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A Theory of Reference Point Formation

¨Ozg¨ur Kıbrıs ∗

Yusufcan Masatlioglu †

Elchin Suleymanov ‡

October 21, 2018

Abstract

We introduce a model of reference-dependent choice where the reference point is en-

dogenously determined through maximization of a conspicuity ranking. This subjective

ranking captures how eye-catching the alternatives are in relation to each other. The

most conspicuous alternative in a choice set serves as its reference point and in turn,

determines the reference-dependent utility the decision-maker will maximize to make a

choice. We show that this conspicuity based endogenous reference model (CER) is char-

acterized by an intuitive and simple behavioral postulate, called Single Reversal, and we

discuss how choice data can be used to reveal information about CER’s parameters. We

additionally analyze special cases where a reference-free utility function, combined with

psychological constraints, is used to make reference-dependent choices.

Keywords: Conspicuity, Reference Point Formation, Reference Dependence, Psycho-

logical Constraints, Revealed Preference, Choice Reversal

JEL Codes: D03, D83

∗Faculty of Arts and Social Sciences, Sabancı University; e-mail: ozgur@sabanciuniv.edu.
†Department of Economics, University of Maryland; e-mail: yusufcan@umd.edu.
‡Department of Economics, University of Michigan; e-mail: elchin@umich.edu.

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1

Introduction

Starting with the seminal works of Markowitz (1952) and Kahneman and Tversky (1979), the

idea of reference-dependence has played a very significant role in economics. Numerous em-

pirical and experimental studies have documented that choices are often reference dependent

With this motivation, researchers have developed a variety of theoretical models in which an

exogenously given reference point affects choice behavior. However, with the exception of

a few studies, this literature remains silent on how the reference point is determined. This

has been recognized as a major drawback (e.g. see Markowitz (1952), Tversky and Kahne-

man (1991), Levy (1992), Wakker (2010), Barberis (2013)). For example, Wakker (2010, p.

245) argues that “If too much liberty is left concerning the choice of reference points, then

the theory becomes too general and is almost impossible to refute empirically. It does not

then yield valuable predictions.” In other words, a full-blown theory of reference-dependence

necessitates a theory of reference point formation.

Many studies informally relate determination of a reference point to some notion of con-

spicuity (or equivalently, salience) and argue that in a choice set the “most conspicuous alter-

native” becomes the reference point (Brickman, Coates, and Janoff-Bulman (1978), Samuel-

son and Zeckhauser (1988), Pratkanis (2007), DellaVigna (2009), Larrick and Wu (2012),

Bhatia and Golman (2015), Bhatia (2017)). To quote Bhatia and Golman (2015), “reference

points are merely options that are especially salient to the decision maker.” For example,

when purchasing an airline ticket, most consumers sort alternatives according a criterion im-

portant to them (say, price), and then use the top of that list (such as the cheapest ticket)

as a reference point when evaluating others.1 Similarly, in online platforms like Amazon,

the best reviewed or the most purchased alternative might serve as a reference point. Our

main objective is to formalize this intuition to offer a theory of endogenous reference point

formation, and analyze its behavioral implications.

In our model, alternatives are ranked according to how conspicuous (equivalently, salient)

they are in relation to others and the most conspicuous alternative serves as the reference

point. The conspicuity ranking captures how eye-catching the alternatives are in relation

to each other.2 In the above example, a cheaper product is more conspicuous for a price-

1Marketing literature establishes price to be a particularly important criterion for conspicuity (e.g. see

Winer (1986), Kalyanaram and Winer (1995), Erdem, Mayhew, and Sun (2001)).

2Empirical findings suggest that conspicuous alternatives are more likely to attract attention and affect
decision-making (e.g. see Lohse (1997), Milosavljevic, Navalpakkam, Koch, and Rangel (2012), Navalpakkam,
Kumar, Li, and Sivakumar (2012)).

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conscious customer. However, in general, conspicuity of a product might depend on features

potentially irrelevant for its valuation such as the size and/or color of its package (e.g. see

Milosavljevic, Navalpakkam, Koch, and Rangel (2012)). Furthermore, individuals can differ

in their criteria for conspicuity, and typically, what a person finds conspicous is not directly

observable from outside.3 This (subjective) conspicuity ranking is the first component of our

model.4

The second component of our model is a set of reference-dependent utility functions

ρ∈X . We do not impose any particular functional form on them. Generality in the

Uρ
{
utility component allows our model to encompass a wide range of reference-dependent utility

}

functions used in the literature, including those of Tversky and Kahneman (1991), Munro

and Sugden (2003), Masatlioglu and Ok (2005, 2014), Sagi (2006), K˝oszegi and Rabin (2006),

and Bordalo, Gennaioli, and Shleifer (2013). This enables us to compare our model to the

existing literature, as discussed below.

We are now ready to define the choice procedure of our agent. Given a choice problem

S, the most conspicuous alternative according to the conspicuity ranking, denoted by r(S),

serves as the reference point. Next, the reference point r(S) induces a utility function Ur(S).

The agent finalizes her choice by maximizing this utility function Ur(S) on S. This model,

summarized in Figure 1, is called the Conspicuity based Endogenous Reference model (here-

after, CER). In what follows, we study the basic properties of CER as well as its economic

implications, and discuss to what extent its ingredients can be inferred from choice data.

Figure 1: Conspicuity based Endogenous Reference model (CER)

The first contribution of our paper is the concept of a conspicuity ranking through which

3Subjectivity of the conspicuity ranking is in line with recent evidence which suggests that different indi-
viduals facing similar decision environments might end up with distinct reference points. For example, Terzi,
Koedijk, Noussair, and Pownall (2016) present experimental data in which there is heterogeneity among
individuals in the reference points that they employ.

4A related yet different notion is discussed in Bordalo, Gennaioli, and Shleifer (2013). In their framework,
each product has different attributes and depending on the context and the reference point, one of the attributes
becomes salient and receives a higher weight on the final evaluation. Thus, in their model, it is the reference
point that determines salience (and of an attribute). Conversely, in our model, conspicuity determines the
reference point. Hence, the two approaches are conceptually different.

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ChoiceConsideration             Set Choice    Set ChoiceChoice Set Reference Point Reference- Dependent UtilityReference-Dependent Utility MaximizationChoiceChoice Set Most Salient AlternativeReference-Dependent Utility Maximization reference point formation is endogenized. To highlight the significance of this innovation, con-

sider the constant loss aversion model of Tversky and Kahneman (1991). Due to its tractable

form, this highly celebrated reference-dependent model is widely used in applications. Yet

it has also been criticized on the basis that it cannot accommodate well-known behavioral

patterns such as the attraction and compromise effects. However, if the reference point is en-

dogenously formed through a conspicuity ranking, as in our model, the constant loss aversion

model can accommodate both. Figure 1 (left) presents an example with three alternatives,

where the conspicuity ranking is A

(cid:29)
λ = 2, as commonly used in the literature.

(cid:29)

B

C and the constant loss aversion parameter is

In the figure, B is chosen over C when only

these two alternatives are available. However, the choice switches from B to C when (and

only when) a third alternative A is added to the colored regions. Note that these areas are

Figure 2: Attraction and compromise effects when conspicuity based reference-point forma-
tion is combined with two standard reference-dependent utility functions. The conspicuity
ranking is A

C and the loss aversion parameter is λ = 2.

B

(cid:29)

(cid:29)

predominantly consistent with the underlying motivation of the attraction and compromise

effects, as detailed in Section 4. Particularly, a choice reversal does not occur when A is

added to a region dominated by B, or to a region that turns B into a “compromise”.

We should also point out that this improvement is a result of the conspicuity ranking

and not just endogenization of the reference point. For example, if we utilize the reference-

dependent utilities introduced by K˝oszegi and Rabin (2006) as our underlying reference-

dependent model, conspicuity based reference-point formation is capable of explaining the

compromise and attraction effect (see Figure 1 (right panel)). On the other hand, if reference-

point formation was based on their own preferred personal equilibrium concept, the implied

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Attribute1Attribute20911.332025.511132024.5CBCompromiseEffectAttractionEffectTverskyandKahneman(1991):V(x,ρ)=u(x1,ρ1)+u(x2,ρ2)whereu(xi,ρi)=(xi−ρiifxi≥ρiλ(xi−ρi)ifxi<ρiAttribute1Attribute209112027.6611152026CBCompromiseEffectAttractionEffectK˝oszegiandRabin(2006):V(x,ρ)=x1+x2+u(x1,ρ1)+u(x2,ρ2)whereu(xi,ρi)=(xi−ρiifxi≥ρiλ(xi−ρi)ifxi<ρi behavior would be identical to the classical model (see Proposition 3 in K˝oszegi and Rabin (2006)). This highlights the importance of reference-point formation for a given underlying reference-dependent choice. Overall, choice of the reference-point formation process has the potential to significantly improve the performance of existing models by allowing them to accommodate additional (seemingly anomalous) choice patterns. Our second contribution is that we allow the conspicuity ranking to be subjective (that is, to depend on the decision maker’s individual characteristics) and we show how this subjective ranking can be inferred from choice data. This inference relies on an important feature of our model concerning choice reversals. A choice reversal is said to occur when the elimination of an unchosen alternative affects the choice.5 In our model such reversals can only be induced by the elimination of the most conspicuous alternative in a choice set. This feature allows us to infer the conspicuity ranking from observed choices. To better understand CER, we explore its behavioral implications. It turns out that one intuitive and simple behavioral postulate, that we call the Single Reversal Axiom, fully characterizes CER. This axiom is motivated by the aforementioned observation on how CER regulates choice reversals. The Single Reversal Axiom requires that if there is a choice problem where an alternative x causes choice reversal when y is available, there cannot be a choice problem where y causes choice reversal when x is available.6 This implies that for a given choice problem, we can observe at most one choice reversal. Since WARP does not allow any choice reversals, the Single Reversal Axiom can be thought of as the minimal deviation from it. Overall, CER enjoys an intuitive and simple axiomatic foundation that provides a clear picture of what type of choice behavior CER can address, and which enables the design of simple experiments to test its validity. Figure 3: Psychologically Constrained CER Model (PC-CER) One important criticism of reference-dependent choice models is that each reference point 5Formally, the statement “x induces a choice reversal in S” can be stated as x (cid:54)= c(S) (cid:54)= c(S x) where c(S) is the choice from S. 6When stated in terms of “revealed conspicuity,” Single Reversal means that if x is revealed to be more conspicuous than y, then y cannot be revealed to be more conspicuous than x. 5 ChoiceConsideration             Sets Choice    Set ChoiceChoice Set Most Salient AlternativeConstrained Utility Maximization induces a new utility function as if there is a new self and that, this makes welfare comparisons across different reference points problematic. To deal with this criticism, Masatlioglu and Ok (2014) proposes a model where there is a single utility function applied under all reference points, but each reference point in turn induces a “psychological constraint” which eliminates certain alternatives. In Section 6, we analyze a special case of CER where choices are made by the procedure of Masatlioglu and Ok (2014). This special case, called the Psychologically- Constrained Conspicuity based Endogenous Reference model (PC-CER), is summarized in Figure 3 (where U is the reference-free utility and Q(r(S)) is the psychological constraint imposed by the reference point r(S)). We analyze the behavioral implications of PC-CER as well. It turns out that a Consis- tency Axiom, together with Single Reversal, fully characterizes PC-CER. Consistency simply states that the revealed (reference-free) preference of this model has no cycles. This charac- terization also helps us to compare our study with the previous literature, including Manzini and Mariotti (2007), Masatlioglu, Nakajima, and Ozbay (2012), Ok, Ortoleva, and Riella (2015), Masatlioglu, Nakajima, and Ozdenoren (2017). To place our paper in the literature, we provide a brief discussion of existing reference- dependent models, a classification of which is presented in Figure 4.7 Columns in Figure 4 follow an approximate historical order to classify existing models in terms of how they treat reference-point formation. Rows, on the other hand, classify models in terms of how choice is made once the reference point is determined. Figure 4: Reference-Dependent Models Studies represented in the first row of Figure 4 all employ reference-dependent utility functions, but differ in terms of reference-point formation. The earliest strand of literature on this specification treats the reference point as exogenous (e.g. Tversky and Kahneman 7For a more detailed discussion, see Section 7. 6 (1991), Munro and Sugden (2003), Sugden (2003), Sagi (2006), Salant and Rubinstein (2008)). Later studies (columns 2 to 5) endogenize reference-point formation. In models of Bodner and Prelec (1994), Kivetz, Netzer, and Srinivasan (2004), Orhun (2009), Bordalo, Gennaioli, and Shleifer (2013), and Tserenjigmid (2015) (column 2), the reference point depends on the structure of the choice set, but is independent of individual characteristics. Thus, these models analyze environments where all decision makers facing the same choice problem nec- essarily have the same reference point. Alternatively, K˝oszegi and Rabin (2006) and Freeman (2017) (see column 3) analyze models where the endogenous reference point can depend on individual characteristics. But in these models, the final choice always coincides with the reference point. CER is related to these earlier studies since it endogenizes reference-point formation, allows it to depend on individual characteristics, and does not restrict choice to coincide with the reference point. The second row of Figure 4 represents another strand of literature that replaces refer- ence dependent utilities with a reference-free utility function combined with psychological constraints. This special case of reference-dependent choice is important for welfare compar- isons, as mentioned earlier (and further discussed in Section 6). Most studies in this strand of the literature treat the reference point as exogenous (e.g. Masatlioglu and Ok (2005, 2014), Apesteguia and Ballester (2009), Masatlioglu and Nakajima (2013), Dean, Kıbrıs, and Masatlioglu (2017)).8 One exception is Ok, Ortoleva, and Riella (2015) where the reference point is determined endogenously but it is required to be distinct from actual choice. PC- CER is closely related to this strand of literature: it endogenizes reference-point formation, employs a reference-free utility and psychological constraints, and does not restrict choice to be distinct from the reference point. As will be discussed in Section 4, our model is consistent with three well-known behavioral patterns frequently observed in empirical studies, namely, Compromise Effect, Attraction Effect, and Cyclical Choice. None of the other studies listed above can accommodate all three. Our study is also unique in the sense that it characterizes the distinction between the two types of models represented in rows 1 and 2 of Figure 4. To elaborate, a comparison of theorems 1 and 2 shows that, (in the confines of our framework) this distinction can be 8Maltz (2017) presents a hybrid model which combines an exogenous reference point (the endowment) with endogenous reference-point formation. In this model, alternatives are partitioned into categories and, given the endowment, the most-preferred feasible alternative in its category serves as the reference point. As far as we know, this is the only model that combines an exogenous reference point with endogenous reference-point formation. 7 characterized by the Consistency axiom, which is closely related to the revealed (reference- free) preference of PC-CER. The paper is organized as follows. In Section 2, we present and discuss our model. In Sec- tion 3, we introduce the Single Reversal Axiom and show that it characterizes CER. In Section 4, we discuss three important behavioral patterns. In Section 5, we show how information is revealed from choice data consistent with CER. In Section 6, we discuss a special case of CER where psychological constraints introduce extra structure on the reference-dependent utility functions. In Section 7, we discuss the related literature. We conclude in Section 8. The Appendix contains all the proofs. 2 Conspicuity and Endogenous Reference Dependence Let X denote a finite set of alternatives and let be the set of all nonempty subsets of X. A choice problem is a set of alternatives S from which the decision maker needs to X X → ∈ X X maps every choice problem S to an ∈ X make a choice. A choice function c : alternative c(S) S.9 We assume that c represents data on the choice behavior of a decision maker (hereafter, DM). ∈ Our model has two components: (i) a family = Uρ ρ∈X of (reference-dependent) } { U utility functions, each associated with a potential reference point, and (ii) a conspicuity ranking In our interpretation, reflects the DM’s perception of how prominent or (cid:29) eye-catching the alternatives are in relation to each other. We assume that is a strict . (cid:29) linear order.10 We theorize that the reference point in a choice set is the most conspicuous alternative in it. Formally, given , the endogenous reference function r : X maps (cid:29) X → (cid:29) each choice set S to the endogenous reference point r(S) S, defined as ∈ r(S) = argmax( , S). (cid:29) Given the reference point r(S) for a choice problems S, the DM uses the induced reference- R to evaluate alternatives in S. The maximizer of dependent utility function, Ur(S) : X Ur(S) in S is the chosen alternative.11 This process is formally stated in the next definition. → 9While we work with choice functions, our results can be extended to choice correspondences, that is, to environments where the decision maker chooses more than one alternative. An extension of our main theorem that allows choice correspondences is available upon request. 10A binary relation R on X is a strict linear order if it is weakly connected, irreflexive and transitive. 11Without loss of generality, we can impose a widely accepted property from the reference dependence 8