Tohoku University Research Center for Policy Design Discussion Paper

TUPD-2024-005

Norms and Emotions

Hiroki Saruya
Economic and Social Research Institute, Cabinet Office, Government of Japan

Masayuki Yagasaki
Graduate School of Economics and Management, Tohoku University

June 2024

TUPD Discussion Papers can be downloaded from:https://www2.econ.tohoku.ac.jp/~PDesign/dp.html

Discussion Papers are a series of manuscripts in their draft form and are circulated for discussion
and comment purposes. Therefore, Discussion Papers cannot be reproduced or distributed without
the written consent of the authors.

Norms and Emotions∗
Hiroki Saruya† and Masayuki Yagasaki‡
June 24, 2024

Click here for the latest version
Abstract
Social norms are an important determinant of behavior, but the
behavioral and welfare effects of norms are not well understood. We
propose and axiomatize a decision-theoretic model in which a reference
point is formed by the decision maker’s perceptions of which actions
are admired (prescriptive norms) and which are prevalent (descriptive
norms), and utility depends on the pride of exceeding the reference
point or the shame of falling below it. The model is simple, yet provides a unified explanation for previous empirical findings, and is usefulfor welfare analysis of norm-evoking policies with a revealed preference
approach.
Keywords: norms, reference dependence, pride, shame, public recognition, norm nudge
JEL classification: D80, D81, D90, D91∗

This paper substantially extends Yagasaki’s previously circulated paper “Pride, Shameand Social Comparisons.” We thank Leonardo Bursztyn, Takaaki Hamada, Yosuke
Hashidate, Hidehiko Ichimura, Ryota Iijima, Michihiro Kandori, Akihiko Matsui, Ran
Spiegler, and Norio Takeoka for their invaluable feedback. Any remaining errors are thesole responsibility of the authors.
†
Economic and Social Research Institute, Cabinet Office, Government of Japan, hiroki.
saruya@aya.yale.edu‡
Graduate School of Economics and Management, Tohoku University, yagamasajp@
gmail.com

1

Introduction

Social norms are receiving increasing attention as a key determinant of behavior in various contexts. Norms can take effect through simple interventions
such as making decisions or outcomes publicly observable1 or providing social
information.2 As a result, policymakers have become increasingly interested
in social norms as a cost-effective policy lever to induce behavioral change.
Despite the growing interest, the behavioral and welfare effects of suchpolicies are not well understood. Norm-evoking policies may produce desired
behavioral outcomes in some cases, but they may fail to do so or even backfire
in others.3 Investigations of the behavioral effects of policies are hindered by
the lack of theoretical foundations on what types of payoffs or constraints are
generated by norms and how they are revealed from choice data. This gap
also makes it unclear how revealed preferences are useful for welfare analysis
in norm-conscious decision-making.
This paper presents a novel decision-theoretic model to describe the behavior of an individual who is concerned with social norms. We consider
a two-stage choice problem (Gul and Pesendorfer 2001; Noor and Takeoka2015) adapted to decisions under social image concerns (e.g. Dillenberger andSadowski 2012; Saito 2015; Evren and Minardi 2017; Hashidate 2021). The
decision maker (hereafter DM) first privately chooses a menu (i.e., choice set)
and then publicly chooses an alternative from the menu. This setting naturally expresses the behavioral effect of norms by the discrepancy between
1

Researchers have studied the effects of publicity, for example, on educational investment(Bursztyn and Jensen 2015), career choice (Bursztyn et al. 2017), tax compliance (PerezTruglia and Troiano 2018), charitable giving (Butera et al. 2022, and see also DellaVigna et
al. 2012), blood donations (Lacetera and Macis 2010), childhood vaccination (Karing 2024),
and voting (Gerber et al. 2008). See Bursztyn and Jensen (2017) for a review of empiricalframework and applications.
2
Providing information about other individuals’ behavior or normative opinions affects
decisions on charitable donation (Frey and Meier 2004), tax compliance (Frey and Torgler
2007; Hallsworth et al. 2017), energy conservation (Schultz et al. 2007; Allcott 2011; Allcott
and Rogers 2014), and female labor participation (Bursztyn et al. 2020).
3
Publicity of decisions may increase or decrease target behavior (Bursztyn and Jensen2015). Providing information about the behavior of others behavior may lead to the avoidance of a choice opportunity (Klinowski 2021) or an undesirable choice (Schultz et al. 2007).

1

preferences in the private (norm-free) and public (norm-conscious) stages, andis also suitable for studying the avoidance of choice opportunities (e.g., Danaet al. 2006) or the welfare effects of norms.4 We axiomatize a utility representation called a pride-shame representation, in which utility depends on an
endogenously derived reference point (cf. Ok et al. 2015; Lleras et al. 2019;
Kıbrıs et al. 2023). The reference point expresses what the DM perceives as“normal” behavior in society.
A key feature of our model is that the reference point is determined by aninteraction of two types of subjective norms, referred to as descriptive norms
and prescriptive norms. Economists typically emphasize descriptive norms,
which express the DM’s perception of what behavior is prevalent or common,i.e., what others choose to do. In contrast, social psychologists also emphasizeprescriptive norms (e.g., Cialdini et al. 1991; Bicchieri 2005; Bicchieri and Dimant 2022), which express the DM’s perception of what behavior is approved ofor admired, i.e., what others think one should do.5 Although economists havestudied prescriptive norms (e.g., Akerlof and Kranton 2000), they have notextensively studied how the two notions of norm interact. We show, througha simple application to prosocial behavior, that interactions between thesenorms can explain a variety of previously documented behavioral patterns.
Crucially, the two types of norms are the subjective beliefs of the DM and areallowed to be biased (Miller and Prentice 1994; Bursztyn et al. 2020).
An essential determinant of behavior is social emotions, such as pride andshame, which arise from comparing one’s own behavior with the typical behavior of others as reference behavior. To illustrate, consider a DM who expects
a donation solicitor to arrive at her home shortly (DellaVigna et al. 2012).
The DM’s satisfaction with donating an amount, say $10, depends on how sheperceives the behavior of others. If she believes that her neighbors donate $0,
then she gains a positive sense of pride from the $10 donation because her
behavior is perceived as normatively superior to that of her neighbors. The
4

The two-stage framework is also useful for studying the welfare effects of a product with
consumption externalities (e.g., Bursztyn et al. 2023).
5
Prescriptive norms are also known as injunctive norms. The terminology and the relationship of our work to the social psychology literature are discussed in Section 5.1.

2

degree of pride depends on the perceived desirability of each action: if donating $10 is considered normatively more desirable than donating $0 by a small(large) amount, then the payoff gain from pride is small (large). In contrast,
if she believes that her neighbors donate $100, she suffers a negative sense of
shame from donating $10 because her behavior is considered normatively inferior. The payoff loss from shame, in turn, depends on the perceived approvalof each action. As this example illustrates, descriptive norms determine which
behavior the DM focuses on as a reference behavior (donating $0 or $100) tocompare her own choice (donating $10) to, and prescriptive norms determine
the payoff from the comparison. The norms then affect the DM’s behavior.
For example, suppose she initially plans to donate $10, but then thinks that
her neighbors are donating $100. If a solicitor is already at her door, she may
increase her planned donation to avoid shame. Alternatively, if the solicitor
has not yet arrived, she may leave the house, thereby avoiding the opportunity
to donate.
Using a simple example of prosocial behavior, we illustrate that our model
provides useful insights for understanding empirical findings documented byprevious studies. First, our model clarifies how the choice of an action depends
on descriptive and prescriptive norms, and when policies such as providing
social information or publicity may be ineffective. For example, if information about others’ behavior (normative opinions) mainly affects the descriptive(prescriptive) norm of the DM, then changing this norm is the main mechanism behind the effect of providing information. The effectiveness of the policy
then depends on how sensitive the perceived norms are to the policy and how
the DM evaluates the resulting pride or shame. Similarly, the effectivenessof public observability depends on how private preferences (preferences oversingleton menus) and public preferences (choices from menus) differ.
Second, the two-stage modeling allows us to study choice avoidance and thewelfare implications of policies directly. For example, if a DM strictly prefers
one menu {$0} over another menu {$0, $10}, this suggests that she is avoiding
an opportunity to donate $10, and a negative welfare effect of publicly making

3a choice.6 In addition, our model illustrates how policies to influence perceived
norms exert differential impacts on the participation in a donation opportunityand on the donation decision conditional on participation. For example, it can
explain the laboratory findings of Klinowski (2021) that informing individuals
about others’ high level of donation after participation increases the amountdonated, but doing so before participation discourages participation.
Third, our model can explain other empirical findings that are not necessarily emphasized in economics. For example, it can rationalize previous
findings that providing information about descriptive or prescriptive norms
is more effective at inducing prosocial behavior when they are aligned thanwhen they are misaligned (Cialdini 2003), and that the descriptive norm hasa greater influence in the latter case (Bicchieri and Xiao 2009; Allcott 2011;
Hallsworth et al. 2017). An individual is more likely to make a charitable donation when others say donations should be done and they do donate, than whenothers say that donations should be done but they do not donate. Intuitively,
when both norms point to prosocial behavior, failure to follow them generates shame. In contrast, if the prescriptive norm points to prosocial behavior
but the descriptive norm points to the opposite, acting prosocially generates
pride. If avoiding shame is a stronger motivator than seeking pride, whichis empirically supported (DellaVigna et al. 2017; Butera et al. 2022), thenaligned norms are more likely to induce prosocial behavior. Our model canalso explain why providing information about the level of prosocial behaviorof others can unintentionally reduce prosocial behavior (Schultz et al. 2007),
illustrating the importance of considering individuals’ perceived norms whenintroducing a policy.
The first step toward deriving the PS representation is to characterize theDM’s subjective reference. Our approach is similar in spirit to that of Masatlioglu et al. (2012) and Kıbrıs et al. (2023), who elicit the DM’s consideration
and reference, respectively, by observing a “choice reversal,” whereby removing
6

Without the menu-selection stage, we might draw a false welfare conclusion: e.g., if
we only observe that the DM chooses a $10 donation from the menu {$0, $10}, we might
mistakenly infer that adding the option to donate $10 is beneficial.

4

an unchosen alternative from a menu affects the choice from the menu. Insteadof requiring a choice reversal, we exploit observations such that removing an
unchosen alternative affects the preference over menus. Suppose we observethat the DM donates $10 whether or not she has the option to decline donation (C({$0, $10}) = C({$10}) = {$10}), but that she strictly prefers todonate with the option to decline ({$0, $10}  {$10}). This suggests that theoption to decline donation improves the DM’s utility from donating by generating pride, which then implies that $0 is the reference choice at the menu{$0, $10}. We generalize this observation to elicit a subjective reference set,i.e., the set of reference alternatives, at each menu.
The second key step is to characterize shame and pride by describing how
preferences for smaller or larger menus emerge depending on the reference set.
Consider first a DM who perceives that her neighbors do not donate. Then,answering the door to meet a solicitor will never hurt ({$0, $10}  {$0}), because she can decline the donation without shame, or she can even feel pride
by choosing to donate. Lemma 2(i), derived from our axioms, formalizes thisidea: the DM will exhibit a preference for larger menus (cf. Evren and Minardi 2017) when the additional alternatives do not enter the reference set.
Next, consider a DM who privately does not want to donate ({$0}  {$10}).
Suppose she notices that some neighbors are donating $10, so $10 enters herreference set when she chooses between donating $10 and not donating. Thenthe option to donate $10 will not improve the DM’s feelings about not donating, because of the shame of falling below her neighbors’ standard. She
then prefers to avoid the donation option ({$0}  {$0, $10}). Lemma 2(ii)characterizes such a preference: the DM will exhibit a preference for smaller
menus (cf. Gul and Pesendorfer 2001; Dillenberger and Sadowski 2012) whenthe extra alternatives enter the reference set.7 Technically, this property relaxes the set-betweenness axiom of Gul and Pesendorfer (2001). These two
properties are crucial for our main result.
Our contribution is to propose a simple, tractable, and axiomatically founded7

To match this example to Lemma 2(ii) precisely, both $0 and $10 donations must enter
the reference set. We leave a formal discussion to Section 3.

5

model of norm-conscious decision-making that can be useful for applied analysis. (1) Our model is simple: it captures the operation of norms through twofunctions, a descriptive norm function and a prescriptive norm function. The
simple model can explain a variety of previous empirical findings. Moreover,distinguishing between the two types of norms clarifies potential mechanisms
behind policy effects and facilitates policy analysis. (2) Our model is tractablein that it does not require the researcher to solve for an equilibrium; instead,the model is directly disciplined by observable choices.8 This also means that
we allow the DM’s perceived norms to be biased. (3) The axiomatic foundation clarifies what kind of observed behavior is a distinctive feature of our PS
model, and can be used as a basis for testing it (cf. Toussaert 2018). Because of
the transparent link between choice and utility representation, the model also
allows the researcher to conduct empirical research in a flexible manner. For
example, we illustrate how the PS model can be used to investigate whether
the DM feels pride or shame using choice data alone, which allows the researcher to discuss the welfare implications of pride and shame without using
a survey (cf. Butera et al. 2022).
The paper is organized as follows. In Section 2, we illustrate our model andits implications by a simple example of prosocial behavior. Section 3 presents
our axioms and the main representation result. Section 4 discusses how ourmodel can be useful for empirical research. We provide a literature review in
Section 5. Section 6 concludes. Proofs and additional results are presented in
the Appendices.

2

Illustration of Results

Denote a typical menu of lotteries by A. A simplified version of our prideshame utility representation has the following form. First, preference  overmenus is represented by
8

Similar to Maccheroni et al. (2012), we do not investigate how norms are formed (e.g.,
as an equilibrium of a game), and instead consider identification of (p)reference regardlessof the norm generation process.

6





VP S (A) = max u(x) − max {w(ϕr (A)) − w(x), 0} +β max {w(x) − w(ϕr (A)), 0} , (1)
|
{z
}
|
{z
}
x∈A
“shame”“pride”

where β > 0, ϕr (A) = arg maxa∈A r(a), which is assumed to be a singleton for
illustrative purpose,9 and u, w, and r are von Neumann-Morgenstern (vNM)
functions.10 Second, the ex-post choice from each menu is determined by
CP S (A) = arg max [u(x) − max {w(ϕr (A)) − w(x), 0} + β max {w(x) − w(ϕr (A)), 0}] . (2)x∈A

The function u represents the DM’s intrinsic utility function, which describes
her private preference ranking.11 The term w(ϕr (A)) represents a social reference point, which consists of two distinct components. First, the function r iscalled the descriptive norm function, which expresses the DM’s perception of
the prevalence of each alternative. ϕr (A) is then interpreted as the alternativethat the DM thinks is typically chosen by other people in her society. Second,
the function w is called the prescriptive norm function, which expresses theDM’s perception of the admirability of each alternative. Together, w(ϕr (A))
represents the normative desirability of the socially prevalent choice, as perceived by the DM. Crucially, we allow the DM to have biased beliefs aboutothers’ behavior or normative opinions.
The last two terms in Eq. (1) represent the utility of social emotions.
If the DM facing menu A chooses an alternative x that is normatively inferior to the reference alternative, she feels shame, which reduces her utility byw(ϕr (A)) − w(x) > 0. Conversely, if she chooses x that is normatively supe9

Our formal theory allows ϕr (A) to contain multiple alternatives.
vNM functions may not well suit some contexts of social decision-making, where theprocesses as well as the consequences of stochastic events matter. The example of Rabin(1995) illustrates this point: “If you have found $100 on the street, and are trying to decide
whether to turn it in, part of your dilemma is figuring out the likelihood that the ownerwill be found, that the local police officials are corrupt, etc.; such cases are captured by thismodel. But part of the moral decision-making may simply be thinking through whether or
not you have a moral obligation to turn it over.” See also Fudenberg and Levine (2012).
11
u may capture not only her self-interest, but also social preferences such as altruism or
moral concerns that are not influenced by social image concerns.
107

rior to the reference alternative, she feels pride, which increases her utility by
β [w(x) − w(ϕr (A))] > 0. By allowing β 6= 1, we allow the DM to care abouta downward deviation from the reference point (shame) differently from an
upward deviation (pride).12 β 6= 1 also allows the choice from menus to bereference-dependent.
Together, U (x; A) ≡ u(x)−max {w(ϕr (A)) − w(x), 0}+β max {w(x) − w(ϕr (A)), 0}
is interpreted as the utility of choosing x from menu A, and the value of A,
VP S (A) = maxx∈A [U (x; A)], is the maximum utility that the DM can obtainfrom the menu.

2.1

A Simple Example of Prosocial Behavior

We illustrate the implications of our model by the following simple example.
Let x ∈ A = {0, 1} denote the DM’s choice of an alternative, where x = 1
indicates the DM engaging in prosocial behavior, and x = 0 indicates nonengagement. Let u(0) = ū > 0 = u(1), w(0) = 0 < w̄ = w(1), and β w̄ < ū <w̄. Thus, the DM privately prefers the non-prosocial choice but believes thatthe prosocial choice is more admired. Also, β < 1 means that the DM is moresensitive to shame than she is to pride.
Benchmark behavior. The DM chooses x = 0 or x = 1 by comparing theutility of each alternative:
U (0; A) = U (0; {0, 1}, u, w, r) = |{z}ū − [w(ϕr ({0, 1})) − 0] = ū − w(ϕr ({0, 1})){z}|
intrinsic

shame

U (1; A) = U (1; {0, 1}, u, w, r) = |{z}0 +β [w̄ − w(ϕr ({0, 1}))] = β [w̄ − w(ϕr ({0, 1}))]{z}|
intrinsic

pride

(3)
These expressions are simpler than the general one in Eq. (1) because x = 0
never causes pride and x = 1 never causes shame, regardless of the reference12

We emphasize the case with β ∈ (0, 1), which expresses shame aversion (cf. Butera etal. 2022), although our theory allows for β ≥ 1. Also, it accommodates β = 0 as a limit
case.

8

alternative ϕr ({0, 1}).
As a benchmark, suppose r(1) < r(0), i.e., the DM believes that other people in her society do not typically engage in prosocial behavior. The reference
alternative is ϕr ({0, 1}) = 0 and the reference point is w(0) = 0. Choosing
x = 0 gives the DM the intrinsic utility ū and no utility from social emotion,
because she chooses the action dictated by the norm. On the other hand,choosing x = 1 gives the DM zero intrinsic utility but gives a positive utility
from pride. Since β w̄ < ū, the DM chooses x = 0.
Perceived norms and behavior. The model predicts how the DM’s choicedepends on the descriptive and prescriptive norms. Consider the following
analysis, where each type of norm shifts toward prosocial behavior relative tothe above benchmark.
(i) Higher descriptive norm. Suppose that the descriptive norm function becomes r0 such that r0 (0) < r0 (1), shifting the reference point to w(ϕr0 ({0, 1})) =w̄. Now, choosing x = 0 gives the DM utility ū− w̄ < 0, whereas choosing
x = 1 yields zero utility. Thus, the DM chooses x = 1.
(ii) Higher prescriptive norm. Suppose that the prescriptive norm function
becomes w0 such that w0 (0) = 0 and w0 (1) = w̄0 > βū . Then, choosing
x = 1 provides a pride benefit of β w̄0 , which exceeds the utility ū fromx = 0. Thus, the DM chooses x = 1.
The DM switches to prosocial behavior x = 1 in both cases, but for different
reasons. In case (i), she chooses x = 1 because she would feel shame if she
stuck to the less admirable choice x = 0 while perceiving that others choose
x = 1. By contrast, in case (ii), she chooses x = 1 because she feels greater
pride from x = 1 perceiving that others choose x = 0.
This analysis is insightful for analyzing the effect of “norm nudges,” which
guide people’s decisions by providing social information. Plenty of economic
research has explored the effect on decisions of information about how others behave (e.g., Frey and Meier 2004; Allcott 2011) or what others think is9the appropriate behavior (e,g, Hallsworth et al. 2017; Bursztyn et al. 2020).
Our model helps clarify the mechanisms underlying such a norm-nudging. For
example, if information about others’ behavior (resp. normative opinions)mainly affects an individual’s perceived descriptive (resp. prescriptive) norm,
then the main mechanism of the effect of such information is described bycase (i) (resp. case (ii)) above.13 This analysis also suggests under what conditions providing each type of information is ineffective for inducing prosocial
behavior. Informing the DM of others’ choice fails to alter behavior if it does
not affect the descriptive norm r because, e.g., the DM does not believe theinformation or because she thinks that the “others” are dissimilar to herself
and out of her reference group. Similarly, informing her of others’ normative
opinions will not alter her behavior if it fails to affect the prescriptive norm w
sufficiently. Alternatively, the latter information is ineffective if the DM does
not derive much utility from pride (β being small).
Public recognition and prosociality. The model illustrates how publicobservability affects the DM’s prosociality. Her choice of action under a private
decision environment is expressed as a choice between two menus {0} and {1},with the utility from each option VP S ({x}) = u(x). By contrast, her choice
under a public environment is expressed as a choice between two actions 0 and 1
from the menu {0, 1}, with the utility from each option U (x; {0, 1}) = u(x) −max{w(ϕr ({0, 1})) − w(x), 0} + β max{w(x) − w(ϕr ({0, 1})), 0}.14 Because
U (x; {0, 1}) is strictly increasing in w(x), the DM becomes more prosocial in
the public environment than in the private environment. The above analysis
also indicates when policies such as public recognition programs are ineffective
for inducing prosocial behavior: they are ineffective when the descriptive andprescriptive norms do not (sufficiently) favor prosocial behavior, or when thedescriptive norm points to non-prosocial behavior and the DM is relatively
13

In reality, information about one type of norms may also affect the perception of theother.
14
The private and public preferences represent norm-free and norm-conscious preferences,respectively. Thus, our model applies to more general settings where some environmentalcue (including publicity as an example) triggers the DM to focus on norms.

10

insensitive to pride.
Perceived norms and avoidance. The two-stage model enables us to study
how norms affect the DM’s decision to participate in the opportunity for prosocial behavior, as well as her decision on prosocial behavior itself. Analysis of
the participation decision is important for two reasons. First, laboratory andfield experiments have documented that a large fraction of individuals avoid
opportunities to engage in prosocial behavior, even if they can choose nonengagement after participation and even if avoidance is costly (e.g., Dana et
al. 2006; Broberg et al. 2007; Lazear et al. 2012; DellaVigna et al. 2012; Andreoni et al. 2017; Klinowski 2021). Our model clarifies how such avoidance
depends on the perceived descriptive and prescriptive norms. Second, theparticipation decision is informative of the DM’s willingness-to-pay (WTP)
for public recognition and can be used to study the welfare impacts of policies such as public recognition programs, assuming that pride and shame arewelfare-relevant. For example, her valuation of the menu {0, 1} relative to thatof the singleton menu {0} is informative of her WTP for public recognition.15
Our model illustrates how perceived norms can differentially impact participation and choice of action. Suppose that the DM first chooses whether toparticipate in the opportunity for prosocial behavior. If she decides to participate, she proceeds to the binary-choice stage described above. Alternatively,
she can decide not to participate and be given a singleton menu {0}, which
gives her utility VP S ({0}) = ū. In the benchmark case, participation gives
utility VP S ({0, 1}) = max{U (0; {0, 1}), U (1; {0, 1})} = ū, and it is indifferent to non-participation. Therefore, the DM can optimally participate in theopportunity and then choose not to engage in prosocial behavior.
Now, suppose that the descriptive norm shifts toward prosocial behavior
15

Butera et al. (2022) use an incentive-compatible mechanism to elicit individuals’ WTP
for public recognition, in the context of charitable behavior. In our framework, the researcher
can infer individuals’ WTP by observing their choices of menus instead of conducting asurvey, which will be useful in some empirical contexts. Although the data requirementfor our approach becomes demanding as the number of alternatives increases, additional
restrictions (e.g., parametric assumption) can replace part of the data requirements.

11

(i.e., the descriptive norm function r changes to r0 ). A possible interpretationis that the DM updates her perception of the norm after she is given information about others’ actions. As the above analysis shows, the DM switches toprosocial behavior conditional on participation. On the other hand, with thedescriptive norm r0 , we have VP S ({0, 1}) = max{ū − w̄, 0} = 0 < VP S ({0}),so the DM avoids the opportunity for prosocial behavior. Thus, the higher(more prosocial) descriptive norm induces the DM to take a prosocial action
if she has no option to avoid the choice occasion, but it induces her to avoid
the occasion if she has the option.
The theoretical predictions match the empirical evidence quite well. In alaboratory experiment, Klinowski (2021) demonstrates that (1) when individuals receive information that others have made a large donation after they
participate in the opportunity, they increase the amount of donation relative
to the no-information benchmark, whereas (2) when they receive the same
information prior to the decision to participate, the participation rate drops
relative to the benchmark. Our model can rationalize these findings by theshift of the descriptive norm caused by the information treatment.16
Aligned vs. misaligned norms. The simple model also explains why thedescriptive and prescriptive norms induce larger behavioral change when they
are aligned than when they are misaligned (Cialdini 2003), and why the descriptive norm tends to trump the prescriptive norm in the latter case (Tyran
and Feld 2006; Bicchieri and Xiao 2009).17 When the prescriptive norm dic16

Klinowski (2021) also examines the effect of providing subjects with information that
others have made a small amount of donation, before or after subjects make the participation
decision. He finds that subjects are more likely to participate in the donation opportunity
than the no-information benchmark if they learn about others’ small donations before theparticipation decision. When they receive the information after participation, they contribute slightly more than the benchmark, though the difference is statistically insignificant.
Although these results can be regarded as evidence for pride seeking, explaining these results
will require some extension of the above simple model, and we omit this exercise.
17
Allcott (2011) and Hallsworth et al. (2017) find evidence that descriptive norm messages
are more effective than prescriptive norm messages for inducing electricity saving and tax
compliance, respectively. Heinicke et al. (2022) find that descriptive norms exhibit a stronger
correlation with behavior than prescriptive norms in the context of mini-dictator games.

12

tates the DM to engage in prosocial behavior (w(0) < w(1)) but the descriptive norm dictates otherwise (r(0) > r(1)), the DM behaves prosocially when
the pride benefit β w̄ is large enough to outweigh the intrinsic benefit of nonengagement, ū. By contrast, if both norms point to the prosocial behavior(w(0) < w(1) and r0 (0) < r0 (1)), then she behaves prosocially as long as theshame cost w̄ of non-engagement is large enough to offset its intrinsic benefit ū.
If the DM is shame-averse (β < 1), which finds some empirical support,18 thealigned norms induce prosocial behavior more effectively than misaligned ones.
In words, when both norms point to prosocial behavior, failing to follow themcauses shame for falling below social expectations, which is a strong motivator of prosocial behavior. By contrast, when the prescriptive norms point toprosocial behavior but the descriptive norms point to the opposite, the social
motivation for prosocial behavior is pride from exceeding social expectations,which may not be so strong (“People say I should behave prosocially, but they
do not live up to their words, so why do I?”).
Other results. In Appendix C, we show that the model can explain otherphenomena observed by previous empirical studies. For example, we argue
that the model can explain why providing information on others’ prosocial
behavior can reduce the amount of prosocial behavior (e.g., Schultz et al.
2007).

3Model

We adopt the framework of Gul and Pesendorfer (2001) (henceforth GP). Let(Z, ρ) be a compact metric space, where Z is a finite set of all prizes, and let∆ ≡ ∆(Z) denote the set of all probability measures on the Borel σ-algebraof Z endowed with the weak topology. Denote by A a set of all closed subsets
18

Butera et al. (2022) estimate social signaling models and find evidence for shame aversion. DellaVigna et al. (2017) find that non-voters in an election sort out of a survey due tothe negative feeling from admitting non-voting or lying about it, while voters do not sort in
to enjoy the positive feeling from saying that they voted.

13

of ∆, and endow A with the topology generated by the Hausdorff metric.19
A typical lottery a ∈ ∆ is called an alternative (or choice), and a typicalelement A ∈ A, a set of alternatives, is called a menu (or choice set). Define
αA + (1 − α)B ≡ {z ∈ ∆ : z = αa + (1 − α)b, a ∈ A, b ∈ B} for A, B ∈ A andα ∈ [0, 1].
We consider a DM who has a preference  over menus, as in the literature (Dillenberger and Sadowski 2012; Saito 2015; Evren and Minardi 2017;
Hashidate 2021), and who also makes a choice from a menu according to achoice rule C. Specifically,  is a binary relation over A and the choice correspondence C : A  ∆ satisfies ∅ =6 C(A) ⊆ A for all A ∈ A. We assume thatboth  and C are observed.
Below, we consider a DM who, prior to making a choice from a menu, forms
a reference point based on her subjective beliefs. Specifically, she focuses on thealternative in the menu which she believes is most commonly chosen by others,
and uses it as the reference alternative. The reference alternative establishes
the reference point, and the DM evaluates her choice positively (negatively)
if she believes it is more (less) admirable than the reference alternative. If
multiple alternatives are perceived to be most common, the DM focuses on the
one that she believes is most admirable. The beliefs on the commonality andadmirability of alternatives shape the descriptive norm and the prescriptive
norm, respectively.

3.1

Axioms

We first introduce some basic axioms.
Axiom 1. (Order)  is complete and transitive.
Axiom 2.
(i) (Lower Semi-Continuity) For any A ∈ A, {B ∈ A : A  B} is closed.
noThat is, dH (A, B) = max maxa∈A minb∈B d(a, b), maxb∈B mina∈A d(a, b) , where d isa metric that metrizes the weak topology.
19

14

(ii) (Upper von Neumann-Morgenstern Continuity) A  B  C implies B  αA + (1 − α) C for some α ∈ (0, 1) .
(iii) (Upper Singleton Continuity) {{b} ∈ A : {b}  {a}} is closed.
Axiom 1 is standard. Axioms 2(i)-(iii), similar to axioms in GP to characterize preferences without self-control, weaken the standard continuity assumption. These axioms yield a reference point that is constrained by a vNM
function,20 which will be interpreted as a social prevalence/commonality ranking. Such a specification seems attractive given that social preferences oftenfeature discontinuities.21
We proceed by introducing a “reference relation” r , which elicits the DM’s
subjective belief about the prevalence/commonality of each alternative fromobserved behavior.
Definition 1. (Reference relation)(i) a ∗ b if there exists A 3 b such that A ∪ {a}  A and a ∈/ C (A ∪ {a}).
(ii) a r b if either of the following conditions holds:
a. a ∗ b.
b. There exists some c ∈ int(∆) such that c ∗ a and c ∗ b.
(iii) a ∼r b if neither a r b nor b r a, and a r b if either a r b or a ∼r b.
To interpret r , suppose a ∗ b. Then, for some menu A 3 b,22 adding a to
A makes the menu more desirable (A ∪ {a}  A) even though a is not chosen(a ∈/ C (A ∪ {a})). This suggests that the unchosen alternative a improves
the value of the menu A by lowering its reference point. We then infer thatthe DM perceives a to be more prevalent than the other alternatives in A,
20

More specifically, it will have a “Strotz representation” (Strotz 1955).
E.g., an equal split of dictator game endowments (cf. Andreoni and Bernheim 2009)
may be a discontinuity point.
22
In Appendix B, we discuss why we do not confine Definition 1(i) to A = {b}.
21

15

including b. Case (ii-b) deals with technical difficulties which arise when a ison the boundary of ∆.23
We elicit the strict reference ranking from preferences for larger menus,
but not from preferences for smaller menus (A  A ∪ {a}). Although thelatter implies that an additional alternative makes the menu unattractive by
raising the reference point, the alternative may be just as prevalent as anotheralternative. Indeed, preferences for smaller menus emerge in the GP model,
without a notion of the prevalence ranking of alternatives. In Section 3.4,
we show that r elicits the true reference ranking r if the choice data are
generated by a PS model that satisfies some weak notion of regularity. Thus,
focusing on preferences for larger menus discards virtually no information.
Next, we define another binary relation which elicits the DM’s subjective
belief about the normative desirability/admirability of each alternative. It is“partial” in that it elicits the ranking only among alternatives with the samereference ranking (see Theorem 2).
Definition 2. (Partial normative relation)(i) a w b if one of the following conditions holds.
a. {b}  {a, b}
b. {b} ∼ {a, b} and C ({a, b}) = {a}.
c. a ∼r b and {a} ∼ {a, b}  {b}.
(ii) a ∼w b if neither a w b nor b w a. a w b if either a w b or a ∼w b.
The elicitation of the normative ranking is similar to that of the temptation
ranking in GP-style models (GP; Noor and Takeoka 2015). Consider the firstcase (i-a), where the presence of alternative a makes the original menu {b}less attractive, which implies that a raises the reference point. Thus, we inferthat the DM believes a is more admirable than b. To interpret case (i-b),
suppose that b sets the reference point at {a, b}. Then, because the reference
point is the same as {b} but the DM does not choose b at {a, b}, she must23

See Figure A2 and discussions in Appendix B.

16

be strictly better off at {a, b}, which contradicts {b} ∼ {a, b}. Thus, thereference point must be higher at {a, b} than at {b}, so the DM believes a ismore admirable than b. Finally, in case (i-c), because both a and b enter thereference set at {a, b}, the reference point is weakly higher at {a, b} than at{b}. Thus, {a, b}  {b} implies that a is the unique choice at {a, b}. Then,{a} ∼ {a, b} implies that a must be weakly higher in the normative ranking
than b (otherwise, the DM prefers to exclude b, so {a}  {a, b}). Because
a PS preference such that the reference and normative rankings are identicalcannot be distinguished from GP’s temptation preference, below we focus on
PS preferences such that they are distinct. Then, a ∼r b means that a must
be strictly higher than b in the normative ranking.
When a w b, a is also revealed to be at least as high as b in the reference
ranking; otherwise, a does not set the reference point at {a, b}, so nothing
about its normative desirability is revealed. However, we do not use the observation a w b to infer the reference ranking, because it does not tell us
whether a is strictly higher than b or just as high as b in the reference ranking.
As Theorem 2 shows, if choice data are generated by a regular PS preference,then any strict reference relation can be elicited by r alone.
A natural process of reference formation is that the DM shapes beliefs
about socially prevalent actions by imagining a “typical person,” and usesthat person’s behavior as a reference. For example, a black woman living
in an urban area may refer to the behavior of another person in a similar
background, than that of a white man in a rural area. Thus, we impose axioms
to rationalize the reference ranking as a preference of some “typical person.”For simplicity, we directly impose axioms on ∗ , r , and w , although we canrewrite these axioms as properties of (, C) .
Axiom 3. (r-EU)(i) If a r b or a w b, then neither b r a nor b w a.
(ii) ∗ is transitive. Also, if a ∼r b ∼r c, a w b, and b w c, then a w c.
(iii)

a. {α ∈ [0, 1] : αA + (1 − α)C  B} is closed in [0,1].
17

b. If there exists a∗ ∈ A such that a∗ r a for all a ∈ A \ {a∗ }, thenfor any {An } such that An → A, an ∈ C (An ) and an → a, we havea ∈ C (A).
(iv) For any α ∈ (0, 1), αa + (1 − α)c ∗ αb + (1 − α)c and a ∈ int(∆) implya ∗ b.
Axiom 3(i) imposes consistency of reference relations and partial normative
relations revealed at different menus.24 Recall a r b reveals that a is perceived
to be more prevalent than b. Also, a w b reveals that a is perceived to be atleast as prevalent as b and more admirable than b. Then, to consistently rankalternatives in prevalence and admirability, the choice data should not reveal
the opposite relations. Axiom 3(ii) states that the directly revealed reference
ranking ∗ is transitive and that w is transitive on the indifference set forr . Axiom 3(iii) expresses Archimedeanity of r . Axiom 3(iv) imposes somelinearity on the reference relation. Axiom 3(iv) is only required to deal withboundary elements.25
We next introduce a weak version of linearity of (, C). Because of themenu effect induced by pride and shame, the standard independence axiom
does not hold. For example, suppose that the reference ranking is $0 r$100 r $10, i.e., the DM believes that donation is uncommon but the donation amount is large conditional on donating. Consider the preference between{$10, $100} and {$100}, with a $100 donation being the reference alternative
at both menus. If the shame from donating a small amount despite the social
expectation of a large donation is strong, the DM will conform to the expectation and donate $100 at both menus, so {$10, $100} ∼ {$100}. Now, consider
two mixture menus, 0.5{$10, $100}+0.5{$0, $10} and 0.5{$100}+0.5{$0, $10},
with the reference alternative 0.5$100 + 0.5$0 at both menus. If adding thepossibility of zero donation lowers the reference point sufficiently, then having
24

Similar axioms appear in Dillenberger and Sadowski (2012) and Kıbrıs et al. (2023).
Note also that the interiority of a is important: if a is on the boundary of ∆, it ispossible that we cannot reveal a ∗ b but can reveal αa + (1 − α)c ∗ αb + (1 − α)c, if
mixing with c brings a to the interior. See the example in Figure A2 and discussions inAppendix B.
25

18

an option to donate a small amount may be valuable because it offers a convenient compromise between self-interest and pride. Thus, the former mixture
may be strictly better than the latter. This phenomenon happens because
the effect of changing the reference point depends on whether each alternative
falls above or below the reference point, so mixing can change the relative
desirability of two alternatives.
The above discussion suggests that independence will hold if we consider
mixtures of menus that preserve the relative desirability of alternatives. We
consider a preference that satisfies independence among menus at which theDM can never feel shame and among menus at which the DM can never feelpride. To formalize the idea, let Lr (a) = {b ∈ ∆ : a r b} denote the set ofalternatives which is strictly below a in the reference ranking r . For an
arbitrary a ∈ ∆, any b r a belongs to exactly one of the following sets:
P (a) = {b ∈ Lr (a) : {a, b}  {b} and C ({a, b}) = {b}}(4)

S (a) = {b ∈ Lr (a) : {a, b} ≺ {b} and C ({a, b}) = {b}}(5)N1 (a) = {b ∈ Lr (a) : {a, b} ∼ {b} and C ({a, b}) = {b}}(6)N2 (a) = {b ∈ Lr (a) : a ∈ C ({a, b})}(7)

I(a) = {b ∈ ∆ : b ∼r a}(8)

P (a) is the set of alternatives b with the reference ranking strictly below a suchthat the DM feels pride by choosing b at {a, b}. We provide formal definitionsof pride and shame in Section 3.2; intuitively, because the DM chooses b at both{a, b} and {b} but is strictly better off at the former, we infer that the unchosenalternative a gives pride by lowering the reference point. Similarly, S (a) is theset of alternatives b ≺r a such that the DM feels shame by choosing b at {a, b}.
This interpretation follows from the observation that the DM chooses the sameb at both {a, b} and {b} but is strictly worse off at the former, implying thata raises the reference point. N1 (a) is the set of alternatives b ≺r a such thatthe DM feels neither pride nor shame at {a, b} because the chosen alternative
b is socially as desirable as the reference alternative a. N2 (a) is the set ofalternatives b ≺r a such that the DM feels neither pride nor shame at {a, b}
19because she chooses the reference alternative.26 Finally, I(a) is the set ofalternatives that is indifferent in the reference ranking to a.
To state linearity axioms, define the following subsets of A:
BP = {{a, b} ∈ A : a = b or b ∈ P(a) ∪ N1 (a)}BS = {{a, b} ∈ A : a = b or b ∈ S(a) ∪ N1 (a) ∪ I(a)}BN = {{a, b} ∈ A : a = b or b ∈ N2 (a)} .
Here, we do not distinguish {a, b} from {b, a}: e.g., a ∈ P(b) implies {a, b} ∈BP . BP is the “pride domain” and BS is the “shame domain.” Specifically,
for any A ∈ BP , the DM may feel pride (or no social emotion) but she cannever feel shame. Similarly, for any A ∈ BS , the DM may feel shame (orzero social emotion) but not pride. Finally, BN is the set of binary menus atwhich we cannot exclude the possibility of either pride or shame. Note wehave ∪j=P,S,N Bj = {{a, b} : a, b ∈ ∆} and ∩j=P,S,N Bj = {{a} : a ∈ ∆}.
We now state linearity axioms on (, C).
Axiom 4. (Weak Independence, WI) For any α ∈ (0, 1),(i) A, B, C ∈ BP and A  ()B imply αA + (1 − α)C  ()αB + (1 − α)C.
(ii) A, B, C ∈ BS and A  ()B imply αA + (1 − α)C  ()αB + (1 − α)C.
(iii) A, B ∈ A, c ∈ ∆ and A  ()B imply αA + (1 − α) {c}  ()αB + (1 −α) {c} .
Axiom 5. (Weak Linearity, WL) For any a, b, c, d ∈ ∆ and α ∈ (0, 1), thefollowing properties hold.
(i) Suppose {a, b} , {c, d} ∈ BP or {a, b} , {c, d} ∈ BS . Then C (α {a, b} + (1 − α) {c, d}) =αC ({a, b}) + (1 − α)C ({c, d}) .
(ii) Let A = α {a, b} + (1 − α) {a, c} and b ∈ N2 (a).
26
Strictly speaking, the DM can feel pride or shame by choosing some b ∈ N2 (a) at {a, b}
because it is possible to have C ({a, b}) = {a, b}.

20

a. If c ∈ P(a), {a, αb + (1 − α)c}  α {b}+(1−α) {a, c}, and C ({a, αb + (1 − α)c}) ={αb + (1 − α)c}, then C (A) = αC ({a, b}) + (1 − α)C ({a, c}).
b. If c ∈ S(a), α {b}+(1−α) {a, c}  {a, αb + (1 − α)c}, and C ({a, αb + (1 − α)c}) ={αb + (1 − α)c}, then C (A) = αC ({a, b}) + (1 − α)C ({a, c}).
(iii) For any A ∈ A, C (αA + (1 − α) {a}) = αC (A) + (1 − α) {a} .
Axiom 4 states that the standard independence property of  holds within
the pride domain and shame domain, and it holds with respect to mixtureswith a singleton. Similarly, Axiom 5(i) states that the linearity of choiceholds within the pride domain and shame domain. Axiom 5(ii) is interpreted
similarly, but requires an additional condition to exclude the possibility that
one of the mixed menus generates pride and the other generates shame. To
interpret (ii-a), note that from b ∈ P(a) and c ∈ N2 (a), we know that a issuperior to b and c in the reference ranking. Suppose we additionally know{a, αb + (1 − α)c}  α {b} + (1 − α) {a, c} and C ({a, αb + (1 − α)c}) = {αb +(1 − α)c}. The former suggests that moving a toward b makes the menu lessdesirable, and the latter suggests that this is not because a is a preferredchoice. We can then infer that a sets a reference point lower than b. Thus,{a, b} can generate pride but not shame, so the linearity of ex-post choice holds
if it is mixed with a menu that never generates shame. Similarly, conditions in(ii-b) suggest that a sets the reference point higher than b, so the linearity of
ex-post choice holds if {a, b} is mixed with a menu that never generates pride.
Finally, Axiom 5(iii) states that linearity holds with respect to mixtures witha singleton.
The next axiom relates preferences  to choice C.
Axiom 6. (Sophistication) Suppose there exists a∗ ∈ A such that a∗ r c
for all c ∈ A ∪ B and a∗ w a for all a ∈ A.
(i) Suppose a∗ w b for all b ∈ B. Then, A ∪ B  A. Moreover, A ∪ B  A
if and only if C (A ∪ B) ∩ A = ∅.
(ii) Suppose there exists b∗ ∈ B such that b∗ w a∗ . Then, A∪B  A implies
C (A ∪ B) ∩ A = ∅.
21

Axiom 6 is analogous to the Sophistication axiom of Noor and Takeoka(2015), with an important difference that our axiom depends on r and w .
Axiom 6(i) is about situations in which some a∗ ∈ A sets the reference point in
A ∪ B. When the menu A is augmented by another menu B, the DM weakly
prefers the larger menu A∪B because it increases options without changing thereference point. Moreover, the larger menu should be strictly more desirable
if and only if the newly added menu contains an option strictly preferred to
all alternatives in A. Axiom 6(ii) concerns a situation in which some newly
added alternative b∗ ∈ B sets a higher reference point than the reference point
at A. With a higher reference point, the DM weakly prefers the larger menu
only if the newly added menu B contains a strictly better alternative to bechosen than alternatives in A.
Our next axiom captures the DM’s shame attitude, i.e., how her social
payoff depends on the size and direction of the deviation of her choice’s admirability from the reference point. We consider a DM whose marginal utility
from pride and that from shame are constant, respectively, but who may careabout pride and shame differently. The following axiom captures such an
attitude toward pride and shame. Let ea,b denote an alternative such that
 a,b
∼ {a, b}. For {a, b} ∈ BS , such ea,b exists by Lemma 12.
e
Axiom 7. (Piecewise linear social payoffs) There exists a unique α ∈(0, 1) such that for any a, b, c, d ∈ ∆ such that c ∈ P(a) ∩ P(b) and d ∈S(a) ∩ S(b), we have α {a, c} + (1 − α) eb,d ∼ α {b, c} + (1 − α) ea,d .
To interpret Axiom 7, suppose c ∈ P(a)∩P(b) and {a, c}  {b, c}. At eachmenu, a or b is the reference alternative and the DM feels pride by choosingc. Therefore, the preference {a, c}  {b, c} must be because a is considered
normatively inferior to b and generates higher pride for choosing the same
alternative c. Suppose also d ∈ S(a) ∩ S(b). Then, similarly, the DM prefers{a, d} to {b, d} because a generates lower shame than b does for choosing thesame alternative d. Now, consider the choice between two lotteries: lottery
1 yields the payoff from the high-pride menu {a, c} or that from the highshame menu {b, d} with probability α and 1 − α, respectively, and lottery 2
22yields the payoff from the low-pride menu {b, c} or that from the low-shamemenu {a, d} with the same mixing rate. As α increases, lottery 1 becomes
more desirable, and the DM will be indifferent between the lotteries at some
α. Such α captures the rate at which the DM trades off the gains from morepride with the loss from more shame. Axiom 7 states that this tradeoff rate isconstant across alternatives involved. Moreover, the tradeoff rate measures thedegree of shame aversion: the higher α, the more pride gain the DM demands
to compensate for the loss from shame.
Definition 3. (Shame attitudes) (i) The DM is α-sensitive to shame if
her preference (, C) satisfies Axiom 7 with α ∈ (0, 1). (ii) The DM who is
α-sensitive to shame is shame-averse if α > 21 ; shame-neutral if α = 21 ; andshame-loving if α < 12 .
Our final axiom imposes some consistency of choices across menus. Consider the donation example above. There, the DM believes that donation is
uncommon but a large donation is common conditional on donating. Then,
she may choose $100 from {$10, $100} to avoid the shame of falling behindthe social expectation of a large donation, whereas she may choose $10 from{$0, $10, $100} because a small donation nicely balances self-interest with pride
from exceeding the social expectation of zero donation. This choice patternviolates the WARP. By contrast, if the reference point remains at $100 after
the no-donation option is added, then the DM at menu {$0, $10, $100} will
not choose $10, because the size of her shame from doing so is unchanged.
She will choose $0 or $100, depending on the relative importance of private
vs. social payoffs. This example suggests the following axiom.
Axiom 8. (Weak WARP, WWARP) For any A, B ∈ A, suppose there
exists a∗ ∈ A ∩ B such that a∗ r c and a∗ w c for all c ∈ A ∪ B. Then,a, b ∈ A ∩ B, a ∈ C (A) and b ∈ C (B) imply a ∈ C (B).
An alternative a∗ ∈ A sets the reference point at menu A if a∗ r a anda∗ w a hold for all a ∈ A.27 Therefore, Axiom WWARP says that the27

Such a∗ exists for any finite A; see Lemma 7.

23

standard WARP property applies to menus A and B which share a common
reference-setting alternative a∗

3.2

Representation Theorem

We show that the above axioms characterize the following utility representation.
Definition 4. (, C) is a pride-shame (PS) preference if there are continuousand linear functions u,w and r and a constant β > 0 such that  is represented
by



VP S (A) = max u(x) − max max w(y) − w(x), 0 +β max w(x) − max w(y), 0x∈A
y∈ϕr (A)y∈ϕr (A)
{z
}
|
{z
}
|




“shame”“pride”(9)
where ϕr (A) = arg maxa∈A r(a), and C coincides with
CP S (A) = arg max u(x) − max
x∈A






max w(y) − w(x), 0 + β max w(x) − max w(y), 0 .
y∈ϕr (A)y∈ϕr (A)(10)
The representation (9)-(10) is called a PS representation.
maxy∈ϕr (A) w(y) is interpreted as the normative desirability which the DM
perceives is expected to achieve, which we simply call the reference point.
The reference point consists of two distinct components. First, the descriptive
norm function r represents the DM’s belief about the prevalence/commonalityof each alternative. The DM’s reference set ϕ(A) consists of alternatives which
she believes is the most prevalent in A. Second, the prescriptive norm function
w represents the DM’s belief about the social desirability/approvedness ofeach alternative. When ϕ(A) contains multiple alternatives, the DM adopts
the highest value of w as the reference point. The second term in Eq. (9)then represents the social utility that the decision maker derives from social
emotion.
24

We say that a DM with a PS preference feels pride by choosing a ∈ A at
A if w(a) − maxy∈ϕr (A) w(y) > 0, and the DM feels shame by choosing a ∈ A
at A if w(a) − maxy∈ϕr (A) w(y) < 0. In words, the DM feels pride (shame) if
she chooses an alternative that she perceives is normatively superior (inferior)
to the reference alternative. Pride (shame) gives the DM a positive (negative)
payoff from social emotion. Because the DM may care about shame differently
than pride (Butera et al. 2022), we allow the DM to be more or less sensitive
to shame than to pride, by allowing β 6= 1.
The PS representation includes GP’s temptation model as a degeneratecase. For our main theorem, however, we focus on nondegenerate PS preference
defined as follows.
Definition 5. (Nondegeneracy) Preference (, C) is nondegenerate if there
exists x, y, y 0 ∈ ∆ such that {x, y}  {y}, C({x, y}) = {y}, {y 0 }  {x, y 0 } andC({x, y 0 }) = {y 0 }.
Nondegeneracy ensures that the DM feels pride at some binary menu andshame at another. The temptation preference of GP is degenerate, because
it never generates pride. In Section 3.4, we show that such a degenerate
preference accommodates non-unique representations. Thus, our main theorem focuses on preferences that accommodates a unique (up to positive affine
transformation) representation, and we treat degenerate cases separately. Notethat nondegeneracy is testable.
We now state our main theorem.
Theorem 1. A nondegenerate preference (, C) satisfies Axioms 1-8 if andonly if it admits a PS representation. Moreover, the decision maker is shameaverse if and only if β < 1, shame-neutral if and only if β = 1, and shameloving if and only if β > 1.

3.3

Sketch of the Proof of Theorem 1

Our proof, formally presented in Appendix A, begins by verifying that thereference relation r admits a linear representation r (i.e., a r b ⇔ r(a) ≥r(b) and r(αa + (1 − α)b) = αr(a) + (1 − α)r(b)).
25

Lemma 1. Suppose Axioms 1-5 hold. Then r admits a linear representationr. The representation is unique up to positive affine transformation.
Define the reference correspondence as ϕr (A) = {a ∈ A : r(a) ≥ r(b) ∀b ∈ A} .
We can then show the following important properties of ϕr .
Lemma 2. Suppose Axioms 1-6 hold. Then ϕ satisfies the following properties.
(i) ϕr (A ∪ B) = ϕr (A) implies A ∪ B  A.
(ii) ϕr (A ∪ B) = ϕr (A) ∪ ϕr (B) and A  B imply A  A ∪ B  B.
Lemma 2(i) states that if augmenting menu A by menu B does not affectthe reference set, then the DM weakly prefers the larger menu. In this case,
the DM exhibits a preference for larger menus (cf. Evren and Minardi 2017)
because the addition will never worsen social emotion. On the other hand,
Lemma 2(ii) states that if the addition of alternatives expands the reference
set, then the set betweenness property (Gul and Pesendorfer 2001) holds.28 In
particular, the DM exhibits a preference for smaller menus (cf. Dillenberger
and Sadowski 2012) because the added alternatives will just raise the bar andwill never improve social emotion.
Lemma 2 implies that the preference over finite menus can be characterized
by at most two elements in each menu.
Lemma 3. Suppose Axioms 1-6 hold. Then, for any finite menu A ∈ A, there
exist a∗ ∈ A and b∗ ∈ ϕr (A) such that A ∼ {a∗ , b∗ }.
The remaining components u and w can be constructed in a way similar
to that of Gul and Pesendorfer 2001, although we address several technicaldifficulties due to the violation of independence and WARP. We first showthat there exists a function VP S that represents  and satisfies some linearity.
Let Af denote the set of all finite menus in A.
28

The two cases analyzed in Lemma 2 are exhaustive.

26

Lemma 4. Suppose Axioms 1-6 hold. Then there exists a function VP S that
represents  on Af and satisfies the following property: A, B ∈ BP or A, B ∈BS implies VP S (αA + (1 − α)B) = αVP S (A) + (1 − α)VP S (B).
Domain-wise linearity of VP S together with nondegeneracy allows us toconstruct a function wP on the pride domain and wS on the shame domain.
Then, Axiom 7 implies that the two functions are proportional: wP (x) =βwS (x) =: w(x) for some β > 0.29
We can show that the PS representation holds for all binary menus. Then,
Lemma 3 allows us to extend the representation to all finite menus, and Axiom
2 further extends the result to all menus in A. For choice C, Axiom 8 extends
the representation to all menus.

3.4

Identification of PS Representation

3.4.1

Uniqueness of Reference Ranking

Given the construction of r , its representation r is unique up to positiveaffine transformation (see also Proposition 1). However, we can ask: is there
an alternative way to construct reference (leading to a different function r0 )to represent the same data by another PS preference? The definition of radopts a specific way to complete the relation, and for general (not necessarily
regular) cases, there can be two PS representations with different references
which both represent the same choice data.
Example 1. Suppose choice data (, C) are generated by the temptation preference of GP: VGP (A) = maxx∈A {u(x) + w(x) − maxy∈A w(y)}. In this case,a strict reference relation a r b never occurs, so the descriptive norm function which rationalizes r is a constant. However, the choice can also be
represented by another PS preference with r = w, because maxy∈ϕw (A) w(y) =maxy∈A w(y). Thus, choice data cannot distinguish the two models.
29

We construct w in a way similar to GP. Another possible approach is to construct it
directly from w . However, w is guaranteed to elicit the normative ranking only between
alternatives with the same reference ranking, so such a proof will involve an incomplete
relation. Instead, we construct w from (, C) and then show in Theorem 2 that w indeed
represents w among alternatives with the same reference ranking.

27

However, we show below that when data are generated by a PS preference
that belongs to some nondegenerate class, r correctly elicits the true reference. Therefore, within that class, the reference function r which is consistent
with observed data is unique up to positive affine transformation. It turns outthat the following weak form of nondegeneracy is sufficient to ensure uniqueand correct elicitation of the true reference by r .
Definition 6. (Weak Nondegeneracy) Preference (, C) is weakly nondegenerate if there exist some ā, b̄ ∈ ∆ such that ā ∗ b̄.
Theorem 2. Suppose the data are generated by a weakly nondegenerate PS
preference. Then the following statements hold. (i) r(a) > r(b) if and only ifa r b. (ii) For a, b ∈ ∆ such that r(a) = r(b), w(a) > w(b) if and only if
a w b.
Because Weak Nondegeneracy is implied by Nondegeneracy, the representation result in Theorem 1 focuses on the case where observed data are consistent
with a unique reference ranking r, which is exactly the relation elicited by r .
Also, under Weak Nondegeneracy, the normative relation w is consistent witha unique normative ranking of w among alternatives with the same reference
ranking. Note that the PS preference that appears in Example 1 is excluded
by Weak Nondegeneracy.
3.4.2

Uniqueness of (u, w, r, β)We can also show that u and w are unique up to affine transformation andthat β is unique, when (, C) satisfies the above axioms.
Proposition 1. Suppose a nondegenerate (, C) satisfies Axioms 1-8. Thenthe following statements are equivalent.
(i) If a PS representation (u, w, r, β) represents , then (u0 , w0 , r0 , β 0 ) also
represents .
(ii) The following properties hold.

28

a. u0 = αu + γu and w0 = αw + γw for some α > 0 and γu , γw ∈ R.
b. r0 = αr r + γr for some αr > 0 and γr ∈ R.
c. β = β 0
Although r is unique up to positive affine transformation, there is no guaranteethat αr = α.

3.5Comparing Shame Aversion

Axiom 7 and Definition 3 yield a notion of a DM being “more shame-averse”
than another DM. For two DMs i = 1, 2, let (i , Ci ) denote the preference ofDM i.
Definition 7. Suppose DM i ∈ {1, 2} is αi -sensitive to shame. Then DM 1
is (weakly) more shame-averse than DM 2 if α1 > (≥)α2 .
The definition captures the idea that, when the degree of shame aversion
increases, the DM who faces a lottery involving pride and shame of equal size
will demand a higher chance of pride to compensate for the utility cost of
shame. We now state how the PS representation and observed behavior are
linked in terms of shame aversion. For i = 1, 2, let Pi and Si denote the setof pride-generating binary menus P and the set of shame-generating binary
menus S defined in Eq. (4) and (5), respectively, for DM i. Also, let ex,y∈∆ix,ybe such that {ei } ∼i {x, y}.
Proposition 2. Suppose DM 1 and DM 2 have a PS preference, with parameters β1 and β2 , respectively. Then the following statements are equivalent.
(i) β1 < (≤)β2 .
(ii) DM 1 is (weakly) more shame-averse than DM 2.
(iii) Take any α ∈ (0, 1) and any a, b, c, d such that c ∈ Pi (a) ∩ Pi (b), d ∈Si (a) ∩ Si (b), {a, nc} oi {b, c} and {a, d} i {b,n d} ofor i = 1, 2. Thena,dα {b, c} + (1 − α) e2
2 α {a, c} + (1 − α) eb,dimplies α {b, c} +
2
n o
n o
(1 − α) ea,d
1 (1 )α {a, c} + (1 − α) eb,d
.
1
1
29

By the equivalence of (i) and (ii), β in the PS representation characterizes
the DM’s shame aversion. Also, the equivalence of (ii) and (iii) implies that
we can compare the shame aversion of two DMs by the following experiment:
Consider two lotteries, lottery 1 giving the payoff of a high-pride menu {a, c}and the payoff of a high-shame menu {b, d} with probability α and 1 − αrespectively, and lottery 2 giving the payoff of a low-pride menu {b, c} and thepayoff of a low-shame menu {a, d} with probability α and 1 − α respectively.
Ask the DMs to choose between the two lotteries at various α. Then, DM1 ismore shame-averse than DM2 if DM1 chooses the “safer” lottery 2 wheneverDM2 does.

4

Empirical Perspective

Although typically available data will be insufficient to completely verify ouraxioms, our PS model can still be used to understand the nature of individuals’
perceived norms and resulting social emotions. In this section, we discuss some
implications of our model that are potentially useful for empirical analysis.

4.1

Testing for the Independence of Norms and Preferences

Researchers may be interested in testing whether the descriptive norm andprescriptive norm are distinct, or whether the choice prescribed by norms
coincides with the intrinsically best alternative.
Claim 1. Suppose the DM has a PS preference. Then, the following statements
hold, where k represents equality up to positive affine transformation.
(i) r ∦ w if Weak Nondegeneracy holds.
(ii) u ∦ w, r if there exist some A ∈ A and a ∈ A such that {a}  A.
Proof. (i) By assumption, there exists A ∈ A and a ∈ ∆ such that A ∪ {a} A and a ∈/ C(A ∪ {a}). By the PS representation, maxy∈ϕr (A∪{a}) w(y) <30

maxy∈ϕr (A) w(y) (otherwise, we would have A  A ∪ {a}). Thus, for b ∈arg maxy∈ϕr (A) w(y), we have r(a) > r(b) and w(a) < w(b). (ii) Without loss ofgenerality, assume {a}  {b} for all b ∈ A. By assumption, maxy∈ϕr (A) w(y) =w(c) > w(a) for some c ∈ A (otherwise, we would have A  {a}). Therefore,
we have u(a) ≥ u(c), w(a) < w(c), and r(a) ≤ r(c). If we further hadu(a) = u(c), then A  {c} ∼ {a}, a contradiction. Thus, u(a) > u(c).
For example, observing {a, b}  {b} and a ∈/ C({a, b}) for some a, b is sufficient to conclude r ∦ w, and observing {a}  {a, b} for some a, b is sufficient
to conclude u ∦ w, r.

4.2

Testing for Pride and ShameThe PS representation allows us to infer whether the DM feels pride or shame(or no social emotion) at a given binary menu {a, b}. The following result
can be used to estimate, e.g., the fractions of people who enjoy a welfare
gain (pride) and those who suffer a welfare loss (shame) from choosing analternative under a public recognition program.
Claim 2. Suppose (, C) is a PS preference such that C({a, b}) = {b}.30 Then,(i) the DM feels pride by choosing b at {a, b} if and only if {b} ≺ {a, b} {a}.
(ii) the DM feels shame by choosing b at {a, b} if and only if {b}  {a, b} {a}.
The proof is straightforward given the PS representation, hence omitted.
Intuitively, in both cases, C({a, b}) = {b} and {a, b}  {b} imply that thereference point is w(a)(6= w(b)). Then, the preference between {a, b} and {b}tells whether the DM feels pride or shame at {a, b}, whereas we necessarily
have {a, b}  {a} because b is a more attractive choice than a and does notaffect the reference point at {a, b}.
30
We preclude C({a, b}) = {a, b}, in which case the social emotion depends on the realization of the ex-post choice.

31

A caveat is that a norm-evoking policy may generate welfare loss even
if the DM does not feel shame of her own choice. Recall the example ofprosocial behavior analyzed in Section 2.1, specifically with norms (r0 , w) suchthat r0 (0) < r0 (1) and w(0) < w(1). In this case, the norms suggest prosocial
behavior x = 1 as a reference alternative and the DM follows the norms,
so she does not feel pride or shame. However, a publicity policy that forces
the DM to make a choice at {0, 1} is welfare reducing relative to the outside
option of avoiding prosocial behavior in private ({0}  {0, 1}). Incorporating
preferences over menus is useful for analyzing the welfare effects of norms
and norm-evoking policies with a revealed preference approach, when socialpressure forces the DM to choose an action that she nonetheless would like tohave removed (cf. Bursztyn et al. 2023).

4.3

Examining Belief Heterogeneity and Treatment Effect

The analysis in Section 2 derives behavioral predictions based on the knowledgeof (u, w, r, β). Conversely, if we observe choice data and make some assumptions on these components, we can infer some properties of the (p)reference.
For example, under a PS preference, the observations {0, 1}  {1} and C({0, 1}) ={1} imply w(0) < w(1) and r(0) > r(1), that is, the DM perceives that 1 isnormatively superior but 0 is more common. On the other hand, if we observe{0, 1} ∼ {1} and C({0, 1}) = {1} and additionally presume w(0) < w(1), then
we infer r(0) ≤ r(1), i.e., the DM believes that 1 is (weakly) more common.
Thus, the assumption that the DM has a PS preference with w(0) < w(1)enables us to examine whether r(0) > r(1) or not, without collecting complete
data on (, C). Similarly, we can make an inference about w by assuming thatthe above observations are generated by a PS preference with r(0) > r(1).
We can conduct these kinds of analyses to see, e.g., whether perceptions aredifferent across demographic groups or are affected by policies.

32

5

Related Literature

5.1

Social Norms and Social Image Concerns

To the best of our knowledge, this paper is the first to formalize and axiomatize the notion of descriptive and prescriptive norms in a decision-theoretic
model. In the social psychology literature (Cialdini et al. 1991; Schultz etal. 2007), descriptive norms commonly refer to norms that dictate individuals
to do what is typically done by others, and prescriptive norms (or injunctive
norms) commonly refer to norms which dictate them to do what people approve of. Bicchieri and Dimant (2022) define a social norm as a behavioral rulethat individuals prefer to follow because they believe that (i) others follow it
and (ii) others think the individuals should follow it. Our model is a formalization of these ideas; social norms are shaped by two functions r and w, wherer represents the belief (perception) about what others do and w represents
the belief about what others think should be done.31 This modeling will facilitate the analysis of how norms are influenced by laws (Bénabou and Tirole
2011; Lane et al. 2023) or social information (Frey and Meier 2004; Schultz etal. 2007; Allcott 2011; Hallsworth et al. 2017) by altering these perceptions.
Also, the DM in our model derives social payoffs/emotions by comparing her
own choice with others’ choice, which closely follows the literature on social
comparisons (Festinger 1954).
Our paper also contributes to the growing literature on social image concerns or social pressure (Bénabou and Tirole 2006), which are considered a key
driver of behavior in various contexts (See footnote 1). Our contribution is to
propose a model with social image concerns which is useful for three reasons.
First, our model distinguishes the descriptive and prescriptive norms in a simple manner. It captures various patterns of social norms’ influences by two
functions, each representing the descriptive norm and the prescriptive norm.
The distinction between the two types of norms also facilitates policy analysis.
31

Bicchieri and Dimant (2022) refer to the former belief as empirical expectations and thelatter as normative expectations, and they define a social norm as a behavioral rule governedby these expectations.

33

For example, we can analyze the effect of altering either the descriptive normor the prescriptive norm on the DM’s prosocial behavior, holding the othernorm constant. Second, our model is tractable because it does not require theresearcher to compute an equilibrium, in contrast to commonly used social
signaling models. Instead of restricting the DM’s beliefs, our model is directlydisciplined by observable choice data. This also allows the perceived norms
to be biased from the objective distribution of choices or normative opinions.
Third, the axioms clarify what types of choice behavior is a distinct featureof the PS model, and can be used for testing the model. The transparent link
between choice and utility representation also allows the researcher to study
the behavioral and welfare effects of norms and norm-evoking policies in a flexible manner. For example, as we show in Section 4, we can use the model toexamine whether the DM feels pride or shame using choice data alone, without
using a survey.

5.2

Axiomatic Decision Theory

Our model relates to the axiomatic two-stage models of choices with temptation (Gul and Pesendorfer 2001; Noor and Takeoka 2015) or social emotions (Dillenberger and Sadowski 2012; Saito 2015; Evren and Minardi 2017;
Hashidate 2021), and axiomatic models of endogenous reference points (Ok et
al. 2015; Lleras et al. 2019; Kıbrıs et al. 2023). The former papers consider
decision problems in which the DM chooses a menu of alternatives in the firststage and then chooses an alternative from the menu in the second stage. In
particular, those with social emotions assume that the first stage is private
whereas the second stage is publicly observed, and that the DM anticipatessocial emotions due to public observability when making the first-stage choice.
We adopt the same observability assumptions, but their domains are different
from ours. Regarding the latter papers on endogenous reference, their models
do not include the menu-choosing stage, and domains of ex-post choice arealso different. Below, we discuss each paper in more detail.
Both Gul and Pesendorfer (2001) and Noor and Takeoka (2015), as we do,34

consider preferences over menus of lotteries. Our PS representation includes
the GP representation as a degenerate case. Noor and Takeoka (2015) use
observations of both preferences  over menus and choice correspondence C
describing choices from each menu, as we do, to derive a Menu-Dependent
Self-Control (MDSC) representation. In the MDSC representation, the selfcontrol cost (similar to the social emotions in our model) of choosing x from
A is specified by ψ (maxy∈A w(y)) (maxy∈A w(y) − w(x)), where ψ(·) ≥ 0 is increasing. Both their and our representations generalize the GP representation
by relaxing the WARP on ex-post choice. Their specification allows for a flexible scaling of the self-control cost, in contrast to our piecewise linear scalingof the “cost” of social emotions. On the other hand, their self-control costis restricted to be non-negative, whereas our emotional cost can be negative,
which is essential for generating pride.
Dillenberger and Sadowski (2012) and Evren and Minardi (2017) study
preferences over menus consisting of social allocations of non-stochastic objects(e.g., dictator games). Dillenberger and Sadowski (2012) characterize shame,
which involves a preference for smaller menus. By contrast, Evren and Minardi(2017) characterizes warm-glow, which involves a preference for larger menus.
Our axioms capture both types of preferences, depending on whether adding
alternatives to a menu expands the reference set (see Lemma 2).
Saito (2015) and Hashidate (2021) study preferences over menus consistingof social allocations p = (pi )i∈{1}∪S of lotteries, where 1 denotes the DM and S
denotes the set of other agents. Saito (2015) derives the generalized utilitarian(GU) representation, which generates the pride β1 maxq∈A α1 (u(q1 ) − u(p1 )) >0 of acting altruistically if the DM compromises her private payoff, and thePPshame −βS maxr∈A ( i∈S αi u(ri ) − i∈S αi u(pi )) < 0 of acting selfishly if the
DM compromises other agents’ private payoffs.32 Hashidate (2021) generalizes
the GU representation by replacing the reference point with a convex combination of the maximum and minimum (instead of maximum), which allows thesigns of the social emotion terms to depend on menus and allows various social
32

Saito (2015) allows for β1 < 0, in which case the term captures the temptation to act
selfishly.

35

emotions to emerge. Their models generate social emotions by the comparison of the DM’s own private payoff or other agents’ private payoffs relative
to reference points. By contrast, pride and shame in our model arise fromthe comparison of the perceived normative desirability of own action against
the reference alternative’s desirability. We adopt our formulation to separate these emotions from private payoffs. We believe this separation is vitalfor two reasons. First, our model formalizes the concept of social norms discussed in Section 5.1 closely. Moreover, we argue that pride and shame should
depend on the degree to which the DM can live up to social expectations(norms), not the degree to which the DM’s or other agents’ private payoffs
are sacrificed.33 Second, our formulation explains empirical findings discussed
in Section 2 straightforwardly. For example, when information about others’
behavior alters the DM’s choice, it is plausibly due to changes in perceived
social expectations, rather than changes in private payoffs.
We conclude this section by discussing the differences between our modeland the aforementioned axiomatic models of endogenous reference dependence.34 Ok et al. (2015) characterize choice behavior which exhibits the“attraction effect.” Specifically, by relaxing the WARP, they derive a model inwhich a dominated alternative serves as a reference alternative and restricts
the choice set to alternatives that dominate it. Therefore, in their model, thereference point is an unchosen alternative, and imposes a mental constraint
on choice sets. By contrast, in our model, the reference alternative may bechosen, and it affects the preference but not choice sets. Lleras et al. (2019)consider a preference over state-contingent contracts (acts) and derive a representation that evaluates an act based on its expected value and expected
gain/loss relative to the expected value. Their representation allows the DM
to derive payoffs from either expected gain or loss, but not both. By contrast,
the DM in our model may feel pride from an alternative and shame from an33

Scheff (1988) discusses how perceived social expectations set a system in which conformity to norms is sustained by the reward of pride and punishment of shame (together withdeference).
34
Köszegi and Rabin (2006) develop a non-axiomatic model of endogenous reference formation. In their model, the agent’s reference point is constrained by rational expectations.

36other. Kıbrıs et al. (2023) consider choices from finite choice sets, and derivea reference-dependent choice model in which the reference point is determinedby an endogenously derived conspicuity ranking, just as an endogenously derived r defines reference in our model. Their representation is quite general,but they do not characterize a specific structure of the reference point in ourmodel, so our axiomatization result is not implied by their work.

6

Conclusion

Despite the growing interest in using social norms for behavioral change, their
behavioral and welfare effects are not well understood. We propose and axiomatize a model of reference-dependent decision-making in which the decision
maker’s perception of others’ choice (“descriptive norm”) and her perceptionof others’ normative opinions (“prescriptive norm”) together shape a reference
point that indicates the approvedness of the socially prevalent behavior. The
key drivers of the decision maker’s behavior are social emotions, such as apositive payoff from pride, which she enjoys if her choice exceeds the reference point, and a negative payoff from shame, which she suffers if her choicefalls short of it. A simple model of prosocial behavior can provide various
useful implications, such as when policies such as public recognition programs
or norm nudges likely induce prosocial behavior, or how policies differentiallyinfluence the choice of an action conditional on participating in an opportunity
for prosocial behavior and the decision on participation. It also rationalizes
empirical findings that aligned descriptive and prescriptive norms are more effective for inducing prosocial behavior compared to misaligned ones, and thatthe descriptive norms have a larger impact than the prescriptive norms when
they are misaligned. Moreover, the axiomatic model is simple enough to clarify
the mechanisms behind policy effects and facilitate policy analysis, tractablebecause it does not impose an equilibrium assumption, and transparent in its
relation to observed choice, which may usefully guide empirical analysis.
There are two limitations to this paper. First, we do not model how individuals’ perceptions are shaped. For example, they may arise as equilibrium37

objects of a game (cf. Bénabou and Tirole 2006), and such an equilibrium
restriction may be necessary to study how norms evolve over time. Also, individuals may form perceptions in a self-serving manner (Heinicke et al. 2022;
Bicchieri et al. 2023), and accounting for this possibility may be crucial for
predicting the effect of an informational intervention. Second, norms may be
specified in a more flexible manner. For example, an individual may compare
her behavior with the behavior of a group of individuals rather than that of asingle “typical person.” This can be modeled by specifying the reference point
to depend on the distribution of descriptive norms of other individuals.35 Eliciting information on individuals’ reference groups based on their choice is aninteresting topic for empirical research.

35

Technically, this will yield a random Strotz representation (Dekel and Lipman 2012).

38

References
Akerlof, George A. and Rachel E. Kranton, “Economics and Identity,”
The Quarterly Journal of Economics, 08 2000, 115 (3), 715–753.
Allcott, Hunt, “Social Norms and Energy Conservation,” Journal of Public
Economics, 2011, 95 (9-10), 1082–1095.
and Todd Rogers, “The Short-Run and Long-Run Effects of Behavioral Interventions: Experimental Evidence from Energy Conservation,” American Economic Review, 2014, 104 (10), 3003–3037.
Andreoni, James and B. Douglas Bernheim, “Social Image and the 5050
Norm: A Theoretical and Experimental Analysis of Audience Effects,”
Econometrica, 2009, 77 (5), 1607–1636.
, Justin M Rao, and Hannah Trachtman, “Avoiding the Ask:
A Field Experiment on Altruism, Empathy, and Charitable giving,”
Journal of Political Economy, 2017, 125 (3), 625–653.
Bénabou, Roland and Jean Tirole, “Incentives and Prosocial Behavior,”
American Economic Review, December 2006, 96 (5), 1652–1678.
and, “Laws and Norms,” Working Paper 17579, National Bureau of Economic Research November 2011.
Bicchieri, Cristina, The Grammar of Society: The Nature and Dynamics ofSocial norms, Cambridge University Press, 2005.
and Erte Xiao, “Do the Right Thing: But Only if Others Do So,”
Journal of Behavioral Decision Making, 2009, 22 (2), 191–208.
and Eugen Dimant, “Nudging with Care: The Risks and Benefitsof Social Information,” Public Choice, 2022, 191 (3-4), 443–464.
,, and Silvia Sonderegger, “It’s not a lie if you believe thenorm does not apply: Conditional norm-following and belief distortion,” Games and Economic Behavior, 2023, 138, 321–354.
Birke, David J, “Anti-Bunching: A New Test for Signaling Motives in Prosocial Behavior,” 2020.
Broberg, Tomas, Tore Ellingsen, and Magnus Johannesson, “Is Generosity Involuntary?,” Economics Letters, 2007, 94 (1), 32–37.

39

Bursztyn, Leonardo, Alessandra L González, and David YanagizawaDrott, “Misperceived Social Norms: Women Working Outside theHome in Saudi Arabia,” American Economic Review, 2020, 110 (10),2997–3029.
and Robert Jensen, “ How Does Peer Pressure Affect Educational
Investments? ,” The Quarterly Journal of Economics, 05 2015, 130 (3),1329–1367.
and, “Social Image and Economic Behavior in the Field: Identifying, Understanding, and Shaping Social Pressure,” Annual Review
of Economics, 2017, 9 (1), 131–153.
, Benjamin R Handel, Rafael Jimenez, and Christopher Roth,“When Product Markets Become Collective Traps: The Case of Social
Media,” Working Paper 31771, National Bureau of Economic Research
October 2023.
, Thomas Fujiwara, and Amanda Pallais, “’Acting Wife’: Marriage Market Incentives and Labor Market Investments,” American
Economic Review, November 2017, 107 (11), 3288–3319.
Butera, Luigi, Robert Metcalfe, William Morrison, and Dmitry
Taubinsky, “Measuring the Welfare Effects of Shame and Pride,”
American Economic Review, 2022, 112 (1), 122–168.
Cialdini, Robert B., “Crafting Normative Messages to Protect the Environment,” Current Directions in Psychological Science, 2003, 12 (4),105–109.
Cialdini, Robert B, Carl A Kallgren, and Raymond R Reno, “A
Focus Theory of Normative Conduct: A Theoretical Refinement andReevaluation of the Role of Norms in Human Behavior,” in “Advances
in Experimental Social Psychology,” Vol. 24, Elsevier, 1991, pp. 201–
234.
Dana, Jason, Daylian M Cain, and Robyn M Dawes, “What You Dont
Know Wont Hurt Me: Costly (but Quiet) Exit in Dictator Games,”
Organizational Behavior and Human Decision Processes, 2006, 100 (2),
193–201.
40

Dekel, Eddie and Barton L. Lipman, “Costly Self-Control and Random
Self-Indulgence,” Econometrica, 2012, 80 (3), 1271–1302.
DellaVigna, Stefano, John A List, and Ulrike Malmendier, “Testingfor Altruism and Social Pressure in Charitable Giving,” The Quarterly
Journal of Economics, 2012, 127 (1), 1–56.
, John A. List, Ulrike Malmendier, and Gautam Rao, “Voting
to Tell Others,” The Review of Economic Studies, 10 2017, 84 (1),
143–181.
Dillenberger, David and Philipp Sadowski, “Ashamed to be Selfish,”
Theoretical Economics, 2012, 7 (1), 99–124.
Evren, Özgür and Stefania Minardi, “WarmGlow Giving and Freedom
to be Selfish,” The Economic Journal, 02 2017, 127 (603), 1381–1409.
Festinger, Leon, “A Theory of Social Comparison Processes,” Human Relations, 1954, 7 (2), 117–140.
Frey, Bruno S and Benno Torgler, “Tax Morale and Conditional Cooperation,” Journal of Comparative Economics, 2007, 35 (1), 136–159.
and Stephan Meier, “Social Comparisons and Pro-social Behavior:
Testing Conditional Cooperation in a Field Experiment,” American
Economic Review, 2004, 94 (5), 1717–1722.
Fudenberg, Drew and David K. Levine, “Fairness, Risk preferences andIndependence: Impossibility Theorems,” Journal of Economic Behavior & Organization, 2012, 81 (2), 606–612.
Gerber, Alan S., Donald P. Green, and Christopher W. Larimer,“Social Pressure and Voter Turnout: Evidence from a Large-Scale Field
Experiment,” American Political Science Review, 2008, 102 (1), 3348.
Gul, Faruk and Wolfgang Pesendorfer, “Temptation and Self-Control,”Econometrica, 2001, 69 (6), 1403–1435.
Hallsworth, Michael, John A List, Robert D Metcalfe, and Ivo
Vlaev, “The Behavioralist as Tax Collector: Using Natural Field Experiments to enhance tax compliance,” Journal of Public Economics,
2017, 148, 14–31.

41

Hashidate, Yosuke, “Social Image Concern and Reference Point Formation,”
The B.E. Journal of Theoretical Economics, 2021, 21 (1), 19–84.
Heinicke, Franziska, Christian König-Kersting, and Robert Schmidt,“Injunctive vs. Descriptive Social norms and Reference Group Dependence,” Journal of Economic Behavior & Organization, 2022, 195, 199–
218.
Karing, Anne, “Social Signaling and Childhood Immunization: A Field Experiment in Sierra Leone,” 2024.
Kıbrıs, Özgür, Yusufcan Masatlioglu, and Elchin Suleymanov, “A
Theory of Reference Point Formation,” Economic Theory, 2023, 75(1), 137–166.
Klinowski, David, “Reluctant Donors and Their Reactions to Social Information,” Experimental Economics, 2021, 24 (2), 515–535.
Köszegi, Botond and Matthew Rabin, “A Model of Reference-Dependent
Preferences,” The Quarterly Journal of Economics, 11 2006, 121 (4),1133–1165.
Lacetera, Nicola and Mario Macis, “Social image concerns and prosocial
behavior: Field evidence from a nonlinear incentive scheme,” Journalof Economic Behavior & Organization, 2010, 76 (2), 225 – 237.
Lane, Tom, Daniele Nosenzo, and Silvia Sonderegger, “Law andNorms: Empirical Evidence,” American Economic Review, May 2023,
113 (5), 1255–93.
Lazear, Edward P, Ulrike Malmendier, and Roberto A Weber, “Sorting in experiments with application to social preferences,” American
Economic Journal: Applied Economics, 2012, 4 (1), 136–163.
Lleras, Juan Sebastián, Evan Piermont, and Richard Svoboda,“Asymmetric Gain–Loss Reference Dependence and Attitudes toward
Uncertainty,” Economic Theory, 2019, 68 (3), 669–699.
Maccheroni, Fabio, Massimo Marinacci, and Aldo Rustichini, “Social
Decision Theory: Choosing within and between Groups,” The Review
of Economic Studies, 01 2012, 79 (4), 1591–1636.

42

Masatlioglu, Yusufcan, Daisuke Nakajima, and Erkut Y. Ozbay, “Revealed Attention,” American Economic Review, May 2012, 102 (5),2183–2205.
Miller, Dale T. and Deborah A. Prentice, “Collective Errors and Errors
about the Collective,” Personality and Social Psychology Bulletin, 1994,
20 (5), 541–550.
Noor, Jawwad and Norio Takeoka, “Menu-dependent Self-control,” Journal of Mathematical Economics, 2015, 61, 1–20.
Ok, Efe A., Pietro Ortoleva, and Gil Riella, “Revealed (P)Reference
Theory,” American Economic Review, January 2015, 105 (1), 299–321.
Perez-Truglia, Ricardo and Ugo Troiano, “Shaming Tax Delinquents,”
Journal of Public Economics, 2018, 167, 120–137.
Rabin, Matthew, “Moral Preferences, Moral Constraints, and Self-serving
Biases,” 1995.
Saito, Kota, “Impure Altruism and Impure Selfishness,” Journal of Economic
Theory, 2015, 158, 336–370.
Scheff, Thomas J., “Shame and Conformity: The Deference-Emotion System,” American Sociological Review, 1988, 53 (3), 395–406.
Schultz, P. Wesley, Jessica M. Nolan, Robert B. Cialdini, Noah J.
Goldstein, and Vladas Griskevicius, “The Constructive, Destructive, and Reconstructive Power of Social Norms,” Psychological Science,
2007, 18 (5), 429–434.
Strotz, Robert H., “Myopia and Inconsistency in Dynamic Utility Maximization,” Review of Economic Studies, 1955, 23 (3), 165–180.
Toussaert, Séverine, “Eliciting Temptation and Self-control through Menu
Choices: A Lab Experiment,” Econometrica, 2018, 86 (3), 859–889.
Tyran, Jean-Robert and Lars P Feld, “Achieving Compliance when Legal Sanctions are Non-deterrent,” Scandinavian Journal of Economics,
2006, 108 (1), 135–156.

43

For Online PublicationA

ProofsBelow, we denote a mixed lottery αa + (1 − α)b ∈ ∆ by aαb and a mixed menu
αA + (1 − α)B ∈ A by AαB, for any a, b ∈ ∆, A, B ∈ A, and α ∈ [0, 1].

A.1

Proof of Theorem 1

A.1.1

Proof of Lemma 1

We first introduce two lemmas.
Lemma 5. Suppose Axioms 4 and 5 hold. Then, for any a, b, c ∈ ∆ andα ∈ (0, 1), a ∗ b implies aαc ∗ bαc.
Proof. Suppose a ∗ b. By definition, there exists A 3 b such that A ∪ {a} A and a ∈/ C (A ∪ {a}). Take any c ∈ ∆ and any α ∈ (0, 1). By Axiom
4(iii), [A ∪ {a}]α {c}  Aα {c}. Also, by Axiom 5(iii), C ([A ∪ {a}]α{c}) =C (A ∪ {a}) α{c}, so aαc ∈/ C ([A ∪ {a}]α{c}). Thus, aαc ∗ bαc.
Lemma 6. Suppose Axioms 1-3 hold. Then, the following statements hold.
(i) If a ∗ b, then for any c ∈ ∆, there exists α ∈ (0, 1) such that aαc ∗ b.
(ii) If b ∗ c, then for any a ∈ ∆, there exists β ∈ (0, 1) such that b ∗ aβc.
Proof. (i) By a ∗ b, we have some A 3 b such that A ∪ {a}  A anda∈/ C (A ∪ {a}). By Axiom 2(i), there exists α1 ∈ (0, 1) such that A∪{aαc} A for all α ∈ (α1 , 1). Also, by Axiom 3(iii-b), we have some α2 ∈ (0, 1)
such that aαc ∈/ C (A ∪ {aαc}) for all α ∈ (α2 , 1).36 Therefore, taking α ∈(max {α1 , α2 } , 1), we have aαc ∗ b.
(ii) By b ∗ c, there exists some B 3 c such that B ∪ {b}  B andb ∈/ C (B ∪ {b}). By Axiom 3(iii), there exists some β ∈ (0, 1) such that[{a} βB] ∪ {b}  {a} βB and b ∈/ C ([{a} βB] ∪ {b}).37 Thus, b ∗ aβc.
36

Suppose not. Then, there is a sequence {αn }n such that αn → 1 and aαn c ∈C (A ∪ {aαn c}). Since a ∗ a0 for all a0 ∈ A, Axiom 3(iii-b) implies a ∈ C (A ∪ {a}), a
contradiction.
37
Following the argument in footnote 36, the latter property holds for all sufficiently small
β > 0. To show that the former property holds for all sufficiently small β > 0, note first that
Axioms 2(i) and 3(iii-a) ensure A ∪ {a}  Ã  A where Ã = Ã(γ) = [A ∪ {a}] γA for someγ ∈ (0, 1). (Otherwise, ΓL = {γ ∈ [0, 1] : A  Ã(γ)} and ΓU = {γ ∈ [0, 1] : Ã(γ)  A ∪ {a}}are nonempty closed sets such that ΓL ∪ ΓU = [0, 1], so Ã(γ)  Ã(γ) for γ ∈ ΓL ∩ ΓU , acontradiction.) Then, for all sufficiently small β, we must have [{a} βB]∪{b}  Ã  {a} βB.

44

Proof of Lemma 1. We prove the following properties.
Completeness. Immediate from the definition of r .
Transitivity. We first show that r is transitive. Suppose a r b r c. If
a ∗ b ∗ c, then a ∗ c by Axiom 3(ii), so a r c. Next, suppose a ∗ b,d ∗ b, and d ∗ c for some d ∈ int(∆). If d ∗ a, then d ∗ b by Axiom
3(ii), a contradiction. Therefore, d ∗ a, hence a r c. Next, suppose d ∗ a,d ∗ b, and b ∗ c for some d ∈ int(∆). Then d ∗ c by Axiom 3(ii), so a r c.
Finally, suppose d ∗ a, d ∗ b, e ∗ b, and e ∗ c for some d, e ∈ int(∆).
If d ∗ c, then c r b, contradicting Axiom 3(i). Therefore, d ∗ c, hencea r c.
Now, suppose a r b r c. If a r b r c, we have a r c by transitivity.
Next, suppose a r b ∼r c. Then, for any d ∈ int(∆), either d ∗ b, cor d ∗ b, c must hold (otherwise, b r c). Now, by a r b, there existsd ∈ int(∆) such that d ∗ a and d ∗ b.38 Therefore, d ∗ c, hence a r c.
Similarly, if a ∼r b r c, then there exists d ∈ int(∆) such that d ∗ b andd ∗ c, and we must have d ∗ a to maintain a ∼r b. Thus, a r c. Finally,
suppose a ∼r b ∼r c. If a r c, then we have c r a ∼r b, so the above
argument yields c r b, a contradiction. Thus, a r c.
Independence. Suppose a r b. If a ∗ b, then we have aαc ∗ bαc by Lemma
5, hence aαc r bαc. Next, suppose there exists some d ∈ int(∆) such thatd ∗ a and d ∗ b. Then Lemma 5 yields dαc ∗ bαc and Axiom 3(iv) yields
dαc ∗ aαc, so aαc r bαc.
Archimedeanity. Suppose a r b r c. If a ∗ b ∗ c, Lemma 6 yields
aαc r b r aβc for some α, β ∈ (0, 1). Next, suppose a ∗ b, d ∗ b,
and d ∗ c for some d ∈ int(∆). By Lemma 6, we have aαc r b andd ∗ aβc, hence b r aβc, for some α, β ∈ (0, 1). Next, suppose b ∗ c,d ∗ a, and d ∗ b for some d ∈ int(∆). We then have b r aβc anddαc ∗ b for some α, β ∈ (0, 1). Also, by Axiom 3(iv), we have dαc ∗ aαc.
Because dαc ∈ int(∆), we have aαc r b. Finally, suppose there exist somed, e ∈ int(∆) such that d ∗ a, d ∗ b, e ∗ b, and e ∗ c. By Axiom 3(iv) andLemma 6, we have dαc ∗ aαc, dαc ∗ b and e ∗ aβc for some α, β ∈ (0, 1).
Therefore, aαc r b and b r aβc.
If a ∗ b, then a ∗ a0.5b ∗ b by Lemma5. Then, taking an arbitrary e ∈ int(∆) andα ∈ (0, 1) sufficiently close to one, Lemma 6 implies d ≡ (a0.5b)αe satisfies the condition.
38

45

Thus, by the Mixture Space Theorem, r admits a linear representation r,
and the representation is unique up to positive affine transformation.
A.1.2

Proof of Lemma 2

The proof is trivial when A ⊃ B, so suppose otherwise. For (ii), we also
assume A 6⊂ B; otherwise the proof is trivial. We first introduce two lemmas
on r and w . Note we impose Axioms 1-5 in the lemmas, so that r admits
a linear representation r.
Lemma 7. Suppose Axioms 1-5 hold. Then, for any finite A ∈ A, there existsa∗ ∈ A such that a∗ r a and a∗ w a for all a ∈ A.
Proof. Because w is complete by definition and transitive on ϕr (A) by Axiom
3(ii), there exists a∗ which maximizes w on ϕr (A). By a∗ ∈ ϕr (A), we musthave a∗ r a for all a ∈ A. Also, for any a ∈ A \ ϕr (A), we have a∗ r a, so
Axiom 3(i) implies a∗ w a. Thus, a∗ r a and a∗ w a for all a ∈ A.
Lemma 8. Suppose that Axioms 1-6 hold and that A and B are finite.
(i) If A  A ∪ B, then there exists b ∈ B \ A such that b r a or b w a for
all a ∈ A.
(ii) If A ∪ B  A and C (A ∪ B) ∩ A 6= ∅, then there exists b ∈ B \ A suchthat b r a for all a ∈ A.
Proof. (i) By Lemma 7, there exists a∗ ∈ A such that a∗ r a and a∗ w a for
all a ∈ A. To prove the contrapositive of the statement, suppose that for anyb ∈ B \ A, there exists a ∈ A such that a r b and a w b. By the transitivityof r and Axiom 3(i)(ii), a∗ r c and a∗ w c for all c ∈ A ∪ B. Therefore,
Axiom 6(i) yields A ∪ B  A.
(ii) Similarly, if the conclusion is false, then we have some a∗ ∈ A such thata∗ r c for all c ∈ A ∪ B and a∗ w c for all c ∈ A. If a∗ w b for all b ∈ B,
then by Axiom 6(i), it is impossible to have A ∪ B  A and C (A ∪ B) ∩ A 6= ∅simultaneously. If there exists b∗ ∈ B such that b w a∗ , then Axiom 6(ii)implies we cannot have both A ∪ B  A and C (A ∪ B) ∩ A 6= ∅. Thus, theclaim holds.
Proof of Lemma 2 (Continued).
(i) If A  A ∪ B, then by Lemma 8(i), there exists b ∈ B \ A such thatb r a or b w a for all a ∈ A. By Axiom 3(i), we have b r a, hencer(b) ≥ r(a), for all a ∈ A. Thus, ϕr (A ∪ B) 6= ϕr (A) .
46

(ii) Suppose ϕr (A ∪ B) = ϕr (A) ∪ ϕr (B) and A  B. If A  B  A ∪ B,
then by Lemma 8(i) there exists some a∗ ∈ A \ B such that a∗ r b or a∗ w bfor all b ∈ B and some b∗ ∈ B \ A such that b∗ r a or b∗ w a for all a ∈ A,
which contradicts Axiom 3(i). Next, suppose A ∪ B  A  B. Note at least
one of C (A ∪ B) ∩ A or C (A ∪ B) ∩ B is nonempty. If C (A ∪ B) ∩ A 6= ∅, thenby Lemma 8(ii) there exists b ∈ B \ A such that b r a, hence r(b) > r(a),for all a ∈ A, which contradicts ϕr (A ∪ B) = ϕr (A) ∪ ϕr (B) . A similar
contradiction results if C (A ∪ B) ∩ B 6= ∅. Thus, A  A ∪ B  B.
A.1.3

Proof of Lemma 3.

For any a ∈ A, pick ba ∈ ϕr (A) such that {a, b}  {a, ba } for all b ∈ ϕr (A).
Let a∗ ∈ A be such that {a∗ , ba∗ }  {a, ba } for all a ∈ A and let b∗ ≡ ba∗ . Then,
iteratively applying Lemma 2(ii), {a∗ } ∪ ϕr (A) = ∪b∈ϕr (A) {a∗ , b}  {a∗ , b∗ }.
Further, because A = ({a∗ } ∪ ϕr (A)) ∪ (A \ ({a∗ } ∪ ϕr (A))) and ϕr (A) =ϕr ({a∗ } ∪ ϕr (A)), applying Lemma 2(i) yields A  {a∗ } ∪ ϕr (A)  {a∗ , b∗ }.
Secondly, by construction, {a∗ , b∗ }  {a, ba } for all a ∈ A. Then iteratively
applying Lemma 2(ii) yields {a∗ , b∗ }  ∪a∈A {a, ba } = A.
A.1.4

Proof of Lemma 4.

We first introduce a lemma.
Lemma 9. Suppose Axioms 3 and 6 hold. Then, b ∈ N2 (a) implies {a, b} ∼{a}.
Proof. Suppose a r b and a ∈ C ({a, b}). By Axiom 3(i), we have a w b.
Thus, by Axiom 6(i), we have {a, b} ∼ {a}.
Now, let(AP =A∈A:A=M
X)
αm Am , Am ∈ BP , M < ∞

m=1

and

(
AS =

A∈A:A=M
X)
αm Am , Am ∈ BS , M < ∞m=1

denote the set of finite mixtures over BP and that of finite mixtures over BS ,respectively. Also, for notational simplicity, define AN = BN . We show that admits a linear representation on each domain Aj , j = P, S, N .

47

Lemma 10. Suppose Axioms 1-6 hold. Then, for each j ∈ {P, S, N }, 
restricted to Aj has a representation V j . Moreover, (i) V P and V S are linear
and (ii) the restrictions of V P , V S and V N to singleton sets are continuous.
Proof. The statements regarding V P and V S follow from Lemma 1 of GP.
 To
construct V N , note that by Lemma 9, for each A∈ AN , we have A ∼ aA
for some known aA ∈ A. By letting V N (A) = V P ( aA ), V N represents  on
AN and its restriction to singleton sets is continuous.
By construction, for any j, k ∈ {P, S, N }, V j ({a}) ≥ V j ({b}) ⇔ {a} {b} ⇔ V k ({a}) ≥ V k ({b}). Because linear representation of  on singleton
sets is unique up to positive affine transformation, we can normalize them sothat V j ({a}) = V k ({a}) ≡ V singleton ({a}) for all a and all j, k.
To obtain a desired representation of  across different domains, we introduce two lemmas. The second lemma relates preferences over AS and AN to
those for singleton menus.
Lemma 11. Suppose Axioms 1-6 hold. If {c, d} ∈ BS , then {c}  {c, d} or{d}  {c, d} holds.
Proof. The conclusion trivially holds if c = d, so we assume c 6= d. Without
loss of generality, let c r d. If d ∈ S(c) ∪ N1 (c), then {d}  {c, d} bydefinition. If d ∈ I(c), then Lemma 2(ii) implies either {c}  {c, d} or {d} {c, d}.
Lemma 12. Suppose Axioms 1-6 hold. If A ∈ AS ∪ AN , then there exists
some e ∈ ∆ such that A ∼ {e}.
Proof. If A ∈ AN , then the conclusion follows from Lemma 9. Suppose A ∈AS . We first note that iteratively applying Lemma 2 yields A  {a} for somea ∈ A.39 If A ∼ {a} for some a ∈ A, the conclusion holds. Next, suppose{a0 }  A  {a} for some a, a0 ∈ A. Then, because {a} , {a0 } , A ∈ AS , we have
V S ({a0 }) > V S (A) > V S ({a}). By linearity, there exists α ∈ (0, 1) such that
Denote A = a1 , · · · , 
a|A| where {a1 }  {a2 }  ·· ·  a|A| . If a|A|−1 r a|A| ,then Lemma 2(i) implies
a|A|−1 , a|A|   a|A|−1  a|A| . If a|A| r a|A|−1 , thenLemma
 2(i) implies a|A|−1 , a|A|  a|A| . If a|A|−1 ∼r a|A| , then Lemma 2(ii) implies
a|A|−1 , a|A|  a
|A| . Next, if a|A|−2 r a|A|−1 , a|A| , then Lemma 2(i) implies
a|A|−2 , a|A|−1 , a|A|  a|A|−2  a|A| . If a|A|−1 r a|A|−2
or a|A| r a|A|−2 , then
Lemma 2(i) implies a|A|−2 , a|A|−1 , a|A|  a|A|−1 , a|A|  a|A| . If a|A|−2 ∼r a|A|−1 aa|A|−2 , a|A|−1 , a|A| 
|A| or a|A|−2 ∼r a|A|  a|A|−1 , then Lemma 2(ii) implies either
a|A|−1 , a|A|  a|A|  or a|A|−2 , a|A|−1 , a|A|  a|A|−2  a|A| . Repeating this finite
times, we obtain A  a|A| .
39

48

V S ({a0 αa}) = αV S ({a0 }) + (1 − α)V S ({a}) = V S (A). Because V S represents
 on AS and {a0 αa} ∈ AS , we have {a0 αa} ∼ A. Finally,P
to see that A  {a}
Afor all a ∈ A does not occur, recall we can write A = M
m=1 αm {a1m , a2m }PMA
Awhere {a1m , a2m } ∈ BS and m=1 αm = 1. By Lemma 11, there exist (em )M
m=1 ,with em ∈P{a1m , a2m } for each m, such that {em }  {a1m , a2m }. By Axiom 4,A
we have { Mm=1 αm em }  A.
We now obtain the desired representation of  on AP ∪ AS ∪ AN , whichcontains all binary menus. For notational simplicity, in Lemma 13 below
we eliminate from AP menus which are contained in AS (i.e., AP denotes
AP \ AS ).40
Lemma 13. Suppose Axioms 1-6 hold. Define V : AP ∪ AS ∪ AN → R as
follows:
X
V (A) =V j (A) × I {A ∈ Aj and |A| > 1} + V singleton (A) × I {|A| = 1} .
j∈{P,S,N }Then, V represents  on AP ∪ AS ∪ AN . Moreover, A, B ∈ BP or A, B ∈ BS
implies V (AαB) = αV (A) + (1 − α)V (B).
Proof. Note that any A ∈ AP ∪AS ∪AN belongs to exactly one of {B ∈ Aj : |B| > 1},
j = P, S, N , or {B : |B| = 1} (recall the re-definition of AP ). Suppose A  B
and let A ∈ Aj and B ∈ Ak . If |A| = 1 and |B| = 1, then V (A) =V singleton (A) > V singleton (B) = V (B) where the inequality follows from V singleton
representing  over singletons. If |A| = 1 and |B| > 1, then V (A) =V singleton (A) = V k (A) > V k (B) = V (B) where the inequality follows fromV k representing  on Ak and A, B ∈ Ak . Similarly, we obtain V (A) > V (B)if |A| > 1 and |B| = 1. Finally, suppose |A| > 1 and |B| > 1. If j = k = P ,then V (A) = V P (A) > V P (B) = V (B). Otherwise, by Lemma 12, there
exists some e ∈ ∆ such that A ∼ {e} or B ∼ {e}. For the former case,
V (A) = V j (A) = V j ({e}) = V k ({e}) > V k (B) = V (B) ,thus V (A) > V (B). Proof for the latter case is analogous. Thus, A  B ⇒V (A) > V (B) holds. Similarly, B  A ⇒ V (B) ≥ V (A) holds.
To prove the last statement, note that for A ∈ Aj , j ∈ {P, S, N },V (A) = V j (A) × I {|A| > 1} + V singleton (A) × I {|A| = 1} = V j (A) .
40

Such duplicates arise from menus {a, b} such that b ∈ N1 (a), which belongs to BP ∩ BS .

49

Because AαB ∈ Aj for any A, B ∈ Bj , we obtain V (AαB) = V j (AαB) =αV j (A) + (1 − α)V j (B) = αV (A) + (1 − α)V (B).
We now obtain the desired representation as follows. Take V from Lemma
13 and define, for each finite A ∈ A, VP S (A) = V ({a∗ , b∗ }) where a∗ = aA∗ ∈A and b∗ = bA∗ ∈ ϕ (A) are constructed as in Lemma 3. ThenA  B ⇔ aA∗ , bA∗  aB∗ , bB∗⇔ V aA∗ , bA∗ ≥ V aB∗ , bB∗⇔ VP S (A) ≥ VP S (A) .
Thus, we have obtained the desired function VP S .
With VP S that represents  over finite menus, we obtain the following
result, analogous to GP.
VP S (A) = max min VP S ({a, b}) = min max VP S ({a, b}).
a∈A b∈ϕr (B)b∈ϕr (B) a∈A(11)

Proof is straightforward given the proof of Lemma 3, hence omitted.
Proof of Theorem 1 (Continued).
For some two-component mixtures A of binary menus, Axioms 5 and 6 identify
a binary subset of A to which A is indifferent.
Lemma 14. Suppose Axioms 1-6 hold. Let A = {a, b} α {c, d} ∈ A. Then,the following statements hold.
(i) If b ∈ P(a) ∪ N1 (a) and d ∈ P(c) ∪ N1 (c), then A ∼ {aαc, bαd}.
(ii) If b ∈ S(a) ∪ N1 (a) and d ∈ S(c) ∪ N1 (c), then A ∼ {aαc, bαd}.
(iii) If b ∈ I(a), {a, b}  {a} and d ∈ S(c), then, {aαc, bαd}  A. If, inaddition, {b}  {a, b}, then {aαc, bαd} ∼ A.
(iv) Suppose b ∈ I(a), {b}  {a, b} ∼ {a}, a ∈ C ({a, b}) and d ∈ S(c).
Then, A ∼ {a} α {c, d}  {aαc, bαd}, and the latter relation is strict if
and only if C ({a, b}) = {a}.
(v) Suppose b ∈ I(a), {a} ∼ {a, b} ∼ {b} and d ∈ S(c). Then,(v-a) C ({a, b}) = {a} implies {bαc, aαd}  A ∼ {a} α {c, d}  {aαc, bαd}and(v-b) C ({a, b}) = {a, b} implies A ∼ {a} α {c, d} ∼ {aαc, bαd}.
50Proof. (i) By Axiom 5(i), C ({a, b} α {c, d}) = {bαd}. Also, by the linearity ofr and Axiom 3(i), we have r(aαc) > r(z), hence aαc w z, for all z ∈ A\{aαc}.
Therefore, Axiom 6(i) implies A ∼ {aαc, bαd}.
(ii) The same argument as (i) yields the result.
(iii) Let VP S be a function that represents  over finite menus in A. By
Eq.(11), there exists z ∈ A such that VP S (A) = minz0 ∈ϕr (A) VP S ({z, z 0 }). If
z = aαc, then by aαc ∈ ϕr (A) and Axioms 3(i) and 6(i), we have VP S (A) ≤VP S ({aαc}) < αVP S ({a, b})+(1−α)VP S ({c, d}) = VP S (A). Thus, we reach acontradiction. If z = aαd, then VP S (A) ≤ VP S ({a} α {c, d}) < αVP S ({a, b})+(1 − α)VP S ({c, d}) = VP S (A), a contradiction. If z = bαc, then VP S (A) ≤VP S ({a, b} α {c}) < αVP S ({a, b}) + (1 − α)VP S ({c, d}) = VP S (A), a contradiction. Thus, VP S (A) = minz0 ∈ϕr (A) VP S ({bαd, z 0 }) ≤ VP S ({aαc, bαd}) . Now,
suppose VP S (b) > VP S ({a, b}) > VP S (a). Recall VP S (A) = VP S ({bαd, z 0 })for some z 0 ∈ ϕr (A). By the linearity of r, we have ϕr (A) = {aαc, bαc}. If
z 0 = bαc, then VP S (A) = αVP S ({b}) + (1 − α)VP S ({c, d}) > αVP S ({a, b}) +(1 − α)VP S ({c, d}) = VP S (A) , a contradiction. Thus, A ∼ {aαc, bαd}.
(iv) By the linearity of r, we have ϕr (A) = {aαc, bαc}. Also, by Axiom
4(i), we have {bαc}  {aαc, bαc}, so aαc w bαc. By Axiom 3(i), aαc w z
for all z ∈ A. Also, by Axiom 5(i), C (A) = C ({a, b}) α {d}. Therefore, Axiom
6(i) yields the desired conclusion.
(v) Let C ({a, b}) = {a}. We first prove the last two relations in (v-a).
By Axiom 5(i), C (A) = {aαd} and C ({a, b} α {c}) = {aαc}. Also, Axiom 4
implies {bαc} ∼ {aαc, bαc} ∼ {aαc} , so aαc w bαc by case (i-b) of Definition
2. Also, by the linearity of r, r(aαc) > r(aαd), r(bαd). Thus, by Axiom 3(i),aαc r z and aαc w z for all z ∈ A. By Axiom 6(i), A ∼ {aαc, aαd} 
{aαc, bαd}. Next, to show the first relation in (v-a), note that bαc r z forall z ∈ A, bαc w aαd (by Axiom 3(i)), and aαc w bαc. Thus, applying
Axiom 6(ii) to Ã = {bαc, aαd} and B̃ = {aαc, bαd}, we obtain {bαc, aαd} Ã ∪ B̃ = A, establishing the conclusion. Finally, to show (v-b), suppose
C ({a, b}) = {a, b}. Then we have C (A) = {aαd, bαd}, and aαc, bαc r z andaαc, bαc w z for all z ∈ A. Thus, applying Axiom 6(i) to Ã = {aαc, bαd}and B̃ = {bαc, aαd} yields A ∼ {aαc, bαd} and applying it to Ã = {aαc, aαd}
and B̃ = {bαc, bαd} yields A ∼ {aαc, aαd}.
Intuitively, previous Lemma 3 says that the value of any finite menu A is
determined by the chosen alternative and reference alternative. Lemma 14(i)considers mixing two binary menus in the pride domain, {a, b} and {c, d}, witha and c setting the reference and b and d being chosen. Then, at the mixture
menu A = {a, b} α {c, d}, the reference alternative is the mixture of originallyreferenced alternatives a and c and the chosen alternative is the mixture of51

originally chosen alternatives b and d. (ii) states an analogous result for the
shame domain. Next, conditions in (iii) imply that b is the chosen alternativeat {a, b}, because the reference point at {a, b} is at least as high as that at{a}. If a sets a lower reference point than b, then {aαc, bαd} is strictly better
than A, as the two menus share the common alternative to be chosen (i.e.,bαd) and the former yields a lower reference point (set by aαc) than the latter(by bαc). Otherwise, the two sets feature the same reference point (set byaαc) and are indifferent. Condition a ∼r b and {b}  {a, b} in particularimplies a sets a strictly higher reference point than b does. Case (iv) considers
the situation where a sets a higher reference point than b. Then, becauseaαc sets the reference at A, a binary subset {aαc, z} of A is indifferent to A
if z is chosen at A and inferior to A otherwise. Finally, when a ∼r b and{a} ∼ {a, b} ∼ {b}, choice C({a, b}) is useful for identifying the reference
setter at {a, b}. Specifically, if C({a, b}) = {a}, then a must set a reference
strictly higher than b does; otherwise, {a, b} would be strictly better than {b}.
The last two relations in (v-a) follows from aαc setting the reference and aαdbeing chosen (bαd not being chosen) at A. Also, the first relation in (v-a) holds
because aαd is chosen at A and bαc sets a reference point strictly lower than
the reference point at A (set by aαc). On the other hand, if C({a, b}) = {a, b},then a and b set the same reference point and are equally desirable choices.
Thus, for both z = a, b, zαc is the reference setter and zαd is the choice at A.
The next lemma is used to construct the normative utility functions separately on the pride domain and shame domain.
Lemma 15. Suppose Axioms 1-5 hold, y ∈ P(x), and y 0 ∈ S(x).
(i) There exists δ ∈ (0, 1) such that y(1 − δ)c ∈ P(x) and y 0 (1 − δ)c ∈ S(x)for all c ∈ ∆.
(ii) y(1 − δ)c ∈ P(x(1 − δ)c) and y 0 (1 − δ)c ∈ S(x(1 − δ)c) for all c ∈ ∆ andδ ∈ (0, 1).
Proof. (i) By definition, {x, y}  {y} and C ({x, y}) = {y}. Because therestriction of  to singleton sets is continuous, and because ∆ is compact,
there exists δ1 ∈ (0, 1) such that {x, y(1 − δ)c}  {y(1 − δ)c} for all c ∈ ∆and δ ∈ (0, δ1 ).41 Also, by Axiom 3(iii-b) and compactness, we have some δ2 ∈(0, 1) such that C ({x, y(1 − δ)c}) = {y(1 − δ)c} for all c ∈ ∆ and δ ∈ (0, δ2 ).
Therefore, by taking δ P = min {δ1 , δ2 }, the first half of the statement holds for
41

Let A = {x, y} 0.5 {y}. By Axiom 4, we have {x, y}  A  {y}. By lower semicontinuity and singleton continuity, we have {x, y(1 − δ)c}  A and A  {y(1 − δ)c} for all
sufficiently small δ.

52

all δ < δ P . An analogous argument yields δ S suchthe second half of the thatS
P
S
satisfies the desired
statement holds for all δ < δ . Thus, δ < min δ , δ
property.
(ii) By Axiom 4(iii), {x(1 − δ)c, y(1 − δ)c}  {y(1 − δ)c} for any c ∈ ∆and δ ∈ (0, 1). Also, by Axiom 5(iii), C({x(1 − δ)c, y(1 − δ)c}) = {y(1 − δ)c}.
Thus, the first half of the statement holds. The second half can be proved
analogously.
We now construct u and w. For any a ∈ ∆, let u(a) = VP S ({a}). By
Nondegeneracy, there exist x, y, y 0 ∈ ∆ such that y ∈ P(x) and y 0 ∈ S(x).
Below, we fix such x, y, y 0 . By Lemma 15, there exists δ ∈ (0, 1) such that,for all c ∈ ∆, we have y(1 − δ)c ∈ P(x) and y 0 (1 − δ)c ∈ S(x). Note that by
Axioms 3(i) and 6(i), b ∈ P(a) ∪ S(a) implies {a, b}  {a} for any a, b ∈ ∆, so
we in particular have {x, y(1 − δ)c}  {x} and {x, y 0 (1 − δ)c}  {x}. Then,define wP and wS by
11−δVP S ({x, y}) − VP S ({c})
wP (c; x, y, δ) = VP S ({x, y(1 − δ)c}) −
δ
δ
1−δ
1
VP S ({x, y 0 }) − VP S ({c}) .
wS (c; x, y 0 , δ) = VP S ({x, y 0 (1 − δ)c}) −δδwP (c; x, y, δ) measures how the utility changes as the reference alternative x
is moved slightly toward c, keeping the ex post choice constant.wS (c; x, y 0 , δ)is interpreted analogously. The next two lemmas show some properties of
wP and wS , including its linearity and independence of the specific choice ofx, y, y 0 ∈ ∆.
Lemma 16. Suppose Axioms 1-6 hold. If y(1 − δ)c ∈ P(x) for all c ∈ ∆, thenthe following statements hold.
(i) If c ∈ P(x) , then wP (c; x, y, δ) = VP S ({x, c}) − VP S ({c}).
(ii) wP (x; x, y, δ) = 0.
(iii) wP (cαc0 ; x, y, δ) = αwP (c; x, y, δ) + (1 − α)wP (c0 ; x, y, δ) for any α ∈(0, 1).
(iv) wP (c; x, y, δ 0 ) = wP (c; x, y, δ) for any δ 0 ∈ (0, δ).
(v) Suppose b(1−δ)c ∈ P(a) for all c ∈ ∆. Then wP (c; x, y, δ) = wP (c; a, b, δ)+wP (a; x, y, δ).

53

Proof. (i) Because y, c ∈ P(x), VP S ({x, y(1 − δ)c}) = VP S ({x, y} (1 − δ) {x, c})by Lemma 14(i). Therefore,1−δ1
VP S ({x, y}) − VP S ({c})
wP (c; x, y, δ) = VP S ({x, y(1 − δ)c}) −δδ1= [(1 − δ)VP S ({x, y}) + δVP S ({x, c}) − (1 − δ)VP S ({x, y}) − δVP S ({c})]δ= VP S ({x, c}) − VP S ({c})
where the second equality follows from {x, y} , {x, c} ∈ BP .
(ii) By {x, y} , {x} ∈ BP , VP S ({x, y(1 − δ)x}) = (1 − δ)VP S ({x, y}) +δVP S ({x}), so11−δwP (x; x, y, δ) = VP S ({x, y(1 − δ)x}) −VP S ({x, y}) − VP S ({x})δδ= 0.
(iii) Because y(1 − δ)c, y(1 − δ)c0 ∈ P(x), Lemma 14(i) implies
VP S ({x, [y(1 − δ)c] α [y(1 − δ)c0 ]}) = VP S ({x, y(1 − δ)c} α {x, y(1 − δ)c0 }) .
Therefore,1wP (cαc0 ; x, y, δ) = VP S ({x, [y(1 − δ)c] α [y(1 − δ)c0 ]})δ
1−δ−
VP S ({x, y}) − VP S ({cαc0 })δ
1−αα
VP S ({x, y(1 − δ)c0 })= VP S ({x, y(1 − δ)c}) +
δ
δ
1−δ
−
VP S ({x, y}) − αVP S ({c}) − (1 − α)VP S ({c0 })δ
=αwP (c; x, y, δ) + (1 − α)wP (c0 ; x, y, δ).

0
δ−δ 0
y
+
1
−
[y(1 − δ)c].
(iv) Let δ 0 ∈ (0, δ). Note that y(1 − δ 0 )c = δ−δδδBecause y, y(1 − δ)c ∈ P(x), Lemma 14(i) implies




δ − δ0δ − δ0
VP S
x,
y+ 1−
[y(1 − δ)c]
δ
δ




δ − δ0δ − δ0{x, y} + 1 −{x, y(1 − δ)c} .
=VP S
δ
δ

54

0

Therefore, VP S ({x, y(1 − δ)c}) = δδ0 VP S ({x, y(1 − δ 0 )c}) − δ−δ
VP S ({x, y}).
δ0
Substituting this into the definition, we have
1−δ1
VP S ({x, y}) − VP S ({c})
wP (c; x, y, δ) = VP S ({x, y(1 − δ)c}) −δδ1δ − δ0= 0 VP S ({x, y(1 − δ 0 )c}) −VP S ({x, y})
δ
δδ 0
1−δ
−
VP S ({x, y}) − VP S ({c})δ=wP (c; x, y, δ 0 ).
(v) What to be shown is
wP (c; x, y, δ) = wP (c; a, b, δ) + wP (a; x, y, δ).
111−δ⇔ VP S ({x, y(1 − δ)c}) = VP S ({a, b(1 − δ)c}) −VP S ({a, b})δδδ
1− VP S ({a}) + VP S ({x, y(1 − δ)a}) .
δ
VP S ({a, b}). SubstiBy (ii), we have VP S ({a}) = 1δ VP S ({a, b(1 − δ)a}) − 1−δ
δtuting this into the above expression, our goal is to show
VP S ({x, y(1 − δ)c} 0.5 {a, b(1 − δ)a}) = VP S ({a, b(1 − δ)c} 0.5 {x, y(1 − δ)a}) .
Because y(1 − δ)c, y(1 − δ)a ∈ P(x) and b(1 − δ)a, b(1 − δ)c ∈ P(a), Lemma
14(i) implies that both sides of this equation equal to
VP S (x0.5a, [(1 − δ)(y + b)] 0.5 [δ(a + c)]) .

Lemma 17. Suppose Axioms 1-6 hold. If y 0 (1 − δ)c ∈ S(x) for all c ∈ ∆,then the following properties hold.
(i) If c ∈ S(x), then wS (c; x, y 0 , δ) = VP S ({x, c}) − VP S ({c}).
(ii) wS (x; x, y 0 , δ) = 0.
(iii) wS (cαc0 ; x, y 0 , δ) = αwS (c; x, y 0 , δ) + (1 − α)wS (c0 ; x, y 0 , δ) for any α ∈(0, 1).
(iv) wS (c; x, y 0 , δ 0 ) = wS (c; x, y 0 , δ) for all δ 0 ∈ (0, δ).

55

(v) Suppose b0 ∈ S(a0 ) for all c ∈ ∆. Then wS (c; x, y 0 , δ) = wS (c; a0 , b0 , δ) +wS (a0 ; x, y 0 , δ).
Proof is analogous to that of Lemma 16 and therefore omitted.
Now, we relate wP to wS using Axiom 7.
Lemma 18. Suppose Axioms 1-7 hold, and suppose y(1−δ)c ∈ P(x) and y 0 (1−δ)c ∈ S(x) for all c ∈ ∆. Then, there exists β > 0 such that wP (c; x, y, δ) =βwS (c; x, y 0 , δ) for all c ∈ ∆.
Proof. Let β = α1 − 1 > 0, where α is as defined in Axiom 7. We then have
δ [wP (c; x, y, δ) − βwS (c; x, y 0 , δ)]= VP S ({x, y(1 − δ)c}) − (1 − δ)VP S ({x, y}) − δVP S ({c})− β [VP S ({x, y 0 (1 − δ)c}) − (1 − δ)VP S ({x, y 0 }) − δVP S ({c})]1= [αVP S ({x, y(1 − δ)c}) + (1 − α)VP S ({x(1 − δ)c, y 0 (1 − δ)c})]α1− [αVP S ({x(1 − δ)c, y(1 − δ)c}) + (1 − α)VP S ({x, y 0 (1 − δ)c})]
α
n
oi
1h
x(1−δ)c,y 0 (1−δ)c=αVP S ({x, y(1 − δ)c}) + (1 − α)VP S
e
α
h
n 0
oi
1
x,y (1−δ)cαVP S ({x(1 − δ)c, y(1 − δ)c}) + (1 − α)VP Se−α=0where the last equality holds because y(1 − δ)c ∈ P(x) ∩ P(x(1 − δ)c) andy 0 (1 − δ)c ∈ S(x) ∩ S(x(1 − δ)c) hold by Lemma 15, so that Axiom 7 applies.
Thus, wP (c; x, y, δ) = βwS (c; x, y 0 , δ) where β > 0.
We now show that the representation holds for binary menus which include
x. Let w(c; x, y, y 0 , δ) = β1 wP (c; x, y, δ) = wS (c; x, y 0 , δ), which is well-definedby Lemma 18.
Lemma 19. Suppose Axioms 1-8 hold. Consider x, b ∈ ∆ such that r(x) ≥r(b) and VP S ({x, b}) ≥ VP S ({x}). Suppose y(1 − δ)c ∈ P(x) and y 0 (1 − δ)c ∈S(x) for all c ∈ ∆. Then VP S is expressed as
 

0
0
VP S ({x, b}) = max g c, 0 max w(c ; x, y, y , δ)c∈{x,b}c ∈ϕr ({x,b})56

and C ({x, b}) coincides with 
CP S ({x, b}) = arg max g c, 0 max

c ∈ϕr ({x,b})c∈{x,b}w(c ; x, y, y , δ)
0

0

where
g(c, R) = g(c, R; x, y, y 0 , δ)= u(c) − max {R − w(c; x, y, y 0 , δ), 0} + β max {w(c; x, y, y 0 , δ) − R, 0}and ϕr (A) = arg maxA r.
Proof. Note first that Axioms 3(i) and 6(i) imply VP S ({x, y(1 − δ)c}) > VP S ({x})
and VP S ({x, y 0 (1 − δ)c}) > VP S ({x}). Consider the following exhaustive
cases.
Case 1. Suppose b ∈ P(x). By Axioms 3(i) and 6(i), VP S ({x, b}) > VP S (x).
By definition, C ({x, b}) = {b}. By Lemma 16(i)(ii) and b ∈ P(x), w(b; x, y, y 0 , δ)−w(x; x, y, y 0 , δ) = β1 wP (b; x, y, δ) = β1 [VP S ({x, b}) − VP S ({b})] > 0. Therefore,g b, 0 max

c ∈ϕr ({x,b})00w(c ; x, y, y , δ)= u(b) + β [w(b; x, y, y 0 , δ) − w(x; x, y, y 0 , δ)]= VP S ({x, b})> VP S ({x})
= g x, 0 max

c ∈ϕr ({x,b})w(c ; x, y, y , δ) .
00Thus, the conclusion holds.
Case 2. Suppose b ∈ S(x) (and therefore VP S ({x, b}) > VP S (x)). By definition, C ({x, b}) = {b}. By Lemma 17(i)(ii) and b ∈ S(x), w(b; x, y, y 0 , δ) −w(x; x, y, y 0 , δ) = wS (b; x, y 0 , δ) = VP S ({x, b}) − VP S ({b}) < 0. Therefore,00g b, 0 max w(c ; x, y, y , δ) = u(b) + w(b; x, y, y 0 , δ) − w(x; x, y, y 0 , δ)c ∈ϕr ({x,b})= VP S ({x, b})> VP S ({x})
= g x, 0 max

c ∈ϕr ({x,b})

57

0

0



w(c ; x, y, y , δ) .

Therefore, the conclusion holds.
Case 3. Suppose b ∈ N1 (x) (so VP S ({x, b}) > VP S (x)). By definition,
C ({x, b}) = {b}. Letting A0 = {x, y 0 } (1 − δ) {x, b}, we also havew(b; x, y, y 0 , δ) − w(x; x, y, y 0 , δ) = wS (b; x, y 0 , δ)11−δ
= VP S ({x, y 0 (1 − δ)b}) −VP S ({x, y 0 }) − VP S ({b})δδ
1= [VP S (A0 ) − (1 − δ)VP S ({x, y 0 }) − δVP S ({b})]δ=0where the third equality follows from Lemma 14(ii) and the last equality follows
from {x, b} ∼ {b}. Therefore,00g b, 0 max w(c ; x, y, y , δ) = VP S ({b})
c ∈ϕr ({x,b})= VP S ({x, b})> VP S ({x})
= g x, 0 max

c ∈ϕr ({x,b})w(c ; x, y, y , δ) .
00Thus, the conclusion holds.
Case 4. Suppose b ∈ N2 (x). By Lemma 9, VP S ({x, b}) = VP S ({x}). By
definition, x ∈ C ({x, b}). Consider first the case where w(b; x, y, y 0 , δ) > 0. By
Axiom 4(iii),
1−δ
1
VP S ({x, y(1 − δ)b}) −VP S ({x, y}) − VP S ({b})δδ= wP (b; x, y, δ)>011−δ= VP S ({x, y} (1 − δ) {b}) −VP S ({x, y}) − VP S ({b}) ,δδ
so VP S ({x, y(1 − δ)b}) > VP S ({x, y} (1 − δ) {b}), which together with Axiom
5(ii-a) implies C (A) = {y} (1 − δ)C ({x, b}) where A = {x, y} (1 − δ) {x, b}.42
Then, by Axiom 6(i), {x, y(1 − δ)x} ∼ A  {x, (1 − δ)y + δb} and the latter42

To apply Axiom 5(ii), recall C({x, y(1 − δ)c}) = {y(1 − δ)c} for all c ∈ ∆.

58

relation is strict if and only if C ({x, b}) = {x}. Therefore,


1
0
0
g b, 0 max w(c ; x, y, y , δ) = u(b) + β · wP (b; x, y, δ)c ∈ϕr ({x,b})β11−δ≤ VP S ({x, y(1 − δ)x}) −VP S ({x, y})δδ= VP S ({x})= VP S ({x, b})00= g x, 0 max w(c ; x, y, y , δ)c ∈ϕr ({x,b})

where the inequality is strict if and only if C ({x, b}) = {x}. Next, consider
the case with w(b; x, y, y 0 , δ) ≤ 0. By Axiom 4(iii),
1−δ
1
VP S ({x, y 0 (1 − δ)b}) −VP S ({x, y 0 }) − VP S ({b})δδ= wS (b; x, y 0 , δ)
≤01−δ1VP S ({x, y 0 }) − VP S ({b}) ,= VP S ({x, y 0 } (1 − δ) {b}) −δδ
so VP S ({x, y 0 (1 − δ)b}) ≤ VP S ({x, y 0 } (1 − δ) {b}), which together with Axiom
5(ii-b) implies C (A0 ) = {y 0 } (1 − δ)C ({x, b}) where A0 = {x, y 0 } (1 − δ) {x, b}.
Then, by Axiom 6(i), {x, y 0 (1 − δ)x} ∼ A0  {x, y 0 (1 − δ)b} and the latter
relation is strict if and only if C ({x, b}) = {x}. Therefore,00g b, 0 max w(c ; x, y, y , δ) = u(b) + wS (b; x, y 0 , δ)c ∈ϕr ({x,b})1−δ
1≤ VP S ({x, y 0 (1 − δ)x}) −VP S ({x, y 0 })δδ
= VP S ({x})= VP S ({x, b})00= g x, 0 max w(c ; x, y, y , δ)c ∈ϕr ({x,b})

where the inequality is strict if and only if C ({x, b}) = {x}.
Case 5. Suppose b ∈ I(x) and VP S ({x, b}) > VP S (x). By Lemma 2, we have

59

VP S (b) ≥ VP S ({x, b}) > VP S (x). Also, Axiom 6 implies C ({x, b}) = {b}.43
By Lemma 14(iii), VP S ({x, y 0 (1 − δ)b}) ≥ VP S (A0 ) where A0 is defined above.
Now, consider first the case with VP S (b) = VP S ({x, b}) > VP S (x). Then,VP S ({x, y 0 (1 − δ)b}) ≥ (1 − δ)VP S ({x, y 0 }) + δVP S ({x, b})= (1 − δ)VP S ({x, y 0 }) + δVP S ({b}) ,so w(b; x, y, y 0 , δ) ≥ 0 = w(x; x, y, y 0 , δ). Therefore,00g b, 0 max w(c ; x, y, y , δ) = VP S ({b})
c ∈ϕr ({x,b})= VP S ({x, b})> VP S ({x})
≥ g x, 0 max

c ∈ϕr ({x,b})w(c ; x, y, y , δ) .
0

0

Next, consider the case with VP S (b) > VP S ({x, b}) > VP S (x). By Lemma
14(iii), we have VP S ({x, y 0 (1 − δ)b}) = VP S (A0 ) . Therefore,1−δ
1−δ
VP S ({x, y 0 }) + VP S ({x, b}) −VP S ({x, y 0 }) − VP S ({b})δδ< 0 = w(x; x, y, y 0 , δ).

w(b; x, y, y 0 , δ) =Thus,g b, 0 max

c ∈ϕr ({x,b})w(c ; x, y, y , δ) = u(b) + w(b; x, y, y 0 , δ)00

= VP S ({x, b})> VP S ({x})
= g x, 0 max

c ∈ϕr ({x,b})w(c ; x, y, y , δ) .
0

0Case 6. Suppose b ∈ I(x) and VP S ({b}) > VP S ({x, b}) = VP S (x). Note wehave x ∈ C({x, b}).44 Then, by Lemma 14(iv), VP S (A0 ) ≥ VP S ({x, y 0 (1 − δ)b}).
43

If x w b, then C({x, b}) = {b} by Axiom 6(i). Suppose b w x. Then, by Axiom 6(ii),x ∈ C({x, b}) implies {x}  {x, b}, a contradiction. Thus, C({x, b}) = {b}.
44
Otherwise, {b}  {x, b} implies x w b whereas {x} ∼ {x, b} and C({x, b}) = {b} implyb w x, contradicting Axiom 3(i).

60Therefore,1−δ1
VP S ({x, y 0 }) − VP S ({b})
w(b; x, y, y 0 , δ) ≤ VP S ({x, y 0 } (1 − δ) {x, b}) −δδ< 0 = w(x; x, y, y 0 , δ),so
g b, 0 max

c ∈ϕr ({x,b})

0

0



= u(b) + w(b; x, y, y 0 , δ)w(c ; x, y, y , δ)11−δ
≤ VP S ({x, y 0 } (1 − δ) {x, b}) −VP S ({x, y 0 })δδ
= VP S ({x, b})= VP S ({x})
00= g x, 0 max w(c ; x, y, y , δ) .
c ∈ϕr ({x,b})

By Lemma 14(iv) the inequality is strict if and only if C ({x, b}) = {x}.
Case 7. Suppose b ∈ I(x) and VP S ({b}) = VP S ({x, b}) = VP S (x).
C ({x, b}) = {x}, then by Lemma 14(v-a), we have
11−δ
VP S ({x, y 0 }) − VP S ({b})
w(b; x, y, y 0 , δ) < VP S ({x, y 0 (1 − δ)x}) −δδ= 0 = w(x; x, y, y 0 , δ).
Therefore,g b, 0 max

c ∈ϕr ({x,b})w(c ; x, y, y , δ) = VP S ({b}) + w(b; x, y, y 0 , δ)00

< VP S ({b})= VP S ({x, b})= VP S ({x})
= g x, 0 max

c ∈ϕr ({x,b})61w(c ; x, y, y , δ) .
0

0

If

If C ({x, b}) = {b}, then Lemma 14(v-a) implies
1−δ1
VP S ({x, y 0 }) − VP S ({b})
w(b; x, y, y 0 , δ) > VP S ({x(1 − δ)b, y 0 (1 − δ)b}) −δδ= 0 = w(x; x, y, y 0 , δ),so
g b, 0 max

c ∈ϕr ({x,b})w(c ; x, y, y , δ) = VP S ({b})
00= VP S ({x, b})= VP S ({x})
> g x, 0 max

c ∈ϕr ({x,b})w(c ; x, y, y , δ) .
0

0

Finally, if C ({x, b}) = {x, b}, then Lemma 14(v-b) implies
11−δw(b; x, y, y 0 , δ) = VP S ({x, y 0 (1 − δ)x}) −VP S ({x, y 0 }) − VP S ({b}) = 0,δδ
so the desired representation holds.
Case 8. Suppose b ∈ I(x) and VP S ({x, b}) > VP S (b). Note we must haveC ({x, b}) = {x} and VP S ({x}) = VP S ({x, b}).45 By Axiom 4, {x} ∼ {x, b} δ {x} {b} δ {x}. Therefore, x w x(1 − δ)b by case (i-c) of Definition 2. Thus,
Axioms 3(i) and 6(i)imply {x, x(1 − δ)b, y 0 (1 − δ)b}  {x, y 0 (1 − δ)b} . Also,
C ({x, y 0 (1 − δ)b}) = {y 0 (1 − δ)b} by construction, and Axiom 8 imply x ∈/C ({x, x(1 − δ)b, y 0 (1 − δ)b}). Therefore, by Axiom 6(ii), {x(1 − δ)b, y 0 (1 − δ)b} {x, x(1 − δ)b, y 0 (1 − δ)b}. Combining these results, {x(1 − δ)b, y 0 (1 − δ)b} {x, y 0 (1 − δ)b}. Then,
11−δ
VP S ({x, y 0 }) − VP S ({b})
w(b; x, y, y 0 , δ) < VP S ({x, y 0 } (1 − δ) {b}) −δδ= 0 = w(x; x, y, y 0 , δ),45

The former can be shown by applying Axiom 6 to each of the cases b w x and x w b.
The latter follows from Lemma 2(ii) and the assumption that VP S ({x, b}) ≥ VP S ({x}).

62

so

g b, 0 max

c ∈ϕr ({x,b})w(c ; x, y, y , δ) = VP S ({b}) + w(b; x, y, y 0 , δ)00

< VP S ({b})< VP S ({x, b})= VP S ({x})
= g x, 0 max

c ∈ϕr ({x,b})w(c ; x, y, y , δ) .
00Now, we can prove that the representation holds for an arbitrary binarymenu {a, b}.
Lemma 20. Suppose Axioms 1-8 and Nondegeneracy hold. Then there exists
continuous and linear functions u, w and r such that VP S is expressed as
0
VP S ({a, b}) = max g c, 0 max w(c )c∈{a,b}c ∈ϕr ({a,b})

and C ({a, b}) coincides with
CP S ({a, b}) = arg max g c, 0 max
c∈{a,b}c ∈ϕr ({a,b})

0



w(c )

where g and ϕ are defined above.
Proof. Take x, y, y 0 ∈ ∆ such that y ∈ P(x) and y 0 ∈ S(x), which exist
by Nondegeneracy. Take an arbitrary {a, b}. First, consider the case witha, b ∈ int(∆). By Lemma 15 and Axioms 3(i) and 6(i), we can choose asufficiently small δ ∈ (0, 1) such that y(1 − δ)b ∈ P(x), VP S ({x, y(1 − δ)b}) >VP S ({x}), y 0 (1 − δ)b ∈ S(x), and VP S ({x, y 0 (1 − δ)b}) > VP S ({x}). Without
loss of generality, suppose r(a) ≥ r(b) and VP S ({a, b}) ≥ VP S ({a}).46 Becausea ∈ int(∆), there exist ā ∈ ∆ and α ∈ (0, 1) such that a = āαx. Definez = āαy and z 0 = āαy 0 . Then, by Lemma 15 and Axioms 3(i), 4, and 6(i), we
have z ∈ P(a), VP S ({a, z}) = αVP S ({ā}) + (1 − α)VP S ({x, y}) > VP S ({a}),z 0 ∈ S(a), and VP S ({a, z 0 }) = αVP S ({ā}) + (1 − α)VP S ({x, y 0 }) > VP S ({a}).
46

If r(a) > r(b), then Lemma 2(i) implies VP S ({a, b}) ≥ VP S ({a}). If r(a) = r(b), thenLemma 2(ii) implies VP S ({a, b}) ≥ VP S ({a}) or VP S ({a, b}) ≥ VP S ({b}), so the assumption
is without loss of generality.

63

By Lemma 15, Axiom 2(i), and compactness, for sufficiently small δ 0 ∈ (0, 1),
we have z(1 − δ 0 )c ∈ P(a), VP S ({a, z(1 − δ 0 )c}) > VP S ({a}), z 0 (1 − δ 0 )c ∈S(a), and VP S ({a, z 0 (1 − δ 0 )c}) > VP S ({a}) for all c ∈ ∆. Defining g(c, R) =g(c, R; a, z, z 0 , δ 0 ) as in Lemma 19, VP S can be expressed as


0
0 0
VP S ({a, b}) = max g c, 0 max w(c ; a, z, z , δ )c∈{a,b}c ∈ϕr ({a,b})

and C ({a, b}) coincides with
CP S ({a, b}) = arg max g c, 0 max
c∈{a,b}c ∈ϕr ({a,b})w(c ; a, z, z , δ ) .
000Now, let δ ∗ = min {δ, δ 0 }. Then, by Lemmas 16 and 17, we have w(·; x, y, y 0 , δ) =w(·; x, y, y 0 , δ ∗ ), w(·; a, z, z 0 , δ 0 ) = w(·; a, z, z 0 , δ ∗ ) and w(·; a, z, z 0 , δ ∗ ) = w(·; x, y, y 0 , δ ∗ )+Constant. Therefore, defining w(·) = w(·; x, y, y 0 , δ) yields the conclusion.47
Next, suppose a ∈ ∆ and b ∈ int(∆). Because aαb ∈ int(∆) for α ∈ (0, 1),
0
VP S ({aαb, b}) = max g c, 0 max
w(c )c∈{aαb,b}c ∈ϕr ({aαb,b})
0⇔ αVP S ({a, b}) + (1 − α)VP S ({b}) = α max g c, 0 max w(c ) + (1 − α)u(b).
c∈{a,b}c ∈ϕr ({a,b})

where the right-hand side follows from the linearity of u, w and r.48 Letting
α → 1 yields the conclusion for VP S . The conclusion for CP S is obtained
analogously using Axiom 5. Proof for the general case with a, b ∈ ∆ is now
straightforward.
47
48

Note δ does not depend on (a, b).
Note that for x ∈ {a, b},max

c0 ∈ϕr ({a,b}α{b})Therefore,g xαb, 0max

c ∈ϕr ({aαb,b})0w(c0 )

w(c ) − w(xαb) = αmax

c0 ∈ϕr ({a,b})0

w(c ) − w(x) .







= αu(x) + (1 − α)u(b) − α max 0 max
w(c0 ) − w(x), 0 + αβ max w(x) − 0 max
w(c0 ), 0c ∈ϕr ({a,b})c ∈ϕr ({a,b})= αg x, 0 max
w(c0 ) + (1 − α)u(b).
c ∈ϕr ({a,b})64

We now extend the representation to any finite menus.
Lemma 21. Suppose Axioms 1-8 and Nondegeneracy hold. Then there exists
continuous and linear functions u, w and r such that VP S is expressed as
0
VP S (A) = max g c, 0 max w(c )c∈Ac ∈ϕr (A)and C (A) coincides with
0
CP S (A) = arg max g c, 0 max w(c )c∈Ac ∈ϕr (A)where g and ϕ are defined above.
Proof. Take any finite set A. By Lemma 3, Lemma 20, and Eq.(11), andbecause g(c, ·) is strictly decreasing at each c ∈ ∆,VP S (A) = min max VP S ({a, b})b∈ϕr (A) a∈A
0= min max max g c, 0 max w(c )b∈ϕr (A) a∈A c∈{a,b}c ∈ϕr ({a,b})
 
 

0
0
= min max max g a, 0 max w(c ) , g b, 0 max w(c )b∈ϕr (A) a∈Ac ∈ϕr ({a,b})c ∈ϕr ({a,b})= min max g a, 0 max w(c0 )b∈ϕr (A) a∈Ac ∈ϕr ({a,b})
0= max g a, 0 max w(c ) .
a∈Ac ∈ϕr (A)where the fourth equality holds because b ∈ ϕr (A) implies0
max g b, 0 max w(c ) ≤ g (b, w(b)) ≤ max g a, 0 max
a∈A

a∈A

c ∈ϕr ({a,b})c ∈ϕr ({a,b})0

w(c ) .

To prove the second result, we first introduce some lemmas. Lemmas 22
and 23 establish Lemma 24, which then establishes Lemma 21.
Lemma 22. Suppose Axioms 1-8 hold, so that the representation in Lemma
20 holds for binary menus. If r(a) = r(b) and w(a) > w(b), then a w b.
Proof. Consider the following exhaustive cases.
65

Case 1. Suppose u(a) + w(a) ≤ u(b) + w(b). Then VP S ({a, b}) = u(b) +w(b) − w(a) < VP S ({b}), so a w b.
Case 2. Suppose u(a) + w(a) > u(b) + w(b) and u(a) < u(b). ThenVP S ({a, b}) = u(a) < VP S ({b}), so a w b.
Case 3. Suppose u(a) + w(a) > u(b) + w(b) and u(a) = u(b). ThenVP S ({a, b}) = u(a) = VP S ({b}) and C ({a, b}) = {a}, so a w b.
Case 4. Suppose u(a) + w(a) > u(b) + w(b) and u(a) > u(b). ThenVP S ({a}) = VP S ({a, b}) > VP S ({b}). Also, since r represents r , a ∼r b.
Thus, a w b.
Lemma 23. Suppose Axioms 1-8 and Weak Nondegeneracy hold, so that the
representation in Lemma 20 holds for binary menus. If r(a) = r(b) and a 6= b,
then w(a) 6= w(b).
Proof. By the proof of Claim 1(i), Weak Nondegeneracy implies r(x) > r(y)and w(x) < w(y) for some x, y ∈ ∆. Now, suppose r(a) = r(b) and w(a) =w(b) for some a 6= b. Since the indifference curves for r and w are parallel
straight lines, this implies that the indifference curves for r and w are also
parallel, which contradicts r(x) > r(y) and w(x) < w(y).
Lemma 24. Suppose Axioms 1-8 and Weak Nondegeneracy hold, so that the
representation in Lemma 20 holds for binary menus. Then, for any A ∈ A
and any b ∈ arg maxc0 ∈ϕr (A) w(c0 ), we have b r a and b w a for all a ∈ A.
Proof. Take any a ∈ A \ {b}. Because b ∈ ϕr (A), b r a. If a ∈/ ϕr (A), thenb r a, so Axiom 3(i) implies b w a. If a ∈ ϕr (A), then r(a) = r(b) andw(a) ≤ w(b) by definition. By Lemma 23, w(a) < w(b). Thus, b w a by
Lemma 22.
Proof of Lemma 21 (Continued). We show the following equality in two steps:
0C(A) = arg max g c, 0 max w(c ) .
c∈Ac ∈ϕr (A)
Step 1. Take a ∈ arg maxc∈A g c, maxc0 ∈ϕr (A) w(c0 ) , b ∈ arg maxc0 ∈ϕr (A) w(c0 )and d ∈ C (A). By Lemma 24, we have b r a0 and b w a0 for all a0 ∈ A.
By the representation of VP S , {a, b, d} ∼ A. By Axiom 8, d ∈ C ({a, b, d}).
Also, by the representation and Lemma 20, we have {a, b} ∼ {a, b, d} anda ∈ C ({a, b}). By Axiom 6(i), we have a ∈ C ({a, b, d}) or b ∈ C ({a, b, d}), so
Axiom 8 implies a ∈ C ({a, b, d}). Thus, again by Axiom 8, a ∈ C (A).

66


Step 2. Suppose d ∈ C (A). Take a ∈ arg maxc∈A g c, maxc0 ∈ϕr (A) w(c0 )and b ∈ arg maxc0 ∈ϕr (A) w(c0 ). If a = d, we are done, so suppose not. By step
1, we have a ∈ C (A). We also have b r a0 and b w a0 for all a0 ∈ A, so
Axiom 8 implies d ∈ C ({a, b, d}). By Axiom 6(i), {b, d} ∼ {a, b, d}. By the
representation and definition of a and b,






0
0
0
max g c, 0 max w(c ) = max g c, 0 max
w(c ) = max g c, 0 max w(c ) .

c∈{b,d}c ∈ϕr ({b,d})c ∈ϕr ({a,b,d})c∈{a,b,d}c∈Ac ∈ϕr (A)00 ∈ϕ ({b,d}) w(c ) .
By Axiom 8, d ∈ C ({b, d}), so Lemma 20 implies
d
∈
arg
max
g
c,
max
c∈{b,d}crThus d ∈ arg maxc∈A g c, maxc0 ∈ϕr (A) w(c0 ) .
To complete the proof of the “only if” part of Theorem 1, we use thefollowing result.
Lemma 25. Suppose Axioms 1-8 hold. Suppose that for any A ∈ A, there
exists a finite subset A0 of A such that (i) maxA r = maxA0 r and (ii) for any
finite A00 such that A0 ⊂ A00 ⊂ A, A00 ∼ A0 . Then we have A ∼ A0 .
Proof. Note that there exists a sequence of finite subsets {An }∞n=1 of A suchthat dH (An , A) → 0 as n → ∞ by Lemma 0 in GP. By (ii), A0 ∼ An ∪ A0 for
all n, so A0 % A ∪ A0 = A by Axiom 2(i). To show the opposite relation, notethat since A is compact, for every  > 0, there are finite x1 , · · · , xn ∈ A suchthat A ⊂ ∪ni=1 N (xi , ). If (A0 ∪ N (xi , )) ∩ A  A for all i = 1, · · · , n, theniteratively applying Lemma 2(ii) yields A = ∪ni=1 {(A0 ∪ N (xi , )) ∩ A}  A,which is a contradiction. Therefore, A % (A0 ∪ N (xi , )) ∩ A for some i. Thus,
10
we can take a sequence {xn }∞n=1 in A such that A % (A ∪ N (xn , n )) ∩ A for
all n = 1, 2, · · · . Since A is compact, there exists a subsequence {xnk }∞k=1 such∗0∗0that xnk → x ∈ A. Then letting k → ∞ yields A % A ∪ {x } ∼ A by Axiom
2(i).
Proof of the Theorem 1 (Continued).

Take any closed set A ∈ A, and take a∗ ∈ arg maxc∈A g c, maxc0 ∈ϕr (A) w(c0 )and b∗ ∈ arg maxc0 ∈ϕr (A) w(c0 ). By construction, max{a∗ ,b∗ } r = maxA r. By
Lemma 21, {a∗ , b∗ } ∼ A00 for any finite A00 such that {a∗ , b∗ } ⊂ A00 ⊂ A.
Therefore, by Lemma 25, we have A ∼ {a∗ , b∗ }. Thus, defining




∗ ∗
0
0
VP S (A) = VP S ({a , b }) = max
g c, 0 max∗ ∗ w(c ) = max g c, 0 max w(c ) ,∗ ∗c∈{a ,b }c ∈ϕr ({a ,b })

67

c∈A

c ∈ϕr (A)VP S represents  on A. Also, following the argument for the proof of Lemma
21, we obtain C (A) = CP S (A) = arg maxc∈A g c, maxc0 ∈ϕr (A) w(c0 ) .
Finally, by construction, β = α1 − 1 if the DM is α-sensitive to shame.
Thus, the DM is shame-averse (α > 12 ) if and only if β < 1, shame-neutral(α = 12 ) if and only if β = 1 and shame-loving (α < 12 ) if and only if β > 1.
Proof of “if” part.
Below, we show that a nondegenerate PS representation implies each of theaxioms. Proofs of Axiom 1 and Axiom 2c are straightforward and omitted.
To proceed to other axioms, we first note thatg(c, R) = u(c) − max {R − w(c), 0} + β max {w(c) − R, 0}andG(A, R) = max g(c, R)
c∈A

are strictly decreasing and continuous in R and continuous in A.
Axiom 2(i) (Lower Semi-Continuity). Suppose A  Bn and Bn → B. Then,because maxc0 ∈ϕr (B) w(c0 ) ≥ limn→∞ maxc0 ∈ϕr (Bn ) w(c0 ),0VP S (A) = G A, 0 max w(c )c ∈ϕr (A)
0≥ lim G Bn , 0 max w(c )n→∞c ∈ϕr (Bn )0= G B, lim 0 max w(c )n→∞ c ∈ϕr (Bn )0≥ G B, 0 max w(c )c ∈ϕr (B)

= VP S (B)where the first inequality follows from A  Bn for all n.

68Axiom 2(ii) (Upper vNM continuity). Suppose A  B  C. Note0
VP S (AαC) = G AαC, 0 max w(c )c ∈ϕr (AαC)
00= G AαC, α 0 max w(c ) + (1 − α) 0 max w(c )c ∈ϕr (A)c ∈ϕr (C)By the continuity of G, VP S (AαC) ≈ VP S (C) < VP S (B) for sufficiently smallα ∈ (0, 1).
We now introduce some lemmas.
Lemma 26. Suppose the data are generated by a nondegenerate PS preference.
If a w b, then r(a) ≥ r(b) and w(a) > w(b).
Proof. Consider the following exhaustive cases.
Case 1. Suppose {b}  {a, b}. Then0u(b) > max g c, 0 max w(c ) ≥ g b, 0 max
c∈{a,b}c ∈ϕr ({a,b})c ∈ϕr ({a,b})
w(c ) ,
0so maxc0 ∈ϕr ({a,b}) w(c0 ) > w(b), yielding the conclusion.
Case 2. Suppose {b} ∼ {a, b} and C({a, b}) = {a}. Then




0
0
u(b) = g a, 0 max w(c ) > g b, 0 max w(c ) ,c ∈ϕr ({a,b})c ∈ϕr ({a,b})

so the conclusion holds as in Case 1.
Case 3. Suppose a ∼r b and {a} ∼ {a, b}  {b}. Then maxc0 ∈ϕr ({a,b}) w(c0 ) =maxc0 ∈{a,b} w(c0 ) ≥ w(b), so we must have C({a, b}) = {a}. Therefore,
0u(a) = max g c, 0max w(c ) .
c ∈{a,b}c∈{a,b}This in turn implies maxc0 ∈{a,b} w(c0 ) = w(a), so w(a) ≥ w(b). By Nondegeneracy and r(a) = r(b), we have w(a) > w(b).
Lemma 27. Suppose the data are generated by a PS preference. If A∪{a}  A
and a ∈/ C (A ∪ {a}), then w(a) = maxc0 ∈ϕr (A∪{a}) w(c0 ) < maxc0 ∈ϕr (A) w(c0 ).
Proof. Suppose A ∪ {a}  A and a ∈/ C (A ∪ {a}). If maxc0 ∈ϕr (A∪{a}) w(c0 ) ≥69

maxc0 ∈ϕr (A) w(c0 ), then
0
VP S (A ∪ {a}) = G A ∪ {a} , 0 max w(c )c ∈ϕr (A∪{a})
0= G A, 0 max w(c )c ∈ϕr (A∪{a})
0≤ G A, 0 max w(c )c ∈ϕr (A)= VP S (A)where the second equality follows from a ∈/ C (A ∪ {a}). This is a contra0
diction. Thus maxc0 ∈ϕr (A∪{a}) w(c ) < maxc0 ∈ϕr (A) w(c0 ), and we must havemaxc0 ∈ϕr (A∪{a}) w(c0 ) = w(a).49
Proof of “if” part (Continued).
Axiom 3(i). If a r b, then Theorem 2 (to be proven below) implies r(a) >r(b). Thus, Theorem 2 again implies b r a, and Lemma 26 implies b w a.
Similarly, if a w b, then Lemma 26 implies r(a) ≥ r(b) and w(a) > w(b), so
we cannot have b r a or b w a.
Axiom 3(ii). Suppose a ∗ b ∗ c. By definition, there exist A 3 b and B 3 csuch that A ∪ {a}  A, a ∈/ C(A ∪ {a}), B ∪ {b}  B, and b ∈/ C(B ∪ {b}).
00Now, let C = A ∪ B. Because r(a) > r(a ) for all a ∈ A and r(b) > r(b0 )for all b0 ∈ B, we have r(a) > r(c0 ) for all c0 ∈ C. Then, together with
Lemma 27, we have w(a) < maxc0 ∈ϕr (A) w(c0 ) = maxc0 ∈ϕr (C) w(c0 ). Therefore,
the representation implies a ∈/ arg maxd∈C∪{a} g (d, w(a)) = C(C ∪ {a}) andC ∪ {a}  C. Thus, a ∗ c.
Next, if a ∼r b ∼r c, Theorem 2 (to be shown below) implies [a w b] ∧[b w c] ⇔ [w(a) ≥ w(b)] ∧ [w(b) ≥ w(c)] ⇒ w(a) ≥ w(c) ⇔ a w c.
Axiom 3(iii-a). Suppose Aαn C  B and αn → α. Because


0
0
0
lim 0 max w(c ) = lim αn 0 max w(c ) + (1 − αn ) 0 max w(c )n→∞ c ∈ϕr (Aαn C)n→∞c ∈ϕr (A)= 0 max0c ∈ϕr (AαC)

49c ∈ϕr (C)w(c ),

Note that for any A, B ∈ A, ϕr (A ∪ B) ∈ {ϕr (A) , ϕr (B) , ϕr (A) ∪ ϕr (B)} .

70

we have



w(c )
0

VP S (AαC) = G AαC, 0 max
c ∈ϕr (AαC)
0= lim G Aαn C, 0 max w(c )n→∞c ∈ϕr (Aαn C)0≥ G B, 0 max w(c )c ∈ϕr (B)

= VP S (B)where the inequality follows from VP S (B) ≤ VP S (Aαn C) for all n.
Axiom 3(iii-b). Suppose a∗ ∈ A is such that a∗ r a for all a ∈ A \ {a∗ }.
Take any (An )n and any (an )n such that An → A, an ∈ C(An ) and an → a.
By Theorem 2, r(a∗ ) > r(a) for all a ∈ A\{a∗ }, so limn→∞ maxc0∈ϕr (An ) w(c0 ) =maxc0 ∈ϕr (A) w(c0 ) = w(a∗ ). By continuity, g c, maxc0 ∈ϕr (An ) w(c0 ) → g (c, w(a∗ ))for all c. By an ∈ C(An ), we have




0
0
g an , 0 max w(c ) = max g c, 0 max w(c ) ,c ∈ϕr (An )c∈Anc ∈ϕr (An )so letting n → ∞ yields g a, maxc0 ∈ϕr (A) w(c0 ) = maxc∈A g c, maxc0 ∈ϕr (A) w(c0 ) .
Thus, a ∈ C(A).
Axiom 3(iv). By Lemma 30 below, aαc ∗ bαc ⇒ r(a) > r(b) ⇒ a ∗ b.
To prove some of the remaining axioms, we need a lemma.
Lemma 28. Suppose the choice data are generated by a PS preference. (i)If b ∈ P(a), then w(a) < w(b). (ii) If b ∈ S(a), then w(a) > w(b). (iii) If
b ∈ N1 (a), then w(a) = w(b).
Proof. (i) By the representation and the definition of P(a), g(b, w(a)) > u(b) =g(b, w(b)), so w(a) < w(b). (ii) If b ∈ S(a), g(b, w(a)) < g(b, w(b)), so w(a) >w(b). (iii) If b ∈ N1 (a), g(b, w(a)) = g(b, w(b)), so w(a) = w(b).
Proof of Theorem 1 (Continued).
Axiom 4(i). We only provide the proof for the case where all menus arebinary; proof is analogous and simpler if a singleton is involved. Suppose71b ∈ P(a) ∪ N1 (a), d ∈ P(c) ∪ N1 (c) and f ∈ P(e) ∪ N1 (e). By Lemma 28(i),r(a) > r(b), w(a) ≤ w(b), r(e) > r(f ) and w(e) ≤ w(f ). Therefore,
VP S (α {a, b} α {e, f })



=
max
u(x) − maxmaxw(y) − w(x), 0x∈{a,b}α{e,f }y∈ϕ({a,b}α{e,f })+β max w(x) −maxw(y), 0y∈ϕ({a,b}α{e,f })=maxx∈{a,b}α{e,f }[u(x) + β (w(x) − w(aαe))]= α max [u(x) + β (w(x) − w(a))] + (1 − α) max [u(x) + β (w(x) − w(e)))]x∈{e,f }x∈{a,b}= αVP S ({a, b}) + (1 − α)VP S ({e, f }).
Similarly, VP S ({c, d} α {e, f }) = αVP S ({c, d}) + (1 − α)VP S ({e, f }). Therefore,{a, b}  () {c, d} implies {a, b} α {e, f }  () {c, d} α {e, f }.
Axiom 4(ii). Take any {a, b} , {e, f } ∈ BS such that a 6= b and e 6= f . By
Lemma 28(ii) we can assume without loss of generality that r(a) ≥ r(b),w(a) ≥ w(b), r(e) ≥ r(f ) and w(e) ≥ w(f ). Then, similarly to above,
VP S ({a, b} α {e, f }) =max

x∈{a,b}α{e,f }[u(x) + w(x) − w(aαe)]= αVP S ({a, b}) + (1 − α)VP S ({e, f }).
Therefore, the conclusion of Axiom 4(ii) holds.
Axiom 4(iii). Note that for any x ∈ A,max

c0 ∈ϕr (Aα{c})Therefore,g xαc, 0 max

c ∈ϕr (Aα{c})0w(c ) − w(xαc) = α0max w(c ) − w(x) .
00c ∈ϕr (A)



w(c )





0
max w(c ) − w(x), 0 + αβ max w(x) − 0 max w(c ), 00

= αu(x) + (1 − α)u(c) − α max
c ∈ϕr (A)= αg x, 0 max w(c0 ) + (1 − α)u(c).
c ∈ϕr (A)72

0c ∈ϕr (A)

Thus,
0
VP S (Aα {c}) = max g xαc, 0 max w(c )x∈Ac ∈ϕr (Aα{c})0= α max g x, 0 max w(c ) + (1 − α)u(c)x∈Ac ∈ϕr (A)= αVP S (A) + (1 − α)VP S ({c}) ,which implies the conclusion.
Axiom 5(i). Consider first the case where b ∈ P(a) ∪ N1 (a) and d ∈ P(c) ∪N1 (c). Following the argument in the proof of Axiom 4(i),C({a, b} α {c, d})= arg max [u(x) + β (w(x) − w(aαc))]x∈{a,b}α{c,d}= α arg max [u(x) + β (w(x) − w(a))] + (1 − α) arg max [u(x) + β (w(x) − w(c))]
x∈{a,b}x∈{c,d}= C({a, b})αC({c, d}).
Proof for the case b ∈ S(a) ∪ N1 (a) ∪ I(a) and d ∈ S(c) ∪ N1 (c) ∪ I(a) is
analogous: letting w(a) ≥ w(b) and w(c) ≥ w(d) without loss of generality
and following the proof of Axiom 4(ii),C({a, b} α {c, d}) = arg max [u(x) + w(x) − w(aαc)]x∈{a,b}α{c,d}= C({a, b})αC({c, d}).
Axiom 5(ii). For (ii-a), suppose A = {a, b} α {a, c}, b ∈ N2 (a), c ∈ P(a),{a, bαc}  {b} α {a, c}, and bαc ∈ C ({a, bαc}). By the PS representation,u(bαc) − max {w(a) − w(bαc), 0} + β max {w(bαc) − w(a), 0}= VP S ({a, bαc})≥ VP S ({b} α {a, c})≥ u(bαc) + β (w(bαc) − w(bαa)) .
Therefore, we must have w(bαa) ≥ w(a), so w(b) ≥ w(a). Because r(a) >r(b), r(c) and w(a) ≤ w(b), w(c), the same argument as the proof of Axiom
5(i) yields C (A) = C ({a, b}) αC ({a, c}). Proof for (ii-b) is analogous: the73

assumptions implyu(bαc) + w(bαc) − w(bαa) ≥ u(bαc) + w(bαc) − w(a)and therefore w(a) ≥ w(b), w(c), so the argument in the proof of Axiom 4(ii)yields the result.
Axiom 5(iii). Proof for is analogous to that of Axiom 4(iii), so omitted.
Axiom 6(i). Suppose there exists a∗ ∈ A such that a∗ r c and a∗ w c for
all c ∈ A ∪ B. Then, by Theorem 2, maxy∈ϕr (A∪B) w(y) = maxy∈ϕr (A) w(y) =w(a∗ ). Therefore,
VP S (A ∪ B) = max g (x, w(a∗ )) ≥ max g (x, w(a∗ )) = VP S (A)
x∈A∪B

x∈A

and the inequality is strict if and only if arg max g (x, w(a∗ )) ∩ A = ∅.
x∈A∪B∗

Axiom 6(ii). Suppose there exists a ∈ A such that a∗ r c for all c ∈ A ∪ B
and a∗ w a for all a ∈ A, and there exists b∗ ∈ B such that b∗ w a∗ . By
Lemma 26 and Theorem 2, r(b∗ ) = r(a∗ ) ≥ r(b) for all b ∈ B and w(b∗ ) >w(a∗ ). Without loss of generality, let b∗ be a maximizer of w on ϕr (B).
Then by Themrem 2, maxy∈ϕr (A∪B) w(y) = w(b∗ ) > w(a∗ ) = maxy∈ϕr (A) w(y).
Therefore, if there exists c ∈ C(A ∪ B) ∩ A, then
VP S (A ∪ B) = g (c, w(b∗ )) < g (c, w(a∗ )) ≤ VP S (A).
Axiom 7. Take any a, b, c, d ∈ ∆ such that c ∈ P(a)∩P(b) and d ∈ S(a)∩S(b).
Then,
VP S ({a, c}) + βVP S ({b, d}) = u(c) + β (w(c) − w(a)) + β [u(d) + w(d) − w(b)]= u(c) + β (w(c) − w(b)) + β [u(d) + w(d) − w(a)]= VP S ({b, c}) + βVP S ({a, d}).

74

1
Therefore, by letting α = 1+β∈ (0, 1),

VP S {a, c} α eb,d = αVP S ({a, c}) + (1 − α)VP S eb,d= αVP S ({a, c}) + (1 − α)VP S ({b, d})= αVP S ({b, c}) + (1 − α)VP S ({a, d})= αVP S ({b, c}) + (1 − α)VP S ea,d
= VP S {b, c} α ea,d .
Axiom 8. Suppose there exists a∗ ∈ A ∩ B such that a∗ r c and a∗ w c for
all c ∈ A ∪ B. By Theorem 2, maxy∈ϕr (A) w(y) = maxy∈ϕr (B) w(y) = w(a∗ ).
Now, suppose a, b ∈ A∩B, a ∈ C (A) and b ∈ C (B). By the PS representation,g (a, w(a∗ )) = g (b, w(a∗ )) ≥ g (c, w(a∗ )) for all c ∈ A∪B. Therefore, a ∈ C (B).

A.2

Proof of Theorem 2

Lemma 29. Suppose the data are generated by a nondegenerate PS preference.
Then,
P1 (a) = {c ∈ ∆ : r(a) > r(c) and u(a) < g(c, w(a))}
andP2 (a) = {c ∈ ∆ : r(a) > r(c) and w(a) < w(c)}are nonempty for all a ∈ int(∆).
Proof. Take ā, b̄ ∈ ∆ such that ā ∗ b̄. Then, there exist A 3 b̄ and c 6= ā suchthat A∪{ā}  A, ā ∈/ C (A ∪ {ā}), and c ∈ C (A ∪ {ā}). By Lemma 27, w(a) =maxy∈ϕr (A∪{ā}) w(y) < maxy∈ϕr (A) w(y), so we must have ϕr (A ∪ {ā}) = {ā},i.e., r(ā) > r(a) for all a ∈ A. Therefore, there exists some d ∈ A such thatw(d) = maxy∈ϕr (A) w(y) > w(ā) and r(d) < r(ā). Thus, d ∈ P2 (ā). Also, thePS representation and ā ∈/ C (A ∪ {ā}) imply g(c, w(ā)) > g(ā, w(ā)) = u(ā),
so c ∈ P1 (ā).
Now, take any a ∈ int(∆). There exist some α ∈ (0, 1] and e ∈ ∆such that a = āαe.50 By the linearity of u, w, and r, we have r(a) >max {r(cαe), r(dαe)}, g(cαe, w(a)) > g(a, w(a)) = u(a), and w(a) < w(dαe).
Therefore, cαe ∈ P1 (a) and dαe ∈ P2 (a).
50

To see this, note first that by taking e = a − ε(āP− a) where ε > 0 is an arbitrary scalar,ε
we obtain a = ā 1+εe. By construction, we have z∈Z e(z) = 1. Because a ∈ int(∆), we
have a(z) > 0 for all z ∈ Z. Because |Z| < ∞, e ∈ int(∆) for sufficiently small ε.

75c ∈ P1 (a) is an alternative which is below a in the descriptive norm ranking
r but which is a choice preferred to a. d ∈ P2 (a) is an alternatives which is
below a in the descriptive norm ranking but above a in the prescriptive norm
ranking w. Therefore, if d ∈ A ∩ P2 (a) sets the reference point at menu A,
then a sets the reference point at A ∪ {a} which is lower than the reference
point at A. Moreover, if c ∈ A ∩ P1 (a), then a is not chosen from A ∪ {a}.
Such c, d are key to establishing a ∗ b for any b ∈ A. Lemma 30 formalizes
the idea. See also the graphical illustration in Figure A1 and discussions inAppendix B.
Lemma 30. Suppose the data are generated by a nondegenerate PS preference.
Then, for any a, b ∈ ∆, a ∗ b implies r(a) > r(b). Moreover, if a ∈ int(∆),
then r(a) > r(b) implies a ∗ b.
Proof. Suppose a ∗ b. Then there exists A 3 b such that A ∪ {a}  A
and a ∈/ C (A ∪ {a}). By Lemma 27, we must have ϕr (A ∪ {a}) = {a},hence r(a) > r(b). Next, suppose a ∈ int(∆) and r(a) > r(b). By Lemma
29, there exists c ∈ P1 (a) and d ∈ P2 (a). Note that d ∈ P2 (a) implies
αd + (1 − α)a ∈ P2 (a) for all α ∈ (0, 1). Therefore, we can assume without
loss of generality that r(a) > r(d) > max {r(b), r(c)}. Then,
VP S ({a, b, c, d}) = G {a, b, c, d} ,maxw(y)y∈ϕr ({a,b,c,d})= G ({a, b, c, d} , w(a))> G ({a, b, c, d} , w(d))≥ G ({b, c, d} , w(d))= VP S ({b, c, d})where the strict inequality follows from w(a) < w(d). Moreover, becausec ∈ P1 (a), we have g (c, w(a)) > u(a) = g (a, w(a)) so a ∈/ C ({a, b, c, d}).
Thus, a ∗ b.
Proof of Theorem 2 (Continued).
(i) Suppose first r(a) > r(b). Take some c ∈ int(∆) such that r(a) > r(c) >r(b).51 By Lemma 30, c ∗ b. Also, if c ∗ a, then r(c) > r(a) by Lemma
30, a contradiction. Therefore, c ∗ a, hence a r b. Next, suppose a r b.
If a ∗ b, we have r(a) > r(b) by Lemma 30. Now, consider the case in
51
Take c0 = aαb for some α ∈ (0, 1). If c0 ∈ int(∆), let c = c0 . Otherwise, take somed ∈ int(∆) and let c = βc0 + (1 − β)d where β < 1 is sufficiently close to 1.

76which we have c ∗ a and c ∗ b for some c ∈ int(∆). By Lemma 30,
r(c) > r(b). Also, if r(c) > r(a), then c ∗ a by Lemma 30, a contradiction.
Thus r(a) ≥ r(c) > r(b).
(ii) By Lemma 26, a w b implies w(a) > w(b). The converse follows from
Lemma 22.

A.3

Other Proofs

A.3.1

Proof of Proposition 1

It is easy to show that (ii) implies (i), so we only prove that (i) implies (ii).
Let VP S and VP0 S denote the function which represents  using (u, w, r, β) and(u0 , w0 , r0 , β 0 ), respectively. Since VP S is unique up to positive affine transformation, u0 (x) = VP0 S ({x}) = αVP S ({x}) + γu = αu(x) + γu for some α > 0 andγu ∈ R. Now, by nondegeneracy, there exists x, y ∈ ∆ such that {x, y}  {y}and C ({x, y}) = {y}. By the construction of w,52 for any z ∈ ∆,1−δ 0
1
VP S ({x, y})u0 (z) + w0 (z) − w0 (x) = VP0 S ({x, (1 − δ)y + δz}) −
δ
δ
1
1−δ
= [αVP S ({x, (1 − δ)y + δz}) + γu ] −[αVP S ({x, y}) + γu ]
δ
δ

1
1−δ
= α VP S ({x, (1 − δ)y + δz}) −VP S ({x, y}) + γuδδ= α {u(z) + w(z) − w(x)} + γu .
Since u0 (z) = αu(z) + γu , w0 (z) = αw(z) − αw(x) + w0 (x) ≡ αw(z) + γw . Next,
by Theorem 2, both r and r0 represent r , so Lemma 1 implies r0 = αr r + γr
for some αr > 0 and γr ∈ R. Finally, because (, C) is α-sensitive to shame
for a unique α ∈ (0, 1), we have β = α1 − 1 = β 0 by construction.
Proof of Proposition 2. Equivalence of (i) and (ii) follows from the constructionof β in the proof of Theorem 1. To show the equivalence of (ii) and (iii), notethat if DM i is αi -sensitive to shame, then the PS representation implies
αi =52

VPi S ({a, d}) − VPi S ({b, d}).
VPi S ({a, c}) − VPi S ({b, c}) + VPi S ({a, d}) − VPi S ({b, d})Recall that the following result does not depend on the choice of x, y.

77

Then
n o
n o
α {b, c} + (1 − α) ea,d

α
{a,
c}
+
(1
−
α)
eb,d
i
i
i
⇔ αVPi S ({a, c}) + (1 − α)VPi S ({b, d}) ≤ αVPi S ({b, c}) + (1 − α)VPi S ({a, d})⇔ α ≤ αi .
n oTherefore, α1 > (≥)α2 if and only if α {b, c} + (1 − α) ea,d
2 α {a, c} +
2
n o
n o
n o
a,d
(1 − α) eb,dimplies
α
{b,
c}
+
(1
−
α)
e

(
)α
{a,
c}
+
(1
−
α)
eb,d
.
1
1
2
1
1

B

Graphical Illustrations of Nondegeneracy andr

We provide graphical illustrations of the nondegeneracy concepts and the elicitation of a r b. Figure A1 illustrates the concepts of nondegeneracy andweak nondegeneracy, providing an example to distinguish the two. It also
shows why, in Definition 1, a ∗ b needs to be elicited using a general menu
A 3 b and not just A = {b}, and presents a graphical illustration of Theorem
2. Figure A2 then demonstrates how Definition 1(ii-b) helps establish a r bwhen we cannot establish the relation via condition (ii-a), i.e., a ∗ b, which
occurs when a is on the boundary of ∆.
Figure A1a illustrates nondegeneracy, which requires that there exist x, y, y 0 ∈∆ such that y ∈ P(x) and y 0 ∈ S(x). For P(x) to be nonempty, we must havesome y ∈ ∆ such that r(x) > r(y), w(x) < w(y), and g(x, w(x)) < g(y, w(x)).
The first two condition ensures that the reference point at {x, y} is lower thanthat at {y}, and adding the third condition ensures that x is not chosen from{x, y}. Similarly, for S(x) to be nonempty, we must have some y 0 ∈ ∆ suchthat r(x) > r(y 0 ), w(x) > w(y 0 ), and g(x, w(x)) < g(y 0 , w(x)), ensuring thatthe reference point at {x, y 0 } is higher than that at {y 0 } and that x is notchosen from {x, y 0 }.
Figure A1b provides an example in which the nondegeneracy property isviolated. To see this, note that for any y ∈ ∆ such that r(ā) > r(y) andw(ā) < w(y), we have g(ā, w(ā)) > g(y, w(ā)), so P(ā) is empty. Since thisobservation holds for any ā ∈ ∆, the nondegeneracy axiom is violated. In thiscase, the reference-lowering alternative ā is also the chosen one, so observing{ā, y}  {y} does not allow us to tell if the larger menu is preferred becauseā lowers the reference point or because ā is a preferred choice.
78However, the preference illustrated in Figure A1b satisfies the weak nondegeneracy axiom. To see this, note that (i) the reference point at  ā, b̄, c, d ,w(ā),
 is lower than the reference point at b̄, c, d , w(d), and (ii) C( b̄, c, d ) =C( ā, b̄, c, d ) = {c}. In this case, ā makes the larger menu more desirable eventhough it is not chosen there,
point lower which implies it sets the reference∗
than the reference point at b̄, c, d . Therefore, we have
 ā  b̄. This example
shows why we cannotconfine Definition 1(i) to A = b̄ : Even if r(ā) > r(b̄),
we may have ā ∈ C( ā, b̄ ) (ā is chosen) or w(ā) ≥ w(b̄) (ā sets a weaklyhigher reference point), preventing us from concluding ā ∗ b̄ with A = b̄ .
Thus, to conclude ā ∗ b̄, we may need a larger menu A which contains a“choice fixer” c ∈ P1 (ā) and a “higher reference setter” d ∈ P2 (ā). The figure
also graphically illustrates Theorem 2, in particular that we observe ā ∗ b̄
whenever r(ā) > r(b̄), as long as P1 (ā) and P2 (ā) are nonempty (which is aquite weak condition).
Figure A2 illustrates how we can elicit ā r b̄ when ā cannot satisfy ā ∗ b̄even though data are generated by a PS preference with r(ā) > r(b̄). Figure
A2a depicts the indifference curves of the same PS preference as in Figure
A1b. However, because ā is on the boundary of ∆, P2 (ā) is empty, and wecannot establish ā ∗ b̄ with any A. Intuitively, when ā is a unique reference
alternative at A ∪ {ā} (i.e., r(ā) > r(y) for all y ∈ A), the reference point isnecessarily higher than that at A, preventing ā ∗ b̄.
However, we can still conclude r(ā) > r(b̄) if there exists some c ∈ ∆ suchthat r(ā) ≥ r(c) > r(b̄). As Figure A2b demonstrates, by taking c ∈ int(∆)
such that r(ā) > r(c) > r(b̄), we can elicit c ∗ b̄. Moreover, we cannot havec ∗ ā, as c cannot set a reference point at A whenever ā ∈ A. Thus, we canconclude ā r b̄ via Definition 1(ii-b).

79

Figure A1: Nondegeneracy and Weak Nondegeneracy(b) Weak Nondegeneracy(a) Nondegeneracy

80
Notes: Panel (a) presents an example of a PS preference which satisfies nondegeneracy, and Panel (b) presents an example of a PS
preference which satisfies weak nondegeneracy but not nondegeneracy. Each dashed or solid straight line represents an indifference
curve of u, w, u + w, u + βw or r, with an allow indicating the increasing direction of the utility function. The bold solid line kinked
at x in Panel (a) (at ā in Panel (b)) denotes the indifference curve of the function g(·, w(x)) (g(·, w(ā))) defined in Lemma 20. In
Panel (a), the black and red shaded areas depict P(x) and S(x), respectively, defined in Eq. (4) and (5) in Section 3.1. In Panel (b),the black and red shaded area depicts P1 (ā) and P2 (ā), respectively, defined in Lemma 29. See the text in Appendix B for details.

Figure A2: Reference Elicitation on the Boundary(a) Non-existence of d ∈ P2 (ā)(b) Mediating alternative c ∈ int(∆)

81
Notes: Panel (a) presents an example of alternatives ā, b̄ ∈ ∆ such that r(ā) > r(b̄) but ā ∗ b̄, and Panel (b) illustrates how we
can establish ā r b̄ via Definition 1(ii-b) using a mediating alternative c ∈ int(∆). Each dashed or solid straight line represents an
indifference curve of u, w, u + w, u + βw or r, with an allow indicating the increasing direction of the utility function. The bold solidline kinked at ā in Panel (a) (at c in Panel (b)) denotes the indifference curve of the function g(·, w(ā)) (g(·, w(c))) defined in Lemma
20. In Panel (a), the black and red shaded areas depict P1 (ā) and P2 (ā), respectively, defined in Lemma 29. In Panel (b), the blackand red shaded areas depict P1 (c) and P2 (c), respectively. See the text in Appendix B for details.

C

Other Predictions from the Simple Model

In Section 2.1, we developed a simple PS model of prosocial behavior andshowed its insightfullness. In this section, we further discuss implications of
the model, such as conformative versus pride-seeking behavior, and boomerang
effects.
Recall that x = 1 denotes engaging in prosocial behavior, and x = 0 denotes
non-engagement. Private and social payoffs are in conflict: u(0) = ū > 0 =u(1) and w(0) = 0 < w̄ = w(1), with β w̄ < ū < w̄. At menu {0, 1}, the DM
chooses an action by comparing the ex post utility of action 0, U (0; {0, 1}) =ū − w(ϕr ({0, 1})), with that of action 1, U (1; {0, 1}) = β [w̄ − w(ϕr ({0, 1}))].
We compare decisions in the benchmark case r(0) > r(1) (prosocial behavior
is perceived to be uncommon) and in a post-intervention case r0 (0) < r0 (1)(prosocial behavior is perceived to be common).
Conformity and pride seeking. The DM conforms to the reference alternative both in the benchmark case (x = ϕr ({0, 1}) = 0) and post-intervention
case (x = ϕr0 ({0, 1}) = 1). By contrast, if we modify the benchmark assumption so that β w̄ > ū, then the DM engages in prosocial behavior under bothscenarios. In the modified benchmark, the DM deviates from the reference
to seek pride. Thus, our model can produce conformative or pride-seeking
behavior depending on β. Typical empirical findings suggest β is small (seefootnote 18); still, in some contexts, individuals may seek to perform better
than a natural reference point (Birke 2020, and see also footnote 16).
Boomerang effect. In a field experiment on electricity consumption, Schultz
et al. (2007) find that providing descriptive information on neighbors’ electricity usage led to desired electricity saving by high-consuming households but
increased consumption by low-consuming households. To explain the latter
result (which Schultz et al. (2007) call a “boomerang effect”) without complicating the model, let x = 0 and x = 1 denote high consumption and low
consumption of electricity, respectively, and suppose that the low-consuming
households originally perceive norms (w, r0 ) but the intervention updates theperceptions to (w, r). By the analysis in Section 2.1, the low-consuming households originally choose x = 1 but the intervention causes them to switch tox = 0. Thus, our model can explain the boomerang effect by an upward shiftof the perceived descriptive norm for those households.
The model can also explain why penalizing certain behavior may induce
the behavior, or rewarding certain behavior can discourage it. An introductionof a fine (reward) is often accompanied by information that the behavior is82

widespread (rare), which may push the descriptive norm in the unintended direction. Thus, changes in perceived norms might be a reason for these counterintuitive empirical findings.
The purpose of the above example is to illustrate the importance of considering the perceived norms of individuals when introducing a policy, rather thandevelop a more thorough model. For example, the reduction in the electricity
consumption by high-consuming households can be explained by the opposite
shift in the perceived descriptive norm. Instead of developing a model whichaccommodates both types of households (possibly requiring more than two
options), we note that even the direction of a policy effect, as well as its magnitude, crucially depends on what norms the households perceive prior to theintervention.

83