Cooperation under the Shadow of the Future: ExperimentalEvidence from Infinitely Repeated GamesBy PEDRO DAL BO´*While there is an extensive literature on the theory of infinitely repeated games,empirical evidence on how “the shadow of the future” affects behavior is scarce andinconclusive. I simulate infinitely repeated prisoner’s dilemma games in the lab witha random continuation rule. The experimental design represents an improvementover the existing literature by including sessions with finite repeated games ascontrols and a large number of players per session (which allows for learningwithout contagion effects). I find that the shadow of the future matters not only bysignificantly reducing opportunistic behavior, but also because its impact closelyfollows theoretical predictions. (JEL C72, C73, C91, C92)The tension between private incentives thatencourage opportunistic behavior and the com-mon good that comes from cooperation is acentral feature of human interaction. The maincontribution of Game Theory to the study of thistension and its remedies is to recognize thatrepeated interaction may enable punishmentand reward schemes that prevent or limit oppor-tunistic behavior and support cooperation.When there is always a future, as in infinitelyrepeated games, the credible threat of futureretaliation casts “the shadow of the future” inevery decision and can overcome opportunisticbehavior and support cooperation, thereby solv-ing the tension between private incentives andthe common good.However, the experimental evidence on infi-nitely repeated games is scarce and in mostcases inconclusive or presents methodologicalproblems. In this paper, I report a series ofexperiments that overcome the shortcomings ofprevious experiments on infinitely repeatedgames. I find that the possibility of future inter-action modifies players’ behavior, resulting infewer opportunistic actions and supporting coop-eration, closely following theoretical predictions.Infinitely repeated prisoner’s dilemma gamesare simulated in the experiment by having arandom continuation rule. The experimental de-sign represents an improvement over the exist-ing literature by, among other things, includingsessions with finitely repeated games as con-trols and a large number of players per session(which allows for learning without contagioneffects).I find strong evidence that the higher theprobability of continuation, the higher the levelsof cooperation. While in the one-shot prisoner’sdilemma games studied here, the cooperationrate is 9 percent, for a probability of continua-tion of3⁄4, it is 38 percent. The effect of theshadow of the future on the levels of coopera-tion is greater than previous studies haveshown.But the finding that increases in the probabil-ity of continuation result in increases in coop-eration is not necessarily evidence in support ofthe theory of infinitely repeated games. It couldbe the case that subjects cooperate more thehigher the number of rounds in which they willinteract; even when there is a final round, pun-ishments are not credible and the future casts noshadow. Then, an increase in the probability ofcontinuation would result in more cooperation,not because of the shadow of the future, but justbecause of the increase in the expected numberof rounds. An innovation of this paper is that Icompare the results from infinitely repeatedgames with the results from finitely repeatedgames to test whether cooperation depends onthe shadow of the future, as theory predicts, or* Department of Economics, Brown University, Provi-dence, RI 02912 (e-mail: [email protected]). I am gratefulto David Levine for invaluable guidance, discussions, andthe financial support for this experiment. The support of thepeople at CASSEL was crucial for this research. I am alsograteful to Anna Aizer, Ernesto Dal Bo´, Alvin Roth, threeanonymous referees, and participants at several seminars forvery useful comments.1591merely on the length of the games. The lengthsof the finitely repeated games were chosen tocoincide with the expected lengths of the infi-nitely repeated ones. I find that the level ofcooperation in the final round of the finitelyrepeated games is similar to the level of coop-eration in one-shot games. In addition, theselevels of cooperation are lower than those ob-served in infinitely repeated games, providingevidence that subjects cooperate less when thereis no future. This seems to be understood by thesubjects at the beginning of the game, resultingin greater levels of cooperation in the first roundof infinitely repeated games than in the firstround of finitely repeated games of the sameexpected length. That is, when the expectednumber of future rounds is the same in bothfinitely and infinitely repeated games, coopera-tion is greater under the latter, as theorypredicts.The findings that cooperation increases withthe probability of continuation and that infi-nitely repeated games result in higher levels ofcooperation than finitely repeated ones of thesame expected length suggests that self-enforcingreward and punishment schemes that limit op-portunistic behavior are important in practice aswell as in theory.But the theory of infinitely repeated gamesprovides more than general comparative staticsresults. It provides precise predictions regardingthe set of equilibrium outcomes. I use the factthat small differences in the payoff matrix mayresult in large differences in the set of equilib-rium outcomes to study how closely the behav-ior of the subjects matches the theoreticalpredictions. I used two different payoff matricesin the experiment with the peculiarity that, for aprobability of continuation of1⁄2, cooperationfor both players is an equilibrium in one but notin the other. I find that the percentage of out-comes in which both subjects cooperate is al-most 19 percent when it is an equilibrium, whileit is less than 3 percent when it is not. Theseexperimental results show that behavior closely,although not perfectly, follows the theoreticalpredictions that are dependent on the payoffdetails of the stage game. This provides furthersupport for the theory of infinitely repeatedgames.Section I summarizes previous experimentalresearch on the topic. Section II describes theexperimental design, and Section III describesthe theoretical predictions that provide thetestable hypothesis. Section IV presents theresults of the experiment, and the last sectionconcludes.I. Related LiteratureThe experimental