2014-9
Leadership in the Prisoner’s Dilemma
With
Inequity-Averse Preferences
Koji Abe Hajime Kobayashi Hideo Suehiro

Leadership in the Prisoner’s Dilemma
withInequity-Averse Preferences∗
Koji Abe†
Yokohama National University

Hajime Kobayashi‡Kansai University

Hideo Suehiro§
Kobe University

May 9, 2014

Abstract
We consider the economic consequences of fairness concerns under the freedom
to choose the timing of moves by developing a new economic theory of leadership.
We study the prisoner’s dilemma in which players are endowed with Fehr andSchmidt preferences with inequity aversion as their private information and thenchoose cooperation or defection once at one of two timings that they prefer. In
this model, we consider an equilibrium in which a leader–follower relationship endogenously emerges as a consequence of players’ heterogeneous inequity aversions.
We present three results. First, we provide a suﬃcient condition for the existenceof a leadership equilibrium. Then, we present a comparative statics analysis of
the equilibrium. Finally, we investigate who takes the leadership, depending on
the game parameters. We provide a characterization of the equilibrium leadership
patterns. These results also hold when agents can choose the timing of moves
from more than two timings.

JEL Classiﬁcation: C72, D03, D82
Keywords: Leadership, Endogenous Timing, Prisoner’s Dilemma, Inequity Aversion∗

We are grateful to Eduard Faingold, Ernst Fehr, Benjamin Hermalin, Hideshi Itoh, Shinsuke
Kambe, Takashi Kunimoto, Eiichi Miyagawa, Yasuyuki Miyahara, Daniele Nosenzo, Hideo Owan,
Larry Samuelson, Klaus Schmidt, and Stephanie Wang for their helpful comments. We thank conference and seminar participants at NASM 2012, EEA/ESEM 2012, GAMES 2012, a seminar at the
University of München, the GCOE workshop at Osaka University, the Kyoto–Hitotsubashi Game
Theory Workshop 2013, and the mini–conference on economics of leadership at Hitotsubashi University. Koji Abe, Hajime Kobayashi, and Hideo Suehiro gratefully acknowledge ﬁnancial support fromthe Japan Society for the Promotion of Science under JSPS KAKENHI Grant Numbers 24730166,
24530207, and 24530198, respectively.
†
Faculty of International Social Sciences, Yokohama National University, 79-4 Tokiwadai,
Hodogaya, Yokohama 240-8501, Japan; abe@ynu.ac.jp.
‡
Faculty of Economics, Kansai University, 3-3-35, Yamate, Suita, Osaka, 564-8680, Japan;khajime@kansai-u.ac.jp.
§
Graduate School of Business Administration, Kobe University, 2-1, Rokkodai, Nada, Kobe, Hyogo,657-8501, Japan; suehiro@kobe-u.ac.jp.

11Introduction

1.1

Introduction

A considerable body of evidence by the experimental research in recent decades indicates that many individuals are concerned with fairness (Fehr and Schmidt (2006),
Cooper and Kagel (2013)). This pushes economists towards examining how fairness
concerns aﬀect the consequences of economic transactions. The research so far has
focused on the implications of the fact that fairness-concerned individuals may choose
diﬀerent alternatives than self-interested individuals in exogenous move games.1 This
paper addresses another issue that fairness concerns may inﬂuence economic outcomes
for the reason that fairness-concerned individuals may choose to take actions at different timings than self-interested individuals when they have the freedom to choose
when to move.2
Economists recognize that fairness concerns can aﬀect economic transactions throughtime. For example, conditional cooperation is one of the central ideas that fairness concerns could cause individuals to behave diﬀerently from self-interest (Gächter (2007)).
Timing is an essential element in the attitude paraphrased as “if you cooperate, I
will cooperate.” Researchers have found that a considerable percentage of people
exhibit the conditional cooperation attitude when placed in an “after you” position(Fischbacher, Gächter and Fehr (2001), Herrmann and Thöni (2009)).
However, when agents have the freedom to choose when to move, the conditional
cooperation is in vain unless the “after you” position is realized as an equilibrium play.
Including conditional cooperation, it remains unexplored how fairness concerns aﬀect
economic transactions by eﬀectively changing individual behaviors taken within a spanof time under the freedom to choose when to move.
We develop a theory of one such mechanism. In particular, we consider the issueof time and social dilemmas. We show that if agents are concerned with fairness,
the freedom to choose the timing of moves could resolve certain social dilemmas. If
individuals are concerned with fairness, conditional cooperation has the potential torealize cooperation when they move sequentially. The issue is how to have agents movesequentially under the freedom to choose the timing of moves and how to have an earlier
mover commit to cooperation, which is necessary for later cooperation by conditional
cooperators. We show that when agents are divergent in their fairness concerns andthere are enough conditional cooperators, they sort themselves into earlier and later
movers by their own will, and earlier movers commit to cooperation so that later
1

Several alternative approaches to modeling fairness concerns have been proposed. One of thepromising models is that of Fehr and Schmidt (1999), which we employ in this paper. Studies of choicebased on this model under exogenously given sequences of moves extend to various ﬁelds, including(1) ultimatum games (Fehr and Schmidt (1999)), (2) moral hazard (Fehr and Schmidt (2004), Itoh(2004), Demougin and Fluet (2006), Demougin, Fluet and Helm (2006), Dur and Glazer (2008), ReyBiel (2008), Englmaier and Wambach (2010), Neilson and Stowe (2010)), (3) partnerships (Bartlingand von Siemens (2010a)), (4) teams (Li (2009)), (5) tournaments (Grund and Sliwka (2005), Bartlingand von Siemens (2010b), Dubey, Geanakoplos, and Haimanko (forthcoming)), (6) contract design(Fehr, Klein, and Schmidt (2007)), (7) ownership (Fehr , Kremhelmer, and Schmidt (2008)), and (8)adverse selection (Desiraju and Sappington (2007) and Rasch, Wambach, and Wiener (2012)).
2Time is an essential ingredient of many economic transactions. In many natural settings, agents
move without an order of moves being formally provided and the consequences of economic transactions
are critically inﬂuenced by the choice of timings of moves by the agents in equilibrium. Under theclassical assumption of self-interested agents, economists have explored how agents guide themselves
to move within a span of time and how the freedom to choose timings of moves aﬀects the eﬃciency oftransactions. Examples are the English auction (Milgrom and Weber (1982)), bargaining (Perry andReny (1993)), and joint projects (Marx and Matthews (2000)).

2

cooperation is realized by later movers who are conditional cooperators. This happens
because agents with diﬀerent fairness concerns have diﬀerent degrees of incentives to
induce conditional cooperation from others. Thus, fairness concerns play a double role
in resolving social dilemmas: the role of generating conditional cooperation and therole of realizing the sequence of moves necessary for conditional cooperation to comeinto eﬀect through leadership.
In our analysis, we speciﬁcally consider the prisoner’s dilemma as a prototypesocial dilemma. We hypothesize that a player has Fehr and Schmidt (1999) preferenceswith inequity aversion over the outcomes of the prisoner’s dilemma. Furthermore, we
assume a variety of preferences among players, under which players may diﬀer in their
sensitivities to inequity measured by an envy parameter α and a guilt parameter β.
We consider a Bayesian model of the prisoner’s dilemma with endogenous moves. A
player is endowed with his type, which is deﬁned by his envy parameter and guiltparameter (α, β). A type is a realization of a continuous random variable and it ishis private information. Each player must choose C (cooperation) or D (defection).
There are two timings for moves: timing 1 and timing 2. Each player makes his choice
between C and D once at one of the two timings and which timing he selects is at his
discretion.
We examine a particular Bayesian strategy, which we will call a three-mode strategy. It divides player’s types into three categories: leader types, defector types, andconditional cooperator types. A leader type chooses C at timing 1, whereas a defector
type and a conditional cooperator type wait.3 A defector type and a conditional cooperator type diﬀer in their responses to their opponent’s behaviors at timing 1. The
former chooses D irrespective of his opponent’s behaviors, while the latter responds to
C with C (and with D otherwise.)
If a three-mode strategy prevails as an equilibrium and a leader type is matchedwith a conditional cooperator type, then the leader type commits to C at timing 1
and then the conditional cooperator type responds with C after postponing his choice
until timing 2. The cooperative outcome (C, C) is realized along an equilibrium pathby having players move sequentially by their own will. In this equilibrium, the on-pathbehavior of the leader type is a leadership behavior (that is, a behavior of taking theleadership).
We show three results. First, we present a suﬃcient condition over a set of payoﬀ
parameters of a prisoner’s dilemma for a three-mode strategy to be a sequential equilibrium of the game. The condition has a simple and straightforward interpretation;a pure materialist who feels no envy and no guilt (α = β = 0) has an incentive tolead by choosing C at timing 1 as long as he expects that his opponent postpones his
choice until timing 2 with certainty. For any prisoner’s dilemma that supports thisincentive, there exists a sequential equilibrium in a three-mode strategy, in which thepure materialist himself may or may not become a leader type.
Second, we examine a comparative statics. Suppose that a payoﬀ from the cooperative outcome becomes higher or a payoﬀ from the defection outcome (D, D) becomes
lower. Then, we can say that the cooperation becomes less diﬃcult, because the incentive to choose D over C against an opponent’s choice of C is weaker under a higher
cooperative payoﬀ; the incentive to choose D over C against an opponent’s choice of D
is weaker under a lower defection outcome payoﬀ; and the degree of the Pareto improvement from the defection outcome to the cooperative outcome is larger in either case.
We show that the leadership behavior is more likely to emerge in a prisoner’s dilemma3

As we mention in the Discussion section, a defector type does not need to wait in general. Here,we consider the simplest case to illustrate our equilibrium strategy.

3with less diﬃculty of cooperation than in a prisoner’s dilemma with more diﬃculty.
Formally, the maximum value of the probabilities that the prior assigns to the leader
type in all the sequential equilibria in a three-mode strategy is higher in the prisoner’s
dilemma with less diﬃculty of cooperation than in the prisoner’s dilemma with morediﬃculty.4 The same is true for the minimum value of the leader-type probabilities
under the suﬃcient condition for the existence of a three-mode equilibrium.
Third, we characterize our equilibrium in terms of who takes the leadership. We
introduce the notion of an incentive to lead and deﬁne a way to compare the strengthof the incentive to lead across types. A type (α, β) who has the strongest incentive tolead is a representative type in the set of leader types in a given prisoner’s dilemma.
We identify the type who has the strongest incentive to lead, and we show how the typevaries according to a change in the parameters of the prisoner’s dilemma. The purematerialist (α = β = 0) has the strongest incentive to lead in a prisoner’s dilemma
with a higher diﬃculty of cooperation, while a particular type who has modest fairness
concerns (modestly high α and β) has the strongest incentive to lead in a prisoner’s
dilemma with a lower diﬃculty of cooperation. Finally, we study how the whole set ofleader types vary according to a change in the parameters of the prisoner’s dilemma.
We show that, as the cooperation becomes less diﬃcult, the set of leader types in acorresponding prisoner’s dilemma consists of more fairness–concerned types.
Furthermore, we show that these three results also hold when agents can choose thetiming of moves from more than two timings. The mechanism of leadership elucidated
by our analysis is also valid and our conclusion on the resolution of social dilemmas
remains unchanged. Our theory of leadership applies to a prisoner’s dilemma withmany timings in general.
The main contribution of this paper is to consider the issue of time and social
dilemmas by taking into account the fact that many individuals are concerned with
fairness. Our theory is a ﬁrst attempt to provide a mechanism by which fairness
concerns aﬀect the consequences of economic transactions under the freedom to choose
the timing of moves.
Our theory advances our understanding of the role that fairness concerns play indynamic transactions. For example, recent experimental results indicate that somesocial dilemmas could be resolved when individuals have the freedom to choose when
to move. Arbak and Villeval (2013) conduct experiments of two-stage voluntary contribution games with a similar time structure to our game, in which each subject isrequested to make a choice of contribution level at one of two timings that he prefers.5
The data display that some of the subjects move with high levels of contributions inadvance of the others and some of those subjects who postpone their choices to timing
2 respond with high levels of contributions.6 In light of the importance for economists
4

We show that there may exist multiple equilibria in a three-mode strategy, depending on a set of
payoﬀ parameters.
5
The game of Arbak and Villeval (2013) is a standard voluntary contribution game, in which payoﬀs
are symmetric across agents and zero contribution is the dominant strategy. This game involves a social
dilemma similar to the prisoner’s dilemma. Nosenzo and Sefton (2011), Rivas and Sutter (2011) andPréget, Nguyen, and Willinger (2012) also conduct similar experiments. The results of Rivas andSutter (2011) and Préget, Nguyen, and Willinger (2012) are less relevant to this paper because they
use the repeated-game setting for their experiments. The experiments of Arbak and Villeval (2013)
and Nosenzo and Sefton (2011) are implemented in the stranger setting and so the subjects in theexperiments face a social dilemma game in the same setting as this paper. However, Nosenzo andSefton (2011) diﬀer from Arbak and Villeval (2013) in that their game is a version of the Varian(1994) game, in which payoﬀs are asymmetric across agents and there exists strategic substitution
eﬀects.
6
See Figure 1 and Table 3 in Arbak and Villeval (2013).

4

to understand how human beings can and do overcome social dilemmas, their ﬁndings
urge economists to develop a theory that explains the role of the freedom to choose
the timing of moves in resolving social dilemmas. Our theory provides a possible
theoretical interpretation for what forces resolve the dilemma in their experiments because their ﬁndings are compatible with our equilibrium prediction for the prisoner’s
dilemma that leading by cooperation occurs with positive probabilities and responding to cooperation with cooperation occurs with positive probabilities, depending on
players’ types.
Our approach to the issue of time and social dilemmas has several features to be
noted. First, we employ the Fehr and Schmidt (1999) model for fairness concerns. Thismodel, which has proved powerful in explaining various experimental results, is one of
the simplest models proposed to express other regarding preferences. The tractabilityof this model enables us to develop not only the condition for equilibrium existence (theﬁrst result) but also two fundamental results: the comparative statics on leadershipprobabilities (the second result) and the characterization of equilibrium in terms ofleader’s types (the third result). These results can be used to induce a set of testablepredictions or hypotheses based on our theory for the purpose of testing the theory
in future experiments. These results may also provide theoretical insights when one
considers the issue of time and other forms of social dilemmas.7
Second, we introduce incomplete information into fairness concerns by agents. If a
player knows that his opponent is endowed with the fairness concerns that make theopponent a conditional cooperator, it is obviously in the interest of the player to leadby taking C at timing 1 and induce the opponent to take C at timing 2. Therefore,
under complete information of fairness concerns by agents, there is no obstacle forthem to resolve a social dilemma by leadership.8 In reality, the way in which people
respond to leadership by others is diverse and it is often observed that a leadership isbetrayed by a follower with uncooperative behaviors. This makes it diﬃcult for people
to resolve a social dilemma by leadership. How this obstacle is overcome by agents isthe central issue that must be studied to understand the mechanism by which fairness
concerns work in resolving a social dilemma under the freedom to choose the timing ofmoves. Hence, it is essential for our theory to consider incomplete information about
fairness concerns by agents.
There are a few papers that discuss incomplete information under the Fehr andSchmidt (1999) preferences. Fehr and Schmidt (1999), (2004), Fehr, Klein, and Schmidt(2007), and Fehr, Kremhelmer, and Schmidt (2008) argue that their experimental ﬁndings on the ultimatum, the multitask, the contract design, and the ownership games
can be explained if they assume that subjects are endowed with one of two or four kindsof parameters in the Fehr and Schmidt (1999) preferences with particular probabilities,
and that a set of parameters for a subject is his private information.9
In contrast to these experimental studies, we examine the whole set of continuous
densities over the domain of the Fehr and Schmidt (1999) preferences. Then, wedevelop our three results in terms of a set of payoﬀs and a density of the preferences.
As Fehr, Kremhelmer, and Schmidt (2008) state, the heterogeneity of fairness concerns
is a well–established fact and thus the question is not whether theory should incorporate
7

For example, if we conﬁne ourselves to prisoner’s dilemmas, we can show that very diﬀerent
behaviors are realized in simultaneous moves and exogenous sequence moves. Our theory helps us
understand why and how they diﬀer. See our Discussion.
8
We will elaborate on this point in the Discussion.
9
They conduct this exercise of choosing particular distributions because their purpose is to showthat the theory based on Fehr and Schmidt (1999) preferences helps us interpret and better understandthe observed data patterns. See footnote 16 in Fehr, Klein, and Schmidt (2007).

5

heterogeneous social preferences but which distribution of preferences theory should
assume.10 In this respect, this paper is one of the ﬁrst attempts to study games played
by Fehr–Schmidt players under general incomplete information and provide robust
results by associating them with conditions on type distribution.
Finally, our theory investigates a three-mode strategy that assigns three diﬀerent behavioral modes to the corresponding three classes of Fehr and Schmidt (1999)
preferences. The Fehr and Schmidt (1999) model is a very simple model with a twodimensional type space: envy parameter and guilt parameter. Fehr and Schmidt(2004), Fehr, Klein, and Schmidt (2007), and Fehr, Kremhelmer, and Schmidt (2008)
successfully illustrate that even such a simple model is enough to explain the experimental data for fairly complex games, such as the multitask, the contract design, andthe ownership with two behavioral modes corresponding to two classes of Fehr andSchmidt (1999) preferences.
In our game, however, it can be shown that cooperation by leadership cannot
be supported in equilibrium if we conﬁne a Bayesian strategy to a class that admits
only two behavioral modes. Our ﬁrst result shows that the Fehr and Schmidt (1999)
model successfully generates three behavioral modes, which support the cooperationin equilibrium. This indicates that in spite of its limited type space, the Fehr andSchmidt (1999) model may be powerful and useful enough to explain diverse behaviors
in complex games.11
The rest of this paper is organized as follows. We review the related literature in
Subsection 1.2. Then, we begin our analysis in Section 2 with an illustrative numericalexample to demonstrate the idea of the leadership mechanism driven by fairness concerns. Section 3 presents the formal model of the prisoner’s dilemma played throughtwo timings by players with inequity-averse preferences. Section 4 explains the notion of a three-mode equilibrium in this game. Section 5 states the ﬁrst result on
the existence of a three-mode equilibrium. Section 6 states the second result on thecomparative statics of the three-mode equilibrium. Section 7 states the third result on
the characterization of leadership patterns with respect to fairness concerns. Section 8shows that the three-results developed in a prisoner’s dilemma with two timings also
hold in a prisoner’s dilemma with many timings in general. Section 9 discusses someissues that we excluded from our analysis. Appendix A supplements Section 5 withrespect to the exact bound on the game parameters for the existence of a three-mode
equilibrium. Furthermore, when this exact bound applies, the results in Sections 6
and 7 become much sharper. Appendix B supplements Section 7 by showing that acertain class of leader patterns is not supported in equilibrium by players with some
distribution of fairness concerns. Appendix C contains proofs.

1.2

Literature

There are many empirical studies showing that observed behaviors are inconsistent withthe classical assumption that all agents are self-interested. An agreed fact is that many
people are concerned with fairness and they are heterogeneous in fairness concerns.
For example, by conducting an Afriat–Diewert–Varian-type revealed preference test onsocial outcomes, Andreoni and Miller (2002) verify it and in addition demonstrate thatmost of the subjects, whether identiﬁed as self-interested or not, are consistent with the10

See footnote 16 in Fehr, Kremhelmer, and Schmidt (2008).
An example of diverse behaviors is the behaviors observed in the ultimatum experiments. Fehrand Schmidt (1999) argue that four classes of the Fehr and Schmidt (1999) preferences is enough to
explain the data.
11

6

standard revealed preference paradigm.12 Such empirical studies have provoked thedevelopment of various economics models that accommodate observed other-regarding
behaviors. Using those models, many researchers have studied how fairness concerns
aﬀect economic outcomes in diﬀerent environments. See Camerer (2003), Fehr andSchmidt (2006), and Cooper and Kagel (2013) for a detailed survey of the empiricaland theoretical studies.
Among the models for studying fairness concerns, the outcome-based social preference models are most closely related to this paper. In the models, an agent withparticular fairness characteristics is modeled as a preference relation over social outcomes. An altruism model (Andreoni and Miller (2002)) assumes that an agent’s
utility depends on the relevant agents’ material payoﬀs proﬁle and is monotonicallyincreasing in the material payoﬀ of the other agent.13 An envy model (Bolton (1991))
assumes that an agent’s utility depends on his material payoﬀ and his relative payoﬀ tothe other player’s payoﬀ and that his utility is strictly increasing in his relative payoﬀ
when it is smaller than 1 and otherwise constant in his relative payoﬀ.14 These models
can capture agents’ unconditional reciprocal behavior and unconditional spiteful behavior, respectively, but not simultaneously. In contrast to these models, the Fehr andSchmidt (1999) model of inequity aversion that we employ in this paper models thetwo-dimensional aspect of social preferences.15 In particular, the two kinds of sourcesof inequity aversions (envy and guilt) are treated separately. This enables us to explain
leadership by three diﬀerent behavioral modes that human beings exhibit in a given
prisoner’s dilemma.16
Other than the outcome-based models of social preferences, there are two dominantmodels of fairness concerns in the literature. In models of intention-based social preferences, a social interaction is modeled by using psychological game theory (Geanakoplos, Pearce, and Stacchetti (1989), Battigalli and Dufwenberg (2009)), and an agent
with particular fairness characteristics is modeled as a preference relation over social
outcomes and (higher-order) beliefs about behavior induced by the supposed strategy proﬁle to capture concerns with agents’ intention of the play. For applications ofintention-based approach, see Rabin (1993), Dufwenberg and Kirchsteiger (2004), Falkand Fischbacher (2006), Segal and Sobel (2007), Charness and Dufwenberg (2006), andBattigalli and Dufwenberg (2007).
In models of interdependent social preferences, an agent with particular fairness
characteristics is modeled as a collection of preference relations over social outcomes,
where interdependency arises in the sense that one of the preference relations is adopted
depending on the fairness characteristics of everyone involved (Levine (1998)). Thispreference interdependency captures agents’ personalities such as “if you are a good12

Of their 142 subjects, there is only one subject who had serious violations of revealed preference.
See also Andreoni (1989), Andreoni (1990) for a related warm-glow model in a public-goods economy. Rotemberg and Saloner (1993) also study a leadership style using a model of preferences thatparameterize a degree of altruism.
14
See Kirchsteiger (1994) for a related model.
15
The ERC model (Bolton and Ockenfels (2000)) assumes that an agent’s utility depends on hismaterial payoﬀ and his payoﬀ relative to the other player’s payoﬀ as in an envy model. In contrast to
an envy model, however, the utility has a ﬂexibility in its partial derivative with respect to the relative
payoﬀ and hence can capture both reciprocity and spitefulness behaviors. The Fehr and Schmidt(1999) and ERC model yield qualitatively similar results for the two-player cases but not in general.
See Fehr and Schmidt (2006) for a related discussion.
16
One can consider more complex models than the Fehr and Schmidt (1999) model by combining
it with other models. For example, Charness and Rabin (2002) and Erlei (2008) develop such hybrid
models. We employ the Fehr and Schmidt (1999) model because the model incorporates the minimum
set of elements in social preferences that we need to explain the leadership mechanism.
13

7

person, then I will treat you kindly.” See Rotemberg (2008, 2011) and Gul and Pesendorfer (2010) for applications of interdependent social preference models.17
Adopting these alternative models of other-regarding concerns may oﬀer diﬀerent
kinds of leadership mechanisms from the one investigated in this paper.18 Although
this may provide interesting research topics, we employ the Fehr and Schmidt (1999)
model for two reasons. First, although this model is simple, it successfully explains
the dynamic behaviors in prisoner’s dilemmas. In this respect, the Fehr and Schmidt(1999) model is certainly a promising candidate model when one considers the issueof time and social dilemmas by taking fairness concerns into account. Second, theFehr and Schmidt (1999) model is tractable. In particular, this model has a simple
and tractable utility functional form.19 This fact enables us to develop our analyses inseveral directions, such as equilibrium existence, comparative statics, and equilibrium
characterization in terms of fairness characteristics. Thus, the tractability of the Fehrand Schmidt (1999) model leads us to a testable theory on time and social dilemmas.
There are also other kinds of other-regarding models in team theory frameworks,
which may be labeled as action-based other-regarding models, because those models
directly assume that an agent feels psychological disutility from the relevant agents’actions. Among action-based other-regarding models, the one most related to ours
is a model of social pressure. In addition to the cost of eﬀort, a member of a teamfeels a disutility when he takes an action in a particular direction.20 Using a pressurefunction, Huck and Rey-Biel (2006) model a conformity pressure on eﬀort choices.
That is, a member feels disutility from the diﬀerence between his eﬀort level and his
opponent’s eﬀort level. They study the relations between leadership and degrees ofconformity inclination in a two-member team and show that endogenous leadership occurs in a team with at least one conformist under a complete information setup. Then,they brieﬂy discuss a private information model with two possible degrees of conformity inclination (conformist and nonconformist) and claim that there are sequential
equilibria with endogenous leadership. In contrast to Huck and Rey-Biel (2006), we
employ an outcome-based social preference model, study endogenous leadership undera general incomplete information setup, and develop our analyses in several directionsin addition to equilibrium existence.
Finally, other than the current paper, there are some attempts to explain leadership17

Gul and Pesendorfer (2010) also provide a foundation for this model in terms of economic primitives.
18
For example, one may consider formalizing betrayal aversion, which is experimentally found by
Bohnet and Zeckhauser (2004) and Bohnet, Greig, Herrmann, and Zeckhauser (2008). As we show in
Section 2, how an agent thinks about his own betrayal (his D to opponent’s C) and the risk of being
betrayed (opponent’s D to his C) is important for the endogenous leadership. The Fehr and Schmidt(1999) model creates a variety of thoughts about it, and this generates the endogenous leadershipwith a positive probability in equilibrium. However, as Bohnet and Zeckhauser (2004) and Bohnet,
Greig, Herrmann, and Zeckhauser (2008) state, betrayal aversion may include elements beyond mere
outcome-based other-regarding preferences. Formalizing this concept by using the alternative models
in the text may be useful for studying other leadership mechanisms.
19
As a result, this model has generated many economic applications as already listed in footnote 1
and has been well studied in terms of its behavioral basis as in Neilson (2006), Sandbu (2008), andRohde (2010). See also Saito (2012) for a recent axiomatic development of the Fehr and Schmidt(1999) model, which proposes an extension of the Fehr and Schmidt (1999) model that accommodates
the ex ante fairness problem presented by Fudenberg and Levine (2012).
20
A seminal work on social pressure is Kandel and Lazear (1992). They modeled peer pressure.
In addition to the cost of eﬀort, a member of a team feels disutility from peer pressure, which may
be social in the sense that it depends on other members’ eﬀort, such as the member feels substantial
disutility when only he shirks. They introduced a general peer-pressure function and studied several
kinds of peer pressure.

8

in social dilemma situations using a model of inequity aversion.21 Duﬀy and MuñozGarcı́a (2011) also employ the model of Fehr and Schmidt (1999) preferences withinequity aversion as in our theory and develop a Bayesian model of the prisoner’s
dilemma. However, they study prisoner’s dilemmas with exogenous sequential moves
for their study of leadership.22 In a later section in this paper, we analyze the samemodel as theirs for the purpose of comparison with our endogenous timing game.

2

An illustrative example

To develop basic intuition about the relation between a diversity of inequity aversions
and endogenous leadership, we illustrate by a degenerate numerical example the ideaof a mechanism by which in equilibrium one player with a particular inequity aversion
moves ﬁrst with C and the other player with a diﬀerent inequity aversion chooses towait and responds with C to the ﬁrst mover’s choice of C.
i/j
C
D

C
2, 2
3, 0

D
0, 3
1, 1

Table 1: Example of a prisoner’s dilemma
Consider the prisoner’s dilemma in Table 1. Suppose that there are three possible types of inequity aversions: Materialist, Envy, and Envy-and-Guilt.23 Each typeevaluates each material payoﬀ proﬁle in Table 1 as in Table 2.
i/j
C
D

C
2
3

D
0
1

Materialist

i/j
C
D

C
2
3
Envy

D
−3
1

i/j
C
D

C
2
0

D
−3
1

Envy-and-Guilt

Table 2: Utilities in the prisoner’s dilemma in Table 1
The Materialist type corresponds to a self-interested agent. His utility from an outcomeof a pair of choices is the material payoﬀ that he receives in the outcome. The othertypes diﬀer from Materialist when there is an inequality in material payoﬀs. When
there is a disadvantageous inequality, the Envy and Envy-and-Guilt types feel envy,
which causes a disutility of the payoﬀ diﬀerence in addition to a utility of own material
payoﬀs. This happens in their (C, D) cells, in which their utilities are 0 + (−3) = −3.
When there is an advantageous inequality, the Envy-and-Guilt type feels guilt, whichcauses a disutility of the payoﬀ diﬀerence. This happens in his (D, C) cell, in which
his utility is 3 + (−3) = 0. We assume that a player is a Materialist type, an Envy
type, and an Envy-and-Guilt type with probabilities 0.2, 0.2, and 0.6, respectively.
Now consider a situation in which two players meet at random and play a game as in
Table 1 without knowledge of the opponent’s type, where players can decide the timing
21
Nosenzo and Sefton (2011) consider a version of the Varian (1994) game. They embed the Varian(1994) game into the same two-period structure as in our game. However, they assume that agents
have common inequity aversion parameters and that the parameter values are common knowledge.
Santos-Pinto (2008) also considers an endogenous timing duopoly market with inequity-averse ﬁrms
under the same assumptions.
22
Their main concern is to study signaling motives about fairness in a twice-repeated simultaneousmove or sequential-move prisoner’s dilemma game.
23
They correspond to (α, β) = (0, 0), (1, 0), and (1, 1) in the Fehr and Schmidt (1999) model.

9

of their choice voluntarily from the possible two timings. In this situation, we can showthat endogenous leadership happens with a positive probability in equilibrium. The key
is how each type thinks about his own betrayal (his D to his opponent’s C) and a risk ofbeing betrayed (his opponent’s D to his C). The diversity of inequity aversions creates
diverse ways of thinking about it; that is, it creates diﬀerent degrees of incentives to
induce conditional cooperation under a risk of being betrayed and/or diﬀerent degrees
of incentives to cooperate after observing the opponent’s cooperation. Then, suchdiverse sensitivities of cooperation naturally generate diverse ways of dynamic decision
making, from which endogenous leadership results.
In what follows, we demonstrate that the following Bayesian strategy constitutes
a sequential equilibrium. The Materialist type behaves as a leader, which means thathe chooses C at timing 1. The Envy type behaves as a defector, which means thathe chooses D at timing 2 irrespective of his observation. The Envy-and-Guilt type
behaves as a conditional cooperator, which means that he moves at timing 2 andchooses C (resp., D) if his opponent chooses C at timing 1 (resp., otherwise).
Let us verify that no type has an incentive to mimic the other types’ behaviors.24
First, we check if the Materialist type behaves as a leader. He is a benchmark case ofthis example in that he is free from envy and guilt; that is, he feels neither disutilityof disadvantageous inequality from being betrayed nor disutility of advantageous inequality from his own betrayal. Given the induced population of the three behavior
modes (leader: 0.2, defector: 0.2, conditional cooperator: 0.6), the Materialist typeenjoys expected utility 1.6 (= 0.2 × 2 + 0.2 × 0 + 0.6 × 2) if he behaves as a leader, 1.4(= 0.2×3+0.2×1+0.6×1) if he behaves as a defector, and 1.2 (= 0.2×2+0.2×1+0.6×1)
if he behaves as a conditional cooperator. Hence, the Materialist type optimally behaves as a leader. This happens because the chance of playing with the Envy-and-Guilt
type, who behaves as a conditional cooperator, is large enough so that the beneﬁt of
inducing the Envy-and-Guilt type to take C exceeds the cost of giving up D against
the Materialist and Envy types.
Second, we check if the Envy type behaves as a defector. Since leading behavior
may create a disadvantageous inequality under the presence of defectors, envy discourages players from the leadership behavior under the betrayal risk. To be precise, theexpected utility earned by the Envy type behaving as a leader reduces from 1.6 to 1.0(= 0.2 × 2 + 0.2 × (−3) + 0.6 × 2). This makes the defector mode optimal for the Envy
type.
Third, we check if the Envy-and-Guilt type behaves as a conditional cooperator.
In addition to the abovementioned eﬀect of envy, guilt plays a role. Since defecting behavior may create an advantageous inequality in the presence of leaders, guilt
discourages players from the betraying behavior. To be precise, the expected utility
earned by the Envy-and-Guilt type behaving as a defector reduces from 1.4 to 0.8(= 0.2 × 0 + 0.2 × 1 + 0.6 × 1). This makes the conditional cooperator mode optimalfor the Envy-and-Guilt type.
Thus, we establish that the abovementioned Bayesian strategy indeed constitutes asequential equilibrium under the speciﬁc type distribution. Now suppose that a playerof the Materialist type and a player of the Envy-and-Guilt type are paired and play theprisoner’s dilemma. Then, the Materialist type chooses C at timing 1 and the Envyand-Guilt type, who moves at timing 2, reacts with C. Thus, a successful leadershipis endogenously realized with a positive probability in equilibrium.
Note that it is critical for the endogenous leadership that all of the three behavior
modes prevail with positive probabilities. First, the leading behavior taken by the24

The reader may verify that he has no incentive for any other deviation.

10

Materialist type is obviously indispensable. Furthermore, the conditional cooperationby the Envy-and-Guilt type is not triggered without the leading behavior. Next, a
conditional cooperator is also indispensable. To see this, suppose that no conditional
cooperator exists. Then, the Materialist type would not behave as a leader because
then behaving as a leader has no eﬀect of inducing a second mover to take C and simply
means giving up the dominant choice of D.25 Finally, defectors must also exist. If there
is no defector, there is no risk of being betrayed. Therefore, the Envy-and-Guilt type
would not wait and see, but rather would behave as a leader. Then, there would be notype who behaves as a conditional cooperator. In this way, the endogenous leadership
cannot aﬀord to lose any of the three behavior modes.
In the above degenerate example, we assume a particular combination of three
types of inequity aversions with a particular distribution and consider a particularleadership pattern by these types. In the general model that we consider in this paper,the degree of incentive for a type to induce conditional cooperation is determined inequilibrium jointly with a distribution of the three behavior modes. As we will show,
there exist three leadership patterns, depending on the parameters.

3

The model

We consider a version of the prisoner’s dilemma game in which players choose thetiming of moves under incomplete information about the preferences of players. This
game is called P D hereafter and deﬁned as follows.
A prisoner’s dilemma is a symmetric game given by Table 3. The parametersa, b, c, d are the material payoﬀs. They are assumed to be b > a > d > c. This payoﬀ
structure exhibits social dilemma, because D is the payoﬀ-maximizing choice for any
given belief about the opponent’s choice although (D, D) is Pareto inferior to (C, C).26
i/j
C
D

C
a, a
b, c

D
c, b
d, d

Table 3: Prisoner’s dilemma
We hypothesize that each player i has Fehr–Schmidt inequity-averse preferences
over the outcomes described by the pairs of material payoﬀs in Table 3. The preferences
are represented by a utility functionu(αi ,βi ) (xi , xj ) = xi − αi max{xj − xi , 0} − βi max{xi − xj , 0}(1)where xi and xj are material payoﬀs to players i and j, respectively. The ﬁrst term
represents the direct utility from the material payoﬀ xi . The second term captures theutility loss from disadvantageous inequality when xi is less than xj . The parameterαi ≥ 0 may be interpreted as the envy of player i. The third term captures the utility
loss from advantageous inequality when xi is larger than xj . The parameter βi ≥ 0
may be interpreted as the sense of guilt of player i.
25

In the general model which we consider in this paper, the eﬀect of no conditional cooperator isthat then behaving as a conditional cooperator dominates behaving as a leader.
26
We do not include the restriction 2a > b + c commonly imposed in repeated prisoner’s dilemmastudies, which guarantees that (C, C) is value-maximizing eﬃcient, because it is irrelevant to ouranalysis.

11

From the pairs of material payoﬀs (xi , xj ) in Table 3, we have a utility representation of the prisoner’s dilemma played by players with Fehr–Schmidt preferences. Thisis shown in Table 4.
i/j
C
D

C
a, a
b − βi (b − c), c − αj (b − c)Dc − αi (b − c), b − βj (b − c)d, dTable 4: Utility representation of the prisoner’s dilemma
We assume that Table 3 is common knowledge among the players, but Table 4 isnot. Player i knows the inequality aversion parameters αi and βi in his utility function,
but he does not know his opponent’s parameters αj and βj .
We introduce a common prior assumption about the belief that a player initiallyholds about his opponent’s utility function. We call the pair (α, β) in the utility
function (1) the type of the player. The type is a realization of the continuous randomvariable (α, β) in the space of possible types given by
T = {(α, β)|0 ≤ α ≤ ᾱ, 0 ≤ β ≤ 1, β ≤ α}.

(2)

The parameter ᾱ is the upper bound of the envy parameter and it can be either ﬁnite
or inﬁnite. The upper bound of the guilty parameter is assumed to be 1 because a
guilty parameter larger than 1 would mean that more material payoﬀs are undesirable
when a player is receiving more than his opponent. The envy parameter α is assumed
to be no less than the guilty parameter β because inequality in material payoﬀs mattersmore when a player receives less than his opponent.27 We assume that a type (α, β) is
realized according to a density f (α, β) with the full support over T . We also assume
that realizations are independent across players. All of these assumptions about initialbeliefs are assumed to be common knowledge.
Under incomplete information about their utility functions, the players play theprisoner’s dilemma in Table 3 in the following sequence. There are two timings: 1 and
2. At timing 1, the players choose either C, D, or ∅ independently and simultaneously,
where ∅ denotes postponing their choices until timing 2 rather than choosing C or
D. At timing 2, a player has a move when and only when he chooses ∅ at timing 1.
Before he moves, he is informed of his opponent’s choice at timing 1 and then he mustchoose either C or D. When both players choose ∅ at timing 1, their choices at timing
2 are made independently and simultaneously. This is the end of a play. Each player
has made a choice over C or D either at timing 1 or 2 once and only once. A player
receives the material payoﬀ in Table 3 corresponding to the pair of their choices.
A P D thus deﬁned is determined by two sets of parameters. One is the material payoﬀs (a, b, c, d) in the underlying prisoner’s dilemma. The other is a characteristic (f, T ) of players’ preferences. We ﬁx T throughout the paper and write
P D((a, b, c, d), f ) when the parameters of the P D need to be explicit.

27
We follow the standard formalization of the parameter space that Fehr and Schmidt (1999) proposed. Several subsequent studies consider the possibility of expanding this parameter space (forexample, status seeking modeled by β < 0). However, we focus on the issue of inequity aversion
captured by this type space.

12

4

The three mode equilibrium

4.1

The strategy

A player has four information sets in P D. One is the information set for choice attiming 1. The other three are the information sets for choice at timing 2 corresponding
to his opponent’s choice at timing 1 being either C, D, or ∅. A (pure) strategy is acomplete plan that assigns to each of these information sets an action available at the
information set. Formally, it is a quadruplet s = (a1 , aC , aD , a∅ ) where a1 ∈ {C, D, ∅}is the prescribed choice at timing 1, and aC , aD , a∅ ∈ {C, D} are the prescribed choices
at timing 2 when his opponent’s choices at timing 1 are C, D, and ∅, respectively. The(pure) strategy space is S = {C, D, ∅} × {C, D} × {C, D} × {C, D}.
A Bayesian strategy is a mapping s : T → S. It assigns to each type (α, β) ∈ T astrategys(α, β) = (a1 (α, β), aC (α, β), aD (α, β), a∅ (α, β)) ∈ S
which this type follows in a play of the P D where ah (α, β) is the prescribed choice ahfor each information set h for type (α, β).

4.2

The three-mode strategy

We examine a Bayesian strategy named three-mode strategy that assigns a strategy
from a particular set of three behavior modes. They are called C-mode, CDD-mode,and DDD-mode and are deﬁned as the following strategies in S where C-mode is
described as a reduced strategy with aC , aD , and a∅ for unreached information setsleft unspeciﬁed.
C = (C, aC , aD , a∅ )CDD = (∅, C, D, D)
DDD = (∅, D, D, D)
These strategies in S are called behavior modes in a three-mode strategy when we
stress the feature that each of them are followed by a mass of types in T according to
the Bayesian strategy.
Formally, for a Bayesian strategy s : T → S, let
TC (s) = {(α, β) ∈ T |s(α, β) = C}
TCDD (s) = {(α, β) ∈ T |s(α, β) = CDD}
TDDD (s) = {(α, β) ∈ T |s(α, β) = DDD}
denote the sets of types who follow C, CDD, and DDD, respectively. Let φ denotethe probability measure over T induced by the prior density f . For a Borel subset B
of T , it assigns the probability of a player being of a type in B by∫
φ(B) =f (α, β)d(α, β).
(3)BThen, the three-mode strategy is deﬁned as follows.
Deﬁnition 1. A Bayesian strategy s : T → S is a three-mode strategy if (TC (s), TCDD (s), TDDD (s))
is a partition of T and φ(TC (s)) > 0, φ(TCDD (s)) > 0, φ(TDDD (s)) > 0.

13

When players play a three-mode strategy s, four kinds of play paths will be realizedwith positive probabilities. These are shown in Table 5. For example, (C, CDD) cellshows a play path in which player i chooses C at timing 1 according to si = C andplayer j waits at timing 1 and chooses C at timing 2 according to sj = CDD. This is
the leadership process.
si / sj
C
CDD
DDD

C
C
,C
∅ → C, C∅ → D, C

CDD
C,∅ → C∅ → D, ∅ → D∅ → D, ∅ → D

DDDC,∅ → D∅ → D, ∅ → D∅ → D, ∅ → D

Table 5: Play paths of three-mode strategies

4.3

The three-mode equilibrium

We consider a symmetric sequential equilibrium in three-mode strategies. We call thisequilibrium a three-mode equilibrium.
We prepare a characterization of the three-mode equilibrium by an associated distribution of the three behavior modes. Let µ = (µC , µCDD , µDDD ) denote a probability
distribution according to which a player follows strategies C, CDD, and DDD. Let∆ ≡ {µ = (µC , µCDD , µDDD )|0 ≤ µC , µCDD , µDDD ≤ 1 and µC + µCDD + µDDD = 1}
be the set of distributions over the three strategies. Then,
Deﬁnition 2. A distribution µ ∈ ∆ is called a three-mode distribution if µC > 0,µCDD > 0, and µDDD > 0.
A three-mode strategy s is associated with a three-mode distribution µ by the relationµC = φ(TC (s)), µCDD = φ(TCDD (s)), and µDDD = φ(TDDD (s)). A three-mode
distribution is called a three-mode equilibrium distribution if a three-mode strategy
that generates the distribution is a three-mode equilibrium strategy.
Imagine that a player holds a belief µ ∈ ∆ according to which his opponent follows
strategies C, CDD, and DDD. Then, the expected utility for a type (α, β) to follow
C given µ is given byU(α,β) (C, µ) = µC a + µCDD a + µDDD (c − α(b − c))because, as is shown in Table 5, the outcome will be (C, C), (C, C), and (C, D) when
the opponent follows C, CDD, and DDD, respectively. Similarly, the expected utilities
from CDD and DDD are:U(α,β) (CDD, µ) = µC a + µCDD d + µDDD dU(α,β) (DDD, µ) = µC (b − β(b − c)) + µCDD d + µDDD d.
We deﬁne the following sets of types that are supposed to describe those sets oftypes for whom C, CDD, and DDD are sequentially rational responses to µ.
TC∗ (µ) = {(α, β) ∈ T |U(α,β) (C, µ) ≥ U(α,β) (CDD, µ), U(α,β) (DDD, µ)}∗
TCDD(µ) = {(α, β) ∈ T |U(α,β) (CDD, µ) ≥ U(α,β) (C, µ), U(α,β) (DDD, µ) and β ≥ β ∗ }∗TDDD(µ) = {(α, β) ∈ T |U(α,β) (DDD, µ) ≥ U(α,β) (C, µ), U(α,β) (CDD, µ) and β ≤ β ∗ },

14

∗
∗
∗
where we deﬁne β ∗ ≡ b−ab−c . The sets TC (µ), TCDD (µ) and TDDD (µ) are called thebest-response type sets. The set TC∗ (µ) is a set of types for whom C is the best among
the three strategies when he evaluates them at the information set at timing 1 given∗
the belief µ. The set TCDD(µ) is a set of types for whom CDD is the best among thethree strategies at two information sets. It is the best at the information set at timing
1 given the belief µ, which is parallel to TC∗ (µ). It is also the best at the information
set at timing 2 where he observes that his opponent chooses C at timing 1. He chooses
between C and D at this information set. He prefers C to D if a > b − β(b − c),he prefers the opposite if a < b − β(b − c), and he is indiﬀerent if a = b − β(b − c).
∗Therefore, the sequential rationality for CDD requires β ≥ β ∗ . The set TDDD(µ) is∗parallel to TCDD (µ).
Now, we show that the three-mode equilibrium is characterized by a three-mode
distribution, as follows.

Lemma 1. Consider a function ψ that assigns to each µ ∈ ∆ a vector ψ(µ) =∗(ψC (µ), ψCDD (µ), ψDDD (µ)) by ψC (µ) = φ(TC∗ (µ)), ψCDD (µ) = φ(TCDD(µ)), and∗
ψDDD (µ) = φ(TDDD (µ)). Then, ψ(µ) ∈ ∆ for every µ ∈ ∆ \ (1, 0, 0) and ψ is continuous on ∆ \ (1, 0, 0). Furthermore,(1) If s : T → S is a three-mode equilibrium strategy, then µ = (φ(TC (s)), φ(TCDD (s)), φ(TDDD (s)))
is a three-mode distribution such that µ = ψ(µ).
(2) If µ ∈ ∆ is a three-mode distribution such that µ = ψ(µ), then any three∗mode strategy s : T → S that satisfies TC (s) ⊆ TC∗ (µ), TCDD (s) ⊆ TCDD(µ),∗
and TDDD (s) ⊆ TDDD (µ) is a three-mode equilibrium strategy. Furthermore,∗φ(TC (s)) = φ(TC∗ (µ)) = µC , φ(TCDD (s)) = φ(TCDD(µ)) = µCDD , and φ(TDDD (s)) =∗φ(TDDD (µ)) = µDDD .
Lemma 1 means that the function ψ generates a distribution over strategies C, CDD,
and DDD by generating probability measures of best-response type sets against abelief µ ∈ ∆ \ (1, 0, 0) and that a three-mode equilibrium strategy is identiﬁed as athree-mode strategy that generates a three-mode distribution as a ﬁxed point of thefunction ψ.

4.4

The best-response type sets

To characterize a ﬁxed point of ψ, let us study the best-response type sets TC∗ (µ),∗∗TCDD(µ), and TDDD(µ). We explore them by examining preferences over C, CDD,
and DDD because the best-response type sets are deﬁned by the preferences. Take
the case of µ with 0 < µC < 1. First, consider a preference over C and CDD. A playerof (α, β) type prefers C to CDD if and only if µC a + µCDD a + µDDD (c − α(b − c)) >µC a + µCDD d + µDDD d. Letα∗ (µ) ≡a − d µCDDd−c−
b − c µDDD
b−c

(4)

where we set α∗ (µ) = ∞ when µDDD = 0. Then, (α, β) type prefers C to CDD if andonly if α < α∗ (µ). He prefers the opposite when α > α∗ (µ), and he is indiﬀerent when
α = α∗ (µ). The threshold α∗ (µ) lies in the interval (0, ᾱ) and partitions the type space
T if and only if
d−c
µCDD
b−c
d−c
<
+
ᾱ.
(5)
<
a−d
µDDD
a−d a−d15Second, consider a preference over CDD and DDD. A player of (α, β) type prefers
CDD to DDD if and only if µC a+µCDD d+µDDD d > µC (b−β(b−c))+µCDD d+µDDD d;that is, β > β ∗ where the threshold β ∗ is deﬁned previously when we introduced thebest-response type sets. He prefers the opposite when β < β ∗ , and he is indiﬀerent
when β = β ∗ . Note that the threshold β ∗ is 0 < β ∗ = b−ab−c < 1 so that it partitions thetype space.
Finally, consider a preference over C and DDD. A player of (α, β) type prefers
C to DDD if and only if µC a + µCDD a + µDDD (c − α(b − c)) > µC (b − β(b − c)) +µCDD d + µDDD d. Let[b − a µ]µDDD
DDD (d − c) − µCDD (a − d)H(α|µ) ≡
α+
+
µC
b−c
µC (b − c)µDDD=(α − α∗ (µ)) + β ∗ .
(6)µCThen, (α, β) type prefers C to DDD if and only if β > H(α|µ). He prefers the opposite
when β < H(α|µ), and he is indiﬀerent when β = H(α|µ).
Note that, as a consequence of transitivity of preferences, the two thresholds α∗ (µ),∗β introduced so far satisfy
U(α∗ (µ),β ∗ ) (C, µ) = U(α∗ (µ),β ∗ ) (CDD, µ) = U(α∗ (µ),β ∗ ) (DDD, µ).
This means that β ∗ = H(α∗ (µ)|µ). Therefore, the best-response type sets to µ with
0 < µC < 1 are characterized immediately, as follows.
Lemma 2. For µ with 0 < µC < 1, the best response type sets areTC∗ (µ) = T ∩ {(α, β)|α ≤ α∗ (µ), β ≥ H(α|µ)}∗TCDD(µ) = T ∩ {(α, β)|α ≥ α∗ (µ), β ≥ β ∗ }∗TDDD(µ) = T ∩ {(α, β)|β ≤ H(α|µ), β ≤ β ∗ }.

(7)
(8)
(9)

An example of the best-response type sets are illustrated in Figure 1. This example isobtained when µ satisﬁes β ∗ < α∗ (µ) < ᾱ and H(0|µ) > 0. All of the best-responsetype sets are nondegenerate in this example.

1

TCDD (µ)⇥ = H( |µ)TC (µ)TDDD (µ)0(µ)

1Figure 1: Inequity concerned leader
16

Consider the case of µ with µC = 0. Then, U(α,β) (CDD, µ) = U(α,β) (DDD, µ) = d
for any (α, β) ∈ T . Therefore, it is straightforward to see that the best-response typesets are as follows.
Lemma 3. For µ with µC = 0, the best-response type sets areTC∗ (µ) = T ∩ {(α, β)|α ≤ α∗ (µ)}∗TCDD(µ) = T ∩ {(α, β)|α ≥ α∗ (µ), β ≥ β ∗ }∗TDDD(µ) = T ∩ {(α, β)|α ≥ α∗ (µ), β ≤ β ∗ }.

5

Existence of the three-mode equilibrium

5.1µC -ﬁxed point and ﬁxed point of ψ

(10)
(11)
(12)

Our ﬁrst result is a characterization of the existence of a three-mode equilibrium.
In light of Lemma 1, we study a ﬁxed point of ψ. Since a three-mode equilibrium
distribution must be fully mixed, we must exclude a ﬁxed point on the boundaries of∆.
We study a ﬁxed point of ψ in two steps.28 First, we identify the set of µC -ﬁxed
points of ψ. A µC -ﬁxed point µ̂ is a belief in ∆ with a property that the probabilityof strategy C remains unchanged by ψ; that is, ψC (µ̂) = µ̂C . The second step is to
identify from the set of µC -ﬁxed points a belief µ̂ with an additional property that theprobability of strategy DDD also remains unchanged by ψ; that is, ψDDD (µ̂) = µ̂DDD .
Now, we begin the ﬁrst step to ﬁnd µC -ﬁxed points of ψ. In particular, we areinterested in those µC -ﬁxed points that are three-mode distributions. For µ̂ to satisfy
ψC (µ̂) = µ̂C and µ̂C > 0, the best-response type set TC∗ (µ̂) must have a positive
probability measure. By Lemma 2, this implies that α∗ (µ̂) > 0 for the threshold (4).
d−cµ̂DDD . Therefore, we study those
By the condition (5), this means that µ̂CDD > a−dµ̂ with ψC (µ̂) = µ̂C in a subset∆0 ≡ {µ̃ ∈ ∆ \ (1, 0, 0)|µ̃CDD ≥d−cµ̃DDD }a−dof ∆ \ (1, 0, 0). The following establishes that those points are an arc in ∆0 .
Lemma 4. There exists a unique µ̄C in (0, 1) such that for each µC ∈ [0, µ̄C ], there
exists a unique µ̂ ∈ ∆0 that satisfies ψC (µ̂) = µ̂C = µC . We denote by µ̂ = µ̂(µC ) afunction that assigns to each µC ∈ [0, µ̄C ] the corresponding µC -fixed point µ̂. Then,d−c a−d(1) for µC = 0, µ̂(0) = (0, a−c, a−c ) and ψ(µ̂(0)) = (0, φ(β > β ∗ ), φ(β < β ∗ )),(2) for µC = µ̄C , µ̂(µ̄C ) = (µ̄C , 1 − µ̄C , 0) and ψ(µ̂(µ̄C )) = (µ̄C , 0, 1 − µ̄C ),(3) for µC ∈ (0, µ̄C ), µ̂(µC ) is in the interior of ∆0 .
Furthermore, the function µ̂(µC ) is continuous.
d−c a−dLemma 4 is illustrated in Figure 2. The distribution µ̂ = (0, a−c, a−c ) is deﬁned
d−cby a boundary µ̂C = 0 of ∆ and a boundary µ̂CDD = a−d µ̂DDD of ∆0 . It is a µC ﬁxed point for µC = 0 because the latter condition (that is α∗ (µ̂) = 0) means that µ̂
28
Alternatively, we can study a ﬁxed point of ψ by a standard method that relies on the Kakutani
ﬁxed-point theorem by extending a function ψ to a correspondence deﬁned over ∆. We use the twostep analysis developed below because this analysis is also useful for the comparative static analysisof the three-mode equilibrium, which we develop later.

17

is mapped by ψ to ψ(µ̂) such that ψC (µ̂) = 0. There exists another µC -ﬁxed pointµ̂ = (µ̄C , 1 − µ̄C , 0) for some µ̄C ∈ (0, 1) on the boundary µDDD = 0 of ∆. To see
it, note that any distribution µ on this boundary is mapped by ψ to ψ(µ) such that
ψCDD (µ) = 0. The reason is that, as we noted in the illustrative example of Section
2, when the opponent is expected to never follow DDD (that is, µDDD = 0), there isno risk of being betrayed by an opponent. This makes C dominate CDD. Therefore,
a best response must be either C or DDD. The distribution µ̂ = (µ̄C , 1 − µ̄C , 0) is a
distribution for which the best response generates the same probability ψC (µ̂) = µ̄C
of C and a probability ψDDD (µ̂) = 1 − µ̄C of DDD, which is equal in size to theprobability µ̂CDD of CDD in µ̂. Lemma 4 means that the set of µC -ﬁxed point is an
arc that connects these two extreme points because a µC -ﬁxed point exists continuously
depending on µC from µC = 0 to µC = µ̄C .

Figure 2: µC -ﬁxed points and their ψ images under f (α, β) = 60(1 − α)β 2
The second step to ﬁnd a ﬁxed point of ψ is to ﬁnd a belief µ̂ along the arcµ̂ = µ̂(µC ) of µC -ﬁxed points for which the probability of strategy DDD also remains
unchanged by ψ; that is, ψDDD (µ̂) = µ̂DDD . Figure 2 also illustrates this exercise.
We map each µC -ﬁxed point µ̂ on the arc µ̂ = µ̂(µC ) to a belief in ∆ by ψ. The
image of the arc by ψ is also an arc in ∆ because ψ is continuous. As is reported in
Lemma 4, it starts at ψ(µ̂(0)) = (0, φ(β > β ∗ ), φ(β < β ∗ )) for µC = 0 and ends at
ψ(µ̂(µ̄)) = (µ̄C , 0, 1 − µ̄C ) for µC = µ̄C . A crossing point of this image arc with thearc µ̂ = µ̂(µC ) is a ﬁxed point of ψ.
For the purpose of this analysis, we deﬁne a function λ that assigns to each µC ∈[0, µ̄C ]λ(µC ) = ψDDD (µ̂(µC )) − µ̂DDD (µC )(13)where we write the components of a µC -ﬁxed point µ̂(µC ) as µ̂(µC ) = (µ̂C (µC ), µ̂CDD (µC ), µ̂DDD (µC )).
Figure 3 shows a graph of this function (13) that corresponds to Figure 2. When aµC -ﬁxed point µ̂ is mapped by ψ to an image ψ(µ̂) located north-west of µ̂ on a linewith the same µC in Figure 2, the corresponding value of λ is negative in Figure 3.
When it is mapped in the opposite direction in Figure 2, λ is positive in Figure 3. The
18

crossing points of the arc and its image in Figure 2 correspond to the solutions to theequation λ(µC ) = 0 in Figure 3.

Figure 3: λ under f (α, β) = 60(1 − α)β 2Now, a three-mode equilibrium distribution is characterized as follows.
Lemma 5. A belief µ ∈ ∆ is a three-mode equilibrium distribution if and only if(1) µ = µ̂(µC )(2) λ(µC ) = 0(3) µC > 0.
Condition (1) is a requirement that µC = ψC (µ). Condition (2) is a requirementthat µDDD = ψDDD (µ). These conditions guarantee that µ is a ﬁxed point of ψin ∆ \ (1, 0, 0). Condition (3) excludes a ﬁxed point at an extreme point µ̂(0) =d−c a−d(0, a−c, a−c ) of the arc µ̂ = µ̂(µC ).
In Lemma 5, we do not need a condition to exclude a ﬁxed point at another extreme
point µ̂(µ̄C ) = (µ̄C , 1 − µ̄C , 0) of the arc µ̂ = µ̂(µC ). From Lemma 4, we already knowthat µ̂(µ̄C ) = (µ̄C , 1 − µ̄C , 0) is mapped to ψ(µ̂(µ̄C )) = (µ̄C , 0, 1 − µ̄C ). Therefore,µ̂DDD (µ̄C ) = 0 < 1 − µ̄C = ψDDD (µ̂(µ̄C )). Hence, µ̂(µ̄C ) is never a ﬁxed point of ψ.

5.2

A suﬃcient condition for the existence of the three-mode equilibrium

Now, we develop a suﬃcient condition for the existence of a three-mode equilibrium.
The suﬃcient condition also conveys a simple and straightforward logic by which leadership should be expected to emerge if a prisoner’s dilemma satisﬁes the condition.
Lemma 5 shows that a three-mode equilibrium exists if there exists a solutionµC > 0 to the equation λ(µC ) = 0. As we noted after the lemma, it is always the casethat λ(µ̄) > 0. Therefore, if λ(0) < 0, the continuity of λ(µC ) guarantees a solutionµC > 0 to the equation λ(µC ) = 0. This leads us to the following suﬃcient condition
for the existence of the three-mode equilibrium.
19

Theorem 1. There exists a three-mode equilibrium in P D((a, b, c, d), f ) ifφ(β > β ∗ ) >

d−c
a−c .

(14)

Recall that β ∗ is a threshold such that a player of type (α, β) with β > β ∗ prefers
CDD to DDD; that is, he is a potential conditional cooperator. Theorem 1 saysthat a three-mode equilibrium exists if the probability φ(β > β ∗ ) of those potentialconditional cooperators is large enough. The condition (14) provides a speciﬁc valued−c
a−c for this probability.
The condition (14) has the following simple meaning. Suppose that no type takes
the leadership behavior. In particular, consider a Bayesian strategy s : T → S in which
all the types (α, β) with β > β ∗ follow CDD and all the types (α, β) with β ≤ β ∗ follow
DDD. Then, the belief µ consistent with this Bayesian strategy is µC = φ(TC (s)) = 0,µCDD = φ(TCDD (s)) = φ(β > β ∗ ), and µDDD = φ(TDDD (s)) = φ(β ≤ β ∗ ). When thecondition (14) is satisﬁed, a player of type (0, 0) prefers C to CDD and DDD given
this belief µ because
U(0,0) (C, µ) − U(0,0) (CDD, µ) = [φ(β > β ∗ )a + φ(β ≤ β ∗ )c] − [φ(β > β ∗ )d + φ(β ≤ β ∗ )d][d − c]>0= (a − c) φ(β > β ∗ ) −a−c[
d − c]
U(0,0) (C, µ) − U(0,0) (DDD, µ) = (a − c) φ(β > β ∗ ) −> 0.
a−cThus, the condition (14) means the situation in which the pure materialist (α = β = 0)
has an incentive to take the leadership behavior if no one takes the leadership, all thetypes with β > β ∗ follow CDD, and all the others follow DDD.
When the condition (14) holds, λ(0) = ψDDD (µ̂(0)) − µ̂DDD (0) < 0 is guaranteed
for the following reason. Lemma 4 states that the µC -ﬁxed point µ̂(0) for µC = 0 is
mapped by ψ to ψ(µ̂(0)) = (0, φ(β > β ∗ ), φ(β < β ∗ )). Then, the above meaning of thecondition (14) states that, in response to this belief ψ(µ̂(0)), strategy C is followed bya set of types with a positive probability measure, that is, ψC (ψ(µ̂(0))) > 0. Therefore,
when we compare µ̂(0) and ψ(µ̂(0)), the property of the belief µ̂(0) that ψC (µ̂(0)) = 0(that is, no one takes the leadership) requires that the belief µ̂(0) places more weighton DDD than the belief ψ(µ̂(0)); that is, µ̂DDD (0) > ψDDD (µ̂(0)).
Given its meaning, the reason why the condition (14) guarantees the existence of athree-mode equilibrium is now transparent, as follows. We consider the arc µ̂ = µ̂(µC )in ∆, along which it holds that ψC (µ̂) = µ̂C . The extreme points of the arc are
d−c a−d, a−c ) and µ̂(µ̄C ) = (µ̄C , 1 − µ̄C , 0). The former corresponds to the beliefµ̂(0) = (0, a−cwith the lowest µC on the arc and the latter corresponds to the belief with the highestµC on the arc. As we explained above, when the condition (14) holds, it holds that
ψDDD (µ̂(0)) < µ̂DDD (0) at the belief µ̂(0) with the lowest µC on the arc; that is, abest response to the belief lowers the probability of DDD-mode. (Figure 2 exhibits
this case.) On the other hand, ψDDD (µ̂(µ̄C )) > µ̂DDD (µ̄C ) at the belief µ̂(µ̄C ) withthe highest µC on the arc; that is, a best response to the belief raises the probabilityof DDD-mode. Hence, there must exist a belief between the two extreme points along
the arc that maintains the probability of DDD-mode under a best response.

5.3

A characterization of the existence of the three-mode equilibrium

Theorem 1 establishes a suﬃcient condition (14) for the existence of the three-mode
equilibrium. Then, we proceed to develop a characterization of the existence of thethree-mode equilibrium.
20

For this purpose, let us restate Theorem 1 as a suﬃcient condition for the existenceof the three-mode equilibrium in a space of prisoner’s dilemma parameters (a, d), as
follows.
Corollary 1. For each a ∈ (c, b), define¯ ≡ φ(β > β ∗ )a + (1 − φ(β > β ∗ ))c.
d(a)

(15)

Then,
¯ < a, d d(a)¯ > 0, lim d(a)¯ = c, lim d(a)¯ = b, and(1) it is c < d(a)
a→c
a→b
da
(2) there exists a three-mode equilibrium in P D((a, b, c, d), f ) if¯
c < d < d(a)(16)

¯ deﬁned by (15) is a value of d for which the condition (14)The upper bound d(a)in Theorem 1 holds in equality, that is,φ(β > β ∗ ) =

¯ −c
d(a)
a−c .

(17)

This gives the area of (a, d) in the space of a normalized prisoner’s dilemma for whicha three-mode equilibrium is guaranteed to exist. It is illustrated by a dashed line in
Figure 4.29

b = '"

a

&#$"
&"
%#$"
%"
!#$"

c = !"
!"

!#$"

%"

%#$"

&"

&#$"

'"

d

Figure 4: Suﬃcient condition for the existence of the three-mode equilibrium¯
Now, we ﬁx f and determine for each (a, d) with d(a)≤ d < a whether there
exists a three-mode equilibrium in P D((a, b, c, d), f ). Note that, in the deﬁnition of29

Figure 4 shows the area of the suﬃciency condition (16) for the case of b = 3, c = 0, andf (α, β) = 27{(α − 23 )4 + β 4 }.

21

the function λ(µC ) in (13), the function µ̂(µC ) depends on a, d and the function ψ(µ)
depends on a, d so that the function λ(µC ) depends on a, d. To be explicit about thisfact, let us rewrite the deﬁnition as follows.
λ(µC , a, d) = ψDDD (µ̂(µC , a, d), a, d) − µ̂DDD (µC , a, d)(18)

Similarly, let us denote µ̄ as µ̄(a, d) to express the fact that the upper bound for µC
that admits µC -ﬁxed points depends on a, d. The function λ(µC , a, d) in (18) and its
domain depend on the prisoner’s dilemma parameters (a, d), as follows.
Lemma 6.
(1) µ̄(a, d) is strictly increasing in a and strictly decreasing in d.
(2) λ(µC , a, d) is strictly decreasing in a and strictly increasing in d.
The intuition for why λ(µC , a, d) is strictly increasing in d is as follows.30 Compare P D((a, b, c, d0 ), f ) and P D((a, b, c, d00 ), f ) such that d0 > d00 . Then, strategies
CDD and DDD, from which a player may receive a payoﬀ d, are more attractive in P D((a, b, c, d0 ), f ) than in P D((a, b, c, d00 ), f ). This makes the probability of
DDD by best response at least as high in P D((a, b, c, d0 ), f ) as in P D((a, b, c, d00 ), f );that is, ψDDD (µ̂(µC , a, d0 ), a, d0 ) ≥ ψDDD (µ̂(µC , a, d00 ), a, d00 ). The increase in attractiveness of CDD and DDD also reduces the incentive for a player to follow strategy C in P D((a, b, c, d0 ), f ). Therefore, for a given µC , a µC -ﬁxed point µ̂ that
achieves the same µC by best response must place less weight on DDD to createa stronger incentive to follow C in P D((a, b, c, d0 ), f ) than in P D((a, b, c, d00 ), f ); thatis, µ̂DDD (µC , a, d0 ) < µ̂DDD (µC , a, d00 ). Hence, λ(µC , a, d0 ) > λ(µC , a, d00 ).
By this monotonicity of a function λ in (a, d), we establish the following characterization that the existence of the three-mode equilibrium is monotone in (a, d).
ˆ ∈ [d(a),¯Theorem 2. For each a ∈ (c, b), there exists d(a)a) such that(1) there exists a three-mode equilibrium in P D((a, b, c, d), f ) ifˆc < d < d(a),(19)(2) there exists no three-mode equilibrium in P D((a, b, c, d), f ) if
ˆ < d < a.
d(a)

(20)

ˆ
ˆ
Furthermore, the bound d(a)is continuous, strictly increasing, lim d(a)= c, anda→cˆ = b.
lim d(a)a→bTheorem 2 is illustrated in Figure 4. A set of prisoner’s dilemma games (a, d) isˆpartitioned by a function dˆ = d(a)(denoted as a bold line) into a north-west partthat supports a three-mode equilibrium and a south-east part that fails to support athree-mode equilibrium.
ˆ
Theorem 2 also shows that the upper bound d(a)for prisoner’s dilemma games(a, d) with a three-mode equilibrium for a given a is bounded away from a. In words,
the nature of the prisoner’s dilemma that there is an opportunity of Pareto improvement from (d, d) to (a, a) does not guarantee successful leadership. The leadership will
not emerge if the gain from the Pareto improvement is limited.
30

The intuitions for the other claims in Lemma 6 are similar.

22

µC

ˆ
The threshold d(a)for the existence of the three-mode equilibrium in Theorem 2¯ in Corollary 1. In Appendix
may or may not coincide with the suﬃciency bound d(a)A, we provide a suﬃcient condition under which they coincide. In general, however,ˆ in Theorem 2 may be strictly higher than the suﬃciency bound d(a)¯
the threshold d(a)¯in Corollary 1, for the following reason. The bound d(a) corresponds to the prisoner’s
dilemma in which a pure materialist (α = β = 0) is indiﬀerent between taking theleadership behavior and following DDD if no type takes the leadership at all. Then,
all the types, other than a pure materialist, are not willing to take the leadershipbehavior either given the same belief. However, although no type has an incentive totake the leadership if the other types are not expected to take the leadership, some
types may be willing to take the leadership if some other types are expected to take¯ < d(a).
ˆ¯ < d(a).
ˆ 31the leadership. Then, d(a)Figure 5 is an example of this case d(a)d¯dFigure 5: Equilibrium locus under f (α, β) = 27{(α − 2/3)4 + β 4 }6Comparative statics of the three-mode equilibrium

Our second result is comparative statics of the three-mode equilibrium. Parallel toour ﬁrst result on the existence of the three-mode equilibrium, we ﬁx f and conduct a
comparative statics analysis with respect to the prisoner’s dilemma parameters (a, d).

6.1

Comparative statics of the three-mode equilibrium distribution

We conduct a comparative statics analysis of the three-mode equilibrium distributionµ. As we noted after Theorem 2, there may exist multiple three-mode equilibria ina prisoner’s dilemma and this is the case shown in Figure 3. Therefore, we pursue
a monotone comparative statics analysis of the three-mode equilibrium distributions
with the maximum of µC and with the minimum of µC . Recall from Lemma 5 thatthe value of µC in the three-mode equilibrium is determined by λ(µC , a, d) = 0. We

31
In addition to Theorem 2, Figure 5 demonstrates that there may exist a three-mode equilibriumˆˆin P D((a, b, c, d(a)),
f ) at the threshold d(a).
In light of Lemma 6, it is easily veriﬁed that this occurs¯¯
if and only if there exists µ > 0 such that λ(µ, a, d(a))= 0 at the suﬃciency bound d(a)in Corollary1.

23

considerµmax
C (a, d) ≡ max{µC ∈ (0, 1)|λ(µC , a, d) = 0}µmin
C (a, d) ≡ min{µC ∈ (0, 1)|λ(µC , a, d) = 0}.
min
The values µmax
C (a, d) and µC (a, d) are well-deﬁned because λ(µC , a, d) is continuous
in µC so that the set of µC in the three-mode equilibrium is compact. Then, we deﬁne
the maximum leadership distribution and the minimum leadership distribution by
max
max
max
µmax (a, d) = (µmaxC (a, d), µCDD (a, d), µDDD (a, d)) ≡ µ̂(µC (a, d))
min
min
min
µmin (a, d) = (µminC (a, d), µCDD (a, d), µDDD (a, d)) ≡ µ̂(µC (a, d)).

First, we conduct a comparative statics analysis of µC . Lemma 6 shows the monotonicity of λ in that λ(µC , a, d) is strictly decreasing in a and strictly increasing in d.
This immediately leads us to the following comparative statics result.32
Theorem 3.
(1) µmaxC (a, d) is strictly increasing in a and strictly decreasing in d.
(2) µminC (a, d) is strictly increasing in a and strictly decreasing in d for a range of¯(a, d) with c < d < d(a),while it is strictly decreasing in a and strictly increasing¯ < d < d(a).
ˆin d for a range of (a, d) with d(a)This is illustrated in Figure 6. Panel A illustrates the monotone comparative statics¯
result for a range of (a, d) with c < d < d(a),while Panel B illustrates the monotone¯ < d < d(a).
ˆcomparative statics result for a range of (a, d) with d(a)32maxTheorem 3 states that µmin
C (a, d) behaves in opposite directions to µC (a, d) for a range of (a, d)¯
ˆwith d(a) < d < d(a). This occurs because we are interested in the three-mode equilibrium. If weconsider the sequential equilibrium that assigns one of C-mode, CDD-mode, and DDD-mode without
max
restricting its distribution to the interior of ∆, the µmin
C (a, d) behaves in a parallel way to µC (a, d).
As we will discuss in Section 9, a no-leadership equilibrium (µC = 0) exists for a range of (a, d) with
¯
ˆ
ˆ
d(a)
< d < d(a)(and d(a)≤ d < a). Therefore, Theorem 3-(2) is rewritten for µmin
C (a, d) over theminentire ∆ as stating that µC (a, d) is (strictly) increasing in a and (strictly) decreasing in d, where the¯strictness applies for a range of (a, d) with c < d < d(a).
The same remark applies to all the resultson the minimum leadership distribution hereafter. This parallel monotonicity of µmin
C (a, d) must be
noticed when we test our theory in experiments using the comparative statics results. We will returnto this issue in Section 9 where we discuss the existence of a no-leadership equilibrium.

24Panel A

Panel B
Figure 6: Graphs of λ when d0 > d00
Next, we consider the comparative statics of µCDD . Note in Lemma 1 that µCDD =∗φ(TCDD(µ)) for a three-mode equilibrium distribution and recall from Lemma 2 that∗
TCDD (µ) is determined by α∗ (µ) and β ∗ as an area of α ≥ α∗ (µ) and β ≥ β ∗ in T . We
can show a comparative statics result of α∗ (µ) and β ∗ .
Lemma 7.
(1) (a) α∗ (µmax (a, d)) is strictly increasing in a and strictly decreasing in d.
(b) α∗ (µmin (a, d)) is strictly increasing in a and strictly decreasing in d for a¯
range of (a, d) with c < d < d(a),while it is strictly increasing in d for a25¯ < d < d(a).
ˆ 33range with d(a)(2) β ∗ is strictly decreasing in a and constant in d.
This comparative statics result immediately leads us to the following comparative
statics of µCDD .
Theorem 4.
∗(1) For a range of (a, d) with α∗ (µmax (a, d)) ≤ β ∗ , µmax
CDD (a, d) = φ(β > β ) and it
is constant in d and strictly increasing in a. µmin
CDD (a, d) is parallel.

(2) (a) For a range of (a, d) with α∗ (µmax (a, d)) > β ∗ , µmax
CDD (a, d) is strictly increasing in d.
(b) For a range of (a, d) with α∗ (µmin (a, d)) > β ∗ , µmin
CDD (a, d) is strictly in¯creasing in d for a range with c < d < d(a),while it is strictly decreasing in¯ < d < d(a).
ˆd for a range with d(a)minNote that µmax
CDD (a, d) ( and µCDD (a, d)) may not be monotone with respect to a∗
for a range of (a, d) with α (µmax (a, d)) > β ∗ ( and α∗ (µmin (a, d)) > β ∗ ) because¯α∗ (µmax (a, d)) ( and α∗ (µmin (a, d)) for a range with c < d < d(a))
is strictly increasing∗
in a while β is strictly decreasing in a.
Finally, we consider the comparative statics of µDDD . As is stated in Theorem 4,
min
∗
µmax
CDD (a, d) and µCDD (a, d) have a constant value φ(β > β ) for a range of (a, d) withα∗ (µmax (a, d)) ≤ β ∗ and α∗ (µmin (a, d)) ≤ β ∗ , respectively. Therefore, the comparative statics of µDDD for this range is obtained from the comparative statics of µC in
Theorem 3 and the comparative statics of β ∗ in Lemma 7, as follows.

Theorem 5.
(1) For a range of (a, d) with α∗ (µmax (a, d)) ≤ β ∗ , µmax
DDD (a, d) is strictly decreasing
in a and strictly increasing in d.
(2) For a range of (a, d) with α∗ (µmin (a, d)) ≤ β ∗ , µmin
DDD (a, d) is strictly decreasing¯in a and strictly increasing in d for a range of (a, d) with c < d < d(a),while it¯ˆis strictly decreasing in d for a range of (a, d) with d(a) < d < d(a).
minNote that µmax
DDD (a, d) (and µDDD (a, d)) may not be monotone with respect to aand d for a range of (a, d) with α∗ (µmax (a, d)) > β ∗ (and α∗ (µmin (a, d)) > β ∗ ).
¯For example, for a range of (a, d) with α∗ (µmax (a, d)) > β ∗ and c < d < d(a),maxmaxµC (a, d) is strictly decreasing in d and µCDD (a, d) is strictly increasing in d so that
max
max
µmax
DDD (a, d) = 1 − µC (a, d) − µCDD (a, d) may or may not be increasing in d. Figuremax
7 below demonstrates that µDDD (a, d) is not monotone in d.34

¯ < d < d(a).
ˆα∗ (µmin (a, d)) may not be monotone with respect to a for a range of (a, d) with d(a)The relation α∗ (µmax (a, d)) > β ∗ applies to the parts “Inequity” and “Hybrid” of the graph in
Figure 7. The three categories of three-mode equilibrium in Figure 7 will be introduced formally in
Section 7.1.
33

34

26

µDDD

d¯
d
Figure 7: Non-monotonic move of µDDD under f (α, β) = 6β6.2Comparative statics of outcome distributions in the three-mode
equilibrium

The comparative statics of the three-mode equilibrium distribution enables us to conduct the comparative statics of the outcome distributions in the three-mode equilibrium. Recall from Table 5 that four kinds of outcomes are realized with positive probabilities in a three-mode equilibrium: (C, C) at timing 1, C → C, C → D, and (D, D) attiming 2. When a three-mode equilibrium with a three-mode equilibrium distribution µprevails, probabilities for outcomes (C, C) at timing 1, C → C, C → D, and (D, D) attiming 2 are given by (µC )2 , 2µC µCDD , 2µC µDDD , and (µCDD + µDDD )2 = (1 − µC )2 .
Therefore, the following is concluded from Theorems 3, 4, and 5.
Theorem 6.
(1) The probability of the outcome of (C, C) at timing 1 under µmax is strictly increasing in a and strictly decreasing in d. The probability of the outcome of(C, C) at timing 1 under µmin is strictly increasing in a and strictly decreasing¯in d for a range of (a, d) with c < d < d(a)and it is strictly decreasing in a and¯ < d < d(a).
ˆstrictly increasing in d for a range of (a, d) with d(a)(2) The probability of the outcome of C → C under µmax is strictly increasing in aand strictly decreasing in d for a range of (a, d) with α∗ (µmax (a, d)) ≤ β ∗ . The
probability of the outcome of C → C under µmin is parallel for a range of (a, d)¯
with α∗ (µmax (a, d)) ≤ β ∗ and c < d < d(a),while it is strictly increasing in d for∗max∗¯ < d < d(a).
ˆa range of (a, d) with α (µ (a, d)) ≤ β and d(a)(3) The probability of the outcome of (D, D) at timing 2 under µmax is strictly decreasing in a and strictly increasing in d. The probability of the outcome of(D, D) at timing 2 under µmin is strictly decreasing in a and strictly increasing¯in d for a range of (a, d) with c < d < d(a),while it is strictly increasing in a
¯ < d < d(a).
ˆ
and strictly decreasing in d for a range of (a, d) with d(a)272µC µCDD

The behaviors of the probabilities of the outcomes that are not stated in Theorem
6 are more complex. For example, the probability of the outcome C → C does notnecessarily move monotonically with respect to a and d for the region that Theorem
6-(2) does not cover. In this region, an increase in the probability of C-mode is partly
realized by having some conditional cooperator become a leader type. Because of thisshift from CDD-mode to C-mode, the matching probability of a leader type and conditional cooperator becomes complex. Figure 8 shows an example of a nonmonotonic
move of the probability of the outcome C → C.35 A similar complexity applies to theoutcome C → D.

dFigure 8: Non-monotonic probability of C → C under f (α, β) = 6β7Who takes the leadership? - Fairness concerns, incentive to lead, and leadership patterns -Our third result is a characterization of leadership patterns. A leadership pattern is a
description of who takes the leadership and it is given by the set of leader types. Whena three-mode equilibrium with a distribution µ prevails in a prisoner’s dilemma game
P D((a, b, c, d), f ), the leadership pattern of this equilibrium is the set TC∗ (µ) and thisset depends on the parameters (a, b, c, d) and f in the prisoner’s dilemma. We characterize the leadership patterns TC∗ (µ) in the three-mode equilibrium. In particular, we
explore how each pattern is generated from the underlying incentive for each type tolead and how the incentive to lead is connected to the fairness concerns of the type.

7.1

The incentive to lead and a leadership pattern under a given
belief

We begin by ﬁxing a prisoner’s dilemma P D((a, b, c, d)f ) and charactering the leadership pattern under a given belief; that is, a best-response type set TC∗ (µ) to an
The relation α∗ (µmax (a, d)) > β ∗ applies to the parts “Inequity” and “Hybrid” of the graph in
Figure 8.
3528

arbitrarily given three-mode distribution µ.
To understand the best-response type set in light of the incentive to lead, let us
introduce an ordering of types by their degrees of incentive to lead.
Deﬁnition 3. Fix a positive number γ > 0. We say that a type (α, β) has a stronger
incentive to lead under the belief ratio γ than a type (α0 , β 0 ) when
(1) under any three-mode distribution µ with µDDD
= γ, if the type (α0 , β 0 ) prefersµC
C to CDD and DDD, then the type (α, β) also prefers C to CDD and DDD,and= γ under which the type(2) there exists a three-mode distribution µ with µDDD
µC
00(α , β ) does not prefer C to CDD and DDD, while the type (α, β) prefers C to
CDD and DDD.
We say that a type (α, β) has the strongest incentive to lead under the belief ratio γ
when no other type in T has a stronger incentive to lead under the belief ratio γ than
the type (α, β). This type is called the strongest incentive to lead type.
The notion of the strongest incentive to lead is important for our understanding ofleadership patterns. First, the best-response type set TC∗ (µ) is not empty if and only if
the strongest incentive to lead type prefers C to CDD and DDD. In words, whether
the leadership behavior occurs with positive probabilities is identiﬁed by whether thestrongest incentive to lead type is willing to take the leadership. Second, when thebest-response type set TC∗ (µ) is not empty, the strongest incentive to lead type serves
as a representative element of the set TC∗ (µ). All the types in the set TC∗ (µ) are orderedaccording to the degree of incentive to lead from the top by the strongest incentive tolead type to the lowest on the boundaries of TC∗ (µ).
For each γ > 0, the set of the strongest incentive to lead types can be identiﬁed in
the following way. A type (α, β) prefers C to CDD and DDD under a belief µ if andonly ifU(α,β) (C, µ) > max[U(α,β) (CDD, µ), U(α,β) (DDD, µ)].
This condition is rewritten asµCDD (a − d) > µDDD (d − (c − α(b − c))) + µC (max[a, b − β(b − c)] − a).

(21)

The left-hand side is a beneﬁt of leading; that is, a utility increase from inducing C fromthe opponent who follows CDD. The right-hand side is a cost of leading, which is asum of a utility decrease from being betrayed by the opponent who follows DDD and autility decrease from losing an opportunity to betray the opponent who follows C. The
beneﬁt of leading is independent of the type because it compares symmetric outcomes= γ are(C, C) and (D, D). Therefore, when beliefs µ with a given belief ratio µDDD
µC
considered, a type has a stronger incentive to lead than another type if and only if thecost of leading is lower for the former type than for the latter type. The type who has
the strongest incentive to lead can be identiﬁed by minimizing the cost of leading, as
follows.
Lemma 8. Let γ > 0 be given. A set of types who has the strongest incentive to leadunder the belief ratio γ is {(0, 0)}, {(α, β) = t(0, 0) + (1 − t)(β ∗ , β ∗ )|0 ≤ t ≤ 1}, and{(β ∗ , β ∗ )} for γ > 1, γ = 1, and γ < 1, respectively.

29

The intuition of Lemma 8 is straightforward. An isocost curve of leading in thetype space is given byµDDD (d − (c − α(b − c))) + µC (max[a, b − β(b − c)] − a) = constant.

(22)

For a type (α, β) with β ≥ β ∗ , it holds that max[a, b − β(b − c)] − a = 0; that is, there
is no cost of losing an opportunity to betray the opponent who follows C because it is
the best for this type not to betray the opponent if he is to move at timing 2. Hence,
the isocost curve of leading (22) is vertical for a range of types (α, β) with β ≥ β ∗ . On
the other hand, for a type (α, β) with β < β ∗ , both kinds of costs of leading matter,and the isocost curve of leading (22) becomesµDDD (d − (c − α(b − c))) + µC ((b − β(b − c)) − a) = constant.
The envy parameter α is the agent’s evaluation of the disadvantageous inequality b − cin the cost of being betrayed and the guilt parameter β is that of the advantageous
inequality b − c in the cost of losing the opportunity to betray. The same size of
inequity b − c is evaluated in the opposite direction. Therefore, the slope of the isocost(b−c)
dβ= µDDD
curve of leading is dα= µDDD= γ. The isocost curve of leading (22)µC
µC (b−c)corresponds to a lower cost of leading when it is located toward a lower α and a higher
β. Hence, the strongest incentive to lead type must lie in a subset of the boundary of
T with α = β and 0 ≤ β ≤ β ∗ . An extreme point (α, β) = (0, 0) of this subset becomesa unique type of the strongest incentive to lead when γ > 1. The other extreme point(α, β) = (β ∗ , β ∗ ) of this subset becomes a unique type of the strongest incentive tolead when γ < 1.
The two candidates for the strongest incentive to lead type, Materialist and thetype (β ∗ , β ∗ ), represent two extreme cases of fairness concerns. The type (β ∗ , β ∗ )maximizes β to minimize the eﬀect of advantageous inequity on the cost of leading.
Maximizing β is accompanied by raising α to β ∗ along the boundary of T with α = β.
This means that the type (β ∗ , β ∗ ) is the type who places the minimum weight on thecost of losing an opportunity to betray the opponent who follows C and the maximum
weight to the cost of being betrayed. In contrast, Materialist minimizes α to minimize
the eﬀect of advantageous inequity on the cost of leading. Minimizing α is accompanied
by lowering β to 0. This means that Materialist is the type who places the maximum
weight on the cost of losing an opportunity to betray the opponent who follows C andthe minimum weight on the cost of being betrayed.
The two canonical forms of fairness concern held by Materialist and the type(β ∗ , β ∗ ) leads us to classify leadership patterns according to which form of fairness concern leads to the leadership behavior. We may call a best-response type set TC∗ (µ) an inequity concerned leader pattern (icl-pattern) when (β ∗ , β ∗ ) ∈ TC∗ (µ) and (0, 0) 6∈ TC∗ (µ).
This occurs when the type (β ∗ , β ∗ ) is the unique strongest incentive to lead type andMaterialist does not take the leadership. An opposite pattern in which (0, 0) ∈ TC∗ (µ)
and (β ∗ , β ∗ ) 6∈ TC∗ (µ) may be called a materialist leader pattern (ml-pattern). Thisoccurs when Materialist is the unique strongest incentive to lead type and the type(β ∗ , β ∗ ) dose not take the leadership. When it happens to be that (0, 0) ∈ TC∗ (µ) and(β ∗ , β ∗ ) ∈ TC∗ (µ), we may call it a hybrid leader pattern (hl-pattern). This occurs ina particular case of γ = 1 in which both Materialist and the type (β ∗ , β ∗ ) have thestrongest incentive to lead. However, this also occurs generically when Materialist is
the unique strongest incentive to lead type and when the type (β ∗ , β ∗ ) is the uniquestrongest incentive to lead type.
Figure 1, which we used to explain best-response type sets in Section 4, illustrates
an inequity concerned leader pattern. The type (β ∗ , β ∗ ) has the strongest incentive to30

lead and receives a surplus µCDD (a − d) − µDDD ((d − c) + (b − a)) by leading. As wemove in the type space T by raising α and keeping β = β ∗ , this surplus is decreased
at a rate of µDDD (b − c). It is decreased to zero when α is increased to α∗ (µ). The
boundary of the set TC∗ (µ) is stretched out from the type (α∗ (µ), β ∗ ) vertically for arange with β > β ∗ and with a positive slope γ for a range with β < β ∗ . The set TC∗ (µ)
is triangle shaped because (0, 0) 6∈ TC∗ (µ).
Figure 9 illustrates a materialist leader pattern. Materialist has the strongest
incentive to lead and receives a surplus µCDD (a − d) − µDDD (d − c) − µC (b − a) byleading. As we move in the type space T by raising α and keeping β = 0, this surplus
is decreased at a rate of µDDD (b − c). It is decreased to zero when α is increased to∗∗α∗ (µ)− βγ . The boundary of the set TC∗ (µ) is stretched out from the type (α∗ (µ)− βγ , 0)with a positive slope γ. The set TC∗ (µ) is also triangle shaped because (β ∗ , β ∗ ) 6∈ TC∗ (µ).

⇥ = H( |µ)1

TCDD (µ)

TDDD (µ)

TC (µ)0(µ)

1Figure 9: Materialist leader
Figure 10 illustrates hybrid leader patterns. The set TC∗ (µ) with γ < 1 is a case inwhich the type (β ∗ , β ∗ ) is the unique strongest incentive to lead type and the boundaryof the set TC∗ (µ) is stretched out from the type (α∗ (µ), β ∗ ). It is quadrilateral shaped
because it also includes Materialist. The set TC∗ (µ) with γ > 1 is a case in which
Materialist is the unique strongest incentive to lead type and the boundary of the set∗
TC∗ (µ) is stretched out from the type (α∗ (µ) − βγ , 0). It is also quadrilateral shaped
because it also includes the type (β ∗ , β ∗ ).

31

1

TCDD (µ)⇥ = H( |µ)TC (µ)TDDD (µ)(µ)

01Panel A: The case of γ < 1⇥ = H( |µ)1

TCDD (µ)

TC (µ)

0

TDDD (µ)(µ)

1Panel B: The case of γ > 1
Figure 10: Hybrid leader patterns

7.2

The incentive to lead and a leadership pattern in the three-mode
equilibrium

We characterize the leadership pattern that prevails in the three-mode equilibrium. In
the previous section, we studied the leadership pattern under a given belief by considering a three-mode distribution arbitrarily. In the three-mode equilibrium, however, theleadership pattern is determined jointly with a three-mode distribution in the equilibrium. Therefore, we explore how a three-mode equilibrium distribution is determined
depending on the underlying incentive for each type to lead and how the incentive to32lead in the equilibrium is connected with the fairness concerns of the type. We explore
it by studying how the leadership pattern in the three-mode equilibrium diﬀers in eachprisoner’s dilemma. For this purpose, we combine the comparative statics results in
Section 6 and the results on leadership patterns under a given belief in the previous
Subsection 7.1, and we answer two particular questions: (1) how the leadership pattern in the three-mode equilibrium diﬀers according to the prisoner’s dilemmas, and(2) how the type of the strongest incentive to lead in the three-mode equilibrium diﬀers
according to the prisoner’s dilemma.
7.2.1

Comparison of the leadership patterns

First, we compare the leadership patterns in the three-mode equilibrium for diﬀerent
prisoner’s dilemmas. Formally, we consider how the leadership patterns TC∗ (µmax (a, d))and TC∗ (µmin (a, d)) diﬀer according to prisoner’s dilemma parameters (a, d). To compare TC∗ (µmax (a, d)) and TC∗ (µmin (a, d)) for diﬀerent values of (a, d), we introduce thefollowing ordering of sets of types who take the leadership.
Deﬁnition 4. Let TC and TC0 be sets of types who take the leadership. We say that TC
is the leadership pattern of a more inequity concerned leader than TC0 if the following
conditions hold.
(1) ∃(α̂, β̂) ∈ TC s.t.∀(α0 , β 0 ) ∈ TC0 ; (α0 , β 0 ) ≤ (α̂, β̂)36 , and(2) ∀(α0 , β 0 ) ∈ TC0 \ TC , ∀(α, β) ∈ TC \ TC0 ; (α0 , β 0 ) ≤ (α, β).
Conditions (1) and (2) deﬁne a set of natural requirements for a set TC of types thatconsists of more inequity concerned types than another set TC0 . Condition (1) requiresthat there is a type (α̂, β̂) in TC that is dominant over TC0 in the sense that the type(α̂, β̂) is more inequity conscious in both envy and guilt for any type in TC0 . This
dominance also implies that a set diﬀerence TC \ TC0 exists and the dominant type(α̂, β̂) lies in this diﬀerence. Condition (2) requires that the set diﬀerences TC \ TC0
and TC0 \ TC are in a speciﬁc relation such that the set diﬀerence TC \ TC0 is dominant
over the set diﬀerence TC0 \ TC in the sense that any type in the set diﬀerence TC \ TC0
is more inequity concerned in both envy and guilt than any type in the set diﬀerence
TC0 \ TC .37 In words, the set TC is obtained by adding types to and extracting types
from the set TC0 in such a way that any added type is more inequity conscious thanany extracted type and some added type is also more inequity conscious than all theremaining types.
Recall from the analysis in Subsection 7.1 that a leadership pattern TC∗ (µ) is determined if we have three values α∗ (µ), β ∗ , and γ ∗ (µ) that deﬁne the boundary ofTC∗ (µ) where we denote γ ∗ (µ) ≡ µDDD
µC . These numbers for the maximum leadership
distribution and the minimum leadership distribution depend on prisoner’s dilemma
parameters (a, d), as follows. (Lemma 9-(1) and (2) are restatements of Lemma 7.)
Lemma 9.
We denote (α0 , β 0 ) ≤ (α̂, β̂) if and only if α0 ≤ α̂, β 0 ≤ β̂, and at least one inequality holds strictly.
Economists have applied various set orderings for the purpose of monotone comparative statics.
The set order advocated by Topkis (1998) and Milgrom and Shannon (1994) is the usual strong order.
Depending on the monotone comparative statics, however, economists sometimes need orderings thatﬁt the objectives of their analysis. The set order by dominance in set diﬀerences is one of the naturally
conceivable set orderings. Indeed, Kukushkin (2013) considers the set ordering by condition (2).
36
37

33

(1) α∗ (µmax (a, d), a, d) is strictly increasing in a and strictly decreasing in d. α∗ (µmin (a, d), a, d)
is strictly increasing in a and strictly decreasing in d for a range of (a, d) with¯c < d < d(a).
(2) β ∗ is strictly decreasing in a and independent of d.
(3) γ ∗ (µmax (a, d)) is strictly decreasing in a and strictly increasing in d. γ ∗ (µmin (a, d))is strictly decreasing in a and strictly increasing in d for a range of (a, d) with¯c < d < d(a).
This characterization of boundaries of the set of types who take the leadership in thethree-mode equilibrium establishes the following comparison of leadership patterns.
Theorem 7.
(1) If a > a0 and d < d0 , then the leadership pattern TC∗ (µmax (a, d), a, d) in the
maximum leadership distribution in prisoner’s dilemma P D((a, b, c, d), f ) is aleadership pattern of more inequity concerned leader than the leadership pattern TC∗ (µmax (a0 , d0 ), a0 , d0 ) in the maximum leadership distribution in prisoner’s
dilemma P D((a0 , b, c, d0 ), f ).
¯ 0 ), then the leadership pattern T ∗ (µmin (a, d), a, d)(2) If a > a0 , d < d0 , and d0 < d(aCin the minimum leadership distribution in prisoner’s dilemma P D((a, b, c, d), f )is a leadership pattern of more inequity concerned leader than the leadership pattern TC∗ (µmin (a0 , d0 ), a0 , d0 ) in the minimum leadership distribution in prisoner’s
dilemma P D((a0 , b, c, d0 ), f ).
Roughly speaking, Theorem 7 states that as a is increased and d is decreased, the leadership pattern in the equilibrium becomes a more inequity concerned leader pattern.
The intuition for this comparison is as follows. For any type (α, β) ∈ T , if a
is increased and d is decreased in the condition (21) for the type (α, β) to leadunder a given belief µ, the beneﬁt of leading is increased and the cost of leading
is decreased. Hence, when we compare prisoner’s dilemmas P D((a, b, c, d), f ) andP D((a0 , b, c, d0 ), f ) with a > a0 and d < d0 and we imagine that the maximum leadership
distribution µmax (a0 , d0 ) of prisoner’s dilemma P D((a0 , b, c, d0 ), f ) prevails as a belief
in both prisoner’s dilemmas, more types in prisoner’s dilemma P D((a, b, c, d), f ) takethe leadership behavior as best-responses to µmax (a0 , d0 ) than in prisoner’s dilemma
P D((a0 , b, c, d0 ), f ), that is, TC∗ (µmax (a0 , d0 ), a0 , d0 ) ⊂ TC∗ (µmax (a0 , d0 ), a, d). Hence, theleadership pattern in P D((a, b, c, d), f ) tends to be of a more inequity concerned leader
than the leadership pattern in P D((a0 , b, c, d0 ), f ) in the sense of an expanding set oftypes who take the leadership. However, the expansion of the set of types who take theleadership causes the probability of C-mode to be increased in P D((a, b, c, d), f ). This0 0
max
is the driving force for the result that µmax
C (a , d ) < µC (a, d), which we established in
Theorem 3. As we discussed in a comparison of incentives to lead between Materialist
and the type (β ∗ , β ∗ ), an increase in the probability of C-mode reduces the incentive
to lead for those types with low guilt parameters including Materialist. The types withlow guilt parameters are more inclined to follow DDD. This eﬀect reduces the belief
ratio γ from γ ∗ (µmax (a0 , d0 )) to γ ∗ (µmax (a, d)). Thus, the change in probabilities of Cmode further shifts the leadership pattern in P D((a, b, c, d), f ) toward a more inequity
concerned leader. The same explanation applies to a comparison of TC∗ (µmin (a, d), a, d)¯
and TC∗ (µmin (a0 , d0 ), a0 , d0 ) for a range of d < d0 < d(a).

34

7.2.2Comparison of who has the strongest incentive to lead

Second, we compare the leadership patterns in the three-mode equilibrium for diﬀerent
prisoner’s dilemmas more speciﬁcally in terms of who has the strongest incentive tolead. Lemma 8 provides the criterion for which of Materialist or the type (β ∗ , β ∗ ) is thetype of the strongest incentive to lead. Together with Lemma 9-(3), we know that when
we compare prisoner’s dilemmas P D((a, b, c, d), f ) and P D((a0 , b, c, d0 ), f ) with a > a0
and d < d0 , the type who has the strongest incentive to lead is more likely to be thetype (β ∗ , β ∗ ) rather than Materialist in P D((a, b, c, d), f ) than in P D((a0 , b, c, d0 ), f ).
We can state this fact more precisely for the maximum leadership distribution and forthe minimum leadership distribution, as follows.
Theorem 8.
maxˆ−1(1) For each d ∈ (c, b), there exists amax
L (d) with d (d) ≤ aL (d) < b such that inˆa prisoner’s dilemma P D((a, b, c, d), f ) for a range c < d < d(a),the strongest
incentive to lead type in the three-mode equilibrium distribution µmax (a, d) is∗∗max
Materialist if a < amax
L (d) and the type (β , β ) if a > aL (d).
min¯−1(2) For each d ∈ (c, b), there exists amin
L (d) with d (d) ≤ aL (d) < b such that in¯a prisoner’s dilemma P D((a, b, c, d), f ) for a range c < d < d(a),the strongest
minincentive to lead type in the three-mode equilibrium distribution µ (a, d) is Ma∗∗
minterialist if a < amin
L (d) and the type (β , β ) if a > aL (d).
min(3) The two functions a = amax
L (d) and a = aL (d) are increasing and related asmaxminaL (d) ≤ aL (d) for any d ∈ (c, b).

Theorem 8 states that a class of prisoner’s dilemma that admits a three-mode equilibrium is divided into a Materialist class (a subclass in which Materialist is the uniquestrongest incentive to lead type) and a β ∗ -class (a subclass in which the type (β ∗ , β ∗ )is the unique strongest incentive to lead type). Furthermore, roughly speaking, theβ ∗ -class is located north-west (a higher a and lower d) of the Materialist class.
Theorem 8 together with Theorem 7 suggests the following relationship between
a prisoner’s dilemma and a leadership pattern in the three-mode equilibrium in thedilemma. Suppose that some prisoner’s dilemma P D((a, b, c, d), f ) has a three-mode
equilibrium with the materialist leader pattern.38 Then, as we move from P D((a, b, c, d), f )by lowering d and raising a, the corresponding leadership patterns in the three-mode
equilibrium turn from a materialist leader pattern into a hybrid pattern and then an
inequity concerned pattern.39 Figure 11 demonstrates that this leadership patterntransition occurs for a particular prior f (α, β) = 6β.

38Note that although Theorem 8 guarantees that the β ∗ -class is nonempty, it does not guarantee¯−1 (d) = aminthat the Materialist class is. If it were the case that dˆ−1 (d) = amax
L (d) and d
L (d) for alld ∈ (c, b), then every prisoner’s dilemma with the three-mode equilibrium is in the β ∗ -class. Then, thematerialist leader pattern would never prevail in the equilibrium.
39
Theorem 8 guarantees that a hybrid pattern prevails in some prisoner’s dilemma P D((a, b, c, d), f )because, as long as a is increased close enough to b, P D((a, b, c, d), f ) belongs to the β ∗ -class. However,as we will explore below in Appendix B, it may be the case for some prior f that an inequity concerned
pattern never prevails in the three-mode equilibrium.

35

b=

a

a

c=

¯ ) a
d(a

d

Figure 11: Leadership pattern transitions
More precisely, we can provide a suﬃcient condition on the prisoner’s dilemma
under which the leadership pattern in the three-mode equilibrium is a materialist
leader pattern.
Theorem 9. Let d¯ml (a) ≡ a − (b − c) minα∗ ∈[0,ᾱ] φ(β < β ∗ |α∗ ≤ α). Suppose thatminα∗ ∈[0,ᾱ] φ(β < β ∗ |α∗ ≤ α) > 0. Then,(1) d¯ml (a) is strictly increasing, d¯ml (a) < a, lim d¯ml (a) = c − (b − c), lim d¯ml (a) = b,a→ca→b
and there exists āml ∈ (c, b) such that d¯ml (āml ) = c.
ˆ(2) for each a ∈ (c, ā0ml ), if max(c, d¯ml (a)) < d < d(a),then the leadership pattern inthe three-mode equilibrium in P D((a, b, c, d), f ) is a materialist leader pattern.
ˆ
Figure 12 illustrates Theorem 9.40 The curve d = d(a),which is identiﬁed in
Theorem 2, divides the space of a normalized prisoner’s dilemma and a three-mode
equilibrium exists in P D((a, b, c, d), f ) if and only if it is located north-west of theˆˆ
curve d = d(a).
The curve d = d¯ml (a) crosses the curve d = d(a).
The area between¯ˆthe curve d = dml (a) and the curve d = d(a) is a subset of a prisoner’s dilemma
in which the leadership pattern in the three-mode equilibrium in P D((a, b, c, d), f ) is
guaranteed to be a materialist leader pattern.

Figure 12 is drawn for the uniform distribution over T = {(α, β)|α ≥ β, 0 ≤ α, β ≤ 1}, for whichminα∗ ∈[0,ᾱ] φ(β < β ∗ |α∗ ≤ α) = β ∗ > 0.
40

36

Figure 12: Suﬃcient condition for materialist leader pattern under f (α, β) = 2
The bound d¯ml (a) identiﬁed in Theorem 9 has the following simple meaning. Take
any α∗ ≥ β ∗ and suppose that all the types (α, β) with α < α∗ follow C, while a type(α, β) with α > α∗ follows CDD if β > β ∗ and DDD if β < β ∗ . Then, the belief µ thatis consistent with this Bayesian strategy s : T → S is µC = φ(TC (s)) = φ(α∗ > α),µCDD = φ(TCDD (s)) = φ(α∗ ≤ α, β ≥ β ∗ ), and µDDD = φ(TDDD (s)) = φ(α∗ ≤α, β < β ∗ ). Consider a player of type (β ∗ , β ∗ ). When d > d¯ml (a), he prefers CDD to
C given this belief µ because
]
[
a−d
∗
∗ ∗
> 0.
U(β ∗ ,β ∗ ) (CDD, µ) − U(β ∗ ,β ∗ ) (C, µ) = (b − c)φ(α ≤ α) φ(β < β |α ≤ α) −b−cThus, the condition d > d¯ml (a) means a situation in which the type (β ∗ , β ∗ ) has anincentive not to take the leadership behavior if all the types with α < α∗ take theleadership, all the types with α ≥ α∗ and β ≥ β ∗ follow CDD, and all the others
follow DDD. In such a situation, the type (β ∗ , β ∗ ) never takes the leadership in
any equilibrium, which would mean α∗ = α∗ (µ) > β ∗ for the corresponding belief µ.
Therefore, only the three-mode equilibrium with a materialist leader pattern exists.
Although a materialist leader pattern prevails in a three-mode equilibrium in someprisoner’s dilemmas for any prior f , there may exist no three-mode equilibrium withan inequity concerned leader pattern in any prisoner’s dilemma, depending on a prior
f . The uniform distribution over a type space T = {(α, β)|α ≥ β, 0 ≤ α, β ≤ 1} isan example (see Appendix B). However, we can provide a suﬃcient condition for theprisoner’s dilemma and a prior under which the leadership pattern in the three-mode
equilibrium is an inequity concerned leader pattern.
Theorem 10. For a, b, c with c < a < b and f given, if()
φ(β > β ∗ )∗φ α < β ∗ (1 − β ∗ )β>β> 1 − β∗,1 − φ(β > β ∗ )(23)then there exists d¯icl (a) ∈ (c, a) such that¯(1) c < d¯icl (a) < d(a)(2) for any d ∈ (c, d¯icl (a)), there exists a three-mode equilibrium in P D((a, b, c, d), f )and any three-mode equilibrium is an inequity concerned leader pattern.
37

The intuition of Theorem 10 is as follows. A type (α, β) with β > β ∗ is the type who
is a potential conditional cooperator. The condition (23) requires that the distribution f over the set of potential conditional cooperators is more concentrated
towardφ(β>β ∗ )∗∗lower envy types and provides an explicit threshold β (1 − β ) 1−φ(β>β ∗ ) for the envy
parameter and an explicit bound 1 − β ∗ for the probability of an envy parameter being
lower than the threshold. Note from Lemma 2 that among the potential conditional
cooperators it is those with lower envy parameters that in fact take the leadershipand choose C at timing 1 rather than behave as conditional cooperators. Therefore,
the condition (23) makes the distribution of C and CDD chosen by the potential conditional cooperators concentrated toward a choice of C. This induces Materialist tofollow DDD and betray the choice of C made by the potential conditional cooperatorswith low envy parameters rather than to follow C and take the leadership himself
in expectation of inducing the conditional cooperator with high envy parameters to
respond with C. Hence, only an inequity concerned leader pattern is possible.

8

An extension to many timings

So far we have considered a prisoner’s dilemma in which a player chooses C or D at oneof two timings he prefers. The theory, however, applies to a prisoner’s dilemma withmany timings in general. The mechanism of leadership elucidated by our analysis isalso valid and our conclusion on the resolution of social dilemmas remains unchanged
even if we allow a player to choose his timing of move from not only two but manytimings.41
Suppose that there are K possible timings from timing 1 to timing K with 2 ≤K < ∞. Consider a prisoner’s dilemma in which a player is allowed to choose C or
D at a timing from K possible timings. We can extend the notion of a three-mode
strategy naturally to this prisoner’s dilemma with K timings. Ck -mode is a strategy
that commands a player to take the leadership behavior at timing k; that is, a strategy
that prescribes ∅ for timings 1 to k − 1 and C for timing k when his opponent chooses∅ at all the timings from 1 through k − 1.42 CDDk -mode is a strategy that prescribes∅ for timings 1 to k − 1, and prescribes C for timing k if his opponent chooses C
before timing k, D if his opponent chooses D before timing k, and D if his opponent
chooses ∅ at all the timings 1 through k − 1. DDDk -mode is parallel to CDDk -mode.
A Bayesian strategy is a three-mode strategy for a prisoner’s dilemma with K timings
if there exists k ∗ with 1 ≤ k ∗ < K such that a player follows either a Ck -mode withk ≤ k ∗ , a CDDk -mode with k > k ∗ , or a DDDk -mode with k > k ∗ , and a probability of some Ck -mode being followed, a probability of some CDDk -mode being followed, and a probability of some DDDk -mode being followed are positive and summed
to one; that is, 0 < φ(∪1≤k≤k∗ TCk ), φ(∪k∗ <k≤K TCDDk ), φ(∪k∗ <k≤K TDDDk ) < 1, andφ(∪1≤k≤k∗ TCk ) + φ(∪k∗ <k≤K TCDDk ) + φ(∪k∗ <k≤K TDDDk ) = 1 where TCk , TCDDk ,TDDDk denote the set of types who follow Ck -mode, CDDk -mode, and DDDk -mode
41

We thank Klaus Schmidt for suggesting this point to us.
Precisely speaking, there are two distinguished Ck -mode, depending on the prescriptions when
the opponent moves before the prescribed leadership timing k. One Ck -mode prescribes the response
C at timing k to his opponent’s choice of C before timing k and the response D to his opponent’s
choice of D. The other Ck -mode prescribes the response with D at timing k to any choice by the
opponent before timing k. When a player follows Ck -mode in a three-mode strategy, a player with
type β ≥ β ∗ follows the former Ck -mode, while a player with type β < β ∗ follows the latter Ck -mode.
This distinction of two distinguished modes of leadership did not arise in a prisoner’s dilemma with
two timings, because there is no pretiming before timing 1, for which C-mode prescribes the leadershipbehavior of choosing C.
42

38

respectively. A three-mode strategy with a threshold timing k ∗ divides K timings intoa span of earlier timings k = 1, · · · , k ∗ in which the leadership behavior occurs and aspan of later timings k = k ∗ + 1, · · · , K in which the follower behaviors occur.
A three-mode strategy in a prisoner’s dilemma with K timings is called a twotiming three-mode strategy if it is a particular three-mode strategy with k ∗ such thata player follows one of Ck∗ -mode, CDDk∗ +1 -mode, or DDDk∗ +1 -mode. A two-timing
three-mode strategy is special in that a player is supposed to move with positive
probabilities at only two adjacent timings among K available timings.
There is a natural correspondence of a two-timing three-mode strategy with athree-mode strategy in a prisoner’s dilemma with two timings. Consider a threemode strategy s in a prisoner’s dilemma with two timings. Then, for each k ∗ with1 ≤ k ∗ < K, the naturally corresponding two-timing three-mode strategy with athreshold k ∗ in a prisoner’s dilemma with K timings assigns Ck∗ -mode, CDDk∗ +1 mode, and DDDk∗ +1 -mode to those types (α, β) ∈ TC (s), (α, β) ∈ TCDD (s), and(α, β) ∈ TDDD (s), respectively. This natural correspondence is shown to preserve theequilibrium condition, as follows.
Theorem 11. Consider a three-mode strategy s in a prisoner’s dilemma with two timings. Then, the corresponding two-timing three-mode strategy in a prisoner’s dilemma
with K timings is a sequential equilibrium if and only if s is a sequential equilibrium.
When a two-timing three-mode strategy with a threshold k ∗ is played in a prisoner’s
dilemma with K timings, there are opportunities for players to take the leadershipbehavior at timings k = 1, · · · , k ∗ − 1 ahead of any planned moves of his opponent.
However, these opportunities are not taken. The reason is that the beneﬁt of leadershipis to induce the choice of C from the opponent who follows CDDk∗ +1 . This beneﬁt
is fully utilized if the leadership behavior is taken before k ∗ + 1. On the other hand,taking the leadership behavior of choosing C before k ∗ opens an opportunity for the
opponent to observe this choice of C irrespective of the opponent’s strategy. This isharmful if the opponent who follows Ck∗ is a type (α, β) with β < β ∗ . This opponent
responds with D to a choice of C made at any timing k before k ∗ , while the opponent
chooses C given that no player moves before k ∗ . Thus, taking the leadership behavior
before timing k ∗ simply increases the risk of being betrayed. This force makes a player
concentrate on timing k ∗ for the leadership behavior.
The force of betrayal risk of early leadership exists when a player considers theleadership in any three-mode strategy. We can show that any three-mode equilibriumwith a threshold k ∗ in a prisoner’s dilemma with K timings is essentially a two-timing
three-mode strategy in the following sense.
Theorem 12. Consider a three-mode strategy with a threshold k ∗ in a prisoner’s
dilemma with K timings. If it is a sequential equilibrium, then(1) φ(TC1 ) = · · · = φ(TCk∗ −1 ) = 0 and φ(TCk∗ ) > 0, and(2) φ(TCDDk∗ +1 ) > 0.
Property (1) means that the leadership behavior must take place at exactly one timingk ∗ in a three-mode equilibrium with a threshold k ∗ . Property (2) means that thefollower behavior based on the conditional cooperation must emerge with a positive
probability at the adjacent timing k ∗ + 1.
An inessential delay of CDDk -mode or DDDk -mode with k > k ∗ + 1 may bepossible in a three-mode equilibrium with a threshold k ∗ . For example, if s is a threemode equilibrium in a prisoner’s dilemma with two timings, then a combination of39

Ck∗ -mode for TC (s), CDDk∗ +1 -mode for TCDD (s), and DDDk -mode for TDDD (s) for
any k with k ∗ + 1 ≤ k ≤ K is supported by a sequential equilibrium in a prisoner’s
dilemma with K timings.
In spite of this kind of possible delay, however, a distribution (φ(TCk∗ ), φ(∪k∗ <k≤K TCDDk ),φ(∪k∗ <k≤K TDDDk )) over C-mode, CDD-mode, and DDD-mode in any three-mode
equilibrium in a prisoner’s dilemma with K timings must correspond to some threemode equilibrium distribution µ in a prisoner’s dilemma with two timings in that
µC = φ(TCk∗ ), µCDD = φ(∪k∗ <k≤K TCDDk ), and µDDD = φ(∪k∗ <k≤K TDDDk ), because it is also a three-mode equilibrium for any type who follows a CDDk -mode or
DDDk -mode with k > k ∗ + 1 to follow CDDk∗ +1 -mode or DDDk∗ +1 -mode, respectively. Hence, as far as an outcome distribution is concerned, our conclusion in a
prisoner’s dilemma with two timings remains unchanged when we allow many timings
in a prisoner’s dilemma.

9Discussion

We studied the leadership in a prisoner’s dilemma by considering a three-mode equilibrium in a Bayesian model of a prisoner’s dilemma with endogenous moves. We brieﬂy
discuss some issues that we excluded from our analysis.

9.1

Other equilibria

Although we focused on the three-mode equilibrium, there may exist other equilibria
in P D.
9.1.1

Other leadership equilibrium

We studied the leadership in a prisoner’s dilemma with the notion of the three-mode
equilibrium in which players follow one of three behavior modes C, CDD, or DDD.
Discussion on other possibilities of the emergence of leadership is in order.
One can show that the leadership is never realized by an equilibrium in which
players follow one particular behavior mode or one of two behavior modes. However,when there exists a three-mode equilibrium in a prisoner’s dilemma, leadership can berealized by an alternative equilibrium in which players follow a behavior mode from adiﬀerent set of three behavior modes: C, CDD, and D that prescribes a choice of D
at timing 1. The type who follows DDD in our three-mode equilibrium follows D inthis alternative equilibrium. The incentive for players to follow D in the alternativeequilibrium is similar to the incentive for them to follow DDD in our three-mode
equilibrium. Therefore, the alternative equilibrium can be regarded as substantiallythe same as our three-mode equilibrium.
There might exist an equilibrium with a more complex set of behavior modes insome prisoner’s dilemma. From the above discussion, an obvious one is a four-mode
equilibrium in which players follow one of four behavior modes: C, CDD, DDD, or D.
This is a combination of our three-mode equilibrium and the alternative three-mode
equilibrium.
Furthermore, one can consider a ﬁve-mode equilibrium in which players follow oneof ﬁve behavior modes: C, CDD, DDD, D, or CDC, which prescribes aC = C,
aD = D, and a∅ = C. The additional behavior mode CDC diﬀers from CDD in thata player chooses C at timing 2 when his opponent postpones his choice to timing 2.

40

We can verify that the abovementioned equilibria exhaust the list of equilibrium
in which leadership is realized. We focused on our three-mode equilibrium in our
analysis of leadership in a prisoner’s dilemma because it is the simplest manner inwhich leadership is realized in a prisoner’s dilemma.
9.1.2

No-leadership equilibrium

We studied the emergence of leadership in equilibrium in a prisoner’s dilemma. However, there may exist in the same prisoner’s dilemma another sequential equilibrium
in which both players choose D.
In particular, consider a wait-and-defect equilibrium in which all the types postpone
their choices to timing 2 and then choose D. From the analysis leading to Lemma 2
and Lemma 3, it is easy to see that this sequential equilibrium must assign CDD to
those types with β > β ∗ and DDD to those types with β < β ∗ . By the discussion
after Theorem 1, such a strategy is a sequential equilibrium strategy if and only if
the (0, 0) type has no incentive to deviate to choosing C at timing 1; that is, if and¯ ≤ d where d(a)¯ is the suﬃciency bound for the existence of a three-mode
only if d(a)equilibrium in Corollary 1.
ˆ
Recall from Theorem 2 that there exists a threshold d(a)such that there existsˆ¯ˆ
a three-mode equilibrium if and only if d < d(a), where d(a) ≤ d(a)and the threshˆ¯
old d(a) may or may not coincide with the suﬃciency bound d(a). Then, we haveconditions for combinations of the (C, C) outcome by the three-mode equilibrium andthe (D, D) outcome by the wait-and-defect equilibrium in terms of payoﬀ parameters¯
a, b, c, d and a type density f . For d < d(a),a three-mode equilibrium exists and thewait-and-defect equilibrium does not exist. Then, it is not possible for all the players¯ ≤ d < d(a),ˆ
to wait and choose D at timing 2. For d(a)both a three-mode equilibrium43
and the wait-and-defect equilibrium exist. Then, the players need to coordinate on
ˆ < d, a three-mode equiliba three-mode equilibrium to achieve cooperation. For d(a)rium does not exist and the wait-and-defect equilibrium exists. Then, it is not possible
to achieve cooperation through the leadership by a three-mode equilibrium.
¯
For a prisoner’s dilemma with d < d(a)in which a three-mode equilibrium exists
and the wait-and-defect equilibrium does not exist, there still remains the possibilityof the (D, D) outcome by a wider class of equilibria s : T → S that assigns to eachtype (α, β) either D, CDD, or DDD. This strategy prescribes an early defection
at timing 1 for those types who are assigned D. The analysis leading to Lemma 2
and Lemma 3 states that CDD must be assigned only to those types with β ≥ β ∗and DDD must be assigned only to those types with β ≤ β ∗ . This means that 0 ≤φ(TCDD (s)) ≤ φ(β > β ∗ ). By the same argument as the discussion after Theorem 1,such a s is a sequential equilibrium strategy if and only if the (0, 0) type has no incentive
to deviate from s by choosing C at timing 1; that is, if and only if φ(TCDD (s))a +(1 − φ(TCDD (s)))c ≤ d. This condition is rewritten as 0 ≤ φ(TCDD (s)) ≤ µ̄CDD (d)d−c¯, φ(β > β ∗ )). When d < d(a),one can ﬁnd such a
by deﬁning µ̄CDD (d) = min( a−c43As we remarked in footnote 32, this fact must be noted when we test our theory by experiments.
The theory of the three-mode equilibrium implies that the leadership behavior should be observed¯ˆ
at a nonnegligible frequency in a prisoner’s dilemma with d(a)< d < d(a).
However, the existenceof a no-leadership equilibrium may cause subjects to rarely take the leadership behavior in such aˆprisoner’s dilemma. Furthermore, when subjects play various prisoner’s dilemmas with c < d < d(a),the observed frequency of the leadership behavior may exhibit a monotone decrease in d over the wholeˆrange (c, d(a)),
although the comparative statics results of the three-mode equilibrium (Theorem 3)¯
states that µmin
C (a, d) for the three-mode equilibrium is strictly decreasing in d over the range (c, d(a))¯ˆ
and strictly increasing in d over the range (d(a),d(a)).
Hence, we should not rush to overthrow our¯theory of leadership even if the data show the overall monotonicity rather than the asymmetry at d(a).

41s that 0 ≤ φ(TCDD (s)) ≤ µ̄CDD (d) < φ(β > β ∗ ). This means that if many types
who would become conditional cooperators (that is, those types with β > β ∗ witha probability more than φ(β > β ∗ ) − µ̄CDD (d) > 0) do not wait until timing 2 andchoose D early at timing 1, the uncooperative outcome (D, D) prevails in equilibrium.
Then, to achieve cooperation through leadership, the players need to coordinate on athree-mode equilibrium.
9.1.3

Cooperation without leadership

We studied the issue of time and social dilemmas by focusing on a mechanism in which
the leadership resolves social dilemmas through dynamic decision making. However,even when agents have the freedom to choose the timing of moves, cooperation may
be realized without leadership.
We say that a Bayesian strategy involves the leadership if it generates a positive
probability for an outcome in which an agent chooses C at timing 1 and the otheragent waits and chooses C at timing 2. We say that a Bayesian strategy supports
cooperation without leadership if it does not involve the leadership and it generates
a positive probability for an outcome in which both agents choose C. Consider thefollowing Bayesian strategy s : T → S which assigns either DDD, CDD, CDC, or D
under a belief µ ∈ ∆({DDD, CDD, CDC, D}).

DDD if β ≤ β ∗ CDD if β ∗ < β ≤ min[ b−d , β ∗ + µCDD +µDDD (α + d−c )]
b−c
µCDC
b−c
s(α, β) =
µCDC
a−d
d−c
∗ + µCDD +µDDD (α + d−c )
CDC
if
α
≤
−
and
β
>
β

µCDD +µDDD d−c
b−c
µCDC
b−c


µCDC

a−d
d−c
b−d
D if α > µCDD
−
and
β
>
.
+µDDD d−c
b−c
b−c
(24)
This Bayesian strategy does not involve the leadership because it does not assign C.
However, when both agents are assigned CDC, they wait and choose C at timing 2.
Hence, this Bayesian strategy supports cooperation without leadership.
One can show that the Bayesian strategy (24) may be a sequential equilibrium, depending on a prior f 44 , and that there is no other sequential equilibrium that supports
cooperation without leadership.
The cooperation occurs by simultaneous choices of C at timing 2 in the equilibriumin the Bayesian strategy (24). However, note that the freedom to choose the timing ofmoves plays a crucial role in realizing this cooperation. Although the types who follow
DDD, CDD, and CDC choose to move at timing 2, the types who follow D choose
to move at timing 1. The types who are assigned D are those types who are endowedwith the highest α and the highest β. They most dislike both being betrayed andbetraying. These types prefer choosing D at timing 1 because, as long as they wait
until timing 2, they cannot avoid either being betrayed by the types who follow DDD
and CDD or betraying the types who follow CDC, so that choosing D at timing 1 isthe only way to guarantee (D, D), which involves neither being betrayed nor betraying.
This behavior by the most fairness concerned types makes the other types refrain fromtaking the leadership. In particular, the types who follow CDC are willing to choose
C at timing 2 in spite of the risk of being betrayed by the types who follow DDD and44

To see this, consider a degenerate prior that assigns a probability 0.2 to the type (0, 0), a probability
0.7 to the type ( 23 , 23 ), and a probability 0.1 to the type (1, 1). Then, consider a special case of theBayesian strategy (24) that assigns DDD to the type (0, 0), CDC to the type ( 23 , 32 ), and D to thetype (1, 1). Then, this strategy is a sequential equilibrium in a prisoner’s dilemma with a = 2, b =3, c = 0, d = 1. Smooth out this degenerate prior over a type space T = {(α, β)|0 ≤ α ≤ 1, β ≤ α}.
Then, a Bayesian strategy (24) under the resulting prior continues to be a sequential equilibrium.

42

CDD. If they choose C at timing 1 instead of timing 2, they would be better oﬀ if
there were no types who follow D, because those types who follow CDD will respondwith C. In fact, however, choosing C at timing 1 also entails an additional risk ofbeing betrayed by the type who follows D, which can be avoided if they stick to theassigned CDC-mode. Thus, it is essential for the cooperation at timing 2 that themost fairness concerned types choose to move at timing 1 and the other types choose
to move at timing 2 under the freedom to choose the timing of moves.

9.2

Other models of a prisoner’s dilemma in the presence of fairness
concerns

To study the issue of time and social dilemmas in the presence of fairness concerns,
we consider a Bayesian model of a prisoner’s dilemma with endogenous moves. In
particular situations, a prisoner’s dilemma may be modeled in rather restricted ways.
9.2.1

Complete information

We studied the leadership in a prisoner’s dilemma under incomplete information about
player’s preferences. However, there are also some cases that are appropriate to bemodeled as complete information games. We brieﬂy discuss leadership in a prisoner’s
dilemma under complete information.
One can show that, under complete information of preferences, (C, C) is realized
by leadership if at least one of the players is a high-guilt type. Namely, if a player
is a high-guilt type and the other player is a low-guilt type, then the low-guilt typetakes the leadership and the high-guilt type becomes a follower in equilibrium. If bothplayers are high-guilt types, then there are two leadership equilibria and each player
becomes a leader in one of the equilibria.
The mechanism of leadership under complete information of preferences is much
simpler than the mechanism of this paper. A player becomes a leader because he knowsthat his opponent is a high-guilt type who becomes a conditional cooperator. Neither
the envy parameter nor the guilt parameter of the leader plays a role in making him
a leader. A follower becomes a follower because he knows that his opponent takes theleadership behavior knowing that the follower becomes a conditional cooperator. The
envy parameter of the follower plays no role in making him a follower.
In contrast, in the leadership mechanism of this paper, both the envy parameterand the guilt parameter play essential roles in making a player a leader and making
a player a follower. This enables us to explain not only how the leadership emerges
endogenously, but also why a particular player becomes a leader when a player’s preferences are his private information.
9.2.2

Exogenous sequence of moves

We studied the possibility of cooperation in a prisoner’s dilemma under the freedom
to choose the timing of moves. We argue that when the players have no freedom tochoose the timing of moves and a sequence of moves is exogenously given, cooperationis also possible depending on the parameters of the game, but how social preferences
make the cooperation realized diﬀers.
Similar to the P D games, we consider a SP D game deﬁned as follows. There are
two roles: leader and follower. Before playing the game, the type of each player isrealized by f independently. Then, under incomplete information about their utility
functions, the players play the prisoner’s dilemma in Table 3 in the following sequence.
43

At timing 1, the leader must choose either C or D. The follower has no move at timing
1 and observes the choice of the leader. Then, the follower chooses C or D at timing
2. This is the end of play. The players receive the payoﬀs in Table 3 corresponding to
the pair of their choices.
The leader has one information set at timing 1. He must choose C or D. The
follower has two information sets corresponding to the leader’s choice at timing 1
being either C or D. His (pure) strategy is a complete plan that assigns either C or
D to each of these information sets. Call a strategy CD when it prescribes C if theleader chooses C and D if the leader chooses D. Call a strategy DD when it prescribes
D always. The Bayesian strategy is deﬁned in the same manner as for the P D game.
The sequential equilibrium in a SP D game is characterized as follows. The sequential equilibrium strategy of the follower is to follow CD if his type (α, β) is β > β ∗ and
DD if it is β < β ∗ . The follower’s strategy generates a distribution of follower’s behaviors. If the leader chooses C, then the follower chooses C with a probability φ(β > β ∗ )and D with a probability 1 − φ(β > β ∗ ). Therefore, the expected utility for the leader
from choosing C is φ(β > β ∗ )a + (1 − φ(β > β ∗ ))[c − α(b − c)]. On the other hand, if
the leader chooses D, then the follower responds with D for sure. The expected utility
is d. Therefore, the leader chooses C if φ(β > β ∗ )a + (1 − φ(β > β ∗ ))[c − α(b − c)] > d.
The leader chooses D if the opposite inequality holds. The leader is indiﬀerent between
C and D if and only if φ(β > β ∗ )a + (1 − φ(β > β ∗ ))[c − α(b − c)] = d. Solve thisequation for α and we obtain
[
]
1
¯ −d
d(a)
α∗∗ ≡
(1 − φ(β > β ∗ ))(b − c)¯ ≡ φ(β > β ∗ )a + (1 − φ(β > β ∗ ))c is a bound deﬁned by (15) in Corollary
where d(a)1. The leader chooses C if α < α∗∗ and D if α > α∗∗ .
The SP D game diﬀers from the P D game in how social preferences make theleadership realized. Two diﬀerences are worth mentioning. First, whether a leader
chooses C or not is solely determined by his envy parameter α. In the SP D game,the opponent of the leader is necessarily a follower. This means that the leader has
no opportunity to betray the opponent. Therefore, in terms of the incentive to lead,which we explored in Section 7, the only tradeoﬀ that the leader faces from choosing C
is between the beneﬁt of inducing C from the follower and the risk of being betrayedby the opponent. Hence, the leadership pattern is independent of the guilt parameterβ in the SP D game.
Second, the leadership is realized with a positive probability in a SP D game if¯and only if α∗∗ > 0; that is, c < d < d(a).
In the SP D game, the leader is the
only player who possibly chooses C at timing 1. This means that the necessary andsuﬃcient condition for the leadership to be realized with a positive probability is thatthe leader of the pure materialist (α = β = 0) is willing to take the leadership when
the follower follows CD if his type (α, β) is β > β ∗ and DD if it is β < β ∗ . Thiscondition corresponds to the condition for the P D game that the pure materialist has
an incentive to take a leadership behavior if no one else takes the leadership (so thatthe opponent is a follower for sure), all the types with β > β ∗ follow CDD, and all the¯ is the necessary and suﬃcient
others follow DDD. Hence, the condition c < d < d(a)condition for leadership in the SP D game, while it is only a suﬃcient condition in the
P D game.
These diﬀerences raise the issue of social dilemmas and social roles in dynamic
decision making in the presence of fairness concerns. The leader role and the follower
role are predetermined and assigned to agents by the rule in the SP D game. In
44contrast, agents are only provided an alternative of taking the role of leader or notin the P D game. The leader role and the follower role are realized by the agents’voluntary will if and only if one of the agents takes the leader role and the other agent
chooses to become a follower. The above diﬀerences in the way of leadership in the
SP D game and the P D game show that how social roles are distributed matters fora possibility of resolving social dilemmas. In particular, one may tend to think that
it is more diﬃcult to support leadership in equilibrium in the P D game, in which anindividual has the freedom not to take the responsibility of leadership, than in the
SP D game, in which the leader agent cannot escape the role of leader. Contrary tothis ﬁrst thought, the second comparison suggests that an agent takes the leadershipbehavior with a positive probability in a wider class of prisoner’s dilemmas in the P D
game than in the SP D game. Note, however, that when we consider a prisoner’s
dilemma that supports a leadership equilibrium both in the SP D game and the P D
game, it is not clear whether the probability of leadership is higher in the SP D game orin the P D game, because the probability of leadership in the P D game is determined
endogenously in equilibrium. This raises an interesting issue for future research.
9.2.3

Simultaneous moves

We studied the possibility of cooperation in a prisoner’s dilemma by the leadership ina model in which the players choose C or D over a span of time. We argue that when
the players have no freedom to choose the timing of moves and they are forced by therules of the game to move simultaneously, cooperation is also possible depending on
the parameters of the game.
Consider a simultaneous-move prisoner’s dilemma that is the same game as P D
except that there is a single timing for moves at which each player must choose C or
D simultaneously and independently. A (pure) Bayesian strategy s assigns to eachtype (α, β) ∈ T a choice s(α, β) ∈ {C, D}. We describe the (symmetric pure) BayesianNash equilibrium condition in terms of payoﬀ parameters a, b, c, d and a type density
f .45
Let µC = φ(s(α, β) = C) denote the consistent belief that a player chooses C.
Then, a player of type (α, β) prefers C to D if and only if µC a+(1−µC ){c−α(b−c)} >µC {b − β(b − c)} + (1 − µC )d, or equivalently:β>

1 − µC d − c b − a
1 − µCeα++=: H(α|µC ).
µCµC b − cb−c(25)

From this, we can derive a ﬁxed-point characterization of symmetric equilibria. The
belief µC supports an equilibrium if and only if:()
eµC = φ β > H(α|µ
)
.
(26)
C
From the linear relation between α and β in (25), it is straightforward to verifyb−c+d−cthat the right-hand side of (26) equals zero for all µC ≤ a−c+b−c+d−c=: µC andmonotonically reaches φ(β > β ∗ ) as µC goes to 1, where β ∗ = b−a
wasdeﬁnedwhenb−c
we introduced the best-response type sets. This implies that (1) µC = 0 always
supports an equilibrium in which all the players choose D irrespective of their types,(2) if a positive µC supports an equilibrium, then µC > µC , and (3) if there exists
a positive µC that supports an equilibrium, then there generically exists at least one
45

See also Duﬀy and Muñoz-Garcı́a (2011).

45

other positive belief µ0C that supports another equilibrium. Figures 13 and 14 below
illustrate these results.
Figure 13 displays the best-response type set given by equation (25). The blue area
describes the set of types who optimally chooses C under the belief µC . CombinedwithC given )µC ; that is,( φ, a belief )µC induces a probability that a player chooses(ee
φ β > H(α|µ).
Figure
14
shows
two
kinds
of
graph
of
φ
β
>
H(α|µ
C
C ) and hence
illustrates the ﬁxed-point characterizations of (26). The left graph displays the casein which no positive µC supports an equilibrium, while the right graph illustrates that
we have two positive beliefs as ﬁxed points. Therefore, cooperation is possible in asimultaneous-move prisoner’s dilemma depending on the parameters (a, b, c, d) and f .

⇤
⇥=H

1
⇤ ⇥

⇥
⇤ |µ)
H(

⇥
|µ

⇥ = H( |µ)⇥ = H ( |1)

0

1

(
)
e
Figure 13: The area of φ β > H(α|µ
C)

1

1
�
�
φ β > β∗µC�
�φ β > β∗0��
�φ β > H(α|µ
C)

µC

�
�
�
φ β > H(α|µ
C)

µC

1

µC

0

µC

1

(
)
e
Figure 14: The graphs of φ β > H(α|µC)
Furthermore, we can discuss how cooperation by the leadership diﬀers from cooperation in simultaneous-move games. In a simultaneous-move prisoner’s dilemma, if
φ(β > β ∗ ) ≤ µC , we never have an equilibrium in which a player cooperates, which is
46µC

illustrated by the left graph of Figure 14. On the other hand, Theorem 1 states that ifd−cφ(β > β ∗ ) > a−c, there exists a three-mode equilibrium in a (endogenous timing) P D
d−cgame. These results imply that if the game parameters satisfy a−c< φ(β > β ∗ ) ≤ µC ,then the cooperation outcome (C, C) is realized as an equilibrium outcome with apositive probability in the (endogenous timing) P D game, whereas we never observe
cooperation in the simultaneous-move prisoner’s dilemma with the same parameters.46
This provides a case in which a prisoner’s dilemma with the chance of voluntary moves
is superior in achieving cooperation to a prisoner’s dilemma without the chance ofvoluntary moves. This case may suggest that working face-to-face is diﬀerent fromworking in isolation in light of advancing cooperation among the players. We leave acomprehensive understanding of this issue for future research.

Appendix A: The case of an exact bound for the existenceof the three-mode equilibrium
We can elaborate Theorem 2 and show that under certain conditions on f , the suﬃ¯ciency bound d(a)for the existence of a three-mode equilibrium in Corollary 1 is infact the exact bound. We say that a prior f of types admits the exact bound for the
existence of a three-mode equilibrium when for any pair (a, d) with c < d < a < b¯there exists a three-mode equilibrium in P D((a, b, c, d), f ) if and only if c < d < d(a).
ˆWhen f admits the exact bound, the threshold d(a) for the existence of a three-mode
equilibrium stated in Theorem 2 is identical to the suﬃciency bound of Corollary 1
ˆ = d(a)¯ for any a ∈ (c, b).47
itself; that is, d(a)We characterize a prior f of types that admits the exact bound for the existence ofa three-mode equilibrium. Let φ(β < β ∗ |α∗ ≤ α) denote the conditional probabilityof β < β ∗ given an event of α∗ ≤ α. We setφ(β < β ∗ |ᾱ ≤ α) = limφ(β < β ∗ |α∗ ≤ α)∗
α →ᾱfor the case of α∗ = ᾱ. Then, we can show that a prior f admits the exact bound forthe existence of a three-mode equilibrium if it satisﬁes the following condition.
Theorem 13. Fix f . Suppose that1 − β̃
φ(β < β̃|α̃ ≤ α)<
1 − β̃ + α̃φ(β < β̃)(27)

holds for any β̃ ∈ (0, 1) and α̃ ∈ (0, ᾱ). Then, there exists a three-mode equilibrium in¯
P D((a, b, c, d), f ) if and only if c < d < d(a).
Theorem 13 says that if the conditional probability of β < β̃ given an event of α̃ ≤ α
for α̃ ∈ (0, ᾱ) is high enough in comparison with the corresponding probability inthe extreme case of α̃ = 0, that is the prior probability φ(β < β̃), there is no three¯mode equilibrium in P D((a, b, c, d), f ) with d ≥ d(a).
The condition (27) provides a1−β̃particular bound 1−β̃+α̃ for a ratio of φ(β < β̃|α̃ ≤ α) against φ(β < β̃). An example
46d−c
To state it by treating the payoﬀ parameters and type distribution separately, if a−c< µC , or2equivalently (b − c)(a − d) > (d − c) , then there exists a type density f such that cooperation outcome(C, C) is realized as an equilibrium outcome with a positive probability in the (endogenous timing)
P D game with those parameters and is never realized in the simultaneous-move prisoner’s dilemma.
47
To be precise, when f admits the exact bound, there exists no three-mode equilibrium inˆ < d < a but also for a boundary case of d = d(a).
ˆP D((a, b, c, d), f ) for an interior case of d(a)47

of a prior that admits the exact bound for the existence of a three-mode equilibriumis the uniform distribution over T = {(α, β)|0 ≤ α, β ≤ 1 and β ≤ α}.
The bound in the condition (27) has the following simple meaning. Set β̃ = β ∗ =b−ab−c for P D((a, b, c, d), f ). Then, for each α̃ given, consider a “C if α < α̃” belief µ̃ suchthat a type (α, β) with α < α̃ follows C, a type (α, β) with α ≥ α̃ and β ≥ β ∗ follows
CDD, and a type (α, β) with α ≥ α̃ and β < β ∗ follows DDD. Then, a thresholdtype (α̃, β)’s preference for C over CDD is written asU(α̃,β) (C, µ̃) − U(α̃,β) (CDD, µ̃)()= φ(α̃ > α)a + φ(α̃ ≤ α)[φ(β ≥ β ∗ |α̃ ≤ α)a + φ(β < β ∗ |α̃ ≤ α)(c − α̃(b − c)]()− φ(α̃ > α)a + φ(α̃ ≤ α)[φ(β ≥ β ∗ |α̃ ≤ α)d + φ(β < β ∗ |α̃ ≤ α)d][()]= φ(α̃ ≤ α) a − φ(β < β ∗ |α̃ ≤ α)(1 − β ∗ + α̃)(b − c) − d .
The bound in condition (27) means that this preference (the value in the brackets) is
maximized at the extreme case of α̃ = 0. In words, the pure materialist (α = β = 0)
is the type who is the most inclined to adopt strategy C over strategy CDD given a“C if α < α̃” belief across all the candidate thresholds α̃ ∈ [0, ᾱ).
From this meaning of the condition (27), it is now straightforward to see why there¯
is no three-mode equilibrium in P D((a, b, c, d), f ) with d ≥ d(a).
Suppose that there
is a three-mode equilibrium distribution µ. As we noted after Theorem 1, the purematerialist is not willing to follow C (or at most is indiﬀerent between C and DDD¯
in the case of d = d(a))
given a “C if α < α̃” belief that corresponds to the extreme
case of α̃ = 0. Then, the condition (27) means that a threshold type (α̃, β) withα̃ = α∗ (µ) > 0 in the presumed equilibrium also strictly prefers CDD to C given a“C if α < α̃” belief µ̃ that corresponds to α̃ = α∗ (µ). Note that this belief µ̃ is abelief that gives greater advantage to strategy C than the presumed distribution µ
because Lemma 2 indicates that when µ is a three-mode equilibrium distribution, aplayer is subject to a risk of being betrayed not only by the opponent of type (α, β)with α ≥ α∗ (µ) and β ≤ β ∗ , but also by the opponent of some type (α, β) with
α < α∗ (µ). Hence, the threshold type (α̃, β) with α̃ = α∗ (µ) must strictly prefer CDD
to C given the presumed distribution µ. Hence, µ cannot be a three-mode equilibrium
distribution.
When a prior f admits the exact bound for the existence of a three-mode equilibrium, our results developed in this paper can be stated in a much sharper way. As forthe results on the existence of a three-mode equilibrium in Theorems 1 and 2, we canuse the explicit formula (15) in Corollary 1 to test whether there exists a three-mode
equilibrium in a prisoner’s dilemma P D((a, b, c, d), f ).
The comparative statics results in Theorems 3 through 10 can be stated as thecomparative statics with respect to the three-mode equilibrium in general by refer¯ring to the case of c < d < d(a)without the necessity of referring to the case of¯ˆd(a) < d < d(a). Then, the comparative statics results with respect to the minimum
leadership equilibrium µmin are much simpler and parallel to the comparative statics
results with respect to the maximum leadership equilibrium µmax . Furthermore, thecharacterization of the materialist leader pattern in Theorem 9 can be strengthened.
First, we can say that the leadership probability µC shrinks to 0 as d approaches the¯
bound d(a).
Lemma 10. Suppose that a prior f admits the exact bound for the existence of athree-mode equilibrium. Then, for any given m > 0, there exists dm (a) with c ≤48¯dm (a) < d(a)such that 0 < µC < m for any three-mode equilibrium distribution¯µ = (µC , µCDD , µDDD ) in any P D((a, b, c, d), f ) with dm (a) < d < d(a).
Then, we can say that any three-mode equilibrium is a materialist leader pattern in¯
any prisoner’s dilemma P D((a, b, c, d), f ) close enough to the bound d(a).
Theorem 14. Suppose that a prior f admits the exact bound for the existence of a¯
three-mode equilibrium. Then, there exists dˆml (a) with c ≤ dˆml (a) < d(a)such thatany three-mode equilibrium in any prisoner’s dilemma P D((a, b, c, d), f ) with dˆml (a) <¯ is of a materialist leader pattern while there exists a three-mode equilibrium ofd < d(a)either a hybrid leader pattern or a inequity concerned leader pattern in any prisoner’s
dilemma P D((a, b, c, d), f ) with c < d < dˆml (a). Furthermore, dˆml (a) is increasing,dˆml (a) ≤ d¯ml (a), and lim dˆml (a) = b where d¯ml (a) is the bound stated in Theorem 9.
a→b

As we noted after Theorem 8, Theorems 7 and 8 suggest that if a prisoner’s dilemma
P D((a, b, c, d), f ) supports a three-mode equilibrium with a materialist leader pattern,then the leadership pattern turns from the materialist leader pattern to a hybrid pattern (and then possibly to an inequity concerned leader pattern) as we move from theprisoner’s dilemma P D((a, b, c, d), f ) by lowering d and raising a. Figure 11 demonstrates this leadership pattern transition for a particular prior f (α, β) = 6β. Theorem
14 means that when a prior f admits the exact bound for the existence of a three-mode¯ inequilibrium, there exists in fact a nondegenerate area along the boundary d = d(a)which any three-mode equilibrium is a materialist leader pattern so that the leadership
pattern transition from a materialist leader pattern to a hybrid pattern (and then possibly to an inequity concerned leader pattern) occurs as we start from any prisoner’s¯
dilemma close enough to the boundary d = d(a)and move by lowering d and raisinga.

Appendix B: An example of no inequity concerned leader
pattern
Consider the uniform distribution over a type space T = {(α, β)|α ≥ β, 0 ≤ α, β ≤ 1}.
Then, for this prior, there exists no three-mode equilibrium with an inequity concerned
leader pattern in any prisoner’s dilemma. Suppose to the contrary that there exists
a three-mode equilibrium distribution µ with an inequity concerned leader pattern insome prisoner’s dilemma. Let γ denote the slope of the boundary of the set TC∗ (µ) for
an area of β < β ∗ in the type space. Then, the three-mode equilibrium requires thatβ∗γ = µDDDµC , and the inequity concerned leader pattern requires that γ ≤ α∗ (µ) . Now,
consider a triangle TC that connects (0, 0), (α∗ (µ), β ∗ ), and (α∗ (µ), α∗ (µ)) in the typespace T . Then, a probability that a type is realized in TC is (α∗ (µ) − β ∗ )α∗ (µ) under
the uniform distribution. This probability is no less than µC because the set TC∗ (µ) is asubset of TC in the inequity concerned leader pattern. Therefore, (α∗ (µ) − β ∗ )α∗ (µ) ≥µC . Similarly, consider a trapezoid TDDD that connects (0, 0), (1, 0), (1, β ∗ ), and(α∗ (µ), β ∗ ). Then, a probability that a type is realized in TDDD is ((1 − α∗ (µ)) + 1)β ∗under the uniform distribution. This probability is no more than µDDD because TDDD∗
is a subset of the set TDDD(µ) in the inequity concerned leader pattern. Therefore,∗∗((1 − α (µ)) + 1)β ≤ µDDD . Hence, we must haveβ∗µDDD((1 − α∗ (µ)) + 1)β ∗≥γ=≥α∗ (µ)µC(α∗ (µ) − β ∗ )α∗ (µ) .
This holds only when α∗ (µ) = 1 and β ∗ = 0. This is a contradiction.
49

Appendix C: Proofs
Proof of Lemma 1[ Step 1 ] Verify that ψ(µ) ∈ ∆ for every µ ∈ ∆ \ (1, 0, 0). This follows from Lemma∗∗
2 and Lemma 3. These lemmas show that TC∗ (µ), TCDD(µ), TDDD(µ) cover T andhave degenerate intersections to each other so that ψC (µ) + ψCDD (µ) + ψDDD (µ) =∗∗∗∗φ(TC∗ (µ)) + φ(TCDD(µ)) + φ(TDDD(µ)) = φ(TC∗ (µ) ∪ TCDD(µ) ∪ TDDD(µ)) = φ(T ) = 1.
48∗
[ Step 2 ] We conﬁrm that ψ(µ) is continuous.
First, the sets TC∗ (µ), TCDD(µ),∗∗
and TCDD (µ) are continuous in µ in the Housdorﬀ metric. Second, the sets TC (µ),∗∗TCDD(µ), and TCDD(µ) are closed subsets in ∆ and the function φ is continuous in
Borel sets in ∆.
[ Step 3 ] We show Lemma 1-(1). Suppose that s : T → S is a three-mode equilibrium strategy. Then, µ = (φ(TC (s)), φ(TCDD (s)), φ(TDDD (s))) is a three-mode distribution. Furthermore, µ is a consistent belief at timing 1. Take (α, β) ∈ TC (s).
Then, it is necessary for the sequential rationality of this type at timing 1 thatU(α,β) (C, µ) ≥ U(α,β) (CDD, µ), U(α,β) (DDD, µ). This means that TC (s) ⊆ TC∗ (µ).
Take (α, β) ∈ TCDD (s). Then, it is necessary for the sequential rationality of thistype at timing 1 that U(α,β) (CDD, µ) ≥ U(α,β) (C, µ), U(α,β) (DDD, µ). As is explainedin the text, β ≥ β ∗ is also necessary for the sequential rationality of this type at∗timing 2 after the opponent chooses C. This means that TCDD (s) ⊆ TCDD(µ). Sim∗∗ilarly, TDDD (s) ⊆ TDDD (µ). Then, φ(TC (s)) = φ(TC (µ)), φ(TCDD (s)) = φ(TC∗ (µ)),∗∗∗
and φ(TDDD (s)) = φ(TDDD(µ)) must hold because TC∗ (µ), TCDD(µ), TDDD(µ) havedegenerate intersections to each other. This means µ = ψ(µ).
[ Step 4 ] We show Lemma 1-(2). Suppose a three-mode distribution µ ∈ ∆ suchthat µ = ψ(µ). Take any three-mode strategy s : T → S that satisﬁes TC (s) ⊆∗∗TC∗ (µ), TCDD (s) ⊆ TCDD(µ), and TDDD (s) ⊆ TDDD(µ). Then, φ(TC (s)) = φ(TC∗ (µ)),∗∗φ(TCDD (s)) = φ(TC∗ (µ)), and φ(TDDD (s)) = φ(TDDD(µ)) because TC∗ (µ), TCDD(µ),∗
TDDD (µ) have degenerate intersections to each other. This means that µ is a belief attiming 1 that is consistent with s because µ = ψ(µ).
Now, we verify that s satisﬁes sequential rationality under µ. Consider the information set at timing 2 after the opponent chooses C. It is sequentially rational for a type∗(α, β) ∈ TCDD (s) to choose C because TCDD (s) ⊆ TCDD(µ), and as is explained in∗
the text, it is sequentially rational for a type (α, β) ∈ TCDD(µ) to choose C. Similarly,
it is sequentially rational for a type (α, β) ∈ TDDD (s) to choose D.49
Consider the information set at timing 2 after the opponent chooses D. Then, it issequentially rational for both a type (α, β) ∈ TCDD (s) and a type (α, β) ∈ TDDD (s)to choose D because the best response to D is D irrespective of a player’s inequity
aversion.
Consider the information set at timing 2 after the opponent chooses ∅. Then, theconsistent belief is that the opponent is either a type (α, β) ∈ TCDD (s) or a type(α, β) ∈ TDDD (s). Both types choose a∅ = D at timing 2 after the player in question
also chooses ∅ at timing 1. Hence, it is sequentially rational for the player to choose D
against the expected choice of D irrespective of whether he is a type (α, β) ∈ TCDD (s)or a type (α, β) ∈ TDDD (s).
Finally, consider the information set at timing 1. First, note that the optimality of48

Note that ψ(µ) also depends on (a, d). The function ψ ia also continuous in (a, d) by the same
argument.
49
C-mode is left unspeciﬁed in timing 2 actions aC , aD , a∅ . To be explicit, a type (α, β) ∈ TC (s)follows (C, C, D, D) if β > β ∗ and (C, D, D, D) if β < β ∗ . Then, it is sequentially rational for a type(α, β) ∈ TC (s) to take these actions.

50

a strategy among those with a1 = ∅ is equivalent to the sequential rationality at timing
2. As is established above, either CDD or DDD is optimal against the three-mode
strategy s : T → S. Second, the strategies with a1 6= ∅ are C and D = (D, aC , aD , a∅ ).
Observe that DDD and D generate the same outcomes given the three-mode strategys : T → S. If the opponent follows C-mode, then both the outcome from following
DDD and the outcome from following D are (D, C). If the opponent follows either
CDD-mode or DDD-mode, then both the outcome from following DDD and theoutcome from following D are (D, D). Therefore, DDD is as good as D given thethree-mode strategy s : T → S. Hence, if a strategy is optimal among C, CDD, and
DDD, then it is optimal in S. This means that it is sequentially rational for a type(α, β) ∈ TC (s) to choose C at timing 1 because TC (s) ⊆ TC∗ (µ), and it is sequentiallyrational for a type (α, β) ∈ TCDD (s) and a type (α, β) ∈ TDDD (s) to choose ∅ because∗∗
TCDD (s) ⊆ TCDD(µ) and TDDD (s) ⊆ TDDD(µ). (Q.E.D.)
Proof of Lemma 4[ Step 1 ] Consider ψ(µ) for µ = (µC , 1 − µC , 0) with µC ∈ [0, 1). Then, there existsµ̄C ∈ ( a−db−d , 1) such that ψC (µC , 1 − µC , 0) > µC for µC ∈ [0, µ̄C ), ψC (µ̄C , 1 − µ̄C , 0) =µ̄C , and ψC (µC , 1 − µC , 0) < µC for µC ∈ (µ̄C , 1).
(Proof)∗Consider µ = (µC , 1 − µC , 0) with µC ∈ [0, 1). Then, TCDD(µ) = ∅ because
U(α,β) (C, µ) − U(α,β) (CDD, µ) = (1 − µC )(a − d) > 0
for any (α, β) ∈ T . Therefore, (α, β) ∈ TC∗ (µ) if and only if
U(α,β) (C, µ) − U(α,β) (DDD, µ) = µC {a − [b − β(b − c)]} + (1 − µC )(a − d) ≥ 0.
Hence, we have()ψC (µ) = φ µC (b − c)β ≥ µC (b − d) − (a − d) .
This means that ψC (µ) = 1 for µC ∈ [0, a−db−d ] and ψC (µ) is strictly decreasing in µC
a−dover µC ∈ ( b−d , 1). Furthermore, ψ : ∆ \ (1, 0, 0) → ∆ is continuous in µ by Lemma
1. Hence, we have the desired µ̄C . ||[ Step 2 ] Consider ψ(µ) for µ = (µC , µCDD , µDDD ) with µC ∈ [0, 1) and µCDD =d−c∗∗a−d µDDD . Then, the threshold α (µ) deﬁned by (4) is α (µ) = 0. Hence, it follows
from Lemma 2 and Lemma 3 that ψ(µ) = (0, φ(β > β ∗ ), φ(β < β ∗ )).
[ Step 3 ] Fix µC ∈ [0, 1) and consider ψ(µ0 ), ψ(µ00 ) for µ0 = (µ0C , µ0CDD , µ0DDD ), µ00 =d−c 00d−c 0(µ00C , µ00CDD , µ00DDD ) with µ0C = µ00C = µC , µ0CDD ≥ a−dµDDD , and µ00CDD ≥ a−dµDDD .
000
000
Then, if µCDD > µCDD (and so µDDD < µDDD ), then(1) ψC (µ0 ) ≥ ψC (µ00 ), and(2) it is never the case that 0 < ψC (µ0 ) = ψC (µ00 ) < 1.
(Proof)Consider µ0 = (µ0C , µ0CDD , µ0DDD ), µ00 = (µ00C , µ00CDD , µ00DDD ) with µ0C = µ00C = µC ∈(0, 1) and µ0CDD > µ00CDD (and so µ0DDD < µ00DDD ). We show that TC∗ (µ0 ) ⊇ TC∗ (µ00 ).
Take (α, β) ∈ TC∗ (µ00 ). Then, U(α,β) (C, µ00 ) ≥ U(α,β) (CDD, µ00 ) and U(α,β) (C, µ00 ) ≥51

U(α,β) (DDD, µ00 ). Then, together with µ0CDD > µ00CDD and µ0DDD < µ00DDD , we haveU(α,β) (C, µ0 ) − U(α,β) (CDD, µ0 ) = µC (a − a) + µ0CDD (a − d) + µ0DDD {[c − α(b − c)] − d}> µC (a − a) + µ00CDD (a − d) + µ00DDD {[c − α(b − c)] − d}= U(α,β) (C, µ00 ) − U(α,β) (CDD, µ00 )≥000U(α,β) (C, µ ) − U(α,β) (DDD, µ ) = µC {a − [b − β(b − c)]} + µ0CDD (a − d) + µ0DDD {[c − α(b − c)] − d}> µC {a − [b − β(b − c)]} + µ00CDD (a − d) + µ00DDD {[c − α(b − c)] − d}= U(α,β) (C, µ00 ) − U(α,β) (DDD, µ00 )≥ 0.
Therefore, (α, β) ∈ TC∗ (µ0 ). Hence, TC∗ (µ0 ) ⊇ TC∗ (µ00 ). This establishes ψC (µ0 ) ≥ψC (µ00 ).
Suppose that 0 < ψC (µ00 ) < 1. Then, it is easily veriﬁed from Lemmas 2 and 3 that∗
there exists (α, β) ∈ TC∗ (µ00 ) ∩ TDDD(µ00 ) such that any open neighborhood N of (α, β)∗contains an open set in (N ∩ TDDD(µ00 )) \ TC∗ (µ00 ). On the other hand, the argument in
the proof of (1) immediately guarantees the existence of an open neighborhood N 0 of
the (α, β) such that C is the best among C, CDD, DDD under µ0 for every (α̃, β̃) ∈ N 0 ;∗that is, N 0 ⊆ TC∗ (µ0 ). N 0 contains an open set in (N 0 ∩ TDDD(µ00 )) \ TC∗ (µ00 ). This open
∗
0
∗
00
0
set is contained in TC (µ ) \ TC (µ ), and hence ψC (µ ) > ψC (µ00 ). ||
d−c
µ̂DDD
[ Step 4 ] For each µC ∈ [0, µ̄C ], there uniquely exists µ̂ ∈ ∆ with µ̂CDD ≥ a−dsuch that ψC (µ̂) = µ̂C = µC . Denote it as µ̂(µC ). Then,d−c(1) µ̂(0) = (0, µ̂CDD (0), µ̂DDD (0)) with µ̂CDD (0) = a−dµ̂DDD (0) and ψ(µ̂(0)) =(0, φ(β > β ∗ ), φ(β < β ∗ ))(2) µ̂(µ̄C ) = (µ̄C , 1 − µ̄C , 0) and ψ(µ̂(µ̄C )) = (µ̄C , 0, 1 − µ̄C )d−c(3) µ̂CDD (µC ) > a−dµ̂DDD (µC ), µ̂DDD (µC ) > 0 for µC ∈ (0, µ̄C )(Proof)d−cConsider µC = 0. Consider µ = (0, µCDD , µDDD ) with µCDD = a−dµDDD . Thisµ satisﬁes the required property for µ̂ that ψC (µ̂) = µ̂C = µC = 0 because ψ(µ) =(0, φ(β > β ∗ ), φ(β < β ∗ )) holds by Step 2. Consider any µ = (0, µCDD , µDDD ) withd−c
µCDD > a−dµDDD . Then,U(0,0) (C, µ)−U(0,0) (CDD, µ) = U(0,0) (C, µ)−U(0,0) (DDD, µ) = µCDD (a−d)+µDDD (c−d) > 0.
Hence, TC∗ (µ) contains some neighborhood of (0, 0). Therefore, ψC (µ) > 0. This µ
fails to satisfy the required property for µ̂ that ψC (µ̂) = µ̂C = µC = 0. Thus, (1) isestablished.
Consider µC = µ̄C . Consider µ = (µ̄C , 1 − µ̄C , 0). This µ satisﬁes the required
property for µ̂ that ψC (µ̂) = µ̂C = µC = µ̄C because ψ(µ) = (µ̄C , 0, 1 − µ̄C ) holds
d−c 00by Step 1. Consider any other µ00 = (µ̄C , µ00CDD , µ00DDD ) with µ00CDD ≥ a−dµDDD .
Then, µDDD = 0 < µ00DDD . Therefore, by Step 3, it is never the case that 0 <ψC (µ) = ψC (µ00 ) = µ̄C < 1. This µ00 fails to satisfy the required property for µ̂ thatψC (µ̂) = µ̂C = µC = µ̄C . Thus, (2) is established.
d−c
To show (3), consider µC ∈ (0, µ̄C ). Note that (1 − µC − µDDD ) ≥ a−dµDDD ifa−da−dand only if µDDD ≤ a−c (1 − µC ). For each µDDD ∈ [0, a−c (1 − µC )], let κ(µDDD ) =ψC ((µC , 1 − µC − µDDD , µDDD )) − µC . Then, κ(0) = ψC ((µC , 1 − µC , 0)) − µC > 052

follows from (1) and κ( a−da−c (1 − µC )) = −µC < 0 follows from (2). The continuity ofd−cψ guarantees that there exists µ̂ ∈ ∆ with 0 < µ̂DDD and µ̂CDD > a−dµ̂DDD suchthat κ(µ̂DDD ) = 0. Then, µ̂ = (µC , 1 − µC − µ̂DDD , µ̂DDD ) satisﬁes the required
property for µ̂ that ψC (µ̂) = µ̂C = µC . Consider any other µ00 = (µC , µ00CDD , µ00DDD )d−c 00with µ00CDD ≥ a−dµDDD . Then, either µ̂CDD > µ00CDD or µ̂CDD < µ00CDD . In either
case, by Step 3, it is never the case that 0 < ψC (µ̂) = ψC (µ00 ) = µC < 1. This µ00 fails
to satisfy the required property for µ̂ that ψC (µ̂) = µ̂C = µC . Thus, (3) is established.
||[ Step 5 ] Note that µ̄C established in Step1 depends on (a, d). Taking this fact into
account, let us denote µ̄C (a, d) explicitly. Then, µ̄C (a, d) is continuous.
(Proof)
Suppose that µ̄C (a, d) is not continuous at some (a, d). Then, there exists a sen nn n ∞quence {(an , dn )}∞n=1 such that lim (a , d ) = (a, d) and {µ̄C (a , d )}n=1 does notn→∞converge to µ̄C (a, d). Then, we can take a convergent subsequence {µ̄C (anm , dnm )}∞m=1
such that there exists µ̄∗C = lim µ̄C (anm , dnm ) and µ̄∗C 6= µ̄C (a, d) because {µ̄C (an , dn )}∞n=1
m→∞

is a bounded sequence. By Step 1, µ̄C (a, d) is a unique solution µ̄C to()µ̄C − φ µ̄C (b − c)β ≥ µ̄C (b − d) − (a − d) = 0.

(28)

Therefore, µ̄∗C 6= µ̄C (a, d) means that()µ̄∗C − φ µ̄∗C (b − c)β ≥ µ̄∗C (b − d) − (a − d) 6= 0.
Then, there exists  > 0 such that
(
)
(
)
˜
˜ > 1 µ̄∗ −φ µ̄∗ (b−c)β ≥ µ̄∗ (b−d)−(a−d) > 0.
µ̄C −φ µ̄C (b−c)β ≥ µ̄C (b−d)−(a−
d)
C
C
C
2
˜ ∈ (µ̄∗ − , µ̄∗ + ) × (a − , a + ) × (d − , d + ) because thefor any (µ̄C , ã, d)CCleft hand side of (28) is continuous in (µC , a, d). Then, there exists m such that(µ̄C (anm , dnm ), anm , dnm ) ∈ (µ̄∗C − , µ̄∗C + ) × (a − , a + ) × (d − , d + ) because
lim µ̄C (anm , dnm ) = µ̄∗C and lim (anm , dnm ) = (a, d). Then,

m→∞

m→∞

(
)
µ̄C (anm , dnm )−φ µ̄C (anm , dnm )(b−c)β ≥ µ̄C (anm , dnm )(b−dnm )−(anm −dnm ) > 0.
This is a contradiction because µ̄C (anm , dnm ) is a unique solution µ̄C to()µ̄C − φ µ̄C (b − c)β ≥ µ̄C (b − dnm ) − (anm − dnm ) = 0.

||

[ Step 6 ] Note that µ̂(µC ) depends on (a, d). Taking this fact into account, let us
denote µ̂(µC , a, d) explicitly. Then, µ̂(µC , a, d) is continuous.
(Proof)nConsider a sequence {(µnC , an , dn )}∞n=1 in [0, 1] × [c, b] × [c, b] such that 0 ≤ µC ≤
n
n
n
n
n
µ̄C (a , d ), c < a < b, and c < d < a for each n. Suppose that the sequence
converges. Let (µ∗C , a∗ , d∗ ) = lim (µnC , an , dn ). Then, it follows from Step 5 thatn→∞0 ≤ µ∗C = lim µnC ≤ lim µ̄C (an , dn ) = µ̄C ( lim an , lim dn ) = µ̄C (a∗ , d∗ ).
n→∞

n→∞

n→∞

Then, there exists the µ̂(µ∗C , a∗ , d∗ ) established in Step 4.

53

n→∞

Examine the corresponding sequence {µ̂(µnC , an , dn )}∞n=1 generated by µ̂. Supnm nm nm ∞pose a convergent subsequence {µ̂(µC , a , d )}m=1 of it arbitrarily. Let µ̂∗ =lim µ̂(µnCm , anm , dnm ). Then,m→∞µ̂∗C = lim µ̂C (µnCm , anm , dnm ) = lim µnCm = µ∗C
m→∞

m→∞

where the second equality holds because µ̂C (µnCm , anm , dnm ) = µnCm holds for each nm
by the deﬁnition of µ̂(µC , a, d). Additionally,ψC (µ̂∗ , a∗ , d∗ ) = lim ψC (µ̂(µnCm , anm , dnm ), anm , dnm )m→∞= lim µ̂C (µnCm , anm , dnm )m→∞= lim µnCm
n→∞
= µ∗C

where the ﬁrst equality holds because ψ(µ, a, d) is continuous by the proof of Lemma
1, and the second and third equalities hold because ψC (µ̂(µnCm , anm , dnm ), anm , dnm ) =µ̂C (µnCm , anm , dnm ) = µnCm holds for each nm by the deﬁnition of µ̂(µC , a, d). Thus,
ψC (µ̂∗ , a∗ , d∗ ) = µ̂∗C = µ∗C .
Then, µ̂(µ∗C , a∗ , d∗ ) = µ̂∗ because µ̂(µ∗C , a∗ , d∗ ) is the unique µ̂ that satisﬁes ψC (µ̂) =µ̂C = µ∗C . Thus, we know that any convergent subsequence of {µ̂(µnC , an , dn )}∞n=1
converges to µ̂(µ∗C , a∗ , d∗ ). Then, the sequence {µ̂(µnC , an , dn )}∞n=1 itself converges toµ̂(µ∗C , a∗ , d∗ ) because it is a sequence in a compact set ∆. This means that µ̂(µC , a, d)is continuous. (Q.E.D.)
Proof of Lemma 5
Suppose that a belief µ is a three-mode equilibrium distribution. Then, µ = ψ(µ)
and µC > 0 by Lemma 1. By the way of construction of µ̂(µC ) and λ(µC ), it holds forµ with µ = ψ(µ) that λ(µC ) = 0 and µ = µ̂(µC ).
Conversely, suppose that µ satisﬁes conditions (1) through (3) in Lemma 5. Then,µ = µ̂(µC ) means that µC = ψC (µ). Together with µ = µ̂(µC ), λ(µC ) = 0 means that
µDDD = ψDDD (µ). Therefore, µ = ψ(µ). Finally, by Lemma 4-(2),λ(µ̄C ) = ψDDD (µ̂(µ̄C )) − µ̂DDD (µ̄C ) = (1 − µ̄C ) − 0 > 0.
This implies that µC 6= µ̄C . Then, by Lemma 4-(3), µC ∈ (0, µ̄C ) implies that µCDD >0 and µDDD > 0. Thus, this µ is a three-mode equilibrium. (Q.E.D.)
Proof of Theorem 1
d−c a−d, a−c ) and ψ(µ̂(0)) =Suppose that condition (14) holds. By Lemma 4, µ̂(0) = (0, a−c∗∗(0, φ(β > β ), φ(β < β )). Then, condition (14) guarantees
λ(0) = ψDDD (µ̂(0)) − µ̂DDD (0) = φ(β < β ∗ ) −

a−d
< 0.
a−c

On the other hand, by Lemma 4, µ̄C < 1 and µ̂(µ̄C ) = (µ̄C , 1 − µ̄C , 0) andψ(µ̂(µ̄C )) = (µ̄C , 0, 1 − µ̄C ). Then,λ(µ̄C ) = ψDDD (µ̂(µ̄C )) − µ̂DDD (µ̄C ) = (1 − µ̄C ) − 0 > 0.
Finally, λ(µC ) is continuous because ψ(µ) is continuous by Lemma 1 and µ̂(µC ) is
continuous by Lemma 4.
54

These establish that there exists µC ∈ (0, µ̄C ) such that λ(µC ) = 0. Set µ = µ̂(µC ).
Then, this µ satisﬁes conditions (1), (2), and (3) in Lemma 5. Hence, this µ is a threemode equilibrium distribution. (Q.E.D.)
Proof of Corollary 1¯
Corollary 1-(2) is straightforward from Theorem 1 and the deﬁnition of d(a).
To
b−a
∗
¯
see 1-(1), recall that β = b−c . Then, c < d(a) < a is immediate by the deﬁnitiond ¯¯of d(a),because 0 < β ∗ < 1 so that 0 < φ(β > β ∗ ) < 1. It holds that dad(a) =
∗
dβ
dβ ∗
d
d
∗
∗
∗
φ(β > β ) + (a − c)( dβ ∗ φ(β > β )) da > 0, because dβ ∗ φ(β > β ) < 0 and da < 0.
¯¯
= c. Finally, it holds that
From c < d(a)< a, it immediately follows that lim d(a)a→c¯ = b because lim β ∗ = 0 so that lim φ(β > β ∗ ) = 1. (Q.E.D.)
lim d(a)
a→b

a→b

a→b

Proof of Lemma 6
To show Lemma 6-(1), recall from the proof of Lemma 4 that µ̄C (a, d) is a uniquesolution µ̄C to()µ̄C − φ µ̄C (b − c)β ≥ µ̄C (b − d) − (a − d) = 0.
()
Note that φ µC (b − c)β ≥ µC (b − d) − (a − d) is 1 for µC ∈ [0, a−db−d ] and strictly
decreasing in µC over µC ∈ ( a−db−d , 1) so that the solution µ̄C lies in the latter regiona−d( b−d , 1). Then, the solution µ̄C is strictly increasing in a and strictly decreasing in dbecause)∂ (φ µ̄C (b − c)β ≥ µ̄C (b − d) − (a − d) > 0∂a)∂ (φ µ̄C (b − c)β ≥ µ̄C (b − d) − (a − d) < 0.
∂d
To show Lemma 6-(2), ﬁx d ∈ (c, b) arbitrarily and compare arbitrary a0 , a00 withc < d < a0 < a00 < b. By Lemma 6-(1) established above, µ̄C (a0 , d) < µ̄C (a00 , d) so thatboth λ(µC , a0 , d) and λ(µC , a00 , d) are deﬁned over [0, µ̄C (a0 , d)]. We will show in two
steps below that for each µC ∈ [0, µ̄C (a0 , d)]
1. µ̂DDD (µC , a0 , d) < µ̂DDD (µC , a00 , d), and2. ψDDD (µ̂(µC , a0 , d), a0 , d) ≥ ψDDD (µ̂(µC , a00 , d), a00 , d).
Then, it is established for any µC ∈ [0, µ̄C (a0 , d)] thatλ(µC , a0 , d) = ψDDD (µ̂(µC , a0 , d), a0 , d) − µ̂DDD (µC , a0 , d)
> ψDDD (µ̂(µC , a00 , d), a00 , d) − µ̂DDD (µC , a00 , d)= λ(µC , a00 , d).
It is proved in a similar way that λ(µC , a, d) is strictly increasing in d. (The proof issimpler because β ∗ is independent of d.)[ Step 1 ] µ̂DDD (µC , a0 , d) < µ̂DDD (µC , a00 , d) for each µC ∈ [0, µ̄C (a0 , d)].
(Proof)a0 −d00
Consider µC = 0. Then, by Lemma 4, µ̂(0, a0 , d) = (0, ad−c
0 −c , a0 −c ) and µ̂(0, a , d) =
00
0
00
a −d
a −d
a −d
0
00
(0, ad−c
00 −c , a00 −c ). Therefore, µ̂DDD (µC , a , d) = a0 −c < a00 −c = µ̂DDD (µC , a , d) when
000
a <a .

55

Consider µC ∈ (0, µ̄C (a0 , d)]. Suppose to the contrary that µ̂DDD (µC , a0 , d) ≥µ̂DDD (µC , a00 , d). Then, on one hand, we haveψC (µ̂(µC , a00 , d), a0 , d) ≥ ψC (µ̂(µC , a0 , d), a0 , d) = µC
= ψC (µ̂(µC , a00 , d), a00 , d)
= µ̂C (µC , a00 , d)where the ﬁrst inequality follows from µ̂DDD (µC , a0 , d) ≥ µ̂DDD (µC , a00 , d) by Step 3of the proof of Lemma 4.
On the other hand, by a0 < a00 , we haveψC (µ̂(µC , a00 , d), a00 , d) > ψC (µ̂(µC , a00 , d), a0 , d)for the following reason. Take (α, β) ∈ TC∗ (µ̂(µC , a00 , d), a0 , d). Then,U(α,β) (C, µ̂(µC , a00 , d), a0 , d) ≥ U(α,β) (CDD, µ̂(µC , a00 , d), a0 , d)U(α,β) (C, µ̂(µC , a00 , d), a0 , d) ≥ U(α,β) (DDD, µ̂(µC , a00 , d), a0 , d).
Then, together with a0 < a00 , we haveU(α,β) (C, µ̂(µC , a00 , d), a00 , d) − U(α,β) (CDD, µ̂(µC , a00 , d), a00 , d)
= U(α,β) (C, µ̂(µC , a00 , d), a0 , d) − U(α,β) (CDD, µ̂(µC , a00 , d), a0 , d)
+ µ̂CDD (µC , a00 , d)(a00 − a0 )≥ U(α,β) (C, µ̂(µC , a00 , d), a0 , d) − U(α,β) (CDD, µ̂(µC , a00 , d), a0 , d)≥0andU(α,β) (C, µ̂(µC , a00 , d), a00 , d) − U(α,β) (DDD, µ̂(µC , a00 , d), a00 , d)
= U(α,β) (C, µ̂(µC , a00 , d), a0 , d) − U(α,β) (DDD, µ̂(µC , a, d), a0 , d)
+ µ̂C (µC , a00 , d)(a00 − a0 ) + µ̂CDD (µC , a00 , d)(a00 − a0 )≥ U(α,β) (C, µ̂(µC , a00 , d), a0 , d) − U(α,β) (DDD, µ̂(µC , a00 , d), a0 , d)
≥ 0.
Therefore, (α, β) ∈ TC∗ (µ̂(µC , a00 , d), a00 , d). Hence, TC∗ (µ̂(µC , a00 , d), a0 , d) ⊆ TC∗ (µ̂(µC , a00 , d), a00 , d).
Furthermore, TC∗ (µ̂(µC , a00 , d), a0 , d) $ TC∗ (µ̂(µC , a00 , d), a00 , d) by a similar argument toStep 3 of the proof of Lemma 4. Hence, ψC (µ̂(µC , a00 , d), a00 , d) > ψC (µ̂(µC , a00 , d), a0 , d).
By combining the above inequalities, we haveψC (µ̂(µC , a00 , d), a00 , d) > ψC (µ̂(µC , a00 , d), a0 , d) ≥ µ̂C (µC , a00 , d).
This contradicts the supposition that ψC (µ̂(µC , a00 , d), a00 , d) = µ̂C (µC , a00 , d). Hence,
it must be the case that µ̂DDD (µC , a0 , d) < µ̂DDD (µC , a00 , d).
[ Step 2 ] ψDDD (µ̂(µC , a0 , d), a0 , d) ≥ ψDDD (µ̂(µC , a00 , d), a00 , d) for each µC ∈ [0, µ̄C (a0 , d)].
(Proof)First, we show that α∗ (µ̂(µC , a0 , d), a0 , d) > α∗ (µ̂(µC , a00 , d), a00 , d). Suppose to thecontrary that α∗ (µ̂(µC , a0 , d), a0 , d) ≤ α∗ (µ̂(µC , a00 , d), a00 , d). Recall from (6) thatµ̂DDD (µC , a0 , d)(α − α∗ (µ̂(µC , a0 , d), a0 , d)) + β ∗ (a0 )µ̂C (µC , a0 , d)µ̂DDD (µC , a00 , d)H(α|µ̂(µC , a00 , d), a00 , d) =(α − α∗ (µ̂(µC , a00 , d), a00 , d)) + β ∗ (a00 )µ̂C (µC , a00 , d)H(α|µ̂(µC , a0 , d), a0 , d) =

56

0

00

b−a
∗ 00
∗
where we denote β ∗ (a0 ) = b−ab−c and β (a ) = b−c to express the fact that β depends
0

00

0

(µC ,a ,d)µ̂DDD (µC ,a ,d)µ̂DDD (µC ,a ,d)on a. Note that the slopes µ̂DDDµ̂C (µC ,a0 ,d) and µ̂C (µC ,a00 ,d) are related as µ̂C (µC ,a0 ,d) <µ̂DDD (µC ,a00 ,d)because µ̂C (µC , a0 , d) = µC = µ̂C (µC , a00 , d) and µ̂DDD (µC , a0 , d) <µ̂C (µC ,a00 ,d)
0
b−a00
∗ 00
0
µ̂DDD (µC , a00 , d) by Step 1. Note also that β ∗ (a0 ) = b−ab−c > b−c = β (a ) when a <
00
∗
0
0
∗
00
00
a . Then, under the supposition that α (µ̂(µC , a , d), a , d) ≤ α (µ̂(µC , a , d), a , d),we haveTC∗ (µ̂(µC , a0 , d), a0 , d) = T ∩ {(α, β)|α ≤ α∗ (µ̂(µC , a0 , d), a0 , d), β ≥ H(α|µ̂(µC , a0 , d), a0 , d)}$ T ∩ {(α, β)|α ≤ α∗ (µ̂(µC , a00 , d), a00 , d), β ≥ H(α|µ̂(µC , a00 , d), a00 , d)}= TC∗ (µ̂(µC , a00 , d), a00 , d).
This means that µ̂C (µC , a0 , d) = φ(TC∗ (µ̂(µC , a0 , d), a0 , d) < φ(TC∗ (µ̂(µC , a00 , d), a00 , d) =µ̂C (µC , a00 , d). This contradicts µ̂C (µC , a0 , d) = µC = µ̂C (µC , a00 , d).
0
b−a00
∗ 00
∗
0
0
Then, together with β ∗ (a0 ) = b−ab−c > b−c = β (a ), the relation α (µ̂(µC , a , d), a , d) >α∗ (µ̂(µC , a00 , d), a00 , d) thus established implies that∗
ψCDD (µ̂(µC , a0 , d), a0 , d) = φ(TCDD(µ̂(µC , a0 , d), a0 , d))= φ(T ∩ {(α, β)|α > α∗ (µ̂(µC , a0 , d), a0 , d), β > β ∗ (a0 )})≤ φ(T ∩ {(α, β)|α > α∗ (µ̂(µC , a00 , d), a00 , d), β > β ∗ (a00 )})∗= φ(TCDD(µ̂(µC , a00 , d), a00 , d))= ψCDD (µ̂(µC , a00 , d), a00 , d).
Hence, together with ψC (µ̂(µC , a0 , d), a0 , d) = µC = ψC (µ̂C (µC , a00 , d), a00 , d), we conclude that
ψDDD (µ̂(µC , a0 , d), a0 , d) = 1 − ψC (µ̂(µC , a0 , d), a0 , d) − ψCDD (µ̂(µC , a0 , d), a0 , d)
≥ 1 − ψC (µ̂(µC , a00 , d), a00 , d) − ψCDD (µ̂(µC , a00 , d), a00 , d)
= ψDDD (µ̂(µC , a00 , d), a00 , d).
(Q.E.D.)
Proof of Theorem 2
ˆ with the desired properties.
[ Step 1 ] Fix a ∈ (c, b). We will show how to ﬁnd d(a)First, recall from Lemma 6-(1) that µ̄C (a, d) is strictly decreasing in d ∈ (c, a).
¯C (a) = limd↓c µ̄C (a, d) because µ̄C (a, d) is a probability andThis allows us to deﬁne µ̄
bounded from above by 1.
¯C (a)), there exists d0 ∈ (c, a) such that µC ∈ [0, µ̄C (a, d0 )] so
For each µC ∈ [0, µ̄
0that λ(µC , a, d ) is deﬁned. Then, λ(µC , a, d) is deﬁned for every d ∈ (c, d0 ) because
d < d0 means µ̄C (a, d) > µ̄C (a, d0 ). Over the interval (c, d0 ), λ(µC , a, d) is increasing
in d by Lemma 6-(2). This allows us to deﬁne λ(µC , a) = limd↓c λ(µC , a, d) because
λ(µC , a, d) is a diﬀerence in two probabilities and bounded from below by −1.
¯C (a))|λ(µC , a) < 0}. Note from the proof of Theorem
Deﬁne MC (a) = {µC ∈ [0, µ̄¯
1 that λ(0, a, d) < 0 for any d ∈ (c, d(a)).
This implies that λ(0, a) < 0. Therefore,
0 ∈ MC (a). Thus, MC (a) is nonempty.
Consider µC ∈ MC (a) \ 0. Then, by the way of construction of λ(µC , a), λ(µC , a) <0 guarantees that there exists some d0 ∈ (c, a) such that λ(µC , a, d0 ) < 0. On the other
hand, we can show the following fact, for which we postpone the proof to Step 3 below.
Claim 1: For µC ∈ MC (a) \ 0, there exists some d00 ∈ (c, a) such that µC ≤ µ̄C (a, d00 )and λ(µC , a, d00 ) > 0.
57

For d0 and d00 thus selected for the given µC ∈ MC (a) \ 0, it immediately follows fromλ(µC , a, d0 ) < 0 and λ(µC , a, d00 ) > 0 that d0 < d00 because λ(µC , a, d) is increasing ind by Lemma 6-(2). Then, there exists a unique d∗ ∈ (d0 , d00 ) in the interval (c, a) suchthat λ(µC , a, d∗ ) = 0 because λ(µC , a, d) is continuous and strictly increasing in d. We
write d∗ (µC , a) to express the fact that this d∗ depends on (µC , a).
¯For µC = 0 ∈ MC (a), note from the proof of Theorem 1 that λ(0, a, d(a))= 0. So∗¯
we set d (0, a) = d(a).
ˆNow deﬁne d(a)≡ supµC ∈MC (a) d∗ (µC , a). We can show the following fact, forwhich we postpone the proof to Step 3 below.
ˆ =Claim 2: There exists µ̂C ∈ MC (a) that achieves supµC ∈MC (a) d∗ (µC , a) so that d(a)∗∗maxµC ∈MC (a) d (µC , a) = d (µ̂C , a).
ˆ is what Theorem 2 claims.
We will show that this d(a)ˆ has the properties claimed in Theorem 2.
[ Step 2] We will show that d(a)ˆˆWe show Theorem 2-(1). Fix d ∈ (c, d(a)).
Then, d < d(a)guarantees that there
0∗0ˆexists some µC ∈ MC (a) such that d < d (µC , a) ≤ d(a).
This means λ(µ0C , a, d) <˜ is strictly increasing in d.
˜ Recall0 because λ(µ0C , a, d∗ (µ0C , a)) = 0 and λ(µ0C , a, d)from the proof of Theorem 1 that λ(µ̄C (a, d), a, d) > 0. Then, there exists µC ∈(µ0C , µ̄C (a, d)) such that λ(µC , a, d) = 0 because λ(µ̃C , a, d) is continuous in µ̃C . Setµ = µ̂(µC ). Then, this µ is a three-mode equilibrium distribution in a prisoner’s
dilemma P D((a, b, c, d), f ).
ˆWe show Theorem 2-(2). It is immediate from the way of construction of d(a)ˆthat if d(a) < d < a, then there is no µC ∈ MC (a) \ 0 that satisﬁes λ(µC , a, d) = 0. IfµC 6∈ MC (a), λ(µC , a, d) = 0 never holds because if it were the case that λ(µC , a, d) = 0,
it must be the case that λ(µC , a, 21 c + 12 d) < 0, a contradiction to µC 6∈ MC (a).
ˆHence, if d(a)< d < a, there is no three-mode equilibrium in a prisoner’s dilemma
P D((a, b, c, d), f ).
ˆ¯
ˆWe show the remaining properties of d(a).
First, we show that d(a)≤ d(a)
< a.
∗
∗
∗
¯
ˆ
¯
Note that d (0, a) = d(a). Therefore, d(a) = supµC ∈MC (a) d (µC , a) ≥ d (0, a) = d(a).
¯ˆˆ¯
If d(a)= d(a),
it is also immediate that d(a)< a because d(a)< a by Corollary¯ˆˆ
1. Consider the other case of d(a) < d(a). Then, d(a) = maxµC ∈MC (a) d∗ (µC , a) =
ˆ < a becaused∗ (µ̂C , a) for some µ̂C ∈ MC (a) \ 0 by Claim 2. Hence, it follows that d(a)∗c < d (µ̂C , a) < a.
¯ ≤ d(a)ˆ < a < b, which we have just established,
Second, from the fact that c < d(a)ˆˆ= b,it immediately follows that lima→c d(a) = c and it also follows that lima→b d(a)¯because lima→b d(a) = b by Corollary 1.
ˆ is strictly increasing. Fix a0 , a00 ∈ (c, b) such that a0 < a00 .
Third, we show that d(a)We will show below that
1. MC (a0 ) ⊆ MC (a00 ), and2. d∗ (µC , a0 ) < d∗ (µC , a00 ) for every µC ∈ MC (a0 ).
Let us show the ﬁrst fact that MC (a0 ) ⊆ MC (a00 ). Recall from Lemma 6-(1)that a0 < a00 implies µ̄C (a0 , d) < µ̄C (a00 , d) for each d ∈ (c, a0 ). This means that¯C (a0 ) = limd↓c µ̄C (a0 , d) ≤ limd↓c µ̄C (a00 , d) = µ̄
¯C (a00 ). Therefore, [0, µ̄¯C (a0 )) ⊆
µ̄
00
0
0
00
¯C (a )) and λ(µC , a ) is deﬁned over µC ∈ [0, µ̄¯C (a )), while λ(µC , a ) is deﬁned over[0, µ̄
000¯¯C (a0 )) and λ(µC , a0 ) < 0.
µC ∈ [0, µ̄C (a )). Now, take µC ∈ MC (a ). Then, µC ∈ [0, µ̄
000¯C (a )) that µC ∈ [0, µ̄¯C (a )) and λ(µC , a00 ) is deﬁned. FurIt follows from µC ∈ [0, µ̄0thermore, if λ(µC , a , d) is deﬁned for d such that µC ∈ [0, µ̄C (a0 , d)], then µ̄C (a0 , d) <µ̄C (a00 , d) guarantees that λ(µC , a00 , d) is also deﬁned for the d, and Lemma 6-(2)58implies that λ(µC , a0 , d) > λ(µC , a00 , d). Hence, λ(µC , a0 ) = limd↓c λ(µC , a0 , d) ≥limd↓c λ(µC , a00 , d) = λ(µC , a00 ). This leads us to conclude that 0 > λ(µC , a0 ) ≥λ(µC , a00 ). Hence µC ∈ MC (a00 ).
Let us show the second fact that d∗ (µC , a0 ) < d∗ (µC , a00 ) for every µC ∈ MC (a0 ). By
way of construction, it holds that λ(µC , a0 , d∗ (µC , a0 )) = 0 and λ(µC , a00 , d∗ (µC , a00 )) =0. It follows from Lemma 6-(2) that a0 < a00 implies 0 = λ(µC , a0 , d∗ (µC , a0 )) >λ(µC , a00 , d∗ (µC , a0 )). Then, it follows from Lemma 6-(2) that λ(µC , a00 , d∗ (µC , a00 )) =0 > λ(µC , a00 , d∗ (µC , a0 )) implies d∗ (µC , a00 ) > d∗ (µC , a0 ).
∗0ˆ 0 ) = supThen, it follows from the second fact that d(aµC ∈MC (a0 ) d (µC , a ) =
∗
0
∗
00
0
d (µ̂C , a ) < d (µ̂C , a ) where µ̂C is the µ̂C for a = a stated in Claim 2. It follows fromthe ﬁrst fact that d∗ (µ̂C , a00 ) ≤ supµC ∈MC (a0 ) d∗ (µC , a00 ) ≤ supµC ∈MC (a00 ) d∗ (µC , a00 ) =ˆ 00 ). These establish that d(aˆ 0 ) < d(aˆ 00 ).
d(aˆFinally, we show that d(a)is continuous. Fix a ∈ (c, b). Consider a sequencen∞nˆ n ) = d(a).
ˆ{a }n=1 such that a ≤ a and limn→∞ an = a. We show that limn→∞ d(an∞Without loss of generality, we consider an increasing sequence {a }n=1 ; that is, an ≤ˆ n ) ≤ d(a)ˆˆˆ n ) exists and limn→∞ d(abecause d(ã)
is strictly
an+1 . Then, limn→∞ d(ann+1nˆ ) ≤ d(aˆˆˆ ) ≥ d(a).
ˆincreasing so that d(a) ≤ d(a).
We show that limn→∞ d(aLet µ̂C∗∗ˆ
achieve d(a) = maxµC ∈MC (a) d (µC , a) = d (µ̂C , a) as in Claim 2. Note that λ(µ̂C , a) <0 because µ̂C ∈ MC (a). Therefore, there exists d ∈ (c, a) such that λ(µ̂C , a, d) is
deﬁned and λ(µ̂C , a, d) < 0. Then, there exists N such that λ(µ̂C , an , d) < 0 for
every n ≥ N because limn→∞ an = a and λ(µ̂C , ã, d) is continuous in ã. This meansthat λ(µ̂C , an ) < 0 for every n ≥ N so that µ̂C ∈ MC (an ) for every n ≥ N . Then,d∗ (µ̂C , an ) is deﬁned for every n ≥ N . Note that d∗ (µ̃C , ã) is continuous in (µ̃C , ã),because d∗ (µ̃C , ã) is a unique solution to λ(µ̃C , ã, d) = 0 and λ(µ̃C , ã, d) is continuousˆ n) =in (µ̃C , ã, d). Therefore, limn→∞ d∗ (µ̂C , an ) = d∗ (µ̂C , a). Hence, limn→∞ d(aˆlimn→∞ supµC ∈MC (an ) d∗ (µC , an ) ≥ limn→∞ d∗ (µ̂C , an ) = d∗ (µ̂C , a) = d(a).
n∞nnConsider a sequence {a }n=1 such that a ≥ a and limn→∞ a = a. Without lossnn+1 . Then,of generality, we consider a decreasing sequence {an }∞n=1 ; that is, a ≥ aˆ n ) exists and limn→∞ d(aˆ n ) ≥ d(a)ˆ because d(ã)ˆ is strictly increasing so thatlimn→∞ d(ann+1ˆ ) ≥ d(aˆˆˆ n ) = d(a).
ˆd(a) ≥ d(a).
We show by contradiction that limn→∞ d(a
Supn
n
n
∗
n
ˆ
ˆ
ˆ
pose that limn→∞ d(a ) > d(a). Let µ̂C achieve d(a ) = maxµC ∈MC (an ) d (µC , a ) =d∗ (µ̂nC , an ) for a = an in Claim 2. Then, a sequence {µ̂nC }∞n=1 has a convergent subsequence because each µ̂nC is in a compact set [0, 1]. Without loss of generality, we asn
sume that {µ̂nC }∞n=1 itself is convergent, and we denote limn→∞ µ̂C = µ̂C . Then, µ̂C ∈
∗
n
n
n
∗
n
n
[0, µ̄C (a, limn→∞ d (µ̂C , a))] because µ̂C ∈ [0, µ̄C (a , d (µ̂C , a ))], limn→∞ µ̂nC = µ̂C ,˜ is continuous in (ã, d).
˜ Therefore, λ(µ̂C , a, limn→∞ d∗ (µ̂n , a))limn→∞ an = a, and µ̄C (ã, d)C
is deﬁned. It must be the case that λ(µ̂C , a, limn→∞ d∗ (µ̂nC , a)) ≤ 0 because if itwere the case that λ(µ̂C , a, limn→∞ d∗ (µ̂nC , a)) > 0, there must exist N such that
1
N
N
∗ n
N ∗ N
λ(µ̂N
C , a , d (µ̂C , a)) > 2 λ(µ̂C , a , limn→∞ d (µ̂C , a)) > 0, a contradiction to theNN∗N
condition that λ(µ̂C , a , d (µ̂C , a)) = 0. We supposed that limn→∞ d∗ (µ̂nC , an ) =ˆ n ) > d(a).
ˆˆlimn→∞ d(aTherefore, λ(µ̂C , a, limn→∞ d∗ (µ̂nC , a)) ≤ 0 implies λ(µ̂C , a, d(a))<∗
0 by Lemma 6-(2). Hence, λ(µ̂C , a) < 0 so that µ̂C ∈ MC (a). Therefore, d (µ̂C , a)ˆ n ) = limn→∞ d∗ (µ̂n , an ) = d∗ (µ̂C , a) because, as we noted
is deﬁned. Then, limn→∞ d(aC∗∗ˆ = supabove, d (µ̃C , ã) is continuous in (µ̃C , ã). This means that d(a)µC ∈MC (a) d (µC , a) ≥ˆ n ) > d(a).
ˆd∗ (µ̂C , a) = limn→∞ d(aThis is a contradiction.
[ Step 3] We prove two claims presented in Step 1.
First, we prove Claim 1. Consider the behavior of µ̄C (a, d) with respect to d. Recall
from Lemma 6-(1) that µ̄C (a, d) is strictly decreasing in d ∈ (c, a). To ﬁnd its limit
as d goes to a, recall from the proof of Lemma 4 (Step 5) that µ̄C (a, d) is a unique59

solution µ̄C to the equation (28), which is rearranged into(b − d a − d)µ̄C = φ µ̄C β ≥ µ̄C
−
.
b−c
b−c
This means that as d goes to a, µ̄C (a, d) approaches(b − a a − a)−= φ(β ≥ β ∗ ).
lim µ̄C (a, d) = φ (lim µ̄C (a, d))β ≥ (lim µ̄C (a, d))
d↑a
d↑a
d↑a
b−c
b−c
¯C (a), there exists someHence, if the given µC ∈ MC (a) \ 0 is φ(β ≥ β ∗ ) < µC < µ̄
d00 ∈ (c, a) such that µC = µ̄C (a, d00 ) because µ̄C (a, d) is continuous in d by the proof
of Lemma 4 (Step 5). Recall from the proof of Theorem 1 that λ(µ̄C (a, d00 ), a, d00 ) > 0.
Therefore, λ(µC , a, d00 ) > 0.
Consider µC in the remaining case of 0 < µC ≤ φ(β ≥ β ∗ ). In this case, it holds forevery d ∈ (c, a) that µC ≤ µ̄C (a, d) so that λ(µC , a, d) is deﬁned. Recall from Lemma
4 that µ̂(µC , a, d) lies in ∆0 so that a−dd−c µ̂CDD (µC , a, d) ≥ µ̂DDD (µC , a, d) holds for anyd ∈ (c, a). This means that limd↑a µ̂DDD (µC , a, d) = 0. On the other hand, note by
Lemma 2 that
ψDDD (µ̂(µC , a, d))∗= φ(TDDD(µ̂(µC , a, d)))= φ(T ∩ {(α, β)|β ≤ H(α|µ̂(µC , a, d)), β ≤ β ∗ })µ̂DDD (µC , a, d)(d − c) − µ̂CDD (µC , a, d)(a − d)µ̂DDD (µC , a, d)
α + β∗ +, β ≤ β ∗ })= φ(T ∩ {(α, β)|β ≤µ̂C (µC , a, d)µ̂C (µC , a, d)(b − c)µ̂DDD (µC , a, d)µ̂DDD (µC , a, d)(d − c) − µ̂CDD (µC , a, d)(a − d)= φ(T ∩ {(α, β)|β ≤α + β∗ +, β ≤ β ∗ })µC
µC (b − c)where the last equality follows from µC = µ̂C (µC , a, d). This means that limd↑a ψDDD (µ̂(µC , a, d)) =φ(β ≤ β ∗ ), because limd↑a µ̂DDD (µC , a, d) = 0 guarantees that
lim

[ µ̂

d↑a

DDD (µC , a, d)µC

α + β∗ +µ̂DDD (µC , a, d)(d − c) − µ̂CDD (µC , a, d)(a − d) ]= β∗.
µC (b − c)Hencelim λ(µC , a, d) = lim ψDDD (µ̂(µC , a, d)) − lim µ̂DDD (µC , a, d) = φ(β ≤ β ∗ ) > 0.
d↑a

d↑a

d↑a

Therefore, there exists d00 close enough to a such that λ(µC , a, d00 ) > 0.
¯
Second, we prove Claim 2. Recall from Step 2 that we noticed that d∗ (0, a) = d(a)∗∗∗
ˆ = sup¯
and we showed that d(a)µC ∈MC (a) d (µC , a) ≥ d (0, a) = d(a). If supµC ∈MC (a) d (µC , a) =¯
ˆ = d(a).
¯d(a),
then µ̂C = 0 achieves maxµC ∈MC (a) d∗ (µC , a) and d(a)∗¯Consider the other case of supd (µC , a) > d(a). Then, there must existµC ∈MC (a)n∗ n∗{µnC }∞
n=1 such that µC ∈ MC (a) and limn→∞ d (µC , a) = supµC ∈I d (µC , a). Thensequence contains a convergent subsequence because each µC is a probability andcontained in a compact set [0, 1]. Without loss of generality, we assume that {µnC }∞n=1
itself is convergent, and we denote µ̂C = limn→∞ µnC .
¯ implies that there exists N such that d∗ (µn , a) >Note that limn→∞ d∗ (µnC , a) > d(a)C¯¯d(a) for any n ≥ N . Therefore, [0, µ̄C (a, d∗ (µnC , a))] ⊂ [0, µ̄C (a, d(a))]
for any n ≥ N by¯
Lemma 6-(1). This guarantees that µ̂C ∈ [0, µ̄C (a, d(a))]
because µnC ∈ [0, µ̄C (a, d∗ (µnC , a))]¯for each n. This means that λ(µ̂C , a, d(a))
is deﬁned. Therefore, λ(µ̂C , a) is deﬁned.

60

Now, we show that λ(µ̂C , a) < 0; that is, µ̂C ∈ MC (a) so that µ̂C achieves
maxµC ∈MC (a) d∗ (µC , a). Suppose to the contrary that λ(µ̂C , a) ≥ 0. Then, it must
be the case that λ(µ̂C , a, d) > 0 for every d ∈ (c, a) for which λ(µ̂C , a, d) is deﬁned.
To see this, suppose that λ(µ̂C , a, d) is deﬁned and λ(µ̂C , a, d) ≤ 0 for some d ∈ (c, a).
Then, µ̂C ∈ [0, µ̄C (a, d)] implies µ̂C ∈ [0, µ̄C (a, 21 c + 21 d)] by Lemma 6-(1) so thatλ(µ̂C , a, 21 c + 12 d) is deﬁned, and λ(µ̂C , a, d) ≤ 0 implies λ(µ̂C , a, 21 c + 12 d) < 0 by
Lemma 6-(2). This means λ(µ̂C , a) < 0, a contradiction. Hence, λ(µ̂C , a, d) > 0 for¯every d ∈ (c, a) for which λ(µ̂C , a, d) is deﬁned. Take d = d(a),for which we showed¯¯
above that λ(µ̂C , a, d(a))
is deﬁned. Then, λ(µ̂C , a, d(a))> 0. Then, there exists
0
1
∗ N0
N0
¯
¯
N 0 such that λ(µNC , a, d (µC , a)) > λ(µC , a, d(a)) > 2 λ(µ̂C , a, d(a)) > 0 because¯
˜ is continuous in µ̃C andlimn→∞ d∗ (µnC , a) > d(a),µ̂C = limn→∞ µnC , and λ(µ̃C , a, d)0∗ N0
˜strictly increasing in d. This contradicts the condition that λ(µNC , a, d (µC , a)) = 0.
Hence, λ(µ̂C , a) < 0 must hold. (Q.E.D.)
Proof of Lemma 7
Lemma 7-(2) is immediate because β ∗ = b−ab−c . As for Lemma 7-(1), we show that¯α∗ (µmin (a, d)) is strictly increasing in a for a range of (a, d) with c < d < d(a).
Theremaining part of Lemma 7-(1) is proved similarly.
Fix d ∈ (c, b) and consider a0 , a00 such that a0 < a00 . Suppose that (a0 , d) lies
¯ 0 ). Then, by Corollary 1, (a00 , d) also lies in a range
in a range with c < d < d(a¯ 0 ) < d(a¯ 00 ). Therefore, we have µmin (a0 , d) < µmin (a00 , d) by Thewith c < d < d(aCCorem 3. We show that α∗ (µmin (a0 , d)) < α∗ (µmin (a00 , d)). Suppose to the contrary
0
that α∗ (µmin (a0 , d)) ≥ α∗ (µmin (a00 , d)). Note by Lemma 7-(2) that β ∗ (a0 ) = b−a
b−c >
b−a00
∗ 00
0
00
min
0
b−c = β (a ) when a < a . Then, together with the fact that µCDD (a , d) =
min
0
0
min
00
min
00
00
ψCDD (µ (a , d), a , d) and µCDD (a , d) = ψCDD (µ (a , d), a , d) in the equilibria,α∗ (µmin (a0 , d)) ≥ α∗ (µmin (a00 , d)) and β ∗ (a0 ) > β ∗ (a00 ) imply that
0
min 0
µmin
(a , d), a0 , d)
CDD (a , d) = ψCDD (µ∗= φ(TCDD(µmin (a0 , d), a0 , d))= φ(T ∩ {(α, β)|α > α∗ (µmin (a0 , d)), β > β ∗ (a0 )})< φ(T ∩ {(α, β)|α > α∗ (µmin (a00 , d)), β > β ∗ (a00 )})∗= φ(TCDD(µmin (a00 , d), a00 , d))= ψCDD (µmin (a00 , d), a00 , d)00
= µminCDD (a , d).
0
min 00min
0min
00Then, µmin
C (a , d) < µC (a , d) and µCDD (a , d) < µCDD (a , d) imply that
0
min 0
min
0
min 00
min
00
min
00
µmin
DDD (a , d) = 1−µC (a , d)−µCDD (a , d) > 1−µC (a , d)−µCDD (a , d) = µDDD (a , d).
µmin (a0 ,d)µmin (a00 ,d)

CDD
CDD
Hence, µmin
< µmin. Then, it must hold for the expression (4) of α∗ that(a0 ,d)(a00 ,d)DDDα∗ (µmin (a0 , d)) =

DDD

0
00
a00 − d µmina0 − d µmin
CDD (a , d) d − c
CDD (a , d) d − c<= α∗ (µmin (a00 , d))−−min (a00 , d) b − c
0 , d) b − c
b − c µmin
b
−
c
(a
µ
DDD
DDD

when a0 < a00 . This contradicts the supposition α∗ (µmin (a0 , d)) ≥ α∗ (µmin (a00 , d)).
(Q.E.D.)
Proof of Theorem 5∗ min (a, d)) ≤We show that µmin
DDD (a, d) is strictly decreasing in a for a range of (a, d) with α (µ¯β ∗ and c < d < d(a).
The remaining part of Theorem 5 is proved similarly.
61

Fix d ∈ (c, b) and consider a0 , a00 such that a0 < a00 . Suppose that (a0 , d) and (a00 , d)
¯ 0 ) < d(a¯ 00 ) and α∗ (µmin (a0 , d)) < α∗ (µmin (a00 , d)) ≤lie in a range with c < d < d(a∗00∗0¯ 0 ) < d(a¯ 00 ) and Lemma 7 guaranβ (a ) < β (a ) where Corollary 1 guarantees d(atees α∗ (µmin (a0 , d)) < α∗ (µmin (a00 , d)) and β ∗ (a00 ) < β ∗ (a0 ). Note by Lemma 2 thatα∗ (µmin (a0 , d)) < β ∗ (a0 ) and α∗ (µmin (a00 , d)) ≤ β ∗ (a00 ) means that∗{(α, β) ∈ T |β < β ∗ (a0 )} = TC∗ (µmin (a0 , d), a0 , d) ∪ TDDD(µmin (a0 , d), a0 , d)∗{(α, β) ∈ T |β < β ∗ (a00 )} = TC∗ (µmin (a00 , d), a00 , d) ∪ TDDD(µmin (a00 , d), a00 , d).

Here, β ∗ (a00 ) < β ∗ (a0 ) implies thatφ({(α, β) ∈ T |β < β ∗ (a0 )}) > φ({(α, β) ∈ T |β < β ∗ (a00 )})so that∗φ(TC∗ (µmin (a0 , d), a0 , d)) + φ(TDDD(µmin (a0 , d), a0 , d))∗> φ(TC∗ (µmin (a00 , d), a00 , d)) + φ(TDDD(µmin (a00 , d), a00 , d)).

¯ 0 ) < d(a¯ 00 ), Theorem 3 applies and we have
In a range with c < d < d(a0
min 00∗min 00φ(TC∗ (µmin (a0 , d), a0 , d)) = µmin
(a , d), a00 , d)).
C (a , d) < µC (a , d) = φ(TC (µTherefore,
0
min
00
∗
min 0
∗
µmin
(a , d), a0 , d)) − φ(TDDD(µmin (a00 , d), a00 , d))
DDD (a , d) − µDDD (a , d) = φ(TDDD (µ> φ(TC∗ (µmin (a00 , d), a00 , d)) − φ(TC∗ (µmin (a0 , d), a0 , d))> 0.
(Q.E.D.)
Proof of Lemma 8
The condition (21) that a type (α, β) prefers C to CDD and DDD under a belief µ is
rewritten asµCDD(a − d) > γ(d − (c − α(b − c))) + (max[a, b − β(b − c)] − a).
µC
Consider (α, β), (α0 , β 0 ) such thatγ(d−(c−α(b−c)))+(max[a, b−β(b−c)]−a) < γ(d−(c−α0 (b−c)))+(max[a, b−β 0 (b−c)]−a).
Then, ifγ(d − (c − α0 (b − c))) + (max[a, b − β 0 (b − c)] − a) <µCDD(a − d),µCγ(d − (c − α(b − c))) + (max[a, b − β(b − c)] − a) <µCDD(a − d).
µC

then
Furthermore, we can take µCDD
µC such thatγ(d−(c−α(b−c)))+(max[a, b−β(b−c)]−a) <µCDD(a−d) < γ(d−(c−α0 (b−c)))+(max[a, b−β 0 (b−c)]−a).
µCµDDD
when we compare
because we can choose any positive value for µCDD
µC given γ = µC00
the incentives to lead between (α, β) and (α , β ). These mean that (α, β) has a stronger

62

incentive to lead under the belief ratio γ than (α0 , β 0 ). Therefore, a type (α, β) has thestrongest incentive to lead under the belief ratio γ if and only if (α, β) is a solution tomin γ(d − (c − α̃(b − c))) + (max[a, b − β̃(b − c)] − a).

(α̃,β̃)∈T

Consider the minimum of γ(d − (c − α̃(b − c))) + (max[a, b − β̃(b − c)] − a) in a subset of
T with β̃ ≥ β ∗ . Note that max[a, b − β̃(b − c)] − a = 0 if β̃ ≥ β ∗ so that the objective
for the minimization reduces to γ(d − (c − α̃(b − c))). The minimum over the set of(α̃, β̃) with α̃ ≥ β̃ ≥ β ∗ is attained at (β ∗ , β ∗ ). This means that the minimum ofγ(d − (c − α̃(b − c))) + (max[a, b − β̃(b − c)] − a) over T is attained by (α, β) in a subset
of T with β̃ ≤ β ∗ . Note that max[a, b − β̃(b − c)] − a = b − β̃(b − c) − a if β̃ ≤ β ∗ sothat the objective for the minimization reduces toγ(d − (c − α̃(b − c))) + (b − β̃(b − c) − a) = (b − c)(γ α̃ − β̃) + γ(d − c) + (b − a).
The minimum of this objective over the set of (α̃, β̃) with 0 ≤ β̃ ≤ β ∗ and β̃ ≤ α̃ isattained by (α, β) = (0, 0) if γ > 1, (β ∗ , β ∗ ) if γ < 1, and {(α, β) = t(0, 0) + (1 −t)(β ∗ , β ∗ )|0 ≤ t ≤ 1} if γ = 1. (Q.E.D.)
Proof of Lemma 9
Lemmas 9-(1) and (2) are restatements of Lemma 7. We prove Lemma 9-(3). In
particular, we prove that γ ∗ (µmin (a, d)) is strictly decreasing in a for a range of (a, d)¯
with c < d < d(a).
The remaining parts of Lemma 9-(3) are proved similarly.
Fix d ∈ (c, b) and consider a0 , a00 such that a0 < a00 . Suppose that (a0 , d) and (a00 , d)
¯ 0 ) < d(a¯ 00 ) where Corollary 1 guarantees d(a¯ 0 ) < d(a¯ 00 ).
lie in a range with c < d < d(a

00
0
µmin
µmin
∗ min (a00 , d)). Suppose
DDD (a ,d)DDD (a ,d)
0 ,d) > µmin (a00 ,d) = γ (µ
µmin
(a
C
C
00
µmin (a0 ,d)
µmin
min 0
DDD (a ,d)to the contrary that µDDD
min (a0 ,d) ≤ µmin (a00 ,d) . Note by Theorem 3 that µC (a , d) <
C
C
0
µmin
00
0
00
DDD (a ,d)
¯ 0
¯ 00
µmin
C (a , d) when a < a for a range with c < d < d(a ) < d(a ). Then, µmin (a0 ,d) ≤
C
00
µmin
min
0
min
00
DDD (a ,d)implies that µDDD (a , d) < µDDD (a , d). On the other hand, note byµmin (a00 ,d)

We show that γ ∗ (µmin (a0 , d)) =C

Lemma 2 that
0
min 0
µmin
(a , d), a0 , d), β ≤ β ∗ (a0 )})DDD (a , d) = φ(T ∩ {(α, β)|β ≤ H(α|µ00
min 00µmin(a , d), a00 , d), β ≤ β ∗ (a00 )})DDD (a , d) = φ(T ∩ {(α, β)|β ≤ H(α|µwhere
H(α|µmin (a0 , d), a0 , d) =0µmin
DDD (a , d)(α − α∗ (µmin (a0 , d), a0 , d)) + β ∗ (a0 )0 , d)µmin(aC

H(α|µmin (a00 , d), a00 , d) =00
µmin
DDD (a , d)(α − α∗ (µmin (a00 , d), a00 , d)) + β ∗ (a00 )00 , d)
µmin
(a
C

0

00

b−a
∗ 00
∗ min (a0 , d), a0 , d) <and β ∗ (a0 ) = b−ab−c , β (a ) = b−c . By Lemma 9-(1) and (2), α (µα∗ (µmin (a00 , d), a00 , d) and β ∗ (a0 ) > β ∗ (a00 ) when a0 < a00 for a range with c < d <min (a0 ,d)µmin (a00 ,d)¯ 0 ) < d(a¯ 00 ). Then, together with µDDD
d(a
≤ DDD
, we must havemin 0
min 00µC (a ,d)µC (a ,d){(α, β)|β ≤ H(α|µmin (a0 , d), a0 , d), β ≤ β ∗ (a0 )} ) {(α, β)|β ≤ H(α|µmin (a00 , d), a00 , d), β ≤ β ∗ (a00 )}0min
00
so that µmin
DDD (a , d) > µDDD (a , d). This is a contradiction. (Q.E.D.)

Proof of Theorem 763

We prove Theorem 7-(2). Theorem 7-(1) is proved similarly.
¯ 0 ). Then,
Consider (a, d) and (a0 , d0 ) such that a > a0 , d < d0 , and c < d0 < d(a0¯¯¯(a, d) also lies in a range with c < d < d(a) because d(a ) < d(a) by Corollary 1 whena > a0 . Therefore, both P D((a, b, c, d), f ) and P D((a0 , b, c, d0 ), f ) admit a three-mode
equilibrium. Recall from Lemma 2 thatTC∗ (µmin (a, d), a, d) = {(α, β)|β ≥ H(α|µmin (a, d), a, d), α ≤ α∗ (µmin (a, d), a, d)}TC∗ (µmin (a0 , d0 ), a0 , d0 ) = {(α, β)|β ≥ H(α|µmin (a0 , d0 ), a0 , d0 ), α ≤ α∗ (µmin (a, d), a, d)}
where
H(α|µmin (a, d), a, d) =H(α|µmin (a00 , d), a00 , d) =µmin
DDD (a, d)(α − α∗ (µmin (a, d), a, d)) + β ∗ (a)minµC (a, d)0 0µmin
DDD (a , d )(α − α∗ (µmin (a0 , d0 ), a0 , d0 )) + β ∗ (a0 ).
min00µC (a , d )We show that TC∗ (µmin (a, d), a, d) is a leadership pattern of a more equity concerned
leader than TC∗ (µmin (a0 , d0 ), a0 , d0 ).
To compare TC∗ (µmin (a, d), a, d) with TC∗ (µmin (a0 , d0 ), a0 , d0 ), consider two lines β =H(α|µmin (a, d), a, d) and β = H(α|µmin (a0 , d0 ), a0 , d0 ), which deﬁne the boundaries of¯TC∗ (µmin (a, d), a, d) and TC∗ (µmin (a0 , d0 ), a0 , d0 ) respectively. Given that c < d < d(a)¯ 0 ), Lemma 9 applies and we have α∗ (µmin (a, d), a, d) > α∗ (µmin (a0 , d0 ), a0 , d0 ),and c < d0 < d(aµmin (a,d)µmin (a0 ,d0 )∗ min (a0 , d0 )). Then,DDD
β ∗ (a) < β ∗ (a0 ), and γ ∗ (µmin (a, d)) = µDDD
min (a,d) < µmin (a0 ,d0 ) = γ (µCC

γ ∗ (µmin (a, d)) < γ ∗ (µmin (a0 , d0 )) implies that there exists a unique (α̃, β̃) in R2 thatsatisﬁes β̃ = H(α̃|µmin (a, d), a, d) and β̃ = H(α̃|µmin (a00 , d), a00 , d) simultaneously. Furthermore, α∗ (µmin (a, d), a, d) > α∗ (µmin (a0 , d0 ), a0 , d0 ) and β ∗ (a) < β ∗ (a0 ) imply that(α∗ (µmin (a, d), a, d), β ∗ (a)) is located south east of (α∗ (µmin (a0 , d0 ), a0 , d0 ), β ∗ (a0 )). Thisin turn implies that α̃ < α∗ (µmin (a, d), a, d), β̃ < β ∗ (a), α̃ < α∗ (µmin (a0 , d0 ), a0 , d0 ), andβ̃ < β ∗ (a0 ).
The three types (α̃, β̃), (α∗ (µmin (a, d), a, d), β ∗ (a)), and (α∗ (µmin (a0 , d0 ), a0 , d0 ), β ∗ (a0 ))identiﬁed above may or may not be included in the type space T . One of the possibilities is a case in which all of them are in T . Figure 15 demonstrates this case.
Then, (α∗ (µmin (a, d), a, d), β ∗ (a)) ∈ T means that when we deﬁne (α̂, β̂) by α̂ =α∗ (µmin (a, d), a, d) and β̂ = max{β|(α∗ (µmin (a, d), a, d), β) ∈ T }, it holds that (α̂, β̂) ∈TC∗ (µmin (a, d), a, d) and that α ≤ α̂ and β ≤ β̂ for any (α, β) ∈ TC∗ (µmin (a, d), a, d); thatis, the type (α̂, β̂) is the most inequity concerned in both envy and guilt among all theleader types in TC∗ (µmin (a, d), a, d). Similarly, (α̂0 , β̂ 0 ) such that α̂0 = α∗ (µmin (a0 , d0 ), a0 , d0 )and β̂ 0 = max{β 0 |(α∗ (µmin (a0 , d0 ), a0 , d0 ), β 0 ) ∈ T } is the most inequity concerned inboth envy and guilt among all the leader types in TC∗ (µmin (a0 , d0 ), a0 , d0 ). Furthermore,(α̂0 , β̂ 0 ) ≤ (α̂, β̂) holds by α∗ (µmin (a, d), a, d) > α∗ (µmin (a0 , d0 ), a0 , d0 ). Hence, the condition (1) for TC∗ (µmin (a, d), a, d) to be a leadership pattern of more equity concerned
leader than TC∗ (µmin (a0 , d0 ), a0 , d0 ) is satisﬁed.
Note also in this case that TC∗ (µmin (a0 , d0 ), a0 , d0 ) \ TC∗ (µmin (a, d), a, d) 6= ∅ and∗TC (µmin (a, d), a, d) \ TC∗ (µmin (a0 , d0 ), a0 , d0 ) 6= ∅. Take (α0 , β 0 ) ∈ TC∗ (µmin (a0 , d0 ), a0 , d0 ) \TC∗ (µmin (a, d), a, d) and (α, β) ∈ TC∗ (µmin (a, d), a, d) \ TC∗ (µmin (a0 , d0 ), a0 , d0 ) arbitrarily. Then, it must hold for (α0 , β 0 ) that β 0 ≥ H(α0 |µmin (a0 , d0 ), a0 , d0 ) and β 0 <H(α0 |µmin (a, d), a, d). This means that α0 < α̃ and β 0 < β̃. On the other hand, it
must hold for (α, β) that β ≥ H(α|µmin (a, d), a, d) and β < H(α|µmin (a0 , d0 ), a0 , d0 ).
This means that α̃ < α and β̃ < β. Hence, we have (α0 , β 0 ) ≤ (α̃, β̃) ≤ (α, β). Thus,
the condition (2) for TC∗ (µmin (a, d), a, d) to be a leadership pattern of more equity
concerned leader than TC∗ (µmin (a0 , d0 ), a0 , d0 ) is also satisﬁed.
64

1

0

1
Figure 15: ordered-leader-set

Similarly, we can show that TC∗ (µmin (a, d), a, d) is a leadership pattern of a more equity concerned leader than TC∗ (µmin (a0 , d0 ), a0 , d0 ) in all the remaining cases of relationsof the three types (α̂, β̂), (α∗ (µmin (a, d), a, d), β ∗ (a)), and (α∗ (µmin (a0 , d0 ), a0 , d0 ), β ∗ (a0 ))to the type set T . (Q.E.D.)
Proof of Theorem 8
We show Theorem 8-(2). Theorem 8-(1) is proved similarly. Fix d ∈ (c, b). Notethat there exists a three-mode equilibrium with the minimum leadership distribution¯ d¯−1 (d)) < d(a)¯ by Corollary 1.
µmin (a, d) for any a ∈ (d¯−1 (d), b) because c < d = d(
∗
min
−1
¯
Suppose that γ (µ (ã, d)) < 1 for any a ∈ (d (d), b). Then, the type of thestrongest incentive to lead in the three-mode equilibrium distribution µmin (a, d) is the(β ∗ , β ∗ )-type for any a ∈ (d¯−1 (d), b) by Lemma 8. Hence, Theorem 8-(2) holds for¯−1amin
L (d) = d (d).
Suppose that γ ∗ (µmin (a0 , d)) ≥ 1 for some a0 ∈ (d¯−1 (d), b). Then, γ ∗ (µmin ( 21 d¯−1 (d)+min
1 0∗ min (a, d)) = µDDD (a,d) for a ∈ (a0 , b). Theo2 a , d)) > 1 by Lemma 9. Examine γ (µµmin
C (a,d)min (a0 , d) for any a ∈ (a0 , b). On the other hand,rem 3 guarantees that µmin
(a,
d)
>
µ
C
C
∗
TDDD
(µmin (a, d), a, d) ⊆ {(α̃, β̃)|β̃ ≤ β ∗ (a)} by Lemma 2. Note that lima↑b β ∗ (a) =min∗min (a, d), a, d)) ≤ lim
lima↑b b−aa↑b φ({(α̃, β̃)|β̃ ≤b−c = 0. Hence, lima↑b µDDD (a, d) = lima↑b φ(TDDD (µ∗0min
β (a)}) = 0. This means that there exists a ∈ (a , b) such that µDDD (a, d) <
min
min 0
0
∗ min (a, d)) = µDDD (a,d) < µC (a ,d) = 1. Thus, we have a0 , aµmin
min (a0 ,d)C (a , d). Then, γ (µ
µmin
(a,d)
µ
C
C
with d¯−1 (d) < a0 < a < b, for which γ ∗ (µmin ( 12 d¯−1 (d) + 21 a0 , d)) > 1 > γ ∗ (µmin (a, d)).
∗ min (a, d)) ≥ 1}, it immediately¯−1Then, when we set amin
L (d) ≡ sup{a ∈ (d (d), b)|γ (µ−1minfollows that d¯ (d) < aL (d) < b, and it holds by Lemma 9 that γ ∗ (µmin (ã, d)) > 1 for∗ min (ã, d)) < 1 for ã ∈ (amin (d), b). This means thatã ∈ (d¯−1 (d), amin
L (d)), while γ (µLthe type of the strongest incentive to lead in the three-mode equilibrium distribution∗∗minµmin (a, d) is Materialist if a < amin
L (d) and the type (β , β ) if a > aL (d).
0 00
We show Theorem 8-(3). We show that amin
L (d) is increasing. Fix d , d such that
0
00
min
0
min
00
c < d < d < b. We show that aL (d ) ≤ aL (d ). Suppose to the contrary that0
min 00min 0∗ min (a, d0 )) ≥
¯−1 0
amin
L (d ) > aL (d ). Recall that we set aL (d ) = sup{a ∈ (d (d ), b)|γ (µ
65

0
¯−1 0
1} if the set on the right-hand side is not empty, and we set amin
L (d ) = d (d ) oth00
min 0
min 00erwise. amin
L (d ) is parallel. Then, the supposition aL (d ) > aL (d ) implies that
min
0
−1
00
∗
min
0
¯
in the deﬁnition of aL (d ) , even {a ∈ (d (d ), b)|γ (µ (a, d )) ≥ 1} is not empty0∗ min (a, d0 )) ≥ 1} because d¯−1 (d0 ) < d¯−1 (d00 )¯−1 00and aminL (d ) = sup{a ∈ (d (d ), b)|γ (µby Corollary 1. Note by Lemma 9 that γ ∗ (µmin (a, d0 )) < γ ∗ (µmin (a, d00 )) for anya ∈ (d¯−1 (d00 ), b). Hence, {a ∈ (d¯−1 (d00 ), b)|γ ∗ (µmin (a, d00 )) ≥ 1} is not empty andsup{a ∈ (d¯−1 (d00 ), b)|γ ∗ (µmin (a, d0 )) ≥ 1} ≤ sup{a ∈ (d¯−1 (d00 ), b)|γ ∗ (µmin (a, d00 )) ≥001} = amin
L (d ). This is a contradiction.
min
Finally, we show that amax
L (d) ≤ aL (d) for any d ∈ (c, b). Fix d ∈ (c, b) and con−1¯
sider a ∈ (d (d), b). We show that γ ∗ (µmin (a, d)) ≥ γ ∗ (µmax (a, d)). Suppose to thecontrary that γ ∗ (µmin (a, d)) < γ ∗ (µmax (a, d)). Then, it must hold that µmin
DDD (a, d) <µmin (a,d)µmax (a,d)∗ min (a, d)) = DDD
minµmax, γ ∗ (µmax (a, d)) = µDDD
max (a,d) , and µC (a, d) ≤DDD (a, d) because γ (µµmin (a,d)
C

C

∗ min
∗ max
µmax
C (a, d). On the other hand, we claim that α (µC (a, d), a, d) < α (µC (a, d), a, d)
∗
min
∗
max
∗
when γ (µ (a, d)) < γ (µ (a, d)). Suppose to the contrary that α (µminC (a, d), a, d) ≥
∗
max
min
α (µC (a, d), a, d). Then, it must follow by Lemma 2 that µCDD (a, d) ≤ µmax
CDD (a, d).
Then, it follows that
min
min
max
max
max
µmin
DDD (a, d) = 1 − µC (a, d) − µCDD (a, d) ≥ 1 − µC (a, d) − µCDD (a, d) = µDDD (a, d).
max
∗ min
This contradicts µmin
DDD (a, d) < µDDD (a, d). Hence, we must have α (µC (a, d), a, d) <α∗ (µmaxC (a, d), a, d). Note by Lemma 2 thatminµmin
(a, d), a, d), β ≤ β ∗ })
DDD (a, d) = φ(T ∩ {(α, β)|β ≤ H(α|µmaxµmax(a, d), a, d), β ≤ β ∗ })
DDD (a, d) = φ(T ∩ {(α, β)|β ≤ H(α|µ

where
µmin
DDD (a, d)(α − α∗ (µmin (a, d), a, d)) + β ∗
µmin
(a,
d)
C
max (a, d)µ
H(α|µmax (a, d), a, d) = DDD(α − α∗ (µmax (a, d), a, d)) + β ∗ .
µmax
(a,
d)
C
H(α|µmin (a, d), a, d) =µmin (a,d)µmax (a,d)∗ min∗ min (a, d)) < γ ∗ (µmax (a, d)) = DDD
Here, µDDD
min (a,d) = γ (µµmax (a,d) and α (µC (a, d), a, d) <
C

C

min
max
min
α∗ (µmaxC (a, d), a, d) imply that µDDD (a, d) > µDDD (a, d). This contradicts µDDD (a, d) <
max
∗
min
∗
max
µDDD (a, d). Hence, it must be the case that γ (µ (a, d)) ≥ γ (µ (a, d)). This∗ min (a, d)) < 1means that γ ∗ (µmax (a, d)) < 1 for any a with a > amin
L (d) because γ (µmax
∗ max (a, d)) ≥ 1} ≤ amin (d).
¯−1if a > amin
L (d). Hence, aL (d) ≤ sup{a ∈ (d (d), b)|γ (µL(Q.E.D.)

Proof of Theorem 9∂∗∗
We show Theorem 9-(1). First, ∂aφ(β < b−ab−c |α ≤ α) < 0 for any α ∈ [0, ᾱ].
∂∗0¯Therefore, ∂aminα∗ ∈[0,ᾱ] φ(β < b−ab−c |α ≤ α) < 0. Hence, (dml (a)) = 1 − (b −
∂
∗
∗ ∗
¯
c) ∂a
minα∗ ∈[0,ᾱ] φ(β < b−ab−c |α ≤ α) > 0. Second, dml (a) < a because φ(β < β |α ≤∗∗∗α) is continuous in α and φ(β < β |α ≤ α) > 0 for each α ∈ [0, ᾱ] so thatminα∗ ∈[0,ᾱ] φ(β < β ∗ |α∗ ≤ α) > 0. Third, lim d¯ml (a) = c − (b − c) because lim φ(β <a→ca→c
β ∗ |α∗ ≤ α) = φ(β < 1|α∗ ≤ α) = 1. Fourth, lim d¯ml (a) = b because lim φ(β <
a→b

a→b

β ∗ |α∗ ≤ α) = φ(β < 0|α∗ ≤ α) = 0. Finally, there exists āml ∈ (c, b) such thatd¯ml (āml ) = c because lim d¯ml (a) = c − (b − c) < c < lim d¯ml (a) = b and d¯ml (a) is
a→c

a→b

continuous and strictly increasing.
66

We show Theorem 9-(2). Fix (a, d) such that a ∈ (c, ā0ml ) and max(c, d¯ml (a)) <ˆd < d(a).
Suppose to the contrary that there exists a three-mode equilibrium and it
is either an inequity concerned leader pattern or a hybrid leader pattern. Then, thethree-mode equilibrium distribution µ must satisfyb−aµCDD a − d d − c= β ∗ ≤ α∗ (µ) =−b−cµDDD b − cb−cwhere the ﬁrst equality is the deﬁnition of β ∗ and the second equality is by (4). Hence,
µCDD
b−c
¯
it must hold that a−c≤ a−da−c (1 + µDDD ). However, dml (a) < d implies
1µCDD )b−ca−d
a − d(1+>≥a−c
a − c φ(β < β ∗ |α∗ (µ) ≤ α)a−c
µDDD
where the second inequality holds for the reason that, as Lemma 2 shows,φ(β > β ∗ |α∗ (µ) ≤ α)
µCDD
≤
µDDD
φ(β < β ∗ |α∗ (µ) ≤ α)in a three-mode equilibrium. This is a contradiction. Therefore, there should be nothree-mode equilibrium of either an inequity concerned leader pattern or a hybrid
leader pattern. (Q.E.D.)
Proof of Theorem 10[Step 1] Fix a ∈ (c, b). Then, we show that under condition (23), there exists d¯icl (a) ∈¯(c, d(a))
such that β ∗ φ(β > β ∗ ) > µCDD for any three-mode equilibrium distributionµ in any P D((a, b, c, d), f ) with d ∈ (c, d¯icl (a)).
(Proof)CDD
when we consider all the three-mode equilibrium
Consider the supremum of µµDDD¯
distributions µ in P D((a, b, c, d), f ) over d ∈ (c, d(a)).
Then, it is ﬁnite. (We postpone
the proof of this fact to Step 3 below.) This fact implies that
[
(
)]
µCDD
lim µCDD − φ α >(1 − β ∗ ) and β > β ∗
d↓c
µDDD
)]
[
(
µCDD
= lim φ(α > α∗ (µ) and β > β ∗ ) − φ α >(1 − β ∗ ) and β > β ∗
d↓c
µDDD
(
)]
[ (
)
)
(
d
−
c
µ
µCDD
µCDD
CDD
= lim φ α >(1 − β ∗ ) −1+
and β > β ∗ − φ α >(1 − β ∗ ) and β > β ∗
d↓c
µDDD
b−c
µDDD
µDDD
=0
where the ﬁrst equality follows from Lemma 2 and the second equality follows from¯
the expression (4) of α∗ (µ). Then, there exists d¯icl (a) ∈ (c, d(a))
such that
(
)
(
)
µCDD
β ∗ φ(β > β ∗ )∗∗µCDD −φ α >(1−β ∗ ) and β > β ∗ < β ∗ φ(β > β ∗ )−φ α >
(1−β
)
and
β
>
β
µDDD
1 − φ(β > β ∗ )for any d ∈ (c, d¯icl (a)), because the condition (23) implies that the right-hand side of
the above inequality is a positive number. Then,
(
)
(
)
µCDD
β ∗ φ(β > β ∗ )µCDD −φ α >(1−β ∗ ) and β > β ∗ < β ∗ φ(β > β ∗ )−φ α >(1−β ∗ ) and β > β ∗
µDDD
µCDD
for any d ∈ (c, d¯icl (a)), because µCDD < 1 − φ(β > β ∗ ) by Lemma 2. This inequality
implies that β ∗ φ(β > β ∗ ) > µCDD , because)(x(1 − β ∗ ) and β > β ∗x−φ α>µDDD67

is strictly increasing in x.
[Step 2] Fix a ∈ (c, b). Take a three-mode equilibrium distribution µ in P D((a, b, c, d), f )with d ∈ (c, d¯icl (a)). Consider the type (α, β) = (0, 0). Then,U(0,0) (DDD, µ) − U(0,0) (C, µ) = (µC b + µCDD d + µDDD d) − (µC a + µCDD a + µDDD c)> µC (b − a) − µCDD (a − c)
> (φ(β > β ∗ ) − µCDD )(b − a) − µCDD (a − c)= (b − c)(β ∗ φ(β > β ∗ ) − µCDD )>0where the ﬁrst inequality holds by d > c, the second inequality holds because µC >φ(β > β ∗ ) − µCDD by Lemma 2, and the last inequality holds by Step 1. This meansthat C is not a best response for the type (α, β) = (0, 0) so that (0, 0) 6∈ TC∗ (µ). Hence,
the three-mode equilibrium must be an inequity concerned leader pattern.
CDD[Step 3] We prove the fact that the supremum of µµDDD
is ﬁnite. Suppose to theCDDcontrary that we can ﬁnd a three-mode equilibrium with an arbitrarily large µµDDD.
n∞¯
Then, there exists a sequence of {d }n=1 in (c, d(a))
and a corresponding sequence ofn∞n{µ }n=1 such that µ is a three-mode equilibrium distribution in P D((a, b, c, dn ), f )and lim µnDDD = 0. Note that lim µnDDD = 0 implies lim φ(α∗ (µn , a, dn ) ≤ α and β <
n→∞

n→∞

n→∞

∗
β ∗ ) = 0 because 0 ≤ φ(α∗ (µn , a, dn ) ≤ α and β < β ∗ ) < φ(TDDD(µn , a, dn )) = µnDDD
by Lemma 2. Note also that lim φ(α∗ (µn , a, dn ) ≤ α and β < β ∗ ) = 0 impliesn→∞that lim µnCDD = lim φ(α∗ (µn , a, dn ) ≤ α and β > β ∗ ) = 0. Hence, we must
n→∞

n→∞

n→∞

n→∞

have lim µnC = 1 − lim µnCDD − lim µnDDD = 1. On the other hand, recall thatn→∞µnC < µ̄(a, dn ) where µ̄(a, dn ) is a unique solution µ̄C to the equation (28) with d = dn
in the proof of Lemma 4. Consider the equation (28) with d = c. Then, it is rewrittenas)(a − c≥ (1 − β)µ̄C .
(29)µ̄C = φb−c
It follows by a similar argument to the proof of Lemma 4 that a unique solution µ̄C (a, c)nto this equation (29) is located as a−cb−c < µ̄C (a, c) < 1. Furthermore, µ̄C (a, d ) <nµ̄C (a, c) follows from c < d by a similar argument to the proof of Lemma 6-(1).
Hence, µnC < µ̄C (a, dn ) < µ̄C (a, c) < 1 for any n. This contradicts lim µnC = 1.
CDD
Hence, it is established that sup µµDDD
is ﬁnite. (Q.E.D.)n→∞

Proof of Theorem 11
Consider a three-mode strategy s in a prisoner’s dilemma with two timings and consider
the corresponding two-timing three-mode strategy in a prisoner’s dilemma with K
timings via the procedure TCk∗ = TC (s), TCDDk∗ +1 = TCDD (s), and TDDDk∗ +1 =TDDD (s).
The necessity is straightforward, because the timing k ∗ along the equilibrium pathin the prisoner’s dilemma with K timings is equivalent to the beginning of play in theprisoner’s dilemma with two timings and no deviation incentive from the two-timing
three-mode strategy within timings k ∗ and k ∗ + 1 in the prisoner’s dilemma with K
timings is equivalent to no deviation incentive from the corresponding three-mode
strategy in the prisoner’s dilemma with two timings.
We show the suﬃciency. Suppose that the three-mode strategy s is a sequential
equilibrium in the prisoner’s dilemma with two timings. Then, by the same argument
as the necessity part, a deviation from the two-timing three-mode strategy within
timings k ∗ and k ∗ + 1 in the prisoner’s dilemma with K timings is not beneﬁcial for68any type. Furthermore, a deviation to moving at timing k 6= k ∗ , k ∗ + 1 is not beneﬁcial
either. First, consider a deviation to choosing C at timing k < k ∗ . The expectedutility from this deviation is given by
φ(TCk∗ ∩ {(α, β) ∈ T |β < β ∗ })(c − α(b − c)) + φ(TCk∗ ∩ {(α, β) ∈ T |β ≥ β ∗ })a+ φ(TCDDk∗ +1 )a + φ(TDDDk∗ +1 )(c − α(b − c))where the ﬁrst term corresponds to the case in which his opponent is a type (α, β) ∈TCk∗ with β < β ∗ , he observes the deviation to C at the earlier timing k, and heresponds with D at the later timing k ∗ to the choice of C. Compare it with the
expected utility from following Ck∗ -mode, which is given by
φ(TCk∗ )a + φ(TCDDk∗ +1 )a + φ(TDDDk∗ +1 )(c − α(b − c)).
The latter utility is higher than the former by φ(TCk∗ ∩ {(α, β) ∈ T |β < β ∗ })[a −(c − α(b − c))] > 0 for any type (α, β) ∈ T . Thus, Ck∗ -mode is strictly better thana deviation to choosing C at timing k < k ∗ , and the equilibrium behavior is at least
as good as Ck∗ -mode by the optimality of the equilibrium behavior within timings k ∗and k ∗ + 1. Hence, a deviation to choosing C at timing k < k ∗ is not beneﬁcial for any
type.
Second, consider a deviation to choosing D at timing k < k ∗ . The expected utility
from this deviation is d because his opponent will observe the deviation to D andwill respond with D for sure. Compare this with the expected utility from following
CDDk∗ +1 -mode, which is given by φ(TCk∗ )a + φ(TCDDk∗ +1 )d + φ(TDDDk∗ +1 )d. The
latter utility is higher than the former by φ(TCk∗ )(a − d) > 0 for any type (α, β) ∈ T .
Thus, CDDk∗ +1 -mode is strictly better than a deviation to choosing D at timingk < k ∗ , and the equilibrium behavior is at least as good as CDDk∗ +1 -mode by the
optimality of the equilibrium behavior within timings k ∗ and k ∗ +1. Hence, a deviation
to choosing D at timing k < k∗ is not beneﬁcial for any type.
Finally, consider a deviation to moving at k > k ∗ +1. The optimal way of deviationis CDDk -mode for a type (α, β) ∈ T with β ≥ β ∗ and DDDk for a type (α, β) ∈ T withβ ≤ β ∗ because his opponent is supposed to ﬁnish choosing either C or D before timing
k and he simply chooses his best response to the observed choice by the opponent.
Note that the expected utilities from CDDk -mode and DDDk -mode are the sameas the expected utilities from CDDk∗ +1 -mode and DDDk∗ +1 -mode respectively. The
equilibrium behavior is at least as good as CDDk∗ +1 -mode and DDDk∗ +1 -mode by
the optimality of the equilibrium behavior within timings k ∗ and k ∗ + 1 and hence it
is at least as good as deviating to moving at timing k > k ∗ + 1. (Q.E.D.)
Proof of Theorem 12
Consider a three-mode strategy with a threshold k ∗ in a prisoner’s dilemma with
K timings. Suppose that a three-mode strategy with a threshold k ∗ is a sequential
equilibrium in a prisoner’s dilemma with K timings. Let TCk (1 ≤ k ≤ k ∗ ), TCDDk(k ∗ + 1 ≤ k ≤ K), and TDDDk (k ∗ + 1 ≤ k ≤ K) denote the sets of types who follow
Ck -mode, CDDk -mode, and DDDk -mode respectively in this equilibrium.
[Step 1] First, we show that φ(∪1≤k≤k∗ TCk ∩{(α, β) ∈ T |β < β ∗ }) > 0. Suppose to thecontrary that φ(∪1≤k≤k∗ TCk ∩ {(α, β) ∈ T |β < β ∗ }) = 0. Then, any type (α, β) ∈ TCk
with 1 ≤ k ≤ k ∗ is a type with β ≥ β ∗ . This type responds with C to a choice of C
by his opponent at an earlier timing k 0 < k than him. This means that choosing C attiming k with 1 ≤ k ≤ k ∗ induces the same outcome; (C, C) and (C, D) are realizedwith probabilities φ((∪1≤k≤k∗ TCk ) ∪ (∪k∗ +1≤k≤K TCDDk )) and φ(∪k∗ +1≤k≤K TDDDk )respectively. Note also that following CDDk with k ∗ + 1 ≤ k ≤ K induces the69

same outcome; (C, C) and (D, D) are realized with probabilities φ(∪1≤k≤k∗ TCk ) andφ((∪k∗ +1≤k≤K TCDDk ) ∪ (∪k∗ +1≤k≤K TDDDk )) respectively. Similarly, following DDDk
with k ∗ + 1 ≤ k ≤ K induces the same outcome; (D, C) and (D, D) are realizedwith probabilities φ(∪1≤k≤k∗ TCk ) and φ((∪k∗ +1≤k≤K TCDDk ) ∪ (∪k∗ +1≤k≤K TDDDk ))respectively. The sequential rationality implies that a type (α, β) ∈ TCk with 1 ≤k ≤ k ∗ prefers the outcome in which (C, C) and (C, D) are realized with probabilities φ((∪1≤k≤k∗ TCk ) ∪ (∪k∗ +1≤k≤K TCDDk )) and φ(∪k∗ +1≤k≤K TDDDk ) to both theoutcome in which (C, C) and (D, D) are realized with probabilities φ(∪1≤k≤k∗ TCk )and φ((∪k∗ +1≤k≤K TCDDk ) ∪ (∪k∗ +1≤k≤K TDDDk )) and the outcome in which (D, C)
and (D, D) are realized with probabilities φ(∪1≤k≤k∗ TCk ) and φ((∪k∗ +1≤k≤K TCDDk ) ∪(∪k∗ +1≤k≤K TDDDk )). The parallel implications hold for a type (α, β) ∈ TCDDk withk ∗ + 1 ≤ k ≤ K and a type (α, β) ∈ TDDDk with k ∗ + 1 ≤ k ≤ K. Now, in0=
duce a two-timing three-mode strategy by setting TC0 k∗ = ∪1≤k≤k∗ TCk , TCDD
k∗ +1
0
0
0
∪k∗ +1≤k≤K TCDDk , and TDDDk∗ +1 = ∪k∗ +1≤k≤K TDDDk where TCk∗ , TCDDk∗ +1 , and0
TDDD
denote the sets of types who follow Ck∗ -mode, CDDk∗ +1 -mode, and DDDk∗ +1 k∗ +1
mode respectively in the induced strategy. Then, following Ck∗ -mode, CDDk∗ +1 -mode,and DDDk∗ +1 -mode against the two-timing three-mode strategy induces the same outcome in the equilibrium. Hence, the implications of the sequential rationality in theoriginal sequential equilibrium guarantee that this two-timing three-mode strategy isalso a sequential equilibrium. Apply Theorem 11 to the induced two-timing three-mode
strategy. Then, the corresponding three-mode strategy s in a prisoner’s dilemma with0
0
two timings where TC (s) = TC0 k∗ , TCDD (s) = TCDD
, and TDDD (s) = TDDD
k∗ +1
k∗ +1
must be a sequential equilibrium. In this three-mode equilibrium, we must haveφ(TC (s) ∩ {(α, β) ∈ T |β < β ∗ }) = 0. This contradicts Lemma 2.
[Step 2] Second, we show that φ(TCk∗ ) > 0. Actually, we show a stronger result thatφ(TCk∗ ∩ {(α, β) ∈ T |β < β ∗ }) > 0. Suppose to the contrary that φ(Ck∗ ∩ {(α, β) ∈T |β < β ∗ }) = 0. Then, by Step 1, we have φ(∪1≤k≤k∗ −1 TCk ∩{(α, β) ∈ T |β < β ∗ }) > 0.
Hence, there is a timing k 0 with 1 ≤ k 0 ≤ k ∗ − 1 such that φ(TCk0 ∩ {(α, β) ∈ T |β <β ∗ }) > 0. Take a type (α, β) ∈ TCk0 ∩ {(α, β) ∈ T |β < β ∗ }. His expected utility fromfollowing Ck0 -mode isφ(∪1≤k≤k0 −1 TCk )(b − β(b − c)) + φ(∪k0 ≤k≤k∗ −1 TCk )a+ φ(∪k∗ +1≤k≤K TCDDk )a + φ(∪k∗ +1≤k≤K TDDDk )(c − α(b − c))where the ﬁrst term corresponds to the case in which his opponent is a type (α, β) ∈ TCk
with 1 ≤ k ≤ k 0 − 1, the opponent chooses C at an earlier timing k < k 0 than him, andhe responds with D at the later timing k 0 because D is the strict best response to C
under β < β ∗ . His expected utility from deviating to Ck∗ -mode isφ(∪1≤k≤k∗ −1 TCk )(b−β(b−c))+φ(∪k∗ +1≤k≤K TCDDk )a+φ(∪k∗ +1≤k≤K TDDDk )(c−α(b−c)).
The latter utility is higher than the former utility by φ(∪k0 ≤k≤k∗ −1 TCk )(b − β(b − c) −a) > 0 under β < β ∗ because φ(∪k0 ≤k≤k∗ −1 TCk ) ≥ φ(TCk0 ∩ {(α, β) ∈ T |β < β ∗ }) > 0.
This is a contradiction to the equilibrium.
[Step 3] Third, we show that φ(TC1 ) = · · · = φ(TCk∗ −1 ) = 0. Suppose to the contrary
that there is a timing k 0 with 1 ≤ k 0 ≤ k ∗ − 1 such that φ(TCk0 ) > 0. Take a type(α, β) ∈ TCk0 . Suppose a case in which β < β ∗ . Then, his expected utility fromfollowing Ck0 -mode isφ(∪1≤k≤k0 −1 TCk )(b − β(b − c)) + φ(TCk0 )a+ φ(∪k0 +1≤k≤k∗ TCk ∩ {(α, β) ∈ T |β ≥ β ∗ })a + φ(∪k0 +1≤k≤k∗ TCk ∩ {(α, β) ∈ T |β < β ∗ })(c − α(b − c))+ φ(∪k∗ +1≤k≤K TCDDk )a + φ(∪k∗ +1≤k≤K TDDDk )(c − α(b − c))70where1. the ﬁrst term corresponds to the case in which his opponent is a type (α, β) ∈ TCk
with 1 ≤ k ≤ k 0 − 1, the opponent chooses C at an earlier timing k < k 0 thanhim, and he responds with D at the later timing k 0 because D is the strict best
response to C under β < β ∗ ,2. the second term corresponds to the case in which his opponent is a type (α, β) ∈TCk0 and chooses C at timing k 0 simultaneously with him,
3. the third term corresponds to the case in which his opponent is a type (α, β) ∈TCk with k 0 + 1 ≤ k ≤ k ∗ such that β ≥ β ∗ , the opponent moves at timing k after
he chooses C at timing k 0 and responds with C because C is a best response to
C under β ≥ β ∗ , and4. the fourth term corresponds to the case in which his opponent is a type (α, β) ∈TCk with k 0 + 1 ≤ k ≤ k ∗ such that β < β ∗ , the opponent moves at timing kafter he chooses C at timing k 0 and responds with D because D is the strict best
response to C under β < β ∗ .
His expected utility from deviating to Ck∗ -mode isφ(∪1≤k≤k0∗ −1 TCk )(b − β(b − c)) + φ(TCk∗ )a+ φ(∪k∗ +1≤k≤K TCDDk )a + φ(∪k∗ +1≤k≤K TDDDk )(c − α(b − c)).
The latter utility is higher than the former utility byφ(TCk0 )[(b − β(b − c)) − a]+ φ(∪k0 +1≤k≤k∗ −1 TCk ∩ {(α, β) ∈ T |β ≥ β ∗ })[(b − β(b − c)) − a]+ φ(∪k0 +1≤k≤k∗ −1 TCk ∩ {(α, β) ∈ T |β < β ∗ })[(b − β(b − c)) − (c − α(b − c))]+ φ(TCk∗ ∩ {(α, β) ∈ T |β < β ∗ })[a − (c − α(b − c))]≥ φ(TCk0 )[(b − β(b − c)) − a] + φ(TCk∗ ∩ {(α, β) ∈ T |β < β ∗ })[a − (c − α(b − c))]>0where the last inequality holds because φ(TCk0 ) > 0 by the supposition and φ(TCk∗ ∩{(α, β) ∈ T |β < β ∗ }) > 0 by Step 2. This is a contradiction to the equilibrium.
Suppose the remaining case in which β ≥ β ∗ for the taken type (α, β) ∈ TCk0 .
Then, his expected utility from following Ck0 -mode isφ(∪1≤k≤k0 −1 TCk )a + φ(TCk0 )a+ φ(∪k0 +1≤k≤k∗ TCk ∩ {(α, β) ∈ T |β ≥ β ∗ })a + φ(∪k0 +1≤k≤k∗ TCk ∩ {(α, β) ∈ T |β < β ∗ })(c − α(b − c))+ φ(∪k∗ +1≤k≤K TCDDk )a + φ(∪k∗ +1≤k≤K TDDDk )(c − α(b − c))where the ﬁrst term is diﬀerent from the corresponding utility for the case of β < β ∗because the type with β ≥ β ∗ responds with C when his opponent is a type (α, β) ∈ TCk
with 1 ≤ k ≤ k 0 − 1 and the opponent chooses C at an earlier timing k < k 0 than him.
His expected utility from deviating to Ck∗ -mode isφ(∪1≤k≤k0∗ −1 TCk )a + φ(TCk∗ )a+ φ(∪k∗ +1≤k≤K TCDDk )a + φ(∪k∗ +1≤k≤K TDDDk )(c − α(b − c))71where the ﬁrst term is diﬀerent from the corresponding utility for the case of β < β ∗for the same reason as his expected utility from following Ck0 -mode. The latter utility
is higher than the former utility by
φ(∪k0 +1≤k≤k∗ −1 TCk ∩ {(α, β) ∈ T |β < β ∗ })[a − (c − α(b − c))]+ φ(TCk∗ ∩ {(α, β) ∈ T |β < β ∗ })[a − (c − α(b − c))]≥ φ(TCk∗ ∩ {(α, β) ∈ T |β < β ∗ })[a − (c − α(b − c))]>0where the last inequality holds because φ(TCk∗ ∩ {(α, β) ∈ T |β < β ∗ }) > 0 by Step 2.
This is a contradiction to the equilibrium.
[Step 4] Finally, we show that φ(CDDk∗ +1 ) > 0. Suppose to the contrary that
φ(CDDk∗ +1 ) = 0. By Step 1 through 3, we have φ(C1 ) = · · · = φ(Ck∗ −1 ) = 0 andφ(Ck∗ ∩ {(α, β) ∈ T |β < β ∗ }) > 0. Take a type (α, β) ∈ Ck∗ ∩ {(α, β) ∈ T |β < β ∗ }.
Then, his expected utility from following Ck∗ isφ(TCk∗ )a + φ(∪k∗ +2≤k≤K TCDDk )a + φ(∪k∗ +1≤k≤K TDDDk )(c − α(b − c)).
Consider a deviation to the following behavior. Choose ∅ at timing k = 1, · · · , k ∗ . At
timing k ∗ + 1, choose D if the opponent chooses C or D before timing k ∗ + 1 andchoose C if the opponent chooses at timing 1 through k ∗ . His expected utility fromthis deviation isφ(TCk∗ )(b − β(b − c)) + φ(∪k∗ +2≤k≤K TCDDk )a + φ(∪k∗ +1≤k≤K TDDDk )(c − α(b − c)).
The latter utility is higher than the former utility by φ(TCk∗ )[(b − β(b − c)) − a] > 0
under β < β ∗ . This is a contradiction to the equilibrium. (Q.E.D.)
Proof of Theorem 13
Suppose that there exists a three-mode equilibrium in P D((a, b, c, d), f ). Then, 0 <α∗ (µ) < ᾱ and a type (α, β) with α = α∗ (µ) is indiﬀerent between C and CDD. Thismeans that µC a + µCDD a + µDDD [c − α∗ (µ)(b − c)] = µC a + µCDD d + µDDD d, whichis rearranged into
µDDD
µCDD
a+
[c − α∗ (µ)(b − c)] = d.
µCDD + µDDD
µCDD + µDDD
Note that µDDD ≥ φ(α∗ (µ) ≤ α, β < β ∗ ) and µCDD ≤ φ(α∗ (µ) ≤ α, β > β ∗ ). Thismeans thatµDDDφ(α∗ (µ) ≤ α, β < β ∗ )≥= φ(β < β ∗ |α∗ (µ) ≤ α).
µCDD + µDDD
φ(α∗ (µ) ≤ α, β > β ∗ ) + φ(α∗ (µ) ≤ α, β < β ∗ )Hence,(1 − φ(β < β ∗ |α∗ (µ) ≤ α))a + φ(β < β ∗ |α∗ (µ) ≤ α)[c − α∗ (µ)(b − c)] ≥ d.

(30)¯
When condition (27) holds, the inequality (30) cannot be satisﬁed if d > d(a).
Supposeto the contrary that the inequality (30) holds. Then,(1−φ(β < β ∗ |α∗ (µ) ≤ α))a+φ(β < β ∗ |α∗ (µ) ≤ α)[c−α∗ (µ)(b−c)] ≥ d > (1−φ(β < β ∗ ))a+φ(β < β ∗ )c,which is rearranged intoφ(β < β ∗ |α∗ (µ) ≤ α)) <1b−c ∗1 + a−cα (µ)

72φ(β < β ∗ ).

This contradicts the condition (27) at α̃ = α∗ (µ) and β̃ = β ∗ . (Q.E.D.)
Proof of Lemma 10¯ is the exact bound for the existence of a three-mode equilibrium. Fix
Suppose that d(a)¯m > 0 for a given a ∈ (c, b). Suppose to the contrary that for any d ∈ (c, d(a))
thereexists a three-mode equilibrium distribution µ = (µC , µCDD , µDDD ) with µC ≥ m
in a prisoner’s dilemma P D((a, b, c, d), f ). Then, we can take a sequence {dn }∞n=1¯¯
such that (1) c < dn < d(a),(2) dn < dn+1 and lim dn = d(a),and (3) µnC ≥ m
n→∞

∗
n
n
n
n+1 implies
where µnC ≡ µmax
C (a, d ). Then, there exists µC ≡ lim µC because d < d
n→∞

n
max
n+1 ) = µn+1 by Theorem 3. Here, µ∗ ≥ m becauseµnC = µmax
C (a, d ) > µC (a, dCC¯¯µnC ≥ m. Then, µ∗C > 0 implies that λ(µ∗C , a, d(a))> 0 because λ(µC , a, d(a))> 0 for¯¯
any µC ∈ (0, µ̄C (a, d(a))) when d(a) is the exact bound for the existence of three-mode¯
equilibrium so that there is no three-mode equilibrium in P D((a, b, c, d(a)),
f ). Then,the continuity of λ(µC , a, d) with respect to (µC , d) guarantees that there exists  > 0¯ − , d(a))¯
such that λ(µC , a, d) > 0 for any d ∈ (d(a)
and any µC ∈ (µ∗C − , µ∗C + ).
∗∗¯ − , d(a))¯
We can ﬁnd N such that dN ∈ (d(a)
and µNC ∈ (µC − , µC + ) because
n
n
∗
N
N
¯
lim d = d(a) and lim µC = µC . We have λ(µC , a, d ) > 0. This is a contradiction

n→∞

n→∞

N
N
because the three-mode equilibrium distribution µN = (µN
C , µCDD , µDDD ) must satisfyN
λ(µNC , a, d ) = 0. (Q.E.D.)

Proof of Theorem 14
Recall that a three-mode equilibrium distribution µ is a materialist leader pattern if
and only if α∗ (µ, a, d) < β ∗ . Note that a similar argument to the proof of Lemma
7 leads us to conclude that, for (a, d) ﬁxed, when there exist multiple three-mode
equilibria µ = (µC , µCDD , µDDD ) and µ0 = (µ0C , µ0CDD , µ0DDD ) in a prisoner’s dilemma
P D((a, b, c, d), f ), α∗ (µ, a, d) < α∗ (µ0 , a, d) if and only if µC < µ0C . Hence, for (a, d)ﬁxed, all the three-mode equilibria in a prisoner’s dilemma P D((a, b, c, d), f ) are of amaterialist leader pattern if and only if α∗ (µmax (a, d), a, d) < β ∗ (a).
¯ is the exact bound for the existence of a three-mode equiNow, suppose that d(a)¯librium. First, we show that for any a ∈ (c, b) ﬁxed, there exists d ∈ (c, d(a))
such thatany three-mode equilibrium in the prisoner’s dilemma P D((a, b, c, d), f ) is of a mate¯
rialist leader pattern. Suppose to the contrary that for any d ∈ (c, d(a))
there exists a
three-mode equilibrium with either a hybrid leader pattern or an inequity concerned
leader pattern in a prisoner’s dilemma P D((a, b, c, d), f ). Then, we can take a pairn ∞nnn+1 and¯of sequences {dn }∞n=1 and {µ }n=1 such that (1) c < d < d(a), (2) d < dn¯lim d = d(a),and (3) α∗ (µn , a, dn ) ≥ β ∗ where µn = µmax (a, d). Then, there exists

n→∞

n
max
n+1 ) = µn+1 byµ∗C ≡ lim µnC because dn < dn+1 implies µnC = µmax
C (a, d ) > µC (a, dCn→∞Theorem 3. Lemma 10 guarantees that µ∗C = lim µnC = 0. By Lemma 2, this impliesn→∞that µ∗CDD = lim µnCDD = φ(β ≥ β ∗ ) and µ∗DDD = lim µnDDD = φ(β < β ∗ ). Hence,
n→∞

n→∞

the expression (4) of α∗ gives
lim α∗ (µn , a, dn ) = lim

n→∞

n→∞

¯ φ(β ≥ β ∗ ) d(a)¯ −cdn − c ] a − d(a)
CDD
=
−
= 0.
−
n
µDDD
b−c
b − c φ(β < β ∗ )b−c[ a − dn µnb−cThis means that there exists N such that 0 ≤ α∗ (µN , a, dN ) < β ∗ . This is a contradiction.
Then, for each a ∈ (c, b) ﬁxed, we can deﬁne¯
dˆml (a) ≡ inf{d ∈ (c, d(a))|
any three-mode equilibrium in P D((a, b, c, d), f ) is of a ml-pattern}.

73¯
From the deﬁnition it follows that c ≤ dˆml (a) < d(a).
Furthermore, any three-mode¯
equilibrium in any prisoner’s dilemma P D((a, b, c, d), f ) with dˆml (a) < d < d(a)is∗maxof a materialist leader pattern because α (µ (a, d), a, d) is strictly decreasing in dand β ∗ is constant in d by Lemma 7 so that if α∗ (µmax (a, d0 ), a, d0 ) < β ∗ (a) holds¯
for d0 ∈ (dˆml (a), d(a)),
then α∗ (µmax (a, d), a, d) < α∗ (µmax (a, d0 ), a, d0 ) < β ∗ (a) holds0¯
for any d ∈ (d , d(a)).
There exists a three-mode equilibrium with either a hybrid
leader pattern or an inequity concerned leader pattern in any prisoner’s dilemma
P D((a, b, c, d), f ) with c ≤ d < dˆml (a) for the same reason.
Second, we show the properties of dˆml (a). Consider a, a0 with c < a < a0 < b.
We show that dˆml (a) ≤ dˆml (a0 ). Consider d ∈ (c, dˆml (a)). Then, there exists a threemode equilibrium with either a hybrid leader pattern or an inequity concerned leader
pattern in P D((a, b, c, d), f ). This means that α∗ (µmax (a, d), a, d) ≥ β ∗ (a). Lemma 7
states that α∗ (µmax (a, d), a, d) is strictly increasing in a and that β ∗ (a) is strictly decreasing in a. Therefore, α∗ (µmax (a0 , d), a0 , d) > α∗ (µmax (a, d), a, d) ≥ β ∗ (a) > β ∗ (a0 ).
Hence, there exists a three-mode equilibrium with either a hybrid leader pattern or an
inequity concerned leader pattern in any prisoner’s dilemma P D((a0 , b, c, d), f ) withd ∈ (c, dˆml (a)). This means that dˆml (a) ≤ dˆml (a0 ).
Third, the relation dˆml (a) ≤ d¯ml (a) is immediate from Theorem 9-(2).
Finally, recall from Theorem 8 that for any d ∈ (c, b) there exists amax
L (d) and, formax
any a ∈ (aL (d), b), the type with the strongest incentive to lead under µmax (a, d) isthe (β ∗ , β ∗ )-type so that α∗ (µmax (a, d), a, d) ≥ β ∗ (a). This means that lim dˆml (a) = b.
a→b

(Q.E.D.)

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