Experiments on Infinitely Repeated Games

by Pedro Dal Bó

The tension between private incentives that encourage opportunistic behavior and the common good that comes from cooperation is a central feature of human interaction. The main contribution of Game Theory to the study of this tension and its remedies is to recognize that repeated interaction may enable punishment and reward schemes that prevent or limit opportunistic behavior and support cooperation. When there is always a future, as in infinitely repeated games, the credible threat of future retaliation casts "the shadow of the future" in every decision and can overcome opportunistic behavior and support cooperation, thereby solving the tension between private incentives and the common good.

In "Cooperation under the Shadow of the Future: experimental evidence from infinitely repeated games" I report a series of experiments on infinitely repeated games. I find that the possibility of future interaction modifies players' behavior resulting in fewer opportunistic actions and supporting cooperation, closely following theoretical predictions.

Infinitely repeated prisoner's dilemma games are simulated in the experiment by having a random continuation rule. The experimental design represents an improvement over the existing literature by, among other things, including sessions with finitely repeated games as controls and a large number of players per session (which allows for learning without contagion effects). I find strong evidence that the higher the probability of continuation, the higher the levels of cooperation. While in the one-shot prisoner's dilemma games studied here the cooperation rate is 9%, for a probability of continuation of (3/4) it is 38%. The effect of the shadow of future on the levels of cooperation is greater than previous studies have shown.

An innovation of this paper is that I compare the results from infinitely repeated games with the results from finitely repeated games to test whether cooperation depends on the shadow of the future, as theory predicts, or merely on the length of the games. The lengths of the finitely repeated games were chosen to coincide with the expected lengths of the infinitely repeated ones. I find that the level of cooperation in the final round of the finitely repeated games is similar to the level of cooperation in one-shot games. In addition, these levels of cooperation are lower than those observed in infinitely repeated games providing evidence that subjects cooperate less when there is no future. This seems to be understood by the subjects at the beginning of the game, resulting in greater levels of cooperation in the first round of infinitely repeated games than in the first round of finitely repeated games of the same expected length. That is, when the expected number of future rounds is the same in both finitely and infinitely repeated games, cooperation is greater under the latter as theory predicts.

The findings that cooperation increases with the probability of continuation and that infinitely repeated games result in higher levels of cooperation than finitely repeated ones of the same expected length, suggests that self-enforcing reward and punishment schemes that limit opportunistic behavior are important in practice as well as in theory.

I am currently designing a new series of experiments on infinitely repeated games that will further our understanding of the determinants of cooperation.

For more on my research please go to http://www.econ.brown.edu/fac/Pedro_Dal_Bo/.