Cognitive Anchor in Other-Regarding Preferences: An Axiomatic Approach

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1 Cognitive Anchor in Other-Regarding Preferences: An Axiomatic Approach Yosuke Hashidate Graduate School of Economics, The University of Tokyo First Draft: October 30, 2016 This Draft: January 30, 2017 Abstract This paper extends the theory of other-regarding preferences to the decision maker whose choice behaviors are reference-dependent. We investigate a decision maker who chooses an allocation between herself and other passive recipients from a menu. The resulting behavior can be related to a criterion for a subjective fairness, which is determined from choice opportunities. Even though, in most studies on reference-dependent preferences, reference points are given exogenously, decision analysts cannot observe reference points directly in general. To overcome this difficulty, by enriching the domain of preference relations, this paper presents a utility representation that identifies a reference point endogenously. The contribution of this paper is that we provide an axiomatic foundation for reference-dependent inequality aversion. The axiomatic foundation accommodates the existence of a subjective reference point, which is called a cognitive anchor, a criterion for social decision-making. This captures pro-social behaviors with menu-dependence. Moreover, in the model, it is shown that the attitude toward pure altruism, menu-dependent inequality aversion, and reference points, are uniquely identified, respectively. Keywords: Axiomatic Foundation for Behavioral Economics; Cognitive Anchor; Inequality Aversion; Other-Regarding Preferences; Reference-Dependent Preferences; Pro-Social Behaviors. JEL Classification Numbers: D01, D03, D63, D64, D81. I am indebted to my adviser Akihiko Matsui for his unique guidance, constant support, and encouragement. I thank Tatsuya Kameda and Daniel Marszalec for their invaluable discussions. I gratefully acknowledge financial support from the Japan Society for the Promotion of Science (Grant-in-Aid for JSPS Fellows ( )). All remaining errors are mine. yosukehashidate@gmail.com

2 Contents 1 Introduction 1 2 Preliminaries: An Extended Domain Endogenous Reference Points: An Axiomatic Approach A Foundation for Revealed Reference Axioms Axioms Standard Preferences Reference-Dependent Preferences Other-Regarding Preferences Results Representation Theorem Uniqueness Result Characterization of No Menu Effects Characterization of Pure Altruism Some Properties of Cognitive Anchor Literature Review: Comparison with Other Models Inequality Aversion Social Image Rawlsian Maximin Principles Warm Glow Concluding Remarks 19 A Proof of Theorem 1 20 A.1 Sufficiency Part B Proofs of Propositions 26 B.1 Proof of Proposition B.2 Proof of Proposition B.3 Proof of Proposition C Other Proofs 29 C.1 Proof of Lemma C.2 Proof of Lemma i

3 1 Introduction It is widely recognized that people often evaluate alternatives relative to a reference point. The notion of reference points reflects on the idea that people react to changes. Even in social contexts such as dictator games, reference points might affect decision-making. Suppose that there are two agents. One is a dictator, and another is a recipient. The dictator is given $10, and asked to divide the $10 between herself and the recipient. Even though game theory predicts that the dictator takes the whole amount for her own, various experimental results show that there is a fraction to give a sum between $0 and $5 to the recipient (Camerer (2003)). In terms of reference-dependent preferences, if the dictator makes a choice to get $6 for herself, and give the recipient $4, then the dictator may have a subjective reference point as a fifty-fifty choice in mind, i.e., the Egalitarian choice ($5, $5). 1 Other-regarding preferences in mind can lead to pro-social behaviors, which intend to benefit and help others. In fact, many experimental results in dictator games are related to pro-social behaviors. In many cases, the study of reference-dependent choices, ranging from status-quo bias (Masatlioglu and Ok (2005)), initial endowments (Masatlioglu and Ok (2014)), aspiration levels (Diecidue and Van De Ven (2008)), to loss aversion (Tversky and Kahneman (1991)), has been studied given a reference point exogenously. As behavioral economics shows, status-quo options, initial endowments, or default options may have the role as reference points for decision makers, which affects their decision-making. However, the fact is that it is not easy for decision analysts (observers) to identify reference points in decision-making. However, in general, the reference point itself is not directly observable. In addition, reference points should be subjective. If reference points are given exogenously, then we cannot study how the decision maker can form a reference point subjectively. The goal of this paper is to provide an axiomatic foundation for preferences for menudependent inequality aversion. The axiomatic foundation accommodates the existence of a subjective cognitive anchor, i.e., a reference point, which is endogenously determined. The menu-dependent cognitive anchor can lead to violations from the postulate of rationality, namely, the Weak Axiom of Revealed Preference (WARP). As the menu-dependent cognitive anchor shows, we only allow for those violations from reference-dependence. To do so, we extend the analysis to provide a foundation for revealed reference-dependence (Section 2). Based on revealed references, we present the postulates for the axiomatization, which are categorized by (i) standard preferences, (ii) reference-dependent preferences, and (iii) other-regarding preferences (Section 3). The main result is stated in Theorem 1 (Section 1 To explain such altruistic behaviors, experimentally, empirically, and axiomatically, various models have been developed. For example, see the following literature: (i) Inequality Aversion (Fehr and Schmidt (1999)), (ii) Rawlsian maximin rule (Charness and Rabin (2002)), (iii) Shame (Neilson (2009), Dillenberger and Sadowski (2012)), (iv) Social Image (Saito (2015)), (v) Ex-Ante and Ex-Post Fairness (Fudenberg and Levine (2012), Saito (2013)), and (vi) Reciprocity (Dufwenberg et al. (2011)). 1

4 4). Moreover, we show that this menu-dependent utility representation is uniquely identified (Proposition 1). The novelty of this paper is that, to incorporate a subjective reference point into otherregarding preferences, i.e., a cognitive anchor, we enrich the domain of binary relations (see Section 2 in detail). 2 Briefly speaking, it is postulated that decision analysts directly observe binary relations of choices from menus, which are primitives of the model in this paper. In this framework, the decision maker such as a dictator, has a taste of choosing an allocation p from a menu A. The decision maker has the rankings of choices from menus. For example, assume that p = ($5, $5) is a fair allocation, q = ($10, $0) is a selfish allocation, r = ($7, $3) is an intermediate allocation between them. In this framework, the decision maker might prefer a choice of an allocation p from a menu {p, q} to a choice of an allocation p from a menu {p, q, r}. In revealed preference theory, choosing an allocation p from a menu is the same as that of another menu. However, we explore some cases such that the decision maker might prefer a choice of an allocation p from a menu {p, q} to a choice of an allocation p from a menu {p, q, r}. Such a binary relation may arise in social contexts from other-regarding preferences including inequality aversion and preferences for fairness. In Section 2, we present an axiomatic foundation for revealed reference-dependence. From choice data, it is hard to justify the difference between choosing p from A and choosing p from B. To provide an axiomatic analysis for reference-dependent preferences, Ok, Ortoleva, and Riella (2015) develop a revealed preference theory of reference-dependent choices. They relax WARP to obtain the existence of reference points endogenously. We follow the choice-theoretic foundation for the primitive of this paper. First, we require that for any pairwise menus such as {p, q}, no cycle condition, which does not allow for preference reversals under pairwise choice problems, holds (Lemma 2). This condition is valid since, in general, we observe some preference reversals when a third alternative is added into the pairwise choice problem. 3 Next, we require that an acyclic condition holds under reference-dependence (Lemma 1). These behavioral foundations from choices make it possible to explore a utility representation of reference-dependent preferences. In Section 3, we present the axioms for the main theorem (Theorem 1). First, we describe the basic axioms that is well-known in decision theory. The basic axioms are related to foundation for revealed references explained in Section 2. Next, we introduce the two key axioms to obtain a reference point (cognitive anchor) endogenously: (i) Separability, and (ii) Comparative Valuations in Menus. The axiom of Separability requires that preferences for allocations and reference-dependent preferences are separable. This axiom guarantees that the reference-dependent preferences happen when the sizes of menus are more than three, i.e., 2 Suzumura and Xu (2001, 2003) study this type of binary relation in social choice theory. 3 See Huber, Payne, and Puto (1982) (Attraction Effects) and Simonson (1989) (Compromise Effects). 2

5 A 3. In addition, the property reflects the existence of reference points. On the other hand, the axiom of Comparative Valuations in Menus captures the property of reference points. This axiom requires that if for any alternatives in a menu, a dominant alternative is added into the menu, then the evaluations of the alternatives in the original menu decrease. Adding the dominant alternative into the menu changes the criterion of decision-making. The criterion, i.e., the cognitive anchor in social contexts increases in comparative valuations. Finally, we provide the axioms for inequality aversion (other-regarding preferences). The two axioms ( (i) Comonotonic Independence, and (ii) Fair Allocations ) are not new. We apply uncertain-averse preferences (Schmeidler (1989), Gilboa and Schmeidler (1989)) into other-regarding preferences. 4 In Section 4, we state the main result: the representation theorem (Theorem 1) and the uniqueness result (Proposition 1). Moreover, we characterize reference-dependent preferences, preferences for allocations, and reference points, respectively. First, we consider the case that there is no menu effects. In this case, the decision maker does not have any reference-dependent preferences, which reduces to the standard utility representation. Next, we characterize the attitude toward pure altruism. By Separability stated above, preferences for allocations and reference-dependent preferences are separable. From preferences for allocations, we can identify the attitude toward pure altruism. In this sense, our model separates pure altruism from reference-dependent preferences. Finally, we characterize the menu-dependent cognitive anchor. We study the attitude toward pro-social behaviors in a positive way or in a negative way with the parameter of the menu-dependent cognitive anchor. In Section 5, we provide a literature review. We compare our model with other models such as (i) inequality aversion, (ii) pride, shame, and temptation, (iii) Rawlsian maximin rules, and (iv) warm glow. In Section 6, we conclude. 2 Preliminaries: An Extended Domain We introduce notation. Let X be a finite set of all alternatives. The elements of X, i.e., alternatives are denoted by x, y, z X. Assume that (X, d) is a separable metric space with the Euclidean distance d. Let A be the set of all non-empty compact subsets of X, endowed with the Hausdorff metric. The Hausdorff metric is defined by { d h (A, B) = max max d(x, y), max min x A y B min x B y A } d(x, y). The elements of A are denoted by A, B, C A. We call the elements of A menus. Menus are opportunity sets. 4 See also Rohde (2010), which is an axiomatic foundation for the Fehr and Schmidt (1999) s inequalityaverse representation. The axiomatic foundation is similar to that in Schmeidler (1989). 3

6 Let us denote a choice correspondence. A set-valued map c : A X is a choice correspondence if c(a) and c(a) A for each A A. We consider the product X A, endowed with the product metric d of the Euclidean metric d and the Hausdorff metric d h. We consider a space: (X A) {(x, A) X A x A}. An element (x, A) (X A) means that a decision maker chooses an alternative x from a menu A. The primitive of the model is a binary relation over (X A). The asymmetric and symmetric parts of are denoted by and, respectively. 2.1 Endogenous Reference Points: An Axiomatic Approach We investigate a binary relation on (X A) as the primitive of the present paper. Take an arbitrary option (x, A) (X A). This option means that a decision maker chooses an alternative x from a menu A. We try to capture the ranking of choices from menus by observing this binary relation. We explicitly analyze menu effects like how sizes of menus or contents in menus affects decision-making. Let us remark that, in general, decision analysts cannot observe the difference between (x, A) and (y, B) from choice data only. If an alternative x is chosen over a menu A, we interpret that alternative as a preferred alternative for that decision maker. Thus, behaviorally, we cannot distinguish the difference between a choice from a menu A and a choice from another menu B. In this sense, our approach is based on non-consequentialism. There is an advantage of considering this binary relation on (X A) directly. This approach makes it possible to characterize reference-dependent preferences directly. We are able to justify which type of axioms the decision maker with reference-dependent preferences do satisfy. In addition, we are able to characterize the decision maker who has no menu effects, i.e., the decision maker who has reference-independent preferences in the standard manner. In addition, we present an axiomatic foundation for reference-dependent choices based on Ok, Ortoleva, and Riella (2015). To derive reference points endogenously, recently, there has been new literature: Ok, Ortoleva, and Riella (2015) and Tserenjigmid (2016), who study a revealed preference theory of reference-dependent choices by analyzing choice correspondences as primitives. The two papers relax the Weak Axiom of Revealed Preference (WARP) suitably. The common feature of the two literature induces the two binary relations from the weaker axioms of WARP. One is a binary relation which represents a standard preference maximization (Debreu (1959), Kreps (1988)), another is related to reference-dependent preferences. Their models are based on, in principle, observed choice data. 4

7 2.2 A Foundation for Revealed Reference We explore an axiomatic analysis of reference-dependent choices. Ok, Ortoleva, and Riella (2015) provide two important notions to identify revealed reference: (i) revealed c-reference and (ii) potential c-reference. We explain about the two notions in the following, one by one. First, we explain about the notion of revealed c-references. Consider an arbitrary pairwise choice problem {x, y}. Suppose that x is not chosen over y in the pairwise choice problem, i.e., {y} = c({x, y}). However, we sometimes observe that when an alternative z X is added into the pairwise choice problem {x, y}, x is chosen, i.e., x c({x, y, z}). In other words, the existence of the third alternative z affects the resulting choice behavior, and then x is chosen. Formally, this notion is stated as follows. Definition 1. (Revealed c-reference): We say that an alternative z is a revealed c-reference for x if there is an alternative y X such that (i) x c({x, y, z}) \ c({x, y}); or (ii) y c({x, y}) and {x, y} c({x, y, z}) = {x}. Second, we consider a related notion for a third alternative z with a pairwise choice problem {x, y}. That is, it is the case that the addition of the alternative z does not change the choice behavior in a choice problem {x, y, z} from the original pairwise choice problem {x, y}. Concretely, if the alternative x is chosen over the alternative y, then x is still chosen when the third alternative z is added. Conversely speaking, if the alternative y is not chosen over x, then y is not chosen even if the third alternative z is added. Definition 2. (Potential c-reference): We say that an alternative z is a potential c-reference for x if, for every y X such that {z} = c({x, y, z}), (i) x c({x, y}) x c({x, y, z}); and (ii) y / c({x, y}) y / c({x, y, z}). The key point is that two definitions focus on pairwise choice problems. We pay attention to observable choice behaviors when a new alternative is added into a pairwise choice problem. The added third alternative is categorized into the two roles: (i) reference-dependence or (ii) reference-independence. We need to observe carefully whether a third alternative for a pairwise choice problem affects the taste on the pairwise choice problem or not. 2.3 Axioms We present some basic axioms for the binary relation on (X A). Plus, we provide a behavioral foundation for the properties of. 5

8 First, we provide a basic postulate on the study of subjective reference-dependence: (i) completeness and (ii) transitivity. Axiom(Weak Order): is complete and transitive: (i) (Completenessr): For any (x, A), (y, B) (X A), (x, A) (y, B) or (x, A) (y, B). (ii) (Transitivity):For any (x, A), (y, B), (z, C) (X A), if (x, A) (y, B) and (y, B) (z, C), then (x, A) (z, C). The axiom of completeness requires that any options are comparable. 5 Also, the property of transitivity is required. The following axiom for choice correspondences is related to the axiom of Reference Acyclicity in Ok, Ortoleva, and Riella (2015). Axiom(Reference Acyclicity): For any integer m 2 and x 1,, x m X, if x i is a revealed c-reference for x i+1 for each i = 1,, m 1, then x 1 is a potential c-reference for x m. The axiom is an acyclic condition under reference-dependence. Without loss of generality, suppose that there exist finitely some alternatives x 1, x 2,, x m X. Then, as the axiom says, x 1 is a revealed c-reference for x 2, x 2 is a revealed c-reference for x 3, and so on. x m 1 is a revealed c-reference for x m. This axiom requires that for any sequences of alternatives, x 1 is a potential c-reference for x m. If x 1 is a revealed c-reference for x m, it immediately leads to a violation of WARP. Lemma 1. Suppose that satisfies Completeness. Then, satisfies Transitivity if and only if c satisfies Reference Acyclicity. Next, we introduce a consistency condition for reference-dependent preferences. To do so, for simplicity, we define on X as follows. Definition 3. For any x, y X, x y if (x, {x}) (y, {y}). The asymmetric and symmetric part of are described by > and, respectively. Axiom(Pairwise Consistency): For any x, y X, if (x, {x, y}) (y, {x, y}), then x y. This axiom says that the ranking of choices from singleton menus should be justified by the ranking of choices from pairwise menus. We provide the following condition for choice correspondences in Ok, Ortoleva, and Riella (2015). 5 See Evren and Ok (2011) for the generalization of incomplete preference relations. 6

9 Axiom(No-Cycle Condition): For any x, y, z X, if x c({x, y}) and y c({y, z}), then x c({x, z}). This condition says that the consistent condition for choice behaviors holds under any pairwise choice problems. Even though pairwise choice problems are extreme, there is no possibility such that a third alternative affects the alternatives in pairwise choice problems. In this case, any preference reversals like cycles do not happen. Lemma 2. Suppose that satisfies Pairwise Consistency. Then, satisfies Weak Order if and only if c satisfies No-Cycle Condition. In addition, we provide a weaker version of continuity. Axiom (Continuity): The sets {(p, A) (X A) (p, A) (q, B)} and {(p, A) (X A) (q, B) (p, A)} are closed (in the product metric d ). Finally, we postulate a basic axiom: Non-Degeneracy. exists an option that is strictly preferred. This axiom requires that there Axiom(Non-Degeneracy): There exists (x, A), (y, B) (X A) such that (x, A) (y, B). 3 Axioms We introduce notation briefly. Let 1 denote the decision maker, and S be the finite set of other agents. Let I = {1} S be the set of all agents. Let Z be a finite set of outcomes. (Z) is the set of all lotteries with finite support. Since Z is finite, the topology generated by the Euclidean metric d is equivalent to the weak topology on (Z). Let ( (Z)) I be the set of all allocations. The elements of ( (Z)) I are denoted by p = (p 1,, p n ) = (p 1, p S ) where S = {2, n}. The lottery p i is an allocation for an agent i. Let A be the set of all non-empty compact subsets of ( (Z)) I endowed with the Hausdorff metric d h. Menus are denoted by A, B, C A. Let us consider a space (Z) I A, endowed with the product metric d by the Euclidean metric d and the Hausdorff metric d h. Consider a space (( (Z) I ) A) {(p, A) (Z) I A p A}. The primitive of the model is a binary relation over (( (Z) I ) A). The asymmetric and symmetric parts of are denoted by and, respectively. 7

10 3.1 Standard Preferences First, we postulate a basic axiom, stated in Section 2: (i) Weak Order, (ii) Pairwise Consistency, (iii) Continuity, and (iv) Non-Degeneracy. Axiom (Standard Preferences): satisfies (i) Weak Order, (ii) Pairwise Consistency, (iii) Continuity, and (iv) Non-Degeneracy. Next, we induce the two preference relations of the decision maker 1: (i) the individual preference, 1 and (ii) the social preference S. For simplicity, for any singleton menus, we write down a binary relation on singleton menus in the following way. We say that p q if (p, {p}) (q, {q}). We define 1 and S as follows. Definition 4. 1 and S are defined as follows: (i) p 1 1 q 1 if (p 1, r S ) (q 1, r S ) for some r S (Z) S. (ii) p S S q S if (r 1, p S ) (r 1, q S ) for some r 1 (Z). In addition, we provide the following axioms: Consistency, Pareto, and Singleton Independence. The first axiom, Consistency, provides a consistency condition between the decision maker 1 s own preference 1 and the social preference S. This axiom follows from the Saito (2015) s consistency axiom (p.344). Remember that the decision maker 1 is not a social planner. The decision maker 1 may not know other agents preference relations directly. This setting is actually valid under dictator games. 6 Axiom (Consistency): For any p S, q S ( (Z)) S, if p i 1 q i for any i S, then p S S q S. The next axiom, Pareto, is standard in economic theory. Axiom (Pareto): For any p, q ( (Z)) I, if p 1 1 q 1 and p S S q S, then (p, A) (q, A) for any A A. Moreover, we introduce another condition for singleton menus. We use convex combinations in the standard manner. For any menus A, B A and λ [0, 1], the convex combination of the two menus is defined by λa + (1 λ)b = {λp + (1 λ)q p A, q A }. The following axiom requires that Independence holds under singleton menus. 6 Axiomatically, this axiom makes it possible to obtain a unique non-constant cardinal utility function u : (Z) R. 8

11 Axiom (Singleton Independence): For any p, q, r ( (Z)) I and λ [0, 1], (p, {p}) (q, {q}) (λp + (1 λ)r, {λp + (1 λ)r}) (λq + (1 λ)r, {λq + (1 λ)r}). 3.2 Reference-Dependent Preferences To obtain a subjective reference point (a cognitive anchor in social decision-making) endogenously, we provide a key axiom: Separability. The axiom, Separability, is a minimal requirement in the framework of this paper to study menu effects. This axiom is a separable condition, which requires that the evaluation of allocations that stems from preferences for allocations and menus preferences including reference-dependent preferences are separable. As the axiom shows, the sizes of menus are more than three, i.e., A, B 3. This axiom captures a type of menu effects of the decision maker. Axiom (Separability): For any p, q A B, (p, A) (p, B) (q, A) (q, B). In the framework of this paper, this axiom captures reference-dependent preferences. The ranking between (p, A) and (p, B) means that if (p, A) is preferred to (p, B), then an allocation p in the menu A is relatively more preferred to a choice of p from the menu B. This type of tastes reflects menu effects. In addition to Separability, we provide another axiom for reference-dependent preferences. Axiom (Comparative Valuations in Menus): Suppose that, for any p A, there exists q ( (Z)) I such that q j j p j for any j {1, S}. Then, (i) (p, A) (p, A {q}); and (ii) (q, A {q}) (q, {q}). Suppose that for any allocations p A, there exists an allocation q (Z) I such that q j j p j for any j {1, S}. That is, q dominates p. The first condition requires that if such an attractive allocation q is added into a menu A, then the comparative valuation of p A decreases due to the existence of q A {q}. This axiom can be interpreted as the notion of satisfaction in Simon (1957). In other words, adding q into A changes the criterion of decision-making. As a result, the criterion in comparative valuations increases. Then, the evaluation of allocations p in the menu A decreases. On the other hand, the second condition requires that the added allocation q is relatively preferred in the menu A {q} more than that in {q} due to the allocations in the original menu A. 9

12 3.3 Other-Regarding Preferences We provide the following two axioms that are related to other-regarding preferences including inequality aversion. In our framework, other-regarding preferences in social contexts can happen when sizes of menus are more than two, i.e., A 2 since the decision maker 1 has to care about acting pro-socially or not. By relaxing the Independence axiom, we provide two axioms: Comonotonic Independence and Fair Allocations. First, we introduce Comonotonic Independence. 7 Let us introduce the notion of comonotonicty of allocations. We say that two allocations p, q are comonotonic if there exists no i S such that p i 1 p 1 and q i 1 q 1. The following axiom requires that, for any menus whose number of alternatives is more than two, Independence only holds when menus are comonotonic each other. Axiom (Comonotonic Independence): For any comonotonic allocations p, q, r, for any menus A, B, C A with A, B, C 2, and for any λ [0, 1], (p, A) (q, B) (λp + (1 λ)r, λa + (1 λ)c) (λq + (1 λ)r, λa + (1 λ)c). Next, we provide a consistency condition for fair allocations. 8 We say that an allocation r (Z) I is a fair allocation if r i = r j for any i, j I. The following axiom says that the mixture with fair allocations does not change the ranking. Axiom (Fair Allocations): For any A, B A and λ [0, 1], (p, A) (q, B) (λp + (1 λ)r, λa + (1 λ)c) (λq + (1 λ)r, λa + (1 λ)c) where r (Z) I such that r i = r j for all i, j I. 4 Results 4.1 Representation Theorem We state the main result. Theorem 1. The following statements are equivalent: (a) satisfies Standard Preferences, Singleton Independence, Separability, Comparative Valuations in Menus, Comonotonic Independence, and Fair Allocations. (b) There exists a tuple (u, α, β( ), γ, θ) where u is a non-constant function u : (Z) R, α 7 This axiom follows from the axiom of comonotonic independence in Schmeidler (1989) in the Anscombe and Aumann framework. 8 See also the axiom of Certainty Independence in Gilboa and Schmeidler (1989), which is a generalization of subjective expected utility theory. 10

13 is a vector such that α 1 > 0, and α i 0 for each i S with i S α i = 1, β( ) is a capacity on {1, S}, γ is the vector such that γ j [0, 1] for any j {1, S}, and θ [0, 1] is a parameter such that is represented by a function V : (( (Z) I ) A) R defined by V (p, A) = [ ] α i u(p i ) + θ β j (A)(u(p j ) u j (A)), i I j {1,S} where u j (A) = γ j max q A u(q j ) + (1 γ j ) min r A u(r j ) for any j {1, S}, and u(p S ) = i S α iu(p i ). We briefly explain about the interpretation of the model in the main theorem. The utility function represents a binary relation on (( (Z) I ) A). The first term, i I α iu(p i ), represents 1 and S, which is a utility representation of a generalized weighted utilitarian. By Consistency, since the decision maker 1 does not know about other agents true preferences, the allocations for other agents are evaluated by the decision maker 1 s utility function. By definition, α 1 > 0 captures the level of (pure) selfishness. To see the interpretation of the first term, take an arbitrary menu A A. For simplicity, assume that θ = 0. Let A = {(u 1 (p 1 ), u S (p S )) R 2 p = (p 1, p S ) A }. In Figure 1, for example, the allocation p A is put as u(p) = (u 1 (p 1 ), u S (p S )). As Figure 1 shows, α 1 captures the level of pure selfishness. If α 1 decreases, then the attitude toward pure altruism increases. us A* u(p) = ( u1(p1), us(ps) ) α1 u1(p1) + us(ps) u1 Figure 1: Attitude toward Pure Altruism Next, let us move on to the explanation about the second term. The parameter θ [0, 1] captures that how much the decision maker reacts to menu effects, especially, a menudependent cognitive anchor. The utility difference u(p j ) u j (A) is a comparative valuation of an allocation in the menu A. (u j (A)) j {1,S} is a cognitive anchor (reference point) of the menu A. The non-additive measure, i.e., the capacity β(a) = (β 1 (A), β S (A)) on {1, S} captures the attitude toward inequality aversion with menu-dependence. 11

14 We explain about the reason why we introduce the non-additive measure on {1, S} to capture inequality aversion. A set-valued function β is called to be a capacity if it is monotonic and normalized. That is, for each j {1, S}, β(j) β({1, S}) holds, and both β({1, S}) = 1 and β( ) = 0 hold. Before stating the interpretation of the second term, let us explain about the Choquet expected utility representation in Schmeidler (1989). Suppose that a finite state space Ω is given (ω Ω). Let Z be a set of outcomes. Also, let (Z) be the set of all probability distributions on Z with finite support. Anscombe=Aumann acts are defined by f : S (Z). Let H be the set of Anscombe=Aumann acts. The primitive is a binary relation on H. The preference relation is represented by a pair (u, v) where (i) u : Z R is a vnm function, and (ii) v is a capacity. 9 Suppose that u(f(ω j )) u(f(ω j+i )) for each j = 1,, n. Let u(f(ω j )) = u j R. Then, the utility of an act f is represented by U(f) = u 1 v(s 1 ) + n j=2 [ ] u j v( j i=1 S i) v( j 1 i=1 S i). To get the intuition of the Choquet integral, consider the following case. Suppose that a finite state space is given by Ω = {E 1, E 2, E 3 }, and that u 1 u 2 u 3. Then, the Choquet expected utility is calculated by u 1 v(e 1 ) + (u 2 u 3 )v(e 1 E 2 ) + (u 3 0)v(E 1 E 2 E 3 ) = u 1 v(e 1 ) + u 2 [ v(e1 E 2 ) v(e 1 ) ] + u 3 [ v(e1 E 2 E 3 ) v(e 1 E 2 ) ]. By arranging the order like u 1 u 2 u 3, we can write down the Choquet expected utility representation (Figure 2). u u1 u2 u3 v (E1 UE2 U E3) v (E1 U E2) v (E1) E3 E1 E2 u3 u2 u1 u Figure 2: Choquet Expected Utility Now, we can move on to explain about the interpretation of the second term. The second term, i.e., the expected utility with the Choquet integral, is written as follows. Given a menu A A, the menu-dependent cognitive anchor of the decision maker is obtained and written by (u 1 (A), u S (A)). For each allocation, the utility with menu-dependence is written by u j (p j ) u j (A) for each j {1, S}. Let û j (p j ) = u j (p j ) u j (A). Then, we have û 1 β 1 (A) + û S (1 β 1 (A)). 9 v(ω) = 1, v( ) = 0, and v is monotonic. 12

15 The menu-dependent capacity β( ) stems from the cognitive anchor u( ) = (u 1 ( ), u S ( )). Since the cognitive anchor in our model depends on menus, the attitude toward inequality aversion also depends on menus. In this sense, pro-social behaviors are determined by the decision maker s subjective reference point, i.e., cognitive anchor in social contexts. The menu-dependent cognitive anchor is formed by menus. For each j {1, S}, the criterion is defined by u j (A) = γ j max q A u(q j ) + (1 γ j ) min r A u(r j ). us us B* A* us(a) B us(a) u1(a) u1 u1(a) B u1 Figure 3: Cognitive Anchor 4.2 Uniqueness Result We state the uniqueness result. Proposition 1. Suppose that the two representations (u, α, β( ), γ, θ) and (u, α, β ( ), γ, θ ) represent the same binary relation. Then, the following statements hold: (i) u = au + b for some a > 0 and b R; (ii) α = α ; (iii) β( ) = β ( ); (iv) γ = γ ; (v) θ = θ. In the uniqueness result (Proposition 1), the axiom of Consistency has an important role for this identification result. Since the decision maker 1 is not a social planner, the decision maker evaluates other agents allocations by using her own preference relations 1. This implies that the allocations for others are also evaluated by her own utility function u : (Z) R. Mathematically speaking, in (i), by the von Neumann Morgenstein Expected Utility Theorem, u is unique up to positive affine transformations. In (ii), since u is unique up to a positive affine transformation, and u satisfies Singleton Independence, α is uniquely 13

16 identified. In (iii), the non-additive measure β( ) is also uniquely identified due to the cardinal utility function u. in (iv), the parameter γ for cognitive anchor is also uniquely identified. Finally, in (v), the parameter θ [0, 1] is uniquely identified. 4.3 Characterization of No Menu Effects We characterize the parameter θ [0, 1]. Especially, we consider an extreme case θ = 0. To characterize this case, we introduce the following axiom. Axiom(No Menu Effects): For any A, B A with p A B, (p, A) (p, B). This axiom says that the decision maker 1 has no menu effects. Choosing an allocation p from any menus produces the same utility for the decision maker 1. Axiomatically speaking, this axiom is a stronger version of Separability. Proposition 2. Suppose that is represented by a tuple (u, α, β( ), γ, θ). Then, exhibits No Menu Effects if and only if θ = 0. In the utility representation, we can write down as follows. For any (p, A) (( (Z) I ) A), V ((p, A)) = U(p) = i I α iu(p i ). The representation reduces to a generalized weighted utilitarian. 4.4 Characterization of Pure Altruism We provide comparative statics on α. α 1 is the level of selfishness for the decision maker 1. In the same way in Saito (2015), we study a comparative attitude toward pure altruism. The attitude toward pure altruism is captured by singleton menus. We investigate a binary relation on (Z) I. Let X be the binary relation of the decision maker X, and Y be the binary relation of the decision maker Y, respectively. Definition 5. For any X and Y on (Z)I such that X j = Y j for each j {1, S}, X is more altruistic than Y if, for any p, q (Z)I with p S h S q S for each h {X, Y }, p Y q p X q. We consider the case that an allocation p is better than q in the social preference: p S h S q S for each h {X, Y }. Under the assumption that X j = Y j for each j {1, S}, if a decision maker Y prefers p to q with p S Y S q S, then another decision maker X also prefers p to q with p S X S q S. Proposition 3. Suppose that is represented by a tuple (u, α, β, γ, θ). Plus, suppose that X and Y are represented by two utility representation (u, αx ) and (u, α Y ), respectively. Then, X is more altruistic than Y if and only if α1 X αy 1. 14

17 us us A* α1 X u1(p1) + us(ps) A* α1 Y u1(p1) + us(ps) u1 u1 Figure 4: Comparative Statics on Pure Altruism 4.5 Some Properties of Cognitive Anchor We examine a menu-dependent cognitive anchor (subjective reference point). We characterize a relationship between γ 1 and γ S. We study pro-social behaviors with reference-dependence (menu-dependence). The key point in our model is that pro-social behaviors can be captured by not only inequality aversion (β( )) but also menu-dependent cognitive anchor (γ). The non-additive measure (capacity) β( ) is put on {1, S} based on the menu-dependent cognitive anchor u( ). As a result, the attitude toward inequality aversion is menu-dependent. We provide an axiomatic characterization for the parameter of the menu-dependent cognitive anchor, since the decision maker privately determines a cognitive anchor, and the cognitive anchor affects resulting choice behaviors. Axiom(Negative Pro-Social Preference): For any p, q (Z) I, if (i) (p, {p}) (q, {q}); and (ii) p 1 1 q 1, then (p, {p, q}) (q, {p, q}). The interpretation of the above axiom is as follows. From the condition (i), the allocations p and q are indifferent. The condition (ii) says that the decision maker 1 prefers p to q in the sense that her own individual preference 1 says that p 1 is strictly preferred to q 1. As a result, the decision maker 1 prefers choosing p from {p, q} to choosing q from {p, q}. Proposition 4. Suppose that is represented by a tuple (u, α, β( ), γ, θ). Then, exhibits Negative Pro-Social Preference if and only if γ 1 γ S. The above result in Proposition 4 says that the decision maker makes use of her advantage of choosing an allocation (γ 1 γ S ). We call such a pro-social behavior a negatively pro-social behavior. To form a cognitive anchor given a menu, the parameter γ = (γ 1, γ S ) is a subjective 15

18 criterion for the decision maker. Suppose γ 1 > γ S > 0. Even though the decision maker feels in making a choice in a selfish way, the decision maker cares about social image deliberately. Next, conversely, we also present the following axiom: Positive Pro-Social Preference. Axiom(Positive Pro-Social Preference): For any p, q (Z) I, if (i) (p, {p}) (q, {q}); and (ii) p S S q S, then (p, {p, q}) (q, {p, q}). In the similar way with Negative Pro-Social Preference, the axiom of Positive Pro-Social Preference is interpreted. By the condition (i), the allocations p and q are indifferent. Contrary to Negative Pro-Social Preference, the condition (ii) says that the decision maker 1 s social preference S means that p S q S. As a result, the decision maker 1 prefers choosing p from {p, q} to choosing q from {p, q}. Proposition 5. Suppose that is represented by a tuple (u, α, β, γ, θ). Then, exhibits Positive Pro-Social Preference if and only if γ 1 γ S. The result of the above Proposition 5 states that the decision maker chooses an altruistic allocation positively (γ S γ 1 ). We call such a pro-social behavior a positively pro-social behavior. The parameter γ S γ 1 implies that a concern for others is subjectively large. Then, decision-making leads to take pro-social behaviors positively. 5 Literature Review: Comparison with Other Models 5.1 Inequality Aversion There are various models for other-regarding preferences. Fehr and Schmidt (1999) present a seminal theory of inequality aversion, which captures a preference for (ex-post) fairness, which only concerns for fair outcomes (allocations). 10 In the model, the decision maker pays attention the difference of payoffs between her own payoff and other agents payoffs. The difference of payoffs can produce a feeling of envy or guilt. Formally, the model is stated as follows. Consider the case that there are two agents: I = 1 S where S = {2}, i.e., I = {1, 2}. In the model, monetary payoffs are considered. The utility of an allocation is defined by U F =S (x) = u(x 1 ) a 1 max{u(x 2 ) u(x 1 ), 0} b 1 max{u(x 1 ) u(x 2 ), 0}, 10 Fudenberg and Levine (2012) argues that the difference between ex-ante fairness and ex-post fairness should be analyzed. Saito (2013) characterizes an axiomatic foundation for social preferences with ex-ante and ex-post fairness. Rohde (2010) provides an axiomatic foundation for the Fehr and Schmidt (1999) s inequality-averse utility model. 16

19 where a i b i, b i [0, 1) for each i {1, 2}. Suppose u(x 2 ) > u(x 1 ). Then, we have U(x) = u(x 1 ) a 1 (u(x 2 ) u(x 1 )). In this case, the decision maker feels dissatisfied with the inequality due to u(x 2 ) u(x 1 ) > 0, which exhibits feeling envy. On the other hand, suppose u(x 2 ) < u(x 1 ). Then, we have U(x) = u(x 1 ) b 1 (u(x 1 ) u(x 2 )). In this case, the decision maker feels guilty with the inequality due to u(x 1 ) u(x 2 ) > 0, which exhibits guilt. Thus, this model captures preferences for (ex-post) fairness. In this paper, the attitude toward inequality aversion is captured by a unique capacity β( ) = (β 1 ( ), β S ( )) with menu-dependence. Because of menu-dependent cognitive anchor and inequality aversion, we allow for preference reversals, deviations from WARP. On the other hand, Fehr and Schmidt (1999) do not allow for any preference reversals. 5.2 Social Image Dillenberger and Sadowski (2012) and Saito (2015) apply the framework of subjective state spaces (Kreps (1979), Dekel et al. (2001)) into social contexts. The two literature investigates two stages in decision-making: first, the decision maker chooses a set of allocations at the ex-ante stage, and then, next, she chooses an allocation at the ex-post stage from the set chosen at the ex-ante stage. Dillenberger and Sadowski (2012) develop the theory of shame of acting selfishly. 11 Saito (2015) develops a theory which uniquely identifies three emotions in social decision-making: (i) pride in acting altruistically, (ii) shame of acting selfishly, and (iii) temptation to act selfishly. First, the model in Dillenberger and Sadowski (2012) is stated as follows. In their model, monetary payoffs are considered. The utility of a menu is given by [ ] V D=S (A) = max α i u(x i ) g(φ(x), max φ(y)), x A y A i I where g : R 2 R captures the disutility of shame of acting selfishly. The function φ : Z I R is a social utility, which captures the decision maker s subjective norm. Then, max y A φ(y) is the optimal social utility in the menu A. Because of the form of g, WARP can be violated. In their model, however, g and φ are not uniquely identified. On the other hand, this paper uniquely identify the parameters for other-regarding preferences β( ) and γ with menu-dependence. Next, the model in Saito (2015) is stated as follows. The utility of a menu is given by [ V Saito (A) = max α i u(x i )+β 1 max α 1(u(q 1 ) u(p 1 )) β S max ( α i u(r i ) ] α i u(p i )), p A q A r A i I i S i S where α 1 > 0, { i 0 and i S α i = 1, β 1 < 1, and β S 0. The first term is the same as that in our model. The difference between Saito (2015) and this paper appears in the second and 11 See also Neilson (2009), which presents a non-axiomatic model of shame. 17

20 third term. The second term captures the utility arising from the pride in acting altruistically if β 1 0, or the disutility arising from temptation of acting selfishly if β 1 0. In the similar way, the third term captures the disutility due to the shame of acting selfishly. The key point in that model is a linear structure, which makes the prediction more restrictive. That is, the Saito (2015) s model satisfies WARP in ex-post choices unlike this paper. 5.3 Rawlsian Maximin Principles Charness and Rabin (2002) shed light on the procedure in other-regarding behaviors, not outcomes. In the model of Rawlsian quasi-maximin principles, the decision maker cares about her own payoffs, the minimum payoff among all agents, and total payoffs. 12 Formally, the model is stated as follows. In their model, monetary payoffs are considered. The utility of an allocation is defined by [ U C=R (x) = δu(x 1 ) + (1 δ) λ i I where δ [0, 1] and λ [0, 1]. ] α i u(x i ) + (1 λ) min u(x i), i I This model does not have an axiomatic foundation, but the Rawlsian maximin principle is similar to the menu-dependent cognitive anchor in other-regarding preferences. 5.4 Warm Glow Evren and Minardi (2015) develop a theory of Warm Glow, which refers to other-serving behavior that is valuable for the decision maker, apart from its social implications. 13 Evren and Minardi (2015) provide an axiomatic foundation for Warm Glow by viewing it as a form of preferences for larger choice sets driven by the decision maker s desire to have freedom to act selfishly. Specifically, the model is based on the idea of Warm Glow to the freedom to be selfish. Formally, the model is stated as follows. In their model, monetary payoffs are considered. The utility of a menu is given by ( ) V E=M (A) = max U α i u(x i ), max u(y 1) u(x 1 ), x A y A i I where U : R 2 R is a weakly increasing function. The key part in their model is the second argument in U. To interpret this part, consider two allocations x and y with y 1 > x 1. Then, V E=M ({x}) = U( i I α iu(x i ), 0). And, consider a menu {x, y}. The interpretation of U( i I α iu(x i ), u(y 1 ) u(x 1 )) U( i I α iu(x i ), 0) is an intrinsic payoff of the decision 12 See also Dufwenberg and Kirchsteiger (2004), which develop a theory of reciprocity for extensive games in which the sequential structure of a strategic situation is made explicit, and propose a new solution concept: sequential reciprocity equilibrium. 13 See also Andreoni (1989, 1990), which are seminal literature of warm glow. 18

21 maker associated with the pro-social behavior to choose the allocation x instead of a selfish allocation y. Compared with our model, in their model, the function U is not uniquely identified even though their model also allows for preference reversals. On the other hand, we uniquely identify the parameters related to other-regarding preferences. 6 Concluding Remarks We have developed a reference-dependent model in other-regarding preferences, which captures an subjective and endogenous reference point as a cognitive anchor, which can be a criterion for decision-making. This reference-dependent model has a foundation for revealed reference-dependence in Ok, Ortoleva, and Riella (2015). The model captures pro-social behaviors arising from reference-dependent preferences. That is, the attitude toward otherregarding preferences depends on menus. The building blocks of the model have a distinction between pure altruism and pro-social behaviors. The pure altruism is captured by α. On the other hand, menu-dependent pro-social behaviors are captured by a pair (β( ), γ) where β( ) captures the menu-dependent inequality aversion, and γ is a paramter for the menu-dependent cognitive anchor. One remark is that the objective of this paper is to identify unique parameters for pure altruism (α) and menu-dependent pro-social behaviors (γ). We do not consider any assumptions for a specific correlation between them. For example, some people may think that if a decision maker has the attitude toward pure altruism is relatively high, i.e., α 1 is low, then her attitude toward pro-social behaviors is also high, i.e., γ S is high. Such a correlation should be studied in empirical studies as a future task. 19

22 A Proof of Theorem 1 A.1 Sufficiency Part We show the sufficiency part. (Theorem 1). Suppose that satisfies the axioms in the main theorem STEP 1: In Step 1, we induce the binary relation on (Z) I to represent the first term. By using 1 and S, we show that any singleton menus are represented by a non-constant function u : (Z) R. That is, for any p (Z) I, an allocation p is represented by U(p) = i I u(p i). First, we show that the primitive of the model, that is, the binary relation is represented by V : (( (Z) I ) A). As Debreu (1959) shows, it suffices to show that (( (Z) I ) A) is separable and connected. Next, remember that we define on (Z) I as follows. Definition 6. For any p, q (Z) I, p q if (p, {p}) (q, {q}). Again, remember the definitions of 1 and S. We can write down the two definitions by using. We say that p 1 1 q 1 if (p 1, r S ) (q 1, r S ) for some r S (Z) S. In the similar way, we say that p S S q S if (r 1, p S ) (r 1, q S ) for some r 1 (Z). The asymmetric and symmetric part of are described by > and, respectively. First, we show that 1 and S are well-defined. Consider 1. We show that p 1 1 q 1 if (p 1, r S ) (q 1, r S ) for any r S (Z) S. Suppose (p 1, r S ) (q 1, r S ) and (p 1, r S ) < (q 1, r S ). Consider 1 2 r S r S (Z)S. Then, by Singleton Independence, (p 1, 1 2 r S r S ) ( 1 2 q p 1, 1 2 r S r S ) and (p 1, r S ) < ( 1 2 q p 1, 1 2 r S r S ). This is a contradiction. Thus, we have p 1 1 q 1 if (p 1, r S ) (q 1, r S ) for any r S (Z) S. In the similar way, we can easily show that S is well-defined. Next, we show that 1 is represented by u : (Z) R. By definition, satisfies Completeness, Transitivity, and Continuity, so it is easily shown that 1 satisfies Completeness, Transitivity, and (Mixture) Continuity. We show that 1 satisfies Independence: For any p 1, q 1, r 1 and λ [0, 1], p 1 1 q 1 if and only if λp 1 + (1 λ)r 1 1 λq 1 + (1 λ)r 1. We show it by using the induced preference relation. Fix p 1, q 1, r 1 (Z) and λ [0, 1]. Then, for any p S, q S (Z) S, p 1 1 q 1 (p 1, p S ) (q 1, p S ) λ(p 1, p S ) + (1 λ)(r 1, q 1 ) λ(q 1, p S ) + (1 λ)(r 1, q S ) λp 1 + (1 λ)r 1 1 λq 1 + (1 λ)r 1. In the similar way, it is shown that S satisfies Independence. Hence, by von Neumann-Morgenstern s Expected Utility Theorem (See Kreps (1988)), there exists a mixture linear utility function u : (Z) R which represents 1. u is unique up to a positive affine transformation. is a binary relation on (Z) I. By Pareto, if p 1 1 q 1, and p S S q S, then p q. By 20