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 10012. E-mail: [email protected]
‡
Department of Economics, New York University.
§
Department of Economics, New York University.
1
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
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 develop-
ment into actuality.
3
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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
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ŽŶƚŚĞƐĞƚ^
z
^
Ě
ƐƉŝƌĂƚŝŽŶƉŽŝŶƚŝŶ
ƚŚĞƉŽƚĞŶƚŝĂůƐĞƚz
ŚŽŝĐĞĨƌŽŵƚŚĞ
ƉƌŽďůĞŵ;^͕zͿ
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
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
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
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
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
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 ¢, 0.0001 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
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.
H
Convergence in X is with respect to this metric and denoted by →. 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
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 6= ∅.
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 6= ∅.
12
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 6= Y or T 6= 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
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 (Sn , Yn ), (S, Y ) ∈ C(X), n = 1, 2, . . . and
H
H
xn ∈ Sn for all n such that Sn → S, Yn → Y and xn → x,
if xn ∈ c(Sn , Yn ) 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
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 ) = argmins∈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
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
,
*
+
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 pG . 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) : pG 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 B1 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 (ax , ay ) and similarly for any other
consumption bundle.
17
frontier rotates to B2 and she starts to consume bundle b on B2 . 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 B1 . 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 B1 to her aspiration.
Formally,
c ∈ argmin d(z, b) = argmin
z∈B2
q
(bx − zx )2 + (by − zy )2 .
z∈B2
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 ∈ B2 such that
bx ≤ tx if and only if cx ≤ ax .
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, tx , 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
KƚŚĞƌ'ŽŽĚƐ
KƚŚĞƌ'ŽŽĚƐ
Ϯ
Ϯ
ϭ
ϭ
ƚ
ď
ƚ
ď
Đ
Ă
Ă
Đ 'ŽŽĚ'
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 ax = 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 vi
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
'ŽŽĚ'
never sells the good for a negative amount of money.
For each agent i, an outcome of the auction is an element of R2 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.
Sp = {(1, p), (0, 0)}, where p ≥ t(v)
So, the decision maker faces the following set of choice problems: {(Sp , 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, p0 )
for any p ≤ p0 ≤ t(v).
20
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) +
p
t(v)2 + 1.
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
where we write v (i) to denote the ith -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
t(v) =
(N − 1)v
N
and
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 u1 (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 a2 ∈ R2+ and uses
the Lp 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
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 isusingthe L1 metric,
1 1
then in the unique SPNE, player 1 offers the allocation
,
and player 2
2 2
1
accepts (x, 1−x) if only if x ≤ . Now, suppose that he uses Lp such that p > 1.
2
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
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, x1 , x2 , . . . ∈ Y with xn → x, if xn ∈ c(S ∪ {xn }, Y ) for each n, then
x ∈ C(S ∪ {x}, Y ).
This postulate states that, if the potential set is held constant and the alternative xn is chosen for each n when it is available in conjunction with all
24
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, Yn ∈
H
X , xn ∈ X, n = 1, 2, . . . such that Yn → Y and xn → x,
if xn ∈ C(Yn , Yn ) 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 xn to all other elements in Yn 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
(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 : RX
+ → R is called monotonic if
va ≤ (resp. <) wa for all a ∈ X implies f (~v ) ≤ (resp. <) f(w̃),
where ~r = (ra )a∈X for any ~r ∈ RX
+.
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 : RX
v ) = va
+ → R is called single agreeing if f (~
for all ~v = (vx )x∈X such that vb = 0 whenever b 6= a15 .
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 : RX
+ → R+ such that φ◦ d(·, A(Y )) is lower
semicontinuous on L (A(Y )) for any Y ∈ X ,
such that
C(S, Y ) = argmin φ(d~(s, A(Y ))) for all (S, Y ) ∈ C(X),
(1)
s∈S
~ A(Y )) := d(x, a)1a∈A(Y ) and A(Y ) := max(Y, ) = C(Y, Y ) for
where d(x,
any x ∈ X and Y ∈ X .
15
Notice that this implies f (~0) = 0.
26
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
~ A(Y )) and d(y,
~ A(Y )) and x, y ∈ Y .
d(x,
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
~ A(Y )) and uses an aggregator φ to surmise
considers the vector of distances d(x,
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
correspondence C satisfies the above monotonicity axiom along with our two
continuity axioms, 5.2 and 5.1, when C is expressed as in equation 1.
Example 1: (Hausdorff Point-Set Distance)
Consider the case where φ(d~(s, A(Y ))) = dH (s, A(Y )) = max d(s, t). In this
t∈A(Y )
example, the decision maker chooses the alternative that minimizes the Hausdorff distance from her aspiration set. In other words, she minimizes the maximal distance between her considered element and all members of the aspiration
set.
Example 2: (Total Distance)
X
If φ(d~(s, A(Y ))) =
d(s, t), then the agent chooses the alternative that
t∈A(Y )
minimizes the total distance from her aspiration set. Notice that the above
formulation is inherently assuming that there are a finite number of aspiration
points. In the case where A(Y ) is not finite, then the above could be done via
integration on an appropriate space.
We now explore a short example of choice where the agent aspires to two
phantom alternatives and chooses from a different set of three available alternatives. We use the standard Euclidean distance and investigate what choices
would be made according to each of the three aggregators outlined above.
In Figure 5, we assume
d(B, P1 ) < d(C, P1 ) < d(D, P1 )
d(D, P2 ) < d(C, P2 ) < d(B, P2 )
We consider an agent facing the following choice problem:
C({B, C, D}, {P1 , P2 , B, C, D}) where A({P1 , P2 , B, C, D}) = {P1 , P2 }.
If the agent chooses according to the aggregator function in Example 1, she
would select the compromise option C. On the other hand, if she uses the
aggregator function provided in Example 2, then B, C, or D are all possible
28
ƚƚƌŝďƵƚĞϮ
Wϭ
WϮ
ƚƚƌŝďƵƚĞϭ
ƚƚƌŝďƵƚĞϭ
Figure 5
choices, that is, her choice depends upon the absolute positioning of these three
elements.
We consider one last example which is a special case of Example 1.
Example 3: (Aspiration Indifference) Consider an agent who chooses in a
manner consistent with both Example 1 and 2. This is only possible if whenever
x ≺ a1 ∼ a2 then d(x, a1 ) = d(x, a2 ), where a1 and a2 are her aspirations . This
is a model of “aspiration indifference” where the agent’s preferences with regard
to an aspiration set A are exactly the same as her preferences with regard to
any member of A. The aspiration points she determines in a potential set are
not only aspirationally indifferent to each other but also each one perturbs her
choice in the actual set in exactly the same way. Hence, taking any arbitrary one
of her aspirations points in a potential set and choosing the available alternative
that minimizes the distance from that particular aspiration point suffices for her
to make a choice.
29
6
Relaxation of Subset Restriction
So far, in any choice problem, we have always taken the actual set to be a
subset of the involved potential set. Although this restriction is in concert
with our interpretation of the model, one can imagine a model without this
restriction as well. In the present section, we provide another extension to our
main characterization result. We relax the requirement that the actual set is a
subset of the potential set.
Consider Joan, a friend of Jane’s, is a graduate student. Unfortunately, Joan
has a very limited budget and hence her actual constraint on choice is due to
cost. Imagine she has gone to many different stores in her city, and has yet to
purchase a pair of shoes. Fed up, tired, and feeling quite depressed, she decides
to go to TDSS, The Dollar Shoe Store, from which she has decided to purchase
a pair of shoes. Their quality is quite bad, but she knows that she will be able to
afford all the shoes there. As before, we denote Joan’s problem (S, Y ), but this
time we consider Y to be the set of shoes- including the expensive ones- that
she has seen in all the previous stores and S to be the set of shoes at TDSS. It’s
quite possible that TDSS may have some of the same shoes as the prior stores
that she has visited and some shoes that are completely new to her. Therefore,
S need not be a subset or a superset of Y . As a result, she faces a different (but
still interesting) choice problem than the one we previously considered.
In this section, a choice problem is again a pair of sets (S, Y ) where S, Y ∈ X ,
but now there is not necessarily a subset or superset relation between S and
17
ˆ
Y . We denote the set of all choice problems by C(X).
As usual, a choice
ˆ
correspondence is a map C : C(X)
→ X such that C(S, Y ) ⊆ S for any (S, Y ) ∈
ˆ
C(X).
Maintaining the condition that C(S, Y ) ⊆ S allows us to continue to
interpret S as the choice set from which the choice is made.
ˆ
Axiom 6.1 (Full Consideration) For any (S, Y ) ∈ C(X),
C(S, Y ) = C(S, S ∪ Y ).
17
ˆ
Notice that C(X)
⊆ C(X), where C(X) is the set of choice problems in which the actual
set is a subset of the potential set.
30
Consider an agent who has previously seen the elements in Y and now
chooses from the set S. The above axiom states that, holding Y fixed, she
should choose the same alternatives in S regardless of whether she had also
seen S before or not. The motivation for this behavior is the following: Any
option s ∈ S that is currently being observed is considered by the agent when
making her choice from S. Hence, her choice does not change even if s was
additionally seen in the past. This axiom is a tautology for problems in C(X)
ˆ
and only matters for problems in C(X)\C(X).
The following representation theorem is very similar to Theorem 3.1. The
difference is that the present one addresses a larger set of choice problems,
ˆ
specifically C(X),
and use one additional axiom, namely Axiom 6.1.
Theorem 6.1 C is an aspirational choice correspondence that satisfies Axioms
3.5 and 6.1 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 d(s, a(S ∪ Y )) for all (S, Y ) ∈ C(X),
s∈S
where a(Y ) is the -maximum element of Y .
According to this representation, the decision maker who is confronted with
a choice problem (S, Y ) deduces her aspiration from the set S ∪ Y , as opposed
to using only Y as in Theorem 3.1. Then, as usual, she chooses the alternative
in S that is closest to her aspiration with respect to her subjective distance
function.
Deducing the aspiration from the union of the actual and potential sets is
very much in line with TDSS example where Joan’s choice problem is such that
the actual set is not necessarily a subset of the potential one. In her story, Y
is the set of shoes she has seen in the previously visited stores and S is the set
of shoes in the store she is currently visiting. It is eminently reasonable that
31
she deduces her aspiration based on the set of all shoes she has observed, both
previously and currently.
7
Conclusion
This paper proposes an axiomatic foundation for aspiration-based choice. Instead of rationalizing choices through a utility maximization procedure, we derive a subjective distance function from choice data and represent choices by a
distance minimization procedure. Alongside the intuitive nature of this characterization, we believe this approach carries other advantages. First, it is
consistent with experimental evidence regarding phantom alternatives, which
cannot be explained by maximization of a fixed preference relation. Second, it
provides a geometrical interpretation to standard problems which may be useful
in several applications. Identifying a subjective distance function from choice
data may also be a useful approach in modeling other phenomena discussed in
the literature. For instance, it can serve as a measure for product differentiation.
8
Appendix
The following theorem is proved in Richter (2010) and we will use it in the proof
of Theorem 3.1.
Theorem 8.1 Take (X, D) to be a compact metric space and suppose φ : X 2 →
R+ is a continuous, reflexive and symmetric function. Then, there exists a
continuous metric Dφ : X 2 → R+ such that
φ(x, y) ≤ φ(z, w) if and only if Dφ (x, y) ≤ Dφ (z, w) for any x, y, z, w ∈ X
Proof See Richter (2010).
Proof of Theorem 3.1
[⇒] Suppose C is a choice correspondence that satisfies Axioms 3.1-3.5. Define
a(Z) := C(Z, Z). By Axiom 3.2, we have that there exists a total order such
32
that a(Z) = max(Z, ). By Axiom 3.4 we have is a total anti-symmetric
order. Hence, is defined as x y if {x} = C({x, y}, {x, y}). Moreover, by
Axiom 3.5, and a are continuous.
We now define the aspiration based preferences. Suppose x, y, z ∈ X. We say
that x z y if x ∈ C({x, y}, {x, y, z}) and C({x, y, z}, {x, y, z}) = {z}. We also
define ∗ as: (x, y) ∗ (z, y) if x y z. We need to show that ∗ satisfies the
conditions for Levin’s (or Nachbin’s) Theorem on the set T := {(x, y) : y x}.
First, we need to show that ∗ is closed-continuous. Take any x, y, z, xn , yn , zn ∈
X and n = 1, 2, . . . such that xn → x, yn → y, zn → z and (xn , yn ) ∗
(zn , yn ) for all n. By the definition of ∗ , we have xn yn zn , i.e. xn ∈
C({xn , zn }, {xn , yn , zn }) and {yn } = C({xn , yn , zn }, {xn , yn , zn }). By Axioms
3.4 and 3.5, we have that x ∈ C({x, z}, {x, y, z}) and {y} = C({x, y, z}, {x, y, z}).
Therefore x y z ⇒ (x, y) ∗ (z, y) and we have shown closed-continuity.
In addition, by Axioms 3.4 and 3.5, T is a closed subset of a compact space
X 2 and hence T is compact as well. Therefore, conditions of Levin’s (or Nachbin’s) Theorem hold, and we conclude that there exists a continuous and strictly
∗ -increasing u : T → R. Define
ˆ y) :=
d(x,


u(y, y) − u(x, y)
if y x,

u(x, x) − u(y, x)
otherwise
ˆ ·) represents the preference ∗ , that is,
Now, we need to show that d(·,
ˆ a(Y )) holds.For ease of notation, denote a(Y ) by a.
C(S, Y ) = argmins∈S d(s,
ˆ a(Y ))”
“C(S, Y ) ⊆ argmins∈S d(s,
ˆ a(Y )). Then, there
Take y ∈ C(S, Y ) and suppose y ∈
/ argmins∈S d(s,
ˆ a) < d(y,
ˆ a). As a is the aspiration element in Y , a exists z ∈ S s.t. d(z,
ˆ we get u(z, a) > u(y, a). So, we have z a y which
z, y. By definition of d,
implies that {z} = C({z, y}, {z, y, a}) and {a} = C({z, y, a}, {z, y, a}). Axiom
3.1 guarantees that y ∈ C({z, y}, Y ). By Axiom 3.3, we get C({z, y}, Y ) =
C({z, y}, {z, y, a}). This is a contradiction since the right-hand side is equal to
{z} but y is an element of the left-hand side.
33
ˆ a(Y )) ⊆ C(S, Y )”
“argmins∈S d(s,
ˆ a(Y )) and assume z 6∈ C(S, Y ). As C is nonConsider z ∈ argmins∈S d(s,
empty valued, there must exist y ∈ C(S, Y ). By Axioms 3.1 and 3.3, we get
{y} = C({z, y}, {a, z, y}). Definition of a gives z ≺a y. Then, we obtain
ˆ a) > d(y,
ˆ a), by definition of d,
ˆ which is a contradiction to z being the
d(z,
alternative in S that is closest to a.
Notice that dˆ is continuous, reflexive and symmetric, but need not satisfy
the triangle inequality. However, by Theorem 8.1, we know that there exists a
continuous metric d : X 2 → R+ such that
ˆ y) ≤ d(z,
ˆ w) if and only if d(x, y) ≤ d(z, w).
d(x,
Hence, we have shown that
C(S, Y ) = argmins∈S d(s, a(Y )) for all (S, Y ) ∈ C(X).
[⇐] Axioms 3.1, 3.2, 3.3, 3.4 all follow trivially.
H
To show Axiom 3.5, suppose xn ∈ C(Sn , Yn ) for all n, where Sn → S,
H
H
Yn → Y and xn → x. Since Sn → S, we know that ∀s ∈ S, there exist
sn ∈ Sn for n = 1, 2, . . . such that sn → s. By xn ∈ C(Sn , Yn ), we have
that d(xn , a(Yn )) ≤ d(sn , a(Yn )) for all n. Continuity of a(·) guarantees that
a(Yn ) → a(Y ) and taking limits of both sides gives d(x, y) ≤ d(s, y). Since s
was chosen arbitrarily, we obtain x ∈ argmins∈S d(s, y).
Proof of Theorem 4.1:
There is a unique y s.t. y − t(v) = d((0, 0), (1, t(v))). Notice that ∀z s.t.
0 ≤ z < y, we have d((1, z), (1, t(v))) < d((0, 0), (1, t(v))). Moreover, ∀z > y,
d((1, z), (1, t(v))) > d((0, 0), (1, t(v))). So, our agent prefers to win the auction
for all price smaller than y and prefers to lose the auction to winning it for a
price above y, therefore it is his dominant strategy to make the bid y. Finally,
34
notice that y := t(v) +
p
t(v)2 + 1 solves for the above unique y, hence this bid
is a weakly dominant strategy of our game.
As for uniqueness, consider if a player of type v makes a bid b̂(v) 6= b(v).
b̂(v) + b(v)
Suppose that all other types make the bid
. Then our agent is worse
2
off having made the bid b̂(v) than bidding b(v). Hence, there are no other weakly
dominant strategies.
Proof of Proposition 4.2:
Overbidding occurs when b(v) > v. Substituting, we get the equation t(v) +
p
t(v)2 + 1 > v. Notice that t(v) < v, hence the above inequality reduces to
v2 − 1
t(v)2 + 1 > (v − t(v))2 ⇔ t(v) >
.
2v
Proof of Proposition 4.1:
Let p1 be the initial price and p2 be the sales-price of good G. Let −m be
the slope of B1 . Now, define t as follows:
1
I − p2 ax − ay
t := a + λ 1,
, where λ =
1
m
p2 + m
.
One can easily verify that t ∈ B2 and cx ≤ ax if and only if bx ≤ tx
Proof of Theorem 5.1
[⇒] Define x y if x ∈ C({x, y}, {x, y}). Since every pair {x, y} is compact,
we get that is a total order. Moreover, by Axiom 5.2, the defined aspiration
choice correspondence A is upper hemi-continuous and the aspiration preference
is continuous.
As before, we will define preferences relative to a single aspiration point.
This definition will differ than the one given before because there may in fact
35
be multiple aspiration points for a given choice problem. However, in the case
of a single aspiration point, these definitions will be the same.
Definition: x z y if x ∈ C({x, y}, {x, y, z}) and z ∈ C({x, y, z}, {x, y, z})
Claim: z represents C(·, Y ) whenever {z} = A(Y ).
Proof: Let x ∈ C(S, Y ) and y ∈ S. Then, x ∈ C({x, y}, Y ) = C({x, y}, {x, y, z})
where the inclusion of x comes from Axiom 3.1 and the equality is due to Axioms
3.2 and 3.3. Since y was generically chosen, x is z -maximal. Now, consider
any w ∈ max(S, z ) and take any x ∈ C(S, Y ). By definition and our earlier
observation {x, w} = C({x, w}, {x, w, z}). Again, due to Axioms 3.2 and 3.3,
we have {x, w} = C({x, w}, Y ) and by Axiom 3.1, we have x, w ∈ C(S, Y ).
We need to show that these preferences are upper semi-continuous. Consider the case where xi z y for all i and suppose xi → x. Then, xi ∈
C({xi , y}, {xi , y, z}) and z ∈ C({xi , y, z}, {xi , y, z}). First, we notice that z ∈
C({x, y, z}, {x, y, z}) by Axiom 5.2. Now, if xi ∈ C({xi , y, z}, {xi , y, z}) happens
infinitely often, then by Axiom 5.2, we have x ∈ C({x, y, z}, {x, y, z}), which implies that x ∈ C({x, y}, {x, y, z}) by Axiom 3.1. If xi ∈
/ C({xi , y, z}, {xi , y, z})
infinitely often, then {z} = C({xi , y, z}, {xi , y, z}) infinitely often because if
y ∈ C({x, y, z}, {x, y, z}), then y ∼ z ⇒ y ∈ C({xi , y, z}, {xi , y, z}) and then
by Axiom 3.1 xi ∈ C({xi , y, z}, {xi , y, z}) for all i which is a contradiction
to xi ∈
/ C({xi , y, z}, {xi , y, z}) infinitely often. So, we can pass to a subsequence where it is only the case that {z} = C({xi , y, z}, {xi , y, z}). Finally,
∞
∞
∞
i=1
i=1
i=1
either x ∈ C( ∪ {xi } ∪ {y, z, x}, ∪ {xi } ∪ {y, z, x}) or {z} = C( ∪ {xi } ∪
∞
{y, z, x}, ∪ {xi } ∪ {y, z, x}). If the previous case holds, then we are done since
i=1
x ∈ C({x, y, z}, {x, y, z}) by Axiom 3.1 and 3.3. If the latter case holds, then
∞
by Axiom 3.3, we get xi ∈ C({xi , y}, ∪ {xi } ∪ {y, z, x}) and Axiom 5.1 im∞
i=1
plies that x ∈ C({x, y}, ∪ {xi } ∪ {y, z, x}). Then, by Axiom 3.3, we obtain
i=1
x ∈ C({x, y}, {x, y, z}).
X is compact, hence separable and therefore L (x) is separable (because
subspaces of separable spaces are separable). Upper semicontinuity of x was
36
just shown above and x is a complete preference relation. Therefore, by
Rader’s Theorem, there is a U : X 2 → R+ such that U (·, z) is upper-semi continuous on L (z) and represents z for any z ∈ X. Hence, U (x, z) ≥ U (y, z) if
and only if x z y.18 Finally, define
d(x, y) =



0







1
if x = y,
x 6= y and x ∼ y
U (x, y)


2 − 1 + U (x, y)




U (y, x)


2 −
1 + U (y, x)
x≺y
xy
The above d is symmetric, reflexive, satisfies the ∆-inequality and d(·, y) is
lower semi-continuous on L (y).
Definition: For any A ∈ X , we define
x A y if x ∈ C({x, y}, {x, y} ∪ A) and A ⊆ C({x, y} ∪ A, {x, y} ∪ A).
Claim: For any (S, Y ) ∈ C(X), C(S, Y ) = max(S, A ) where A = A(Y ).
Proof : Let x ∈ C(S, Y ) and y ∈ Y . By Axioms 3.2, 3.1 and 3.3, we get
x ∈ C({x, y}, {x, y} ∪ A). By Axiom 3.3, we have A = C({x, y} ∪ A, {x, y} ∪ A).
Hence, by definition of A , we obtain x A y. As y is an arbitrary element in Y ,
we get x ∈ max(S, A ). For the other inclusion, let x ∈ max(S, A ) and suppose
further that x ∈
/ C(S, Y ). Then there must exist y ∈ C(S, Y ). By definition
of A , we have x ∈ C({x, y}, {x, y} ∪ A) and A ⊆ C({x, y} ∪ A, {x, y} ∪ A).
Applying Axioms 3.2, and 3.3 gives x ∈ C({x, y}, Y ). By Axiom 3.2, we get
x ∈ C(S, Y ), which is a contradiction.
Notice that we are only concerned with A such that ∀a, b ∈ A, a ∼ b. For
any Y , consider the situation where A = A(Y ) and we have a sequence xi → x
18
Rader’s theorem does not guarantee non-negativity of U . If U takes negative values, we
can always consider eU instead of U , which is non-negative and order-preserving. Hence,
WLOG, we assume non-negativity of U function.
37
and a z such that ∀i, xi A z. Suppose there exists a ∈ A such that z a.
Then, it must be the case that xi z a and xi ∈ A(Y ∪ xi ∪ z). Then by
Axiom 5.2, we have x ∈ A(Y ∪ x ∪ z) and x A z. Otherwise, consider the
case where xi a z infinitely often. Then, again by Axiom 5.2, we have that
x a and x A z. Finally, suppose that A(Y ∪ {xi , z}) = A infinitely often.
Then, by Axiom 5.2, we have that A(Y ∪ x) = A or A ∪ x. In the first case,
we can apply Axiom 5.1 and in the second, by virtue of x being an aspiration
element, we have x A z.
Now, for any set A of this type, A is upper semi-continuous and satisfies the
other conditions for Rader’s Representation Theorem on L (A) by analogous
arguments to those in the previous paragraph where A = {z}. Also, by our last
claim, we have uA representing C(·, Y ) whenever A(Y ) = A.
Hence, Rader’s Representation Theorem yields there exists uA : L (A) →
R+ an upper semicontinuous function representing A .19
Suppose that A = C(Y, Y ) for some Y ∈ X and x, y ∈ Y . Define
φ̂((x, A)) =




d(x, y)






d(x, y)


1






5 −

|A| = {y}
|A| = 2, x, y ∈ A, x 6= y
x ∈ A, |A| > 2
1
1 + uA (a) − uA (x)
x∈
/ A, a ∈ A and |A| =
6 1
Notice that uA (a) is constant across all a ∈ A, so the fourth case above is
well-defined. It can be checked that the above function is lower semi-continuous
due to the upper semicontinuity of uA and the fact that 0 ≤ d ≤ 2 < 4 ≤
1
.
5−
1 + uA (a) − uA (x)
Now, we have only defined φ̂ for certain tuples (x, A). This is because only
certain choice problems arise. More formally we make the following definition.
Definition: Let CP = {(x, A) : x ∈ Y for some Y ∈ X such that A(Y ) = A}
19
For the non-negativity of uA , please refer to footnote 18.
38
݀Ԧ
ƒ
W
y
෡ )
)
ƒ
Figure 6: Commuting Graph
~ A) =
Claim: For any (x, A), (y, B) ∈ CP , where A = A(Y ), B = A(Z), if d(x,
~ B), then φ̂(x, A) = φ̂(y, B).
d(y,
~ A) = ~0 = d(y,
~ B) then φ̂(x, A) = φ̂(y, B) = 0. Next, we
Proof : First, if d(x,
show that A = B. Suppose not. Then, WLOG, ∃z ∈ B\A. Since,
0,
1z∈A =
1z∈B = 1, it must be the case that y = z. So, there can be at most one
element in B\A. If B contains only one element, then we are in the previous
case, so, let’s take another element b ∈ B ∩ A. Now, y = z ∼ b ⇒ d(y, b) = 1.
Therefore, it must be the case that d(x, b) = 1. Thus x ∼ b. But, then 0 =
d(x, x) = d(y, x) when we consider the (now known to be aspirational) element
x which implies that y = x. But, now we have a contradiction because z ∈ B\A
and z = x ∈ A.
If x = a for some a ∈ A, then 0 = d(x, a) = d(y, a) ⇒ x = y. Otherwise,
~ A) = d(y,
~ A), means d(x, a) = d(y, a) for any a ∈ A. This
x, y ≺ a and d(x,
means that A({x, y, a}) = a and {x, y} = C({x, y}, {x, y, a}). If |A| = 1, then
φ̂(x, A) = d(x, a) = d(y, a) = φ̂(x, A) for A = {a}. Otherwise, by Axiom 5.3,
{x, y} = C({x, y}, A ∪ {x, y}) ⇒ x ∼A y ⇒ uA (x) = uA (y) and the last case of
φ̂ applies, so φ̂(x, A) = φ̂(y, A). 39
So, we now know that φ̂ : CP → R and d~ : CP → RX and the equivalence
relations defined by the inverse image of d~ is a refinement of φ̂. So, by a standard
argument, there exists a φ : RX → R that makes the diagram above commute.
~ A)) = φ̂(x, A). The only two cases in which a vector can
Therefore φ(d(x,
arise with at most one non-zero entry is if |A| ≤ 2. In addition, if |A| = 2,
then x must be one of the two aspiration points. Either way, φ̂ and hence φ
are defined by one of the first two cases. Both of these cases assign the unique
non-zero distance in the relevant vector (including the case where the vector is
~0). Therefore, φ has the “single-agreement” property.
~ A(Y ))).
For representability, we must show that C(S, Y ) = argmin φ(d(s,
s∈S
First, let us note that φ ◦ d~ is lower semi-continuous, S is compact, and
S ⊆ L (A(Y )). Hence the above argmin will exist.20 For notational ease, let
A = A(Y ).
~ A(Y )))”
“C(S, Y ) ⊆ argmin φ(d(s,
s∈S
~ A)) < φ(d(y,
~ A)). We consider
Take y ∈ C(S, Y ), suppose ∃z ∈ S s.t. φ(d(z,
the following cases:
1. A = {x}. Then the above reduces to d(z, x) < d(y, x). But, then y ∈
/
C(S, Y ).
2. A = {z, y}. Then the above becomes d(z, y) < d(y, z)
3. A = {w, y}, w 6= z. Then the above becomes 4 < 5−
1
<
1 + uA (y) − uA (z)
d(w, y) < 2
4. z ∈ A, y ∈
/ A, then y ∈
/ C(S, Y )
5. z, y ∈ A, |A| > 2, then 1 < 1
~ A)) < 1
6. z ∈
/ A, y ∈ A, |A| > 2. Then the above becomes φ(d(z,
7. y, z ∈
/ A, |A| > 2. Then the above becomes
1
1
5−
<5−
⇒ uA (y) < uA (z)
1 + uA (a) − uA (z)
1 + uA (a) − uA (y)
20
By definition, recall A(Y ) = argmax(Y, ) and therefore L (Y ) ⊇ Y ⊇ S.
40
~ A(Y ))) ⊆ C(S, Y )”
“argmins∈S φ(d(s,
~ A(Y ))) and assume z 6∈ C(S, Y ), y ∈ C(S, Y ).
Consider z ∈ argmins∈S φ(d(s,
We consider the following cases:
1. Suppose z ≺ a ∈ A, |A| > 1. Then, it must be y ≺ a ∈ A and
1
1
≤ 5−
⇒ uA (y) ≤ uA (z) and
5−
1 + uA (a) − uA (z)
1 + uA (a) − uA (y)
therefore y ∈ C(S, Y ) = max(S, A ) ⇒ z ∈ max(S, A ) = C(S, Y )
2. Suppose z ∈ A, then by Axioms 3.2 and 3.3, z ∈ C(S, Y )
3. Suppose z ∈
/ A, |A| = {a}. Then we have d(z, a) ≤ d(y, a) ⇒ z a y and
since y was chosen generically, we have z ∈ max(S, a ) ⇒ z ∈ C(S, Y )
[⇐] Axioms 3.1, 3.2, 3.3, 5.3 all follow trivially. Axiom 5.2 follows from the
continuity of the aspiration preference .
To show Axiom 5.1, let us consider the situation xn ∈ C(S ∪ {xn }, Y ) and
~ n , A(Y ))) ≤ φ(d(s,
~ A(Y ))). By
xn , x ∈ Y and xn → x, we have ∀s ∈ S, φ(d(x
~ A(Y ))) we have φ(d(x,
~ A(Y ))) ≤ lim inf φ(d(x
~ n , A(Y ))) ≤
lower semicontinuity of φ(d(·,
n→∞
~ A(Y ))). Therefore x ∈ C(S ∪ {x}, Y ).
φ(d(s,
Now, we consider the case of monotonicity.
“⇐” If we have x ∈ C({x, z}, {x, y, z}), then we either have x ∈ A({x, y, z})∀y ∈
A(Y, Y ) ⇒ x ∈ C({x, z}, Y ) or y = A{x, y, z}∀y ∈ C(Y, Y ). In this case, we
~ A) ≤ d(z,
~ A) ⇒ φ(d(x,
~ A)) ≤
have d(x, y) ≤ d(z, y)∀y ∈ A(Y, Y ) and hence d(x,
~ A)) ⇒ x ∈ C({x, z}, Y ). The strict inequality follows likewise.
φ(d(x,
~ A) ≤ d(z,
~ A). Fix a ∈ A.
“⇒” Take A = A(Y ). Suppose we have vectors d(x,
Then, it must be the case d(x, a) ≤ d(z, a) which implies x ∈ C({x, z}, {x, a, z}).
Since this holds for all a ∈ A, by Axiom 5.4, we have x ∈ C({x, z}, Y ). Therefore
~ A)) ≤ φ(d(z,
~ A)).
φ(d(x,
Proof of Theorem 6.1
[⇒] As the same axioms apply to choice problems in C(X), we can apply Theorem 3.1 when restricting our attention to those problems. We derive d as before
41
from those problems and need to show that this representation is valid for our
new problems as well.
ˆ
Consider a choice problem (S, Y ) ∈ C(X)\C(X).
In other words, S 6⊆ Y .
By Axiom 6.1 C(S, Y ) = C(S, S ∪ Y ) and we know that the right-hand side is
equal to argmin d(s, A(S ∪ Y )) since it is a choice problem in C(X).
s∈S
[⇐] Axioms 3.1 - 3.5 are true because, for any (S, Y ) ∈ C(X), our agent behaves
the same as our main model, Theorem 3.1. To show Axiom 6.1 C(S, Y ) =
argmin d(s, A(S ∪ Y )) = C(S, S ∪ Y ).
s∈S
42
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