Aspiration-Based Choice Theory

Transcription

1 Aspiration-Based Choice Theory Begum Guney Michael Richter Matan Tsur JOB MARKET PAPER February 6, 2011 Abstract We model choice environments in which an unobservable, and possibly unavailable, aspiration alternative influences choices. A choice problem in this paper is a pair of sets (S, Y ) such that S Y X, where X is the grand set of alternatives, Y is the set of alternatives that are potentially available to choose and S is the set of alternatives that are actually available to choose. Our revealed preference approach enables us to construct an endogenous (subjective) distance function as well as a preference relation and to view choices as arising from a distance minimization procedure. According to our theory, when confronted with a choice problem (S, Y ), the decision maker first deduces her aspiration by maximizing her preference relation over Y and then chooses the alternative in S that is closest to her aspiration with respect to her subjective distance function. This representation captures the intuitive idea that our choices resemble our aspirations. We also consider some extensions and applications of this model. In our first application, we examine sales/discounts and demonstrate that our model allows for different pre- and post-sale consumption amounts of a good, providing an alternative explanation of the post-promotion dip phenomenon observed in the marketing literature. In our second application, we consider a second price auction where bidders are aspirational agents and show that overbidding may be observed in equilibrium. In the last application, we find that incorporating aspirations into the standard framework may lead to more equal divisions in the ultimatum game than the standard theory predicts. We are indebted to Efe A. Ok for his continuous support and guidance throughout this project. We also thank Andrew Caplin, Vicki Morwitz, David Pearce, Debraj Ray, Ariel Rubinstein, Andrew Schotter, Kemal Yildiz, and the participants of Decision Theory Seminar at NYU for helpful comments. Corresponding Author: Department of Economics, New York University, 19 West 4th Street New York, NY begum@nyu.edu Department of Economics, New York University. Department of Economics, New York University. 1

2 1 Introduction It is surely a familiar observation that our aspirations shape our choices and influence countless aspects of the economic part of our daily lives. For instance, aspirations for a dream job may influence which career path an individual chooses to follow and eventually which job offer she accepts even if that dream job is currently unattainable. Similarly, one s choice of which college to attend to may depend on her aspirations concerning prestige or certain ideals. On a more tangible front, aspirations for luxury brands which are unaffordable may lead consumers to buy fake products resembling the originals, resulting in a worldwide demand for counterfeit products. Aspirations may even play a role in partner selection as individuals could have a definition in their minds of an ideal partner and then look for someone similar to that ideal. One can come up with many other real life situations where possibly unavailable aspirations play crucial roles in decisions. Yet, surprisingly, the literature on decision theory does not provide much way of incorporating the effects of aspirations into a choice setting. The present paper sets out to do precisely this by developing an extended choice model that addresses how aspirations arise and how they affect and even guide choices. In standard choice theory, a decision maker is characterized by the way she makes choices from a given set of feasible alternatives. The following example highlights that aspirations may affect choices in a way that such a theory cannot possibly capture: Jane is looking for a pair of shoes. After observing the shoes set out in the store, she finds a pair which she likes the best and asks for them. Unfortunately, those shoes are unavailable in her size. She then realizes that only a subset of the shoes set out in the store are available in her size. Now, she must choose from this set of available shoes. Is Jane s choice problem the same as that of the rational man, who only considers the available elements when choosing? Should we expect Jane s choices 2

3 be the same in two stores with the same set of shoes in her size but with different shoes set out? If the answer to the first question is yes, then so is the answer to the second question because if Jane were to act according to the standard model, shoes set out in the store which are not in Jane s size should not influence her choices. However, experimental evidence suggests that phantom products, alternatives which seem real but are not available at the time of decision, may affect choices (Farquhar and Pratkanis (1992)). Furthermore, it stands to reason that the shoes set out in the store may influence choices through various channels. We focus on one specific channel: aspirations. Different shoes set out in the store may give rise to different aspirations, which in turn may influence the choice. But then how do aspiration affect choices? Consider the next example: Bob and Alice are both applying to colleges, Bob aspires to go to Wesleyan and Alice aspires to go to Harvard. They both get accepted to Berkeley and Yale. If Bob perceives Berkeley s ideology as resembling Wesleyan s, his aspirations may affect his choice in Berkeley s favor. Likewise, if Alice perceives Yale s status as more resembling Harvard s she may view Yale more favorably. This exemplifies the intuitive notion that our choices resemble our aspirations through some subjective notion of similarity. To be able to deal with such examples, we extend the basic setup of the standard choice theory by including the data of choice problems which exist in the background but not necessarily available. A choice problem in this paper is, therefore, a pair of sets (S, Y ) where S Y X, with X denoting the grand set of alternatives. In the choice problem (S, Y ), we will refer to Y as the potential set, which consists of all alternatives that are potentially 1 available to choose, to S as the actual set which consists of all alternatives that are actually available to choose, and to Y \S as the phantom set which consists 1 Merriam-Webster defines potential to mean: existing in possibility, capable of development into actuality. 3

4 Figure 1: A Choice Problem with Phantoms (Left) and No Phantoms (Right) of phantom 2 alternatives. A choice correspondence attaches a subset of the actual set to each possible choice problem. To wit, in the shoe store example, Y is the set of all shoes set out in the store and S is the set of shoes set out in the store that are available in Jane s size. She needs to choose an alternative from the set S while she observes the set Y. In the college example, Y is the set of schools which have been applied to and S is the set of schools that give offer. 3 We follow a revealed preference approach. First, we restrict type of effects that a potential set may have on choice by requiring choice to be WARP consistent for two types of problems: first, when the potential set is fixed and the actual set varies, and second, when the actual set coincides with the potential 2 We adopt the name phantom from a strand of psychology and marketing literature which define phantom alternatives to be options that seem real but are unavailable at the time of decision. 3 In the first example, X can be taken to be the set of all shoes in the world and in the second example, it can be thought as the set of all colleges in the world. 4

5 Figure 2: Aspirational Choice from the problem (S,Y) set. Next, we impose an axiom that restricts how different potential sets may affect choice. In addition, a technical axiom that we relax in subsequent sections stipulates that a single element is chosen whenever all potential alternatives are actual, that is, whenever there are no phantom alternatives. Finally, we also posit two standard continuity restrictions on choice correspondence. These axioms together imply our main representation which characterizes the choice process of a decision maker with a fixed preference relation and a distance function in her mind. When facing the choice problem (S, Y ), the decision maker first forms her aspiration by maximizing her preference relation over the potential set Y and then chooses the alternatives from the actual set S that are closest to her aspiration with respect to her distance function. This representation coincides with the intuitive notion that our choices resemble our aspirations as exemplified in the college application example. The main contribution is a representation based on a subjective distance function 5

6 which captures the notion of resemblance between alternatives as perceived by the decision maker. The rest of the paper is organized as follows. In section 2, we provide a review of the experimental and theoretical literature. In Section 3, we introduce our framework, axioms, and we present our main representation result. In Section 4, we present three types of applications for our model. In the first application, we consider a dynamic model of discounts and argue that pre- and post-sales consumption of a good may be different if the decision maker is an aspirational chooser. In our second application, we find that overbidding may be observed in the equilibrium of a second-price auction when one bidder is an aspirational chooser. In our final application, we consider the ultimatum game where one of the agents is an aspirational chooser and we show that aspirations may lead to a decrease in the bargaining power of the proposer. In Section 5, we extend our main characterization result to the case where multiple aspiration points may be chosen from a set. In Section 6, we relax the restriction that for any choice problem the actual set is a subset of the potential set. Finally, we conclude in Section 7 and all proofs are given in the Appendix. 2 Literature Review Aspirations have drawn great attention in behavioral sciences. In the psychology literature, an aspiration is informally defined to be a subjective goal or target, and any outcome below this target is considered to be a failure while any outcome above it to be a success. 4 Siegel (1957) suggests a formalization by associating aspiration with the alternative where the agent s marginal utility is maximized. Contrary to these definitions, in the present paper we define an aspiration in a choice situation to be the alternative with the highest utility 4 In particular, Simon s (1955) Satisficing procedure describes the choice behavior of a decision maker who first determines how good an alternative she seeks to find and then stops searching as soon as she finds one at least as good as that. Simon calls this threshold aspiration. 6

7 when the decision maker faces no unavailability restrictions. Hence, we consider aspiration more of an ideal that the decision maker is dreaming about rather than a threshold. In the experimental research about effects of one s aspirations on her choice, different methods are used to extract aspirations of a decision maker. In some studies, the subject is assigned a goal and hence aspiration is exogenously determined, while in other studies she is asked to form an aspiration herself and experimenters figured out what her aspiration is either by using Siegel s definition (Harnett (1967); Becker and Siegel (1959)) or by asking directly what her goal is (Lant (1992); Larrick, Heath and Wu (2009)). Our theoretical work, however, suggests an alternative way that enables an outside agent to understand what a decision maker s endogenously determined aspiration is. Effects of aspirations on choice have been empirically studied in various contexts. For instance, Payne, Laughhunn and Crum (1980) examine the effect of exogenously determined aspiration on risky choice; Harnett (1967) analyzes how aspirations affect group decision making; and Lant (1992) explores the organizational formation and adjustment of aspirations (goals) over time. In the present paper, to provide a theoretical study of aspiration formation and its influence on choice, we incorporate phantom options into the choice framework. In contrast to the standard theory which says that unavailable options should not influence the choice between available options, previous research in psychology and marketing has empirically documented that the addition of phantom alternatives to a choice set can change the choice share (probability) of the existing options in a systematic way (Farquhar and Pratkanis (1993), Highhouse (1996), Pettibone and Wedell (2000)). It has been found that an asymmetrically dominant (dominated) phantom alternative increases the relative choice share of the alternative that it dominates (is being dominated by) (Doyle et al. (1999)). In a recent experimental study, Ge et al. (2009) finds that consumers exhibit a significant tendency to seek out information about sold-out products and the information they obtain, helping them understand the product distribution, make them more likely to choose the available option that fall 7

8 in the middle of product distribution, thereby resulting in compromise effect even when one of the extreme options is phantom. They also find that sold-out products, conveying a sense of immediacy, prompt customers to expedite their purchases. There is a distinction between known and unknown phantoms. Unknown phantoms are options whose unavailability is initially unknown, such as fully booked hotels. Known phantoms, however, are alternatives whose unavailability is known from the beginning, such as preannounced new products. In previous studies, some disagreeing evidence has been found about these two types of phantom alternatives. Min (2003) shows that a highly preferred phantom is more likely to induce a decision maker to choose a similar alternative when it is a known phantom rather than an unknown one. He argues that a decision maker who learns about the unavailability at a later stage may exhibit reactance that leads her to choose a dissimilar alternative to reduce her negative feelings. In contrast, in a recent study by Hedgcock et al (2009), the choice share of a similar alternative is found to be higher when the phantom is unknown rather than known. As a reason for this, they suggest that an unknown phantom is more likely to receive careful thought before its unavailability is revealed, resulting in a change in attribute weights, while this effect does not occur in situations where the phantom is known at the outset. Our model does not make a distinction about when the unavailability of an alternative is revealed but the procedure we characterize is in accordance with the finding that a highly preferred phantom alternative may lead decision makers to choose the alternative that is most similar to it. Alongside phantom alternatives, numerous other factors, which are not accounted for in the standard choice model, have been documented to affect choices. Prominent examples are status quo and default options (Samuelson and Zeckhauser (1988)), asymmetrically dominated alternatives (Huber, Payne and Puto (1982)), presentation order of alternatives (Wilson (1997), Bruine de Bruin (2005)). One theoretical approach to incorporate these factors has been to extend the standard choice problem and posit axioms which characterize 8

9 choice procedures that are consistent with these findings. Rubinstein and Salant (2006a) and Masatlioglu and Ok (2005, 2010) take this approach to address order effects and status quo bias, respectively. Rubinstein and Salant (2008) (RS) and Bernheim and Rangel (2009) (BR) develop a more general framework of extended choice problems, (A, f) where A is the set of available alternatives and f stands for any factor affecting choice but irrelevant in the standard model. RS refer to f as a frame while BR refer to f as an ancillary condition. RS identify conditions for which choices can be rationalized and examine the effects of different type of frames. BR use this framework to generalize welfare analysis to a broader set of choice models. In line with these papers, we consider extended choice problems with a particular type of frame, namely potential sets. 5 Rubinstein and Salant (2006b) propose a choice model, called Triggered Rationality, that shares some common features with ours. In their model, a reference point is determined endogenously via the maximization of an ordering over a set of available alternatives and it then affects the choice by perturbing the preferences. Even though the reference point is always attainable, it need not necessarily be chosen from the set. In our model, however, reference points may not be attainable but is always be chosen whenever they are. This is in concert with our interpretation of the reference point as an aspiration. Another difference is that we characterize a choice procedure based on distance functions as opposed to preferences. Another related paper is Rubinstein and Zhou (1999). They characterize a distance-minimization procedure which always selects the available alternative that is closest to an exogenously given reference point. Their paper differs fundamentally from ours because the distance function used in their model is the standard Euclidean metric. For the Euclidean metric to make sense in 5 We say that potential sets is a particular type of frame with a bit of abuse of the definition of a frame. Because if Y constitutes a frame, then we should consider (S, Y ) for all S, even for those S that do not lie in Y, as legitimate choice problems. However, in a later section of this article, we present a model that corresponds to this case as well. 9

10 their model, the grand alternative space must be identifiable with a subset of a finite dimensional Euclidean space. One cannot use a typical multi-utility representation because they are typically not finite-dimensional. However, even if it were, we here caution that it may not be natural to assign attributes to the objects under consideration. For example, consider the problem of choosing between different girlfriends. This is a case where each girlfriend may not be naturally understood as a vector of attributes. Another drawback of using the standard Euclidean metric is that scale matters. Consider an agent in a single good economy where the good sells for $1 and she has an endowment of $1. Fix her reference point to be ($1, 1 unit). Her Euclidean distance minimizing choice is ($1/2, 1/2 units). Now, what if we considered her problem measured in cents? Then, her reference point will be (100, 1 unit) and the point in her budget that is at minimum distance to her reference is (99.99, units), a quite different result. In contrast, our endogenously derived distance function allows for the same choices to be made, regardless of scales. Influences of aspirations on outcomes are studied in various economic models. For instance, in game theoretic models of aspiration-based reinforcement learning, aspirations affect players strategy choices and these aspirations are either fixed or evolve according to transition rule which might depend on different factors such as the past payoffs or the experience of peers (Borgers and Sarin (2001), Bendor et al. (1995, 1998, 2000, 2001)). Finally, Kalai and Smorodinsky (1975) consider a two-person bargaining problem (S, a) where S is a feasible set of outcomes and a is disagreement point. For any problem, they identify a utopia point, which may not be attainable, and the bargaining outcome is the intersection of the the pareto frontier with the the line connecting this point with the disagreement point. The existence of a point that may not be attainable and that affects the outcome is reminiscent to our model even though the way in which the point arises and how it affects the outcome are both formally and conceptually different. 10

11 3 The Model We fix a grand set of alternatives X, and posit that X is a compact metric space. Let X denote the set of all nonempty closed subsets of X. Throughout this paper, when we refer to a metric on X, we mean the Hausdorff metric. Convergence in X is with respect to this metric and denoted by. H By a preference relation, we mean a reflexive, transitive binary relation and by a linear order, we mean a complete anti-symmetric preference relation. Choice problems in our setting consist of a pair of sets (S, Y ) where S, Y X and S Y. The set of all choice problems is denoted by C(X). C stands for a choice correspondence by which we mean a map C : C(X) X such that C(S, Y ) S holds for all (S, Y ) C(X). Notice that a choice correspondence on C(X) must be non-empty valued by definition. We interpret the choice problem (S, Y ) in the following manner: A decision maker observes the set Y and makes a choice from the set S. We refer to Y as the potential set, which consists of all alternatives that are potentially available. Any element y Y may either be available for the decision maker to choose, that is y S, in which case we call it an actual alternative, or unavailable, that is y Y \S, in which case we call it a phantom alternative. We refer to y S as an actual alternative because it is physically possible to choose y in the choice problem (S, Y ). Likewise, we refer to z Y \S as a phantom alternative because it is physically impossible for the decision maker to choose z in the choice problem (S, Y ). Since S is a collection of actual alternatives, we refer to S as the actual set of (S, Y ) and similarly, we refer to the collection of phantom alternatives, Y \S, as the phantom set 6 of that choice problem. We shall refer to the choice problem (Y, Y ) as being phantom free. For this choice problem, the actual set of alternatives is equal to the potential set, that is, every alternative that is potentially available is indeed an actual alternative. 6 We are motivated by a strand of psychology and marketing literature, which uses the word phantom to denote alternatives that are perceived by the decision maker but can never be chosen. 11

12 To illustrate, consider again the decision problem of Jane, who is looking for a pair of shoes in a given store. According to our notation, all shoes set out in the store is the potential set of Jane s problem while the set of all pairs in the store that are available in her size is the actual set. The problem that Jane faces is to choose among the shoes available in her size after observing all the shoes set out in the store. In the rest of this section, we take an axiomatic approach to conceptualize one s choice behavior. Throughout this article, we consider the situation where both the actual and phantom sets in a choice problem are observable to the external observer and this is reflected in our axioms. Our first axiom places a restriction across choice problems that share the same potential set. Specifically, it considers choice problems of the form C(, Y ) for any fixed Y X. It stipulates that our agent satisfies the Weak Axiom of Revealed Preference (WARP) when choosing within this restricted class of problems. Axiom 3.1 (WARP Given Potential Set) For any (S, Y ), (T, Y ) C(X) such that T S, C(T, Y ) = C(S, Y ) T, provided that C(S, Y ) T. Referring to our example, this postulate says that if all stores have an identical set of shoes set out, then Jane s choices are rational across these stores, in the sense of being consistent with WARP. This guarantees that, for a given potential set, choices of the involved decision maker can be viewed as the outcome of maximization of a complete preference relation. Our next axiom states that across phantom free choice problems, observed choices should be rational as well. Axiom 3.2 (WARP for Aspirations) For any S, Y X such that S Y, C(S, S) = C(Y, Y ) S, provided that C(Y, Y ) S. 12

13 This axiom allows us to identify a preference relation on X which rationalizes choices over phantom free choice problems. In our motivating example, if Jane were to only visit stores where all the shoes set out are also available in her size, then her observed choices across these stores should be rational. Axioms 3.1 and 3.2 together say that when observing choices C(S, Y ) and C(T, Z) where S T, WARP can be violated only if the two potential sets Y, Z are different and at least one of the actual sets differs from its frame, that is, S Y or T Z. Consider a choice problem in which the involved actual set and the potential set are equal to each other, formally (Y, Y ) for some Y X. Since this choice problem is phantom free,that is, any potential alternative is actually available, we consider that the agent will choose her favorite alternative from the set Y. Put differently, the element she prefers the most in Y is available, and hence she is able to and will choose that alternative. Hence, we interpret C(Y, Y ) as the aspiration set of Y. In line with this interpretation, Axiom 3.2 can also be interpreted as guaranteeing aspiration sets to arise from the maximization of a complete preference relation. Moreover, Axiom 3.1 implies that whenever elements of an agent s aspiration set are available, then she will choose them. The following axiom builds a link between choices under different potential sets. Specifically, it posits that if the decision maker makes the same choice from two sets Y, Z in the absence of any phantom alternatives, then the choice behavior induced by Y and Z on a smaller actual set is identical. Axiom 3.3 (Independence of Non-Aspirational Alternatives) For any (S, Z), (S, Y ) C(X), C(S, Z) = C(S, Y ), provided that C(Z, Z) = C(Y, Y ). This postulate implies that what is relevant for the decision maker s choice from S is not Y or Z specifically, but rather the aspiration sets that Y, Z generate. In other words, it stipulates that as long as Y and Z give rise to the same 13

14 aspiration sets, then the choice behavior on subsets of Y Z should be the same with respect to either Y or Z as the frame. Definition 3.1 A choice correspondence C on C(X) is called aspirational if it satisfies Axioms 3.1, 3.2, and 3.3. We now introduce an axiom that requires the decision maker to choose a single element whenever she faces a phantom free choice problem. In the next section, we will examine the behavior of an aspirational agent without this simplifying axiom and present a more general characterization. Axiom 3.4 (Single Aspiration Point) C(Y, Y ) = 1 for all Y X. The above axiom states that the decision maker necessarily chooses only one alternative when there are no phantom alternatives. Put differently, in any given set the agent is assumed to identify a single aspiration point. When dealing with choice problems with infinitely many alternatives, we need to utilize some continuity properties. This situation is analogous to that in the standard choice theory. In fact, the next axiom is a modification of the standard continuity axioms for choice correspondences. It says that the decision maker chooses similarly in similar choice problems. Axiom 3.5 (Continuity) For any (S n, Y n ), (S, Y ) C(X), n = 1, 2,... and H H x n S n for all n such that S n S, Yn Y and xn x, if x n c(s n, Y n ) for every n, then x c(s, Y ). Notice that any finite grand set can be considered a compact metric space when endowed with the discrete metric. In that case, the above continuity axioms trivially hold true, that is, Axiom 3.5 becomes operational only when X is an infinite set. The above axioms together yield our representation of the choice behavior of an agent who endogenously forms aspirations which she then incorporates into her choice from the relevant actual set. 14

15 Theorem 3.1 C is an aspirational choice correspondence that satisfies Axioms 3.4 and 3.5 if and only if there exist 1. a continuous linear order, 2. a continuous metric d : X 2 R + such that C(S, Y ) = argmin s S d(s, a(y )) for all (S, Y ) C(X), where a(y ) is the -maximum element of Y. The interpretation of the choice behavior characterized in Theorem 3.1 is as follows: When a decision maker is confronted with a choice problem (S, Y ), she first forms her aspiration, call it a(y ), by maximizing her aspiration preference in the potential set Y. She then uses the subjective metric d to choose the alternatives in S that are closest to her aspiration a(y ). The agent s distance-minimizing choice behavior implies that she chooses her aspiration a(y ) uniquely whenever it is an element of S. Because a(y ) is the unique minimum-distance element from itself. Notice that the metric d is endogenously defined and could possibly be quite different than the metric that the space X is endowed with. Each decision maker who makes different choices possibly give rise to her own distinct distance function. Hence, we may interpret this metric as the psychological distance between different alternatives in the mind of the decision maker. We think that minimizing a distance more naturally represents the behavior of an agent who has an unavailable (phantom) aspiration and seeks to choose the closest actual element. Hence, this behavioral model of distance minimization adds additional insight in this context as opposed to a utility maximization model. We make no claims that the element chosen when an agent faces the choice problem (S, Y ) gives her more utility than any other available element. Rather, her choice behavior is better understood procedurally, trying to get as close as she can to some endogenously-formed ideal point. 15

16 Remark: Consider an agent who possesses a complete preference order and, in any given choice problem (S, Y ), she chooses the alternative that maximizes, regardless of the potential set. Put differently, her choices are such that C(S, Y ) := max(s, ) for every (S, Y ) C(X). This agent clearly satisfies all of our axioms. Hence, our theory accepts rational choice as a special case. 4 Applications In this section, we consider three applications of our model. In the first application, we examine sales/discounts. We argue that, as opposed to what standard theory predicts, an agent s pre- and post-sale consumption amounts of a good may be different if aspirations play a role in an agent s decisions. This observation may provide an alternative explanation for the post-promotion dip phenomenon observed in the marketing literature (Van Heerde, Leeflang and Wittink (2000)). Our second application analyzes a second-price auction where one of the bidders is assumed to be aspirational. We find that the aspirational agent may overbid in equilibrium, which is a consistent outcome with the findings in the experimental literature. Explanations provided for the overbidding observed in second-price auctions include a joy of winning motivation (Cooper and Fang (2008)) and the illusion that bidding above valuation increases the chance of winning at no cost (Kagel and Levine (1993)). Our theoretical result suggests that aspirations may provide an alternative explanation for overbidding. In our final application, we consider the ultimatum game played between a standard utility maximizer and an aspirational chooser. We find that, in contrast to what the standard theory predicts, aspirations (together with distance functions) may work to reduce the bargaining power of the proposer if the other player aspires for all the dollar for himself and nothing for the other player. This result can be viewed as an alternative explanation of more equal divisions observed in many experimental studies such as Guth, Schmittberger and Schwarze (1982) and Thaler (1988). 16

17 Figure 3: Price of Good G decreases 4.1 Sales/Discounts Consider an economy where a firm produces exactly one good. Call this good G and denote its price as p G. Imagine an agent who makes consumption decisions in this economy. She chooses a pair (x, y) where we think of the first coordinate representing her consumption of G and the second coordinate representing how much she spends on consumption of other goods. Suppose that she behaves as an aspirational chooser when a natural aspiration point arises in her decision problem. Furthermore, assume she has a fixed income level I so that makes a choice from the budget set B = {(x, y) : p G x+y = I}. We consider consumption to be desirable for our agent. Hence, she consumes a bundle on her budget frontier. Our aim is to examine how this agent s consumption changes from before the sale on good G to after the sale. Now, suppose that our decision maker initially faces a budget frontier B 1 and consumes bundle a. 7 Imagine that, in the next period, the seller creates a sale on good G so that the agent s budget 7 We refer to the coordinates of the choice bundle a by (a x, a y) and similarly for any other consumption bundle. 17

18 frontier rotates to B 2 and she starts to consume bundle b on B 2. The question that we are interested in is what would happen to the amount of G that the agent consumes when the sale on good G ends, that is, when the budget frontier rotates back to B 1. Thinking that she could have continued consuming bundle b if sales had never ended, our agent may still have a desire to continue consuming bundle b. Hence, it is not unreasonable to imagine that b may now act as a natural aspiration point for her. 8 In that case, following the aspirational choice procedure, she chooses c which is the closest bundle in B 1 to her aspiration. Formally, c argmin z B 2 d(z, b) = argmin z B 2 (b x z x ) 2 + (b y z y ) 2. The following proposition states when the bundle consumed during the sale induces our agent to consume more or less of G after the sale relative to her pre-sale consumption (Figure 4). Proposition 4.1 There exists t B 2 such that b x t x if and only if c x a x. According to standard utility maximization theory, an agent would go back to her original consumption habit and start to consume bundle a again after the sale ends. However, the empirical literature finds a difference between pre- and post-sale consumption. In fact, one particular example of such a difference is the post-promotion dip phenomenon which refers to post-sales consumption of a good being lower than its pre-sales consumption (Van Heerde, Leeflang and Wittink (2000)). In the marketing literature, this observed dip is often explained by stockpiling or brand switching. Our model, however, suggests an alternative explanation in terms of aspirations. We find that if, during the sale, the decision maker consumes less of good G than the threshold level, t x, and 8 In her post-sale consumption choice problem, the budget set she faces during the sale is her potential set; the bundle she chooses from that potential set is her aspiration; and the budget set she faces after the sale ends is her actual set. 18

19 Figure 4: Different Pre- and Post-sale Consumption Amounts uses this consumption as an aspiration, then she will consume less of good G relative to her original consumption after the sale ends. Hence, in this situation, we observe the post-promotion dip. Now, consider a decision maker who does not consume any G initially, that is a x = 0, but consumes some positive amount of G during the sale. The above proposition says that this decision maker will continue to consume a positive amount of good G after the sales end. This points to the potential usefulness of sales upon the introduction of a new product to create an aspiration or taste for consumption of this new product. In sum, this application shows that, in contrast to the predictions of standard utility theory, pre- and post-sales consumption quantities of a good may be different if aspirations play a role in an agent s decisions. This allows us to offer a novel explanation for an anomaly observed in the marketing literature. 4.2 Second-Price Auction In the present section, we analyze an independent private value second-price auction with N bidders. Let F be a distribution which each agent s value v i is drawn from and suppose F (0) = 0, that is, no agent s valuation is negative. Furthermore, we require that the seller s valuation of the good is 0, hence she 19

20 never sells the good for a negative amount of money. For each agent i, an outcome of the auction is an element of R 2 where (x, y) is such that x is 0 if agent i does not receive the good and 1 if she does and y represents how much agent i paid to receive the good. Hence, an outcome (x, y) can either be (0, 0) or (1, y) where y R +. Now, consider the case where all bidders but one are standard utility maximizers. 9 Suppose for any v there exists a price t(v) where 0 t(v) v such that when she is of type v, the non-standard bidder would be happy to get the good as long as she pays a price lower than t(v). 10 However, she does not feel comfortable if the price is above t(v) and therefore her problem reduces to determining a bidding strategy in case the maximum bid of the other agents is above t(v). The potential set of outcomes she faces in this reduced problem is Y = {(1, p) p t(v)} {(0, 0)} However, the actual set of this reduced problem is not precisely known at the time this agent chooses his strategy. So, for any p t(v), there is one actual set of outcomes. S p = {(1, p), (0, 0)}, where p t(v) So, the decision maker faces the following set of choice problems: {(S p, Y ) p t(v)} and her aspiration in each of these reduced problems is getting the good at price t(v), that is, the outcome (1, t(v)). Assume further that this agent uses the standard Euclidean metric as her distance function. Therefore, she evaluates an outcome with respect to its distance from her aspiration (1, t(v)) and prefers outcomes which are closer to her aspiration. The following proposition shows that, for situations in which this agent is willing to participate, she has a unique strategy that will benefit her no matter 9 The utility of a standard agent with valuation v is 0 if he loses the auction and v p if wins the auction at a price p. 10 Formally, her preferences are such that (1, p) (0, 0) for any p t(v), and (1, p) (1, p ) for any p p t(v). 20

21 what the maximum bid of other players is and therefore which actual set is realized at the end of the auction. Theorem 4.1 When she participates, it is a weakly dominant strategy for an aspirational agent of type v to bid b(v) := t(v) + t(v) This is, in fact, the only weakly dominant strategy. Therefore, there is a unique dominant strategy equilibrium where she bids b(v) and all other agents bid their value. The experimental literature on second price auctions has noticed frequent instances of overbidding (Kagel and Levine (1993)). In the literature, there are several explanations for this regularity, most notably, a joy of winning motivation (Cooper and Fang (2008)) and the illusion that bidding above valuation increases the chance of winning at no cost (Kagel and Levine (1993)). 11 As an alternative explanation, our theory suggests that overbidding may be observed if aspirations play a role in bidders decisions. The optimal bidding function we present in Theorem 4.1 may in fact have agents overbid, that is, it may be the case that b(v) > v. We obtain this as an equilibrium outcome and the following proposition establishes when overbidding occurs. Proposition 4.2 An aspirational agent of type v overbids if and only if t(v) > v2 1. 2v In the standard setting, each agent bids her valuation since it is a dominant strategy to do so. Hence, the winner of the auction on average pays E[v (2) v (1) = v], 11 In addition, there are papers that provide a theoretical foundation for overbidding in first price auctions. For instance, Filiz-Ozbay, Ozbay (2007) explains this through agents fear of regret while Crawford and Irriberi (2007) uses level-k thinking (as a non-equilibrium outcome). 21

22 where we write v (i) to denote the i th -highest valuation among N agents. 12 One may suggest that, in our model, the above average price can serve as a natural aspiration at which the aspirational agent desires to buy the good. Our first example illustrates this situation and demonstrates that overbidding occurs whenever there are least three bidders in the auction. Example: Suppose F = U[0, c], where c > 0, and t(v) = E[v (2) v (1) = v]. Then, one can easily show that and t(v) = (N 1)v N t(v) > v2 1 2v 2v(N 1) 2 v 2 > (v 2 1)N 2. If N 3, then 2(N 1) 2 > N 2 and hence, we have overbidding if v 3 v 2 1. Notice that the latter inequality holds true for any positive v. Therefore, when values are uniformly distributed and there are at least three bidders, the aspirational agent overbids. 4.3 The Ultimatum Game We consider the ultimatum bargaining game played between two agents. Player 1 makes a proposal to player 2 about how to divide the dollar, that is, she offers (x, 1 x) where x [0, 1]. Player 2 accepts or rejects. If he accepts, then player 1 s proposal is adopted. Otherwise, the disagreement allocation (0, 0) is adopted. We assume that player 1 who makes the initial offer is a standard utility maximizer with u 1 (x) = x for any x [0, 1]. So, her aim is to get as much as possible from the bargaining game. As opposed to the standard bargaining games where both agents are assumed to be utility maximizers with the above utility function, we now consider that player 2 is an aspirational agent whose choices satisfy our axioms. Suppose that he has an aspiration a 2 R 2 + and uses the L p metric for 1 p < as her the distance function. 12 Due to the independent private values assumption, t(v) is an increasing function of v. 22

23 There are several natural points that the second player may have. Let us first consider the case where player 2 aspires for an equal divison which is either barely feasible or unfeasible, that is, player 2 s aspiration is (a, a) where a 1/2. One can easily show that the unique Subgame Perfect Nash Equilibrium (SPNE) is independent of such a and p. In equilibrium, player 1 offers the allocation (1, 0) and player 2 accepts any offer. Therefore, player 2 receives nothing when he follows our aspirational procedure with such an aspiration. We know that in the standard ultimatum bargaining game where player 2 is also a standard utility maximizer, player 1 has a first mover advantage and gets the entire dollar. The above example shows that aspiring for equal divisions in an aspirational choice procedure does not help player 2 beat (even partially) the first mover advantage player 1 has in the bargaining game. Now, let us examine the situation where player 2 aspires for a higher monetary payoff for himself, say his aspiration is (0, 1). If he is( using the L 1 metric, 1 then in the unique SPNE, player 1 offers the allocation 2, 1 ) and player 2 2 accepts (x, 1 x) if only if x 1 2. Now, suppose that he uses Lp such that p > 1. Then, there still exists a unique SPNE of this game where the player 1 offers the allocation (f(p), 1 f(p)) and player 2 accepts (x, 1 x) if and only if x f(p). This f function denotes how much player 1 receives and is strictly increasing in p, that is, player 2 s negotiating power decreases as his p increases. Yet, player 2 still has some negotiating power as f(p) < 1 for any p <. Hence, we see that when the second player aspires for a high payoff for himself, he does not accept an offer as easily as in the case where he is a standard utility maximizer and therefore player 1 partially loses her power. 13,14 This outcome is in fact consistent with a considerable amount of experimental evidence (Guth, Schmittberger and Schwarze (1982), and Thaler (1988)). Contrary to the predictions 13 This effect can be strengthened or moderated based upon which metric is used by the second player. 14 Suppose player 2 s aspiration is (a, 1 a) where a (0, 1/2). For a given p 1, player 1 s power increases (weakly) as a increases. If a 1/2, then player 1 gets all the dollar as in the standard ultimatum game. 23

24 of the standard theory, experimental studies have well-established that the ultimatum game outcomes are accumulated around equal divisions and this is often explained by fairness (Fehr and Schmidt (1999)) or fear of rejection (Lopomo and Ok (2001)). Our theory, however, suggests that experimental regularities in the ultimatum game may be due to aspirations. As we see from the examples above, aspirations may play a crucial role in determining the outcome of the ultimatum game. A player s bargaining power may also depend on her opponent s aspiration and distance function which (perhaps weakly) works to reduce the first mover advantage in the bargaining game. These observations constitute the initial step in examining the impact of aspirational choices in other bargaining games including the ones where players have imperfect information about each others aspirations and distance functions. 5 Multiple Aspiration Points Axiom 3.4 in the previous section examines the case where the decision maker chooses a single alternative from every phantom free choice problem. Hence, this agent can be modeled as one who is only affected by a single aspiration point. In this section, by dropping this single aspiration point axiom, we consider the case in which an agent may aspire to multiple elements. Put differently, in the previous section, we considered a( ) to be a choice function, and in this section, we allow it to be a correspondence. For our representation, we need to introduce two continuity axioms, which are modifications of the standard continuity axioms for choice correspondences. The first one imposes a continuity requirement along a fixed potential set. Axiom 5.1 (Upper Hemicontinuity Given Aspiration) For any (S, Y ) C(X) and x, x 1, x 2,... Y with x n x, if x n c(s {x n }, Y ) for each n, then x C(S {x}, Y ). This postulate states that, if the potential set is held constant and the alternative x n is chosen for each n when it is available in conjunction with all 24

25 elements of S, then so is their limit x. Our next axiom is a weaker version of Axiom 3.5 in that it requires Axiom 3.5 to be satisfied only in choice problems where the involved actual and potential sets coincide. Axiom 5.2 (Upper Hemi-continuity of the Aspiration Preference) For any Y, Y n H X, x n X, n = 1, 2,... such that Y n Y and xn x, if x n C(Y n, Y n ) for each n, then x C(Y, Y ). This property requires that the agent s choices should be upper hemi-continuous when she is choosing from a phantom free choice problem. In other words, if she (weakly) prefers element x n to all other elements in Y n and if these elements and choice sets are converging to x and Y, respectively, then she (weakly) prefers x to every element in Y. Finally, we introduce an indifference and a monotonicity axiom relating how an agent chooses relative to how she would choose in the single aspiration point case. Axiom 5.3 (Indifference) For any Y X and x, z Y, if {x, z} = C({x, z}, {x, y, z}) y C(Y, Y ), then {x, z} = C({x, z}, Y ). Axiom 5.4 (Monotonicity) For any Y X and x, z Y, (i) x C({x, z}, {x, y, z}) for all y C(Y, Y ) implies x C({x, z}, Y ). (ii) C({x, z}, {x, y, z}) = {x} for all y C(Y, Y ) implies {x} = C({x, z}, Y ). The above axioms give a way of tying together the agent s preferences with regard to different aspiration sets. The first axiom states that if an agent is indifferent between x and z with respect to every element in her aspiration set, then she is indifferent between x and z when considering that aspiration set as a whole. The second axiom states that if there is instead a weak (resp. strict) preference for x over z with respect to each aspiration point, then that weak 25

26 (resp. strict) preference holds when consider the entire aspiration set. Note that Axiom 5.4 implies Axiom 5.3. We now present a definition of monotonicity for functions with vector inputs. Definition 5.1 A function f : R X + R is called monotonic if v a (resp. <) w a for all a X implies f( v) (resp. <) f( w), where r = (r a ) a X for any r R X +. According to the above definition, if a vector v dominates another vector w in every coordinate, then a monotonic function assigns a higher value to v than to w. Our next definition calls a function single agreeing if, for any vector with at most one non-zero entry, the function takes the value of that non-zero entry. Definition 5.2 : A function f : R X + R is called single agreeing if f( v) = v a for all v = (v x ) x X such that v b = 0 whenever b a 15. Theorem 5.1 C is an aspirational choice correspondence that satisfies Axioms 5.1, 5.2, and 5.3 if and only if there exist 1. a continuous complete preference relation, 2. a metric d : X 2 R +, where d(, a) is lower semi-continuous on L (a), 3. a single agreeing function φ : R X + R + such that φ d(, A(Y )) is lower semicontinuous on L (A(Y )) for any Y X, such that C(S, Y ) = argmin s S φ( d(s, A(Y ))) for all (S, Y ) C(X), (1) where d(x, A(Y )) := [ d(x, a)1 a A(Y ) ] and A(Y ) := max(y, ) = C(Y, Y ) for any x X and Y X. 15 Notice that this implies f( 0) = 0. 26

27 Moreover, if C satisfies Axiom 5.4 as well, then the above theorem remains true with φ taken to be monotonic when comparing vectors of the form d(x, A(Y )) and d(y, A(Y )) and x, y Y. We interpret the choice model obtained above in the following manner: When a decision maker is confronted with a choice problem (S, Y ), she first forms her set of aspiration points, A(Y ), by maximizing her aspiration preference over Y. In addition, she has an endogenous distance function d that she uses to measure the distance between the alternatives she is considering and an individual aspiration point. When making her decision, for any x S, she considers the vector of distances d(x, A(Y )) and uses an aggregator φ to surmise this vector of distances into a single dissimilarity value. 16 She then chooses the alternative(s) in S with the smallest dissimilarity value relative to A(Y ). The single agreeing requirement of φ is so that our aggregator reduces to the endogenous distance function when there is a single aspiration point. Put differently, aggregation only takes place when it needs to and the agent s choices follow this distance in the case of a single aspiration point. Let us notice that Axiom 5.3 is fundamental for the above representation. If, for every element of an aspiration set, an agent is indifferent between two options x and z, then x, z will both give rise to the same vector of distances with respect to that aspiration set. Hence, any aggregator must view these two elements indifferently in that situation. Finally, Axiom 5.4 strengthens Axiom 5.3 in understanding how preferences with respect to individual aspiration points may relate to preferences with respect to an entire aspiration set. This restriction seems very natural because it stipulates that an agent who considers x closer than y to every aspiration element would also consider x to be closer than y with respect to the relevant aspiration set. We now provide several examples of aggregator functions for which the choice 16 Note that this value is how dissimilar the agent finds an alternative x to an aspiration set A(Y ). 27