The Evolution of Cooperation in Infinitely Repeated Games: Experimental Evidence

Dal Bó, Pedro ; Fréchette, Guillaume R. (2011) “The Evolution of Cooperation in Infinitely Repeated Games: Experimental Evidence." The American Economic Review, Volume 101, Number 1, February 2011 , pp. 411-429(19). DOI:; 提供的 [PDF] ; Download Data Set (355.92 KB) | Online Appendix (284.08 KB);

==notes by yinung==

此文用的 PD game,進行無限重覆賽局的實驗。see also 另一篇也在 AER 2005 年刊出的和有限重覆 PD game 之比較(或參目本站的另一篇 note)。

          合作     背叛
 合作    (R, R)   (12, 50)
 背叛    (50,12)  (25, 25)
繼續玩之機率:1/2, 3/4


The 18 experimental sessions were conducted between July 2005 and March 2006. A total of 266 New York University undergraduates participated in the experiment, with an average of 14.78 subjects per session, a maximum of 20 and a minimum of 12. The subjects earned an average of $25.95, with a maximum of $42.93 and a minimum of $16.29. In the treatments with δ =
1/2 and δ = 3/4 the average number of rounds per match was 1.96 and 4.42 respectively, and the maximum was nine and 23 respectively.


…each subject participated in between 23 and 77 infinitely repeated games

…Previous experimental evidence has shown that subjects often fail to coordinate on a specific equilibrium when they play a small number of infinitely repeated games



…the level of cooperation decreases with experience and converges to low levels

…the level of cooperation does not necessarily increase and may remain at low levels even after significant experience is obtained.

…this evidence suggests that while being an equilibrium action may be a necessary condition for cooperation to arise with experience, it is not sufficient.

若 cooperation 是 risk dominant, 則合作隨著經驗增加而上升

If we consider together all sessions for which cooperation is risk dominant, we find that cooperation increases on average as subjects gain experience….Risk dominance has been used as a selection criterion in the study of coordination games.


…These results show how difficult it is for experienced subjects to sustain high levels of cooperation. They cast doubt on the common assumption that agents will make the most of the opportunity to cooperate whenever it is possible to do so in equilibrium.

…In fact the impact of repetition on rates of cooperation was rather modest, leading Roth to conclude that the results are equivocal (Roth 1995). (註:這些結果皆來自於隨機結束的賽局 randomly terminated game. All of these papers used games with a randomly determined length.)


A usual criticism of the theory of infinitely repeated games is that it does not provide sharp predictions since there may be a multiplicity of equilibria. To address this issue, we present experimental evidence on the evolution of cooperation in infinitely repeated prisoner’s dilemma games as subjects gain experience. We show that cooperation may prevail in infinitely repeated games, but the conditions under which this occurs are more stringent than the subgame perfect conditions usually considered or even a condition based on risk dominance.


  • Bereby-Meyer,Yoella, and Alvin E. Roth. 2006. “The Speed of Learning in Noisy Games: Partial Re-inforcement and the Sustainability of Cooperation.” American Economic Review, 96(4): 1029–42.
  • Dal Bó, Pedro. 2005. “Cooperation under the Shadow of the Future: Experimental Evidence from Infinitely Repeated Games.”American Economic Review, 95(5): 1591–1604.
    這篇比較無限和有限重覆 PD game, 發現無限重覆賽局,在同樣條件下會有較高的合作率

    [this paper] compares infinitely repeated and finitely repeated prisoner’s dilemma games of the same expected length and finds that cooperation is larger in the former as theory predicts.
  • Dal Bó, Pedro. 2007. “Tacit Collusion under Interest Rate Fluctuations.” RAND Journal of Economics, 38(2): 533–40.