Choosing with the worst in mind: A reference-dependent model

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Choosing with the worst in mind: A reference-dependent modelR Gerelt Tserenjigmid Department of Economics, Virginia Tech, 3016 Pamplin Hall, 880 West Campus Drive, Blacksburg, VA 24061, USA

a r t i c l e

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Article history: Received 5 August 2018 Revised 1 November 2018 Accepted 2 November 2018 Available online xxx JEL classification: D01 D11 D81 D91

a b s t r a c t This paper focuses on choices from menus of two-attribute alternatives and develops a reference-dependent model with endogenous reference points. The reference point for a given menu is a vector that consists of the minimum of each dimension of the menu. The model is characterized by a novel weakening of the Weak Axiom of Revealed Preference in addition to separability axioms. We show an equivalence between the attraction and compromise effects, two well-known choice reversals, and diminishing sensitivity, a widelyused behavioral property. We provide bounds on choice reversals, theoretically justify using minimums as references, and apply the model to intertemporal choice. © 2018 Elsevier B.V. All rights reserved.

Keywords: Reference dependence Menu-dependent reference WARP Attraction effect Compromise effect Diminishing sensitivity Intertemporal choice

1. Introduction Rational choice theory does not allow for choice reversals, which occur when an agent chooses alternative x over alternative y in some cases but chooses y over x in other cases. However, many experimental, marketing, and field studies suggest that the introduction of additional alternatives which are not chosen could cause a choice reversal. Choice reversals are often explained by models of reference-dependent behavior in which the agent’s basis for decision making, her reference point, changes between choice problems. In this paper, we axiomatically develop a reference-dependent model with endogenous reference points. The model is not just consistent with two well-known choice reversals, the compromise effect and the attraction effect (to be defined below), but it also establishes an equivalence between the two effects and diminishing sensitivity (to be defined below), a widely-used behavioral property in economics.

R This paper is a revised version of the second chapter of my dissertation at Caltech. I am indebted to Federico Echenique and Pietro Ortoleva for their advice, guidance, and encouragement. I also thank Colin Camerer, Kota Saito, Leeat Yariv, Yuval Salant, Marcelo Fernandez, Matthew Kovach, Simone CerreiaVioglio, Tom Cunningham, Larry Samuelson, John Quah, Nicola Gennaioli, Kwok Ping Tsang, Rahul Bhui, Adam Dominiak, and the audiences of the Fall 2014 Midwest Economic Theory Conference, SWET 2015, BRIC 2015, RUD 2015, and 11th World Congress of the Econometric Society. E-mail address: [email protected]

https://doi.org/10.1016/j.jebo.2018.11.001 0167-2681/© 2018 Elsevier B.V. All rights reserved.

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Fig. 1. Compromise and attraction effects.

Before introducing our modeling approach, let us discuss the compromise and attraction effects.1 Throughout the paper, we focus on alternatives with two attributes and assume the agent’s preference is increasing in both attributes. For example, the first dimension could be the inverse of the price (cheapness) of the good and the second dimension could be the quality of the good. In both the compromise and attraction effects, the agent first compares two alternatives x = (x1 , x2 ), a cheap and low-quality good, and y = (y1 , y2 ), a medium-price and medium-quality good. Since x1 > y1 and x2 < y2 , the agent faces a tradeoff between price and quality. Suppose x is chosen over y. Then the two effects relate to the consequences of adding a very expensive third alternative. Compromise effect (henceforth, CE): the introduction of a very expensive, high-quality good z = (z1 , z2 ) causes a choice reversal; that is, y is preferred over x in the presence of z (the left-hand side of Fig. 1). The common explanation is that the very expensive, high-quality good z makes the cheap, low-quality good x seem like an extreme alternative. Therefore, people compromise by choosing y since they have a tendency to avoid extremes.2 Attraction effect (henceforth, AE): the introduction of a very expensive, but medium-quality good z = (z1 , z2 ) causes a choice reversal; that is, y is preferred over x in the presence of z (the right-hand side of Fig. 1). The common explanation is that since the medium-quality, but expensive good z is dominated by the medium-quality, medium-price good y, but not by x, the third alternative makes y seem more attractive than x. In other words, z works as a decoy for y. Some suggest that the CE and the AE are not separate, but rather two manifestations of the same behavior. Consistent with this suggestion, in the seminal paper by Simonson (1989), the same group of subjects exhibited the two effects in roughly the same magnitude. One of the aims of this paper is to develop a model that can provide a unified explanation for the two effects using diminishing sensitivity (DS): the marginal values (of attributes) decrease with their distance from the reference point. In fact, to the best of our knowledge, this is the first paper that theoretically shows that the two effects are not only the implications of DS, but they are also equivalent to DS under our reference-dependent model (Propositions 1, 3, and 4). We focus on a choice-theoretic environment in which an agent makes choices from menus of two-attribute alternatives. To introduce our model, we use the following formulation of reference dependence by Tversky and Kahneman (1991): the total utility of a two-attribute alternative x for an exogenously given reference point r = (r1 , r2 ) is













U x|r = f 1 u1 ( x1 ) − u1 ( r1 ) + f 2 u2 ( x2 ) − u2 ( r2 ) ,

(1)

where u1 and u2 are strictly increasing utility functions and f1 and f2 are strictly increasing distortion functions. In other words, the utility of the ith dimension xi is evaluated with respect to the reference for the ith dimension ri and distorted by the distortion function fi . Despite their explanatory power, reference-dependent models with exogenous reference points can be consistent with essentially any choice behavior by appropriately choosing reference points. In order to have a model that makes testable predictions for observed behavior, we need to explicitly model reference points in a way that they are determined by observable factors. Therefore, in this paper we propose a model in which reference points are endogenously determined by the menu that the agent faces. In particular, the agent uses a vector that consists of the minimum of each dimension of the menu as a reference point; that is, mA is the reference point of a menu A where mAi is the minimum of the ith dimension of A. Since utility functions are strictly increasing, mAi is the worst attribute of the ith dimension of A. We also require some symmetry of the two dimensions (i.e., f1 = f2 ) since the utility distance from the reference point on each dimension is taken 1 The compromise effect and the attraction effect (also called the asymmetric dominance effect or the decoy effect) are first documented in the experimental studies of Simonson (1989) and Huber et al. (1982), respectively and later confirmed by many studies in consumer choice (e.g., Ariely and Wallsten, 1995; Chernev, 2004; Doyle et al., 1999; Herne, 1998; Sharpe et al., 2008; Simonson and Tversky, 1992; Tversky and Simonson, 1993). These effects are also demonstrated in the contexts of choice over risky alternatives (Herne 1999), choice over policy issues (Herne 1997), and choice over political candidates (Sue O’Curry and Pitts 1995), among others. 2 Here x1 is the lowest price and z1 is the highest price while x2 is the lowest quality and z2 is the highest quality.

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into account. Then in our model, the total utility of an alternative x of a given menu A is









UA (x ) = f u1 (x1 ) − u1 (mA1 ) + f u2 (x2 ) − u2 (mA2 ) . It is very natural to think that people care about how attractive a given attribute is compared to the worst attributes. In economics, the Nash bargaining solution is precisely modeling people who use minimums as references by the functional form log(u1 (x1 ) − u1 (mA1 )) + log(u2 (x2 ) − u2 (mA2 )). In psychology, there is a long tradition of evaluating attributes with respect to the best and worst attributes (e.g., the range-frequency theory of Parducci, 1965).3 The main result of this paper provides an axiomatic foundation for the above reference-dependent model with endogenous reference points. In particular, the model is characterized by a novel weakening of the Weak Axiom of Revealed Preference (WARP), called Dominated WARP (D-WARP), in addition to relatively standard separability axioms (Theorem 1). D-WARP captures the essence of the model (using minimums as references) and it can be tested quite easily in experiments. In Section 4, we also propose a generalization of the CE, called the general compromise effect. We then show that the general compromise effect is not just (almost) equivalent to DS in our model, but it also justifies use of minimums as references for more general class of models.4 In particular, the general compromise effect can pin down minimums from all possible (well-behaved) references.5 In Section 5, we apply the model to intertemporal choice. The main implication of the model is that borrowing constraints produce a psychological pressure to move away from the constraints even if they are not binding. It is common to explain deviations from the standard model of consumer choice such as the excess sensitivity of consumption to current income and a hump-shaped consumption profile with liquidity or borrowing constraints (see Attanasio and Weber 2010). But empirically it cannot be determined whether constraints were binding at the time of the decision. Our model gives a justification for those explanations since liquidity constraints can have an effect on choices even if they are not binding. In Section 6, we discuss existing models of reference-dependent preferences including two models that endogenize reference points (Koszegi and Rabin, 20 06; 20 07; Ok et al., 2014). A challenge to many existing reference-dependent models that can rationalize the CE and the AE is that they do not predict the symmetric dominance effect, which states that a choice reversal is unlikely when the third alternative is dominated by both x and y symmetrically.6 However, our model can predict the symmetric dominance effect as well as the following implication of the AE and DS, called the two-decoy effect. Recall the situation in which x is chosen over y, but y is chosen over x when a decoy z for y is added. When there is the fourth alternative which is a decoy for x, then x should be chosen over y. Intuitively, the decoy effect of z for y should be canceled out by the decoy effect of the fourth alternative for x. The symmetric dominance and two-decoy effects allow us to easily distinguish our model from many prominent models in the literature (see Section 3.2). The paper is organized as follows. In Section 2, we formally define the model and provide the behavioral foundation of it. In Section 3, we discuss behavioral implications of the model and DS. Section 4 also discusses the relation between the general compromise effect, DS, and minimums as references. Section 5 discusses applications to intertemporal choice. Section 6 discusses the related literature. The proofs, four extensions of the model (e.g., extensions to n-attribute alternatives), and applications to risky choice are collected in the appendices. 2. Model Let X = X1 × X2 ≡ R2+ be the set of all alternatives and A ⊂ 2X \ {∅} be a collection of compact subsets (menus) of X. Suppose A includes all menus with two and three alternatives. An alternative with two attributes can have many different interpretations: (i) a single consumer good with two attributes (price and quality), (ii) a consumption bundle of two goods, (iii) an allocation to two agents, (iv) a state-contingent prospect, (v) a binary lottery, or (vi) a consumption stream.7 For each menu A ∈ A , we denote the  vector that consists of the minimum of each dimension (i.e., the meet of A) by mA = (mA1 , mA2 ) ≡ minx∈A x1 , minx∈A x2 . For all x, y ∈ X, we write x > y if x = y, x1 ≥ y1 , and x2 ≥ y2 , and x >> y if x1 > y1 and x2 > y2 . We also say x dominates y if x > y. 3 In our model, the two dimensions are evaluated separately, which is motivated by actual decision making procedures (e.g., see Arieli et al., 2011; Tversky, 1969). This separation may lead to a reference point mA = (mA1 , mA2 ) that is not an element of A. However, this does not mean that the agent has to evaluate alternatives with respect to something imaginary. It simply means that price x1 and quality x2 are evaluated with respect to the highest price mA1 and the lowest quality mA2 , which are not difficult to observe from A. Moreover, in the CE there is no obvious alternative in the menu to be a reference. The literature of habit formation is another example that studies agents who use objects (habit stocks) that are not necessarily consumed or available as reference points. 4 A special case of our model with a CRRA distortion function and linear utility functions is discussed in Kivetz et al. (2004). They show that the strict concavity of the CRRA distortion function explains the CE and the AE. However, the aim of our paper is not just to explain the two effects, but also to establish the equivalence between the two effects and DS. Moreover, axiomatic foundations, characterizations of DS, justifications of minimums, and economic applications are not discussed in Kivetz et al. (2004). 5 Notice that models that uses maximums as references cannot explain the AE. Moreover, models in which reference points are determined by several factors in the menu will have a much weaker predictive power. For example, if we allow any kind of menu dependence for reference points, then the model will not have any testable restriction on observed behavior (Proposition 6 in Appendix C). Hence we need to restrict our attention to a particular kind of menu dependence such as dimension-by-dimension minimums. In Appendix E, we will consider a model in which reference points depend on both the maximums and the minimums. 6 See Masatlioglu and Uler (2013) for an experimental evidence and the detailed discussion of the challenge that existing reference-dependent models face. 7 It is not difficult to extend our model to the context of choices from menus with n-attribute alternatives. See Appendix B.

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The primitive is a choice correspondence C : A ⇒ X where C(A) is non-empty and C(A)⊆A for each A ∈ A . For notational simplicity, we write C (A ) = x instead of C (A ) = {x} when C(A) is a singleton. First, we define our model. Definition 1. A choice correspondence C is a Min-Min Reference Dependent Choice (MRDC) if there are strictly increasing functions f, u1 , u2 : R+ → R+ such that for any menu A ∈ A ,

 





C (A ) = arg max f u1 (x1 ) − u1 (mA1 ) + f u2 (x2 ) − u2 (mA2 ) x∈A



.

(2)

Note that the calculations of the utilities rely on a procedure with the following three components: (i) evaluations of alternatives are relative, (ii) dimensions are evaluated separately, and (iii) attributes are evaluated with respect to minimums. The first component is not only the essence of reference dependence, but it is also psychologically appealing. In fact, the CE and the AE are well-known examples that suggest people evaluate alternatives in relative terms. The second component, the separation between two dimensions, is motivated by actual decision making procedures (e.g., see Arieli et al., 2011; Tversky, 1969). For example, by using an eye-tracking experiment in the context of choice over binary lotteries, Arieli et al. (2011) found that subjects compare prizes (the first dimension) and probabilities (the second dimension) separately. The third component, using minimums as references, is consistent with the range-frequency theory of Parducci (1965) in psychology and will be theoretically justified in Sections 2.1 and 4.8 Denote a MRDC with functions f, u1 , u2 by C( f, u1 , u2 ) . Throughout the paper, we assume that f, u1 , u2 are continuous and



surjective. Note that when f is linear, the MRDC reduces to the standard additive utility model: C (A ) = arg maxx∈A u1 (x1 ) +



u2 ( x2 ) . Now we define two induced preferences  and t for a given C. Let and t (∼ and ∼t ) denote the asymmetric (resp. symmetric) parts of  and t . Definition 2 (Induced preferences). For all x, y, t ∈ X, (i) xy if and only if x ∈ C({x, y}), and (ii) xt y if and only if x ∈ C({x, y, t}).

Rational choice theory relies on WARP, which requires that a comparison of alternatives x and y of A is not affected by other alternatives in A. Therefore, under WARP, xt y implies xy. In this paper, we weaken WARP and allow for choice reversals such as x y and y t x. Finally, note that for any MRDC C( f, u1 , u2 ) ,  satisfies transitivity (i.e., xy and yz imply xz) as illustrated in the following remark. Remark 1. Take any binary menu {x, y} with x1 > y1 and x2 < y2 . Notice that the reference point of {x, y} is (y1 , x2 ). Therefore, we have







((



((



x  y ⇔ f u1 ( x1 ) − u1 ( y1 ) + ( f ( u2 (( x2( ) −(u( f ( u1 (( y1( ) −(u( 2 ( x2 ) ≥ ( 1 ( y1 ) + f u2 ( y2 ) − u2 ( x2 ) ⇔ u1 ( x1 ) + u2 ( x2 ) ≥ u1 ( y1 ) + u2 ( y2 ).



2.1. Axioms and representation theorem We now axiomatically characterize our model by two novel axioms called Dominated WARP (D-WARP) and Reference Separability in addition to three standard axioms. The first axiom, Standardness, is a collection of three properties that guarantee a well-behaved representation. Axiom 1. (Standardness) Let C be a choice correspondence on A . i) (Monotonicity) For all A ∈ A and x, x ∈ A, if x > x, then x ∈ C(A). ii) (Continuity)  and {t }t ∈ X are continuous; that is, for all x, t ∈ X, {y ∈ X: yx}, {y ∈ X: xy}, {y ∈ X: yt x}, and {y ∈ X: xt y} are closed. iii) (Solvability) For all x, y, t ∈ X, for each i ∈ {1, 2}, there exists xi ∈ Xi such that (xi , x j ) y and (xi , x j ) t y. Monotonicity and continuity are standard properties. Solvability requires that (xi , x j ) can be preferred over y as long as  xi is large enough. Solvability is rather technical, but it is frequently used in the literature. We also require transitivity of binary comparisons.9

8 Although our representation theorem proves the uniqueness of f, u1 , u2 under standard separability axioms, to a certain extent using minimums has an advantage of not relying on cardinality of attributes. For example, consider a menu of personal computers, A = {x = (10 0, 90 0 ), y = (40 0, 40 0 ), z = (90 0, 10 0 )}, where the first dimension represents speed and the second dimension represents memory by some measures of speed and memory. Suppose that A is equivalent to A˜ = {x˜ = (10, 30 ), y˜ = (20, 20 ), z˜ = (30, 10 )} by different measures of speed and memory. In the MRDC, the reference point is de˜ ˜ termined by the lowest speed mA1 = 100 (or mA1 = 10) and the lowest memory mA2 = 100 (or mA2 = 10). In fact, u1 and u2 can be adjusted appropriately to generate equivalent behaviors. However, when the reference point is determined by the average speed and memory, y is strictly worse than the reference point (500, 500) of A while y˜ is equivalent to the reference point (20, 20) of A˜ . 9 In Appendix F, we weaken transitivity and obtain a general representation with two different distortion functions.

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Fig. 2. Cancellation (i).

Axiom 2 (Transitivity). For all x, y, z ∈ X, if xy and yz, then xz. The next axiom, D-WARP, is our main axiom and a weakening of WARP. Recall that WARP is stated as follows: for all A, B ∈ A and x, y ∈ A ∩ B, if x ∈ C(A) and y ∈ C (B ), then x ∈ C (B ). To capture the idea of the worst (or minimums) in mind, for any menu A, we define a set of alternatives that are dominated by all alternatives in A:

D(A ) = {x ∈ X : x ≤ a for any a ∈ A}. D-WARP is a postulate that is very similar to WARP and can be stated in the following way: WARP is satisfied when the sets of alternatives that are dominated by all alternatives in A and all alternatives in B, respectively, are the same. Axiom 3 (Dominated WARP (D-WARP)). For all A, B ∈ A , x, y ∈ A ∩ B,

if x ∈ C (A ), y ∈ C (B ), and D(A ) = D(B ), then x ∈ C (B ). D-WARP essentially captures the idea of using minimums as references. In fact, Lemma 3 shows that under Monotonicity and D-WARP, for all x, y ∈ A, if x ∈ C (A ), then x mA y. Therefore, under Standardness and D-WARP, we obtain the following general representation: for any A ∈ A ,

C (A ) = arg max V (x, mA ), x∈A

(3)

where V(x, mA ) is strictly increasing and continuous in x. The last two axioms impose more structure on V (x, mA ) in the representation (3). With the next axiom, called Cancellation, in addition to Transitivity, we can use existing methods to obtain additive representations. It is well known that to have an additive representation two axioms of Transitivity and Cancellation are necessary and sufficient (see Fishburn and Rubinstein, 1982; Krantz et al., 1971; Tversky and Kahneman, 1991; Wakker, 1988).10 In particular, we use Cancellation for  and 0 where 0 = (0, 0 ). Axiom 4 (Cancellation). For all x, y, z ∈ X, i) if (x1 , x2 ) ∼ (y1 , z2 ) and (y1 , y2 ) ∼ (z1 , x2 ), then (x1 , y2 ) ∼ (z1 , z2 ); ii) if (x1 , x2 ) ∼ 0 (y1 , z2 ) and (y1 , y2 ) ∼ 0 (z1 , x2 ), then (x1 , y2 ) ∼ 0 (z1 , z2 ). Fig. 2 illustrates Cancellation for . The solid and dashed curves are indifference curves. Intuitively, it requires that if the relative advantage of x1 over y1 is equivalent to that of z2 over x2 (dashed intervals) (i.e., (x1 , x2 ) ∼ (y1 , z2 )) and the relative advantage of y1 over z1 is equivalent to that of x2 over y2 (dotted intervals), then the relative advantage of x1 over z1 is equivalent to that of z2 over y2 (combined intervals). The last axiom, Reference Separability, is a novel modification of the standard Separability axiom. Separability requires that if (x1 , x2 ) ∼ (y1 , y2 ), (x1 , x2 ) ∼ (y1 , y2 ), (x1 , x2 ) ∼ (y1 , y2 ), then (x1 , x2 ) ∼ (y1 , y2 ).11 Separability is motivated by the idea that two dimensions are evaluated separately (see Arieli et al., 2011; Tversky, 1969 for experimental evidence). Reference Separability extends Separability by requiring separate evaluations of two dimensions in the presence of a third alternative. Formally, 10

Cancellation is sometimes called the Thomsen condition or double cancellation. Separability is used in the theories of additive utility representation. See Keeney and Raiffa (1976), Wakker (1988), and Tserenjigmid (2015). Separability is also called the corresponding tradeoff condition in Keeney and Raiffa (1976) and triple cancellation in Wakker (1988). There are a large literature on Separability (See Blackorby et al. 2008). 11

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Fig. 3. Reference Separability.

Axiom 5 (Reference Separability). For all i, j ∈ {1, 2} with i = j and for all x, y, x , y , z ∈ X with xj > yj > z j and xi , xi < zi , if

( xi , x j ) ∼ ( yi , y j ), (xi , x j ) ∼ (yi , y j ), and   (xi , x j ) ∼z (yi , y j ), then (xi , xj ) ∼z (yi , yj ). Reference Separability is illustrated in Fig. 3. Note that x and y are on the same indifference curve. Similarly, (x1 , x2 ) and (y1 , y2 ) are also on the same indifference curve. Therefore, roughly speaking, the relative advantage of y2 over x2 is equal

to that of y2 over x2 . Reference Separability requires that the relative advantage of y2 over x2 in the presence of z is equal to that of y2 over x2 in the presence of z. In other words, (x1 , x2 ) is indifferent to (y1 , y2 ) in the presence of z (solid curve Iz ) if and only if x is indifferent to y in the presence of z (dashed curve Iz ). Alternatively, Reference Separability says that if the relative advantage of y2 over x2 is equal to that of y2 over x2 , then the advantages of the compromise alternatives (y1 , y2 ) and (y1 , y2 ) caused by the third alternative z against (x1 , x2 ) and (x1 , x2 ) are the same. It turns out that, under Reference Separability and Cancellation (i), we show that there are functions W, u1 , u2 such that





C (A ) = argmax{ W u1 (x1 ) − u1 (mA1 ), u2 (x2 ) − u2 (mA2 ) },

(4)

x∈A

where W (t, 0 ) = W (0, t ) for any t ∈ R+ (Lemma 5). Therefore, Reference Separability precisely captures the separate evaluations of two dimensions. Moreover, under Cancellation (ii), we show that W is additive. Finally, we can state our main theorem. Theorem 1 characterizes the representation (2) and also provides a uniqueness result, which guarantees that utility functions have cardinal meaning. Theorem 1. A choice correspondence C satisfies Standardness, Transitivity, D-WARP, Cancellation, and Reference Separability if and only if there exist strictly increasing, continuous, and surjective functions f, u1 , u2 : R+ → R+ such that for any menu A ∈ A ,









C (A ) = arg max{ f u1 (x1 ) − u1 (mA1 ) + f u2 (x2 ) − u2 (mA2 ) }. x∈A

Moreover, for any two vectors of strictly increasing, continuous, and surjective functions (f, u1 , u2 ) and ( f  , u1 , u2 ) such that f (1 ) = f  (1 ) and u1 (1 ) = u1 (1 ), if C = C( f, u1 , u2 ) = C( f  , u , u ) , then ( f, u1 , u2 ) = ( f  , u1 , u2 ). 1

2

The uniqueness result in Theorem 1 can be stated in two steps. First, by Remark 1 in Section 2, we have xy if and only if u1 (x1 ) + u2 (x2 ) ≥ u1 (y1 ) + u2 (y2 ). It is known that u1 and u2 are unique up to a positive linear transformation (e.g., see Krantz et al. 1971). Second, for given u1 and u2 , f is also unique up to a positive linear transformation. In other words, after fixing f(1) and u1 (1), functions f, u1 , and u2 are unique.

3. Behavioral implications of diminishing sensitivity in MRDC Diminishing sensitivity (henceforth, DS) is a widely-used behavioral property in economics and will be the driving force behind our results in Sections 3–5. We define DS in terms of functional properties on f as in Tversky and Kahneman (1991). It requires that marginal utility is decreasing in the distance from the reference point. Please cite this article as: G. Tserenjigmid, Choosing with the worst in mind: A reference-dependent model, Journal of Economic Behavior and Organization, https://doi.org/10.1016/j.jebo.2018.11.001

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Fig. 4. The effect of a shift in reference points when f (t ) =

7

√ t.

Definition 3 (Diminishing sensitivity). A MRDC C( f,u1 ,u2 ) satisfies diminishing sensitivity (DS) if, for all x1 , y1 , r1 , r1 ∈ X1 such that x1 > y1 ≥ r1 > r1 ,12

















f u1 (x1 ) − u1 (r1 ) − f u1 (y1 ) − u1 (r1 ) > f u1 (x1 ) − u1 (r1 ) − f u1 (y1 ) − u1 (r1 ) .

(5)

Note that DS is essentially equivalent to the strict concavity of f.13 Moreover, it can be phrased in the following way: the relative value of the ith dimension increases when its reference increases. Fig. 4 demonstrates how indifference curves Ir change as the reference point r changes under DS. Here, I(1, 1) (solid) and I(3, 1) (dashed) are indifference curves such that









x1 − 1 + x2 − 1 = 2.7 and x1 − 3 + x2 − 1 = 2, respectively. As the reference for the first dimension r1 = 1 increases to r1 = 3 (i.e., the reference point (1, 1) shifts to (3, 1)), the indifference curve I(1, 1) (solid) becomes steeper since the first dimension is more valuable now and shifts to I(3, 1) (dashed). On the other hand, as the reference for the second dimension r2 = 1 increases to r2 = 3 (i.e., the reference point (1, 1) shifts to (1, 3)), the indifference curve I(1, 1) becomes flatter since the second dimension is more valuable now and shifts to I(1, 3) (dotted).  Finally, note that the relative value of x1 over y1 in the presence of r1 decreases when r1 is added since f u1 (x1 ) −















u1 (r1 ) − f u1 (y1 ) − u1 (r1 ) > f u1 (x1 ) − u1 (r1 ) − f u1 (y1 ) − u1 (r1 ) . Therefore, we can rephrase DS in the following way: adding r1 hurts x1 relative to y1 in the presence of r1 ∈ (r1 , y1 ]. By setting r1 = y1 and r1 = z1 , we can obtain the following implication of DS: adding z1 hurts x1 relative to y1 . We call this implication Weak DS and it will explain the CE and the AE. Definition 4 (Weak DS). A MRDC C( f,u1 ,u2 ) satisfies Weak DS if, for all x1 , y1 , z1 ∈ X1 with x1 > y1 > z1 ,













f u1 ( x1 ) − u1 ( y1 ) + f u1 ( y1 ) − u1 ( z1 ) > f u1 ( x1 ) − u1 ( z1 ) .

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3.1. Compromise and attraction effects In this subsection, we first show the equivalence between the CE and the AE and Weak DS (Proposition 1). A further connection between the two effects and DS is provided in Propositions 3 and 4, where we axiomatically characterize DS. Second, we show that more concavity (of f) means that we are more likely to observe the CE and the AE (Proposition 2). Third, we discuss the effects of adding a fourth alternative and a symmetrically dominated third alternative, and their relation to the CE and the AE. Explicit modeling of reference points (especially using minimums) will be useful to have predictions which are either consistent with existing experimental results or can be tested easily in a lab. We now state the first implication of DS. Proposition 1. Suppose C = C( f,u1 ,u2 ) . Take alternatives x, y, z, and z = (z1 , z2 ) ∈ X such that x1 > y1 > z1 and x2 < z2 < y2 < z2 . Suppose C ({x, z} ) = x. Then the following statements are equivalent. i) (Compromise effect) C ({x, y} ) = x and C({x, y, z})  y. ii) (Attraction effect) C ({x, y} ) = x and C({x, y, z })  y. 12







By the additive nature of the model, DS can be defined only using f and u1 . Indeed, DS is equivalent to requiring f u2 (y2 ) − u2 (r2 ) − f u2 (x2 ) −











u2 (r2 ) > f u2 (y2 ) − u2 (r2 ) − f u2 (x2 ) − u2 (r2 ) for all y2 > x2 ≥ r2 > r2 . 13 DS may be an implication of a general law of human perception in psychology called the Weber-Fechner law. The law states that perceived intensity is proportional to the logarithm of the stimulus (or to more general strictly concave functions). In other words, the law states that people exhibit DS to stimuli in general. This connection between DS in economics and the Weber-Fechner law in psychology is suggested by Bruni and Sugden (2007) and also discussed in Bordalo et al. (2012, 2013).

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Fig. 5. Compromise effect, attraction effect, and diminishing sensitivity.

iii) (Weak diminishing sensitivity) C exhibits Weak DS at x1 , y1 , z1 in the following way:

















f u1 ( x1 ) − u1 ( y1 ) > f u2 ( y2 ) − u2 ( x2 ) ≥ f u1 ( x1 ) − u1 ( z1 ) − f u1 ( y1 ) − u1 ( z1 ) . The first two statements (i) and (ii) are the formal definitions of the CE and the AE, respectively.14 Note that in the first case, adding z to {x, y} makes x an extreme option and y a compromise option. In the second case, z = (z1 , z2 ) is dominated by y since y1 > z1 and y2 > z2 . Indeed, using minimums as references is the reason we can obtain this equivalence since in the MRDC z and z have the same effect on references.15 We omit the proof of Proposition 1 since it is not difficult to check. The intuitive argument behind Proposition 1 is given in Fig. 5. Here I(y1 , x2 ) (solid) is the indifference curve for the reference point (y1 , x2 ) of {x, y} and I(z1 , x2 ) (dashed) is the indifference curve for the reference point (z1 , x2 ) of {x, y, z} (also {x, y, z }). In both the CE and the AE, adding a third alternative changes the reference point (y1 , x2 ) to (z1 , x2 ). Then by DS, the second dimension becomes more valuable since the reference of the first dimension has decreased. Therefore, the indifference curve I(y1 , x2 ) (solid) becomes flatter and shifts to the left of I(z1 , x2 ) (dashed), which helps y to be chosen over x. Our explanations are consistent with common rationales for the two effects. A common rationale for the CE is that people avoid extreme options. Similarly, in our model, the agent also avoids extremes because of the strict concavity of f. In the AE, a decoy z makes y more attractive. However, it must be that y becomes more attractive because of the second dimension rather than the first dimension since y1 < x1 . Similarly, in our model, the second dimension becomes more valuable compared to the first dimension because of DS. As long as minimums are references, the equivalence between the two effects does not rely on the additive-separable and utility-difference structures. In particular, as in Proposition 1, under the representation (3), we can show that (i) C ({x, y} ) = x and C({x, y, z})  y if and only if (ii) C ({x, y} ) = x and C({x, y, z })  y. A connection between reference-dependent behaviors and the two effects is very intuitive. The main idea of reference dependence is that people evaluate objects in relative terms instead of in absolute terms. On the other hand, the AE and the CE provide well-known evidence that people do not evaluate things in absolute terms (or do not maximize utilities) as WARP suggests. In fact, Simonson and Tversky (1992) and Tversky and Simonson (1993) suggest a connection between the two effects and a psychological hypothesis called tradeoff contrast. The main idea behind tradeoff contrast is that perception and judgment of objects are affected by other objects and its surroundings,16 consistent with menu-dependent reference points. Moreover, evaluating objects with respect to the best and worst attributes is psychologically appealing (e.g., see Parducci, 1965). In this paper, we formally connect a reference dependence with respect to minimums, the two effects, and DS. Next, we will show that the CE and the AE are more likely to be observed when f is more concave.17 Proposition 2 (More concave, more choice reversals). Take any two C( f, u1 , u2 ) and C( f  , u1 , u2 ) and their induced preferences {t } and {t }, respectively. If f is more concave than f, then for all x, y, z, and z = (z1 , z2 ) ∈ X such that x1 > y1 > z1 and x2 < z2 < y2 < z2 , i) x ≺z y implies x ≺z y, and ii) x≺z y implies x ≺z y. This proposition shows that C( f  ,u1 ,u2 ) must exhibit the CE or the AE (x ≺z y and x  y or x ≺z y and x  y) whenever C( f,u1 ,u2 ) exhibits the CE or the AE (x ≺z y and x y or x≺z y and x y). Indeed, when f is linear, the MRDC reduces to the standard additive utility model which does not allow for choice reversals. Because of continuity, there is no essential difference between having C ({x, y} ) = x and C({x, y, z})  y and having C ({x, y} ) = {x, y} and C ({x, y, z} ) = y. Proposition 1requires the first coordinates of z and z to be the same. However, the equivalence between the CE and the AE is still true when the first coordinates of z and z are close enough. 16 For example, the same circle appears larger when surrounded by small circles. 17 We say a situation S1 is “more likely” to be observed than a situation S2 if S1 is observed whenever S2 is observed. We also say S1 is “strictly more likely” to be observed than S2 if S1 is “more likely” to be observed than S2 , and there is a case in which S1 is observed but S2 is not observed. 14

15

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Fig. 6. Two-decoy effect and two-compromise effect.

3.2. Two-compromise and two-decoy effects, and bounds on preference reversals We now discuss three novel predictions of the MRDC. When can one say that the MRDC is a good model of the CE and the AE? We argue that the MRDC needs to be consistent with (i) common explanations of the CE and the AE and (ii) implications of the two effects. In the previous subsection, we have argued that (i) is satisfied. We now will argue that (ii) is also satisfied by introducing three implications of the two effects.18 3.2.1. Implication of the AE – two-decoy effect The usual explanation of the AE says that z works as a decoy for y, so y is chosen over x in the presence of z . Then we can consider the following obvious implication of the AE: If we introduce a fourth alternative which is a decoy for x, then the decoy effect of z for y should be cancelled out by the decoy effect of the fourth alternative for x. We call it the twodecoy effect, which is confirmed by the experimental result of Teppan and Felfernig (2009). In fact, our model is consistent with the two-decoy effect, and the intuition is given on the left-hand side of Fig. 6. To illustrate, suppose that we add t to {x, y, z } where t is dominated by x (i.e., t < x), but not by y (i.e., y1 < t1 ). Note that when t is added, the reference point shifts down to (z1 , t2 ) since t2 < x2 . So the indifference curve I(z1 , x2 ) (dashed)

becomes steeper and shifts to I(z1 , t2 ) (dotted) as the reference point (z1 , x2 ) shifts to (z1 , t2 ).19 In other words, although z1 hurts x1 relative to y1 , the decoy effect should be cancelled out by t2 since t2 also hurts y2 relative to x2 . Therefore, it is more likely that x is chosen from {x, y, z , t}.20 More formally, Observation 1: Suppose C( f, u1 , u2 ) satisfies Weak DS. For all x, y, z , t ∈ X with x > t, y > z , not x > z , and not y > t, if C ({x, y, z } )  x, then C ({x, y, z , t} ) = x. Moreover, there exist x, y, z , t ∈ X with x > t, y > z , not x > z , and not y > t such that

C ({x, y} ) = x, C ({x, y, z } ) = y, and C ({x, y, z , t} ) = x. 3.2.2. Implication of the CE – two-compromise effect The usual explanation of the CE says that z makes x an extreme option, so people compromise by choosing y. Consider the following obvious implication of the CE: If we add a fourth alternative which is more extreme than x, then x is not an extreme option anymore, so people would not avoid x. We call it the two-compromise effect, which is confirmed by the experimental evidence of Manzini and Mariotti (2010). In fact, our model is consistent with the two-compromise effect, and the intuition is given on the right-hand side of Fig. 6. To illustrate, suppose we add t = (t1 , t2 ) to {x, y, z} where x1 < t1 and t2 < x2 . We will obtain an argument which is very similar to the two-decoy effect. When t is added, the reference point shifts down to (z1 , t2 ) since t2 < x2 . Notice that the three indifference curves on the right-hand side of Fig. 6 are identical to the three indifference curves on the left-hand side. Therefore, it is more likely that x is chosen from {x, y, z, t }. Formally, Observation 2: Suppose C( f, u1 , u2 ) satisfies Weak DS. For all x, y, z, and t = (t1 , t2 ) ∈ X with t1 > x1 > y1 > z1 and t2 < x2 < y2 < z2 , if C ({x, y, z} )  x, then C ({x, y, z, t } ) = x. Moreover, there exist x, y, z, t = (t1 , t2 ) ∈ X with t1 > x1 > y1 > z1 and t2 < x2 < y2 < z2 such that

C ({x, y} ) = x, C ({x, y, z} ) = y, and C ({x, y, z, t } ) = x. 18 One may notice that our explanations of implications of DS will not rely on the additive structure of the MRDC. Hence, all the implications of DS we will discuss are true even if we weaken the additive structure. 19 Note that I(z1 , t2 ) (dotted) is very similar to I(y1 , x2 ) (solid) compared to I(z1 , x2 ) (dashed). 20 Indeed, the decoy effect of t should be strong enough to cancel out the decoy effect of z ; i.e., t2 must be small enough.

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Fig. 7. (i) Symmetric dominance, (ii) Third alternatives that cause a choice reversal.

Notice that the two-decoy and two-compromise effects violate the following commonly used axiom in the literature, called Weak WARP: for all A and x, y ∈ A,

if x = C ({x, y} ), x = C (A ), then y = C (B ) for any B such that {x, y} ⊂ B ⊂ A. Therefore, the two-decoy and two-compromise effects are important implications of the MRDC that allow us to distinguish our model from many prominent models, including models of Koszegi and Rabin (20 06, 20 07), Manzini and Mariotti (2007, 2012a), Ehlers and Sprumont (2008), Lombardi (2009), and Cherepanov et al. (2013), that satisfy Weak WARP.21 3.2.3. Symmetric dominance The above arguments also suggest that if we add a symmetrically dominated alternative (z1 , t2 ) to the binary menu {x, y}, we are less likely to observe a choice reversal, compared to the CE and the AE (see Fig. 7). We call this implication the symmetric dominance effect, which is also consistent with the experimental results of Masatlioglu and Uler (2013). Formally. Observation 3: Suppose C( f, u1 , u2 ) satisfies Weak DS. For all x, y, z = (z1 , z2 ) ∈ X and t2 ∈ X2 with x1 > y1 > z1 and y2 >  z2 > x2 >t2 , if x ≺(z1 , t2 ) y, then x ≺(z , z ) y. Moreover, there exist x, y, z = (z1 , z2 ) ∈ X and t2 ∈ X2 with x1 > y1 > z1 and y2 > z2 > x2 >t2 such that

1

2

C ({x, y} ) = x, C ({x, y, z } ) = y, and C ({x, y, (z1 , t2 )} ) = x. Let us conclude the discussion of the symmetric dominance effect by arguing that Observation 3 can be strengthened in some cases. The idea is that when the attributes z1 and t2 of the third alternative are small enough, the utility differences u1 (x1 ) − u1 (z1 ) and u1 (y1 ) − u1 (z1 ) and u2 (y2 ) − u2 (t2 ) and u2 (x2 ) − u2 (t2 ) are large. Moreover, by DS, f is close to linear at large values. Therefore, similar to the fact that we cannot obtain a choice reversal when f is linear, we cannot obtain a choice reversal when z1 and t2 is small enough because f is close to linear. The above argument is illustrated in the following numerical example. √ This numerical example also demonstrates the predictive power of the model. Suppose u1 (t ) = u2 (t ) = t, f (t ) = t , x = (20, 11 ), and y = (9, 20 ). Note that x y. Then Fig. 7 (ii) illustrates all possible third alternatives that can cause a choice reversal for the pair x and y. Note that in order to cause a choice reversal, the first dimension of the third alternative should ∗ ∗ be smaller than z1∗ = 80 9 and the second dimension of the third alternative should be in the interval (t2 , z2 ) ≈ (5.61, 47 ). In particular, the top shaded area is the set of z that can cause the CE. The middle shaded area is the set of all asymmetrically dominated alternatives z (by y) that can cause the AE. Finally, the bottom shaded area is the set of all symmetrically dominated alternatives (z1 , t2 ) that can cause a choice reversal.22 In other words, any alternative in the non-shaded area (including all alternatives around the origin) cannot cause a choice reversal.23 21 Manzini and Mariotti (2007, 2012a), Ehlers and Sprumont (2008), Lombardi (2009), and Cherepanov et al. (2013) directly use Weak WARP to characterize their models. Moreover, Freeman (2017) shows that the personal preferred equilibrium of Koszegi and Rabin (20 06, 20 07), the most popular version of their models, satisfies Weak WARP. 22 , 11 )) that bounds the SD area from below is the set of symmetrically dominated alternatives that make x and y The curve (from (0, 5.61) to ( 80 9 indifferent. 23 One may notice that if the alternative (0, 0) is always available, then the MRDC reduces to the standard additive utility model. However, we can modify the model in the following way: the reference point is formed after eliminating alternatives that are dominated by all undominated alternatives. Under this modification, we still can obtain the AE, since z would not be eliminated because it is not dominated by x. We did not follow this approach for three reasons. First, as illustrated by Fig. 7, our model can predict the symmetric dominance effect. Second, in reality such obviously worse alternatives are not available in menus (e.g., for some sellers it is not their best interest to have or produce such low-quality, but expensive products). Third, the MRDC complements models with the default or status quo options.

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11

3.3. Characterizing diminishing sensitivity In Section 3.1, we have shown that DS is sufficient to rationalize the CE and the AE. In this subsection, we show that DS is in fact “necessary” for the AE by behaviorally characterizing DS. This result in addition to Propositions 1 and 4 establishes an equivalence between the two effect and DS. In order to prove the necessity direction, we will impose two novel axioms to characterize weak and strong versions of DS. Then we argue that these axioms are essentially equivalent to imposing bounds on the AE. Axiom 6 (Bound on the AE (BAE)). For all x, y ∈ X and z1 , t1 ∈ X1 such that x1 > y1 > z1 > t1 , if x ∼(z1 , x2 ) y, then x ≺(t1 , x2 ) y and x y. Roughly speaking, BAE requires that (z1 , x2 ) helps y over x, but (t1 , x2 ) helps more when t1 < z1 . Now we illustrate a connection between BAE and the AE without assuming our model. Suppose x ∼(z1 , x2 ) y and BAE is satisfied. By BAE, we have x ≺(t1 , x2 ) y and x y. By continuity and monotonicity, there is  > 0 such that (x1 +  , x2 ) (z1 , x2 ) y, (x1 +  , x2 ) ≺(t1 , x2 ) y, and (x1 +  , x2 ) y. Now we argue that BAE is summarzied by two properties. First, notice that we obtained the AE at (x1 +  , x2 ), y, and (t1 , x2 ). Second, (x1 +  , x2 ) (z1 , x2 ) y implies that (t1 , x2 ) is more likely to cause a choice reversal compared to (z1 , x2 ) since z1 > t1 . Therefore, BAE is equivalent to observing the AE in the neighborhood of a triple x, y, and (t1 , x2 ) while not in the neighborhood of a triple x, y, and (z1 , x2 ). In other words, BAE restricts the collection of triples at which the AE is observed. In our model, since DS is equivalent to BAE, DS is equivalent to observing the AE in the neighborhood of x, y, and (t1 , x2 ) but not in that of x, y, and (z1 , x2 ). We now define the second axiom. Axiom 7 (Weak Bound on the AE (WBAE)). For all x, y ∈ X and z1 ∈ X1 such that x1 > y1 > z1 , if x ∼(z1 , x2 ) y, then x y. Similar to the discussion of BAE, WBAE is equivalent to observing the AE in a neighborhood of any triple x, y, and (z1 , x2 ). Finally, we show that (W)BAE characterizes (Weak) DS in the MRDC. Proposition 3. Suppose C = C( f, u1 , u2 ) for some strictly increasing, continuous, and surjective functions f, u1 , and u2 . Then i) C satisfies WBAE if and only if C satisfies weak diminishing sensitivity. ii) C satisfies BAE if and only if C satisfies diminishing sensitivity. This equivalence between (W) BAE and (Weak) DS only relies on the additive-separable structure of the model. In other words, Proposition 3 is true when

C (A ) = arg max V1 (x1 , mA1 ) + V2 (x2 , mA2 ),

(7)

x∈A

where DS is defined as follows: Vi (xi , ri ) − Vi (yi , ri ) > Vi (xi , ri ) − Vi (yi , ri ) when xi > yi ≥ ri > ri .24 4. Why minimums as references? So far, we have shown that using minimums as references in addition to DS does not just help us to explain the CE and the AE, but it also provides many implications that are either consistent with existing experimental findings or can be tested easily in a lab. One might wonder if there are testable behaviors that can pin down minimums as references and DS. It turns out that, a generalization of the CE called the general compromise effect is not only (almost) equivalent to DS, but it also justifies the use of minimums as references. The general compromise effect states that if there is a choice reversal in favor of y after adding some alternatives to a menu, then y must be a compromise alternative in the new menu. Formally, Definition 5 (General compromise effect). A choice correspondence C satisfies the (weak) general compromise effect if, for all A, B ∈ A and x, y ∈ A ⊂ B (resp., A = {x, y}), if x ∈ C(A) and y ∈ C (A ), but y ∈ C(B), then y >> mB . Notice that the AE and the CE are consistent with the general compromise effect since in the both effects y >> m{x, y, z}  and y >> m{x, y, z } . We first establish a relation between the general compromise effect and DS in the MRDC. Proposition 4. Suppose that C = C( f, u1 , u2 ) . i) C satisfies the general compromise effect if and only if for all x1 , y1 , r1 , r1 ∈ X1 with x1 > y1 ≥ r1 > r1 ,

















f u1 (x1 ) − u1 (r1 ) − f u1 (y1 ) − u1 (r1 ) ≥ f u1 (x1 ) − u1 (r1 ) − f u1 (y1 ) − u1 (r1 ) .

(8)

ii) C satisfies the weak general compromise effect if and only if for all x1 , y1 , z1 ∈ X1 with x1 > y1 > z1 ,













f u1 ( x1 ) − u1 ( y1 ) + f u1 ( y1 ) − u1 ( z1 ) ≥ f u1 ( x1 ) − u1 ( z1 ) .

(9)

24 However, WBAE implies Weak DS under a more general sub-additive model: C (A ) = arg maxx∈A W (V1 (x1 , mA1 ), V2 (x2 , mA2 )) where W (t1 , t2 ) ≤ W (t1 , 0 ) + W (0, t2 ) for any t ∈ R2+ and Weak DS is defined as follows: W (V1 (x1 , y1 ), 0 ) + W (V1 (y1 , z1 ), 0 ) > W (V1 (x1 , z1 ), 0 ) when x1 > y1 > z1 . Note that in both this model and (7), we can assume that V1 (x1 , x1 ) = V2 (x2 , x2 ) = 0 without loss of generality.

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Notice that Eqs. (8) and (9) are very similar to Eqs. (5) and (6) (i.e., DS and Weak DS), respectively. However, piecewiselinear distortion functions are not strictly concave, but they can satisfy (8) and (9). Moreover, Proposition 4 has the following useful corollary. Corollary 1. Suppose C( f, u1 , u2 ) satisfies (Weak) DS. For all A, B ∈ A and x, y ∈ A ∩ B (resp., A = {x, y}), if x = C (A ), but y ∈ C(B) and y = C(B), then C (B \ {y} ) = C (B ) \ {y}. The property on C in Corollary 1 is indeed weaker than WARP. Since it does not depend on the two-attribute structure of X, it can be tested in more general environments. Second, we discuss implications of the general compromise effect in general reference-dependent models. Consider the following reference-dependent model with menu-dependent references: for any A ∈ A ,

 





C (A ) = arg max h u1 (x1 ) − u1 (r1 (A )) + h u2 (x2 ) − u2 (r2 (A )) x∈A



,

(10)

where h : R → R is strictly increasing and for each i ∈ {1, 2}, ri : A → R+ is a mapping such that ri ({x} ) = xi for each x ∈ X. Without assumptions on ri (A) the representation (10) does not have any testable implication (see Proposition 6). In fact, without loss of generality, C admits the representation (10) and ri ({x} ) = xi . However, under some reasonable assumptions on r1 and r2 , the general compromise effect is satisfied only when ri (A) is determined by minimums and maximums. Moreover, models in which reference points only depend on the maximums cannot explain the AE. Therefore, the general compromise effect and the AE justify our use of minimums.25 Consider the following condition on r1 and r2 to rule out cases in which minimums and maximums are used as references. Sensitivity: for all A ⊂ X with |A| ≤ 2 and z ∈ X \ A, min{zi , ri (A )} ≤ ri (A ∪ {z} ) ≤ max{zi , ri (A )}, with strict inequalities when zi = ri (A). Intuitively, Sensitivity requires that when a new alternative z is added to a menu A, the old reference point r(A ) should move towards z. It is satisfied as long as r (A ∪ {x} ) is a (strict) convex combination of r(A) and x. For example, the mean  x of attributes (i.e., ri (A ) = x|A∈A| i ) satisfies Sensitivity. However, when ri (A) is determined by only minimums and maximums and minx∈A xi < zi < maxx∈A xi , Sensitivity is violated since adding z does not change ri (A). It is important to note that Sensitivity only restricts ri on menus with three or fewer alternatives. Our next result shows that Sensitivity is not compatible with the weak general compromise effect and DS. Proposition 5 (Characterizing minimums as references). Suppose C is standard and admits the representation (10) with functions h, u1 , u2 , r1 , r2 . Suppose r1 , r2 satisfy Sensitivity and h satisfies (6). Then there exist x, y, z ∈ X such that x = C ({x, y} ), {x, y, z} y ∈ C ({x, y, z} ), and y1 = m1 . Moreover, similar to Proposition 5, when minimums and maximums are not used as references, we can find a violation of the property in Corollary 1. That is, when Sensitivity is satisfied, there exist x, y, z, t ∈ X such that C ({x, y} ) = x and C ({x, y, z, t} ) = {x, y}, but C ({x, z, t} ) = {x, z}. 5. Application: intertemporal choice with non-binding borrowing constraint Explicit modeling of reference points allows us to apply our model to many different contexts. In order to demonstrate the usefulness of the model, we apply our model to the standard intertemporal consumption choice. The main implication of the model is that if there is a borrowing constraint and we assume that the consumer always exhausts her lifetime income, then she prefers to consume away from the constraint even if the constraint is not binding.26 Let us consider a two-period intertemporal choice model. A consumer lives two periods and her utility function is

UB (c1 , c2 ) = (c1 − c1B )γ + β (c2 − c2B )γ , where β ∈ (0, 1) is a discount factor, ci is consumption in period i ∈ {1, 2}, and ciB is the possible minimum consumption level in period i for the given budget set B. Suppose γ ∈ (0, 1), so we will have DS.27 The consumer earns income yi in period i and she can borrow at most b¯ in the first period at the interest rate r. Since y the consumer’s preference is strictly increasing, we assume that she exhausts all her discounted total income M ≡ y1 + 1+2r 25 Nevertheless, we discuss models with reference points which are determined by not only by minimums, but also by maximums in Appendix E. Adding maximums to our model weakens behavioral predictions of the model, but some implications of DS stay the same. 26 In Appendix D, we also apply the model to risky choice. Our model can explain contradicting risk behavior (in the sense of the expected utility). More precisely, the model can allow for behaviors such that an agent needs to have both high and low degrees of risk aversion, i.e., the agent prefers riskier options in some cases and safer options in other cases. 27 We can modify our model to apply it to cases with more than two periods. In order to obtain the modification of our model, notice that any standard T t−1 u(ct ) also maximizes consumption problem can be written as a utility maximization with two periods. That is, a consumer who maximizes t=1 β V(M2 ) is the continuation value for the rest of the income M2 = (M − c1 )(1 + r ). Then the MRDC version of u(c1 ) + β V (M2 ) will u(c1 ) + β V (M2 ) where  be f (u(c1 ) − u(r1 ) + β f (V (M2 ) − V (r2 )). With this modification the effect of non-binding borrowing constraints in the first period persists even if there are many periods. See Appendix I for a detailed discussion.

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13

Fig. 8. The Effect of a Non-Binding Borrowing Constraint.

(see Footnote 23 for a related discussion). Then the budget set is

B = {(c1 , c2 ) ∈ R2+ | c1 +

c2 = M and c1 ≤ y1 + b¯ }. 1+r

We now calculate the optimal consumption profile for two cases, without and with a borrowing constraint, and compare them. 5.1. Without a borrowing constraint Suppose there is no borrowing constraint; that is, b¯ ≥ γ

y2 1+r .

Then the consumer’s maximization problem is:

γ

max UB (c1 , c2 ) = c1 + β c2 ,

(c1 ,c2 )∈B

where B = {(c1 , c2 ) ∈ R2+ | c1 +

c2 = M } and (0, 0 ) is the reference point. 1+r

By a direct calculation, the optimal consumption levels are

c1∗ =

M 1

γ

1 + β 1−γ (1 + r ) 1−γ

and c2∗ =

M·



β (1 + r )



1 1−γ

γ

1

1 + β 1−γ (1 + r ) 1−γ

.

When there is no borrowing constraint, similar to the standard lifetime consumption model, the optimal consumption levels are proportional to M. 5.2. With a borrowing constraint Now suppose there is a borrowing constraint; that is, 0 ≤ b¯ <







γ max UB (c1 , c2 ) = c1 + β c2 − y2 − (1 + r )b¯

(c1 ,c2 )∈B

where B = {(c1 , c2 ) ∈ R2+ | c1 +

y2 1+r .

γ,

c2 = M and c1 ≤ y1 + b¯ } and (0, y2 − (1 + r )b¯ ) is the reference point. 1+r

By a direct calculation, the new optimal consumption levels are

c1∗∗

=

y2 M − ( 1+ − b¯ ) r

1+β

1 1−γ

γ

(1 + r ) 1−γ

=

Then the consumer’s maximization problem is:

y1 + b¯ 1+β

1 1−γ

γ

(1 + r ) 1−γ

and

c2∗∗

=

M·



β (1 + r )



1+β

1 1−γ

1 1−γ

+ (y2 − b¯ (1 + r )) γ

(1 + r ) 1−γ

.

In the standard model, the optimal consumption profiles are the same when the constraint is not binding; i.e., y1 + b¯ > c1∗ . However, in our model, the two cases are different even if the constraint is not binding (see Fig. 8). The first implication of the model and DS is the effect of non-binding constraints on the optimal consumption profile. ∂ c∗∗ Observation 4: Borrowing constraints decrease first period consumption: if y1 + b¯ < M, then 1 > 0 and c∗ > c∗∗ , even ∂ b¯

1

1

if the borrowing constraint is not binding (c1∗ < y1 + b¯ ). The intuition behind Observation 4 is as follows. The borrowing constraint decreases the consumer’s maximum consumption level for today and increases minimum consumption level for tomorrow. Then by DS (γ < 1), the relative value Please cite this article as: G. Tserenjigmid, Choosing with the worst in mind: A reference-dependent model, Journal of Economic Behavior and Organization, https://doi.org/10.1016/j.jebo.2018.11.001

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Fig. 9. Average income and consumption by education (Attanasio and Weber, 2010).

of tomorrow’s consumption with respect to today’s consumption will increase. Therefore, the consumer will decrease her consumption today and increase consumption tomorrow. It is common to explain deviations from the standard model of consumer choice such as the excess sensitivity of consumption to current income and a hump-shaped consumption profile with liquidity or borrowing constraints (see Attanasio, 1999; Attanasio and Weber, 2010). However, there is no strong empirical evidence that constraints were actually binding at the time of the decision. For example, Jappelli (1990) directly asks consumers whether they applied for and were denied credit and only 12.5 percent of 1982 respondents answered that they were denied credit. Moreover, Deaton (1991) shows that liquidity constraints are rarely binding by simulation. Our model gives a justification for those explanations since liquidity constraints can have an effect on choices even if they are not binding.28 Observation 5: For a given M, the optimal consumption is increasing in the current income: ci∗∗ = α yi + βi (b¯ , M ) for some α > 0 and β i . In contrast to the permanent income hypothesis, it is well known that consumption depends on current income. For example, Fig. 9, borrowed from Attanasio and Weber (2010), reports life-cycle profiles for two education groups in the UK. The left-hand side of Fig. 9 shows income and consumption paths for the group with compulsory education; the dotted curve is disposable income and the solid curve is nondurable consumption. Note that the consumption path closely follows the income path. Moreover, Campbell and Mankiw (1991) show that in many different countries, a large number of consumers follow a “rule of thumb” and set their consumption proportional to their income. Observation 5 shows that our model is ∂ c∗∗

consistent with the above empirical regularities on consumption paths (note that ∂ yi = α is independent of i). i We can also make another interesting observation: when b¯ is small, c1∗∗ is more dependent on y1 .29 In fact, Zeldes (1989) observes that consumers with a low level of assets (low b¯ ) more tightly link their consumption to their income. Koszegi and Rabin (2009) and Pagel (2017) apply the expectation-based reference dependent model of Koszegi and Rabin (2006) to lifetime consumption choices. They explain empirical observations about consumption profiles, including the excess sensitivity of consumption to current income and a hump-shaped consumption profile. The main difference is that we rely on a non-binding borrowing constraint. However, since the expectation-based reference dependent model reduces to the standard model when there is no uncertainty, their models rely on uncertainty.30 6. Related literature The notion of reference dependence was first introduced in economics by Markowitz (1952) and was formalized by Kahneman and Tversky (1979) in the context of risky choice and by Tversky and Kahneman (1991) in the context of riskless choice. Tversky and Kahneman (1991) is further developed by Munro and Sugden (2003) to the n-dimensional commodity space, Sugden (2003) to subjective expected utility, and Koszegi and Rabin (20 06, 20 07) to expectation-based reference points. 28 A similar behavior (a non-binding constraint matters) is obtained by Hayashi (2008) in the context of choice under ambiguity (i.e., choice over Anscombe-Aumann acts). He also considers a menu-dependent choice, and in his model, an agent responds to non-binding constraints because of anticipated ex post regrets. It is not obvious how to relate our results with Hayashi (2008) since regret aversion is defined in the context of choice under uncertainty and naturally has a dynamic interpretation, while DS is defined in a deterministic environment. 29 is decreasing in σ (b¯ ). Imagine that b¯ and y1 are independent random variables. Then cor r (c1∗∗ , y1 ) = √ 2 σ (y1 ) 2 30

σ (y1 )+σ (b¯ )

Moreover, in some of Pagel (2017)’s results, she also needs to have a hyperbolic discounting agent.

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The main objective of this paper is to develop an axiomatic model of reference-dependent preferences with endogenous reference points. However, in most of the previous work, reference points are exogenously given. To the best of our knowledge, Ok et al. (2014) is the only paper that axiomatically develops a model with purely endogenous reference points. Ok et al. (2014) also weaken WARP and can accommodate the AE. The main difference is that we explicitly model reference points, while they identify reference points from observed choices in the spirit of classical revealed preference analysis. The explicit modeling gives us more flexibility to apply our model to different contexts and to use DS to provide bounds on choice reversals.31 Moreover, Ok et al. (2014) cannot capture the CE.32 The literature related to reference-dependent behavior and prospect theory is too large to be discussed here (see Barberis 2013). We will now narrow our focus and discuss related literature in three parts. First, we discuss two main approaches to model reference points in the literature. Second, we discuss papers that accommodate behavioral phenomena we study in this paper. Finally, we attempt to place our model in the broad choice theory literature. 1. Two main approaches to model reference points: In the first approach, an agent uses an exogenously given alternative as a reference point. For example, that alternative is purely exogenous (Rubinstein and Zhou 1999), the status quo, the default option (Masatlioglu and Ok 2005, Sagi 2006, and Apesteguia and Ballester 2009), or the initial endowment (Masatlioglu and Ok 2013). However, there are many choice situations where no alternative is exogenously given as the status quo or the initial endowment. For example, there is no status quo or initial endowment in the CE and the AE. Since in our model the reference point can be defined for any given menu, our approach complements their approach in situations where there is no default option or initial endowment. In the second approach, an agent has probabilistic beliefs over possible outcomes and she uses her expectations of the outcome as a reference point (Koszegi and Rabin 20 06; 20 07). This approach builds on Tversky and Kahneman (1991), as ours does. However, there are two crucial differences. First, in their approach the reference point is determined by probabilistic beliefs, whereas in our model the reference point is completely determined by the menu. Second, their model reduces to the standard rational model when there is no underlying uncertainty. On the other hand, our model can be applied to both risky and riskless environments (Section 5 and Appendix D). A general expectation-based model of Koszegi and Rabin (20 06, 20 07) does not provide any testable implications. Therefore, they introduce three solution concepts that restricts their model, the personal equilibrium (PE), the preferred personal equilibrium (PPE), and the choice-acclimating personal equilibrium (CPE), for modeling the endogenous determination of expectation-based reference points. Freeman (2017) axiomatically characterizes the PE, PPE, and CPE, and shows that the PE, PPE, CPE all satisfy an axiom called Expansion, which does not admit the CE and the AE. 2. Papers that discuss behavioral phenomena we study: The AE and the CE are usually used as motivations for menudependent or context-dependent preferences. Therefore, the two effects can be explained by other menu-dependent preferences or context-dependent preferences (e.g., Simonson and Tversky, 1992; Wernerfelt, 1995, Bordalo et al. 2013; Kamenica 2008).33 However, in this paper we show an “equivalence” between the two effects and DS and provide formal justifications for using minimums as references.34 In the salience theory of Bordalo et al. (2013), the averages of attributes affect evaluations of alternatives. The salience theory is consistent with the CE and the AE but it is not consistent with the two-decoy and symmetric dominance effects ˝ and cannot provide bounds on choice reversals. In the focusing model of Koszegi and Szeidl (2012), the utility differences between the maximum and minimum attributes affect evaluations of alternatives. However, the objective of this model is completely different from ours. That is, the focusing model generates the present bias and time inconsistencies in the context of intertemporal choice, but it is not consistent with the CE and the AE and it also does not generate the effect of non-binding borrowing constraints. It might be useful to compare our model with information-theoretic models of consumer choice (Kamenica, 2008; Wernerfelt, 1995) that are consistent with the two effects. For example, Kamenica (2008) studies a model in which there is a market with rational consumers who learn values of attributes of a good from a product line. Consumers obtain menudependent utility as a result of a market equilibrium that leads to choice reversals. However, there is evidence of the CE and the AE that information-theoretic models cannot fully explain. First, in information-theoretic models, it is important that consumers face the binary menu first and the tripleton menu second in order to have different information in two choice situations (because consumers cannot unlearn information after facing the tripleton menu). However, the two effects are observed even if a decision maker sees the tripleton menu first and the binary menu second (e.g., see Sivakumar and Cherian, 1995; Wiebach and Hildebrandt, 2012). Second, the two effects are robustly exhibited in non-market situations (e.g.,

31 A very recent paper Kibris et al. (2018) also proposes a reference-dependent model, in which the reference point for a given menu is the maximal element of the menu by a linear order. This model can explain the AE and the CE, but it is not consistent with the two-decoy and two-compromise effects. Moreover, they do not discuss DS. 32 In their model, an alternative that causes a choice reversal (e.g., z of {x, y, z} in the CE) should be unambiguously worse than the chosen alternative (y of {x, y, z}). See Proposition 2 of Ok et al. (2014). 33 Moreover, there are a number of recent studies on stochastic choice that are consistent with the two effects. For example, see Natenzon (2018), Echenique et al. (2018), and Fudenberg et al. (2015). 34 De Clippel and Eliaz (2012) rationalize the AE and the CE as the result of a single bargaining protocol (to be discussed in part 3). However, behavioral postulates leading to the two effects are different.

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choice over binary lotteries, Herne 1999, and choice over bundles of chewing gum and chocolate cookies, Herne 1998), and learning the values of attributes from a product line does not seem to be the main factor of choice reversals. 3. Choice theory literature: In this paper, we weaken WARP in order to obtain a model that is consistent with observed behavioral phenomena. This approach is in the same spirit as a body of work that seeks to characterize models of nonstandard choice in terms of direct axioms on choice behavior. In addition to models discussed in Section 3.2, this body of work includes, for example, choosing from a subset of a menu (Masatlioglu et al. 2012, Manzini et al. 2013), choosing two finalists (Eliaz et al. 2011), choosing an alternative that survives after a sequential elimination (Manzini and Mariotti 2012b), multiself (De Clippel and Eliaz 2012), and comparing all pairs by certain orders (Apesteguia and Ballester 2013).35 The main difference between our paper and the above papers is that, except for the fact that we study reference-dependent behavior, attributes of alternatives are known as in consumer theory, which allows us to obtain stronger predictions (e.g., bounds on choice reversals as in Section 3.2), while the above papers study alternatives with unknown attributes (as in abstract choice theory). Moreover, only Masatlioglu et al. (2012), De Clippel and Eliaz (2012), and Manzini et al. (2013) are consistent with both the AE and the two-decoy effect. The paper by De Clippel and Eliaz (2012) is especially relevant here; it axiomatically models choices as a solution to an intrapersonal bargaining problem among two selves of an individual. Each self is endowed with a preference relation, so having two preference relations for two selves is similar to having two-attribute alternatives. However, their model requires that C ({x, y} ) = {x, y} for all x1 > y1 and x2 < y2 . Therefore, technically, they allow for only “weak” forms of the AE and the CE: the original two alternatives (x and y) are indifferent and the third alternative breaks the tie (in favor of y).36 Appendix A. Proofs A1. Implications of Standardness and D-WARP We first prove the following useful implications of Standardness and D-WARP. Lemma 1. Suppose C satisfies Standardness. For all i, j ∈ {1, 2} with i = j and for all y, t ∈ X and xj ∈ Xj with y > t and xj < yj , there exist xi and xi such that (xi , xj ) ∼ y and (xi , x j ) ∼t y. The proof of Lemma 1 is essentially identical to the proof of Lemma 1 of Tserenjigmid (2015). We repeatedly use the following implication of D-WARP. Implication of D-WARP: for all A ∈ A and x ∈ A, if C (A ∪ {x} ) = x and x ≥ mA , then C (A ) = C (A ∪ {x} ) \ {x}. This holds because D(A ) = D(A ∪ {x} ) when x ≥ mA . Axiom 8 (Transitivity∗ ). For all x, y, z, t ∈ X with x, y, z > t, if xt y and yt z, then xt z. Lemma 2. If C satisfies D-WARP and Monotonicity, then it satisfies Transitivity∗ . Proof of Lemma 2. Take any x, y, z, t ∈ X with x, y, z > t and xt y and yt z. If C({x, y, z, t})  x, then since y > t and C({x, y, z, t}) = y, by D-WARP, we have C({x, z, t})  x. In other words, xt z. Now suppose C({x, y, z, t})⊆{y, z}. If C({x, y, z, t})  y, then since z > t and C({x, y, z, t}) = z, by D-WARP, we have C ({x, y, t} ) = y. In other words, we have y t x, a contradiction. Finally, suppose C ({x, y, z, t} ) = z. Then since x > t and C({x, y, z, t}) = x, by D-WARP we have C ({y, z, t} ) = z. In other words, we have z t y, a contradiction.  Lemma 3. Suppose C satisfies D-WARP and Monotonicity. Then for any A ∈ A , if x ∈ C (A ), then x mA y for any y ∈ A. Proof of Lemma 3. Take any A ∈ A , and x, y ∈ A with x ∈ C (A ). Consider a menu B = {x, y, mA }. Note that D(A ) = D(B ) = {t ∈ X : t ≤ mA }. Therefore, by D-WARP and Monotonicity, we have x ∈ C (B ); i.e., x mA y.  A2. Useful Lemmas for the proof of Theorem 1 Now we turn to the proof of Theorem 1. First, we focus on binary choices. Lemma 4. If C satisfies Standardness, Transitivity, and Cancellation (i), then there exist strictly increasing, continuous, and surjective functions u1 : X1 → R+ and u2 : X2 → R+ such that for all x, y ∈ X,

x  y if and only if u1 (x1 ) + u2 (x2 ) ≥ u1 (y1 ) + u2 (y2 ).

(11)

35 Some researchers study violations of WARP in the context of preferences over menus. For example, Barbos (2010, 2013) consider (reference-dependent) preferences over menus of lotteries. They allow for the AE, but not the CE. 36 More precisely, they only allow for C ({x, y} ) = {x, y} and C ({x, y, z} ) = {y}. In other words, they do not allow for the CE and the AE such that C ({x, y} ) = {x} and C ({x, y, z} ) = {y}.

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Since Lemma 4 is a relatively well-known result, we omit the proof. For example, see Krantz et al. (1971). See also the online appendix for a more direct proof using the results of Suppes and Winet (1955) and Tserenjigmid (2015). Now we assume that we have strictly increasing, continuous, and surjective functions u1 and u2 that satisfy (11). Lemma 5. If C satisfies Standardness, Transitivity, Cancellation (i), D-WARP, and Reference Separability, then there exists a strictly increasing and continuous function W : R2+ → R+ such that W (t, 0 ) = W (0, t ) = t for any t ∈ R+ and for any A ∈ A , (4) is satisfied. Proof of Lemma 5. First, take a menu B = {x, y, z} with x1 > y1 > z1 and x2 < y2 < z2 . Let WB (x ) ≡ u1 (x1 ) − u1 (z1 ) and WB (z ) ≡ u2 (z2 ) − u2 (x2 ). Moreover, by Lemma 1, there exists x1 such that (x1 , x2 ) ∼(z1 , x2 ) y. Then let WB (y ) ≡ u1 (x1 ) − u1 ( z1 ).  Fact 1: C (B ) = arg maxx ∈B {WB (x )}. To prove Fact 1, we consider two cases. Case 1: When C(B)  x, we shall prove that

WB (x ) = u1 (x1 ) − u1 (z1 ) ≥ max{WB (z ) = u2 (z2 ) − u2 (x2 ), WB (y ) = u1 (x1 ) − u1 (z1 )}. By D-WARP, C (B )  x implies xz; equivalently, u1 (x1 ) − u1 (z1 ) ≥ u2 (z2 ) − u2 (x2 ) by (11). Moreover, by Lemma 3, C(B)  x implies x (z1 , x2 ) y. Since WB (y ) = u1 (x1 ) − u1 (z1 ) where (x1 , x2 ) ∼(z1 , x2 ) y, by Transitivity∗ , we have (x1 , x2 ) (z1 , x2 ) (x1 , x2 ); equivalently, x1 ≥ x1 . Therefore, u1 (x1 ) − u1 (z1 ) ≥ u1 (x ) − u1 (z1 ). Case 2: When C(B)  y, we shall prove that

WB (y ) = u1 (x1 ) − u1 (z1 ) ≥ max{WB (z ) = u2 (z2 ) − u2 (x2 ), WB (x ) = u1 (x1 ) − u1 (z1 )}. By Lemma 3, C(B)  y implies y (z1 , x2 ) x. Since WB (y ) = u1 (x1 ) − u1 (z1 ) where (x1 , x2 ) ∼(z1 , x2 ) y, by Transitivity∗ , we have (x1 , x2 ) ∼(z1 , x2 ) x; equivalently, x1 ≥ x1 . Therefore, u1 (x1 ) − u1 (z1 ) ≥ u1 (x1 ) − u1 (z1 ). Similarly, by Lemma 3, C(B)  y implies y (z1 , x2 ) z. Since WB (y ) = u1 (x1 ) − u1 (z1 ) where (x1 , x2 ) ∼(z1 , x2 ) y, by Transitivity∗ ,  we have (x1 , x2 ) (z1 , x2 ) z. Since (z1 , x2 ) ≥ m{(x1 ,x2 ),z} , by D-WARP, (x1 , x2 ) (z1 , x2 ) z implies (x1 , x2 )  z. Therefore, u1 (x1 ) − u1 (z1 ) ≥ u2 (z2 ) − u2 (x2 ) by (11). Now we construct W : R2+ → R+ in the following way: for any menu B = {x, y, z} with x1 > y1 > z1 and x2< y2 < z2 , let W (u1 (x1 ) − u1 (z1 ), 0 ) ≡ WB (x ) = u1 (x1 ) − u1 (z1 ), W (0, u2 (z2 ) − u2 (x2 )) ≡ WB (z ) = u2 (z2 ) − u2 (x2 ), and W u1 (y1 ) −    u1 (z1 ), u2 (y2 ) − u2 (x2 ) ≡ WB (y ). Finally, note that W u1 (y1 ) − u1 (z1 ), u2 (y2 ) − u2 (x2 ) is a function of y1 , z1 , x2 , y2 because WB (y) does not depend on x1 and z2 . Now we shall prove that W is well-defined.  Fact 2: W is well-defined; that is, for all yi , zi , x j , y j , xj , yj , if u j (y j ) − u j (x j ) = u j (yj ) − u j (xj ), then W ui (yi ) −    ui (zi ), u j (y j ) − u j (x j ) = W ui (yi ) − ui (zi ), u j (yj ) − u j (xj ) . Take for all yi , zi , x j , y j , xj , yj such that u j (y j ) − u j (x j ) = u j (yj ) − u j (xj ). Since ui is surjective, there is xi such that ui (xi ) − ui (0 ) = u j (y j ) − u j (x j ); i.e., (xi , xj ) ∼ (0, yj ) by (11). Since u j (y j ) − u j (x j ) = u j (yj ) − u j (xj ), we also have (xi , xj ) ∼ (0, yj ) by (11). By Lemma 1, there exists xi such that (xi , x j ) ∼(zi ,x j ) (yi , y j ). Therefore, by Reference Separability we have (xi , x j ) ∼(zi ,x j )

(yi , y j ) iff (xi , xj ) ∼(zi ,x j ) (yi , yj ).   By the definition of W, (xi , x j ) ∼(zi ,x j ) (yi , y j ) is equivalent to W ui (yi ) − ui (zi ), u j (y j ) − u j (x j ) = ui (xi ) − ui (zi ),    and (xi , xj ) ∼(zi , x j ) (yi , yj ) is equivalent to W ui (yi ) − ui (zi ), u j (yj ) − u j (xj ) = ui (xi ) − ui (zi ). Therefore, W ui (yi ) −    ui (zi ), u j (y j ) − u j (x j ) = W ui (yi ) − ui (zi ), u j (yj ) − u j (xj ) . Finally, we will prove that we have obtained desired W. Fact 3: For any menu A ∈ A , (4) is satisfied. By (11), it is obvious when |A| = 2. Nowconsider the cases where |A| ≥ 3. Take  x, y ∈ A with x ∈ C (A ). We shall prove that W u1 (x1 ) − u1 (mA1 ), u2 (x2 ) − u2 (mA2 ) ≥ W u1 (y1 ) − u1 (mA1 ), u2 (y2 ) − u2 (mA2 ) .

By Lemma 3, x ∈ C (A ) implies that x mA y. By the definition of W, we have W (u1 (x1 ) − u1 (mA1 ), u2 (x2 ) − u2 (mA2 )) = u1 (x1 ) − u1 (mA1 ) where (x1 , mA2 ) ∼mA x and W (u1 (y1 ) − u1 (mA1 ), u2 (y2 ) − u2 (mA2 )) = u1 (x1 ) − u1 (mA1 ) where (x1 , mA2 ) ∼mA y. Then by Transitivity∗ , we have (x1 , mA2 ) ∼mA x mA y ∼mA (x1 , mA2 ). Therefore, we have x1 ≥ x1 . In other words, W (u1 (x1 ) − u1 (mA1 ), u2 (x2 ) − u2 (mA2 )) = u1 (x1 ) − u1 (mA1 ) ≥ W (u1 (y1 ) − u1 (mA1 ), u2 (y2 ) − u2 (mA2 )) = u1 (x1 ) − u1 (t1 ). A3. Proof of Theorem 1 We focus on the sufficiency part of Theorem 1. See the online appendix for the necessity part of Theorem 1. By Lemma 5, there exist strictly increasing, continuous, surjective functions u1 : X1 → R+ , u2 : X2 → R+ , and a strictly increasing, continuous function W : R2+ → R+ such that W (t, 0 ) = W (0, t ) = t for each t ∈ R+ and for any A ∈ A , (4) is satisfied. Fact Cancellation (ii), then there exists a strictly increasing, continuous function f such that W (t1 , t2 ) =  4: If C satisfies  f −1 f (t1 ) + f (t2 ) . Let us consider a binary relation 0 . By (4), x0 y if and only if W(u1 (x1 ), u2 (x2 )) ≥ W(u1 (y1 ), u2 (y2 )). Similar to Lemma 4, by Cancellation (ii) and Transitivity∗ , there exist strictly increasing, continuous, and surjective functions f 1 : X1 → R+ and f2 : X2 → R+ such that x0 y if and only if f1 (x1 ) + f2 (x2 ) ≥ f1 (y1 ) + f2 (y2 ). See Krantz et al. (1971) and the online appendix. Please cite this article as: G. Tserenjigmid, Choosing with the worst in mind: A reference-dependent model, Journal of Economic Behavior and Organization, https://doi.org/10.1016/j.jebo.2018.11.001

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Take any x, y ∈ X such that x ∼ 0 y. Equivalently, f1 (x1) + f2 (x2 ) = f1 (y1 ) + f2 (y2 ). Given that f1 is continuous and strictly increasing, we obtain x1 = f1−1 f1 (y1 ) + f2 (y2 ) − f2 (x2 ) . Therefore, W (u1 (x1 ), u2 (x2 )) = W (u1 (y1 ), u2 (y2 )) implies that

 











W u1 f1−1 f1 (y1 ) + f2 (y2 ) − f2 (x2 ) , u2 (x2 ) = W u1 (y1 ), u2 (y2 ) . Since the right-hand side does not depend on x2 , the left-hand side also should not depend on x2 . Therefore, there exists a continuous and strictly increasing function F such that

 















W u1 f1−1 f1 (y1 ) + f2 (y2 ) − f2 (x2 ) , u2 (x2 ) = F f1 (y1 ) + f2 (y2 ) = W u1 (y1 ), u2 (y2 ) .









Since W (t, 0 ) = W (0, t ) = t, if we set yi = 0, then the above equality implies that F f j (y j ) = W u j (y j ), 0 = u j (y j ). There-







fore, f j = F −1 ◦ u j . Therefore, W (t1 , t2 ) = F f1 (u−1 (t1 )) + f2 (u−2 (t2 )) = F (F −1 (t1 ) + F −1 (t2 ) . Let f = F −1 , then we have W (t1 , t2 ) = f −1 ( f (t1 ) + f (t2 )). Now it is easy to see that for any A ∈ A , (2) is satisfied. Uniqueness: Take any two vectors of continuous functions (f, u1 , u2 ) and ( f  , u1 , u2 ) such that C = C( f,u1 ,u2 ) = C( f  ,u ,u ) , 1

2

u1 (1 ) = u1 (1 ), and f (1 ) = f  (1 ). We prove the uniqueness in two steps. Step 1. u1 = u1 and u2 = u2 . Take any y ∈ X. By Lemma 1, there exists x1 ∈ X1 such that (x1 , 0) ∼ y. Since all the functions above are strictly increasing and continuous, we have

u1 (x1 ) = u1 (y1 ) + u2 (y2 ) ⇔ u1 (x1 ) = u1 (y1 ) + u2 (y2 ); equivalently,









x1 = u−1 u1 (y1 ) + u2 (y2 ) = u1−1 u1 (y1 ) + u2 (y2 ) . 1 First, let y1 = 0. Then we have u−1 ◦ u2 = u1−1 ◦ u2 ; equivalently, u1 ◦ u−1 = u2 ◦ u−1 . 1 1 2  −1   Second, let t1 ≡ u1 (y1 ), t2 ≡ u2 (y2 ), and h(t ) ≡ u1 u1 (t ) . Since u1 and u1 are strictly increasing, continuous, and surjective, h : R+ → R+ is also strictly increasing, continuous, and surjective. Then we have













 −1 h(t1 + t2 ) = u1 u−1 1 (t1 + t2 ) = u1 u1 (u1 (y1 ) + u2 (y2 )) , (by the definitions of h, t1 , t2 )

= u1 u1−1 (u1 (y1 ) + u2 (y2 )) = u1 (y1 ) + u2 (y2 )

















 −1 = u1 u−1 1 (t1 ) + u2 u2 (t2 ) , (by the definitions of t1 , t2 )  −1 = u1 u−1 1 (t1 ) + u1 u1 (t2 ) = h (t1 ) + h (t2 ).

In other words, we have obtained a typical Cauchy functional equation (see Kuczma 2008) for h. Therefore, there exists α > 0 such that h(t ) = α t; that is, u1 = α u1 and u2 = α u2 . Since u1 (1 ) = u1 (1 ), we have u1 = u1 and u2 = u2 . Step 2. f  = f . Take any y ∈ X. By Lemma 1, there exists x1 ∈ X1 such that (x1 , 0 ) ∼0 y. Then we have

f (u1 (x1 )) = f (u1 (y1 )) + f (u2 (y2 )) ⇔ f  (u1 (x1 )) = f  (u1 (y1 )) + f  (u2 (y2 )). Since u1 , u2 , f, f are strictly increasing and continuous, similar to Step 1, we will obtain a Cauchy functional equation for f ◦ f −1 . Therefore, there exists β > 0 such that f = β · f  . Since f  (1 ) = f (1 ), we have f  = f . A4. Proof of Proposition 2  Suppose f is more concave than f. Then there is a strictly increasing concave function h such   that f = h ◦ f . Since f (0 ) = f  (0 ), h(0 ) = 0. By C( f, u1 , u2 ) , x ≺(z ,z ) y (x≺z y) is equivalent to f u1 (x1 ) − u1 (z1 ) − f u1 (y1 ) − u1 (z1 ) <





1

2

f u2 (y2 ) − u2 (x2 ) . Since h is strictly increasing, we have

 





h f u1 ( x1 ) − u1 ( z1 ) − f u1 ( y1 ) − u1 ( z1 )



 

< h f u2 ( y2 ) − u2 ( x2 )







= f  u2 ( y2 ) − u2 ( x2 ) .

Then by C( f  , u1 , u2 ) , in order to obtain x ≺(z , z ) y (x ≺z y), it is enough to prove that 1

 

h f u1 ( x1 ) − u1 ( z1 )



 

2

− h f u1 ( y1 ) − u1 ( z1 )



 







≤ h f u1 ( x1 ) − u1 ( z1 ) − f u1 ( y1 ) − u1 ( z1 ) .

The above inequality is true since h(a ) = h(a ) − h(0 ) ≥ h(a + b) − h(b) for all a, b > 0 when h is concave. A5. Proof of Proposition 3 First we prove (i). Take any x1 , y1 , z1 ∈ X1 such that x1 > y1 > z1 . Since and  f,u1 , u2 are continuous,   strictly increasing,  surjective functions, there exist x2 , y2 ∈ X2 such that f u1 (x1 ) − u1 (z1 ) = f u1 (y1 ) − u1 (z1 ) + f u2 (y2 ) − u2 (x2 ) ; that is, x ∼(z1 , x2 ) y. By WBAE, we have x y; that is, u1 (x1 ) − u1 (y1 ) > u2 (y2 ) − u2 (x2 ). Therefore, we have





















f u1 ( x1 ) − u1 ( z1 ) = f u1 ( y1 ) − u1 ( z1 ) + f u2 ( y2 ) − u2 ( x2 ) < f u1 ( y1 ) − u1 ( z1 ) + f u1 ( x1 ) − u1 ( y1 ) . Please cite this article as: G. Tserenjigmid, Choosing with the worst in mind: A reference-dependent model, Journal of Economic Behavior and Organization, https://doi.org/10.1016/j.jebo.2018.11.001

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Second we prove (ii). Take any x1 , y1 , z1 , t1 ∈ X1 with x1 > y1 ≥ z1 > t1 . Since f, u1 , u2 are continuous, strictly increasing, and surjective functions, there exist x2 , y2 ∈ X2 such that f (u1 (x1 ) − u1 (z1 )) − f (u1 (y1 ) − u1 (z1 )) = f (u2 (y2 ) − u2 (x2 )); equivalently, x ∼(z1 , x2 ) y. By BAE, we have x ≺(t1 , x2 ) y; equivalently,









f u1 (x1 ) − u1 (t1 ) − f u1 (y1 ) − u1 (t1 ) < f (u2 (y2 ) − u2 (x2 )) = f (u1 (x1 ) − u1 (z1 )) − f (u1 (y1 ) − u1 (z1 )). A6. Proof of Proposition 4 Eq. (8) implies the general compromise effect. Take A, B ∈ A and x, y ∈ A ⊂ B such that x ∈ C(A), y ∈ C (A ), and y ∈ C(B). Without loss of generality, suppose x1 > y1 and x2 < y2 . Then we shall will prove that y1 > mB1 . By way of contradiction, suppose y1 = mB1 . Note that C(A)  x and C (A )  y imply that f (u1 (x1 ) − u1 (mA1 )) − f (u1 (y1 ) − u1 (mA1 )) > f (u2 (y2 ) − u2 (mA2 )) − f (u2 (x2 ) − u2 (mA2 )). Moreover, C(B)  y and y1 = mB1 imply that f (u2 (y2 ) − u2 (mB2 )) − f (u2 (x2 ) − u2 (mB2 )) ≥ f (u1 (x1 ) − u1 (y1 )). Since A ⊂ B, we must have mA ≥ mB . Combining the above two inequalities, by Eq. (8), we have

f (u1 (x1 ) − u2 (mA1 )) − f (u1 (y1 ) − u1 (mA1 )) > f (u2 (y2 ) − u2 (mA2 )) − f (u2 (x2 ) − u2 (mA2 )) ≥ f (u2 (y2 ) − u2 (mB2 )) − f (u2 (x2 ) − u2 (mB2 )) ≥ f (u1 (x1 ) − u1 (y1 )). In other words, we have f (u1 (x1 ) − u2 (mA1 )) − f (u1 (y1 ) − u1 (mA1 )) > f (u1 (x1 ) − u1 (y1 )), a violation of Eq. (8). Eq. (9) implies the weak general compromise effect. Take B ∈ A and x, y ∈ B such that x = C ({x, y} ) and y ∈ C(B). Without loss of generality, suppose x1 > y1 and x2 < y2 . Then we shall will prove that y1 > mB1 . By way of contradiction, suppose y1 = mB1 . Note that C (A ) = x implies that f (u1 (x1 ) − u1 (y1 )) > f (u2 (y2 ) − u2 (x2 )). Moreover, C(B)  y and y1 = mB1 imply that f (u2 (y2 ) − u2 (mB2 )) − f (u2 (x2 ) − u2 (mB2 )) ≥ f (u1 (x1 ) − u1 (y1 )). Combining the above two inequalities, we obtain f (u2 (y2 ) − u2 (mB2 )) − f (u2 (x2 ) − u2 (mB2 )) > f (u2 (y2 ) − u2 (x2 )), which is a violation of Eq. (9) since {x, y} ⊂ B implies x2 ≥ mB2 . The general compromise effect implies Eq. (8). By way of contradiction, suppose that there are x2 , y2 , r2 , r2 ∈ X2 with y2 > x2 ≥ r2 > r2 such that f (u2 (y2 ) − u2 (r2 )) − f (u2 (x2 ) − u2 (r2 )) < f (u2 (y2 ) − u2 (r2 )) − f (u2 (x2 ) − u1 (r2 )). There exist x1 , y1 ∈ X1 such that f (u1 (x1 ) − u1 (y1 )) = f (u2 (y2 ) − u2 (r2 )) − f (u2 (x2 ) − u1 (r2 )). Since y > (y1 , r2 ), (y1 , r2 ), the previous equality implies that y ∈ C ({x, y, (y1 , r2 ), (y1 , r2 )} ). Moreover, f (u2 (y2 ) − u2 (r2 )) − f (u2 (x2 ) − u2 (r2 )) < f (u1 (x1 ) − u1 (y1 )) implies that x = C ({x, y, (y1 , r2 )} ). Therefore, we have obtained x = C ({x, y, (y1 , r2 )} ), y ∈ C ({x, y, (y1 , r2 ), (y1 , r2 )} ), and {x,y,(y ,r ),(y ,r )}

1 2 1 2 y1 = m1 , a violation of the general compromise effect. The weak general compromise effect implies Eq. (9). By way of contradiction, suppose that there are x2 , y2 , z2 ∈ X2 with y2 > x2 > z2 such that f (u2 (y2 ) − u2 (x2 )) < f (u2 (y2 ) − u2 (z2 )) − f (u2 (x2 ) − u2 (z2 )). There exist x1 , y1 ∈ X1 such that f (u1 (x1 ) − u1 (y1 )) = f (u2 (y2 ) − u2 (z2 )) − f (u2 (x2 ) − u2 (z2 )). Since y > (y1 , z2 ), the previous equality implies that y ∈ C ({x, y, (y1 , z2 )} ). Moreover, f (u2 (y2 ) − u2 (x2 )) < f (u1 (x1 ) − u1 (y1 )) implies that x = C ({x, y} ). Therefore, we have obtained {x,y,(y1 ,z2 )} y ∈ C ({x, y, (y1 , r2 )} ), y ∈ C ({x, y, (y1 , z2 )} ), and y1 = m1 , a violation of the weak general compromise effect.

A7. Proof of Proposition 5 Since C is standard, we can find x, y ∈ X such that {x, y} = C ({x, y} ) and x1 > y1 and x2 < y2 . Let r ≡ r({x, y} ). By Sensitivity, r1 > y1 and r2 < y2 . Let r1 ≡ r({x, y, (y1 , r2 )} ). Then by Sensitivity, y1 < r11 < r1 and r21 = r2 . Therefore by (6) and monotonicity, we have y = C ({x, y, (y1 , r2 )} ). Since C is continuous, there exists  > 0 such that (y1 , y2 −  ) = C ({x, (y1 , y2 −  ), (y1 , r2 )} ). Moreover, {x, y} = C ({x, y} ) implies x = C ({x, (y1 , y2 −  )} ). In other words, we have obtained {x, (y ,y − ), (y1 ,r2 )} x = C ({x, (y1 , y2 −  )} ), (y1 , y2 −  ) = C ({x, (y1 , y2 −  ), (y1 , r2 )} ), and y1 = m1 1 2 , which is a violation of the weak general compromise effect. Appendix B. Alternatives with n-attributes In this section, we briefly discuss how to generalize our characterization result to the n-dimension case. Let X ≡ Rn+ be the set of all alternatives and A ⊂ 2X \ {∅} be a collection of compact subsets of X. Let us consider the following model: for any A ∈ A ,

C (A ) = arg max x∈A

n





f ui (xi ) − ui (mAi ) ,

(12)

i=1

where mA = (minx∈A xi )ni=1 . Similar to Theorem 1, the representation (12) can be characterized by Standardness and D-WARP in addition to the following modifications of Transitivity, Cancellation, and Reference Separability. Axiom 9 (Cancellation∗ ). For any x, y, z ∈ X and i ∈ {1, . . . , n}, i) if (xi , x−i ) ∼ (yi , z−i ) and (yi , y−i ) ∼ (zi , x−i ), then (xi , y−i ) ∼ (zi , z−i ); ii) if (xi , x−i ) ∼0 (yi , z−i ) and (yi , y−i ) ∼0 (zi , x−i ), then (xi , y−i ) ∼0 (zi , z−i ). Please cite this article as: G. Tserenjigmid, Choosing with the worst in mind: A reference-dependent model, Journal of Economic Behavior and Organization, https://doi.org/10.1016/j.jebo.2018.11.001

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Axiom 10 (Transitivity∗∗ ). For any i, j ∈ {1, . . . , n} and x, y, z ∈ X, if

(xi , x j , 0−i, − j )  (yi , y j , 0−i, − j ) and (yi , y j , 0−i, − j )  (zi , z j , 0−i, − j ), then (xi , x j , 0−i, − j )  (zi , z j , 0−i, − j ). Axiom 11 (Reference Separability∗ ). For any x, y, x , y , z ∈ X and i ∈ {1, . . . , n} such that xi , xi < zi and z−i < y−i < x−i , if

(xi , x−i ) ∼ (yi , y−i ), (xi , x−i ) ∼ (yi , y−i ), and   (xi , x−i ) ∼z (yi , y−i ), then (xi , x−i ) ∼z (yi , y−i ). Now we state our characterization theorem for the representation (12). Theorem 2. A choice correspondence C satisfies Standardness, Transitivity∗∗ , D-WARP, Cancellation∗ , and Reference Separability∗ if and only if there exist strictly increasing, continuous, and surjective functions f, u1 , . . . , un : R+ → R+ such that for any menu A∈A,

C (A ) = arg max

n

x∈A





f ui (xi ) − ui (mAi ) .

i=1

Moreover, for any two vectors of strictly increasing, continuous, and surjective functions ( f, {ui }ni=1 ) and ( f  , {ui }ni=1 ) such that f (1 ) = f  (1 ) and u1 (1 ) = u1 (1 ), if C = C( f,{ui }n ) = C( f  ,{u }n ) , then ( f, {ui }ni=1 ) = ( f  , {ui }ni=1 ). i=1

i i=1

Let us briefly discuss the proof of Theorem 2. First, identical to Lemma 3, under Standardness and D-WARP, for any x, y ∈ A, x ∈ C(A) implies x mA y. Therefore, we can focus on {t }t∈Rn . Second, under Cancellation∗ and Transitivity∗∗ (techni+

cally in addition to Reference Separability∗ ), we can find functions w1t , . . . , wnt such that for any x, y with x, y > t,

x t y iff

n

wit (xi ) ≥

i=1

n

wit (yi ).

i=1





Without loss of generality, for any wit , we can write that wit (xi ) = fti ui (xi ) − ui (ti ) where ui is a utility function that is   n i   n i consistent with . Therefore, we have x t y iff i=1 f t ui (xi ) − ui (ti ) ≥ i=1 f t ui (yi ) − ui (ti ) . Third, under Reference Separability∗ , we can prove that fti is independent of t. Therefore, we have x t   n i   n i ∗∗ i y iff i=1 f ui (xi ) − ui (ti ) ≥ i=1 f ui (yi ) − ui (ti ) . Finally, under Transitivity , we can prove that f = f for some f. Therefore, we have

x t y iff

n





f ui (xi ) − ui (ti ) ≥

i=1



n



f ui (yi ) − ui (ti ) .

i=1

Appendix C. General menu-dependent references As we mentioned in the introduction, (1) is too general to have testable implications on observed choice behavior since reference points are exogenously given. The objective of this paper is to endogenize reference points. Therefore, we have focused on menu-dependent reference points. In this section, we show that, in fact, we do not lose any generality by focusing on menu-dependent reference points (e.g., see (10)). In other words, we will show that any observed choice data can be rationalized by a model with menu-dependent reference points. Therefore, we should have more specific models for reference points in order to obtain testable predictions. For the sake of simplicity, suppose we observe choices from a finite number of finite menus A1 , A2 , . . . , AN . We also assume that C is a choice function; i.e., C(An ) ∈ An for each n ∈ {1, . . . , N}.37 The next result shows that any such choices can be rationalized by a reference-dependent model with menu-dependent reference points as long as monotonicity is satisfied. Proposition 6. If C satisfies monotonicity, then there exist λ > 1 and reference points r1 , r2 , . . . , rN such that for each n, rn ∈ An and

C (An ) = arg max f (x1 − x∈An

r1n

) + f ( x2 −

r2n

) where f (t ) =

t

λt

if t ≥ 0. if t < 0.

Notice that we can rationalize observed choices by a reference-dependent model with linear utility functions and piecewise-linear distortion functions, applied in Kahneman and Tversky (1979) and Tversky and Kahneman (1991). Therefore, Proposition 6 suggests that specific functional forms may not give us more predictive power without the further specification of menu-dependence. Proof of Proposition 6. Let A ≡ ∪N n=1 An and

  x − y1 y2 − x2  λ¯ ≡ max max{ 1 ; } x, y ∈ A with x1 > y1 and x2 < y2 . y2 − x2 x1 − y1

37

This is observationally without loss of generality in most consumer choice settings.

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Moreover, let rn ≡ C(An ) and

f (t ) ≡

t

(λ¯ + 1 ) t

if t ≥ 0 if t < 0.

Then we shall prove that rn = C (An ) = arg maxx∈An f (x1 − r1n ) + f (x2 − r2n ) for each n. In other words, we shall prove that for any n and x ∈ An \ {rn }, f (r1n − r1n ) + f (r2n − r2n ) = 0 > f (x1 − r1n ) + f (x2 − r2n ). It is obvious when rn > x since 0 ≥ f (x1 − r1n ) and 0 ≥ f (x2 − r2n ), with at least one strict inequality. Moreover, by monotonicity, we cannot have rn ≤ x. ¯ + 1 )(r n − x2 ) < 0, Therefore, we consider two cases. When x1 > r n and x2 < r n , f (x1 − r n ) + f (x2 − r n ) = (x1 − r n ) − (λ 1

2

1

2

1

2

¯ . Similarly, when x1 < r n and x2 > r n , f (x1 − r n ) + f (x2 − r n ) = −(λ ¯ + 1 )(r n − x1 ) + (y2 − r n ) < 0, by by the definition of λ 1 2 1 2 1 2 ¯ the definition of λ.  Supplementary material Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.jebo.2018.11.001. References Apesteguia, J., Ballester, M.A., 2009. A theory of reference-dependent behavior. Econ. Theory 40 (3), 427–455. Apesteguia, J., Ballester, M.A., 2013. Choice by sequential procedures. Games Econ. Behav. 77 (1), 90–99. Arieli, A., Ben-Ami, Y., Rubinstein, A., 2011. Tracking decision makers under uncertainty. Am Econ. J. 3 (4), 68–76. Ariely, D., Wallsten, T.S., 1995. Seeking subjective dominance in multidimensional space: an explanation of the asymmetric dominance effect. Organ. Behav. Hum. Decis. Process. 63 (3), 223–232. Attanasio, O.P., 1999. Consumption. In: Handbook of Macroeconomics, vol. 1, pp. 741–812. Attanasio, O.P., Weber, G., 2010. 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Please cite this article as: G. Tserenjigmid, Choosing with the worst in mind: A reference-dependent model, Journal of Economic Behavior and Organization, https://doi.org/10.1016/j.jebo.2018.11.001