Discussion Paper No.314
Asymmetric Volunteer’s Dilemma Game:
Theory and ExperimentToshiji Kawagoe
Future University Hakodate

Taisuke Matsubae
Chuo University

Hirokazu Takizawa
Chuo University

Tetsuo Yamamori
Dokkyo University

May 2019

INSTITUTE OF ECONOMIC RESEARCH
Chuo University
Tokyo, Japan

Asymmetric Volunteer’s Dilemma Game: Theory andExperimenta
Toshiji Kawagoeb, Taisuke Matsubaec, Hirokazu Takizawad and Tetsuo
Yamamorie

Abstract
This study theoretically and experimentally explores asymmetric volunteers’ dilemma(VOD) games where costs for volunteering is different among players. Diekmann (1993)
conjectures that an S-equilibrium, in which a player with less costs contributes, is more
likely to be played if it is risk dominant. We re-examined this hypothesis experimentally,
as well as via Diekmann’s data, to find that even though an S-equilibrium is risk dominant,it does not necessarily hold. Conducting an econometric comparison among models
including inequality aversion, level-k, and quantal response equilibrium (QRE), we findthat the QRE model fits the data best.
Keywords: Volunteers’ dilemma, risk dominance, quantal response equilibrium (QRE),
level-k model, inequality aversion, experiment
JEL classification number: C72, C92, D72, D74, H41a

We thank Thomas Palfrey and other participants in 2018 ESA World Meeting at Berlin for theirinterests and comments. This paper is a result of research conducted at “contemporary policy
research group,” Institute of Economic Research Chuo University. We would also like to thank
Editage (www.editage.jp) for English language editing.
b
Department of Complex and Intelligent Systems, Future University Hakodate
116-2 Kameda Nakano cho, Hakodate, Hokkaido, 041-8655, Japan
Tel. +81-138-66-4481 E-mail: kawagoe@fun.ac.jpc
I Faculty of Economics, Chuo University. 42-1 Higashi-nakano, Hachioji, Tokyo 192-0393, Japan.
Tel. +81-42-67-3355 E-mail: matsubae@tamacc.chuo-u.ac.jpdFaculty of Economics, Chuo University. 742-1 Higashi-nakano, Hachioji, Tokyo 192-0393, Japan.
Tel. +81-42-67-3355 E-mail: taki@tamacc.chuo-u.ac.jpeFaculty of Economics, Dokkyo University. 1-1 Gakuen-cho, Soka, Saitama 340-0042, Japan.
Tel. +81-48-943-2217 E-mail: yamamori@dokkyo.ac.jp
11. Introduction
Natural disasters often devastate a country and ravage the lives of many. Still fresh in ourmemories are the huge earthquake and the resulting tsunami that hit the North-easternpart of Japan in 2011, as well as the ensuing tragic accident of the Fukushima Daiichi
Nuclear Power Plant. The associated social and economic losses were huge, and therecovery from them has taken a long time. Such incidences make us acutely aware of theimportance of volunteers, as we hear many stories about people who help others even at
the risk of their own lives. However, the question of what kind of people volunteer andwhat their motivations still remain unanswered.
Given that, by definition, no monetary rewards are given to volunteers,
pecuniary incentives do not explain why they are so engaged. Sympathy or empathy is astrong candidate to serve as an explanans. If someone was hurt, we would feel, more or
less, uneasy; we will be urged to take some action and gain relief if the victim is,ultimately, saved. However, this explanation is also not conclusive. Since volunteering isa costly action, even empathetic people might think of free-riding on others, anticipatingthat someone else may do the job.
A famous parable in the New Testament best illustrates this complicatedsituation (The Parable of the Good Samaritan, Luke 10:25-37). Aware of an injured
fellow citizen who fell on the street, a priest and a Levite passed him by. Then a Samaritan,
who is, of course, a stranger, took action and saved him. The priest and the Levite, whilefeeling uncomfortable, ultimately, free-rode on the Samaritan’s helping behavior. This
story is even impressive given that the Israelites and the Samaritans were in an adversarial
relationship in those days. This meant that helping an Israelite was costlier for the
Samaritan not only in monetary terms but also politically and psychologically. Even today,
we see that many people help victims from very distant areas rather than nearby ones, and
it is often the case that they themselves were once victims in other disasters. How can weunderstand such a phenomenon?
This study attempts to address this issue in terms of game theory as well as
behavioral game theory1. The situations cited above are well captured by a game called
1

Some studies deals with essentially the same problem outside of game theory. Darly and Latané(1968) examine the helping behavior of people witnessing an accident or crime, as best exemplified
by the murder case of Kitty Genovese. It is said that her life could have been saved if only one of the
bystanders had paid a small amount of cost (such as making an emergency call to the police). A large
amount of evidence has also been accumulated by political scientists and psychologists to investigate
factors affecting the “bystander effect” and the effect of the group size on the tendency to cooperateor contribute in similar situations (Latané and Nida, 1981).
2

Volunteer’s Dilemma (VOD) game, which was first formulated by Diekmann (1985) toelucidate “social dilemmas” or “social traps,” which are broader than those covered by
the prisoner’s dilemma.
While symmetric versions of VOD conduce to the elucidation of suchinteresting issues like the so-called “bystander effect,”2 asymmetric versions of VOD are
useful for exploring the issue of who most likely contribute (help) in the aforementioned
situations. Diekmann (1993) introduced asymmetry in terms of costs of contribution
among players and showed, theoretically, that in the completely mixed strategy
equilibrium, a player with more costs volunteers more often (the behavior of the goodSamaritan is, thus, explained!).
While this theoretical prediction seems to support some observations, we arestill puzzled with this conclusion. For example, if we consider a bystander’s rescue
decisions in emergencies, is it not the case that the one with the least cost will help thevictim? This intuition, naturally, leads us to further investigations. With the same
motivation of study, Diekmann (1993) and Healy and Pate (2009) conducted laboratory
experiments and confirmed that the converse of the prediction by the mixed strategy
equilibrium was really the case; a player with less cost volunteers more often. But, how
can one provide a theoretical justification for such experimental findings? Diekmann(1993) suggested in a footnote an explanation based on risk dominance proposed by
Harsanyi and Selten (1988), but his ‘analysis’ seems to be incomplete.
Thus, we first provide a rigorous theoretical analysis of the game based on risk
dominance. We confirm Diekmann (1993)’s above-mentioned conjecture and give somecharacterizations of equilibria in a more general setting by utilizing an extension of therisk dominance concept to general n-person games (Güth 1990). Then, we analyze thegame with other-regarding preference and bounded rationality. We use inequality
aversion (Fehr and Schmidt, 1999), level-k model (Stahl and Wilson 1995; Crawford etal. 2013), and Quantal response equilibrium (QRE: McKelvey and Palfrey 1995) as
representative models.
Based on these analyses, S-equilibrium (where a player with less costs
volunteers) is shown to be risk dominant for any treatment of Diekmann (1993)’s.
However, depending on parameter values of the games (QRE’s prediction depends on its
noise parameter), the inequality aversion and the level-k model give different predictions
than risk dominance. Thus, if Diekmann (1993)’s conjecture were right, as S-equilibrium
was risk dominant in every treatment, we would observe S-equilibrium more frequently
2

Kawagoe et al. (2018) characterize all the symmetric equilibria of this class of games.
3

in his data. Our analysis showed that predictions by both mixed strategy and risk
dominance were rejected. Other models captured important characteristics of the results,
but no model explained all the data consistently. Finally, we, then, conducted an
econometric horse race to determine which model better fits the data. The estimation
results showed that QRE best explains Diekmann (1993)’s data. Thus, Diekmann(1993)’s conjecture that a player with less cost volunteers more often was falsified.
We, then, extended his experiment by adding a treatment to differentiate theprediction by inequality aversion from other models. We also added a more complicatedthree-person version of the game. Diekmann (1993)’s conjecture was also falsified in our
experiment, and it was confirmed, again, that QRE best fits the data. Thus, Diekmann(1993)’s conjecture is not supported by his and our results.
Of course, this does not mean free-riding among volunteering players with lesscost is dominant in the data. Rather, in some treatments, the majority of players with lesscost volunteer more often. What we show, empirically, is that the condition that Sequilibrium is risk dominant is not a source of volunteering by players with less cost.
Inequality aversion also fails. No single theory can explain the data. Roughly speaking,the fact that QRE was the best fit means that the subject behaviors might be a combinationof mixed strategy equilibrium and random play, while its proportion depends on theparameters of the game. However, the understanding of volunteering decision by playerswith less cost needs a more careful scrutiny.
The organization of the study is as follows. In the next section, we formulate
the VOD game with an asymmetric cost structure and present several candidate models
that may explain the behavior of subjects in the experiments. This part includes extensive
mathematical characterizations of the models with some new results. Section 3 reexamines the data reported by Diekmann (1993) with those models. In Section 4, we turnto the analyses of our experiments. We conclude in the final section.
2. ModelIn the VOD game, if at least one of n players volunteers, certain public goods are provided.
The benefit from the public goods for player i is denoted as 𝑉" , and the cost of
volunteering for player i is 𝐾" . When public goods are not provided, the payoff for player
i is 𝐿" . Assume 𝑉" − 𝐿" > 𝐾" > 0 for all i. Thus, without any strategic considerations,
all players have an incentive to volunteer. Each player chooses between volunteering (C)and not volunteering (N). Obviously, there are multiple pure strategy Nash equilibria
where one and only one player chooses C, and the rest of the players choose N. Note thata strategy profile where all players choose 𝑁 is not a Nash equilibrium.
42.1 Mixed strategy equilibria
Next, we consider the mixed strategy equilibria of the game. The probability that player
𝑖 chooses C is denoted by 𝑝" , and the probability of choosing N is denoted by 𝑞" = 1 −𝑝" . Then, the expected payoff of choosing C for player 𝑖 is𝐸" (𝐶 ) = 𝑉" − 𝐾" ,while the expected payoff of choosing N is𝐸" (𝑁) = 𝑉" 41 − 5 𝑞6 8 + 𝐿" 5 𝑞6 .
67"

67"

If player 𝑖 chooses C with a probability of one, then 𝐸6 (𝑁) > 𝐸6 (𝐶 ) for any
other player 𝑗 ≠ 𝑖 by 𝐾6 > 0. Furthermore, if all the other players than 𝑖 choose N
with a probability of one, then 𝐸" (𝐶 ) > 𝐸" (𝑁) by 𝑉" − 𝐿" > 𝐾" . Therefore, in a mixed
strategy equilibrium, 𝑞" = 0 if and only if 𝑞6 = 1 for any 𝑗 ≠ 𝑖. This implies that if aplayer takes a completely mixed strategy (mixed strategy with full support) in a mixed
strategy equilibrium, then there are no other players who choose C with a probability of
one in this equilibrium.
Thus, the volunteer’s dilemma game may have three kinds of mixed strategy
equilibria. The first is that only one player chooses C and the rest of players choose N
with a probability of one (that is, a pure strategy Nash equilibrium). The second is thatsome players (two or more players) take completely mixed strategies, and the rest of theplayers choose N with a probability of one. The third is the completely mixed strategy
equilibrium (mixed strategy equilibrium where all players take completely mixed
strategies).
Diekmann (1993) only considers the first and the last types of mixed strategy
equilibrium and shows that the completely mixed strategy equilibrium always exists in asymmetric case (𝑉" = 𝑉, 𝐾" = 𝐾, and 𝐿" = 0 for any 𝑖 = 1,2, . . 𝑛) and in the case with𝑛 = 2. However, the completely mixed strategy equilibrium does not always exist in
more general cases. Hence, we identify the completely mixed strategies of the players ina mixed strategy equilibrium (which is not necessary completely mixed).
Given any mixed strategy profile 𝜃, let 𝐶(𝜃) be a set of players who do notchoose 𝑁 with a probability of one in 𝜃. Then, we have the following proposition.
Proposition 1. Suppose that player 𝑖 takes a completely mixed strategy in a mixed
strategy equilibrium 𝜃 (𝑖 ∈ 𝐶(𝜃)). Then5𝑞" = A𝐾6
𝑉" − 𝐿"BC 5 D
EH
𝐾"
𝑉6 − 𝐿6I#F(G)KI.

(1)

6∈F(G)

where #𝐶(𝜃) is the number of 𝐶(𝜃).
Proof. In Appendix A.
It follows from this proposition that the completely mixed strategy equilibriumis unique if it exists, and the strategies of the players are given by equation (1) if 𝐶(𝜃)is the set of all players.
Hereafter, we consider a special case in which there are only two different
values for benefit 𝑉" and cost 𝐾" , and characterize the set of the mixed strategy
equilibria. Following Diekmann (1993), we call the players with low cost “strong players”(hereafter S-player) and the players with high cost “weak players” (hereafter W-player).
That is, the cost for an S-player 𝐾L is strictly smaller than that for a W-player 𝐾M (𝐾L <𝐾M ). Here, let 𝑉L and 𝑉M be the benefit for an S-player and a W-player, respectively.
𝐿L and 𝐿M are defined similarly. We also, then, assume that these parameters satisfy theinequality
𝑉L − 𝐿L 𝑉M − 𝐿M
>
,
𝐾L
𝐾M

(2)

which implies that the marginal per capita cost for volunteering for an S-player is lessthan the one for a W-player. Diekmann (1993) considers the case where 𝐾L < 𝐾M ,0 < 𝑉M ≤ 𝑉L , and 𝐿L = 𝐿M = 0, which satisfies condition (2). Moreover, as for thenumber of S-players, m, only the case of m = 1 is considered. We generalize thoseanalyses to an arbitrary 𝑚 ≤ 𝑛.
Note that Proposition 1 implies that if two players are of the same type, andthey take completely mixed strategies in equilibrium, then their probabilities ofvolunteering are the same. Furthermore, by condition (2), we have the following.
Corollary 1. Suppose that an S-player 𝑖 and a W-player 𝑗 take completely mixed
strategies in a mixed strategy equilibrium, then the probability that S-player 𝑖 will
volunteer is less than that of a W-player 𝑗; that is, 𝑞" > 𝑞6 .
As mentioned above, Diekmann (1993) shows that the completely mixed
strategy equilibrium always exists in a symmetric case (𝑚 = 𝑛 or 𝑚 = 0) and in thecase with 𝑛 = 2. The following proposition gives the necessary and sufficient condition
6

for the existence of the completely mixed strategy equilibrium when 𝑛 ≥ 2 and 1 ≤𝑚 ≤ 𝑛.
Proposition 2. For any 𝑛 ≥ 2 and 1 ≤ 𝑚 ≤ 𝑛, the volunteer’s dilemma game has thecompletely mixed strategy equilibrium if and only if
𝑉M − 𝐿M RKS
𝑉L − 𝐿L RKSKI
A
B
>A
B
.
𝐾M
𝐾L

(3)

Proof. In Appendix A.
For any 𝑛 ≥ 2 and 𝑚 ≤ 𝑛 , let Θ(𝑛, 𝑚) be the set of mixed strategy
equilibria of the VOD game, where the number of players is 𝑛 and the number of Splayers is 𝑚. By the above arguments, we can easily identify all mixed strategy profilesin Θ(2,1) (and also Θ(2,2) and Θ(2,0)), which has only two kinds of Nash equilibria.
One is that one player chooses C and the other player chooses N with a probability of one,and the other is the completely mixed strategy equilibrium in which the strategies of theplayers are given by (1) with #𝐶(𝜃) = 2. By the following proposition, we can also
identify Θ(𝑛, 𝑚) where 𝑛 > 2, inductively.
Proposition 3. Suppose that 𝜃 ∈ 𝛩(𝑛, 𝑚) is the completely mixed strategy equilibrium,and 𝑞L (𝑛, 𝑚) and 𝑞M (𝑛, 𝑚) are the probabilities of choosing 𝑁 by an S-player and a
W-player in 𝜃, respectively. For any two non-negative integers, 𝑙 and ℎ, consider thevolunteer’s dilemma game where the number of players is 𝑛 + 𝑙 + ℎ, and the number ofS-players is 𝑚 + 𝑙. In this game, a mixed strategy profile 𝜃′such that 𝑙 S-players andℎ W-players are choosing 𝑁 with a probability of one, 𝑚 S-players take 𝑞L (𝑛, 𝑚),and 𝑛 − 𝑚 W-players take 𝑞M (𝑛, 𝑚) is a mixed strategy equilibrium; that is, 𝜃′ ∈𝛩(𝑛 + 𝑙 + ℎ, 𝑚 + 𝑙).
Proof. In Appendix A.
For example, Θ(3,1) consists of two or three kinds of mixed strategy
equilibria. First is that a player chooses C and the other two players choose N with aprobability of one. Second is that a W- (an S-) player chooses N with a probability of one,and the other two players take completely mixed strategies in the completely mixed
strategy equilibrium in Θ(2,1) ( Θ(2,0) ). Third is the completely mixed strategy
7equilibrium in which two W-players take 𝑞M (3,1) and the S-player takes 𝑞L (3,1) if (3)is satisfied for 𝑛 = 3 and 𝑚 = 1.
We now focus on the completely mixed strategy equilibrium. As in the above
proposition, let 𝑞L (𝑛, 𝑚) and 𝑞M (𝑛, 𝑚) be the probabilities of choosing 𝑁 of S-player
and W-player in the completely mixed strategy equilibrium in Θ(𝑛, 𝑚), respectively. The
following proposition states how these probabilities change with the group size and thenumber of S-players in it. Goeree et al. (2017) explored the relationship between groupsize and volunteering, where the Nash equilibrium predicts the probability of volunteering
to be a decreasing function of group size and that the probability of a no-volunteer
outcome increases with the number of players. The following propositions considers theirresults in more detail. Thus, it shows that the probability of a volunteer outcome increases
with the number of S-players, and the probability of a no-volunteer outcome increases
with the number of W-players.
Proposition 4. Suppose that 𝛩(𝑛, 𝑚), 𝛩(𝑛 + 1, 𝑚), and 𝛩(𝑛 + 1, 𝑚 + 1) have thecompletely mixed strategy equilibria. Then(i) 𝑞L (𝑛 + 1, 𝑚) > 𝑞L (𝑛, 𝑚),(ii) 𝑞M (𝑛 + 1, 𝑚) > 𝑞M (𝑛, 𝑚),(iii) 𝑞L (𝑛 + 1, 𝑚) > 𝑞L (𝑛 + 1, 𝑚 + 1), and(iv) 𝑞M (𝑛 + 1, 𝑚) > 𝑞M (𝑛 + 1, 𝑚 + 1).
Proof. In Appendix A.
2.2 Equilibrium selection based on risk dominanceIn the volunteer’s dilemma, there are two kinds of pure strategy Nash equilibria: Sequilibrium where only one of the S-players chooses C and W-equilibrium where only
one of the W-players chooses C. For the case of m = 1, Diekmann (1993) suggests that Sequilibrium is only a risk dominant equilibrium. Suppose, for example, m = 1 and n = 2.
Then, the game becomes as follows (Table 1).
Table 1. Asymmetric volunteer’s dilemma when m = 1 and n = 2
Weak

C

N

C

𝑉L − 𝐾L , 𝑉M − 𝐾M

𝑉L − 𝐾L , 𝑉M

N

𝑉L , 𝑉M − 𝐾M

𝐿L , 𝐿M

Strong8

In this game, there are two pure strategy Nash equilibria, (C, N) and (N, C), since 𝑉" −𝐿" > 𝐾" for all i. If (C, N) risk dominates (N, C), the product of deviation loss for (C, N)
is greater than that for (N, C) (Harsanyi and Selten, 1988). This implies(𝑉L − 𝐿L − 𝐾L )𝐾M > 𝐾L (𝑉M − 𝐿M − 𝐾M ).
Note that this condition is equivalent to condition (2). Thus, we have the following
proposition.
Proposition 5. In the case of 𝑛 = 2 and 𝑚 = 1, S-equilibrium is risk dominant.
We now consider a risk dominant equilibrium in the case of many players. Herewe consider Güth (1990)’s notion of unilateral deviation stability (UDS). UDS satisfies
Harsanyi and Selten’s axioms for characterizing risk dominance.3 Suppose that an 𝑛person game has several strict pure Nash equilibria. Take two of them, 𝑠 = (𝑠I , ⋯ , 𝑠R )and 𝑡 = (𝑡I , ⋯ , 𝑡R ) . Let M(𝑠, 𝑡) be a set of players whose equilibrium strategy isdifferent in 𝑠 and 𝑡. For players 𝑖 and 𝑗 in M(𝑠, 𝑡), construct a comparison game𝐺"6 (𝑠, 𝑡), where the set of strategies is 𝑣 = (𝑠_ , 𝑡_ ) for 𝑘 = 𝑖, 𝑗, and the payoff functionis also restricted by this set of strategies (players other than 𝑖 and 𝑗 use the same
strategies in both 𝑠 and 𝑡 by the assumption of M(𝑠, 𝑡) .). Thus, 𝐺"6 (𝑠, 𝑡) can be
represented by the following game in Table 2.
Table 2. Comparison game 𝐺"6 (𝑠, 𝑡)
𝒔𝒋

𝒕𝒋

𝒔𝒊

𝜋" (𝑠), 𝜋6 (𝑠)𝜋" f𝑠" , 𝑡6 g, 𝜋6 (𝑠" , 𝑡6 )𝒕𝒊

𝜋" f𝑡" , 𝑠6 g, 𝜋6 (𝑡" , 𝑠6 )𝜋" (𝑡), 𝜋6 (𝑡)The relative strength of equilibrium 𝑠 against 𝑡, 𝑅"6 (𝑠, 𝑡), is defined as follows.
i𝜋6 (𝑠) − 𝜋6 (𝑠" , 𝑡6 )ji𝜋" (𝑠) − 𝜋" (𝑡" , 𝑠6 )j
𝑅"6 (𝑠, 𝑡) =i𝜋6 (𝑡) − 𝜋6 (𝑡" , 𝑠6 )ji𝜋" (𝑡) − 𝜋" (𝑠" , 𝑡6 )jThus, the product of losses resulting from the unilateral deviation of players 𝑖 and 𝑗from equilibrium 𝑠 and 𝑡 is compared. Finally, the aggregated value of 𝑅"6 (𝑠, 𝑡) for
any pair of players in M(s, t) is given by
The notion of p-dominance, introduced by Morris, Rob, and Shin (1995), generalizes the notion ofrisk dominance in Harsanyi and Selten (1988) differently than Güth (1990); that is, 1/2-dominancecoincides with the latter in symmetric 2x2 games. Peski (2010), also, proposes a related concept for
n-person symmetric games based on 1/2-dominance. However, the game we study is not symmetric;
we did not use these concepts.
93𝑅∗ (𝑠, 𝑡) =5

𝑅"6 (𝑠, 𝑡)",6∈n(o,p)"q6Then, if 𝑅∗ (𝑠, 𝑡) > 1, equilibrium 𝑠 risk dominates 𝑡. With this notion, we have ageneralization of Proposition 5.
Proposition 5’. In the case of 𝑛 ≥ 2 and 1 ≤ 𝑚 < 𝑛, S-equilibrium risk dominates Wequilibrium in the sense of unilateral deviation stability.
Proof. In Appendix A.
2.3 Alternative theories
In what follows, we also examine the explanatory power of alternative theories that take
into account the possibility that agents have a non-selfish motivation or act irrationally.
In fact, we will see that the explanatory power of both mixed strategy equilibria and risk
dominance is not satisfactory concerning the explanation of the experimental data,
Diekmann (1993)’s and ours. Among many alternative models so far proposed for
defining non-selfish motivation, we choose inequality aversion (Fehr and Schmidt, 1999)
as the most appropriate in the present context. As for irrational behavior, we investigate
the level-k model (Stahl and Wilson 1995; Crawford et al. 2013) and the Quantal response
equilibrium (QRE: McKelvey and Palfrey 1995), which are both simple, but known tohave been very successful in the experimental economics literature.
A. Inequity aversion
The concept of inequity aversion (resistance to inequitable outcomes) in explaining
experimental regularities was developed in Fehr and Schmidt (1999).4 They postulated
that people make decisions to minimize inequity in outcomes. Specifically, consider asetting where player 𝑖 (𝑖 = 1,2, ⋯ , 𝑛) receives a pecuniary payoff 𝑥" . Then the utility of
an inequity averse player 𝑖 for the allocation (𝑥I, 𝑥s , … , 𝑥R ) is given by
𝑈" f𝑥I, 𝑥s , … 𝑥R g = 𝑥" −𝛼𝛽w 𝑚𝑎𝑥i𝑥6 − 𝑥" , 0j −w 𝑚𝑎𝑥i𝑥" − 𝑥6 , 0j,
𝑛−1
𝑛−1
67"

(4)

67"

Bolton and Ockenfels (2000), also, proposed a similar kind of concept. One of the distinguishedfeatures in Fehr and Schmidt (1999)’s model as compared to Bolton and Ockenfels (2000)’s is that itallows a distinction between advantageous and disadvantageous inequalities. As this feature plays arole in explaining our data, we adopted Fehr and Schmidt (1999)’s model.
104where 𝛼 is player 𝑖’s disutility of having less than the others and 𝛽 is player 𝑖’s
disutility of having more than the others. Fehr and Schmidt (1999) assume that 0 ≤ 𝛽 <1, and 𝛽 ≤ 𝛼.
If there is a player having the above utility function, S-equilibrium or Wequilibrium would no longer be a Nash equilibrium, and other outcomes would be a Nashequilibrium depending on values of the parameters 𝛼 and 𝛽. We will list equilibrium
conditions for each outcome later in examining Diekmann (1993)’s data.
However, the following proposition implies that the outcome that every playerchooses C, All-C, cannot be an equilibrium irrespective of the values of the parameters.
Proposition 6. Suppose that player 𝑖 is an inequity averse player. Then player 𝑖 neverchooses C when all other players choose C.
Proof. In Appendix A.
B. Level-k model
Level-k model is a non-equilibrium model that reflects strategic thinking by boundedly
rational players. It assumes that each player adopts a strategy that corresponds to somelevel of strategic thinking. Level-k models have so far been applied to many games, andhave succeeded in explaining a number of anomalous behaviors found in the laboratory(for a survey, see Crawford et al., 2013).
Assume that L0 player, who is the least rational among the players, chooses C
and N with probability 1/2, respectively. In the level-k model, L𝑘 player chooses the best
response to the actions taken by L(𝑘 − 1) players. The outcome the level-k model
predicts also depends on the payoff structure of the game; we will show its predictions
later in analyzing Diekmann (1993)’s data.
C. Quantal response equilibrium (QRE)
QRE is an equilibrium concept based on boundedly rational strategic behavior, assumingthat players play a noisy best response (McKelvey and Palfrey, 1995). For player 𝑖, thestochastic best response in terms of his choice probability of C is given by𝑝" =exp (𝜆 ∙ 𝐸" (𝐶))
1
=
,
(𝐶))
(𝑁))
[𝜆
{𝐸
exp (𝜆 ∙ 𝐸"
+ exp (𝜆 ∙ 𝐸"
1 + exp ∙ " (𝑁) − 𝐸" (𝐶)}]11where parameter λ ∈ [0, ∞) represents the degree of rationality such that 𝜆 = 0
implies complete randomizing over pure strategies. QRE is a fixed point of this mapping.
If 𝜆 = 0, then 𝑝" = 1/2 for any 𝑖, which is usually called the centroid of the simplex of
the strategy space.
McKelvey and Palfrey (1995) showed that for any normal-form game, (i) the
correspondence QRE(λ) is upper hemi-continuous, (ii) the number of QREs is odd forgeneric values of λ, and, (iii) generically, the graph fλ, QRE(λ)g contains a uniquebranch which starts at the centroid and converges to a unique Nash equilibrium as λapproaches infinity. The limiting point of this principal branch is called limiting (logit)
QRE. Thus, limiting QRE can serve as an equilibrium selection criterion in generic cases.
Turocy (2005) showed that with the homotopy method (under an intuitive
monotonicity assumption), the limiting QRE is the risk dominant equilibrium in generic
2x2 games with two strict Nash equilibrium.
However, the cases with more than two players may be problematic. It is easy
to see that these games have multiple S-equilibria, depending on which S-player finallycontribute. Thus, the principal branch of the QRE graph that starts from the centroid may
bifurcate somewhere on the way, meaning that these cases are not generic. This actually
happens in our case. However, in the statistical estimation that follows, the log-likelihood
function is maximized well before the bifurcation occurs.
3. Re-examination of Diekmann’s experiments
3.1 The experiments
We begin with the reexamination of Diekmann (1993)’s experiments. Diekmann (1993)
concludes that the predictions of the risk-dominance theory accord better with his data
than that of the mixed-strategy equilibrium. Thus, unlike the mixed strategy equilibrium
prediction which states that a player with less cost contributes less often, an S-player inthe experiment contributes a significantly higher proportion.
However, once we closely consider his experimental result, it seems the
explanatory power of the risk-dominance theory is not so high. Therefore, it is worthwhile
to reexamine the data with econometric comparisons, including the alternative theories,
which was not available when his research was conducted. This was explained in theprevious section.
His experiment consists of ten different sessions, including symmetric andasymmetric versions of VOD. The total of 328 subjects were recruited and allotted
randomly into the sessions. However, as a result of the preliminary test, twenty-seven12subjects were excluded from the proper experiment. In what follows, we will only focus
on the part of his experiment that concerns asymmetric VOD.
Table 3 shows the sessions in his experiment that corresponds to asymmetricversions of VOD. It is important to notice that the subjects really played the game; thatis, they were not matched with one another to play the games. Instead, subjects were
simply asked to make a decision as a W-player or an S-player before the experiment ended.
Each subject could participate in only one session. How the game outcome wasdetermined, and, thus, how the rewards were determined, is not reported in the study.
Table 3 Diekmann's experimental games (Sessions)
Games

D-1

D-2

D-3

D-4

Sessions# of players

B
2

D
2

C
2

E
2

F
2

G
2

I
5

J
5

# of S-players
V(common)

1
100

1
100

1
100

1
100

1
100

1
100

1
100

1
100

𝑲𝑺

40

40

10

10

20

20

40

40

𝑲𝑾

50

50

50

50

80

80

50

50

Subjects’ role

W

S

W

S

W

S

W

S

# of subjects

29

30

32

39

27

32

27

27

We classify each session by games and give each game a name, from D-1 to D4. Note that Sessions D-1 to D-3 are two-player games, and only Session D-4 is a fiveplayer game. In every session, the number of S-players is one. Note also that for parametervalues 𝑉, 𝐾L and 𝐾M (𝐿L = 𝐿M = 0), the condition in Proposition 5 and 5’ is satisfied;that is, S-equilibrium is risk dominant.
3.2 Hypotheses
Testing Diekmann’s conjecture is the primary purpose here. As S-equilibrium is risk
dominant in all the session, it is expected that S-equilibrium should be observed mostfrequently. If it is not the case, our next task is to judge whether the mixed strategy
equilibrium prediction is observed in his experiment.
As shown in Corollary 1, if players adopt completely mixed strategies, theprobability that an S-player volunteers is less than that of a W-player. However, it iscounterintuitive because the marginal per capita cost for volunteering, 𝐾L /(𝑉L − 𝐿L ), for
an S-player is less than the one for a W-player, 𝐾M /(𝑉M − 𝐿M ).
13

Thus, our first hypothesis concerns whether S-equilibrium is attained in thelaboratory.
Hypothesis 1. S-equilibrium is more frequently observed than W-equilibrium.
Of course, subjects in the experiment may not always play rationally. Sometimes they
deviate from the equilibrium either consciously or unconsciously. The former results fromdifferent motivations subjects have; for example, other-regarding preferences. The latteris due to the subjects’ misunderstanding or confusion, lack of concentration, among others.
Thus, we cannot expect that a particular equilibrium is played with a probability of one.
If it is the case, a mixed strategy might predict the subjects’ behavior better than the ones
predicted by risk dominance.
Therefore, is the subjects’ choice frequency of volunteering close to the mixed
strategy equilibrium prediction in Section 2? This is our second hypothesis to be tested.
Hypothesis 2. A subject’s choice frequency of volunteering coincides with the mixed
strategy equilibrium.
Table 4 summarizes completely mixed strategy equilibria in Diekmann (1993)’s
experiment.5
Table 4 Symmetric mixed strategy equilibrium prediction for Diekmann's Games
D-1

D-2

D-3

D-4

𝑷𝑺

0.500

0.500

0.200

0.006

𝑷𝑾

0.600

0.900

0.800

0.205

As both hypotheses are mutually inconsistent, it is an easy task to judge which
theory, mixed strategy equilibrium and risk dominance, predicts better subject behavior
in the experiment.

Strictly speaking, there are two types of mixed strategy equilibria other than pure strategy Nashequilibria. The first type is completely mixed strategy equilibria in which all players choose C and Nwith non-zero probabilities. The second type is mixed strategy equilibria where some players choose
N with a probability of 1 and the other players choose the completely mixed strategies. As the latter
exists only in D-4 (five-person game) and our objective is to differentiate counter intuitive
completely mixed strategy prediction from that of risk dominance, we use the former in testing
Hypothesis 2.
14
5

If the date supports neither hypothesis, we have to resort to different theories
other than rational ones such as mixed strategy equilibrium and risk dominance. Thus, toobtain theoretical predictions by inequity aversion for each experimental session, we
assume that all players have the same utility function given by (4). Then, for each session,
either S-equilibrium, W-equilibrium, or All-N can be a Nash equilibrium depending on
the values of the parameters. The ranges of the values of 𝛼 and 𝛽 that each outcome
will be a Nash equilibrium for each session are summarized in Table 5.
Table 5. The prediction of inequality aversion
D-1

D-2

D-3

𝛼 ≤ 9,𝛼 ≤ 4,𝛽≤5𝛽≤4𝛼 ≤ 1,𝛼 ≤ 1,𝛼 ≤ 1,𝛼 ≤ 8,𝛽 ≤ 4/5

𝛽 ≤ 1/5

𝛽 ≤ 1/4

𝛽 ≤ 16/5

α≥1

α≥9

α≥4

α≥6

𝛼 ≤ 3/2

S-eq.
W-eq.
All-N

D-4
𝛼≤6

As for the level-k model, the best responses of L𝑘 S-player and L𝑘 W-player to theactions of L(𝑘 − 1) players up to 𝑘 = 4 are summarized in Table 6.
Table 6. Best response of Lk to L(k-1)6

L1

L2

L3
L4

D-1

D-2

D-3

D-4

S

C

C

C

N

W

C,N

C,N

N

N

S

𝐶

C

C

C

W

N

N

N

C

S

C

C

C

N

W

N

N

N

N

S

C

C

C

C

W

N

N

N

C

We, then, econometrically measure the goodness-of-fit of these alternative theories,
including QRE.
6

“C, N” means that both strategies are indifferent. The identification of a higher-levelstrategy assumed that both are played with equal probability.
15

3.3 Results
Table 7 shows Diekman’s experimental data, where n is the number of groups, 𝑛L and𝑛M are the number of C chosen by an S-player and a W-player, respectively, and 𝑓L and𝑓M are the choice frequency of C by an S-player and a W-player respectively. If
Diekmann’s conjecture is right, as S-equilibrium is risk dominant in every session, we
have to observe 𝑓L > 𝑓M . In two-person game sessions (D-1, D-2 and D-3) 𝑓L > 𝑓M isobserved (chi squared test, p < 0.05 except Session D-3). However, in Session D-4, afive-player game, we have a counter fact: 𝑓L = 0.30 < 𝑓M = 0.56 (chi squared test, p＜0.05). Thus, we conclude that Diekmann’s conjecture is only applicable to two-person
games. Therefore, Hypothesis 1 is only partially confirmed.
Table 7. Choice Frequencies of Each Strategy
D-1

D-2

D-3

D-4

𝒏𝑺

20

37

27

8

𝒏𝑾

13

6

2

15

𝒇𝑺

0.67

0.95

0.84

0.30

𝒇𝑾

0.45

0.19

0.07

0.56

As for mixed strategy equilibrium, when we compare Tables 4 and 7, there arehuge discrepancies between them. Significant differences are visible in D-2, D-3, and D4 Sessions. The null hypothesis stating that the choice frequency of C and the probabilities
obtained from the completely mixed strategy equilibrium are equal is rejected for Splayers in Sessions D-2, D-3, and D-4 (binomial test, p < 0.01 for each). On the contrary,W-players chose C less frequently than the mixed strategy equilibrium. The null
hypothesis stating that the choice frequency of C and the probabilities obtained fromcompletely mixed strategy equilibrium are equal is rejected for W-players in Sessions D2, D-3, and D-4 (binomial test, p < 0.01 for each). Thus, the mixed strategy equilibrium
prediction also fails. 7
Furthermore, to find the model that best fits Diekmann (1993)’s data among alternative
theories, we will resort to an econometric horse race; that is, conducting maximum
likelihood estimation for obtaining the parameter values such as 𝛼, 𝛽, and 𝜆 in thesemodels and comparing explanatory powers among them.

Note that in Session D-1, the null hypothesis is not rejected (binomial test, p < 0.05 for S-players
and W-players).
167Moreover, to make a comparison, we need to make the models “statistical” inthe sense that they “contain adjustable parameters” (Sober 2008, p.79). Recall that thelevel-k analysis in Section 2.3B usually gives a “point prediction” with a probability of
one assigned to a specific strategy. This can be problematic when we conduct an
econometric comparison because the value of the log-likelihood of a probability of zerois minus infinity. Thus, we decided to introduce the same kind of noises across all themodels, which may be simply interpreted as the errors in the implementation of theactions agents choose. Specifically, we assume that the choice probability of C for eachplayer 𝑖 is given by the following logit form,
𝑝" =

exp A
exp A

𝐸𝑈" (𝐶)𝜇 B

𝐸𝑈" (𝐶)𝐸𝑈" (𝑁)𝜇 B + exp A 𝜇 B,

𝑖 = 1,2, … , 𝑛,(5)where 𝜇 is the noise parameter, and 𝐸𝑈" (𝐶 ) (𝐸𝑈" (𝑁)) is the expected utility of player
𝑖 for choosing C (𝑁). In the model of inequality aversion, player 𝑖’s expected utility is
calculated by (4); that is, the utility of an inequality averse player. In the level-k model,
if player 𝑖 is L𝑘 player, his expected utility is the expected payoff when all other players
are L(𝑘 − 1) players. In our estimation of QRE, we also use 𝜇 = 1/λ for ease ofcomparison.
In each model, the probabilities of each player’s choices are obtained in a fixed
point of the above 𝑛 equations for any given values of parameters. Let 𝜔 be the vectorof parameters for each model (for example, 𝜔 = (𝜇, 𝛼, 𝛽 ) in inequality aversion),—6—6

𝑝L (𝜔) be the choice probability of the S-player, and 𝑛L be the number of S-players—6—6
who chose C in game D- 𝑗. 𝑝M (𝜔), 𝑛M , and notations for the other sessions are definedsimilarly. Then, for each model, the maximum likelihood estimates are obtained by
maximizing the following log likelihood function with respect to 𝜔.
œ

—

𝐿𝑜𝑔𝐿

œ
—6
—6
—6
—6
—6
(𝜔) = w 𝑛L ln f𝑝L (𝜔)g + wf𝑔L − 𝑛L gln f1 − 𝑝L (𝜔)g
6•I
6•I
œ

œ
—6
—6
—6
—6
—6
+ w 𝑛M ln f𝑝M (𝜔)g + wf𝑔ž − 𝑛M gln f1 − 𝑝M (𝜔)g,
6•I
6•I
—6

—6

where 𝑔L (𝑔M ) is the number of the S-players (W-players) in D- 𝑗. Since the subjects—6in Diekmann (1993) were not matched with one another to play the games, 𝑔L is not—6—Inecessarily the same as 𝑔M . For example, 𝑔L—I = 30, whereas 𝑔M= 29.
Table 8. Estimated parameters based on Diekmann (1993)17

SymmetricInequality

mixed

aversion

Level-1

Level-2

QRE

strategy
𝜶

-----

0.145

-----

-----

-----

𝜷

-----

0.778

-----

-----

-----

𝝁

-----

30.829**28.590**78.750**32.755**LogL-280.382-124.989-143.165-157.774-125.957

AIC

560.764255.978288.330

317.548253.915**, *, and + denote significance at the 1%, 5%, and 10% levels, respectively.
Table 8 shows the estimated parameters.8 In Level-k, we estimate two types ofmodels. In Level-1, it is assumed that all players are L1 players. Another model is Level2 where all players are assumed to be L2-players. The log likelihood for symmetric mixed
strategy equilibrium (in Table 4) is also shown in this table. For model comparisons, thefollowing Akaike information criterion (AIC) is used,
AIC = −2𝐿𝑜𝑔𝐿 + 2𝑘,
where k is the number of parameters in the model. The smaller the value of AIC, the betterthe model fit the data.
First, note that parameters 𝛼 and 𝛽 in the inequality aversion model are notsignificant. Thus, altruistic motivation does not play a role in explaining the data. Then,among the rest of the models, the value of AIC for QRE is minimum. Thus, we conclude
that QRE is the best fit model in the data reported in Diekmann (1993). This result andthe fact that S-equilibrium is frequently observed for two-person games are mutuallyconsistent because limiting QRE converges to risk dominant S-equilibrium in two-person
games according to Turocy (2005).
4. Our Experiment
In the previous section, we showed that Diekmann’s conjecture (which states that whenS-equilibrium is risk dominant, it is frequently observed) is only confirmed in twoperson games. However, our theoretical analysis in Section 2 proved that S-equilibrium
was risk dominant in the five-person game in his experiment. Thus, the explanatory
power of the risk dominance theory is not universal. Rather, QRE is the best fit modelin his data. Interestingly, inequality aversion also fails. As Diekmann’s experiment is
not an experiment in the exact sense, and for generalizing our findings, we need to run a
8

Our estimation procedures follow the method described in Chapter 16 in Moffatt (2016).
18

proper experiment with more variations. Thus, we decided to conduct our own
experiment.
4.1 Design and procedures
The experiments were conducted in November 2016 at Chuo University in Japan. The
subjects were undergraduates of the university and were recruited via a university e-maillist. Most of them were not from the Economics Department; only a few knew of gametheory, and none had previously participated in any experiment.
The games we used in the experiment were two-person and three-person
Volunteer’s dilemma games. In the two-person game, one player was an S-player, andthe other was a W-player. In the three-person game, we had two variations: In one, there
were one S-player and two W-players; In the other, there were two S-players and one Wplayer. We call the two-person game session as Session A, the first variation of the threeperson game session as Session B, and the second variation of the three-person gamesession as Session C.
In every session, 𝐿L = 𝐿M = 0, and the cost for volunteering is common forplayers of the same type; that is, 𝐾L = 200 and 𝐾M = 400. There were three sessions
for each of two-person and three-person games, which have different pair of benefits (forS- and W-players) from the public goods. Note that the condition (2) assumed in themodel in Section 2 is always satisfied with these parameter values. Thus, S-equilibriumis always risk dominant. Table 9 summarizes the details of each Session.
The experiments were conducted manually. Each subject was randomly
assigned to a seat in the room. There was sufficient physical distance between seats toprevent eye contact, and no oral communication was allowed among the subjects. The
experimental instructions were distributed, and the experimenter read them aloud in frontof the subjects.
Table 9. Summary of experimental sessions
Session

A-1

A-2

A-3

B-1

B-2

B-3

C-1

C-2

C-3

# of players

2

2

2

3

3

3

3

3

3

# of S-player

1

1

1

1

1

1

2

2

2

𝑽𝑺

600

600

350

1000

1000

600

1000

1000

600

𝑽𝑾

1000

600

600

1800

1000

1000

1800

1000

1000

19

Some subjects played the two-person game first and, then, without knowing theresults, played three-person game. The others played the three-person game first and, then,
the two-person game. Each game was played only once in each session.
A total of 30 subjects participated in each two-person game session, forming 15 groups.
A total of 36 subjects participated in each three-person game with one S-player session,and 12 groups were created. For the three-person game with two S-players session, thenumber of participants differs in each session due to no-show subjects: 13 groups in two
sessions and 10 groups in one session.
At the end of the experiment, the total payoffs each participant earned in bothgames was paid in cash. The conversion rate was one-to-one: each point earned was
exchanged for JPY 1 at the end of the experiments. No participation fee was paid to eachsubject. For the one-hour experiment, the average reward was around JPY1,276(approximately 10 US dollars at the time). Details of the experimental procedure andinstructions are given in Appendix B.
Table 10 summarizes the type-1 symmetric mixed strategy equilibrium
predictions in our experiment, where the same type of players choose C with the sameprobability.
Table 10. Symmetric mixed strategy equilibrium prediction in the experiment.
Eq.

Session

A-1

A-2

A-3

B-1

B-2

B-3

C-1

C-2

C-3

1

𝑝L

0.600

0.333

0.333

0.503

0.106

0.307

0.529

0.368

0.368

𝑝M

0.667

0.667

0.429

0.553

0.553

0.423

0.576

0.684

0.473

𝑝L

---

---

---

0.000

0.000

0.000

0.800

0.800

0.667

𝑝M

---

---

---

0.778

0.600

0.600

0.000

0.000

0.000

2

4.2 Results
Table 11 summarizes our experimental data, where n is the number of groups, 𝑛L and𝑛M are the number of C chosen by an S-player and a W-player, respectively, and 𝑓L and𝑓M are the choice frequency of C by an S-player and a W-player, respectively.9 Notethat in Sessions B-1, B-2, and B-3, there are one S-player and two W-players in a group,and that in Sessions C-1, C-2, and C-3, there are two S-players and one W-player in agroup.

9

Raw data is given in Appendix B.
20

Table 11. Choice frequencies of each strategy.
Session

A-1

A-2

A-3

B-1

B-2

B-3

C-1

C-2

C-3

𝒏

15

15

15

12

12

12

13

13

10

𝒏𝑺

12

13

5

4

9

4

13

14

9

𝒏𝑾

6

2

7

13

9

11

8

3

5

𝒇𝑺

0.800

0.867

0.333

0.333

0.750

0.333

0.500

0.538

0.450

𝒇𝑾

0.400

0.133

0.467

0.542

0.375

0.458

0.615

0.231

0.500

Figure 1 shows the difference between the probabilities obtained from the completely
mixed strategy equilibrium and the choice frequencies. In most Sessions, S-players chose
C more frequently than the mixed strategy equilibrium. Significant differences are visible
in Sessions A-2 and B-2, and, in fact, the null hypothesis which states that the choicefrequency of C and the probabilities obtained from the completely mixed strategy
equilibrium are equal is rejected for S-players in Sessions A-2 and B-2 (binomial test, p< 0.01 for each).
Moreover, S-players chose C most often in the second treatment in every
Session (86.7%, 75.0%, and 53.8% in Session A-2, B-2, and C-2 respectively). This fact
contradicts the mixed strategy equilibrium prediction since the lowest probabilities(33.3%, 10.6%, and 36.8% in Session A-2, B-2, and C-2 respectively) should be assignedin the equilibrium to the second treatment among the three treatments.
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
-0.1

A-1

A-2

A-3

B-1

B-2

-0.2
-0.3

(a) S-player
21

B-3

C-1

C-2

C-3

0.1
0
A-1

A-2

A-3

B-1

B-2

B-3

C-1

C-2

C-3

-0.1
-0.2
-0.3
-0.4
-0.5
-0.6

(b) W-player
Figure 1. Difference between mixed strategy and the choice frequencies.
On the contrary, W-players chose C less frequently than the mixed
strategy equilibrium. The null hypothesis stating that the choice frequency of C and theprobabilities obtained from completely mixed strategy equilibrium are equal is rejected
for W-players in Sessions A-1, A-2, and C-2 (binomial test, p < 0.01 for each). Especially,
W-players chose C least often in the second treatment in every Session (13.3%, 37.5%,
and 23.1% in Session A-2, B-2, and C-2 respectively). However, in the mixed strategy
equilibrium prediction, the highest probabilities (66.7%, 55.3%, and 68.4% in Session A2, B-2, and C-2 respectively) should be assigned to the second treatment among the threetreatments.
From these, we conclude that the mixed strategy equilibrium prediction fails inour data. Thus, we reject Hypothesis 2.
Then, how about equilibrium selection from risk dominance? We count the
number of occurrences for each equilibrium and summarize them in Table 12. Here wecount it as an S-equilibrium (W-equilibrium) when exactly one S-player (W-player)choose C and the other players choose N. Otherwise, the subjects made coordination
failures, which are either over-cooperation where more than one player chooses C
including every player choosing C (“All-C”) or played mutually non-cooperatively bychoosing N (“All-N”). This is never an equilibrium in our setting.
22

Table 12. The percentage of each outcome in the experiment.
Session

A-1

A-2

A-3

B-1

B-2

B-3

C-1

C-2

C-3

S-eq.

60.0

73.3

13.3

8.3

41.7

8.3

15.4

30.8

30.0

W-eq.

13.3

0.0

26.7

41.7

0.0

25.0

15.4

15.4

20.0

All-C

26.7

13.3

20.0

0.0

16.7

8.3

7.7

0.0

10.0

All-N

0.0

13.3

40.0

0.0

8.3

25.0

7.7

15.4

10.0

Others

---

---

---

50.0

33.3

20.0

53.8

38.4

30.0

While S-equilibrium always risk dominates W-equilibrium, theoretically, inevery Session, sometimes the percentages of W-equilibrium are higher than that of Sequilibrium in our data (Sessions A-3, B-1, and B-3). More so, the percentages of noncooperative outcomes (“All-N”) are higher than that of S-equilibrium in some Sessions(Sessions A-3 and B-3). In fact, the null hypothesis stating that the frequency of Sequilibrium is equal among three treatments (such as among A-1, A-2, and A-3) isrejected in both Sessions A and B (Chi-squared test, p < 0.01 for Session A and p < 0.05
for Session B).
If the sum of the percentages of W-equilibrium and the non-cooperative
outcome is seen as counter-evidence against S-equilibrium prediction, risk dominanceprediction fails in more than a half of Sessions (except for Sessions A-1, A-2, B-2 and C3). Thus, we have to conclude from these data that the explanatory power of the risk
dominance concept is very limited. Therefore, we reject Hypothesis 1.
Finally, as both Hypotheses 1 and 2 are rejected, we will examine whether
asymmetric mixed strategy equilibrium can explain our data. In Sessions B-1, B-2, andB-3, there are asymmetric mixed strategy equilibria in which two W-players choose C
with different probabilities. In these cases, one of the W-players chooses N with aprobability of 1 and the other W-players and S-players choose C with a probability of lessthan 1. Likewise, in Sessions C-1, C-2, and C-3, there are asymmetric mixed strategy
equilibria in which two S-players choose C with different probabilities. In these cases,
one of the S-players chooses N with a probability of 1 and the other S-players and Wplayers choose C with a probability of less than 1.
Asymmetric mixed strategy equilibrium predictions in the experiment aresummarized in Table 13. The percentages of every possible outcome implied by theasymmetric mixed strategy equilibrium are shown in Table 14. Differences of outcome

23frequencies between predictions of asymmetric mixed strategy equilibrium in theexperiment are also presented in Figure 2.
Table 13. Asymmetric mixed strategy equilibria.
B-1

B-2

B-3

C-1

C-2

C-3

𝒑𝑺

0.778

0.600

0.600

𝒑𝑺

0.000

0.000

0.000

𝒑𝑾

0.000

0.000

0.000

𝒑𝑺

0.778

0.600

0.600

𝒑𝑾

0.800

0.800

0.667

𝒑𝑾

0.800

0.800

0.667

Table 14. The percentages of each outcome implied by asymmetric mixed strategy
equilibria.
B-1

B-2

B-3

C-1

C-2

C-3

S-eq.

15.6

12.0

20.0

0.0

0.0

0.0

W-eq.

17.8

32.0

26.7

33.4

44.0

46.7

All-C

0.0

0.0

0.0

0.0

0.0

0.0

All-N

4.4

8.0

13.3

4.4

8.0

13.3

Others

62.2

48.0

40.0

62.2

48.0

40.0

S-eq.

W-eq.

40
30
20
10
0
All C

All N

-10
-20
-30
-40
B-1

B-2

B-3

(a) Session B

24

Others

40
30
20
10
0
-10

S-eq.

W-eq.

All C

All N

Others

-20
-30
-40
C-1

C-2

C-3

(b) Session C
Figure 2. Differences of outcome frequencies between predictions of asymmetric
mixed strategy equilibria in the experiment.
In Session B-1, the frequency of S-equilibrium in the experiment is slightly
lower than theoretical predictions, while the frequency of W-equilibrium is significantly
higher. The null hypothesis stating that the frequency of W-equilibrium is equal to theprobability implied by asymmetric mixed strategy equilibrium is rejected for Session B1 (binomial test, p < 0.01), but the null hypothesis that the frequency of S-equilibrium isequal to the probability implied by the asymmetric mixed strategy equilibrium is not
rejected. The opposite is the case in Session B-2 where the frequency of S-equilibrium inthe experiment is significantly higher than the theoretical predictions (binomial test, p <0.01), while the frequency of W-equilibrium is significantly lower (binomial test, p <0.01). In Session B-3, non-equilibrium outcomes, All-C, All-N, and others, due tocoordination failures, are prominent.
In Session C-1, C-2, and C-3, frequencies of S-equilibrium in the experimentare significantly higher than the theoretical predictions (p < 0.01 for each), while thefrequencies of W-equilibrium are significantly lower. The null hypothesis stating that thefrequency of W-equilibrium is equal to the probability implied by asymmetric mixed
strategy equilibrium is not rejected for every session. In summary, we must conclude thatthe asymmetric mixed strategy equilibrium prediction also fails in our experimental data.
25

Thus, Diekmann’s conjecture that S-equilibrium is played more frequently
when it is risk dominant also fails in our experiment. The mixed strategy equilibrium
prediction fails as well. Therefore, we have to examine alternative theories with altruistic
motivation or bounded rationality.
4.3 Econometric estimation
As in Section 3, we examine three alternative theories, inequality aversion, level-k
mode, and QRE as representative models in behavioral game theory. Model structuresand estimation strategy are identical in Section 3.
A. Inequity aversion
Ranges of parameter values consistent with each equilibrium are shown in Table 15.
Moreover, remember that the most prominent regularity in our data is that the frequencies
of S-equilibrium vary among sessions, unlike the prediction of risk dominance.
In fact, from Table 14, S-equilibrium was observed most frequently in thesecond treatment in every Session (73.3%, 41.7% and 30.8% in Session A-2, B-2 and C2 respectively), while W-equilibrium was observed most frequently in the third treatment
in Sessions A and C (26.7% and 20.0% in Session A-3 and C-3 respectively) and the firsttreatment in Session B (41.7%).
Table 15. The prediction of inequity aversion.10
A-1

A-2

A-3

B-1

B-2

B-3

C-1

C-2

C-3

2
3

𝛼≤2

𝛼≤

1
3

𝛼≤

4
5

𝛼≤4

𝛼≤

2
3

𝛼≤

4
3

𝛼≤4

𝛼≤1

1
2

𝛼≤

4
3

3
2

any 𝛼

S-eq.

𝛼≤

W-eq.

any 𝛼

𝛼≤

All-N

--

2≤𝛼

4
3

≤𝛼

𝛼+𝛽
≤7
7
≤ 𝛼+𝛽

3
2

𝛼≤3

any 𝛼

𝛼≤

4≤𝛼

3≤𝛼

--

4≤𝛼

𝛼≤

--

In fact, the most frequent outcome was an S-equilibrium in sessions A-1, A-2,
B-2, C-1, C-2, and C-3. However, All-N was the most frequent in sessions A-3 and B-3,
and W-equilibrium was the most frequent in session B-3.

10

The symbol “—“ implies that the outcome cannot be a Nash equilibrium irrespective of the valuesof 𝛼 and 𝛽.
26

But these regularities are not consistent with inequality aversion for any
parameter values shown in Table 15.
B. Level-k model
Assume that L0 player, who is the least rational player, chooses C and N with a probabilityof 1/2 respectively. In the level-k model, L𝑘 player chooses the best response to theactions taken by L(𝑘 − 1) players. If we assume 𝐿L = 𝐿M = 0 as in the experimental
sessions, the best responses of L𝑘 S-player and L𝑘 W-player to the actions of L(𝑘 − 1)players up to 𝑘 = 4 are summarized in Table 16.
Table 16. Best response of Lk to L(k-1)

L1
L2
L3
L4

A-1

A-2

A-3

B-1

B-2

B-3

C-1

C-2

C-3

S-player

C

C

N

C

C

N

C

C

N

W-player

C

N

N

C

N

N

C

N

N

S-player

N

𝐶

C

N

C

C

N

N

C

W-player

N

N

C

N

N

C

N

N

C

S-player

C

C

N

C

C

N

C

C

N

W-player

C

N

N

C

N

N

C

C

N

S-player

N

C

C

N

C

C

N

N

C

W-player

N

N

C

N

N

C

N

N

C

As L1 and L3, as well as L2 and L4, choose exactly the same responses in every
Session, it is enough to consider up to L2 in the analysis. Then, suppose that both an Splayer and a W-player have the same level of strategic thinking. This assumption isnatural and justifiable, as the subjects were in the same population, and each player’s role
was randomly determined in the experiment.
Remember that S-equilibrium was observed most frequently in the second
treatment in every session in our experiment. The fact can be obtained if all the players
are L1 (In this case, an S-player chooses C and a W-player chooses N.). Thus, the level-k
model can explain a part of the regularities found in our experiment.
Note, however, that whatever level the players find themselves in, the level-k
model predicts that a W-equilibrium where an S-player chooses N and a W-player chooses
C never occur in every session. This contradicts the fact that the percentages of Wequilibrium are higher than that of S-equilibrium in some sessions in our experimental
data.
27

C. Estimation results
Similar to the previous section, we conduct a maximum likelihood estimation for
obtaining the parameter values for inequality aversion, the level-k model, and QRE. For
each model, the maximum likelihood estimates are obtained by maximizing the following
log-likelihood function with respect to 𝜔.
𝐿𝑜𝑔𝐿(𝜔) = 𝐿𝑜𝑔𝐿¨ (𝜔) + 𝐿𝑜𝑔𝐿© (𝜔) + 𝐿𝑜𝑔𝐿F (𝜔),where each term is defined similarly to 𝐿𝑜𝑔𝐿— (𝜔) in Section 3.11 Table 17 shows theestimated parameters. The log likelihood for symmetric mixed strategy equilibrium(especially, Equation (1) in Table 10) is also shown in this table.12
From Table 17, one can see that the value of AIC for QRE is minimum among
them. Thus, we conclude that QRE is the best fit model in our experimental data. However,since λ = 1/𝜇, a relatively higher value of 𝜇̂ = 25.781 means that players’ choices areclose to the ones made via a low level of rationality, as λ = 0.039. Next, the value of
AIC for inequality aversion is lowest, but none of the parameters concerning inequality
aversion, 𝛼 and 𝛽 , is significant, and only 𝜇 is significant. Finally, Level-1’s
performance is better than Level-2. All these facts indicate that our data is generated by
players with a relatively low level of rationality.
Table 17. Estimated parameters
Symmetric

Inequality

Level-1

Level-2

QRE

mixed strategy

aversion

𝜶-----0.0241

-----

-----

-----

𝜷

-----

0.0547

-----

-----

-----

𝝁

-----

25.738**22.752**

186.228+

25.781**LogL-237.324-202.188-205.784-210.384-202.351

AIC

474.648410.377

413.569422.767406.702**, *, and + denote significance at the 1%, 5%, and 10% levels, respectively.

11

In this definition, it is implicitly assumed that players of the same type have the same choiceprobabilities in a fixed point. Even if we redefine the log likelihood function to consider the possibility
in which players of the same type have different choice probabilities, no result of the estimation
changes.
12
As shown in Tables 10 and 13, there are the other mixed strategy equilibria in Session B and C.
However, the AIC when assuming Equation (1) in all sessions is smaller than those for othercombinations of equilibria.
28But this does not necessarily imply that subject choice were random. QRE
captures some of the regularities in our experimental data. In fact, Table 18 shows choiceprobabilities implied by the estimated parameter values, which are closer to the data than
the symmetric mixed strategy equilibrium prediction.
Table 18. Choice probabilities implied by the estimated parameters
SessionActual Choice

Symmetricfrequencies

mixed strategy

QRE

A-1

𝑝𝑆

0.800

0.600

0.591

A-1

𝑝𝑤

0.400

0.667

0.509

A-2

𝑝𝑆

0.867

0.333

0.703

A-2

𝑝𝑤

0.133

0.667

0.298

A-3

𝑝𝑆

0.333

0.333

0.510

A-3

𝑝𝑤

0.467

0.429

0.399

B-1

𝑝𝑆

0.333

0.503

0.542

B-1

𝑝𝑤

0.542

0.553

0.507

B-1

𝑝𝑤

0.542

0.553

0.507

B-2

𝑝𝑆

0.750

0.106

0.768

B-2

𝑝𝑤

0.375

0.553

0.287

B-2

𝑝𝑤

0.375

0.553

0.287

B-3

𝑝𝑆

0.333

0.307

0.518

B-3

𝑝𝑤

0.458

0.423

0.396

B-3

𝑝𝑤

0.458

0.423

0.396

C-1

𝑝𝑆

0.500

0.529

0.537

C-1

𝑝𝑆

0.500

0.529

0.537

C-1

𝑝𝑤

0.615

0.576

0.487

C-2

𝑝𝑆

0.538

0.368

0.588

C-2

𝑝𝑆

0.538

0.368

0.588

C-2

𝑝𝑤

0.231

0.684

0.290

C-3

𝑝𝑆

0.450

0.368

0.493

C-3

𝑝𝑆

0.450

0.368

0.493

C-3

𝑝𝑤

0.500

0.473

0.365

29

6. Conclusion
We have analyzed a generalized version of an asymmetric Volunteers’ Dilemma (VOD)
game where the cost for volunteering is different among players. In this game, there is Sequilibrium where a player with less cost contributes more often. S-equilibrium is
intuitively appealing, and, in fact, it is an efficient outcome. Under certain plausible
conditions, S-equilibrium is risk dominant for a general n-person game, which was, firstly,
proved in this study. As many researchers accept the risk dominance concept as anequilibrium selection criterion, the plausibility of S-equilibrium is reinforced by that fact.
Thus, our first task was to closely examine Diekmann’s conjecture which states that whenS-equilibrium is risk dominant, it is observed more frequently. However, this conjecture
was not fully confirmed in both Diekmann’s and our experiments, even though Sequilibrium was risk dominant in every session.
As for the prediction by the mixed strategy equilibrium, which sometimes
contradicts with our intuition, if we consider people’s volunteering decision in the timeof a natural disaster, we can say that it is not an unacceptable prediction. Sometimes aplayer with more cost contributes more often, as the Parable of the Good Samaritan
illustrates. However, again, it failed in explaining Diekmann’s and our data.
Thus, we further analyzed the game with alternative theories including
inequality aversion, level-k model, and quantal response equilibrium (QRE) as
representative models in behavioral game theory. Interestingly, our econometriccomparison results show that altruistic motivation reflected by the inequality aversion
model play no role in explaining the data. Instead, QRE best fits the data. In fact, choiceprobabilities implied by QRE is closer than the mixed strategy equilibrium prediction.
For two-person games, it is shown that limiting QRE converges to risk
dominant equilibrium. Thus, for two-person games, the frequency of S-equilibrium isrelatively higher when S-equilibrium is risk dominant. However, for a general n-person
game, whether limiting QRE converges to risk dominant equilibrium is unknown.
Especially, if there are more than two S-players in the game, as in our experiment, theprincipal branch of QRE correspondence bifurcates because there exits at least twosymmetric S-equilibria. Analyzing, theoretically, such complicated nonlinear dynamics
is quite difficult at this moment. The numerical simulation also shows quite exotic
behaviors and no convergence. Fortunately, as the best fit parameter of QRE was found
before the principal branch of QRE correspondence bifurcates, our conclusion is intact, aclose examination of limiting QRE in this game requires more elaboration in the future
research.
30

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American Economic Review, 90, 166-193.
Crawford, V. P., M. A. Costa-Gomes, and N. Iriberri (2013) “Structural models of
nonequilibrium strategic thinking: Theory, evidence, and applications.” Journal of
Economic Literature, 51, 5-62.
Darley, J. M. and B. Latané (1968) “Bystander intervention in emergencies. Diffusionof responsibility.” Journal of Personality and Social Psychology, 8, 377-383.
Diekmann, A. (1985) “Volunteer’s Dilemma” Journal of Conflict Resolution, 29, 605610.
Diekmann, A. (1993) “Cooperation in an asymmetric volunteer’s dilemma game: theory
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Fehr, E. and K. M. Schmidt (1999) “A theory of fairness, competition and cooperation.”
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Güth, W. (1992) “Equilibrium selection by unilateral deviation stability.” in R. Seltened. (1992) Rational Interaction Essays in Honor of John C. Harsanyi, Springer Verlag,
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Harsanyi, J. C. and R. Selten (1988) A General Theory of Equilibrium Selection inGames. The MIT Press: Cambridge.
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31

Kawagoe, T., T. Matsubae and H. Takizawa (2018) “Quantal Response Equilibria in a
Generalized Volunteer’s Dilemma and Step-level Public Goods Games with Binary
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McKelvey, R. D. and T. R. Palfrey (1995) “Quantal response equilibrium for normalform games.” Games and Economic Behavior, 10, 6-38.
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32

Appendix A. Proofs
Proposition 1. Suppose that player 𝑖 takes a completely mixed strategy in a mixed
strategy equilibrium 𝜃 (𝑖 ∈ 𝐶(𝜃)), then𝑞" = A𝐾6
𝑉" − 𝐿"BC 5 D
EH
𝐾"
𝑉6 − 𝐿6I#F(G)KI(1).

6∈F(G)

where #𝐶(𝜃) is the number of 𝐶(𝜃).
Proof. If some players take completely mixed strategies, then the rest of players choose
N with probability one in an equilibrium 𝜃 . Thus 0 < 𝑞" < 1 for any 𝑖 ∈ 𝐶(𝜃) .
Therefore, 𝐸" (𝐶 ) = 𝐸" (𝑁) must hold, and we have𝐾"
5 𝑞6 =,
𝑉" − 𝐿"67"

for any 𝑖 ∈ 𝐶(𝜃). Therefore, we have
5 45 𝑞6 8 = 5 A"∈F(G)

67""∈F(G)𝐾"
B.
𝑉" − 𝐿"

Note that for any player ℎ ∈ 𝐶(𝜃), 𝐶 (𝜃) ∖ {ℎ} ≠ ∅ since at least two players belong to𝐶(𝜃). Solving the left-hand side of the above equation for 𝑞± , we have
5 45 𝑞6 8 = 45 𝑞6 8"∈F(G)

67"

67±

5

4𝑞± 5 𝑞6 8"∈F(G)∖{±}

67",±= 𝑞± #F(G)KI 45 𝑞6 867±

5

4 5 𝑞6 8"∈F(G)∖{±}

67",±#F(G)Ks= 𝑞±#F(G)KI 4567±

𝑞6 8 45 𝑞6 8
67±

= 𝑞±

#F(G)KI#F(G)KI𝐾±
A
B.
𝑉± − 𝐿±The third equality follows from the fact that 𝑞_ = 1 for any 𝑘 ∉ 𝐶 (𝜃). Thus, we haveequation (1) for any player ℎ ∈ 𝐶(𝜃).
Q.E.D.
Proposition 2. For any 𝑛 ≥ 2 and 1 ≤ 𝑚 ≤ 𝑛, the volunteer’s dilemma game has thecompletely mixed strategy equilibrium if and only if
33

A

𝑉M − 𝐿M RKS
𝑉L − 𝐿L RKSKI
B
>A
B
.
𝐾M
𝐾L

(3)

Proof. By Proposition1, the probability of choosing N of a W-player, 𝑞M , in thecompletely mixed strategy equilibrium is given byIS𝐾o𝑉ž − 𝐿ž SKI RKI´
𝑞M = ³A
B A
B
𝑉o − 𝐿o𝐾žThis is always a positive number. Moreover, 1 > 𝑞M holds by (2) since
𝑉L − 𝐿L S
𝑉M − 𝐿M SKI
1 > 𝑞M ⟺ A
B >A
B
.
𝐾L
𝐾M
The probability of choosing N of a S-player, 𝑞L , in the completely mixed strategy
equilibrium is given byI

RKS RKI
𝑉L − 𝐿L RKSKI
𝐾M´ .
𝑞L = ³A
B
A
B
𝐾L
𝑉M − 𝐿M

This is always a positive number. Moreover,
𝑉M − 𝐿M RKS
𝑉L − 𝐿L RKSKI
1 > 𝑞L ⟺ A
B
>A
B
.
𝐾M
𝐾L
Thus, 𝑞M always satisfy 1 > 𝑞M > 0 and (3) gives the necessary and sufficientcondition for 1 > 𝑞L > 0.
Q.E.D.
Proposition 3. Suppose that 𝜃 ∈ 𝛩(𝑛, 𝑚) is the completely mixed strategy equilibrium,and 𝑞L (𝑛, 𝑚) and 𝑞M (𝑛, 𝑚) are the probabilities of choosing 𝑁 of S-player and Wplayer in 𝜃, respectively. For any two non-negative integers 𝑙 and ℎ, consider thevolunteer’s dilemma game that the number of players is 𝑛 + 𝑙 + ℎ and the number of Splayers is 𝑚 + 𝑙. In this game, a mixed strategy profile 𝜃′ such that 𝑙 S-players andℎ W-players are choosing 𝑁 with probability one, 𝑚 S-players take 𝑞L (𝑛, 𝑚), and𝑛 − 𝑚 W-players take 𝑞M (𝑛, 𝑚) is a mixed strategy equilibrium, i.e., 𝜃′ ∈ 𝛩(𝑛 + 𝑙 +ℎ, 𝑚 + 𝑙).
Proof. For any S-player 𝑖 with 𝑞" = 𝑞L (𝑛, 𝑚) in 𝜃′,

34

SKI

RKS

RKS RKIS
𝑉L − 𝐿L RKSKI
𝐾M
𝐾L
𝑉M − 𝐿M SKI RKI´³A´
5 𝑞6 = ³A
B
A
B
B A
B
𝐾L
𝑉M − 𝐿M
𝑉L − 𝐿L
𝐾M
67"

=A

𝐾L
B.
𝑉L − 𝐿L

Therefore, 𝐸" (𝐶 ) = 𝐸" (𝑁). Similarly, for any W-player 𝑖 with 𝑞" = 𝑞M (𝑛, 𝑚) in
𝜃′, 𝐸" (𝐶 ) = 𝐸" (𝑁). For any player 𝑖 with 𝑞" = 1 in 𝜃′,IS
RKS RKI𝐾L
𝐾M´ .
5 𝑞6 = ³A
B A
B
𝑉L − 𝐿L
𝑉M − 𝐿M
67"

Therefore, for any S-player 𝑖 with 𝑞" = 1 in 𝜃′,𝐸" (𝑁) > 𝐸" (𝐶 ) ⇔ 5 𝑞6 < A
67"

𝐾L
B
𝑉L − 𝐿LIS
RKS RKI
𝐾L
𝐾M
𝐾L
´
⇔ ³A
B A
B
<A
B
𝑉L − 𝐿L
𝑉M − 𝐿M
𝑉L − 𝐿L

𝑉M − 𝐿M RKS
𝑉L − 𝐿L RKSKI
B
>A
B
.
𝐾M
𝐾L
The last inequality is always true by (3) since Θ(𝑛, 𝑚) has the completely mixed strategy⇔Aequilibrium. For any W-player 𝑖 with 𝑞" = 1 in 𝜃′,𝐸" (𝑁) > 𝐸" (𝐶 ) ⇔ 5 𝑞6 < A
67"

𝐾M
B
𝑉M − 𝐿MISRKS RKI
𝐾L
𝐾M
𝐾M
´
⇔ ³A
B A
B
<A
B
𝑉L − 𝐿L
𝑉M − 𝐿M
𝑉M − 𝐿M

𝑉L − 𝐿L S
𝑉M − 𝐿M SKI
⇔A
B >A
B
.
𝐾L
𝐾M
The last inequality is always true by (2).
Q.E.D.
Proposition 4. Suppose that 𝛩(𝑛, 𝑚), 𝛩(𝑛 + 1, 𝑚), and 𝛩(𝑛 + 1, 𝑚 + 1) have thecompletely mixed strategy equilibria, then(i) 𝑞L (𝑛 + 1, 𝑚) > 𝑞L (𝑛, 𝑚),(ii) 𝑞M (𝑛 + 1, 𝑚) > 𝑞M (𝑛, 𝑚),(iii) 𝑞L (𝑛 + 1, 𝑚) > 𝑞L (𝑛 + 1, 𝑚 + 1), and(iv) 𝑞M (𝑛 + 1, 𝑚) > 𝑞M (𝑛 + 1, 𝑚 + 1).
35

Proof. Suppose that 𝜃 ∈ Θ(𝑛, 𝑚) is the completely mixed strategy equilibrium. Then,
we have𝑛−𝑚−1
𝑉L − 𝐿L𝑛−𝑚
𝑉M − 𝐿M
ln 𝑞L =
ln A
B−
ln A
B
𝑛−1
𝐾L
𝑛−1
𝐾M
Differentiate both sides by 𝑛, we haved𝑞L𝑞L
𝑉L − 𝐿L
𝑉M − 𝐿M=·𝑚 ln AB − (𝑚 − 1) ln A
B¸ > 0.
sd𝑛(𝑛 − 1)𝐾L
𝐾M
The last inequality follows from the fact that
𝑉L − 𝐿L 𝑉M − 𝐿M
>
> 1.
𝐾L
𝐾M
On the other hand, differentiate both sides by 𝑚, we haved𝑞L𝑞L
𝑉L − 𝐿L
𝑉M − 𝐿M=−·ln A
B − ln A
B¸ < 0d𝑚(𝑛 − 1)𝐾L
𝐾M
For W-player, we haveln 𝑞M = −𝑚
𝑉L − 𝐿L
𝑚−1
𝑉M − 𝐿M
ln A
B+
ln A
B.
𝑛−1
𝐾L
𝑛−1
𝐾M

Differentiate both sides by 𝑛, we haved𝑞M
𝑞M
𝑉L − 𝐿L
𝑉M − 𝐿M=·𝑚 ln AB − (𝑚 − 1) ln A
B¸ > 0sd𝑛(𝑛 − 1)𝐾L
𝐾M
On the other hand, differentiate both sides by 𝑚, we haved𝑞M
𝑞M
𝑉L − 𝐿L
𝑉M − 𝐿M=−·ln A
B − ln A
B¸ < 0.
d𝑚(𝑛 − 1)𝐾L
𝐾M
Q.E.D.
Proposition 5’. In the case of 𝑛 ≥ 2 and 1 ≤ 𝑚 < 𝑛, S-equilibrium risk dominants Wequilibrium in the sense of unilateral deviation stability.
Proof. Note that there are 𝑚 S-equilibrium and n − 𝑚 W-equilibrium. For any pair(𝑠, 𝑡) where 𝑠 is a S-equilibrium and 𝑡 is a W-equilibrium, 𝑀(𝑠, 𝑡) = {𝑖, 𝑗} where 𝑖is a S-player and 𝑗 is a W-player, respectively, and players other than 𝑖 and 𝑗 choose
N in both 𝑠 and 𝑡. Therefore, 𝐺"6 (𝑠, 𝑡) is similar to Table 1, and we have𝑅∗ (𝑠, 𝑡) = 𝑅"6 (𝑠, 𝑡) =𝐾M (𝑉L − 𝐿L − 𝐾L ).
𝐾L (𝑉M − 𝐿M − 𝐾M )Thus, 𝑅∗ (𝑠, 𝑡) > 1 if and only if
36

A

𝑉L − 𝐿L
𝑉M − 𝐿M
B>A
B.
𝐾L
𝐾M
Q.E.D.

Proposition 6. Suppose that player 𝑖 is an inequity averse player. Then player 𝑖 neverchoose C when all other players choose C.
Proof. For player 𝑖, let 𝐻(𝑖) be the set of players such that 𝑉6 − 𝐾6 ≥ 𝑉" , 𝐿(𝑖) be theset of players such that 𝑉" − 𝐾" > 𝑉6 − 𝐾6 , and 𝑀(𝑖) be the set of players other than 𝑖such that 𝑉" > 𝑉6 − 𝐾6 ≥ 𝑉" − 𝐾" (the players of same type as 𝑖 belong to 𝑀(𝑖)). Notethat 𝐻 (𝑖 ) ∩ 𝐿(𝑖 ) = ∅ 𝑎𝑛𝑑 𝐻 (𝑖 ) ∩ 𝑀(𝑖 ) = ∅ by their definition. Then, if all players
choose C, the utility of player 𝑖 is given by(𝑉" − 𝐾" ) −𝛼𝑛−1

f𝑉6 − 𝐾6 − 𝑉" + 𝐾" g −w
6∈½(")∪n(")

𝛽w f𝑉" − 𝐾" − 𝑉6 + 𝐾6 g.
𝑛−1
6∈¿(")On the other hand, if player 𝑖 chooses N and the other players choose C, then player 𝑖’s
utility is
𝑉" −

𝛼
𝛽
w f𝑉6 − 𝐾6 − 𝑉" g −𝑛−1
𝑛−1
6∈½(")wf𝑉" − 𝑉6 + 𝐾6 g.

6∈n(")∪¿(")

Therefore, if player 𝑖 deviates from All-C, player 𝑖’s utility will increase by𝛼𝐾" +w if𝑉6 − 𝐾6 − 𝑉" + 𝐾" g − f𝑉6 − 𝐾6 − 𝑉" gj𝑛−1
6∈½(")+

𝛽w if𝑉" − 𝐾" − 𝑉6 + 𝐾6 g − f𝑉" − 𝑉6 + 𝐾6 gj
𝑛−1
6∈¿(")+

𝛼𝛽w f𝑉6 − 𝐾6 − 𝑉" + 𝐾" g −w f𝑉" − 𝑉6 + 𝐾6 g.
𝑛−1𝑛−1
6∈n(")

6∈n(")

If 𝛽 = 0, then the above equation is positive since 𝐾" > 0 and the other terms arenonnegative. Suppose that 𝛽 > 0 and rearrange this equation, we have,37

𝐾" D1 −(1 + 𝛼 )#𝑀(𝑖 )#𝑀(𝑖 ) 𝛼#𝐻 (𝑖 ) 𝛽#𝐿(𝑖 )
𝛼+𝛽
+
−
E+
𝐾" −
w f𝑉" − 𝑉6 + 𝐾6 g
𝑛−1
𝑛−1
𝑛−1
𝑛−1
𝑛−1
6∈n(")≥ 𝐾" À1 −−(1 + 𝛼 )#𝑀(𝑖 )#𝑀(𝑖 )𝛽f#𝐻(𝑖) − #𝐿(𝑖 )gÁ ++𝐾"
𝑛−1 𝑛−1𝑛−1

𝛼+1
w f𝑉" − 𝑉6 + 𝐾6 g𝑛−1
6∈n(")= 𝐾" À1 −+#𝑀(𝑖 )𝛽f#𝐻(𝑖) − #𝐿(𝑖 )gÁ
+𝑛−1 𝑛−1

1+𝛼w f𝐾" − 𝑉" + 𝑉6 − 𝐾6 g𝑛−1
6∈n(")≥ 𝐾" À1 −#𝑀(𝑖 )𝛽f#𝐻(𝑖) − #𝐿(𝑖 )gÁ,+𝑛−1 𝑛−1

where #𝐻 (𝑖 ), #𝐿(𝑖 ), 𝑎𝑛𝑑 #𝑀(𝑖) be the number of their set, respectively. The firstinequality follows from 𝛽 < 1 and 𝛽 ≤ 𝛼. The last inequality follows from 𝐾" − 𝑉" +𝑉6 − 𝐾6 ≥ 0 for any 𝑗 ∈ 𝑀(𝑖 ). Since there are only two types of players, S and W, if#𝐻 (𝑖 ) > 0 then #𝐿(𝑖 ) = 0, and vice versa. Suppose that #𝐻 (𝑖 ) > 0, then the lastequation is positive since n − 1 ≥ #𝑀(𝑖 ) and 𝛽 > 0. Suppose that #𝐿(𝑖 ) > 0, then wehave

𝐾" À1 −

#𝑀(𝑖 )𝛽#𝑀(𝑖 ) 𝛽#𝐿(𝑖 )f#𝐻(𝑖) − #𝐿(𝑖 )gÁ = 𝐾" D1 −+−E𝑛−1 𝑛−1𝑛−1
𝑛−1> 𝐾" D1 −#𝑀(𝑖 ) #𝐿(𝑖 )#𝑀(𝑖 ) + #𝐿(𝑖 )−E = 𝐾" D1 −E ≥ 0,
𝑛−1 𝑛−1𝑛−1since 𝛽 < 1 and 𝑛 − 1 ≥ #𝑀(𝑖 ) + #𝐿(𝑖 ).
Q.E.D.

38

Appendix B. Instructions (originally in Japanese)
Session A-1: Instructions
In this experiment, you are paired with another player and make a decision.
Pairing is randomly determined and you are not informed of whom you are paired withduring and after the experiment.
Your role in the experiment is either X or Y, which is randomly determined.
As a result, there are a player whose role is X and a player whose role is Y in each pair.
Please check in the recording sheet about which role do you play in the experiment.
In the experiment, each player in a pair chooses option A or B simultaneously
and independently. If a player X chooses A, he/she has to pay 200 JPY. If a player Y
chooses A, he/she has to pay 400 JPY. If each player chooses B, no one needs to pay
anything.
Depending on the number of players who chose A, each player's payoff isdetermined by the following table.
Payoff table# of player who chose

0

1

Option A

400

400

Option B

0600

Option A

600

600

Option B

01000

A other than you
Player X
Player Y

Thus, if one member in your pair chooses A, a player X gains 600 JPY and a player Y
1000 JPY. Please note that if your role is X and choose A, as you have to pay 200 JPY,
your payoff becomes 400 JPY and that if your role is Y and choose A, as you have to
pay 400 JPY, your payoff becomes 600 JPY.
If no one chooses A including yourself, each player gains nothing.
Then, once you decide your choice, draw a circle on A or B in the recording
sheet. After everyone’s choice is made, experimenters collect your recording sheet.
If this is your first experiment today, please wait for a while for next
experiment. Your outcome in the experiment is not informed before all the experiments
ends. If this is your second experiment today, we sum up your outcomes in bothexperiments and pay the total to you in cash.
39

Session A-1: Recording sheet
Your identity：Pair ()

Role () Player ()

Payoff table# of player who chose

0

1

Option A

400

400

Option B

0600

Option A

600

600

Option B

01000

A other than you
Player X

Player Y

Your choice

A

40

or

B

Session B-1: Instructions
In this experiment, you are paired with other two players and make a decision.
Pairing is randomly determined and you are not informed of whom you are paired withduring and after the experiment.
Your role in the experiment is either X or Y, which is randomly determined.
As a result, there are a player whose role is X and two players whose role are Y in eachgroup. Please check in the recording sheet about which role do you play in theexperiment.
In the experiment, each player in a group chooses option A or B
simultaneously and independently. If a player X chooses A, he/she has to pay 200 JPY.
If a player Y chooses A, he/she has to pay 400 JPY. If each player chooses B, no one
needs to pay anything.
Depending on the number of players who chose A, each player's payoff isdetermined by the following table.
Payoff table# of player who

0

1

2

Option A

800

800

800

Option B

0

1000

1000

Option A

1400

1400

1400

Option B

0

1800

1800

chose A otherthan you
Player X

Player Y

Thus, if one member in your group chooses A, a player X gains 1000 JPY and a player
Y 1800 JPY. Please note that if your role is X and choose A, as you have to pay 200
JPY, your payoff becomes 800 JPY and that if your role is Y and choose A, as you haveto pay 400 JPY, your payoff becomes 1400 JPY.
If no one chooses A including yourself, each player gains nothing.
Then, once you decide your choice, draw a circle on A or B in the recording
sheet. After everyone’s choice is made, experimenters collect your recording sheet.
If this is your first experiment today, please wait for a while for next
experiment. Your outcome in the experiment is not informed before all the experiments
ends. If this is your second experiment today, we sum up your outcomes in bothexperiments and pay the total to you in cash.
41

Session B-1: Recording sheet
Your identity：Pair ()

Role () Player ()

Payoff table# of player who

0

1

2

Option A

800

800

800

Option B

0

1000

1000

Option A

1400

1400

1400

Option B

0

1800

1800

chose A otherthan you
Player X

Player Y

Your choice

A

42

or

B