron content of 36.5 which precisely corresponds to the composition Ca4Ga4.5Sn3.5 determined from analysis of our single crystal. Within the energy range from 35.6 to 36 electron-filling, the bands mainly result from pz atomic orbitals contributions and display rather nonbonding character. These bands are able to accommodate the electron content variations and are greatly involved in puckering of the hexagon layers. The Sn-Ga interactions within layers display a weak antibonding character between 34.5 and 36 electrons while Ga-Ga interactions between the layers are bonding all over the energy range with a maximum overlap population just above 35 electrons. Note that with 37 electrons the Sn-rich composition Ca4Ga3Sn5 would display clearly antibonding overall interactions. Actually, CASTEP band structures calculated at the DFT level for Ca4Ga4Sn4 and Ca4Ga5Sn3 are globally similar and characteristic of a metallic behavior (see in supplementary material). No band crosses the Fermi level at the HK segment of the irreducible Brillouin zone (also at ML segment for Ca4Ga5Sn3), thus compound Ca4Ga4+xSn4-x is expected to display some anisotropy in its electrical conduction.

Conclusion

The structure of compounds SmGaSn2 and Ca4Ga4.9Sn3.1 has been solved and refined from X-ray single crystal data. Cubic SmGaSn2 is characterized with Sn/Ga atomic mixing at all positions and may be seen as derived from SmSn3 by atomic substitution. Moreover, analogies in their band structures allows to expect intermetallic heavy fermion properties for SmGaSn2. Hexagonal Ca4Ga4.9Sn3.1 is isostructural with Ca4Ga4Sn4 and also displays atomic mixing. The significant differences noted in their geometries are most probably due to a poor quality of the powder data structure determination of Ca4Ga4Sn4 [6]. The information collected in this work particularly from DFT calculations tends to prove the existence of a composition domain for Ca4Ga4+xSn4-x in the range 0 ≤ x ≤ 1, excluding a possible extension on the Sn-rich side. It should also be stressed that partial atomic ordering occurs within the (Sn,Ga) sub-lattice in Ca4Ga4+xSn4-x as also described in a ternary intermetallic compound with samarium [45].

Figure 1 :

 1 Figure 1: Representation of cubic SmGaSn2 and hexagonal Ca4Ga4.9Sn3.1 structures.

Figure 2 :

 2 Figure 2: Schematic representation of the packing of hexagon layers in

Figure 3 :Figure 4 :

 34 Figure 3: Typical parameters (angles and distances) to define the geometry of hexagon layers

Figure 5 :

 5 Figure 5: Packing modes within hexagonal lattices. The models are built with four layers of A-

Figure 6 : 4 . 6 .

 646 Figure 6: Total energy (empty bars) and formation enthalpy (filled bars) calculated for various

Figure 7 :

 7 Figure 7: left -density of states calculated within a rigid model; right -crystal orbital overlap

Table 1 .

 1 Crystal data and structure refinement

	Compound	SmGaSn2	Ca4Ga4.9Sn3.1
	Initial composition (alloy)	SmGa3Sn2	Ca5Ga9Sn4
	Refined composition	SmGa3Sn2.00(1)	Ca4Ga4.91(3)Sn3.09(3)
	Crystal system	cubic	hexagonal
	Space group	Pm m	P63/mmc
	Lattice parameters (Å)	4.5778(8)	4.4233(9), 17.601(7)
	Volume (Å 3 )	95.93(5)	298.24(17)
	Crystal size (mm)	0.05×0.06×0.10	0.05×0.10×0.10
	F(000)	193	387
	Absorption coefficient (mm -1 )	34.63	18.9
	 range (deg)	4.45-45.35	2.31-29.88
	Diffractometer	Oxford Xcalibur CCD	Nonius CAD4
	Extinction coefficient (× 10 -4 )	0.011(1)
	Independent reflections	114 [R(int) = 0.0470]	204 [R(int) = 0.0475]
	Goodness-of-fit on F 2	1.074	1.191
	Final R1/wR2 indices [I > 2(I)]	0.0084/0.0207	0.0418/0.1042
	R1/wR2 indices (all data)	0.0118/0.0212	0.0673/0.1142
	Residual densities (e. Å -3 )	0.95/-0.74	3.60/-2.46

Table 2 .

 2 Positional and atomic equivalent displacement parameters. Ueq is defined as one third of the trace of orthogonalized Uij tensor.

	SmGaSn2
	Position	occupation	x	y	z	Ueq ( Å 2 )
	Ga	3c	0.334(4)	0	1/2	1/2	0.0180(1)
	Sn		0.666(4)
	Sm	1a	1	0	0	0	0.01296(7)
	Ca4Ga4.9Sn3.1
		Position	occupation	x	y	z		Ueq ( Å 2 )
	M1 Sn	4f	0.773(2)	1/3	2/3	0.38497(6)	0.0077(4)
	Ga		0.227(2)
	M2 Ga	4f	1	2/3	1/3	0.3414(2)	0.0250(7)
	Ca1	2a	1	0	0	0		0.0135(9)
	Ca2	2b	1	0	0	0.25		0.0146(9)

  leading in two-by-two interconnection of the layers through Ga-Ga bonds of 2.78 Å, slightly longer than bonds within layers (2.68 Å).

	Note that the
	angles α are 109.5° in Li3Ga2 and 109.3° in Li5Ga4 and very close to the tetrahedral angle. A more
	thorough condensation of hexagon layers is achieved in the structure of ternary compound
	Li2Ga2Sn in which Sn and Ga are statistically disordered in 1:2 proportion or in other words mixed
	at all sites [40]. In this structure, a three dimensional network is built with bonds between layers
	(2.71 -2.81 Å) that are just slightly longer than bonds within layers (2.67 -2.72 Å). With α angles
	Ga bond length of 2.667 Å in Ca4Ga4.9Sn3.1 is typical for covalently bonded atoms and must be
	compared with sum of covalent radii of the elements, respectively 2.52, 2.66 and 2.80 Å for Ga-
	Ga, Sn-Ga and Sn-Sn bonds. Even if the organization of layers in Ca4Ga4.9Sn3.1 reminds that in
	Li5Ga4, the layers are too distant to allow interconnections. The shortest distance of 3.22 Å between
	layers occurs at Ga pairs and compares well with the alike distances of 3.30 or 3.35 Å, calculated

ranging from 108.2 to 111.2°, the mean deviation to flatness remains of the same order as in the previous binary compounds. Layers in these Ga-containing structures may be regarded as built with 3-bonded atoms involved in homoatomic connections. Actually, Ga-Ga bonding occurs in binary combinations, and M-M bonding occurs in Li2Ga2Sn within layers composed of M pseudo-atoms (Sn and Ga distributed in 1:2 proportion at all M sites). Note that additional bonding occurs between layers in Li5Ga4 and Li2Ga2Sn.

Things are somewhat different in Ca4Ga4Sn4 (or CaGaSn) as in isostructural Sr4Ga4Sn4 where Sn and Ga atoms are found with a totally ordered distribution, the layers being there characterized by true heteroatomic Sn-Ga bonding. Despite the presence of some atomic mixing in Ca4Ga4.9Sn3.1, the repartition of atoms within the layers is so that only M-Ga bonds are formed that involve M pseudo atoms (Sn and Ga in 4:1 proportion) giving a heteroatomic character to bonding. The M-