Impure Altruism and Impure Selfishness
Kota SAITO∗
California Institute of Technology
September 5, 2013
Abstract
Altruism refers to a willingness to benefit others, even at one’s own expense. In
contrast, selfishness refers to prioritizing one’s own interests with no consideration
for others. However, even if an agent is selfish, he might nevertheless act as if he
were altruistic out of selfish concerns triggered when his action is observed; that is,
he might seek to feel pride in acting altruistically and to avoid the shame of acting
selfishly. We call such behavior impurely altruistic. Alternatively, even if an agent is
altruistic, he might nevertheless give in to the temptation to act selfishly. We call such
behavior impurely selfish. This paper axiomatizes a model that distinguishes altruism
from impure altruism and selfishness from impure selfishness. In the model, unique
real numbers separately capture altruism and the other forces, or pride, shame, and
∗
Email: [email protected] This paper is a revised version of “Role Conflict and Choice” (2011), which
no longer exists as a separate paper. I am indebted to my adviser, Eddie Dekel, for continuous guidance,
support, and encouragement. I would like to thank David Dillenberger for discussion and comments that
have led to the improvement of the paper. I would also like to thank Federico Echenique, Pietro Ortoleva,
Leeat Yariv, and Chris Shannon for their comments. I am also grateful to James Andreoni, Pierpaolo Battigalli, Colin Camerer, Mark Dean, Jeff Ely, Ozgur Evren, Ernst Fehr, Takashi Hayashi, Michihiro Kandori,
Edi Karni, Peter Klibanoff, Barton Lipman, Yusufcan Masatlioglu, Stefania Minardi, Jawwad Noor, Efe
Ok, Tom Palfrey, Wolfgang Pesendorfer, David Pearce, Andrew Postlewaite, Matthew Rabin, Joel Sobel,
Marciano Siniscalchi, Simon Wilkie, and seminar participants at the Brown University Spring Conference
2011, California Institute of Technology, University of Southern California, University of Texas Austin, University of California San Diego, New York University, University of Pennsylvania, Hitotsubashi University,
the University of Tokyo, the Johns Hopkins University, and RUD 2012 at Northwestern University. I gratefully acknowledge financial support from the Center for Economic Theory of the Economics Department of
Northwestern University.
1
the temptation. We show that the model can describe recent experiments on dictator
games with an exit option. In addition, we describe two empirical puzzles involving
charitable donations: (i) government spending only partially crowds out consumers’
donations and (ii) redistribution of income by the government affects the total donation
of consumers, contrary to the prediction based on standard consumer theory.
Keywords: Impure altruism, altruism, warm glow, pride, shame, temptation, dictator
games, and preferences on sets.
JEL Classification Numbers: D03, D63, D64, D81.
1
Introduction
Altruism refers to a willingness to benefit others, even at one’s own expense. Selfishness,
the opposite of altruism, refers to prioritizing one’s own interests, with a concomitant lack
of consideration for others. While these definitions may seem clear, it is difficult to detect
from an agent’s actions alone whether he is truly altruistic or selfish.
Even if an agent is indifferent to the welfare of others and therefore selfish by definition,
he might nevertheless act as if he were truly altruistic (i.e., act to benefit others) out of
selfish concerns triggered when his action is observed; that is, he might seek to feel pride in
having acted altruistically and to avoid the shame of having acted selfishly.1
In contrast to pure altruism, we say that an agent exhibits impure altruism if he chooses
an action that benefits others in order to feel pride in acting altruistically and to avoid the
shame of acting selfishly. (In this paper, the terms altruism and pure altruism will be used
interchangeably, as will the terms selfishness and pure selfishness.)
On the other hand, even if an agent is willing to benefit others and is therefore by
definition altruistic, he might nevertheless give in to the temptation to act selfishly. In
contrast to pure selfishness, we say that an agent exhibits impure selfishness if the temptation
1
Note that being motivated to avoid the shame of acting selfishly could also be described as “the temptation to act altruistically”. But since these two descriptions are behavioral equivalent, we will use just the
expression the shame of acting selfishly for the sake of simplicity.
2
to act selfishly motivates him to depart from his tendency to act altruistically.2 An agent
can exhibit impure selfishness, especially when an immediate payoff is at stake.3
Moreover, these three forces–pride, shame, and the temptation to act selfishly–could
interact in conflicting ways in affecting an agent’s actions.4 For example, an agent could
behave impurely altruistically even if he feels a temptation to act selfishly, when his desire
to avoid the shame has overwhelmed the temptation. Conversely, an agent could behave
impurely selfishly even if he knows that he will feel shame at doing so, when the temptation
has overwhelmed such shame. Therefore, existing models capturing only one of the forces,
temptation or shame, cannot capture an agent’s impure altruism and impure selfishness.
Given that altruism is a fundamental source of human cooperation, it is crucial to distinguish altruism from impure altruism and selfishness from impure selfishness. The purpose of
this paper is to axiomatizes a model that makes the distinction behaviorally. In the model,
unique real numbers separately capture altruism as well as pride in acting altruistically,
shame of acting selfishly, and the temptation to act selfishly. The model can capture the
trade-off between the three forces and, thereby, distinguish altruism from impure altruism
and selfishness from impure selfishness. For example, we can show that the agent could
behave impurely altruistically if his index of shame is larger than that of the temptation;
otherwise, an agent could behave impurely selfishly.
The rest of the paper is organized as follows. In Section 2, we provide preview of results.
In Section 3, we present the axioms. In Section 4, we present a representation theorem,
2
Of course, such an agent could also be said to exhibit impure altruism, since his altruistic tendencies
could be seen as rendered “impure” whenever he gives in to the temptation to act selfishly. However, we feel
that the term impure altruism is more suitable for the selfish person who acts altruistically when motivated
by his inherent selfishness. As a result, we use the term impure selfishness for the opposite situation, namely,
a typically altruistic agent who sometimes gives in to the temptation to act selfishly.
3
Indeed, in experiments on dictator games, Noor and Ren (2011) found that the average donation from
dictators to receivers is 65 percent of the total endowment when the payment to subjects is made one month
later, but only 38 percent when the payment is made immediately after the dictators’ choices.
4
Throughout this paper, when we use the terms, pride in acting altruistically, shame of acting selfishly,
and the temptation to act selfishly, we are referring to individual acts of an agent, and not to an agent’s
personality or general tendencies. For example, if we say that an agent seeks to “feel pride in acting
altruistically,” that description will refer to the pride in one particular altruistic act, rather than to the
agent’s typical or habitual behavior.
3
the uniqueness of the representation, and characterization of the parameters. In Section 5,
we show that the model is consistent with recent experimental evidence. In Section 6, we
provide two applications. Finally, in Section 7, we discuss limitations of the model and the
related literature. The proofs of results in Section 4 are in Appendix A.
2
Preview of Results
2.1
The model
We investigate a decision maker who determines an allocation between himself and other
passive agents. The decision maker’s choice consists of two stages. In the first ex-ante stage,
the decision maker chooses a set of allocations. We assume that the other agents do not
know that the decision maker has such choices. His ex-ante choices are private.
In the ex-post stage, the decision maker chooses an allocation from the set that he chose
ex-ante. We assume that other agents can observe which allocation the decision maker
chooses from the set, even though they do not know that he chose the set ex-ante.
Ex-post
Ex-ante
Choose a set A of allocations privately
Choose an allocation from the set A publicly
Figure 1: Two-Stage Decision Problem
Because his choices will be public at the ex-post public stage, the decision maker could
anticipate that he will feel pride in acting altruistically, or shame of acting selfishly at
that stage. In addition, the decision maker could anticipate that he will suffer from the
temptation to act selfishly ex-post. In light of these potential ex-post feelings, the decision
maker chooses a set ex-ante. (We, henceforth, call these three feelings simply, pride, shame,
and temptation, when there is no danger of confusion.)
To introduce the model, we first define some notation. The decision maker is denoted by
1 and the finite set of other agents is denoted by S. Define I = {1} ∪ S to be the set of all
agents. A payoff profile p ≡ (pi )i∈I is called an allocation. An allocation p is also denoted
4
by (p1 , pS ), where p1 is a payoff to the decision maker and pS ≡ (pi )i∈S is a payoff profile to
the other agents.
In the model, the decision maker is a utilitarian. As we explain in the next section, when
the decision maker chooses an allocation at either stage, he maximizes the weighted sum of
∑
his utility α1 u(p) and the other agents’ utilities uS (pS ) ≡ i∈S αi u(pi ), where α1 > 0 and
∑
i∈S αi = 1. Hence, we call the model a generalized utilitarian (GU) model.
When the decision maker chooses a set of allocations at the ex-ante private stage, he
maximizes the following utility function. The utility of a set A is :
[
]
V (A) = max α1 u(p1 ) + uS (pS ) + β1 max α1 (u(q1 ) − u(p1 )) − βS max(uS (rS ) − uS (pS )) , (1)
p∈A
q∈A
r∈A
where β1 < 1 and βS ≥ 0.
In the model, the maximizer p over A is the decision maker’s ex-post public choice from
∑
A. The first term of the GU model, α1 u(p1 ) + uS (pS ) ≡ i∈I αi u(pi ), captures the decision
maker’s utilitarian evaluation of his ex-post choice of p.
The second term, β1 maxq ∈A α1 (u(q1 ) − u(p1 )), captures the utility arising from the pride
of acting altruistically if β1 ≥ 0 or the disutility arising from the temptation to act selfishly if
β1 ≤ 0. The utility and disutility are proportional to the difference between the maximum
selfish utility maxq∈A α1 u(q1 ) and the actual utility α1 u(p1 ) attained by the decision maker’s
ex-post choice of p. The difference captures how much the decision maker controls himself
so as to keep himself from acting selfishly (i.e., from maximizing his selfish interest).
Similarly, the third term of the GU model, βS maxr∈A (uS (rS ) − uS (pS )), captures the
disutility of shame of acting selfishly. The disutility is proportional to the difference between
∑
the maximum social utilities maxr∈A uS (rS ) ≡ maxr∈A i∈S αi u(ri ) (i.e., the maximum sum
∑
of the utilities of the other agents) and the actual social utilities uS (pS ) ≡
i∈S αi u(pi )
attained by the decision maker’s ex-post choice of p. Hence, the difference captures how
much utility the other agents lose because of the decision maker’s ex-post choice.
5
2.2 α captures altruism/selfishness; β captures impure altruism/selfishness
In the GU model, 1/α1 captures the level of pure altruism. To see this interpretation of α1 ,
note that whether the decision maker is purely altruistic or not is determined through his
ex-ante private choices. This is because in private choices, the the decision maker would feel
neither pride nor shame. Moreover, in ex-ante choices, he would not feel temptation.
In the GU model, ex-ante private choices between allocations, say, between p and q , are
formalized as choices between the singleton sets, such as {p} and {q }. Hence, the level of
altruism is captured by 1/α1 as follows:
V ({p}) = α1 u(p1 ) + uS (pS ).
(2)
Therefore, the smaller α1 is, the more the decision maker is willing to sacrifice his own
allocation p1 to improve the others’ allocations pS .
βS
Shame
Impure Altruism
βS > −β1
Impure Selfishness
βS < −β1
Temptation
Pride β1
0
β1 < 0
β1 > 0
Figure 2: Pride, Shame, and Temptation Cause Impure Altruism and Impure Selfishness
In contrast to the ex-ante private choices, ex-post public choices can be affected by pride,
shame, and the temptation to act selfishly. To see this effect, note that (1) can be expressed
as V (A) = maxp∈A [(1 − β1 )α1 u(p1 ) + (1 + βS )uS (pS )] + β1 maxq∈A α1 u(q1 ) − βS maxr∈A uS (rS )
for all A ∈ A . This representation shows that the ex-post public choice p (i.e., the maximizer
over A) maximizes
U (p) ≡ (1 − β1 )α1 u(p1 ) + (1 + βS )uS (pS ).
6
(3)
Note that since β1 < 1, the function U is monotonic in the decision maker’s utility u(p1 ) and
the social utility uS (pS ). Hence, the decision maker will not derive utility from decreasing his
own utility (or social utility) unless it increases social utility (or his own utility, respectively).
A comparison between (3) and (2) shows that the decision maker’s choices become more
altruistic at the ex-post public stage (i.e., in (3)) than they are at the ex-ante private stage
(i.e., in (2)) if and only if βS > −β1 . To see this note that the relative weight on the social
utility uS at the ex-post public stage is (1 + βS )/((1 − β1 )α1 ). The relative weight at the
ex-ante private stage is 1/α1 . Hence, the relative weight at the ex-post public stage is higher
than the relative weight at the ex-ante private stage if and only if 1 + βS > 1 − β1 .5
In this way, βS > −β1 captures impure altruism caused by pride in acting altruistically or
shame of acting selfishly. In the same way, βS < −β1 captures impure selfishness motivated
by the temptation to act selfishly.
2.3
Experimental evidence and applications
The GU model can describe recent experimental evidence on dictator games with an exit
option. In the experiments, when a dictator exited, he obtained the whole endowment, while
the receivers obtained nothing without knowing that this was a consequence of the dictator’s
choice. About one-third of subjects exited, but when the same subjects played a standard
dictator game without the exit option, they donated a positive amount to the receivers. (See
Dana et al. (2006), Broberg et al. (2007), and Lazear et al. (2012).)
As we observe in Section 5, this tendency to exit is inconsistent with Andreoni’s (1989,
1990) well-known model of warm glow as well as any models of inequality aversion that are
monotonic on constant allocations.
The GU model can describe not only the choice of exit but also the tendency for both
low-level and high-level donors to play (i.e., not exit) the dictator games more often than
medium-level donors. Capturing this tendency would be important because the tendency
has an important implication in increasing the amount of donations: screening is necessary
5
We provide behavioral characterizations of impure altruism and impure selfishness in Section 4.2.
7
to induce high-level donations.
In addition, we show that the GU model is consistent with two classical empirical puzzles involving charitable donations. First, standard consumer theory, which assumes that
consumers’ preferences are solely defined on allocations, predicts that government spending
for charity should completely crowd out their donation. However, empirical evidence suggests that the crowding out is far from complete, and is at most about 50 percent. (See
Andreoni (2006) for a survey of the evidence.) By applying the GU model, we describe the
partial crowding out under the condition that consumers’ pride dominates their shame (i.e.,
β1 > βS ).
Second, standard consumer theory predicts that redistribution of income by the government does not affect the total donation of consumers. However, an empirical study by
Hochman and Rodgers (1973) shows that total donation of consumers is sensitive to distribution of income across consumers. By applying the GU model, we show that the redistribution
between a consumer whose pride dominates their shame (i.e., β1 > βS ) and a consumer who
does not exhibit neither pride nor shame (i.e., β1 = 0 = βS ) affects the sum of donations.
2.4
Relationship with Andreoni’s (1989, 1990) model of warm
glow and models of inequality aversion
In Andreoni’s (1989, 1990) model of warm glow, the decision maker can obtain positive utility
by his donation even if his donation is private to the other agents and does not improve the
welfare of the other agents. Hence, donations captured by Andreoni’s (1989, 1990) model
are essentially different from donations captured by the GU model. (We discuss further
differences between the model of warm glow and the GU model in Section 6 and 7.)
The GU model does not incorporate inequality aversion because the GU model implies
that the decision maker is a utilitarian. This feature may not be so restrictive, given our
purpose. Since the model of inequality aversion is defined solely on allocations, the model
inherently cannot distinguish altruism from impure altruism and selfishness from impure
selfishness, which is the main purpose of our paper. To incorporate inequality aversion, we
8
axiomatize an extended GU model where uS is a maxmin utility function, in Appendix B.
3
Axioms
First, remember that the decision maker is denoted by 1 and the finite set of other agents
is denoted by S. Hence, I ≡ {1} ∪ S is the set of all agents. Let Z be a finite set. A lottery
is a probability distribution over Z. We denote the set of lotteries by ∆(Z). For simplicity,
we assume that a payoff for each agent is a lottery. Hence, the set of allocations is (∆(Z))I .
Note that outcomes of lotteries are not correlated across agents. We denote the set of all
nonempty closed subsets of (∆(Z))I by A . An element of A is called a set.
The primitive of our model is a binary relation ≿ on A that describes the decision maker’s
ex-ante private preference. Eliciting such preferences would be possible in double-blinded
experiments. In typical double-blinded experiments, experimenters (as well as receivers)
cannot know the identify of subjects but the experimenters can observe the subjects’ choices.
We denote the asymmetric and symmetric parts of ≿ by ≻ and ∼, respectively. We
endow A with the topology generated by the Hausdorff metric.6
We use the following notation. Sets are denoted by A, B, and C ∈ A . Allocations are
denoted by p, q, l, and r ∈ (∆(Z))I . Lotteries are denoted by p, q, l, and r ∈ ∆(Z). In
particular, lotteries for agent i ∈ I are denoted by pi , qi , li , and ri ∈ ∆(Z). We define convex
combinations of two sets, two allocations, and two lotteries in the usual manner.7 For a
singleton set, we write p instead of {p}.
The first three axioms are standard ones.
Axiom (Ex-ante Rationality): ≿ is a nondegenerate, complete, transitive, and continuous
binary relation.8
6
dh (A, B) = max{maxp∈A minq∈B d(p, q ), maxp∈B minq∈A d(p, q )}, where d is the Euclidean metric. (Remember that p and q are |Z|·|I| dimensional real-valued vectors. Hence, d is the Euclidean metric on R|Z|·|I| .)
7
For all pi , qi ∈ ∆(Z) and α ∈ [0, 1], αpi + (1 − α)qi is a lottery such that (αpi + (1 − α)qi )(z) =
αpi (z) + (1 − α)qi (z) for each z ∈ Z. For all p, q ∈ (∆(Z))I and α ∈ [0, 1], αp + (1 − α)q is an allocation
such that (αp + (1 − α)q ) = (αpi + (1 − α)qi )i∈I . For all A, B ∈ A and α ∈ [0, 1], αA + (1 − α)B =
{αp + (1 − α)q|p ∈ A and q ∈ B}.
8
Formally, the continuity is defined as follows: the sets {B ∈ A |B ≿ A} and {B ∈ A |A ≿ B} are closed
in the Hausdorff metric topology.
9
Axiom (Independence): A ≿ B if and only if αA + (1 − α)C ≿ αB + (1 − α)C.
We define the decision maker’s risk preference ≿1 on ∆(Z) and social preference ≿S on
(∆(Z))S as follows:
Definition: (i) For all p1 , q1 ∈ ∆(Z), p1 ≿1 q1 if (p1 , lS ) ≿ (q1 , lS ) for some lS ∈ (∆(Z))S ;
(ii) for all pS , qS ∈ (∆(Z))S , pS ≿S qS if (l1 , pS ) ≿ (l1 , qS ) for some l1 ∈ ∆(Z).
The binary relation ≿1 captures the decision maker’s individual risk preference, and ≿S
captures his social preference (i.e., his preference on other agents’ allocations), which reflects
the decision maker’s belief about the other agents’ preferences.9 To see these interpretations,
note that in part (i) of the definition, the two allocations differ only with respect to the
decision maker’s allocations, so that his choice between the two allocations does not affect
other agents’ allocations. Hence, the decision maker would allow himself to choose one based
solely on his selfish preference.
Similarly, in part (ii) of the definition, the two allocations differ only with respect to
other agents’ allocations, so that the decision maker’s choice between the two allocations
does not affect his own allocation. Hence, the decision maker should choose one to maximize
the anticipated welfare of other agents.
The next axiom requires that ≿ satisfy a Pareto condition.
Axiom (Pareto): If pi ≿1 qi for all i ∈ I, then p ≿ q , moreover if p1 ≻1 q1 or pi ≻1 qi for
all i ∈ S, then p ≻ q.
This Pareto condition is different from the standard Pareto condition in the sense that
the decision maker uses his own selfish preference ≿1 to evaluate the other agents’ allocations
pi̸=1 . This is because the decision maker here is not a social planner and might not know
the other agents’ preferences.
To define our key axiom, Dominance, we introduce the following definition:
Definition: (i) An allocation p is respectively dominated in A if there exist allocations q
9
For each i ∈ {1, S}, we write the asymmetric and symmetric parts of ≿i by ≻i and ∼i , respectively.
10
and l in A such that q1 ≿1 p1 and lS ≿S pS ; (ii) p is jointly dominated in A if there exists an
allocation q in A such that q1 ≿1 p1 and qS ≿S pS .
If an allocation p is dominated in a set A for the social preference ≿S , then not choosing
p from A ∪ p does not cause the decision maker to feel shame of acting selfishly. In addition,
if p is dominated among A for the selfish preference ≿1 , then p would not be tempting
in A ∪ p. Moreover, adding such an allocation p would not be able to cause the decision
maker to feel pride in acting altruistically. This is because not choosing p does not reveal his
altruism since even a purely selfish agent, who maximizes the selfish preference ≿1 , would
not choose p.
Therefore, if p is respectively dominated in A, then adding p to A does not cause the
disutilities arising from the shame and the temptation nor the utility arising from the pride.
Hence, the decision maker should weakly prefer A ∪ p to A because p might be optimal expost. However, if p is jointly dominated by a single allocation both for his selfish preference
and for the social preference, then the decision maker should not choose p ex-post. Hence,
the decision maker should be indifferent between A∪p and A. The following axiom expresses
the above observations formally:
Axiom (Dominance): If p is respectively dominated in A, then A ∪ p ≿ A. Moreover, if p
is jointly dominated in A, then A ∪ p ∼ A.
We identify the decision maker’s ex-post preference as follows: if p is respectively dominated in A but the decision maker still prefers A ∪ p to A, then he must prefer p to q ∈ A
ex-post. This is because the only reason that the decision maker could prefer A ∪ p to A is
his ex-post preference since adding p to A does not cause the utility arising from pride in
acting altruistically. Formally,
Definition: For all p, q ∈ (∆(Z))I , p ≻′ q if there exists a set A ∈ A such that q ∈ A, p is
respectively dominated in A, and A ∪ p ≻ A.
We assume the following weak rationality condition on the ex-post preference:
11
Axiom (Ex-post Rationality): ≻′ is nondegenerate and asymmetric.
The last axiom captures shame of acting selfishly: the decision maker feels shame when
his ex-post choice is judged inferior by the social preference ≿S . Hence, the decision maker
might have a preference for commitment in order to exclude the socially superior allocation
from his choice set as follows:
Axiom (Shame of Acting Selfishly): If p ≻′ q and qS ≻S pS , then {p} ≿ {p, q }.
We provide the axioms of pride and the temptation in the next section. Our representation is general enough to allow both phenomena.
4
Theorem
Theorem: The following statements are equivalent:
(a) ≿ satisfies Ex-ante Rationality, Independence, Pareto, Dominance, Ex-post Rationality,
and Shame of Acting Selfishly.
(b) There exist a real-valued nonconstant function u on ∆(Z) and real numbers (α, β1 , βS )
such that ≿ is represented by
[
]
V (A) = max α1 u(p1 ) + uS (pS ) + β1 max α1 (u(q1 ) − u(p1 )) − βS max(uS (rS ) − uS (pS )) ,
q∈A
p∈A
where α1 > 0,
∑
i∈S
r∈A
αi = 1, β1 < 1, βS ≥ 0, and uS (pS ) ≡
∑
i∈S
αi u(pi ). Moreover, the
maximizer p over A is a maximal element of the ex-post preference ≻′ in A (i.e., there is no
q ∈ A such that q ≻′ p).
By the definitions, the theorem trivially implies that ≿1 and ≿S are represented by u1
and uS , respectively. In the GU model, the parameters α and β are unique.
Remark: If two GU models with (u, α, β) and (u′ , α′ , β ′ ) represent the same ≿, then β1 =
β1′ , βS = βS′ , αi = αi′ for all i ∈ I, and there exist real numbers a, b such that a > 0 and
u = au′ + b.
12
4.1
Pride in Acting Altruistically and Temptation to Act Selfishly
The next axiom captures a decision maker’s temptation to act selfishly (i.e., the temptation
to maximize his own allocation). Because of this temptation, the decision maker might prefer
to commit himself to a smaller choice set in order to minimize the cost of self-control.
Axiom (Temptation to Act Selfishly): p ≻′ q and q1 ≻1 p1 ⇒ {p} ≿ {p, q }.
In contrast, by not choosing such selfish allocations, a decision maker might feel pride in
acting altruistically. Hence, such a decision maker might have a preference for flexibility so
that he could publicly keep himself from choosing such selfish allocations.
Axiom (Pride in Acting Altruistically): p ≻′ q and q1 ≻1 p1 ⇒ {p, q } ≿ {p}.
Proposition 1 Suppose that ≿ is represented by the GU model with (u, α, β).
≿ exhibits pride in acting altruistically if and only if β1 ≥ 0.
(i)
(ii) ≿ exhibits the temptation to act selfishly if and only if β1 ≤ 0.
4.2
Impure Altruism and Impure Selfishness
We say that a decision maker exhibits impure altruism if he acts to benefit others’ allocations
because of pride in acting altruistically and shame of acting selfishly. If such a decision maker,
at the ex-ante private stage, weakly prefers an allocation that is ranked superior by his social
preference ≿S , then at the ex-post public stage, he should strictly prefer the same allocation
because the pride and the shame are triggered by the publicness of the choice.
Axiom (Impure Altruism): For all p, q ∈ (int.∆(Z))I , {p} ≿ {q} and pS ≻S qS ⇒ p ≻′ q.
In contrast, we say that a decision maker exhibits impure selfishness if the temptation to
act selfishly motivates him to diverge from his ex-ante choices. If such a decision maker, at
the ex-ante stage, weakly prefers an allocation that is ranked superior by his selfish preference
≿1 , then he should strictly prefer the same allocation at the ex-post stage.
Axiom (Impure Selfishness): For all p, q ∈ (int.∆(Z))I , {p} ≿ {q} and p1 ≻1 q1 ⇒ p ≻′ q.
13
Proposition 2 Suppose that ≿ is represented by the GU model with (u, α, β).
(i) ≿ exhibits impure altruism if and only if βS > −β1 .
(ii) ≿ exhibits impure selfishness if and only if βS < −β1 .
5
Consistency with Experiments
In this section, we show that the GU model is consistent with recent experimental evidence
on dictator games with an exit option. Before describing the evidence, we clarify the meaning
of public in the experiments. We call a dictator’s choice public if playing dictator games is
common knowledge among subjects, even though a receiver does not know the identity of
his paired dictator. Given the common knowledge, the dictator would consider the receiver’s
wish that the dictator should behave altruistically.10 Hence, the dictator could feel pride
in acting altruistically by living up to the receiver’s wish or shame of acting selfishly by
denying that wish. (Indeed, in Appendix C we observe that anonymity does not change
major tendencies of choices in some experiments.)
In the experiments conducted by Dana et al. (2006), Broberg et al. (2007), and Lazear
et al. (2012), the exit option could be costly (i.e., playing the dictator game is subsidized)
but the option ensured that receivers never knew the choice of exit. So, by using the exit
option, dictators could consume the whole endowment (minus the cost of exit, if any) and
leave nothing to receivers–without feeling shame of acting selfishly.11 In these experiments,
about one-third of subjects used the exit option privately, but when the same subjects played
a standard dictator game without the exit option, they donated a positive amount to the
receivers.
Moreover, the most recent experiments conducted by Lazear et al. (2012) found two
interesting correlations between the choice of exit and the proportion donated: (i) when
playing the dictator game is subsidized, the medium-level donors exit more often than both
10
Given the common knowledge, the dictator knows that the receiver knows that a dictator determines
the receiver’s allocation. Hence, the dictator would consider the receiver’s wish.
11
An experimenter observed the choice of exit. This is consistent with our model because the experimenter
is an outside observer (i.e., the subjects’ choices should not affect the experimenter’s welfare).
14
Figure 3: Tendency (i) observed in Lazear et al. (2012) (left) and Dana et al. (2006) (right)
low-level and high-level donors; (ii) when playing the dictator game is costly, the lower-level
donors exit more often than the higher-level donors.
Tendency (i) has the important implication in implementing higher donations. The
tendency means that both low-level and high-level donors tend to participate in donation.
Hence, screening is important to implement high-level donations. Other experiments found
consistent evidence for the two tendencies (i) and (ii). (See Figure 3 for tendency (i) and
Appendix C for details.)
The theory of inequality aversion and Andreoni’s (1989, 1990) model of warm glow are
inconsistent with this robust tendency to exit, not to mention tendencies (i) and (ii). In
both theories, whether the dictator’s choice is private or public does not make any difference
in his utility. Moreover, by playing the dictator game, the dictator can allocate the whole
endowment and the subsidy arbitrarily between himself and the receiver. Hence, the theory of
inequality aversion and Andreoni’s (1989, 1990) model of warm glow predict that any subjects
should not use the exit option. By the same reason, the tendency to exit is inconsistent with
any other-regarding preferences that are monotonic on constant allocations.
To see that the GU model is consistent with these tendencies, note that the singleton
set {(w, 0)} corresponds to exiting with endowment w; the set {(c, d) ∈ R2+ |c + d ≤ w + τ }
15
corresponds to playing the dictator game publicly with total endowment w + τ , where τ > 0
is the subsidy and τ < 0 is the cost of playing the game. Hence, subjects exit if and only if
V ({(w, 0)}) > V ({(c, d) ∈ R2+ |c + d ≤ w + τ }).
Note also that the donation decreases as the index α1 of selfishness increases. Hence, the
medium-level donors correspond to subjects whose α1 is medium level and the lower-level
donors correspond to subjects whose α1 is higher level.
τ >0
τ <0
Figure 4: Tendency (i) (left) and Tendency (ii) (right) captured by the GU model. (The
graph shows the utility difference between exiting and playing the dictator game depending
on the value of α1 . Exiting is optimal if and only if the graph is above the horizontal line.)
Proposition 3 Suppose that u′ > 0, u′′ < 0, u(0) = 0, and α1 ≥ 1. Then, there exist
positive numbers βS and τ for which the following implication holds:12 If
(a) the decision maker is pride-loving (i.e.,β1 > 0) and shame-averse enough (i.e.,βS > βS ),
(b) the cost and the subsidy to play the dictator game are not too large (i.e., |τ | < τ ),
then the decision maker exhibits tendencies (i) and (ii). That is,
(i) when playing the dictator game is subsidized (i.e., τ > 0), there exist unique α1 and α1
such that he exits if and only if α1 ∈ [α1 , α1 ], where 1 < α1 < α1 ,
(ii) when playing the dictator game is costly (i.e., τ < 0), there exists unique α′1 such that
he exits if and only if α1 ≥ α′1 , where α′1 > 1.13
The assumption that α1 ≥ 1 is consistent with the robust evidence that most dictators donate less than
half of the endowment in dictator games.
13
For simplicity, we assume that the decision maker exits if he is indifferent.
12
16
Proof of Proposition 3: Define f (α1 , β1 , βS , τ ) = V ({(w, 0)}) − V ({(c, d) ∈ R2+ |c + d ≤
w + τ }). To show the proposition. It suffices to show the following:
Claim: (I) f is strictly concave in α1 . (II) If β1 > 0 and βS > 0, then limα1 →∞ f (α1 , β1 , βS , τ ) <
0 for any τ > 0 and limα1 →∞ f (α1 , β1 , βS , τ ) > 0 for any τ < 0. (III) There exist positive
numbers βS and τ satisfying the following: If βS > βS , |τ | < τ , and β1 > 0, then (III-a)
f (1, β1 , βS , τ ) < 0 and (III-b) f (α1′ , β1 , βS , τ ) > 0 for some α1′ ∈ (1, ∞).
Note that (II) means that extremely selfish subjects exit if and only if τ < 0; (III-a)
means that the altruistic subject with α1 = 1 always plays the game; and (III-b) means that
some subjects exit the game. Hence, (II) and (III) together with the strict concavity of f
(i.e.,(I)) imply the proposition. (See Figure 4 for an illustration.14 )
To show the claim, first note that since u′ > 0, α1 ≥ 0, β1 < 1, and βS ≥ 0, the
budget constraint is binding (i.e., c + d = w + τ ). Hence, V ({(c, d) ∈ R2+ |c + d ≤ w + τ }) =
V ({(w+τ −d, d)|d ∈ [0, w+τ ]}) = (1−β1 )α1 u(c∗ )+(1+βS )u(w+τ −c∗ )+(β1 α1 −βS )u(w+τ ),
where c∗ is the maximizer. (I) and (II) follow from direct calculations.15 To prove claim
(III), suppose that α1 = 1 and β1 > 0. Then, V ({(c, d) ∈ R2+ |c + d ≤ w}) ≥ (1 − β1 )u(0) +
(1 + βS )u(w) + (β1 − βS )u(w) = u(w) + u(0) + β1 (u(w) − u(0)) > u(w) + u(0) = V ({(w, 0)}).
Hence, f (1, β1 , βS , 0) < 0 for any βS . Moreover, since β1 < 1, there exists α1′ such that
α1′ > (1 + β1 α1′ ). Since c∗ → 0 as βS → ∞, then limβS →∞ f (α1′ , β1 , βS , 0) = α1′ u(w) − (1 +
β1 α1′ )u(w) > 0. Since f is continuous in τ and βS , there exist desirable βS and τ .
■
Conclusion (i) in Proposition 3 claims that the subjects whose α1 is medium-level tend to
exit when playing the dictator game is subsidized. Conclusion (ii) claims that the subjects
whose α1 is higher-level tend to exit when playing the game is costly. Since the donation
decreases as α1 increases, conclusion (i) and (ii) capture tendencies (i) and (ii), respectively.
14
In Figure 4, we assume that u(x) = log(x + 1), β1 = .8, βS = 5, and w = 10. We also assume that τ = 1
in the left figure and τ = −1 in the right figure.
d2 f
15
′
∗ dd∗
To prove claim (I), note that by the envelop theorem, dα
< 0, where d∗ is
2 = (1 − β1 )u (w + τ − d ) dα
1
1
∗
the optimal donation and strictly decreasing in α1 . To prove claim (II), note that since d → 0 as α1 → ∞.
Hence, by a direct calculation, limα1 →∞
then (II) holds.
V ({(c,d)∈R2+ |c+d≤w+τ })
V ({(w,0)})
17
=
u(w+τ )
u(w) .
Since limα1 →∞ V ({(w, 0)}) = ∞,
6
Applications
In this section, we describe two empirical puzzles involving charitable donations: (i) government spending only partially crowds out consumers’ donations, even though standard
consumer theory predicts that the crowding out should be complete; (ii) redistribution of
income by the government affects the total donation of consumers, contrary to the prediction
based on standard consumer theory.
In his well-known paper, Andreoni (1989, 1990) has obtained conditions that capture
these two puzzles. However, his conditions are not closed-form expressions: the conditions
are imposed on derivatives of the first-order conditions. On the other hand, in the following,
we obtain conditions on the unique parameters, namely α and β, to capture these two
puzzles.
Moreover, Andreoni’s (1989, 1990) approach is essentially different from ours. In his
model of warm glow, the decision maker donates even if it is private to the other agents and
does not improve the welfare of the other agents. Hence, Andreoni’s (1989, 1990) approach
does not need inter-temporal frameworks, in contrast to our following approach.
6.1
Partial Crowding Out
We consider the following two-period decision problem. Period 1 consists of ex-ante private
and ex-post public stages. At the ex-ante private stage, the decision maker divides his income
e between the saving s for Period 2 and the budget w for the ex-post public stage. At the
ex-post public stage, the decision maker divides his budget w between his donation d and
his consumption c. At Period 2, the decision maker consumes the saving s privately.16
We assume that the decision maker’s preference is separable across the two periods and
that his utility at each period is represented by the GU model. Then, for any time-discount
factor δ ∈ (0, 1) and government spending g and tax τ , the decision maker’s problem at the
ex-ante stage of Period 1 is:
16
The two-period framework is adopted here for simplicity. The crucial assumption here is that the decision
maker can consume some of his income privately.
18
s: saving
e: income
d: donation
w: budget
c: consumption
Ex-ante
Ex-post
Figure 5: Decision Making in Period 1
max V ({(c, d + g)|c + d ≤ w − τ }) + δV ({(s, 0)}),
s,w
(4)
subject to s + w ≤ e.17 Given the optimal budget w∗ (g, τ ), the decision maker’s problem at
the ex-post stage of Period 1 is:
max U (c, d + g),
d,c
(5)
subject to d + c ≤ w∗ (g, τ ) − τ . We denote the optimal donation (i.e., the solution d to
(5)) by d∗ (g, τ ). We say that the decision maker’s donation is crowded out completely if
d∗ (g, τ ) + g is constant in g and that his donation is crowded out partially if d∗ (g, τ ) + g is
strictly increasing in g.
Proposition 4 Suppose that (a) government spending is financed by tax (i.e., g = τ ) and
(b) u′ > 0, u′′ < 0, u′′′ ≥ 0, u′ (0) = +∞, and α1 ≥ 1. Then, the following holds:
(i) if the decision maker’s index (i.e., β1 ) for pride loving is larger than that (i.e., βS ) of
his shame aversion, then his donation is crowded out partially.
(ii) if he does not exhibit the pride, the shame, and the temptation (i.e., β1 = 0 = βS ), then
his donation is crowded out completely.
Proof of Proposition 4: Define dˆ ≡ d∗ + τ = d∗ + g for τ = g, where d∗ is the optimal
donation. Since u′ > 0, the budget constraints must be binding (i.e., s + w = e and
c + d = w − τ ). Hence, the decision maker’s problem is as follows: maxw V ({(w − τ − d∗ , d∗ +
Our results hold for any δ ∈ (0, 1). It is easy to axiomatize this extended representation (4) by considering
ˆ over A × (∆(Z))I . We could axiomatize this extended representation by assuming
extended preferences ≿
ˆ on A satisfies our axioms in the theorem as well as standard conditions,
that the conditional preference of ≿
including the Independence axiom on A × (∆(Z))I .
17
19
ˆ
ˆ
g)|d∗ ≤ w − τ }) + δV ({(e − w, 0)}) = maxd,w
ˆ h(w, d, τ ), where h(w, d, τ ) = (1 − β1 )α1 u(w −
ˆ + (1 + βS )u(d)
ˆ + β1 α1 u(w − τ ) − βS u(w) + δα1 u(e − w).
d)
If β1 = 0 = βS , then h does not depend on τ (i.e., g), so that the solution dˆ is constant
ˆ τ ) = arg maxw h(w, d,
ˆ τ ),
in g. Hence, (ii) holds. To show (i), assume β1 > βS . Define w∗ (d,
ˆ τ ) = h(w∗ (d,
ˆ τ ), d,
ˆ τ ), and d(τ
ˆ ) = arg max ˆ f (d,
ˆ τ ).
f (d,
d
ˆ ) is strictly increasing. Given β1 α1 > βS ≥ 0 and u′′′ > 0, direct
We show that d(τ
calculation shows
ˆ )
∂ 2 h(w,d,τ
∂w2
< 0.18 Moreover,
ˆ )
∂ 2 h(w,d,τ
∂w∂τ
= −β1 α1 u′′ (w − τ ) > 0, for β1 α1 >
βS ≥ 0. Therefore, by the standard result on monotone comparative statics,
ˆ )
∂w∗ (d,τ
∂τ
> 0.
(See Milgrom and Shannon (1994, Theorem 5 and 6) for the result.) Hence, by the envelop
theorem,
ˆ )
∂ 2 f (d,τ
ˆ
∂ d∂τ
ˆ )
ˆ τ ) − d)
ˆ ∂w∗ (d,τ
= −(1 − β1 )α1 u′′ (w∗ (d,
> 0.
∂τ
By the implicit function theorem, a direct calculation shows that
ˆ )
∂ 2 f (d,τ
∂ dˆ2
ˆ )
∂w∗ (d,τ
ˆ
∂d
< 1.19 Hence,
ˆ )
ˆ τ ) − d)(
ˆ ∂w∗ (d,τ
ˆ < 0. Therefore, by the
= −(1 − β1 )α1 u′′ (w∗ (d,
− 1) + (1 + βS )u′′ (d)
∂ dˆ
monotone comparative statics again, dˆ is strictly increasing in τ (i.e., g).
■
To see the result intuitively, note that V ({(c, d + g)|c + d ≤ w − τ }) = (1 − β1 )α1 u(c∗ ) +
(1 + βS )u(d∗ ) + β1 α1 (u(w − τ ) − u(c∗ )) − βS (u(w) − u(d∗ )), where c∗ and d∗ are the optimal
consumption and the optimal donation respectively. Under the assumption that α1 ≥ 1, if
β1 > βS , then β1 α1 > βS . Hence, the net effect β1 α1 (u(w − τ ) − u(c∗ )) − βS (u(w) − u(d∗ ))
caused by pride and shame is increasing in w.20 Therefore, when τ increases, the decision
maker would increase w to compensate this increase of τ . This increase of w maintains the
level of donations, which implies that the crowding out will be only partial.
2
ˆ )
18 ∂ h(w,d,τ
=
∂w2
′′
′′
ˆ + β1 α1 u′′ (w − τ ) − βS u′′ (w) + δα1 u′′ (e − w). Since u′′′ ≥ 0, then
(1 − β1 )α1 u′′ (w − d)
2
ˆ
d,τ )
u (w − τ ) ≤ u (w) < 0. Hence, β1 α1 > βS implies ∂ h(w,
< 0.
∂w2
ˆ )
d,τ
19
∗ ˆ
= (1 −
To see this, note that w (d, τ ) is characterized by the first order condition: 0 = ∂h(w,
∂w
ˆ + β1 α1 u′ (w − τ ) − βS u′ (w) − δα1 u′ (e − w) ≡ k(w, d,
ˆ τ ). By the implicit function theorem,
β1 )α1 u′ (w − d)
(
′′
ˆ )
∂w∗ (d,τ
)−βS u′′ (w)+δα1 u′′ (e−w) )−1
∂k/∂ dˆ
= 1 + β1 α1 u (w−τ
< 1 because β1 α1 u′′ (w − τ ) − βS u′′ (w) < 0.
= − ∂k/∂w
ˆ
∂ dˆ
(1−β1 )α1 u′′ (w−d)
20
By the envelop theorem, the derivative of the difference is β1 α1 u′ (w − τ ) − βS u′ (w). By the concavity,
′
u (w − τ ) > u′ (w). Hence, the derivative of the difference is positive because β1 α1 > βS .
20
6.2
Redistribution Non-Neutrality
To capture the redistribution non-neutrality, we consider two agents X and Y who make the
same decisions as in the previous section simultaneously at the ex-ante private stage. For
simplicity, we assume that agent Y is purely altruistic (i.e., β1Y = 0 = βSY ) and each agent’s
utility depends on his selfish consumption and (common) public goods but not on the other
agent’s selfish consumption.
Given the government redistribution policy (τX , τY ) such that τX + τY = 0 and the
donation dj̸=i of the other agent, agent i’s problem at the ex-ante stage of Period 1 is:
max Vi ({(ci , di + dj )|ci + di ≤ wi + τi }) + δVi ({(si , 0)})
si ,wi
subject to wi + si ≤ e. Given the optimal budget wi∗ (dj , τi ), the decision maker’s problem at
the ex-post stage of Period 1 is
max Ui (ci , di + dj )
ci ,di
(6)
subject to ci + di ≤ wi∗ (dj , τi ) + τi . We denote the optimal donation by d∗i (dj , τi ). Nash
equilibrium {(wi∗ , d∗i )}i∈{X,Y } is characterized by wi∗ (d∗j , τi ) = wi∗ and d∗i = d∗i (d∗j , τi ) for each
i, j ∈ {X, Y } such that i ̸= j.
Proposition 5 Let u′ > 0, u′′ < 0, u′ (0) = +∞, and α1X ≥ 1.
(i) If agent X’s index (i.e., β1X ) of pride loving is larger than that (i.e., βSX ) of shame
aversion, then the redistribution between agent X and purely altruistic agent Y affects
the total equilibrium donation d∗X + d∗Y . In particular,
(a) redistribution from agent X to agent Y increases the total equilibrium donation,
(b) redistribution from agent Y to agent X decreases the total equilibrium donation.
(ii) If agent X is also purely altruistic (i.e., β1 = 0 = βS ), then the redistribution does not
affect the total equilibrium donation.
Proof of Proposition 5: Nash equilibria are characterized by the first order conditions:
(1 − β1i )α1i u′ (wi∗ + τi − d∗i ) = (1 + βSi )u′ (d∗X + d∗Y ) and (1 − β1i )α1i u′ (wi∗ + τi − d∗i ) + (β1i α1 −
21
βSi )u′ (wi∗ + τi ) = δα1i u′ (e − wi∗ ) for all i ∈ {X, Y }. Denote τY by τ . Since τX + τY = 0,
τX = −τ . Let dˆ = d∗X + d∗Y . By arranging, we obtain



0 =




 0 =


0 =




 0 =
∗
ˆ
(1 − β1X )α1X u′ (wX
− τ − d∗X ) − (1 + βSX )u′ (d)
∗
ˆ
≡ f1 (wX
, wY∗ , d∗X , d),
ˆ + (β X αX − β X )u′ (w∗ − τ ) − δαX u′ (e − w∗ ) ≡ f2 (w∗ , w∗ , d∗ , d),
ˆ
(1 + βSX )u′ (d)
1
1
1
S
X
X
X
Y
X
ˆ
(1 − β1Y )α1Y u′ (wY∗ + τ − dˆ + d∗X ) − (1 + βSY )u′ (d)
∗
ˆ
≡ f3 (wX
, wY∗ , d∗X , d),
ˆ + (β Y αY − β Y )u′ (w∗ + τ ) − δαY u′ (e − w∗ )
(1 + βSY )u′ (d)
1 1
1
S
Y
Y
∗
ˆ
≡ f4 (wX
, wY∗ , d∗X , d).
∗
ˆ = 0.
, wY∗ , d∗X , d)
Let f = (f1 , f2 , f3 , f4 ) : R4 → R4 . Then, equilibria are characterized by f (wX
( ∗
)
(
)
∗
∗
∗
ˆ ∂wX , ∂wY , ∂dX , ∂ dˆ t = ∂f1 , ∂f2 , ∂f3 , ∂f4 t .21
, wY∗ , d∗X , d)
By the implicit function theorem, −∇f (wX
∂τ
∂τ
∂τ ∂τ
∂τ
∂τ
∂τ
∂τ
∂d
Then by the Cramer’s rule, ∂τ
= |−∇f (w∗|A|
∗
∗ ˆ , where the matrix A is obtained by subX ,wY ,dX ,d)|
( ∂f1 ∂f2 ∂f3 ∂f4 )t
∗
ˆ Therefore, by
stituting ∂τ , ∂τ , ∂τ , ∂τ
, wY∗ , d∗X , d).
to the fourth column of −∇f (wX
ˆ
∗
ˆ < 0. Moreover, if β X > β X then |A| < 0 and if β X = β X
calculation, | − ∇f (wX
, wY∗ , d∗X , d)|
1
1
S
S
then |A| = 0.22 Hence, if β1X > βSX then
∂ dˆ
∂τ
∂ dˆ
∂τ
> 0. This shows result (i). If β1X = βSX then
■
= 0. This shows result (ii).
To see the result intuitively, note that as in Proposition 4, if β1X > βSX , the net effect
β1X α1X (u(wX − τ ) − u(c∗X )) − βSX (u(wX ) − u(d∗X )) caused by pride and shame is increasing in
wX . Therefore, when τ increases, the decision maker would increase wX to compensate this
increase of τ . This increase of wX maintains the level of donations dX . In contrast, agent
Y ’s utility depends only on allocations. Hence, when τ increases, Y does not decrease wY ,
( ∂w∗ ∂w∗ ∂d∗ ∂ dˆ)t
(
) (
)
∗
ˆ −1 ∂f1 , ∂f2 , ∂f3 ∂f4 t .
By the implicit function theorem, ∂τX , ∂τY , ∂τX , ∂τ
= − ∇f (wX
, wY∗ , d∗X , d)
∂τ
∂τ
∂τ ∂τ
22
By calculation,




λX
0
−λX
−µX
−λX
0
λX −λX
 η X + ζX


0
0
µX
0
0
−ζX 
 , A =  −ηX − ζX
,
∇f =



0
λY
λY
−λY − µY
0
−λY
−λY
λY 
0
η Y + ζY
0
µY
0
−ηY − ζY
0
ζY
21
∗
ˆ
where λX = (1 − β1X )α1X u′′ (wX
− τ − d∗X ), λY = (1 − β1Y )α1Y u′′ (wY∗ + τ − dˆ + d∗X ), µi = (1 + βSi )u′′ (d)
i ′′
X X
X
′′
∗
for each i ∈ {X, Y }, ηi = δα1 u (e − wi ) for each i ∈ {X, Y }, ζX = (β1 α1 − βS )u (wX − τ ), and
ζY = (β1Y α1Y − βSY )u′′ (wY∗ + τ ). Note that λX < 0, λY < 0, µi < 0 and ηi < 0 for each i ∈ {X, Y }.
Moreover, ζX < 0, and ζY = 0. Hence, | − ∇f | = (−1)4 |∇f | = −λX λY µY (ηX + ζX ) − [λY µX (ηX + ζX ) +
λX {µY (ηX + ζX ) + λY (µX + ηX + ζX )}](ηY + ζY ) < 0 and |A| = −λX λY ηY ζX . Since α1X > 1, if β1X > βSX
then β1X α1X > βSX , so that ζX > 0. Hence, |A| < 0. Similarly, if β1X = βSX then |A| = 0.
22
so that dY increases. Therefore, the total donation dX + dY increases.23
7
Concluding Remarks
In this paper, we axiomatized the GU model, in which unique real numbers separately
captures altruism as well as pride in acting altruistically, shame of acting selfishly, and the
temptation to act selfishly. We can distinguish altruism from impure altruism and selfishness
from impure selfishness by using the unique real numbers. In this section, we discuss the
limitations of the model and related literature.
First, in the GU model, the decision maker evaluates the other agents’ allocations (pi )i∈S
by using his own utility function u.24 Whether this feature is restrictive or not depends on
the context where we apply the GU model. For example, evaluating allocations by linear
utility function would be a good approximation in experiments with small stakes because
subjects tend to be risk neutral in such experiments. Moreover, by using additional primitive
preferences (≿i )i∈S of the other agents, we could immediately obtain an extension of the GU
model in which each agent’s allocation pi is evaluated by the agent’s utility function ui
representing ≿i .
Second, in the GU model, only the most selfish allocation and the most altruistic allocation cause pride, shame, and the temptation to act selfishly for the decision maker. Because
of this feature, the GU model could not fully incorporate menu-dependent choices, such as
violations of the independence of an irrelevant alternative (IIA) axiom. At the same time,
this feature makes the model tractable and facilitates distinguishing altruism from impure
altruism and selfishness from impure selfishness, which is the main purpose of our paper.
In the following, we discuss the related literature. This paper is not the first to study
impure altruism. Andreoni (1989, 1990) has proposed the celebrated model of warm glow.
23
This observation suggests that the similar results hold even when agent Y is not purely altruistic as
long as β1Y and βSY are small enough relative to β1Y and βSY . In such a case, when τ increases, agent Y
would decrease dY . However, the decrease of dY should be smaller than the increase of dX , so that dX + dY
increases.
24
This limitation originates from the Pareto axiom in which the decision maker uses his own selfish
preference ≿1 to evaluate the other agents’ allocations.
23
By using the model, Andreoni (1989, 1990) has obtained conditions that capture the partial
crowding out and the redistribution non-neutrality, although the conditions are not closedform expressions. Providing axiomatic foundation is outside the scope of Andreoni (1989,
1990).
Donations captured by Andreoni’s (1989, 1990) model are essentially different from donations captured by the GU model. In his model of warm glow, the decision maker donates
even if it is private to the other agents and does not improve the welfare of the other agents.
Hence, Andreoni (1989, 1990) does not need inter-temporal frame works (such as the private stage and the public stage) to capture the decision maker’s donations. Because of this
feature, at the same time, Andreoni’s (1989, 1990) model of warm glow is inconsistent with
the recent experimental evidence in Section 5.
Dufwenberg, Heidhues, Kirchsteiger, Riedel, and Sobel (2011) have studied menu-dependent
other-regarding preferences. Their purpose is to analyze competitive market outcomes in
economies where agents have such preferences. They proposed a general utility function
that depends on the budge sets (opportunities, in their terminology), although they do not
provide an axiomatization
In the axiomatic literature, no paper provides an inclusive model that distinguishes altruism and impure altruism and selfishness and impure selfishness. The different forces causing
impure altruism and impure selfishness, namely temptation and shame, have been studied
separately.
Gul and Pesendorfer (2001) propose a general model of temptation. However, they do not
study social decision making in particular. In their model, moreover, the self-control problem
is captured by the difference between two von Neumann-Morgenstern utility functions. In
contrast, we study a specific temptation in a social context: the temptation to act selfishly.
Moreover, the GU model captures the self-control problem by the unique nonnegative number
β1 , which facilitates distinguishing selfishness from impure selfishness.
Dillenberger and Sadowski (2012) has proposed the first model on social decision making
within the literature on preferences over menus. The purpose of their paper is to capture
24
shame of acting selfishly; the other forces, namely pride and temptation, are outside of
their scope. In their model, shame is captured by two nonunique functions. Because of this
generality, their model can allow violations of IIA. In contrast, the GU model captures shameaversion by the unique nonnegative number βS , which facilitates distinguishing altruism from
impure altruism.
Independent of our paper, three recent papers address related phenomena. Noor and Ren
(2011) study a menu-dependent temptation to act selfishly. Noor and Ren (2011) extend the
model of Gul and Pesendorfer (2001) into preferences over menus of menus of allocations.
Menu dependence of temptation is an importance issue but orthogonal to the main issue in
our paper. As a result, we have not studied the menu dependence in our paper.
Evren and Minardi (2011) study warm glow. Their model is based on an interesting idea
of relating warm glow to the freedom to the be selfish. In Evren and Minardi’s (2011) main
result, their primitive preference is different from ours: they focus a preference over sets of
Pareto undominated allocations. In addition, Evren and Minardi’s (2011) utility function
implies that a set A can be worse than another set B even if A contains an allocation that
Pareto dominates any allocation in B. Their model is general enough to allow for violations
of IIA. On the other hand, due to this generality, Evren and Minardi’s (2011) representation
lacks uniqueness properties.
Feddersen and Sandroni (2009) also provide a model of warm glow. Their approach is
unique and different from the other papers including ours: They study a decision maker’s
choice function and an aspiration function that captures the decision maker’s normative
concerns. Advantage of their approach is that they dispense with ex-ante choices of sets.
However, they need the aspiration functions instead.
25
Appendix
A
Proofs
In this section, we prove the theorem. First we show the sufficiency of the axioms. Fix ≿
that satisfies the axioms in the theorem. The next lemma provides representations for ≿1
and ≿S .
Lemma 1 There exist a mixture-linear function u1 on ∆(Z) and positive numbers {αi }i∈S
such that (i) u1 represents ≿1 on ∆(Z), (ii) there exist z, z ∈ Z such that u1 (z) = 1 ≥
∑
u1 (p) ≥ 0 = u1 (z) for all p ∈ ∆(Z), and (iii) uS ≡ i∈S αi u1 on (∆(Z))S represents ≿S
∑
and i∈S αi = 1.
Proof of Lemma 1: First, we show that ≿1 is well defined (i.e., if (l1 , pS ) ≿ (r1 , pS ) for some
pS ∈ (∆(Z))S , then (l1 , qS ) ≿ (r1 , qS ) for all qS ∈ (∆(Z))S ). Suppose by way of contradiction
that (l1 , pS ) ≿ (r1 , pS ) and (l1 , qS ) ≺ (r1 , qS ). By Independence, (l1 , 21 pS + 12 qS ) ≿ ( 12 r1 +
1
l , 1p
2 1 2 S
+ 12 qS ) and (l1 , 12 qS + 12 pS ) ≺ ( 21 r1 + 12 l1 , 12 qS + 12 pS ). This is a contradiction. By the
same way, we can show that ≿S is well defined.
To show ≿1 satisfies Independence, fix p1 , q1 , l1 ∈ ∆(Z) and α ∈ [0, 1]. Then, for any
pS , qS ∈ ∆(Z), p1 ≿1 q1 ⇔ (p1 , pS ) ≿ (q1 , pS ) ⇔ α(p1 , pS ) + (1 − α)(l1 , qS ) ≿ α(q1 , pS ) + (1 −
α)(l1 , qS ) ⇔ αp1 +(1−α)l1 ≿1 αq1 +(1−α)l1 . By the same way, we can show that ≿S satisfies
Independence. Then, by the standard argument with von Neumann-Morgenstern’s Theorem,
there exist mixture-linear utility functions u1 and ûS , which are unique up to positive affine
transformation. Then, (ii) follows from the finiteness of Z and a normalization. Finally,
Pareto shows that if pi ≿ qi for all i ∈ S, then pS ≿ qS . Hence, by Fishburn (1984), there
∑
exist a positive numbers {αi }i∈S and a real number γ such that ûS (pS ) = i∈S αi u1 (pi ) + γ
∑
and i∈S αi = 1. By defining uS = ûS − γ, we obtain (iii).
■
For all A ∈ A , define u(A) = {(u1 (p1 ), uS (pS )) ∈ R2 |p ≡ (p1 , pS ) ∈ A} and A ∗ =
{u(A)|A ∈ A }. Since A ∈ A is closed, u(A) is also closed by the continuity of u1 and uS .
Since u1 (∆(Z)) = [0, 1] and uS ((∆(Z))S ) = [0, 1], A ∗ is a set of compact subsets of [0, 1]2 . We
endow A ∗ with the Hausdorff metric dh (A∗ , B ∗ ) = max{maxu∈A∗ minv∈B ∗ d(u, v ), maxu∈B ∗
26
minv∈A∗ d(u, v )}, where d is the Euclidean metric. We define a mixture on A ∗ as follows: for
all A∗ , B ∗ ∈ A ∗ and α ∈ [0, 1], αA∗ + (1 − α)B ∗ = {u ∈ R2 |u = αv + (1 − α)w for some v ∈
A∗ , w ∈ B ∗ }. Define ≿∗ on A ∗ as follows:
A∗ ≿∗ B ∗ ⇔ A ≿ B,
where u(A) = A∗ and u(B) = B ∗ . Next lemma shows the properties of ≿∗ .
Axiom (Independence*): A∗ ≿∗ B ∗ if and only if αA∗ + (1 − α)C ∗ ≿∗ αB ∗ + (1 − α)C ∗ .
Definition: (i) u is respectively dominated in A∗ if there exist elements v and w in A∗ such
that v1 ≥ u1 and wS ≥ uS . (ii) u is jointly dominated in A∗ if there exists an element v in
A∗ such that v1 ≥ u1 and vS ≥ uS .
Axiom: (Strong Dominance*) (i) If any element u in B ∗ is respectively dominated in A∗ ,
then A∗ ∪ B ∗ ≿∗ A∗ . (ii) if any element u in B ∗ is jointly dominated in A∗ , then A∗ ∪ B ∗ ∼∗ A∗ .
Lemma 2 ≿∗ is a well-defined complete, transitive, and continuous binary relation that
satisfies Independence* and Strong Dominance*.
Proof of Lemma 2: To show ≿∗ is well defined, first we show the following claim:
Claim: (i) If any element p in B is respectively dominated in A, then A ∪ B ≿ A and (ii)
if any element p in B is jointly dominated in A, then A ∪ B ∼ A.
Proof of Claim: Fix A, B ∈ A . Since ∆(Z) is separable, there exists a countable set
B ′ = {p1 , p2 , p3 , . . . } such that cl.B ′ = B. Let B0 = ∅ and Bn = {p1 , . . . , pn }, so that
′
Bn ⊂ Bn+1 for all n and B ′ = ∪∞
n=1 Bn . Then, Bn → cl.B = B in the Hausdorff metric
topology.25 To show (i), suppose that any pn ∈ B is respectively dominated in A. Then, by
Dominance, A ∪ Bn ≡ (A ∪ Bn−1 ) ∪ pn ≿ A ∪ Bn−1 . By the transitivity, A ∪ Bn ≿ A for all
n ∈ N. By the continuity, hence, A ∪ B ≿ A. (ii) can be proved in the same way.
□
To show ≿∗ is well defined, it suffices to show that if u(A) = u(B), then A ∼ B. If
u(A) = u(B), then for any p ∈ A there exists q ∈ B such that q1 ∼1 p1 and qS ∼S pS .
25
cl.A denotes the closure of A for all A ∈ A .
27
Therefore, the claim (ii) shows that A ∼ A ∪ B. In the same way, we can show B ∼ A ∪ B.
Hence, A ∼ B.
By the definition of ≿∗ , the claims (i) and (ii) show that ≿∗ satisfies Strong Dominance*.
Since ≿ is a complete and transitive binary relation that satisfies Independence, so is ≿∗ .
In the following, we show that ≿∗ is continuous. Choose any A∗ ∈ A ∗ to show {B ∗ ∈
A ∗ |B ∗ ≿∗ A∗ } and {B ∗ ∈ A ∗ |A∗ ≿ B ∗ } are closed.
Let {Bn∗ } be a sequence such that Bn∗ ≿∗ A∗ and Bn∗ → B ∗ to show B ∗ ≿∗ A∗ . By
definition, there exists a sequence {Bn } such that u(Bn ) = Bn∗ and Bn ≿ A. Since Bn ∈ A
and A is compact, there exists a convergent subsequence {Bk′ } such that Bk′ → B ′ . Since
Bk′ ≿ A for all k, then the continuity of ≿ shows that B ′ ≿ A. Since u is continuous, then
u(Bk′ ) → u(B ′ ). Since {u(Bk′ )} is a subsequence of {Bn∗ } and Bn∗ → B ∗ , u(B ′ ) = B ∗ . Since
B ′ ≿ A, B ∗ ≿∗ A∗ . In the same way, we can show that {B ∗ ∈ A ∗ |A∗ ≿∗ B ∗ } is closed.
■
For all u, v ∈ [0, 1]2 , define ≻′∗ on [0, 1]2 as follows:
u ≻′∗ v ⇔ p ≻′ q,
where u(p ) = u and u(q ) = v. Then by the definition of ≻′ , u ≻′∗ v if there exists A∗ ∈ A ∗
such that v ∈ A∗ , u is respectively dominated in A∗ , and A∗ ∪ u ≻∗ A∗ . Next lemma shows
the properties of ≻′∗ .
Axiom (Ex-post Independence*): If u ≻′∗ v then αu + (1 − α)w ≻′∗ αv + (1 − α)w for all
α ∈ (0, 1] and w ∈ [0, 1].
For any x ∈ R2 and ε ∈ R++ , we denote {y ∈ R2 |∥x − y∥ ≤ ε} by Bε (x ).
Axiom (Lower Continuity*): If u ≻′∗ v then there exists a positive number ε such that
u ≻′∗ w for all w ∈ Bε (v).
For all u, v ∈ [0, 1]2 , define u ≿′∗ v if v ̸≻′∗ u.
Lemma 3 (i) ≻′∗ is a nondegenerate and asymmetric binary relation satisfies Ex-post Independence* and Lower Continuity*. (ii) For any A∗ ∈ A ∗ , there exists u ∈ A∗ such that
u ≿′∗ v for all v ∈ A∗ .
28
Proof of Lemma 3: First, we show (i). Since ≻′ satisfies nondegeneracy and asymmetry, so
does ≻′∗ . To show Ex-post Independence*, let u ≻′∗ v. Choose any w ∈ [0, 1]2 and α ∈ (0, 1].
By definition, there exists A∗ ∈ A ∗ such that v ∈ A∗ , u is respectively dominated in A∗ ,
and A∗ ∪ u ≻∗ A∗ . Since ≻∗ satisfies Independence*, (αA∗ + (1 − α)w ) ∪ (αu + (1 − α)w ) ≡
α(A∗ ∪ u ) + (1 − α)w ≻∗ αA∗ + (1 − α)w. Since u is respectively dominated in A∗ , so
is αu + (1 − α)w in αA∗ + (1 − α)w. Since αv + (1 − α)w ∈ αA∗ + (1 − α)w, then
αu + (1 − α)w ≻′∗ αv + (1 − α)w.
To show Lower Continuity*, let u ≻′∗ v. Then, there exists A∗ ∈ A ∗ such that v ∈ A∗ ,
u is respectively dominated in A∗ , and A∗ ∪ u ≻∗ A∗ . Since v ∈ A∗ and ≿∗ is continuous,
there exists ε > 0 such that A∗ ∪ u ∪ w ≻∗ A∗ ∪ w for all w ∈ Bε (v). Since u is respectively
dominated in A∗ ∪ w, it follows that u ≻′∗ w for all w ∈ Bε (v).
To show (ii), fix A∗ ∈ A ∗ . First, we prove the lemma under the assumption that there
exists u ∈ co(A∗ ) such that u ≿′∗ v for all v ∈ co(A∗ ), where co(A∗ ) is the convex hull
of A∗ . If u ∈ A∗ then the lemma holds. If u ̸∈ A∗ then by the Caratheodory’s theorem,
there exist u1 , u2 , u3 ∈ A∗ and α1 , α2 ∈ [0, 1) such that u = α1 u1 + α2 u2 + (1 − α1 − α2 )u3 .
Since u ≿′∗ v for all v ∈ co(A∗ ), then u ≿′∗ ui for all i ∈ {1, 2, 3}. If u ≻′∗ ui for some
i ∈ {1, 2, 3}, then Ex-post Independence* shows that α1 u + α2 u2 + (1 − α1 − α2 )u3 ≻′∗
α1 u1 + α2 u2 + (1 − α1 − α2 )u3 ≡ u. This is a contradiction. It follows that ui ∼′∗ u for all
i ∈ {1, 2, 3}. Hence, ui ≿′∗ v for all v ∈ A∗ .
Now we show that there exists u ∈ co(A∗ ) such that u ≿′∗ v for all v ∈ co(A∗ ). For all
∗
v ∈ [0, 1]2 , define A (v ) = {u ∈ co(A∗ )|u ≻′∗ v} and A∗ (v ) = {u ∈ co(A∗ )|v ≻′∗ u }. By
∗
Sonnenschein (1971, Theorem 4), it suffices to show (i) v ̸∈ co(A (v )) for all v ∈ [0, 1]2 and
(ii) if u ∈ A∗ (v ), then u ∈ int.A∗ (v′ ) for some v′ ∈ [0, 1]2 .26
Lower Continuity* implies that A∗ (v ) is open for any v ∈ [0, 1]2 . Hence, A∗ (v ) =
∗
int.A∗ (v ). Therefore, we establish (ii) by letting v′ = v. By the asymmetry of ≻′∗ , v ̸∈ A (v ).
Sonnenschein (1971, Theorem 4) proves the following result: Let K ⊂ Rm be a compact and convex and
ˆ
ˆ
let ≻ be a binary relation on K satisfying the following : (i) x ∈ co{y ∈ K|y ≻x}
for all x ∈ K and (ii) if
′ˆ
′
ˆ
y ∈ {z ∈ K|x≻z}, then y ∈ int.{z ∈ K|x ≻z} for some x ∈ K (possibly x = x′ ). Then, there exists x ∈ K
ˆ
such that there is no y ∈ K such that y ≻x.
26
29
∗
∗
Hence to show (i), it suffices to show that A (v ) is convex. Choose u1 , u2 ∈ A (v ) and α ∈
[0, 1]. By Ex-post Independence*, αu1 +(1−α)u2 ≻′∗ αv+(1−α)u2 and αv+(1−α)u2 ≻′∗ v.
∗
∗
By the transitivity, αu1 + (1 − α)u2 ≻′∗ v. Hence, αu1 + (1 − α)u2 ∈ A (v ). Hence, A (v )
is convex. Therefore, it follows from Sonnenschein (1971, Theorem 4) that there exists
u ∈ co(A∗ ) such that u ≿′∗ v for all v ∈ co(A∗ ).
■
Now, we show a general representation for ≿∗ .
∑ (
Lemma 4 There exists a function µ : R → R such that V ∗ (A∗ ) = λ∈R maxu∈A∗ λu1 +
)
(1 − λ)uS µ(λ) represents ≿∗ . Moreover, supp(µ) ≡ {λ ∈ R|µ(λ) ̸= 0} is finite.
Proof of Lemma 4:
Step 1: (i) There exist finite sets K and L of mixture-linear utility functions on [0, 1]2 such
∑
∑
that ≿∗ is represented by V ∗ (A∗ ) = U ∈K maxu∈A∗ U (u ) − U ∈L maxu∈A∗ U (u ). (ii) For
any U ∈ K ∪ L, there exist no U ′ ∈ K ∪ L \ {U }, a > 0, and b ∈ R such that U = aU ′ + b.
Proof of Step 1: By Lemma 2, ≿∗ is a continuous, complete, and transitive relation that
satisfies Independence*. To show Step 1, by Kopylov (2009, Theorem 2.1), it suffices to
show that ≿∗ satisfies his Finiteness axiom, that is, for any sequence {A∗n } of A ∗ there
∪
∪N +1 ∗ 27
∗
∗
exists a positive integer N such that N
To show that ≿∗ satisfies
n=1 An ∼
n=1 An .
this axiom, choose any A∗1 , A∗2 , A∗3 , A∗4 ∈ A ∗ . Let u∗ = arg maxu∈A∗1 ∪A∗2 ∪A∗3 ∪A∗4 u1 and v∗ =
arg maxv∈A∗1 ∪A∗2 ∪A∗3 ∪A∗4 vS . (Such u∗ and v∗ exist because each A∗i is compact.) In addition,
denote by w∗ a maximal element of ≻′∗ in A∗1 ∪ A∗2 ∪ A∗3 ∪ A∗4 . (Such w∗ exists by Lemma 3
(ii).) Without loss of generality, assume u∗ ∈ A∗1 , v∗ ∈ A∗2 , and w∗ ∈ A∗3 . In the following,
we show A∗1 ∪ A∗2 ∪ A∗3 ∪ A∗4 ∼∗ A∗1 ∪ A∗2 ∪ A∗3 .
Since u∗ ∈ A∗1 and v∗ ∈ A∗2 , Strong Dominance* (i) shows that A∗1 ∪ A∗2 ∪ A∗3 ∪ A∗4 ≿∗
A∗1 ∪ A∗2 ∪ A∗3 . Suppose by way of contradiction that A∗1 ∪ A∗2 ∪ A∗3 ∪ A∗4 ≻∗ A∗1 ∪ A∗2 ∪ A∗3 .
Since [0, 1] is separable, there exists a countable set A′4 = {u1 , u2 , u3 , . . . } such that cl.A′4 =
A∗4 . Let B0∗ = A∗1 ∪ A∗2 ∪ A∗3 and Bn∗ = A∗1 ∪ A∗2 ∪ A∗3 ∪ {u1 , . . . , un } for each n. Then
Bn∗ → A∗1 ∪ A∗2 ∪ A∗3 ∪ cl.A′4 = A∗1 ∪ A∗2 ∪ A∗3 ∪ A∗4 in the Hausdorff metric topology. Since
Kopylov (2009, Theorem 2.1) shows that Step 1 holds if and only if ≿∗ is a continuous, complete, and
transitive relation that satisfies Independence* and Finiteness.
27
30
∗
A∗1 ∪ A∗2 ∪ A∗3 ∪ A∗4 ≻∗ B0∗ , by the continuity of ≿∗ , there exists n such that Bn+1
≻∗ Bn∗ .
Note that un+1 is respectively dominated in Bn∗ because u∗ ∈ A∗1 and v∗ ∈ A∗2 . Since
∗
Bn∗ ∪ un+1 = Bn+1
and w∗ ∈ Bn∗ , then un+1 ≻′∗ w∗ . This contradicts that w∗ is a maximal
element of ≻′∗ in A∗1 ∪ A∗2 ∪ A∗3 ∪ A∗4 . Hence, A∗1 ∪ A∗2 ∪ A∗3 ∪ A∗4 ∼∗ A∗1 ∪ A∗2 ∪ A∗3 .
□
Normalize each U ∈ K ∪ L by adding a constant number so as to obtain U (0, 0) = 0.
Step 2: For all U ∈ K ∪ L, there exist (a1 (U ), aS (U )) ∈ R2 such that U (u ) = a1 (U )u1 +
aS (U )uS . Moreover, for any U, U ′ ∈ K ∪ L,
a1 (U )
a1 (U )+aS (U )
̸=
a1 (U ′ )
.
a1 (U ′ )+aS (U ′ )
Proof of Step 2: For all U ∈ K ∪ L, define a1 (U ) = U (1, 0) and aS (U ) = U (0, 1). Fix
1
u ≡ (u1 , uS ) ∈ [0, 1]2 . Consider the case where u1 + uS ≥ 1. Then, u1 +u
U (u1 , uS ) + (1 −
S
(
)
1
1
1
1
S
)U (0, 0) = U u1u+u
, uS
= u1u+u
U (1, 0)+ u1u+u
U (0, 1) = u1 +u
(a1 (U )u1 +aS (U )uS ).
u1 +uS
S u1 +uS
S
S
S
Since U (0, 0) = 0, then U (u) = a1 (U )u1 + aS (U )uS . The other case where u1 + uS ≤ 1 can
be proved in the same way.
Suppose by way of contradiction that
Let c =
U (1,0)
.
U ′ (1,0)
a1 (U )
a1 (U )+aS (U )
=
a1 (U ′ )
a1 (U ′ )+aS (U ′ )
for some U, U ′ ∈ K ∪ L.
Then, a1 (U ) = ca1 (U ′ ), aS (U ) = caS (U ′ ), and U (u ) = u1 a1 (U ) + uS aS (U ) =
c(u1 a1 (U ′ ) + uS aS (U ′ )) = cU ′ (u ) for all u ∈ [0, 1]2 . This contradicts with Step 1 (ii).
□
For all λ ∈ R define

1 (U )


a + aS
if λ = a1 (Ua)+a
for some U ∈ K,

S (U )
 1
1 (U )
µ(λ) =
for some U ∈ L,
−(a1 + aS ) if λ = a1 (Ua)+a
S (U )



 0
otherwise.
Note that µ is well defined because for any U, U ′ ∈ K ∪ L,
a1 (U )
a1 (U )+aS (U )
̸=
a1 (U ′ )
.
a1 (U ′ )+aS (U ′ )
Therefore, by Step 1 and Step 2, we establish Lemma 4. Moreover, since K and L are finite,
■
supp(µ) is finite.
Fix ε < 12 . For all λ ∈ R, define
u∗1 (λ) =
1
ελ
1
ε(1 − λ)
+
, u∗S (λ) = +
, and u∗ (λ) = (u∗1 (λ), u∗S (λ)).
2 ∥(λ, 1 − λ)∥
2 ∥(λ, 1 − λ)∥
Then u∗ (λ) ∈ Bε ( 21 , 12 ). The next lemma is useful to characterize µ.
Lemma 5 (i) For all λ ∈ R, if λ′ ̸= λ, λu∗1 (λ) + (1 − λ)u∗S (λ) > λu∗1 (λ′ ) + (1 − λ)u∗S (λ′ )
31
for any λ′ ̸= λ. (ii) For all λ ∈ [0, 1], u∗1 (1) ≥ u∗1 (λ) and u∗S (0) ≥ u∗S (λ). (iii) If λ ̸∈ [0, 1],
|λ|
|λ|
u∗1 ( |λ|+|1−λ|
) ≥ u∗1 (λ) and u∗S ( |λ|+|1−λ|
) ≥ u∗S (λ).
Proof of Lemma 5: To show (i), choose any u ∈ Bε ( 21 , 21 ) and any λ ∈ R. Then, λu∗1 (λ) +
(1 − λ)u∗S (λ) = 21 + ε∥(λ, 1 − λ)∥ ≥ 21 + ∥u − ( 21 , 12 )∥∥(λ, 1 − λ)∥ ≥ 12 + (u − ( 21 , 12 )) · (λ, 1 − λ) =
λu1 + (1 − λ)uS , where the first inequality holds because u ∈ Bε ( 12 , 12 ) and the second
inequality holds by Cauchy-Scharz’s inequality. The two inequalities hold with equalities if
and only if u−( 12 , 12 ) =
ε
(λ, 1−λ),
∥(λ,1−λ)∥
or equivalently u = u∗ (λ). Moreover, u∗ (λ) = u∗ (µ)
if and only if λ = µ. Therefore, (i) holds. (ii) and (iii) follow from direct calculations.28 ■
Lemma 6 (i) For any λ ̸∈ [0, 1], then µ(λ) = 0. (ii) If λ ∈ (0, 1), then µ(λ) ≥ 0. (iii)
There exists unique λ∗ ∈ (0, 1) such that µ(λ∗ ) > 0. (iv) µ(0) ≤ 0.
Proof of Lemma 6: To show (i), suppose by way of contradiction that there exists λ′ ̸∈
{
}
{
|λ′ |
[0, 1] such that µ(λ′ ) ̸= 0. Define A∗ = u∗ (λ) ∈ R2 λ ∈ 0, 1, |λ′ |+|1−λ
∪ supp(µ) \
′|
}
{λ′ } . Since supp(µ) is finite, both A∗ and A∗ ∪ u∗ (λ′ ) are closed. Therefore, V ∗ (A∗ ) and
′
|λ |
∗
∗ ′
V ∗ (A∗ ∪ u∗ (λ′ )) are well defined. Since u∗ ( |λ′ |+|1−λ
′ | ) ∈ A , Lemma 5 (iii) shows that u (λ )
is jointly dominated in A∗ . Strong Dominance* (ii) shows that A∗ ∼∗ A∗ ∪ u∗ (λ′ ), so that
V ∗ (A∗ ) = V ∗ (A∗ ∪ u∗ (λ′ )).
However, Lemma 5 (i) shows that maxu∈A∗ ∪u∗ (λ′ ) λu1 + (1 − λ)uS = λu∗1 (λ) + (1 −
}
{
|λ′ |
∪ supp(µ) \ {λ′ }. Moreover,
λ)u∗S (λ) = maxu∈A∗ λu1 + (1 − λ)uS for all λ ∈ 0, 1, |λ′ |+|1−λ
′|
maxu∈A∗ ∪u∗ (λ′ ) λ′ u1 +(1−λ′ )uS = λ′ u∗1 (λ′ )+(1−λ′ )u∗S (λ′ ) > maxu∈A∗ λ′1 u1 +(1−λ′ )uS . There(
)
fore, V ∗ (A∗ ∪u∗ (λ′ ))−V ∗ (A∗ ) = λ′ u∗1 (λ′ )+(1−λ′ )u∗S (λ′ )−maxu∈A∗ λ′1 u1 +(1−λ′ )uS µ(λ′ ) ̸=
0 because µ(λ′ ) ̸= 0. This is a contradiction. Hence, (i) holds.
To show (ii), choose any λ′ ∈ (0, 1). Define A∗ = {u∗ (λ)|λ ∈ {0, 1} ∪ supp(µ) \ {λ′ }}.
Since λ′ ∈ (0, 1), Lemma 5 (ii) shows that u∗ (λ′ ) is respectively dominated by u∗ (0) and
u∗ (1) in A∗ . Since supp(µ) is finite, then A∗ is closed. Hence, A∗ ∈ A ∗ . Then, Strong
(ii) holds because u∗1 (1) = 12 + ε ≥ u∗1 (λ) and u∗S (0) = 12 + ε ≥ u∗S (λ) for all λ ∈ R. To see (iii) holds,
|λ|
|λ|
∈ [0, 1]. Hence, u1 ( |λ|+|1−λ|
) ≥ 21 > u1 (λ). Moreover, by a
consider the case where λ < 0. Then |λ|+|1−λ|
28
|λ|
direct calculation, uS ( |λ|+|1−λ|
)=
can be proved in the same way.
1
2
+
ε|1−λ|
∥(λ,1−λ)∥
= uS (λ) because 1 − λ > 0. The other case where λ > 1
32
Dominance* (i) shows V ∗ (A∗ ∪ u∗ (λ′ )) ≥ V ∗ (A∗ ).
By Lemma 5 (i), for all λ ̸= λ′ , maxu∈A∗ λu1 + (1 − λ)uS = λu∗1 (λ) + (1 − λ)u∗S (λ) =
(
maxu∈A∗ ∪u∗ (λ′ ) λu1 +(1−λ)uS . This equality shows V ∗ (A∗ ∪u∗ (λ′ ))−V ∗ (A∗ ) = maxu∈A∗ ∪u∗ (λ′ )
)
λ′ u1 +(1−λ′ )uS −maxu∈A∗ (λ′ u1 +(1−λ′ )uS ) µ(λ′ ). By Lemma 5 (i) again, maxu∈A∗ ∪u∗ (λ′ ) λ′ u1 +
(1 − λ′ )uS = λ′ u∗1 (λ′ ) + (1 − λ′ )u∗S (λ′ ) > maxu∈A∗ λ′ u1 + (1 − λ′ )uS . Since V ∗ (A∗ ∪ u∗ (λ′ )) −
V ∗ (A∗ ) ≥ 0, it follows that µ(λ′ ) ≥ 0. Hence, (ii) holds.
To show (iii), suppose by way of contradiction that there exist no λ ∈ (0, 1) such that
µ(λ) > 0. Then, by (ii), µ(λ) = 0 for all λ ∈ (0, 1). Therefore, V ∗ (u ) = µ(1)u1 + µ(0)uS for
all u ∈ [0, 1]2 . Since ≿ satisfies Pareto, it must hold that µ(1) > 0 and µ(0) > 0. However,
this implies that if u is respectively dominated in A∗ (i.e., v1 ≥ u1 and wS ≥ uS for some
v, w ∈ A∗ ) then A∗ ∪ u ∼∗ A∗ because V (A∗ ∪ u) = µ(1) maxu′ ∈A∗ u′1 + µ(0) maxu′ ∈A∗ u′S =
V (A∗ ). Therefore, there exist no u, v ∈ [0, 1]2 such that u ≻′∗ v. This contradicts with the
nondegeneracy of ≻′∗ .
Hence, to show (iii), it suffices to show that there exists at most one λ ∈ (0, 1) such
that µ(λ) > 0. Suppose by way of contradiction that there exist distinct elements ξ, η ∈
(0, 1) such that µ(ξ) > 0 and µ(η) > 0. Define A∗ = {u∗ (λ)|λ ∈ {0, 1} ∪ supp(µ) \
{ξ}}. Since supp(µ) is finite, A∗ is closed. Hence, A∗ ∈ A ∗ . Lemma 5 (i) shows that
maxu∈A∗ ∪u∗ (ξ) ξu1 +(1−ξ)uS = ξu∗1 (ξ)+(1−ξ)u∗S (ξ) > maxu∈A∗ ξu1 +(1−ξ)uS . For all λ ̸= ξ,
maxu∈A∗ ∪u∗ (ξ) λu1 + (1 − λ)uS = λu∗1 (λ) + (1 − λ)u∗S (λ) = maxu∈A∗ λu1 + (1 − λ)uS . Hence,
(
)
V ∗ (A∗ ∪u∗ (ξ))−V ∗ (A∗ ) = maxu∈A∗ ∪u∗ (ξ) ξu1 +(1−ξ)uS −maxu∈A∗ (ξu1 +(1−ξ)uS ) µ(ξ) > 0.
Therefore, A∗ ∪ u∗ (ξ)≻∗ A∗ . By Lemma 5 (ii), u∗ (ξ) is respectively dominated by u∗ (0) and
u∗ (1) in A∗ . Since u∗ (η) ∈ A∗ , u∗ (ξ)≻′ ∗ u∗ (η). However, by defining A∗ = {u∗ (λ)|λ ∈
{0, 1} ∪ supp(µ) \ {η}}, we obtain u∗ (η)≻′ ∗ u∗ (ξ) in the same way, which contradicts with
the asymmetry of ≻′ ∗ . Hence, (iii) holds.
Finally, to show (iv), note that the axiom of Shame of Acting Selfishly immediately implies the following: If u ≻′∗ v and uS < vS then {u } ≿∗ {u, v}. Choose any u1 , v1 , uS ∈ (0, 1)
such that u1 > v1 . Choose positive numbers ε and η such that ε < min{u1 , uS } and
∗
λ
∗
η < min{ 1−λ
is the unique λ ∈ (0, 1) such that µ(λ) > 0.
∗ (u1 − v1 ), 1 − uS }, where λ
33
Let A∗ = {(v1 , uS + η), (u1 − ε, uS ), (u1 , uS − ε)}. Since ε < min{u1 , uS }, then u1 − ε > 0
and uS − ε > 0. Since η < 1 − uS , then uS + η < 1. Therefore, A∗ ⊂ [0, 1]2 . Hence,
A∗ ∈ A ∗ .29 Since η <
λ∗
(u1
1−λ∗
− v1 ), then λ∗ u1 + (1 − λ∗ )uS > λ∗ v1 + (1 − λ∗ )(uS + η).
(
Hence, V ∗ (A ∪ u ) − V ∗ (A) = µ(λ∗ ) λ∗ u1 + (1 − λ∗ )uS − max{λ∗ v1 + (1 − λ∗ )(uS + η), λ∗ (u1 −
)
ε) + (1 − λ∗ )uS , λ∗ u1 + (1 − λ∗ )(uS − ε)} > 0. Therefore, A∗ ∪ u ≻∗ A∗ . Since u is respectively dominated in A∗ then u ≻′∗ (v1 , uS + η). Then by Shame of Acting Selfishly,
V ∗ ({u }) ≥ V ∗ ({u, (v1 , uS + η)}). Moreover, maxu′ ∈{u,(v1 ,uS +η)} u′1 = u1 = maxu′ ∈{u } u′1
and maxu′ ∈{u,(v1 ,uS +η)} λ∗ u′1 + (1 − λ∗ )u′S ≥ maxu′ ∈{u } λ∗ u′1 + (1 − λ∗ )u′S . Then by (iii),
V ∗ ({u, (v1 , uS + η)}) − V ∗ ({u }) ≥ (uS + η − uS )µ(0) = ηµ(0). It follows that µ(0) ≤ 0. ■
By using Lemma 4 and 6, we can show the sufficiency of the axioms. By the lemmas,
V ∗ (A∗ ) = µ(1) max∗ u1 + µ(λ∗ ) max∗ (λ∗ u1 + (1 − λ∗ )uS ) + µ(0) max∗ uS .
u∈A
u∈A
u∈A
(7)
Hence, for all u ∈ [0, 1]2 , V ∗ (u ) = (µ(λ∗ )λ∗ + µ(1))u1 + (µ(λ∗ )(1 − λ∗ ) + µ(0))uS . Normalize
µ so as to hold µ(λ∗ )(1 − λ∗ ) + µ(0) = 1. By Pareto, µ(λ∗ )λ∗ + µ(1) > 0. Define
α1 = µ(λ∗ )λ∗ + µ(1), β1 =
µ(1)
, and βS = −µ(0).
+ µ(1)
µ(λ∗ )λ∗
(8)
Then, α1 > 0. By Lemma 6 (iii), β1 < 1. By Lemma 6 (iv), βS ≥ 0. By substituting
(
)
(8) to (7), we obtain V ∗ (A∗ ) = maxu∈A∗ (1 − β1 )α1 u1 + (1 + βS )uS + β1 α1 maxu∈A∗ u1 −
βS maxu∈A∗ uS .30 For all A ∈ A , define V (A) = V ∗ (A∗ ). Then, A ≿ B ⇔ A∗ ≿∗ B ∗ ⇔
V ∗ (A∗ ) ≥ V ∗ (B ∗ ) ⇔ V (A) ≥ V (B). Hence, V represents ≿. By arranging the terms and
substituting u1 = u1 (p1 ) and uS = uS (pS ), we obtain the GU model.
To show that the maximizer p over A is a maximal element of ≻′ in A, we show the
following.
Lemma 7 If p ≻′ q, then U (p) > U (q).
Remember u1 (∆(Z)) = [0, 1] = uS ((∆(Z))S ). Then, for any (u1 , uS ) ∈ [0, 1]2 , there exist p1 ∈ ∆(Z)
and pS ∈ (∆(Z))S such that u1 (p1 ) = u1 and uS (pS ) = uS .
(
)
µ(1)
30
By the normalization and the definitions, µ(1) = β1 α1 , µ(λ∗ )λ∗ = (µ(λ∗ )λ∗ + µ(1)) 1 − µ(λ∗ )λ
=
∗ +µ(1)
∗
∗
α1 (1 − β1 ), and µ(λ )(1 − λ ) = 1 − µ(0) = 1 + βS .
29
34
Proof of Lemma 7: Let p ≻′ q. There exist A ∈ A such that (i) q ∈ A, (ii) p is respectively
dominated in A, and (iii) A ∪ p ≻ A. Then by (ii) and (iii), 0 < V (A ∪ p) − V (A) =
maxr∈A∪p U (r) − maxr∈A U (r). Hence, by (i), U (p) > U (q).
■
Fix A ∈ A and p ∈ arg maxp∈A U (p). Suppose by way of contradiction that there exists
q ∈ A such that q ≻′ p. Then by Lemma 7, U (q) > U (p). This is a contradiction. This
completes the proof of the sufficiency of the axioms.
The asymmetry of ≻′ follows from Lemma 7. By the lemma, p ≻′ q and q ≻′ p implies
U (p) > U (q) and U (q) > U (p), which is a contradiction. The necessity of the other axioms
is straightforward.
A.1
Proof of Remark and Propositions
Proof of Remark: By the standard uniqueness result on (∆(Z))I and the normalization
∑
(i.e., i∈S αi = 1), αi = αi′ for all i ∈ I and u = au′ +b for some real numbers a > 0 and b. By
the nondegeneracy, z ≻ z for some z, z ∈ Z. Normalize u such that u(z) = 1 and u(z) = −1.
(
)
For all x ∈ [0, 1] define p(x) = xδz + (1 − x) 21 δz + 12 δz ∈ ∆(Z), pS (x) = (p(x))i∈S , and
p(x) = (p(x), pS (x)). Then, u(p(x)) = x and uS (pS (x)) = x.
Now, we show β1 = β1′ . Since 1 − β1 > 0, 1 + βS > 0, α1 > 0, and αi > 0 for some
i ∈ S, then for all a ∈ [0, 1], there exists εa ∈ [0, a) such that (1 − β1 )α1 εa + (1 + βS )εa =
(
)
(1 − β1 )α1 a. Since u(p(x)) = x = uS (pS (x)) for all x ∈ [0, 1], then (1 − β1 )α1 u p(εa ) + (1 +
(
)
βS )uS pS (εa ) = (1 − β1 )α1 u(p(a)) + (1 + βS )uS (pS (0)).
({
(
)})
∑
For all a ∈ [0, 1], define g(a) = V p(εa ), p(a), pS (0) . Since i∈S αi = 1, then
(
)
g(a) = (α1 + 1)u p(εa ) + β1 α1 (a − εa )(u(z) − u(l0 )). Since ε0 = 0, then g(0) = V (p(0)).
Since g is continuous and V (z) > g(0) > V (z), there exists a positive number a such that
{
(
)}
V (z) > g(a) > V (z), so that z ≻ p(εa ), p(a), pS (0) ≻ z. By the continuity, there exists
{
(
)}
η ∈ [0, 1] such that ηδz + (1 − η)δz ∼ p(εa ), p(a), pS (0) . To make notation simple,
let r = ηδz + (1 − η)δz .31 Since εa < a and αi = αi′ , then β1 =
(α1 +1)(u′ (r)−u′ (p(εa ))
(a−εa )α′1 (u′ (z)−u′ (p(0)))
31
(α1 +1)(u(r)−u(p(εa ))
(a−εa )α1 (u(z)−u(p(0)))
= β1′ .
Then, the last indifference shows (α1 + 1)u(r) = (α1 + 1)u(p(εa )) + β1 α1 (a − εa )(u(z) − u(p(0)).
35
=
Next, we show βS = βS′ . Since 1 − β1 > 0, 1 + βS > 0, and αi > 0 for some i ∈ I, then
for all a ∈ [0, 1], there exists ξa ∈ [0, a) such that (1 − β1 )α1 u(p(ξa )) + (1 + βS )uS (pS (ξa ))) =
(1 − β1 )α1 u(p(0)) + (1 + βS )uS (pS (a)). Given this ξ, we can show βS = βS′ by considering
{
}
{
(
)}
(p(ξa ), (p(0), pS (a)) , instead of p(εa ), p(a), pS (0) , in the same way.
Proof of Proposition 1: Let p ≻′ q and q1 ≻1 p1 . Then, by Lemma 7, 0 < U (p) −
U (q ) = (1 − β1 )α1 (u(p1 ) − u(q1 )) + (1 + βS )(uS (pS ) − uS (qS )). Since u1 (q1 ) > u1 (p1 ),
then uS (pS ) > uS (qS ). Hence, V ({p, q }) − V ({p}) = β1 α1 (u(q1 ) − u(p1 )). It follows that
V ({p, q }) ≥ V ({p}) ⇔ β1 ≥ 0. Hence, Proposition 1 holds.
Proof of Proposition 2: In the following, we show (i). (ii) can be proved in the same way.
Fix l, l ∈ int.∆(Z) such that l ≻ l. For simplicity, normalize u by u(l) = 1 and u(l) = 0.
Consider the case where α1 ≥ 1. Define q1 =
1
l + αα1 −1
l,
α1
1
p = (l, (l)i∈S ), and q = (q1 , (l)i∈S ).
Hence, p, q ∈ (int.∆(Z))I . By a direct calculation, p ∼ q and p ≻S q.32 Hence, Impure
∑
Altruism imply p ≻′ q. Hence, by Lemma 7, 0 < U (p) − U (q) = i∈I αi (u(pi ) − u(qi )) +
βS (uS (pS ) − uS (qS )) + β1 α1 (u(q1 ) − u(p1 )) = βS + β1 , so that βS > −β1 . In the case where
α1 ≤ 1, we can show the result in the same way by defining q1 = α1 l + (1 − α1 )l. This
completes the proof of (i).
Next, we show the converse (i.e., βS > −β1 implies Impure Altruism). Choose any
p, q ∈ (int.∆(Z))I such that p ≿ q and pS ≻S qS . First, we consider the case where
∑
∑
p1 ≺1 q1 , so that u(p1 ) < u(q1 ). Since p ≿ q, then i∈I αi u(pi ) ≥
i∈I αi u(qi ). Note
∑
that U (p) − U (q) =
i∈I αi (u(pi ) − u(qi )) + βS (uS (pS ) − uS (qS )) + β1 α1 (u(q1 ) − u(p1 )).
Therefore, if β1 ≥ 0 then U (p) − U (q) > 0. If β1 ≤ 0, by assumption, β1 > −βS . Hence,
∑
U (p) − U (q) > i∈I αi (u(pi ) − u(qi )) + βS (uS (pS ) − uS (qS )) − βS α1 (u(q1 ) − u(p1 )) = (1 +
∑
βS ) i∈I αi (u(pi ) − u(qi )) ≥ 0. Therefore, U (p) > U (q). In the other case where p1 ≿1 q1 ,
U (p) > U (q) because pS ≻S qS .
Now, we show that U (p) > U (q) implies p ≻′ q. Since p ∈ (int.∆(Z))I , there exists
p′1 ∈ ∆(Z) and p′S ∈ (∆(Z))S such that u(p′1 ) < u(p1 ) and uS (p′S ) < uS (pS ). Define
32
Then, u(l) = 0, uS (l) = 1, u(q1 ) = 1/α1 , and uS (qS ) = 0. Hence,
36
∑
i∈I
αi u(pi ) = 1 =
∑
i∈I
αi u(qi ).
A = {q, (p1 , p′S ), (p′1 , pS )}. Then, V (A∪p )−V (A) = U (p )−max{U (q ), U (p1 , p′S ), U (p′1 , pS )}.
Hence, U (p ) > U (q ) implies A ∪ p ≻ A. Since p is respectively dominated in A, then p ≻′ q.
B
Extension
In this section, to incorporate inequality aversion, we axiomatize an extended GU model, in
which uS is a maxmin utility function. We consider a decision maker who is inequality-averse
among other agents’ allocations.
It is well known that the independence axiom may fail in social context because mixtures
among allocations can offset inequality in the mixed allocation. However, mixing with constant allocations does not offset inequality. Hence, we keep the following weaker version of
the independence axiom:
Definition: A set C ∈ A is called constant over S if pi = pj for any i, j ∈ S and p ∈ C.
Axiom (Weak Independence): Suppose that C is constant over S. Then A ≿ B if and only
if αA + (1 − α)C ≿ αB + (1 − α)C.
The next axiom captures inequality aversion among other agents’ allocations.
Axiom (Quasi-Concavity) If pS ∼S qS , then 21 pS + 12 qS ≿S pS .
Corollary: The following statements are equivalent:
(a) ≿ satisfies Quasi-Concavity, Weak Independence, as well as the axioms in the theorem
except Independence.
(b) There exists an extended GU model in which uS (pS ) = minαS ∈C
∑
i∈S
αi u(pi ) for some
C ⊂ ∆(S).
Proof: It is easy to see the necessity of the axioms. To show the sufficiency it suffices to
show the following two lemmas. First, instead of Lemma 1, we prove the next lemma by
using the standard argument with the von Neumann-Morgenstern’s theorem and Gilboa and
Schmeidler’s (1989) theorem.
Lemma 8 There exist a mixture linear function u1 on ∆(Z) and a closed subset C of ∆(S)
such that (i) u1 represents ≿1 on ∆(Z), (ii) there exist z, z ∈ Z such that u1 (z) = 1 ≥
37
u1 (p) ≥ 0 = u1 (z) for all p ∈ ∆(Z), and (iii) uS (pS ) ≡ minα∈C
∑
i∈S
αi u1 (pi ) represents ≿S .
Given the above u1 and uS , we define ≿∗ in the same way as in the proof of theorem.
Weak Independence of ≿ on A implies Independence of ≿∗ on A ∗ .
Lemma 9 ≿∗ satisfies Independence*.
Proof of Lemma 9: Fix C ∗ ∈ A ∗ . For all x ∈ [0, 1], define p(x) = xδz + (1 − x)δz and
pS (x) = (p(x))i∈S . Then, for all u ≡ (u1 , uS ) ∈ [0, 1]2 , u1 (p(u1 )) = u1 and uS (pS (uS )) = uS .
Define C = {(p(u1 ), pS (uS ))|u ≡ (u1 , uS ) ∈ C ∗ }. Then, C is constant over S and u(C) = C ∗ .
Therefore, by Weak Independence, A∗ ≿∗ B ∗ ⇔ A ≿ B ⇔ αA+(1−α)C ≿ αB +(1−α)C ⇔
αA∗ + (1 − α)C ∗ ≿∗ αB ∗ + (1 − α)C ∗ .
□
Since ≿∗ satisfies the same properties as in the proof of the theorem, Lemma 2–7 hold
in the same way. Hence, the sufficiency of the axioms holds with u1 = u1 (p1 ) and uS =
∑
minα∈C i∈S αi u1 (pi ). Therefore, Corollary holds.
■
C
Discussion on Experiments
Since two tendencies (i) and (ii) have not been extensively discussed in the experimental
papers, we describe these tendencies in detail in this section. We also observe that the
tendencies are robust even in experiments in which receivers could not identify dictators.
In the experiments conducted by Lazear et al. (2012), we could observe tendency (i),
namely, that medium-level donors exit more often than low-level and high-level donors,
when playing the dictator game is subsidized. In the experiments, 96 subjects (48 dictators)
participated in five sequential sessions of dictator games with an exit option. Lazear et
al. (2012) provided dictators with $10 as baseline endowment and, on top of that, added
subsidies of $0, $1, $3, $6, and $10 to the baseline endowment in order. For each subsidy
value, dictators decided whether to play the dictator game or exit. Then, the dictators
decided the donation amount publicly if they did not exit. For each dictator, the left figure
in Figure 3 (p. 14) of Section 5 shows the minimal subsidy needed to play the dictator game
38
and the dictator’s average donated proportion.33 Clearly, the figure shows tendency (i).34
We found consistent evidence for tendency (i) in the earliest experiments on dictator
games with an exit option conducted by Dana et al. (2006). Dana et al. (2006) provided
dictators with $10 as an endowment and asked dictators the donation amount before the
dictators knew that they could exit privately. When the dictators exited, they obtained
$9 privately and receivers obtained $0 without knowing that this is a consequence of the
dictators’ choice. The right figure in Figure 3 shows the percentage of dictators who exited
and their (intended) donated proportion, which clearly exhibits tendency (i).35
In the experiment conducted by Dana et al. (2006), dictators were anonymous, while in
the treatment conducted by Lazear et al. (2012), receivers could identify dictators. Hence,
the consistency between these two experiments, as captured by Figure 3, would support our
hypothesis: as long as playing dictator games is common knowledge among subjects, the
dictator would consider the receiver’s wish that the dictator should act altruistically. Hence,
the dictator could feel pride in acting altruistically by living up to the receiver’s wish and
ashamed of acting selfishly by denying their wish, even though the receiver could not identify
the dictator.
In another treatment, Lazear et al. (2012) imposed costs instead of providing subsidies.
Then, they found tendency (ii), namely, that the lower-level donors exit more often than the
higher-level donors. In the treatment, if dictators exited then they received $11; otherwise,
they played the dictator game with a total endowment of $10. Lazear et al. (2012) found that
among the dictators who did not exit, the average donation from dictators to receives was
35 percent of the total endowment. On the other hand, the proportion was only 22 percent
in their baseline standard dictator game without the exit option. This implies tendency (ii).
Moreover, in a field experiment, Della Vigna et al. (2011) found consistent evidence for
33
We regressed donated proportion on subsidy size. The estimated coefficient on the subsidy size is
−1.6 · 10−4 (p = 0.887), which is not significantly different from zero. Hence, the donated proportion is
statistically constant across the treatments.
34
We made the left figure of Figure 3 based on the no-anonymity treatment in Experiment 2 in Lazear et
al. (2012).
35
We made the right figure of Figure 3 based on Figure 1 (p.197) in Dana et al. (2006).
39
tendency (ii).36
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36
Della Vigna et al. (2011) designed a door-to-door fundraiser where some households had an option to
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40
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