Abstract
This experiment investigates a stochastic version of the infinitely repeated prisoner’s dilemma. The stochastic element introduces the importance of beliefs about the future for supporting cooperation as well as cooperation and defection on the equilibrium path. There is more cooperation in treatments where beliefs predict cooperation after subjects gain sufficient experience. There is some evidence for cooperation and defection as predicted by equilibrium, but there is stronger evidence for behavior conditioning on past actions that is not consistent with equilibrium play. In particular, subjects continue cooperating even when it is no longer possible in equilibrium for the realized game.
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Notes
Grim Trigger is the strategy cooperate until any unilateral defection and then defect thereafter. It is well-known that cooperation is possible in an infinitely repeated prisoner’s dilemma if and only if Grim Trigger is an equilibrium.
Each expected payoff is expressed as the payoff in period 0 plus \(\delta /(1-\delta )=(2/3)/(1/3)=2\) times the expected payoff in each period \(t\ge 1\).
The value of \(q^*\) is less than or equal to 1 only when cooperation is possible in equilibrium \((p_A\le 14/19)\). When it is not, then defection is the best response for all beliefs and \(q^*\) will be denoted \(>1\) by convention.
That is, replacing the 2’s with \(\delta /(1-\delta )\) and the 3’s with \(1/(1-\delta )\).
Standard arguments show that these are equivalent if players are risk neutral. This is the most common way to implement an infinite horizon in the laboratory.
Except for the last set which was used only for the fifth session of Treatments AB and BB. There were only four sessions for Treatments AA and BA because it was expected (and later verified) that variation in behavior would be small.
For example, Engle-Warnick and Slonim (2006) show that realized match lengths affect behavior. So if match lengths differed across treatments then this could be a confounding factor.
Additionally, all terms were neutrally framed and the experiment was programmed and run in z-tree (Fischbacher 2007). Instructions for Treatment AA are in Appendix C in electronic supplementary material and instructions for other treatments were identical except for the differing parameter values.
This was a job market paper containing several experiments that has now been separated into two papers; this one and a second paper (Kloosterman 2019).
The closest treatment in terms of the parameter \(q^*\) is from Dal Bó and Fréchette (2011) where they found a cooperation rate of about .55.
An alternative way to do hypothesis testing is with non-parametric Wilcoxon ranksum tests where the unit of observation is the average cooperation rate in a session. There are 18 sessions and thus 18 units of observation (8–10 when making a comparison between two treatments) and so significant results are hard to obtain unless all the cooperation rates from sessions where the variable of interest is one value are greater than the cooperation rates from sessions where the variable is the other value. See footnote 14.
For subject random effects with errors clustered at the session level, the last comparison is only marginally significant (\(.58>^*.23\)).
A Wilcoxon ranksum test of average cooperation rates by session is significant only for the comparison of Treatment AB to AA. Nevertheless, if the 18 sessions are separated by the half with largest cooperation rates and the half with the smallest cooperation rates, 8/9 sessions from the half with larger cooperation rates are sessions in Treatments AB or BB and 7/9 sessions from the half with the smaller cooperation rates are sessions in Treatments AA or BA indicating that it may be a lack of power that leads to insignificance.
Both results are also robust to using a later match as the cutoff for experienced subjects. All comparisons between treatments where B is likely to treatments where A is likely are significant at (at least) the 5% level if the data is restricted to matches 41–50, matches 45–50, or just match 50. And all comparisons between treatments where the period 0 game is the same are not significant when restricted to matches 41–50, matches 45–50, or just match 50. See also Fig. 1.
In this case, standard errors clustered at the session level actually improve significance in each treatment except for Treatment BB (\(.63>^{**}.44\), \(.63>^{**}.47\), \(.65>^{**}.44\)).
Lastperiod1A and lastperiod1B are not collinear, because some matches end in one period.
There is no complementary significant result for period0A in the early matches. This is probably because, as Fig. 1 shows, learning occurs quite quickly.
The analysis is similar if all matches are considered. The last half is presented here to be consistent with period 0 results.
The case where the subject cooperated in period 0 while his/her opponent defected is denoted (C,D) while the case where the subject defected while his/her opponent cooperated is denoted (D,C). In other words, all subjects are treated as Player 1.
The comparison for Treatment BA is only marginally significant when errors are clustered at the session level (\(.30>^{*}.23\)).
The results also look reminiscent of the Semi-Grim strategy proposed by Breitmoser (2015) where subjects randomize after the outcomes (C, D) and (D, C). While it may look like randomization in aggregate here, the evidence for individual strategies does not look like randomization at all. The total number of subjects (out of 224) that chose cooperate and defect at least once each in the last 25 matches is 16 for period 1 game A after (C, D), 22 for period 1 game B after (C, D), 22 for period 1 game A after (D, C), and 15 for period 1 game B after (D, C). And of these already less than 10% of subjects, just 2, 6, 3, and 5 of them chose each strategy at least twice for the 4 respective cases.
All 6 of those comparisons are not significant when errors are clustered at the session level, but the 18 others remain significant. However, 2 of those 18 comparisons are not significant for the subject random effects and session-clustered errors method.
Each treatment’s cooperation rate when A is realized in period 1 after the period 0 outcome (C, C) is compared to the cooperation rate when B is realized after (D, D) in period 0 in Treatments AA and BA and to the defection rate when A is realized after (C, C) in period 0 in Treatments AB and BB. For two of the comparisons, the cooperation rate to the defection rate in Treatments AB and BB, the cooperation rate equals 1 minus the defection rate and so the test is simply a test of whether the cooperation rate was greater than .5. These two tests were done with OLS regressions of cooperate on just a constant with standard errors clustered at the subject level and then an F-test of whether the constant was equal to .5. The test of .69 to .5 in Treatment BB is insignificant when errors are clustered at the session level.
As a robustness test of the findings here, Table 11 in Appendix B in electronic supplementary material considers cooperation rates for periods 1 and greater. One might suspect that perhaps cooperation rates decrease considerably over the course of the game when A is realized after (C, C) as cooperation can not be supported there. The table shows that this is not the case, and the results noted in this section are robust to considering all periods after period 0.
References
Aoyagi, M., & Fréchette, G. R. (2009). Collusion as public monitoring becomes noisy: Experimental evidence. Journal of Economic Theory, 144(3), 1135–1165.
Bagwell, K., & Staiger, R. W. (1997). Collusion over the business cycle. RAND Journal of Economics, 28(1), 82–106.
Blonski, M., Ockenfels, P., & Spagnolo, G. (2011). Equilibrium selection in the repeated prisoner’s dilemma: An axiomatic approach and experimental evidence. American Economic Journal: Microeconomics., 3(3), 164–192.
Bolton, G., & Ockenfels, A. (2000). ERC: A theory of equity, reciprocity, and competition. The American Economic Review, 90(1), 166–193.
Breitmoser, Y. (2015). Cooperation, but no reciprocity: Individual strategies in the repeated prisoner’s dilemma. American Economic Review, 105(9), 2882–2910.
Cabral, L., Ozbay, E. Y., & Schotter, A. (2014). Intrinsic and instrumental reciprocity: An experimental study. Games and Economic Behavior, 87, 100–121.
Charness, G., & Genicot, G. (2009). Informal risk sharing in an infinite-horizon experiment. The Economic Journal, 119(537), 796–825.
Dal Bó, P., & Fréchette, G. R. (2011). The evolution of cooperation in infinitely repeated games: Experimental evidence. American Economic Review, 101(1), 411–429.
Dal Bó, P., & Fréchette, G. R. (2018). On the determinants of cooperation in infinitely repeated games: A survey. Journal of Economic Literature, 56(1), 60–114.
Engle-Warnick, J., & Slonim, R. L. (2006). Learning to trust in indefinitely repeated games. Games and Economic Behavior, 54(1), 95–114.
Falk, A., & Fischbacher, U. (2006). A theory of reciprocity. Games and Economic Behavior, 54, 293–315.
Fehr, E., & Schmidt, K. (1999). A theory of fairness, competition and cooperation. Quarterly Journal of Economics, 114, 817–868.
Feinberg, R. M., & Snyder, C. M. (2002). Collusion with secret price cuts: An experimental investigation. Economics Bulletin, 3(6), 1–11.
Fischbacher, U. (2007). z-Tree: Zurich toolbox for ready-made economic experiments. Experimental Economics, 10(2), 171–178.
Fudenberg, D., Rand, D. G., & Dreber, A. (2012). Slow to anger and fast to forgive: Cooperation in an uncertain world. American Economic Review, 101(2), 720–749.
Haltiwanger, J., Jr, H., & Joseph, E. (1991). The impact of cyclical demand movements on collusive behavior. RAND Journal of Economics, 22(1), 89–106.
Holcomb, J. H., & Nelson, P. S. (1997). The role of monitoring in duopoly market outcomes. The Journal of Socio-Economics, 26(1), 79–93.
Kandori, M. (1991). Correlated demand shocks and price wars during booms. The Review of Economic Studies, 58(1), 171–180.
Kartal, M., & Müller, W. (2018). A new approach to the analysis of cooperation under the shadow of the future: Theory and experimental evidence. In Working paper.
Kloosterman, A. (2013). An experimental study of public information in Markov games. In Working paper.
Kloosterman, A. (2019). An experimental study of public information in the asymmetric partnership game. Southern Economic Journal, 85(3), 663–690.
Peysakhovic, A., & Rand, D. (2015). Habits of virtue: Creating norms of cooperation and defection in the laboratory. Management Science, 62(3), 631–647.
Rojas, C. (2012). The role of demand information and monitoring in tacit collusion. RAND Journal of Economics, 43(1), 78–109.
Rotemberg, J. J., & Saloner, G. (1986). A supergame-theoretic model of price wars duing booms. American Economic Review, 76(3), 390–407.
Roy, N. (2014). Cooperation without immediate reciprocity: An experiment in favor exchange. In Working paper.
Ruffle, B. J. (2013). When do large buyers pay less? Experimental evidence. Journal of Industrial Economics, 61(1), 108–137.
Saijo, T., Sherstyuk, K., Tarui, N., & Ravago, M.-L. (2015). Games with dynamic externalities and climate change experiments. Journal of the Association of Environmental and Resource Economists, 3, 247–281.
Vespa, E. (2015). An experimental investigation of strategies in the dynamic common pool game. In Working paper.
Wilson, A., & Vespa, E. (2018). Experimenting with the transition rule in dynamic games. In Working paper.
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A. Kloosterman: I wish to thank Andrew Schotter, Guillaume Fréchette, Emmanuel Vespa, and Jack Fanning for their help on this project, as well as participants at the North American ESA Conference 2012, CESS-Amsterdam Graduate Student Conference 2014, the SEA Conference 2015, and the UVA Experimental Social Science Conference 2016 and two anonymous referees for insightful comments. Also, thanks to NSF Doctoral Dissertation Research grant SES-1260840 and UVA for financial support.
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Kloosterman, A. Cooperation in stochastic games: a prisoner’s dilemma experiment. Exp Econ 23, 447–467 (2020). https://doi.org/10.1007/s10683-019-09619-w
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DOI: https://doi.org/10.1007/s10683-019-09619-w