Attribute-Based Inferences: An Axiomatic Analysis of Reference-Dependent Preferences

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1 Attribute-Based Inferences: An Axiomatic Analysis of Reference-Dependent Preferences Yosuke Hashidate Graduate School of Economics, The University of Tokyo First Draft: January 2015 This Draft: February 14, 2017 Abstract This paper provides an axiomatic foundation for reference-dependent preferences in attribute-based inferences. As an interpretation of reference-dependent preferences, it is argued that people often evaluate alternatives relative to some reference points, which are subjective criteria for decision-making. However, in general, decision analysts cannot observe reference points directly in terms of revealed preference theory. In this paper, we identify a menu-dependent reference point, which is called an attribute-based aspiration level, from behaviors. This axiomatic foundation accommodates the existence of a subjective aspiration level for decision-making, which can produce some deviations from the postulates of rationality. Moreover, we show that the building blocks of the model are uniquely identified. We analyze a relationship between our model and the two well-known behavioral regularities, the attraction effect and the compromise effect. Keywords: Foundation for Behavioral Economics; Attribute-Based Inferences; Attraction Effects; Compromise Effects; Reference-Dependent Preferences; Aspiration Levels. JEL Classification Numbers: D01, D03, D11, D81. I am indebted to my adviser Akihiko Matsui for his unique guidance, constant support, and encouragement. The earlier version of the paper was presented at Okayama University (Contemporary Economics Seminar), and Doshisha University (Asian Meeting of Econometric Society 2016). I would like to thank Takao Asano, Kazuhiro Hara, Youichiro Higashi, Kazuya Hyogo, Daisuke Oyama, Koji Shirai, Norio Takeoka, and Keisuke Yoshihara for their insightful suggestions and 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 Related Literature 3 3 Axioms Standard Preferences Reference-Dependent Preferences Relaxing Independence of Irrelevant Alternatives Results Representation Theorem Uniqueness Result Characterization of Attitude toward Reference-Dependence Characterization of Diminishing Sensitivity A Relationship between Preference and Choice Revealed Reference-Dependence Behavioral Foundation Preference Reversals Violations of IIA Dissatisficing Aversion and Compromise Effects Concluding Remarks 14 A Proof of Theorem 1 16 A.1 Sufficiency Part A.2 Necessity Part B Other Proofs 20 B.1 Proof of Lemma B.2 Proof of Lemma C Proof of Propositions 21 C.1 Proof of Proposition C.2 Proof of Proposition C.3 Proof of Proposition i

3 1 Introduction In many cases, it is not a simple task to choose one alternative from a choice problem (a menu). When we evaluate alternatives, there are little cases that people make choices with a single attribute for evaluating feasible alternatives. For example, when searching for a hotel for participating a conference, the ranking of the feasible and available hotels is often determined by the attributes of hotels we make use of such as prices, locations, service, safety, and convenience. Even though it is important to determine which attributes are used for decision-making, we often face a difficult task to overcome the trade-off among attributes. Intentionally or unintentionally, people often evaluate alternatives relative to some reference points, which are subjective criteria for decision-making. If we face a choice problem which has enormous feasible alternatives, we intentionally eliminates some available alternatives to narrow down them to simplify the choice problem. As in the example of booking a hotel, we often focus more attention on the hotels that satisfy some criterion in terms of attributes like prices, location, and service. In this case, a subjective reference point can be a key criterion in decision-making. The study of reference-dependent choices ranging from statusquo 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. However, in general, the reference point itself is not directly observable for decision analysts. In addition, reference points should be subjective, as mentioned above. If reference points are given exogenously for decision analysis, then we cannot study how the decision maker can form a reference point subjectively. This paper proposes an axiomatic framework to analyze reference-dependence directly. We especially apply this framework into attribute-based inferences, and we provide an axiomatic foundation for reference dependent preferences. To obtain endogenously subjective reference points, i.e., menu-dependent aspiration levels in this paper, we enrich the domain of preference relations. That is, we postulate that decision analysts directly observe binary relations of choices from menus, which are primitives of the model in this paper. In this framework, decision makers have a taste of choosing an alternative x from a menu A. They have the rankings of choices from menus. For example, when we search for hotels, we may prefer a choice of a hotel from a smaller menu since there are many hotels in the city the decision maker visits. In other words, in this framework, it is interpreted that the decision maker prefers a choice of a hotel x from a menu {x, y} to a choice of a hotel x from a menu {x, y, z}. We present the contributions of this paper. Instead of giving a reference point exogenously, we identify a menu-dependent aspiration level endogenously. To do so, we introduce a key axiom, Separability. This axiom requires that the absolute evaluations of alternatives and the menu-based relative evaluations of alternatives are separable. Mathematically speaking, take 1

4 arbitrary two alternatives, x and y. Suppose that the two alternative include the following two menus, A and B. That is, x, y A B. Then, if a decision maker prefers a choice of x from a menu A to a choice of x from a menu B, then this axiom requires that the decision maker prefers a choice of y from a menu A to a choice of y from a menu B. This accommodates the existence of a menu-dependent criteria for decision-making. Moreover, we identify the attitude toward changes of menus, which can produce deviations from the postulate of rationality, namely, the Weak Axiom of Revealed Preference (WARP). Our characterization allows for the well-known robust violations of WARP such as the attraction effect and the compromise effect. In Section 3, we present the axioms for the main result (Theorem 1). First, we describe the basic axioms that is well-known in decision theory. The basic axioms are related to a foundation for revealed reference-dependence. Next, we introduce the two key axioms to obtain a reference point (aspiration level) endogenously: (i) Separability, and (ii) Comparative Valuations in Menus. As described above, the axiom of Separability requires that preferences for alternatives 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., A 3. In addition, this 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 aspiration level in attribute-based inferences increases in comparative valuations. Finally, we provide the axioms for a weaker version of Independence of Irrelevant Alternatives (IIA). The two axioms ( (i) Weak IIA and (ii) Weak Translation Invariance ) are introduced. The first axiom is a weaker version of IIA. The second axiom is a weak consistency condition. 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 alternatives when there is no menu effects, and aspiration levels, 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. In Section 6, we explain about how our model is related to the two well-known preference reversals: the attraction effect and the compromise effect. This is related to the axiomatization of the model, i,e, the way of weakening the axiom of Independence of Irrelevant Alternatives (IIA). We also mention that preference reversals happen due to not only the attitude toward changes of menus, but also the position of reference points, aspiration levels. In some cases, even though we can observe compromise effects, but attraction effects do not happen. 2

5 The rest of the paper is organized as follows. In Section 2, we provide a literature review. In Section 3, we present the axioms in this paper. In Section 4, we state the main result, that is, the representation theorem and the uniqueness result. In addition, we characterize some properties of the building blocks of the model. In Section 5, we study a relationship between preference and choice. To study reference-dependent preferences, we enrich the domain of preference relations. As a result, in some sense, choice behaviors are not consequential. We justify the identification of reference-dependent preferences from the notion of revealed preference theory. In Section 6, we explain about violations from rationality. Finally, we conclude. 2 Related Literature We provide a literature review to make the contributions of this paper. The notion of referencedependence has a wide range of the study even in economics. In this review, we focus on the theoretical approach to obtain endogenous reference points and the choice-theoretic approach in revealed preference theory. The first papers on the study of endogenous reference-dependence are Kőszegi and Rabin (2006, 2007). In the Kőszegi and Rabin approach, it is postulated that the decision maker has a belief over outcomes, and one s reference point is a belief about future returns, which is determined by one s expectation. That reference point is not an alternative. In this paper, on the other hand, reference points are formed with both menu-dependence and attribute-based inferences. In Kőszegi and Rabin (2006, 2007), reference points are meaningful in choice environments under uncertainty, so their papers are not related to robust violations of WARP such as attraction effects and compromise effects. This paper only considers choice environments under certainty, but it is easily applicable to choices under uncertainty. To explain about both attraction and compromise effects, de Clippel and Eliaz (2012) provides a theory of multi-selves in decision-making as a intrapersonal bargaining problem. Attraction effects and compromise effects are well-known as the most important and robust violations of the postulate of rationality. Both behavioral regularities might stem from the difficulty of making a choice in the trade-off among attributes. de Clippel and Eliaz (2012) requires that in the control treatment, the alternative x and y are indifferent. The new alternative z breaks the tie. On the other hand, we do not require that the two original alternatives are indifferent. Masatlioglu, Nakajima, and Ozbay (2012) provides a theory of limited attention in the spirit of revealed preference theory. Due to the limitation of cognitive ability, decision makers might not be able to consider all feasible alternatives. Masatlioglu, Nakajima, and Ozbay (2012) reveals limited consideration as a result of unawareness of some alternatives. On the other hand, in this paper, the decision maker intentionally narrow downs feasible alternatives 3

6 with attribute-based aspirations. Cherepanov, Feddersen, and Sandroni (2013) also captures choice procedures to eliminate some feasible alternatives intentionally. The decision maker chooses the best alternative from the narrowed down alternatives. In this paper, the decision maker s taste for alternatives is induced by the primitive of the model, so our model is nested in decision-making processes. Unlike ours, their model is not nested. Manzini and Mariotti (2007, 2012) present axiomatic foundations for sequential choice procedures. The Manzini and Mariotti approach also weaken WARP. Ok, Ortoleva, and Riella (2015) provide a seminal framework of reference-dependent choices by using revealed preference theory. Relaxing WARP, Ok, Ortoleva, and Riella (2015) identify reference points endogenously. In addition, to form a subjective reference point, Ok, Ortoleva, and Riella (2015) obtain the decision maker s subjective attribute space endogenously. This approach is new since decision analysts are not able to observe which attributes are made use of in decision-making. That subjective attributes are only made use of to narrow down feasible alternatives. That is, this process capture a psychological constraint, and preferences for alternatives are not related to attributes. One remark is that their models are restrictive in the sense that an endogenous reference point is an alternative in a given menu. Then, Ok, Ortoleva, and Riella (2015) cannot capture compromise effects. Tserenjigmid (2016) explicitly model reference points to allow for both attraction effects and compromise effects in terms of diminishing sensitivity. To keep predictive power, Tserenjigmid (2016) puts the reference point as a vector of the minimums of each dimension the given menu. That reference point does not have the role of narrowing down feasible alternatives. On the other hand, in this paper, the parameter of the criterion for decision-making is uniquely identified. 3 Axioms We introduce notation. Let X be a finite set of all alternatives. To extend the axiomatic analysis into attribute-based inferences, assume X Π n i=1 X i where A = {1,, n} be a finite set of attributes for alternatives. The domain X i is regarded as an attribute i s evaluation of alternatives. The elements of X i are denoted by x i, y i, z i X i. 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 min x A y B d(x, y), max 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

7 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. 3.1 Standard Preferences We present some basic axioms for the binary relation on (X A). 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. 1 Also, the property of transitivity is required. Next, we introduce a consistency condition for reference-dependent preferences. To do so, for simplicity, we define on X as follows. Definition 1. 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 are justified by the ranking of choices from pairwise menus. In addition, we provide an axiom of continuity. Axiom (Continuity): The sets {(x, A) (X A) (x, A) (y, B)} and {(x, A) (X A) (y, B) (x, A)} are closed (in the product metric d ). 1 See Evren and Ok (2011) for the generalization of incomplete preference relations. 5

8 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). We postulate Standard Preferences: (i) Weak Order, (ii) Pairwise Consistency, (iii) Continuity, and (iv) Non-Degeneracy. Axiom (Standard Preference): satisfies (i) Weak Order, (ii) Pairwise Consistency, (iii) Continuity, and (iv) Non-Degeneracy. Next, we present a simple extension, which makes it possible to induce attribute-byattribute evaluations. We introduce the following definition. We define the binary relation on X. By using the definition of, we define the ranking of an attribute i by i on X i. The ranking i for each i A is induced by. Definition 2. We say that x i i y i if (x i, z i ) (y i, z i ) for some z i X i. We have the following lemma, which states that if an option is desirable with respect to all attribute-based evaluations than another option, then the option is preferred irrespective of menus. We can obtain the monotonic property for the ranking on X, i.e., attribute-based evaluations. Lemma 1. For any x, y X, if x i i y i for any i A, then x y. Following Lemma 1, we introduce the following axiom. This axiom is a monotonic condition for the aggregation of attribute-based evaluations. Axiom (Attribute-Based Monotonicity): For any x, y X, if x i i y i for any i A, then (x, A) (y, A) for any A A. The axiom says that if an option is desirable than another option in terms of attribute-based evaluations, then the option is preferred to the other in the choice of that option from any menus. 3.2 Reference-Dependent Preferences To obtain a subjective reference point (an aspiration level in attribute-based inferences) endogenously, we introduce a key axiom in our framework: Separability. This axiom is a minimal requirement to study menu effects including reference-dependence. 6

9 Axiom (Separability): For any x, y A B, (x, A) (x, B) (y, A) (y, B). This axiom says that, for some option x A B, if choosing the option from a menu A is preferred to choosing the option from the menu B, this type of menu preference holds for any options y A B. In the framework of this paper, this axiom captures reference-dependent preferences. The ranking between (x, A) and (y, B) means that if (x, A) is preferred to (y, B), then x in the menu A is relatively more preferred to a choice of x 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 x A, there exists y X such that y i i x i for any i A. Then, (i) (x, A) (x, A {y}); and (ii) (y, A {y}) (y, {y}). Suppose that for any altenatives x A, there exists an alternative y X such that y i i x i for any i A. That is, y dominates x. The first condition requires that if such a desirable alternative y is added into a menu A, then the comparative valuation of x A decreases due to the existence of y A {y}. This axiom can be interpreted as the notion of satisfaction in Simon (1957). In other words, adding y into A changes the criterion of decision-making. As a result, the criterion in comparative valuations increases. Then, the evaluation of alternatives x in the menu A decreases. On the other hand, the second condition requires that the added alternative y is relatively preferred in the menu A {y} more than that in {y} due to the alternatives in the original menu A. 3.3 Relaxing Independence of Irrelevant Alternatives Reference-dependent preferences may allow for preference reversals, which deviates from the postulate of rationality. We introduce a consistency condition, i,e., Independence of Irrelevant Alternatives. Axiom (Independence of Irrelevant Alternatives (IIA) ): For any x, y A B, (x, A) (y, A) (x, B) (y, B). This axiom requires that the ranking of alternatives is not affected by menus. The axiom of IIA can be written in the following way: (x, A) (y, A) (x, A {z}) (y, A {z}) for 7

10 any z X. This statement implies that irrelevant alternatives do not change any tastes of choices from menus. To allow for some preference reversals like attraction effects or compromise effects, we need to relax IIA. Given a menu A A, attraction effects and compromise effects can happen when a new alternative z / co(a) is added. Axiom (Weak IIA): For any x, y A, (x, A) (y, A) (x, A {z}) (y, A {z}), for any z co(a) where co(a) is the convex hull of a menu A. This axiom says that IIA only holds when an alternative z co(a) is added. The convex hull of a menu is the smallest convex set that contains A. Even if an alternative z co(a) is added into A, the attention of the decision maker does not change. Next, we provide a weaker version of translation invariance. Axiom (Weak Translation Invariance): For any x, y A, z X, and λ [0, 1], (x, A) (y, A) (λx + (1 λ)z, λa + (1 λ){z}) (λy + (1 λ)z, λa + (1 λ){z}). The justification to introduce a property of translation invariance is that there is no menu effect under singleton menus. Then, the mixture of menus does not change the decision maker s taste of choice from menus. 4 Results 4.1 Representation Theorem We obtain the representation result of attribute-based inferences with reference-dependence. Theorem 1. The following statements are equivalent: (a) satisfies Standard Preferences, Attribute-Based Monotonicity, Separability, Comparative Valuations in Menus, Weak IIA, and Weak Translation Invariance. (b) There exists a tuple (U, f, θ, α) where U is the set of attribute-based functions denoted by U = {u 1,, u i,, u n }, u i : X i R, f = (f 1,, f n ) is a set of continuous and strictly increasing functions defined by f i : R R for each i {1,, n}, θ [0, 1] is a real number, and α = (α i ) i {1,,n} is a vector such that α i [0, 1] for any i {1,, n} such that is represented by a function V : (X A) defined by n n ) V (x, A) = u i (x i ) + θ f i (u i (x i ) u i (A), i=1 i=1 where u i (A) = α i max y A u i (y i ) + (1 α i ) min z A u i (z i ). 8

11 4.2 Uniqueness Result Proposition 1. Suppose that the two attribute-based utility representation (U, f, θ, α) and (U, f, θ, α ) represent the same binary relation. Then, the following statements hold: (i) u i = au i + b i for some a > 0 and b i R; (ii) f = f ; (iii) θ = θ ; (iv) α i = α i for any i A. 4.3 Characterization of Attitude toward Reference-Dependence We characterize the parameter θ. The parameter θ 0 captures how much the decision maker reacts to aspiration levels. Here, we characterize a special case where θ = 0, i.e., there is no aspiration levels. The decision maker s choices are menu-dependent. To analyze this case, consider the following axiom: Axiom (No Aspiration): For any A, B A such that x A B, (x, A) (x, B). Proposition 2. Suppose that is represented by a tuple (U, f, θ, α). Then, exhibits No Aspiration if and only if θ = 0. This result corresponds to the additively separable attribute-based utility representation in Krantz et al. (1971). There is no menu effects, so we can write down the representation as follows: For any A A, V (x, A) = U(x) = i A u i(x i ). 4.4 Characterization of Diminishing Sensitivity We characterize the property of the function f i : R R for each i A. The function f i captures how sensitively the decision maker reacts to aspiration levels. First, we characterize a special case where the decision maker has no reaction for aspiration levels, which reduces to the standard utility maximization. In this case, aspiration levels including reference points do not lead to any preference reversals. Remember that, Axiom (Independence of Irrelevant Alternatives (IIA)): For any x, y A B, (x, A) (y, A) (y, B) (y, B). This axiom says that menu effects do not affect the ranking of alternatives. In other words, irrelevant alternatives do not change the ranking of options, which implies that there is no effect of aspiration levels behaviorally. 9

12 We provide a behavioral characterization. We say that f is linear if for any a R, f(a) = a. Proposition 3. Suppose that is represented by a tuple (U, f, θ, α). Then, exhibits IIA if and only if f i is linear for each i A. Next, we explore the properties of f i for i A. As in Tversky and Kahneman (1991), diminishing sensitivity says that decision makers sensitively react to change from some reference points. Proposition 4. Suppose that is represented by a tuple (U, f, θ, α). If satisfies the axioms in Theorem 1, then f i is concave for each i A. 5 A Relationship between Preference and Choice 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. 5.1 Revealed Reference-Dependence We investigate 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 may 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 3. (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} in the similar way. 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. 10

13 Definition 4. (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}). 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) referenceindependence. We need to observe carefully whether a third alternative for a pairwise choice problem affects the taste on the pairwise choice problem or not. 5.2 Behavioral Foundation 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 2. Suppose that satisfies Completeness. Then, satisfies Transitivity if and only if c satisfies Reference Acyclicity. (2015). Next, we provide the following condition for choice correspondences in Ok, Ortoleva, and Riella 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. 11

14 Lemma 3. Suppose that satisfies Pairwise Consistency. Then, is a weak order if and only if c satisfies No-Cycle Condition. 6 Preference Reversals 6.1 Violations of IIA As we have seen, we relax IIA (Independence of Irrelevant Alternatives) to allow for referencedependent preferences behaviorally. We study the two behavioral regularities, attraction effects and compromise effects. To do so, throughout this section, for simplicity, we assume that the number of attributes is two, i.e., A = 2. Attraction Effects Attraction effect is one of the well-known behavioral regulaities as a violation of rational choice theory (Huber et al. (1982)). In the framework of this paper, attraction effects are defined in the following way (see also Figure 1). Let A = {x, y}. Definition 5. (Attraction Effects) Suppose that (x, A) (y, A), and that there exists z X such that (i) x dominates z; and (ii) z is not dominated by y. Then, (x, A {z}) (y, A {z}). Attribute 2 y x Attribute 2 y x z y 2 c({x, y}) Attribute 1 Attribute 1 x 2 c({x, y, z}}) Figure 1: Attraction Effects As the definition of attraction effects says, IIA is violated. In the case that z is added, the menu-dependent aspiration level decreases. As a result, x can be relatively attractive. 12

15 Compromise Effects Compromise effect is also one of the well-known behavioral regulaities as a violation of rational choice theory (Huber et al. (1982)). In the framework of this paper, compromise effects are defined in the following way (see Figure 2). Let A = {x, y}. Definition 6. (Compromise Effects) Suppose that (x, A) (y, A), and that there exists z X such that (i) z i i x i and z i i y i ; and (ii) z j j x j and z j j y j, for some i, j A where i j. Then, (x, A {z}) (y, A {z}). Attribute 2 y x Attribute 2 y x z y 2 c({x, y}) Attribute 1 Attribute 1 x 2 c({x, y, z}}) Figure 2: Compromise Effects If compromise effects happen, then IIA is violated. This behavioral regularity exhibits an aversion to extremeness. In the same way, when z is added, the menu-dependent aspiration level decreases. As a result, x can be relatively a complementary alternative. 6.2 Dissatisficing Aversion and Compromise Effects Even if the decision maker exhibits diminishing sensitivity, attraction effects may not happen. To capture this, let us focus on the position of aspiration levels. Assume that the decision maker has a criterion as the maximum of menus, i.e., α i = 1 for all i A. Consider a pairwise problem {x, y}. And, an alternative z that is dominated by both x and y is added. The maximum of menus is a criterion for decision-making, so adding z does not change aspiration levels. As a result, attraction effects do not happen. To characterize this case, we present the following axiom. 13

16 Attribute 2 y x z Attribute 1 Figure 3: Dissatisficing Aversion Axiom (Dissatisficing Aversion): Suppose that there exists z X such that for any i A, there exists x A with x i i z i. Then, (x, A) (x, A {z}). This axiom says that even if an alternative z that is dominated by some alternatives in a menu is added, the evaluations of alternatives in the original menu do not change. Proposition 5. Suppose that is represented by a tuple (U, f, θ, α). Then, exhibits Dissatisficing Aversion if and only if α i = 1 for any i A. 7 Concluding Remarks We have developed a reference-dependent model in attribute-based inferences, which captures a subjective and endogenous reference point as an aspiration level. To identify the menu-dependent aspiration level endogenously, we enriched the domain of preference relations. In terms of revealed preference theory, this reference-dependent model has a foundation for revealed reference-dependence in Ok, Ortoleva, and Riella (2015). In this sense, our characterization has testable implications. In addition, if we add a stronger axiom, No Menu Effect, then our model reduces an additively separable attibute-based utility representation. This is a standard preference maximization in economics. In this sense, the study of referencedependent preferences are easily connected to the study of utility representation in economic theory. In addition, this axiomatic model can explain about both the attraction effect and the compromise effect, which are well-known as preference reversals. We have presented how this model captures such behavioral regularities. One remark is that if the menu-dependent aspiration level exhibits the axiom of Dissatisficing Aversion, then the attraction effect does not happen. 14

17 A Proof of Theorem 1 A.1 Sufficiency Part We show the sufficiency part of Theorem 1 in the following steps. Step 1: In Step 1, we induce the set of attribute-based rankings ( i ) i A from the primitive of the model:. First, we prove Lemma 1. Next, we show that each attribute-based ranking i, i = 1,, n, is represented by a non-constant utility function u i : X i R. Finally, we show that the set of attribute-based rankings ( i ) i A is an additively separable representation. Remember that the definition of is stated as follows (Definition??). For any x, y X, we say that x y if (x, {x}) (y, {y}) By using this definition, we define the ranking of each attribute i by i on X i. The ranking i for each i A is induced by (Definition 2). We say that x i i y i if (x i, z i ) (y i, z i ) for some z i X i. Proof of Lemma 1: We prove Lemma 1, which states that if an alternative is better with respect to all attribute-based evaluations than another alternative, then the alternative is preferred. Formally, for any x, y X, if x i i y i for any i A, then x y. Notice that we use the convex combination in the standard manner. Proof. It is sufficient to show that ( i ) i A is well-defined. Consider a ranking of an attribute i: i. Suppose (x i, z i ) (y i, z i ) and (x i, z i ) < (y i, z i ). Consider 1 2 z i z i X i. Then, (x i, 1 2 z i z i ) (y i, 1 2 z i z i z i ) and (x i, z i ) < (y i, 1 2 z i z i ). This is a contradiction. Thus, we have x i i y i if (x i, z i ) (y i, z i ) for any z i X i. We can verify that other attributes rankings are well-defined. Hence, by definition, if x i i y i for any i A, then x y. is a binary relation on X Π n i=1 X i. We can obtain an additively separable utility representation by the Theorem 13 in Krantz et al. (1971) (See p.302). We verify that satisfies (i) weak ordering, (ii) independence, (iii) restricted solvability, (iv) every strict bounded standard sequence is finite, and (v) at least three components are essential. Let us state the axioms, respectively. (i) weak ordering: is complete and transitive. (ii) independence: A relation on X = Π n i=1 X i is independent if and only if, for every 15

18 B A, the ordering B for fixed x i, i A \ B, is unaffected by those choices. (iii) restricted solvability: A relation on X = Π n i=1 X i satisfies restricted solvability if and only if for each i A, whenever (x 1,, x i,, x n ) (y 1,, y i,, y n ) (x 1,, x i,, x n ), then there exists x i X i such that (x 1,, x i,, x n ) (y 1,, y i,, y n ). (iv) every strictly bounded standard sequence is finite: Suppose that is an independent weak ordering. For any set Z of consecutive integers, a set {x j i x j i X i, j Z} is a standard sequence if and only if there exist z i, z i X i such that (z i i z i ) and for all j, j + 1 Z, (x j i, z i) (x j+1 i, z i ). A standard sequence {xj i j Z} is strictly bounded if and only if there exist y i, z i X i such that for all j Z, y i i x j i i z i. This holds for the other attributes rankings. (v) at least three components are essential: A component in X i is essential if and only if there exist x i, y i X i and z i X i such that ((x i, z i ) (y i, z i )). This holds for at least three components. We can easily verify that the axioms (i) - (v) hold. In (i), by definition, satisfies completeness. By the transitivity of, satisfies transitivity. In (ii), it is easily shown by the definition of ( i ) i A. (iii) and (iv) are shown by Continuity of since this holds under any singleton menus. (v) is shown by Non-Degeneracy in Standard Preferences. There exists a set of attribute-based utility functions (u i ) i A where u i : X i R such that is represented by U(x) = i A u i(x i ). Hence, for any x, y X, x y if and only if U((x) U((y). Step 2: In Step 2, we construct a menu-dependent aspiration level by attributes, i.e., (u i ) i A. By using Separability and Comparative Valuations in Menus, we consider a menu-based binary relation on A. In addition, we induce an attribute-based menu binary relation i on A for each attribute i A. We construct a menu-dependent aspiration levels (u i ) i A. First of all, we define the two binary relations: and ( i ) i A. Define the binary relation on A. We say that for any A, B A, A B if for any x A B, (x, A) (x, B). This follows from Separability. Notice that the definition of presumes A 3. In addition, we define the following ranking of 16

19 each attribute i based on menus. We say that for any A, B A, A i B if for any x A and y B, we have x i i y i. Denote the asymmetric and symmetric part of i by i and i, respectively. We show that ( i ) i A satisfies Weak Order (Completeness and Transitivity) and Continuity. Axiom(Continuity): {A A A j B} and {A A A i B} are closed (in the Hausdorff metric topology). Remark. i (i A) is a weak order. As stated above, since i is well-defined and i is a weak order, i is also a weak order. Next, we show that i satisfies Continuity. To do so, it suffices to show that on A is continuous. For any A A, let us define u(a) = A = {(u 1 (x 1 ),, u n (x n )) R n x = (x 1,, x n ) A}. In addition, let A = {u(a) A A}. Since A A is compact, u(a) is also compact by the continuity of u i for each i A. A is a compact subset of R n. We endow A with the Hausdorff metric d h defined by d h (A, B ) = max{max u A min u B d(u, u ), max u B min u A d(u, u )}, where d is the Euclidean metric. We use the convex combination in the standard manner. For any A, B A and λ [0, 1], λa +(1 λ)b = {u R n u = λu +(1 λ)u for some u A, u B }. Define on A as follows: A B if A B. The asymmetric and symmetric parts of are described by and, respectively. Consider i. We show that i is continuous. Take an arbitrary menu A A to show {B A B A } and {B A A B } are closed (in the Hausdorff metric topology). Let {Bn} be a sequence of menus such that Bn A and Bn B. We show B A. By the definition of, we have a sequence {B n } such that u(b n ) = Bn and B n A. Since B n A and A is compact, there exists a convergent sub-sequence {B m} with B m B. We have B m A for all m. Since both u 1 and u S are continuous, u is continuous. Then, we have u(b m) u(b ). Since {u(b m)} is a sub-sequence of {Bn} and Bn B, u(b ) = B. Remember that B A, so we can obtain B A. In the similar way, it is easily shown that {B A A B } is closed. Hence, by definition, since and i are continuous, i is continuous. In the same way, we can show that j for any j A with j i is continuous. To construct the menu-dependent aspiration level, we use the axiom of Comparative Valuations in Menus. Fix an attribute i A. i satisfies weak order and continuity. Then, for any A, B A, A i B if and only if u i (A) u i (B) for some u i : A R. By the first condition of Comparative Valuations in Menus, suppose that for any x A, there exists z X such 17

20 that z i i x i. Then, we obtain (x, A) (x, A {z}) if and only if u i (A) u i (A {z}). On the other hand, take z X and suppose that there exists x A such that x i i z i. By Comparative Valuations in Menus, we have (x, A {z }) (x, A). Then, u i (A {z }) u i (A). By the definition of i, given A A, we define u i (A) = α i max x Au i (x) + (1 α i ) min x Au i (x) for some α i [0, 1]. This holds for any A A. Hence, we obtain a menu-dependent aspiration level (u i ( )) i A with α = (α 1,, α n ). Step 3: We show that on A is represented by a strictly increasing and continuous function f : R R. By Step 2, there exists a set of menu-dependent attribute-based aspiration levels (u i ) i A. Each menu-dependent attribute-based aspiration level is unique, so, without loss of generality, we can consider, for any x A, u i (x i ) u i (A). For any x, y A, x i i y i u i (x i ) u i (A) u i (y i ) u i (A). By definition, A B if for any x A B, (x, A) (x, B). By Step 1 and Attribute- Based Monotonicity, we have an additively separable utility representation with a strictly increasing and continuous function f i : R R for each i A. That is, for any A, B A, A B if and only if i A f i(u i (x i ) u i (A)) f i (u i (x i ) u i (B)). By the condition (i) in Weak Continuity, f i is continuous. By Attribute-Based Monotonicity, f i is strictly increasing. Step 4: We complete the utility representation. In other words, is represented on the whole domain. By Attribute-Based Monotonicity, if x y, then (x, A) (y, A) for any A A. Then, for any (x, A) (X A), define V : (X A) R by V (x, A) = i A u i(x i )+θ i A f i(u i (x i ) u i (A)) where u i (A) = α i max x A u i (x i ) + (1 α i ) min x A u i (x i ) and θ [0, 1]. First, take x, y X such that x i i y i for any i A. Consider an arbitrary menu A A such that x, y A. Then, we obtain i A u i(x i ) i A u i(y i ), and i A f i(u i (x i ) u i (A)) i A f i(u i (y i ) u i (A)). Hence, by Attribute-Based Monotonicity, V (x, A) V (y, A) for any A A. Next, by Separability, suppose (x, A) (x, B). Then, we obtain i A f i(u i (x i ) u i (A)) i A f i(u i (x i ) u i (B)). Thus, for any (x, A), (y, B) (X A), (x, A) (y, B) if and only if V (x, A) V (y, B). V is well-defined. For any binary choice problems {x, y} A, by Pairwise Consistency, (x, {x}) (y, {y}) if and only if (x, {x, y}) (y, {x, y}). It is easily verified that θ [0, 1]. Therefore, we obtain the desired utility representation. 18

21 A.2 Necessity Part We show the necessity part. We show that the utility representation satisfies Separability. Other axioms are easily verified. First, we show that the representation satisfies Separability of. Take two menus A, B A such that x, y A B. Suppose that V (x, A) V (x, B). Then, we obtain V (y, A) V (y, B). Hence, for any x, y A B, if (x, A) (x, B), then (y, A) (y, B). satisfies Separability. B Other Proofs B.1 Proof of Lemma 2 First, we show the necessity part. Suppose that c satisfies Reference Acyclicity. Consider a finite sequence of alternatives: x 1,, x m X where m 2. Let A i = {x i, y i }. By Reference Acyclicity, for i = 1,, m 1, we have x i c(a i {x i+1 }). Define x i c(a i {x i+1 }) if and only if (x i, A i {x i+1 }) (y i, A i {x i+1 }). Moreover, by Reference Acyclicity, y m c(a m {x 1 }) if and only if (y m, A m {x 1 }) (y m, A m {x 1 }). Suppose that (x i 1, A i 1 {x i }) (x i, A i {x i+1 }) for each i = 2,, m 1. Then, we have (x 1, A 1 {x 2 }) (x 2, A 2 {x 3 }) (x m 1, A m 1 {x m }). If (x 1, A 1 {x 2 }) (x m, A m {x 1 }), this violates Reference Acyclicity. Hence, (x 1, A 1 {x 2 }) (x m, A m {x 1 }). This implies that satisfies Transitivity. Next, we show the sufficiency part. Suppose that satisfies Transitivity. Let A i = {x i, y i } for i = 1,, m. Suppose that (x i, A i ) (y i, A i ) for i = 1,, m, and that (x i, A i {x i+1 }) (y i,, A i {x i+1 }) for i = 1,, m 1. Let us define (x i, A i ) (y i, A i ) if and only if y i c (A i ). Suppose that (x i, A i {x i+1 }) (x i+1, A i+1 {x i+2 }) for i = 1,, m 1. By Transitivity of, c (A m {x 1 }) y m. Then, c satisfies Reference Acyclicity. B.2 Proof of Lemma 3 First, we show the necessity part. Suppose that c satisfies No-Cycle Condition. Let us define (x, {x, y}) c (y, {x, y}) if x c({x, y}). Since c is non-empty, this definition satisfies completeness. Moreover, since c is non-empty, for any x X, we have c({x}) =}x{. if c satisfies No-Cycle Condition, then we can obtain (x, {x, y}) c (y, {x, y}) if and only if (x, {x}) c (y, {y}). Thus, Pairwise Consistency is satisfied. Since c satisfies No-Cycle Condition, c satisfies Transitivity. Next, we show the sufficiency part. satisfies Completeness, Transitivity, and Pairwise Consistency. By Pairwise Consistency, let us define a choice correspondence induced by for binary choice problems: c ({x, y}) = {x {x, y} (x, {x}) (y, {y}), y {x, y} }. By definition, it is easily shown that c satisfies No-Cycle Condition. 19

22 C Proof of Propositions C.1 Proof of Proposition 1 We show the uniqueness result of Theorem??. In (i), for any i A, u i is unique up to positive affine transformations: i.e., u i = au i + b i for some a > 0 and b i R. This result follows from Krantz et al. (1971). It is immediately shown that u i, for i A, is unique up to a posistive affine transformation. (ii) is shown by the uniqueness property of (u i ) i A. Fix an attribute i A. Take an option x A. Let û i (x i ) = u i (x i ) u i (A). Since u i is unique up to a positive affine transformation, there exists γ > 0 such that f i u i = γ f i u i for each i A. Hence, we obtain f i = f i. f i : R R is unique. To show (iii), we use the uniqueness result of (u i ) i A again. C.2 Proof of Proposition 2 Suppose that is represented by a tuple (U, f, θ, α). First, we show the necessity part. Suppose that satisfies No Aspiration. Then, for any x A B, (x, A) (x, B). Then, V (x, A) = (x, B). Therefore, θ = 0. Next, we show the sufficiency part. Suppose θ = 0. Take an option x from a menu A. Then, V (x, A) = U(x) = i A u i(x i ). Then, we have V (x, A) = (x, B). That is, (x, A) (x, B). Thus, satisfies No Aspiration. C.3 Proof of Proposition 3 Suppose that is represented by a tuple (U, f, θ, α). First, we show the necessity part. Suppose that satisfies IIA. Then, if (x, A) (y, A), then, (x, {x}) (y, {y}). This holds for any A x, y. That is, i u i(x i ) u i (A) i u i(y i ) u i (A) holds. Hence, f is linear. Next, we show the sufficiency part. Suppose that f is linear. By the utility representation, for any A A, V (x, A) V (y, A) U(x) U(y). Then, V (x, A ) V (y, A ) holds for A A A. Hence, satisfies IIA. 20

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