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An experimental investigation of a third-person enforcement in a prisoner’s dilemma game
Economics Letters 117 (2012) 704–707
Contents lists available at SciVerse ScienceDirect
Economics Letters journal homepage: www.elsevier.com/locate/ecolet
An experimental investigation of a third-person enforcement in a prisoner’s dilemma game Takao Kusakawa a , Kazuhito Ogawa b,∗ , Tatsuhiro Shichijo c a
Faculty of Economic Sciences, Hiroshima Shudo University, 1-1-1, Ozuka-Higashi, Asaminami-ku, Hiroshima 731-3195, Japan
b
Faculty of Sociology, Kansai University, 3-3-35, Yamate-cho, Suita, Osaka 564-8680, Japan
c
Graduate School of Economics, Osaka Prefecture University, 1-1, Gakuen-cho, Naka-ku, Sakai, Osaka 599-8531, Japan
article
info
Article history: Received 2 April 2012 Received in revised form 31 July 2012 Accepted 10 August 2012 Available online 23 August 2012
abstract We experimentally study the effect of a third-person enforcement on a one-shot prisoner’s dilemma game played by two persons, with whom the third person plays repeated prisoner’s dilemma games. We find that when the third person can observe these two persons’ play, the possibility of the third person’s future punishment causes them to cooperate in the one-shot game. © 2012 Elsevier B.V. All rights reserved.
JEL classification: C73 C91 L14 Keywords: Third person Infinitely repeated prisoner’s dilemma Cooperation Experiment
1. Introduction Theoretically, mutual cooperation is not realized when only two players play the finitely repeated prisoner’s dilemma (PD) game. However, a third person (hereafter, we call this person M) can promote cooperation between the players. As a real-life example, suppose that there are two mutually unacquainted persons (X1 and X2 ) whom M knows well. M introduces them to each other and recommends that they start a joint business. In this case, they may cooperate in their business so that M, observing their behavior, will not develop distrust of them. This effect of a common acquaintance on the cooperative behavior of two other persons has been theoretically investigated by Kandori (1992), and Dal Bó (2007) in more general settings, but with two extreme information structures. One is perfect monitoring, in which players receive information about what actions all the players choose. The other is private monitoring, in which players only receive information about their own decision. To apply their
∗
Corresponding author. Tel.: +81 6 6368 0700; fax: +81 6 6368 0082. E-mail addresses: [email protected] (T. Kusakawa), [email protected], [email protected] (K. Ogawa), [email protected] (T. Shichijo). 0165-1765/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.econlet.2012.08.014
theory to our context, we need a slight modification to their information structures. We adopt an intermediate information structure between the two. The intermediary M can observe all the actions X1 and X2 choose because M knows both of them. However, Xi cannot observe the results of the stage games played between Xj and M because Xi is not an acquaintance of Xj . We also introduce another modification to simplify the setting by reducing the number of players to three. We investigate the effect of M on the cooperative relationship between X1 and X2 through laboratory experiments with these modifications. To examine whether M, who observes the behavior of X1 and X2 and interacts with them, promotes their mutual cooperation, our experimental session adopts one of the following two treatments. In a control treatment called ‘‘Non-observable’’, M cannot observe the actions played in the stage game between X1 and X2 . On the other hand, in a treatment called ‘‘Observable’’, M can observe them. Comparing the cooperation rate in the stage game played between X1 and X2 , we find that cooperation occurs more frequently in Observable than in Non-observable. Our contribution is to experimentally verify that the third person M serves as a cooperation facilitation device for X1 and X2 . Because most experimental studies on social dilemmas or the PD do not account
T. Kusakawa et al. / Economics Letters 117 (2012) 704–707 Table 1 Payoff table for the stage game.
Cooperate Defect
Cooperate
Defect
75, 75 100, 10
10, 100 45, 45
Table 2 Descriptive data for sessions. Observable Date Participants Total rounds Number of stages: average [min, max] Earnings (yen): average [min, max]
Non-observable
12/1/10 27 12
1/22/11 21 9
12/11/10 21 9
2/23/11 21 9
3.92 [1, 16]
6.00 [2, 15]
4.00 [2, 8]
7.11 [1, 21]
2964 [2760, 3225]
3410 [2750, 3860]
2640 [2085, 2980]
3619 [2600, 4505]
for the third person, our findings provide a different perspective on infinitely repeated PD experiments. 2. Experimental settings We consider two treatments: Observable and Non-observable. The game structure in each treatment is common knowledge among all participants. Each treatment consists of multiple identical rounds. In each round, participants are put in groups of three such that they never play with participants with whom they have played before. Each round consists of one stage or multiple stages depending on the results of the experimenter’s throw of the dice at the end of each stage, with a probability of 3/4 that all participants proceed to the next stage and a probability of 1/4 that all participants end with this round. These probabilities remain unchanged throughout the sessions. The game structure above is very similar to that used in one treatment of the Dal Bó (2005) experiment. The main differences are the number of participants in each group and the roles played by them. In each round, they are assigned one of three roles: X1 , X2 , or M. The participants who are assigned the roles of X1 or X2 play a PD game (Table 1) in the first stage of a round. In the first stage, M does nothing. In Observable, right after the decision-making in the first stage, M observes the decisions and payoffs of X1 and X2 in the same group. In Non-observable, M cannot observe them. This is the sole difference between the two treatments. If the second stage starts as a result of a throw of the dice, M and Xi play a PD game, the payoff of which is the same as that in the first stage. From the second stage, M plays the game until the round terminates, while X1 or X2 is randomly chosen as a player of the game in each stage. The unchosen participant does nothing during this stage and is not informed of the results of this stage. When a round terminates at a stage as a result of a throw of the dice, each group of three participants separates. After that, groups of three participants are newly formed, each participant is newly assigned a role, and the next round starts. The experiment was computerized by z-Tree (Fischbacher, 2007), and all sessions were conducted at Kyoto Sangyo University. We conducted two sessions for each treatment. Table 2 shows the descriptive statistics.
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stage. We will show that the trigger strategies of M and Xi —M (Xi ) defects to players whose defection M (Xi ) has observed, and to players to whom M (Xi ) has defected; M (Xi ) cooperates with the other players—constitute a perfect Bayesian equilibrium in Observable. Hence, mutual cooperation is realized in every stage on the equilibrium path. First, we focus on the second and later stages played between M and Xi . Here, we calculate only the expected payoff that M and Xi obtain in ‘‘subgames’’ between them, because none of their actions in the subgames affects the expected payoff obtained in stage games played with Xj (j ̸= i). We show that a deviation is not beneficial when the other players follow the abovementioned strategies. When a player is supposed to offer cooperation according to the strategy, ∞the expected payoff for offering cooperation is 75 + (75/2) · t =1 (3/4)t = 187.5, which is larger than the maximum expected payoff for offering defection, ∞ i.e., 100 +(45/2)· t =1 (3/4)t = 167.5. When a player is supposed to offer defection according to the abovementioned strategy, because the counterpart is also supposed to offer the ∞ defection, t expected payoff for offering defection is 45 · t =0 (1/2) = 90, which is larger ∞than the expected payoff for offering cooperation, i.e., 10 + 45 · t =1 (1/2)t = 55. Thus, the abovementioned strategy profile generates an equilibrium in the second and later stages. Next, we focus on the play between X1 and X2 in the first stage. The expected payoff is 187.5 according to the abovementioned strategies, while it is 167.5 when defecting all the time. Thus, the strategy constitutes a perfect Bayesian equilibrium in Observable. Although there is an equilibrium where mutual cooperation in the first stage is realized in Non-observable,1 we expect that the impossibility of applying the abovementioned strategy makes it difficult for Non-observable to realize the mutual cooperation in the first stage.2 From the discussion above, we propose the following three hypotheses. In the first stage, the percentage of cooperation offered by X1 and X2 is higher in Observable than in Non-observable (Hypothesis 1). In the second stage in Observable, M is more likely to cooperate against ‘‘cooperators’’ who offered cooperation in the first stage than against ‘‘defectors’’ who offered defection in the first stage (Hypothesis 2), and the cooperators are more likely to cooperate against M than are the defectors (Hypothesis 3). 4. Results First, we investigate Hypothesis 1. The cooperation rates pooled over all rounds in the first stage are shown in the first column of Table 3. Fig. 1 indicates the evolution of these cooperation rates. The number of observations in the first stage for each round is presented at the top of the figure for Observable and at the bottom for Non-observable. The first column of Table 4 shows the results of a logistic regression of individual choice to cooperate, clustered by individual. These results support Hypothesis 1. Second, we examine Hypotheses 2 and 3. In Table 3, the cooperation rates pooled over all rounds in the second stage— M’s play against cooperators and defectors in the middle of the table, and the cooperators’ and defectors’ play against M on the right—are reported. Fig. 2 (Fig. 3) shows the evolution of M’s (Xi ’s) cooperation rates in the second stage. The number of second stages played between M and the cooperators for each round is represented at the top of these figures, and that between M and the defectors at the bottom. These and the regression results in Table 4 support Hypotheses 2 and 3.
3. Theoretical predictions Only in Observable can M choose actions in the second and/or later stages depending on the actions of X1 and X2 in the first
1 See Shichijo (2012). 2 On the possibility of community enforcement without observability of others’ action histories, see Duffy and Ochs (2009) and Camera and Casari (2009).
706
T. Kusakawa et al. / Economics Letters 117 (2012) 704–707 Table 3 Cooperation rates pooled over all rounds. Treatment
Observable Non-observable
Cooperation rate First stage
Second stage
Xi ’s play
M’s play
33.0% (113/342) 13.1% (33/252)
Xi ’s play
Against cooperator
Against defector
Xi is cooperator
Xi is defector
50.0% (27/54) 26.7% (4/15)
3.7% (3/81) 19.6% (19/97)
77.8% (42/54) 80.0% (12/15)
12.3% (10/81) 16.5% (16/97)
Fig. 1. Xi ’s cooperation rate in the first stage. Table 4 Logistic regression of cooperation. Dependent variable
Observable Cooperator Constant Number of observations Number of participants
Play in the first stage
Play in the second stage in Observable
Xi ’s play: Hypothesis 1
M’s play: Hypothesis 2
Xi ’s play: Hypothesis 3
26.00 (16.19)* 0.04 (0.02)* 135 48
24.85 (12.15)* 0.14 (0.04)* 135 47
3.27 (1.28)* 0.15 (0.05)* 594 90
Odds ratio estimates from logistic regression models clustered by individual (cluster-robust standard errors in parentheses). Dependent variable = 1 for cooperation, otherwise 0; Observable = 1 for the Observable treatment, otherwise 0; Cooperator = 1 for the stages where Xi player is a cooperator, otherwise 0. * Significant at the 1% level.
Fig. 2. M’s cooperation rate in the second stage in Observable.
Fig. 3. Xi ’s cooperation rate in the second stage in Observable.
T. Kusakawa et al. / Economics Letters 117 (2012) 704–707
707
5. Conclusions
Appendix. Supplementary data
We experimentally show that, because of the threat of a third person’s future punishment, mutual cooperation between two persons in a one-shot PD game is facilitated by the observation of the third person who will play with them later. To our knowledge, this is the first study that demonstrates this in a laboratory. Therefore, we experimentally identify a cooperation facilitation device in the infinitely repeated PD game.
Supplementary material related to this article can be found online at http://dx.doi.org/10.1016/j.econlet.2012.08.014.
Acknowledgments This work was supported by KAKENHI (20200042). Takao Kusakawa was in residence at the Department of Resource Economics at the University of Massachusetts-Amherst while conducting this research. He would like to thank the Department for their hospitality.
References Camera, G., Casari, M., 2009. Cooperation among strangers under the shadow of the future. American Economic Review 99, 979–1005. Dal Bó, P., 2005. Cooperation under the shadow of the future: experimental evidence from infinitely repeated games. American Economic Review 95, 1591–1604. Dal Bó, P., 2007. Social norms, cooperation and inequality. Economic Theory 30, 89–105. Duffy, J., Ochs, J., 2009. Cooperative behavior and the frequency of social interaction. Games and Economic Behavior 66, 785–812. Fischbacher, U., 2007. z-Tree: Zurich toolbox for ready-made economic experiments. Experimental Economics 10, 171–178. Kandori, M., 1992. Social norms and community enforcement. The Review of Economic Studies 59, 63–80. Shichijo, T., 2012. Third person enforcement in a Prisoner’s dilemma game. Mimeo. Available at http://ssrn.com/abstract=2133313.
Contents lists available at SciVerse ScienceDirect
Economics Letters journal homepage: www.elsevier.com/locate/ecolet
An experimental investigation of a third-person enforcement in a prisoner’s dilemma game Takao Kusakawa a , Kazuhito Ogawa b,∗ , Tatsuhiro Shichijo c a
Faculty of Economic Sciences, Hiroshima Shudo University, 1-1-1, Ozuka-Higashi, Asaminami-ku, Hiroshima 731-3195, Japan
b
Faculty of Sociology, Kansai University, 3-3-35, Yamate-cho, Suita, Osaka 564-8680, Japan
c
Graduate School of Economics, Osaka Prefecture University, 1-1, Gakuen-cho, Naka-ku, Sakai, Osaka 599-8531, Japan
article
info
Article history: Received 2 April 2012 Received in revised form 31 July 2012 Accepted 10 August 2012 Available online 23 August 2012
abstract We experimentally study the effect of a third-person enforcement on a one-shot prisoner’s dilemma game played by two persons, with whom the third person plays repeated prisoner’s dilemma games. We find that when the third person can observe these two persons’ play, the possibility of the third person’s future punishment causes them to cooperate in the one-shot game. © 2012 Elsevier B.V. All rights reserved.
JEL classification: C73 C91 L14 Keywords: Third person Infinitely repeated prisoner’s dilemma Cooperation Experiment
1. Introduction Theoretically, mutual cooperation is not realized when only two players play the finitely repeated prisoner’s dilemma (PD) game. However, a third person (hereafter, we call this person M) can promote cooperation between the players. As a real-life example, suppose that there are two mutually unacquainted persons (X1 and X2 ) whom M knows well. M introduces them to each other and recommends that they start a joint business. In this case, they may cooperate in their business so that M, observing their behavior, will not develop distrust of them. This effect of a common acquaintance on the cooperative behavior of two other persons has been theoretically investigated by Kandori (1992), and Dal Bó (2007) in more general settings, but with two extreme information structures. One is perfect monitoring, in which players receive information about what actions all the players choose. The other is private monitoring, in which players only receive information about their own decision. To apply their
∗
Corresponding author. Tel.: +81 6 6368 0700; fax: +81 6 6368 0082. E-mail addresses: [email protected] (T. Kusakawa), [email protected], [email protected] (K. Ogawa), [email protected] (T. Shichijo). 0165-1765/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.econlet.2012.08.014
theory to our context, we need a slight modification to their information structures. We adopt an intermediate information structure between the two. The intermediary M can observe all the actions X1 and X2 choose because M knows both of them. However, Xi cannot observe the results of the stage games played between Xj and M because Xi is not an acquaintance of Xj . We also introduce another modification to simplify the setting by reducing the number of players to three. We investigate the effect of M on the cooperative relationship between X1 and X2 through laboratory experiments with these modifications. To examine whether M, who observes the behavior of X1 and X2 and interacts with them, promotes their mutual cooperation, our experimental session adopts one of the following two treatments. In a control treatment called ‘‘Non-observable’’, M cannot observe the actions played in the stage game between X1 and X2 . On the other hand, in a treatment called ‘‘Observable’’, M can observe them. Comparing the cooperation rate in the stage game played between X1 and X2 , we find that cooperation occurs more frequently in Observable than in Non-observable. Our contribution is to experimentally verify that the third person M serves as a cooperation facilitation device for X1 and X2 . Because most experimental studies on social dilemmas or the PD do not account
T. Kusakawa et al. / Economics Letters 117 (2012) 704–707 Table 1 Payoff table for the stage game.
Cooperate Defect
Cooperate
Defect
75, 75 100, 10
10, 100 45, 45
Table 2 Descriptive data for sessions. Observable Date Participants Total rounds Number of stages: average [min, max] Earnings (yen): average [min, max]
Non-observable
12/1/10 27 12
1/22/11 21 9
12/11/10 21 9
2/23/11 21 9
3.92 [1, 16]
6.00 [2, 15]
4.00 [2, 8]
7.11 [1, 21]
2964 [2760, 3225]
3410 [2750, 3860]
2640 [2085, 2980]
3619 [2600, 4505]
for the third person, our findings provide a different perspective on infinitely repeated PD experiments. 2. Experimental settings We consider two treatments: Observable and Non-observable. The game structure in each treatment is common knowledge among all participants. Each treatment consists of multiple identical rounds. In each round, participants are put in groups of three such that they never play with participants with whom they have played before. Each round consists of one stage or multiple stages depending on the results of the experimenter’s throw of the dice at the end of each stage, with a probability of 3/4 that all participants proceed to the next stage and a probability of 1/4 that all participants end with this round. These probabilities remain unchanged throughout the sessions. The game structure above is very similar to that used in one treatment of the Dal Bó (2005) experiment. The main differences are the number of participants in each group and the roles played by them. In each round, they are assigned one of three roles: X1 , X2 , or M. The participants who are assigned the roles of X1 or X2 play a PD game (Table 1) in the first stage of a round. In the first stage, M does nothing. In Observable, right after the decision-making in the first stage, M observes the decisions and payoffs of X1 and X2 in the same group. In Non-observable, M cannot observe them. This is the sole difference between the two treatments. If the second stage starts as a result of a throw of the dice, M and Xi play a PD game, the payoff of which is the same as that in the first stage. From the second stage, M plays the game until the round terminates, while X1 or X2 is randomly chosen as a player of the game in each stage. The unchosen participant does nothing during this stage and is not informed of the results of this stage. When a round terminates at a stage as a result of a throw of the dice, each group of three participants separates. After that, groups of three participants are newly formed, each participant is newly assigned a role, and the next round starts. The experiment was computerized by z-Tree (Fischbacher, 2007), and all sessions were conducted at Kyoto Sangyo University. We conducted two sessions for each treatment. Table 2 shows the descriptive statistics.
705
stage. We will show that the trigger strategies of M and Xi —M (Xi ) defects to players whose defection M (Xi ) has observed, and to players to whom M (Xi ) has defected; M (Xi ) cooperates with the other players—constitute a perfect Bayesian equilibrium in Observable. Hence, mutual cooperation is realized in every stage on the equilibrium path. First, we focus on the second and later stages played between M and Xi . Here, we calculate only the expected payoff that M and Xi obtain in ‘‘subgames’’ between them, because none of their actions in the subgames affects the expected payoff obtained in stage games played with Xj (j ̸= i). We show that a deviation is not beneficial when the other players follow the abovementioned strategies. When a player is supposed to offer cooperation according to the strategy, ∞the expected payoff for offering cooperation is 75 + (75/2) · t =1 (3/4)t = 187.5, which is larger than the maximum expected payoff for offering defection, ∞ i.e., 100 +(45/2)· t =1 (3/4)t = 167.5. When a player is supposed to offer defection according to the abovementioned strategy, because the counterpart is also supposed to offer the ∞ defection, t expected payoff for offering defection is 45 · t =0 (1/2) = 90, which is larger ∞than the expected payoff for offering cooperation, i.e., 10 + 45 · t =1 (1/2)t = 55. Thus, the abovementioned strategy profile generates an equilibrium in the second and later stages. Next, we focus on the play between X1 and X2 in the first stage. The expected payoff is 187.5 according to the abovementioned strategies, while it is 167.5 when defecting all the time. Thus, the strategy constitutes a perfect Bayesian equilibrium in Observable. Although there is an equilibrium where mutual cooperation in the first stage is realized in Non-observable,1 we expect that the impossibility of applying the abovementioned strategy makes it difficult for Non-observable to realize the mutual cooperation in the first stage.2 From the discussion above, we propose the following three hypotheses. In the first stage, the percentage of cooperation offered by X1 and X2 is higher in Observable than in Non-observable (Hypothesis 1). In the second stage in Observable, M is more likely to cooperate against ‘‘cooperators’’ who offered cooperation in the first stage than against ‘‘defectors’’ who offered defection in the first stage (Hypothesis 2), and the cooperators are more likely to cooperate against M than are the defectors (Hypothesis 3). 4. Results First, we investigate Hypothesis 1. The cooperation rates pooled over all rounds in the first stage are shown in the first column of Table 3. Fig. 1 indicates the evolution of these cooperation rates. The number of observations in the first stage for each round is presented at the top of the figure for Observable and at the bottom for Non-observable. The first column of Table 4 shows the results of a logistic regression of individual choice to cooperate, clustered by individual. These results support Hypothesis 1. Second, we examine Hypotheses 2 and 3. In Table 3, the cooperation rates pooled over all rounds in the second stage— M’s play against cooperators and defectors in the middle of the table, and the cooperators’ and defectors’ play against M on the right—are reported. Fig. 2 (Fig. 3) shows the evolution of M’s (Xi ’s) cooperation rates in the second stage. The number of second stages played between M and the cooperators for each round is represented at the top of these figures, and that between M and the defectors at the bottom. These and the regression results in Table 4 support Hypotheses 2 and 3.
3. Theoretical predictions Only in Observable can M choose actions in the second and/or later stages depending on the actions of X1 and X2 in the first
1 See Shichijo (2012). 2 On the possibility of community enforcement without observability of others’ action histories, see Duffy and Ochs (2009) and Camera and Casari (2009).
706
T. Kusakawa et al. / Economics Letters 117 (2012) 704–707 Table 3 Cooperation rates pooled over all rounds. Treatment
Observable Non-observable
Cooperation rate First stage
Second stage
Xi ’s play
M’s play
33.0% (113/342) 13.1% (33/252)
Xi ’s play
Against cooperator
Against defector
Xi is cooperator
Xi is defector
50.0% (27/54) 26.7% (4/15)
3.7% (3/81) 19.6% (19/97)
77.8% (42/54) 80.0% (12/15)
12.3% (10/81) 16.5% (16/97)
Fig. 1. Xi ’s cooperation rate in the first stage. Table 4 Logistic regression of cooperation. Dependent variable
Observable Cooperator Constant Number of observations Number of participants
Play in the first stage
Play in the second stage in Observable
Xi ’s play: Hypothesis 1
M’s play: Hypothesis 2
Xi ’s play: Hypothesis 3
26.00 (16.19)* 0.04 (0.02)* 135 48
24.85 (12.15)* 0.14 (0.04)* 135 47
3.27 (1.28)* 0.15 (0.05)* 594 90
Odds ratio estimates from logistic regression models clustered by individual (cluster-robust standard errors in parentheses). Dependent variable = 1 for cooperation, otherwise 0; Observable = 1 for the Observable treatment, otherwise 0; Cooperator = 1 for the stages where Xi player is a cooperator, otherwise 0. * Significant at the 1% level.
Fig. 2. M’s cooperation rate in the second stage in Observable.
Fig. 3. Xi ’s cooperation rate in the second stage in Observable.
T. Kusakawa et al. / Economics Letters 117 (2012) 704–707
707
5. Conclusions
Appendix. Supplementary data
We experimentally show that, because of the threat of a third person’s future punishment, mutual cooperation between two persons in a one-shot PD game is facilitated by the observation of the third person who will play with them later. To our knowledge, this is the first study that demonstrates this in a laboratory. Therefore, we experimentally identify a cooperation facilitation device in the infinitely repeated PD game.
Supplementary material related to this article can be found online at http://dx.doi.org/10.1016/j.econlet.2012.08.014.
Acknowledgments This work was supported by KAKENHI (20200042). Takao Kusakawa was in residence at the Department of Resource Economics at the University of Massachusetts-Amherst while conducting this research. He would like to thank the Department for their hospitality.
References Camera, G., Casari, M., 2009. Cooperation among strangers under the shadow of the future. American Economic Review 99, 979–1005. Dal Bó, P., 2005. Cooperation under the shadow of the future: experimental evidence from infinitely repeated games. American Economic Review 95, 1591–1604. Dal Bó, P., 2007. Social norms, cooperation and inequality. Economic Theory 30, 89–105. Duffy, J., Ochs, J., 2009. Cooperative behavior and the frequency of social interaction. Games and Economic Behavior 66, 785–812. Fischbacher, U., 2007. z-Tree: Zurich toolbox for ready-made economic experiments. Experimental Economics 10, 171–178. Kandori, M., 1992. Social norms and community enforcement. The Review of Economic Studies 59, 63–80. Shichijo, T., 2012. Third person enforcement in a Prisoner’s dilemma game. Mimeo. Available at http://ssrn.com/abstract=2133313.