Discussion Paper No.245

Quantal Response Equilibria in a Generalized
Volunteer’s Dilemma and Step-level Public Goods
Games with Binary Decision

Toshiji Kawagoe

Taisuke Matsubae Hirokazu Takizawa

February 2015

INSTITUTE OF ECONOMIC RESEARCH
Chuo University
Tokyo, JapanQuantal Response Equilibria in a Generalized
Volunteer’s Dilemma and Step-level Public Goods
Games with Binary Decision§Toshiji Kawagoe, Taisuke Matsubae, and Hirokazu TakizawaFebruary 3, 2015
Abstract
The present paper fully characterizes equilibria of a generalized Volunteer’s Dilemma
game, which is an integration of the volunteer’s dilemma game and the step-level public
goods game with binary decision. We also examined the explanatory power of a widely
accepted model with bounded rationality, the quantal response equilibrium (QRE). It isshown that the performance of the QRE model is better in explaining laboratory data.

Keywords: Volunteer’s dilemma, public goods, binary decision, quantal response
equilibrium
JEL classification number: C72, D72, D74, H41

We thank Andreas Diekmann and Axel Franzen for sharing raw data of theirexperiments. We also thank Sacha Bourgeois-Gironde for drawing our attention to the
Volunteer’s Dilemma game during our discussion.
 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.jp
 Institute of Economic Research, Chuo University. 42-1 Higashi-nakano, Hachioji,
Tokyo 192-0393, Japan. Tel. +81-42-67-3355 E-mail: fwin6544@nifty.com
 Faculty 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.jp§

1. Introduction
The Volunteer’s Dilemma game (VOD) was first formulated by Diekman (1993) toelucidate “social dilemmas” or “social traps” broader than those covered by theprisoner’s dilemma. A typical social situation is the helping behavior of people
witnessing an accident or crime, as best exemplified by the murder case of Kitty
Genovese examined by Darly and Latané (1968). It is said that her life could have beensaved if only one bystander had paid a small amount of cost (e.g., making an emergency
call to the police).
An interesting issue concerning VOD is the effect of the group size on thetendency to cooperate or contribute, the so called “bystander effect.” A large amount ofevidence has been accumulated by political scientists and psychologists to investigate
factors affecting this effect (Latané and Nida, 1981).
Biologists have also found that similar situations arise in groups of animals. It
is known that one member of the group occasionally looks up and checks for a predator
for protecting the group as a whole. Archetti and Scheuring (2010) extend VOD to thesituation where more than one volunteer is necessary (we call it a generalized VOD) and
derive an approximation formula for the equilibrium probability of contribution.
Technically speaking, a generalized VOD is a special case of the step-level public goods
game with binary decision (Croson and Marks, 2000). However, to the best of ourknowledge, the formal characterization of the mixed strategy equilibria in this class ofgames has not yet appeared.
Against these backdrops, this paper is first seen as an attempt to derive thenecessary and sufficient condition for the existence of the mixed strategy equilibrium ofgeneralized VOD. Furthermore we will analyze a generalized VOD with a widely
accepted behavioral model with bounded rationality, quantal response equilibrium(QRE) . Goeree et al. (2005a) report their QRE analysis of this game, but their analysis
does not seems to cover all cases. We will provide almost complete characterization of
the QRE of this class of games. Level-k analysis of a generalized VOD is brand new.
Finally, we will give the econometric estimation for the two models using laboratory
data in previous research. Our results show that QRE is better than level-k model in the
explanation of the data.

12. Model
2.1 General environment and pure strategy equilibria
Each of n players simultaneously decides whether to contribute (denoted by C), or not
to contribute (denoted by N). If more than or equal to m players contribute, public goods
is provided and a fixed amount of public goods benefit,
0, goes to all the players
regardless of their decisions. Otherwise, every player receives a fixed amount of payoff,
0. Cost of contribution is
0. Assume that.
There are two kinds of pure strategy Nash equilibria, cooperative andnon-cooperative. If contribution by m players is necessary for the provision of the
public goods, exactly m players’ choosing C constitutes a pure strategy Nashequilibrium. For
1, there exists a non-cooperative equilibrium in which no player
contributes. For m = 1, no such equilibrium exists.
2.2 Mixed strategy equilibria in normal-form
The following analyses focus on symmetric equilibria. Suppose that every player
contributes with probability p. Then, the expected payoff for C is
1
1
1
,
where
1

≡
is the probability that at least z out of
payoff for N is1

1 players choose C. Similarly, the expected1.
In equilibria in totally mixed strategy, the expected payoffs to C and N shouldbe equal. Thus we have
(1)

1

The left-hand side of (1) is the “marginal probability,” the difference of probability
between the event that at least
1 players contribute and the event that at least mplayers contribute.
The equation (1) can be simplified as follows.
≡

1
1

(2)

1

2

Solving the equation (2) numerically gives the mixed strategy probability. For the caseswith m = 1 or m = n, a unique solution can be explicitly solved as below (for m =1 case,Diekmann (1993)).
Proposition 1. For
1 or
1, there exists a unique symmetric mixed
strategy equilibrium. The probability of contribution in the symmetric equilibrium inmixed strategy is decreasing in n when m = 1 and increasing in n when m = n.
Proof. When m = 1, the equation (2) becomes1

1,

and the solution is obviously unique. It is also obvious that p is decreasing in n. When m= n, the equation (2) becomes.
In this case too, the solution is unique and p is increasing in n asassumption.

by
Q.E.D.

The reason that the equilibrium is unique when
1 or
1 is thatismonotone. When 1 < m < n,is bell shaped with0
1
0 as thefollowing calculation shows. Differentiating the left side of equation (2) with respect top yields
1
1
Obviously,

1

1,

0 for 0

1

(3)

.

1 and

0 for fortakes the maximum value when1.
1
In such cases, we have the following proposition.
∗

Proposition 2. Suppose 1

∗

. Let

∗

. (1 ) if

there is no symmetric equilibrium in totally mixed strategy. (2) if
then∗is the unique symmetric equilibrium. (3) if

3

∗

/
∗

, then//,, thenthere are two symmetric equilibria in totally mixed strategy, one with
other
Proof.

and the.
First suppose that∗/
Equation (2) has obviously no solution. If∗

there are two equilibria. If
equilibrium.

/
∗

.

(4)

,

(5)
,

/

∗

is the unique symmetric
Q.E.D.

With the parameter values V = 1.0, L = 0.2, and K = 0.2 used by Goeree et al.
(2005b), the numerical solutions of mixed strategies for (n, m) = (6, 3) and (n, m) = (14,7) are presented in Figure 1. The former has two solutions but the latter has no solution.

Figure 1. Mixed strategy probability for (n, m) = (6, 3) and (n, m) = (14, 7).
2.3 Quantal response equilibrium (QRE)
Quantal response equilibrium (QRE) is an equilibrium concept based on boundedly
rational strategic behavior, assuming that players play a noisy best response (McKelvey
and Palfrey, 1995). We focus on symmetric equilibria. For a parameter λ ∈ 0, ∞ , the4stochastic best response in terms of choice probability of C is given by
exp
exp

∙

1

∙
exp

1

∙

exp

∙

.

(6)

QRE is a fixed point of this mapping.
The parameter λ ∈ 0, ∞ represents the degree of rationality such that0
implies complete randomizing over pure strategies. If
0, then
1/2, which isusually called the centroid of the simplex of the strategy space. McKelvey and Palfrey(1995) show that, for almost all games, the graph λ, QRE λ contains a uniquebranch which starts at the centroid and converges to a unique Nash equilibrium as λgoes to infinity. The limit point of this principal branch is called limiting (logit) QRE.
Thus, limiting QRE can be interpreted as an equilibrium selection criterion.
Assuming

allows us to rearranging Equation (6) as
log

1
.

(7)

Note that the numerator log 1/ is monotone decreasing, and zero at
1/2.
It is positive in 0, 1/2 , and negative in 1/2,1 . With this in mind, we need toconsider the behavior of the denominator to analyze equation (7). As the proof of
Proposition 1 shows, the value ofsatisfying
/
when
1 or
2. Let the value be denoted as ̂ .

exists and unique,

Proposition 3. Suppose
1. Then the limiting logit QRE is ∗̂ (the uniquesymmetric Nash equilibrium in totally mixed strategy). For
1, the limiting∗∗∗
logit QRE is
0 if ̂ 1/2,
1 if ̂ 1/2 and
1/2 if ̂ 1/2.
Proof. Consider the case with
1. First consider the case that ̂ 1/2. Thenconsidering the branch starting at the centroid, we observe it is disconnected at ̂ , atwhich λ goes to infinity as
approaches ̂ from 1/2. This establishes the statement.
In the case ̂ 1/2, the branch starting at the centroid is connected over the interval0, ̂ ) and λ goes to infinity as
approaches ̂ from 1/2. When ̂ 1/2, thenstays at 1/2 along the graph of QRE.
Next suppose
1. Then. Over the interval 0, ̂ ), thedenominator of equation (7) is positive. If ̂ 1/2, then the branch starting at thecentroid is connected in 0, ̂ . Furthermore, λ goes to infinity as
approaches zero.
If ̂ 1/2, the branch starting at the centroid is connected in ̂ , 1 . λ goes to infinity
asapproaches unity. If ̂ 1/2,
1/2 is compatible with any λ 0. Other5branches are separated from the centroid.

Q.E.D.

(a)(b)
Figure 2. Typical QRE graphs when m = 1. (a) n = 2, m = 1, V = 1, L = K = 0.2 , (b)n = 2, m = 1, V= 100, L = 0, K = 50.
6

Diekman (1993), Franzen (1995) and Goeree et al. (2005b) all consider cases
with1, where the above proposition applies. Figure 2 (a) depicts the QRE graphin the case of Goeree et al. (2005b) where
2,
1,
1,
0.2,
0.2. The
denominator of the right side of Equation (7) is negative iff ∈ 0, 3/4 and thenumerator is nonpositive iff∈ 1/2, 1 . Thus only the interval 1/2, 3/4 iscompatible with λ 0. Figure 2 (b) shows the QRE graph in the case of Diekman(1993) and Frazen (1995), where
2,
1,
100,
0,
50. In this case,only
1/2 is compatible with nonnegative value of λ. It is also compatible withany λ 0.
Now consider the case with 1
as in Dawes et al (1986) and Rapoportand Ehed-Levy (1989). From Proposition 2, if inequality (4) holds, only pure strategy
equilibria exists. If inequality (5) holds, there exist two values ofsuch that/. Denote the smaller of these as
and the larger one asin such a case.
Otherwise, there exists unique mixed strategy equilibrium. For each of these cases, wehave following characterizations for the limiting QRE.
Proposition 4. If Inequality (4) holds, the limiting QRE is∗0.

Proof. Inequality (4) ensures the denominator of the right hand side of Equation (7) is
positive. Then the only possible principal branch starting at the centroid for λ 0 is
the graph of Equation (7) restricted on ∈ 0, 1/2 . This corresponds to the branchleading to the limiting QRE. Obviously λ → ∞ as → 0.
Q.E.D.
, the limiting QRE is
Proposition 5. Suppose Inequality (5) holds. If 1/2∗
1/2, the limiting QRE is ∗. If
1/2, then
1/2.
Proof. If Inequality (5) and1/2

Equation (7) is positive in 0,

∗

0. If

hold, the denominator of the right hand side ofand the numerator is positive in 0, 1/2 . Focusing

on the branch starting from the centroid, the same logic applies as the previous case.
Suppose Inequality (5) holds. First consider the case where
1/2. The righthand side of Equation (7) is undefined atand , between which 1/2 lies. This allows
us to restrict attention to ,for finding the branch starting at the centroid. In thisinterval, 1/2
is compatible with λ 0. Obviously λ → ∞ as → . Next
consider the case where
1/2. The branch starting from the centroid obviously liesin , 1/2 . Obviously λ → ∞ as → along this branch. Finally if
1/2, then71/2 is compatible with any λ0. Other branches are separated from the centroid.
Q.E.D.

Proposition 6. Suppose that there exist unique value ofsuch that/and Letbe that value. If
1/2, then the limiting QRE is ∗
. If
1/2,
∗
∗
then the limiting QRE is
1/2. If
1/2, then the limiting QRE is0.
Proof. If
1/2, then the branch starting at the centroid is connected in , 1/2 andλ goes to infinity as
approachfrom 1/2. This proves the first claim..If
1/2,
then the branch starting at
1/2 is a straight horizontal line. Although there is abranch in 0,1/2 , it is disconnected from the centroid. If
1/2, then along thebranch starting at the centroid, λ goes to infinity as
approach 0.
Q.E.D.
Figure 3 shows the QRE graph for the case in Rapoport and Ehed-Levy (1989),
where
5,
3,
5,
0,
2. Note that the correspondence QRE λ is
upper hemi-continuous and one-dimensional manifold, as shown in McKelvey and
Palfrey (1995).

Figure 3. A typical QRE correspondence when 1 < m < n (n = 5, m = 3, V = 5, L = 0,
K = 2).
8Remark: Our Propositions 3-6 correspond to Propositions 3 (Volunteer’s Dilemma) and4 (step-level public goods game with binary decision) in Goeree et al. (2005a). They
assert that the choice probability of contributing, p, in the limiting QRE in both cases
unconditionally converges to 0. But it is not the case with 1
wheninequality (4) doesn’t hold. So, our analysis adds some qualification to their assertion.

3. Estimation
We now conduct econometric comparisons to check the explanatory power of QRE.
Data are taken from the experiments reported in Diekmann (1993), Franzen (1995) and
Goeree et al. (2005b) for m = 1 (Volunteer’s Dilemma), and from those in Dawes et al.
(1986) and Rapoport and Ehed-Levy (1989) for m > 1 (step-level public goods gamewith binary decision). In Diekmann (1993), Franzen (1995) and Dawes et al. (1986), thegame was played once and for all. In Goeree et al. (2005b), the game is repeated twenty
times and Rapoport and Ehed-Levy (1989) twenty-five times. Tables 1 and 2 show thedata, Nash equilibria in mixed strategies and the estimation results.
Tables 1 and 2 here.
In the QRE estimation, the following log-likelihood function is maximized
with respect to :
log
log 1
,
where
is the choice probability of C in QRE corresponding to ,is thenumber of instances in which C is chosen,
is the total number of choices.
Then, as for the Volunteer’s Dilemma game, substantial deviation from mixed
strategy equilibria is observed in the experiments of Diekmann (1993), Franzen (1995),
while the experimental result is very close to mixed strategy equilibria in Goeree et al.
(2005b). In n = 2 in Diekmann (1993) and Franzen (1995) and n = 3 in Goeree et al.
(2005b), as the choice probability of contributing is constant, i.e., 0.5, for any value of(see again Figure 2 (b)), the goodness-of-fit of the QRE cannot improve no more thanthe mixed strategy equilibria. So, except for those cases, the QRE model gives better
explanation for the data in general.
Next, we check the data in step-level public goods games with binary decision.
Note that in the games examined by Dawes et al. (1986) and Rapoport and Ehed-Levy(1989), there is no Nash equilibrium in totally mixed strategy (0 < p < 1), that is, there
9are only pure strategy equilibria. The limiting QRE selects the non-cooperativeequilibrium, p = 0, in these cases. However, substantial deviation from these equilibria
was observed. Subjects actually played more cooperatively. We see that, thechoice probability of C based on the estimated , is very close to the observed data,
except for one of the sessions with (n, m) = (7, 5) in Dawes et al. (1986), where theproportion of C substantially exceeds 0.5. As the upper bound of the QRE probability is
0.5 (see the proof of Proposition 4), the goodness-of-fit of the QRE cannot improve any
more than that at the centroid. But as there is no mixed strategy equilibrium in this case,
the QRE model gives better prediction than pure strategy equilibrium.
In sum, QRE’s prediction is (weakly, at least) better than equilibrium prediction
in pure and mixed strategies both in Volunteer’s Dilemma and step-level public goods
games.

4. Conclusion
We have characterized equilibria in a generalized Volunteer Dilemma game, and havederived predictions with a boundedly rational model, quantal response equilibrium(QRE). It is found that the QRE model better performed in econometric analysis of thelaboratory data. In this paper, we only focus on symmetric game where cost forcontribution is the same among players. Analyzing the game with asymmetric cost will
be our next research agendum. Finally, our approach is also applicable to a certain kindof voting games. Such an application is also of great interest.

References
Archetti, M. and I. Scheuring (2010) “Coexistence of cooperation and defection in
public goods games.” Evolution, 65, 1140-1148.
Croson, R. T. A. and M. B. Marks (2000) “Step returns in threshold public goods: ameta- and experimental analysis.” Experimental Economics, 2, 239-259.
Darley, J. M. and B. Latané (1968) “Bystander intervention in emergencies: diffusion of
responsibility.” Journal of Personality and Social Psychology, 8, 377–383.

10Dawes, R. M., J. M. Orbell, R. T. Simmons, and A. J. C. Van de Kragt (1986)“Organizing groups for collective action.” American Political Science Review, 80,
1171-1185.
Diekmann, A. (1985) “Volunteer’s Dilemma” Journal of Conflict Resolution, 29,
605-610.
Franzen, A. (1995) “Group size and one-shot collective action.” Rationality and Society,7, 183-200.
Goeree, J. K. and C. A. Holt (2005a) “An explanation of anomalous behavior inbinary-choice games: entry, voting, public goods, and the volunteers’ dilemma.”
American Political Science Review, 99, 201-213.
Goeree, J. K., C. A. Holt and A. K. Moore (2005b) “An experimental examination of thevolunteers’ dilemma.”mimeo.
Latané, B. and S. Nida (1981) “Ten years of research on group size and helping.”
Psychological Bulletin, 89, 308-324.
McKelvey, R. D. and T. R. Palfrey (1995) “Quantal response equilibrium fornormal-form games.” Games and Economic Behavior, 10, 6-38.
Rapoport, A. and D. Ehed-Levy (1989) “Provision of step-level public goods: effects ofgreed and fear of being gypped.” Organizational Behavior and Human Processes, 44,
325-344.

11

Franzen (1995) 1)

Diekmann (1993)

Goeree et al. (2005b) 2)

n

2

5

2

3

5

7

9

21

51

2

3

6

9

12

(V, L, K)(100, 0,(100, 0, 50)(100, 0, 50)(100, 0, 50)(100, 0, 50)(100, 0, 50)(100, 0, 50)(100, 0, 50)(100, 0, 50)(1, 0.2, 0.2)

(1, 0.2, 0.2)

(1, 0.2, 0.2)

(1, 0.2, 0.2)

(1, 0.2, 0.2)

20
13
0.610

7
18
0.280

15
8
0.652

18
13
0.580

9
12
0.429

6
18
0.250

12
22
0.353

7
16
0.304

5
20
0.200

333
347
0.490

281
439
0.390

298
662
0.310

137
583
0.190

182
778
0.190

0.5

0.159

0.5

0.293

0.159

0.109

0.083

0.034

0.014

0.750

0.500

0.242

0.159

0.118

0.500

0.160

0.500

0.293

0.160

0.110

0.083

0.034

0.014

0.750

0.500

0.242

0.159

0.119

0.500

0.280

0.500

0.500

0.4286

0.250

0.353

0.3043

0.200

0.500

0.500

0.310

0.190

0.190

0, ∞
22.87
4

0.041
14.8240, ∞15.9420.000
21.4880.007
14.341

0.034
13.5000.013
22.0740.017
14.1340.028
12.5100.000
471.3400.990
499.066

10.606594.656

27.762350.370

12.029466.19350)# of C# of N
Proportionof C
Equilibrium
Prob.(C)limiting
QRE
Prob. (C) in
QRE

1) In Frenzen (1995)’ s experiment, there was a case where n = 101. As the computation does not converge in that case, we omit it here.
2) As the game was repeated 20 times in Goeree et al. (2005b)’s experiment, here we show their aggregate data.

Table 1. Experimental data and the estimation results for m = 1 (Volunteer’s Dilemma game).

12Dawes et al. (1986)

Rapoport andEhed-Levy (1989) 1)(n, m)(V, L, K)(7, 3)(10, 0, 5)(7, 5)(10, 0, 5)(7, 5) 2)(10, 0, 5)(5, 3)(5, 0, 2)# of C

36

45

9

548

# of N

34

25

26

952

Proportion of C

0.514

0.643

0.257

0.365

Prob.(C) in mixed
strategy 3)

N/A

N/A

N/A

N/A

limiting QRE

0

0

0

0

in QRE

0.5

0.5

0.257

0.365

0.000

0.000

0.229

1.427

48.52048.52019.952

984.638

1) As the game was repeated 25 times in Rapoport and Ehed-Levy (1989)’s experiment, here we show their aggregate data.
2) This is the follow-up experiment.
3) In the games examined by Dawes et al. (1986) and Rapoport and Ehed-Levy (1989), there is no completely mixed strategy (0 < p < 1).

Table 2. Experimental data and the estimation results for m > 1 (step-level public goods game with binary decision).

13