Tacit Cooperation, Strategic Uncertainty, and Coordination Failure

Tacit Cooperation, Strategic Uncertainty, and Coordination Failure:
Evidence from Repeated Dominance Solvable Games.

John B. Van Huyck, John M. Wildenthal, and Raymond C. Battalio

Revised January 2000

Abstract: This paper reports an experiment designed to discover how the prospect of future interaction influences peoples ability to tacitly cooperate in repeated dominance solvable games. The experiment varies two treatment variables: whether the constituent game is solvable by strict or iterated dominance and whether prospective interaction is finitely or randomly terminated. Specifically, we introduce a special repeated matching protocol consisting of an initial phase terminated randomly and a terminal phase of T periods. We call this protocol T-death. The T-death protocol allows us to observe a pair's behavior in both a sequence of infinite continuation games and a sequence of finite continuation games.

Key Words: Cooperation, Coordination, Dominance, Repeated Games, Prisoners' Dilemma, Duopoly, Public Goods, Experiment.

JEL Classification: c720, c920, l120, l400.

Acknowledgments: An associate editor provided a succinct summary that shortened the paper. The National Science Foundation and the Texas Advanced Research Program provided financial support.

Introduction

Reciprocity amongst patient players is often used to explain why an apparent incentive problem when analyzed in a static game does not prevent tacit cooperation in a repeated game. For example, it is often argued that oligopolists will be able to tacitly collude on the monopoly solution since they interact repeatedly. However, almost "anything" is an equilibrium of a repeated game if players are sufficiently "patient." So in applying repeated game theory, economists typically select an equilibrium that is efficient and symmetric.

Using field data to investigate the psychological salience of deductive selection principles, like efficiency and symmetry, is difficult. An alternative approach is to use the experimental method. Van Huyck, Battalio, and Beil (1990,1991) present evidence that security can undermine the salience of efficiency in repeated coordination games. Van Huyck, et al. (1995) present evidence that security can undermine the salience of symmetry in repeated bargaining games.

The relevance of these discoveries for repeated cooperation problems is an open question. Repeated game theory suggests that the prospect of future interaction amongst patient players converts a cooperation problem into a coordination problem. However, the resulting strategy coordination problem is more difficult than those arising in repeated bargaining and coordination games. Not only are there multiple equilibria in repeated cooperation games, but the equilibria with desirable strategic properties, like efficiency, are constructed using history contingent strategies. Consequently, history must be used not only to focus expectations on an equilibrium assignment, but also to monitor compliance with the equilibrium assignment.

Since history contingent strategies are not observable by other players in the game, one wonders if observing the history of play would allow players to solve the strategy coordination problem in the continuation game. Surprising play by other strategically rational players may be due either to a sincere disagreement about the equilibrium assignment or to an opportunistic defection from a commonly understood equilibrium assignment. In the first case one should try to teach the assignment, and in the second case one should try to enforce the assignment. This dual use of history does not arise in repeated bargaining and coordination games.

In this paper, we report evidence on how the prospect of future interaction influences the ability of people to tacitly cooperate. The experiment uses repeated dominance solvable games that vary two treatment variables: whether the constituent game was solvable by strict dominance or iterated dominance and whether prospective interaction is finitely or randomly terminated. Specifically, we introduce a special repeated game protocol consisting of an initial phase terminated randomly and a terminal phase of T periods. The T-death protocol allows us to observe a pair's behavior in both a sequence of infinite continuation games and a sequence of finite continuation games. Subjects were given experience with the T-death protocol by the use of an evolutionary repeated game matching protocol.

Our principle finding is that efficiency and symmetry need not be psychological salient in the repeated dominance solvable games considered. In fact, it turns out to be quite difficult for subjects to learn to coordinate on the symmetric payoff-dominant play path under the T-death protocol, but it does occur in some cases.

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Literature Review

There is a large experimental literature on repeated dominance solvable games, which can be divided into matrix game experiments, oligopoly experiments, and public goods experiments. Our review of this literature focuses on the influence of prospective interaction in promoting tacit cooperation. There is also a smaller evolutionary tournaments (strategy method) literature, which is of interest here due to its use of explicit strategies.

Rapoport, Guyer, and Gordon (1976) survey the early 2×2 game experiments. This literature emphasized the relatively high levels of cooperation observed in finitely repeated games for which the theory of the day predicted none. Under complete information, game theory predicts backwards unraveling in the finitely repeated Prisoners' Dilemma. [4] Kreps, et al. (1982) explained these early results for the Prisoners' Dilemma by analyzing incomplete information in finitely repeated games. Their analysis bounds the number of periods of backwards unraveling when one player may be a "tit-for-tat" automaton. [5]

Selten/Stoecker (1986) report an experiment in which 35 subjects participated in 25 repeated Prisoners' Dilemma contests of ten periods each against randomly and anonymously assigned opponents. The typical behavior of experienced subjects involves cooperation until shortly before the end of the contest. They conclude that this behavior is inconsistent with the Kreps, et al. model because "subjects first have to learn cooperation and only afterwards do they discover the end effect. (p.48)" They conclude that it is not clear whether this decay would have continued in a much longer sequence of contests. [6]

The backwards unraveling argument in the finitely repeated Prisoners' Dilemma breaks down when the repeated game is terminated randomly. Roth/Murnighan (1978) report an experiment using a randomly terminated repeated Prisoners' Dilemma, where the probability the game continues varied over 0.105, 0.5, and 0.895. Tacit Cooperation is consistent with many equilibrium strategy combinations for continuation probabilities 0.5 and 0.895, but not 0.105. They find that the continuation probability has a statistically significant effect on the incidence of cooperation observed in the first period. Given that subjects were playing against the "tit-for-tat" automaton, it is interesting to note how little cooperation actually emerges overall: 31 percent for 0.5 and 41 percent for 0.895.

Their results illustrate the importance of allowing subjects to acquire experience in the repeated game. Surely, cooperation rates would have been much higher had subjects learned they were playing against the "tit-for-tat" automaton in Roth/Murnighan. In retrospect the puzzle about this early literature is not why there is so much cooperation, but rather why so little cooperation is observed in repeated dominance solvable games. [7] Of course, cooperation in "one-play" experiments cannot be explained by modern repeated game theory, but perhaps these results simply demonstrate either that subjects don't realize they have a strictly dominant strategy or that the experimenter failed to induce the preferences assumed in the formal analysis. [8]

Cooper, et al. (1992) conduct what is probably the strongest test to date of the decision theoretic prescription--don't use strictly dominated strategies--in that subjects were allowed to acquire experience in a "one-shot" Prisoners' Dilemma but were prevented from interacting with the same player more than once. In fact, their matching protocol even prevents a subject from interacting with anyone who has interacted with someone that a subject has already meet, which rules out contagion equilibria in the evolutionary game. With nineteen periods of experience, only about 12 percent of the subjects played the symmetric efficient, but strictly dominated, action, that is, only 12 percent cooperate.

They also report sessions in which subjects participate in two repeated Prisoners' Dilemma contests of 10 periods each. For the second contest, they observe cooperation rates less than 75 percent even in the early periods of the contest, which violates the Kreps, et al. (1982) bound for plausible amounts of incomplete information. A comparison of the "one-shot" and repeated pairing treatments does suggest that repeated interaction promotes cooperation, but, like the results reported by Roth/Murnighan, the observed cooperation rates are well below the theoretical prediction suggesting a lot of confusion and the need for more experience.

The once-repeated Prisoners' Dilemma is special in that it can be solved by a single deletion of strictly dominated strategies. Duopoly experiments provide an interesting contrast in that the stage game is usually solvable by iterated dominance. Unlike the deletion of strictly dominated strategies, iterated dominance requires a player to actually think about the behavior of his opponent. Hence, one would expect games solvable by iterated dominance to be more confusing to inexperienced subjects.

Fouraker/Siegel (1963) report sessions with both a price setting and quantity setting duopolists under "complete information." The price setting duopoly session allowed subjects to choose from 10 actions and the resulting matrix game is solvable by iterated dominance in 7 iterations. The quantity setting duopoly session allowed 24 actions and could be solved in 2 iterations. One problem with fitting these sessions into a repeated versus random matching classification is that, while the experimental design called for a finite number of periods (14 and 24 periods respectively), subjects did not know this. The symmetric efficient outcome was achieved by 11 percent of the pairs in the terminal period of the repeated price setting duopoly session and by 12 percent of the pairs in the terminal period of the repeated quantity setting duopoly sessions. In comparison to the finitely repeated Prisoners' Dilemma, these are low rates of tacit cooperation.

Holt (1985) reports a quantity setting duopoly experiment with 18 actions resulting in a payoff matrix solvable by iterated dominance in 4 iterations. Holt carefully distinguishes between random and repeated matching protocols. The repeated pairings lasted for at least seven periods. After seven periods, the pairing continued into the next period with probability 5/6th. The random pairings consisted of 12 subjects randomly repaired in each of ten periods. In the terminal period of the repeated pairing session, 25 percent of the pairs achieved the symmetric efficient outcome (collusion). In the terminal period of the random pairing sessions, none of the pairs achieved the symmetric efficient outcome. While the repeated pairing treatment significantly increased observed cooperation, the reported level of cooperation is still well below the level predicted by the theory of repeated games when combined with the selection principles of efficiency and symmetry.

The Handbook of Experimental Economics includes two chapters in which the influence of prospective interaction on cooperation is discussed. Holt surveys the experimental industrial organization literature and Ledyard surveys the experimental public goods literature. Prospective interaction is found to increase cooperation in the industrial organization literature, but to reduce cooperation in the public goods literature. On the surface this appears puzzling, because public goods experiments using a voluntary contributions mechanism typically have more than 10 actions and are solvable by strict dominance. Upon reflection, this difference seems to be attributable to an interaction effect between contest size--public goods experiments usually have four or more subjects and industrial organization experiments usually have four or fewer subjects--and repeated play.

Since repeated game strategies are not observed, it is hard to know if the experimental literature is reporting equilibrium or disequilibrium behavior. [9] An alternative approach is to have subjects actually write programs, which are then run against each other and scored, see Axelrod (1984) and Selten, Mitzkewitz, and Uhlich (1988). These tournaments tend to select strategies that have robust disequilibrium properties. For example, Axelrod (1984) concludes that the "tit-for-tat" automaton does well in his repeated prisoners' dilemma tournaments, because it is nice, provocable, forgiving, and clear.

The "tit-for-tat" automaton is nice in that it will coordinate on the efficient symmetric play path if matched with another nice automaton. It is provocable and, hence, willing to "punish" deviations from the efficient symmetric play path, which can make cooperate until the terminal period a best response to "tit-for-tat" if the incentive to defect from the cooperative assignment is not to large. It is forgiving and clear, which allows it to teach cooperation. Note that all these characteristics, except provocability, concern the strategy coordination problem rather than the enforcement problem. [10]

Conclusion

The experiment varied two treatment variables: whether the constituent game was solvable by strict dominance or iterated dominance and whether prospective interaction was finitely or randomly terminated. Both treatments had an economically significant influence on behavior.

The difference between strict and iterated dominance had a large influence on behavior initially. Our results suggest a low order of reasoning about the reasoning of others, see also Nagel (1995) and Stahl/Wilson (1995). Nevertheless, with experience behavior does appear to be converging to the dominance solvable equilibrium, which is the same for both payoff tables, under a once-repeated random matching protocol, '(0,1), as predicted by adaptive learning theories.

While reciprocity amongst patient players explains why an apparent incentive problem when analyzed in a static game does not prevent tacit cooperation in theory, our results suggest that the resulting strategy coordination problem is difficult and should not be assumed away in practice. Our subjects were not able to coordinate on the symmetric payoff-dominant equilibrium play path initially. In fact, if one only studied the first (or second) match, as is typical, one would conclude that prospective interaction, whether with a known or unknown termination, didn�t influence behavior.

The use of an evolutionary repeated game protocol allowed subjects to learn to discriminate between the randomly terminated initial phase and the finite terminal phase of the T-death matching protocol in the S(5/6,2) treatment, but not in I(5/6,2). We attribute this to the difference between the depth of reasoning needed to solve the stage games. In S(5/6,2), there are only two salient actions and this clarity allowed subjects to learn to coordinate on the symmetric payoff-dominant equilibrium play path.

References

Dan Alger, Investigating Oligopolies within the Laboratory, Bureau of Economics, Federal Trade Commission, January 1986.

J. Andreoni, "Cooperation in Public Goods Experiments: Kindness or Confusion," laser-script, March 1993.

Robert Axelrod, The Evolution of Cooperation, (New York: Basic Books, Inc., Publishers, 1984).

Ken Binmore, Playing Fair, (Cambridge, MA: MIT press, 1994).

K. Binmore, C. Proulx, S. Hsu, and J. Swiezbinski, "Focal Points and Bargaining," laser-script, 1994, forthcoming International Journal of Game Theory.

G.P. Cachon and C. Camerer, "The Sunk Cost Fallacy, Forward Induction, and Behavior in Coordination Games," laser-script, July 1991.

W.J. Conover, Practical Nonparametric Statistics, 2nd edition, (New York, NY: John Wiley & Sons, 1980).

R. Cooper, D.V. DeJong, R. Forsythe, and T.W. Ross, "Cooperation with-out Reputation: Experimental Evidence from Prisoners' Dilemma Games," laser-script, August 1992.

R. Cooper, D.V. DeJong, R. Forsythe, and T.W. Ross, "Selection Criteria in Coordination Games: Some Experimental Results," American Economic Review, 80(1), March 1990.

Robyn M. Dawes and Richard H. Thaler, "Anomalies: Cooperation," Journal of Economic Perspectives, 2(3), Summer 1988, 187-197.

Lawrence E. Fouraker and Sidney Siegel, Bargaining Behavior, (New York, NY: McGraw-Hill Book Company, 1963).

Drew Fudenberg and Jean Tirole, Game Theory, (Cambridge, MA: MIT Press, 1991).

Charles A. Holt, "An Experimental Test of the Consistent-Conjectures Hypothesis," The American Economic Review 75(3), June 1985, 314-25.

John Kagel and Alvin Roth, Handbook of Experimental Economics, laser-script, 1993.

M.G. Kendall and A. Stuart, The Advanced Theory of Statistics, (New York,NY: Hafner, 1969).

David M. Kreps, Paul Milgrom, John Roberts, and Robert Wilson, "Rational Cooperation in the Finitely Repeated Prisoners' Dilemma," Journal of Economic Theory 27, 1982, 245-52.

Richard D. McKelvey and Thomas R. Palfrey, "Quantal Response Equilibria for Normal Form Games," laser-script, revised October 1993.

Paul Milgrom and John Roberts, "Adaptive and Sophisticated Learning in Normal Form Games," Games and Economic Behavior, 3(1), February 1991.

Rosemarie Nagel, "Experimental Results on Interactive Competitive Guessing," laser-script, April 1993.

A. Rapoport, M.J. Guyer, and D.G. Gordon, The 2×2 Game, (Ann Arbor, MI: University of Michigan Press, 1976).

Alvin E. Roth and J. Keith Murnighan, "Equilibrium Behavior and Repeated Play of the Prisoner's Dilemma," Journal of Mathematical Psychology, 17, 1978, 189-98.

Alvin E. Roth and Francoise Schoumaker, "Expectations and Reputations in Bargaining: an experimental study," American Economic Review, 73, 1983, 362-373.

Dale O. Stahl, II and Paul W. Wilson, "On Players' Models of other Players: a new theory and experimental evidence," laser-script, August 1993.

Reinhard Selten, Michael Mitzkewitz, and Gerald R. Uhlich, "Duopoly Strategies Programmed by Experienced Players," laser-script, December 1988.

R. Selten and R. Stoecker, "End behavior in sequences of finite repeated prisoner's dilemma supergames," Journal of Economic Behavior and Organization, 7, 1986, 47-70.

Paul G. Straub, "Understanding Coordination Failures," laser-script, October 1990.

John B. Van Huyck, Raymond C. Battalio, and Richard O. Beil, "Tacit Coordination Games, Strategic Uncertainty, and Coordination Failure," The American Economic Review 80(1), March 1990, 234-248.

John B. Van Huyck, Raymond C. Battalio, and Richard O. Beil, "Strategic Uncertainty, Equilibrium Selection, and Coordination Failure in Average Opinion Games," The Quarterly Journal of Economics, 106(3), August 1991, 885-910.

John Van Huyck, Raymond Battalio, Sondip Mathur, Andreas Ortmann, and Patsy Van Huyck, "On the Origin of Convention: Evidence from Symmetric Bargaining Games, " laser-script, April 1993, forthcoming International Journal of Game Theory.

John B. Van Huyck, Joseph P. Cook, and Raymond C. Battalio, "Adaptive Behavior and Coordination Failure," laser-script, August 1993.

Footnotes

[1] See Fudenberg/Tirole (1991, p.160).

[2] See also Cachon/Camerer (1991), Cooper et al. (1990), and Straub (1990).

[3] See also Binmore et al. (1994), Roth/Schoumaker (1983), Van Huyck, Cook, and Battalio (1993).

[4] At the time it seems to have been taken for granted that game theory failed to predict human behavior: "...most of the game-theoretic prescriptions cannot be taken as serious contenders in a behavioral theory because, as a rule, they are disconfirmed...(Rapoport, Guyer, Gordon, 1976, p.72)". Social psychologists began to investigate non-strategic details, which sometimes produced dramatic differences, but these differences are hard to organize into a coherent theory. Our favorite was an experiment designed to see if thinking about the Prisoners' Dilemma improved cooperation. The treatment variable was whether the choice had to be made immediately after reading the instructions or the next day. The finding was that sleeping on one's decision didn't promote cooperation. See Binmore (1994, chapter 3) for a critical survey of the extraordinary amount of intellectual energy devoted to justifying the use of a strictly dominated strategy in the once-repeated Prisoners' Dilemma.

[5] A "tit-for-tat" automaton plays the action consistent with the symmetric efficient outcome in the first period of a match and the action played by its opponent in the previous period in all subsequent periods.

[6] It is interesting to note the cohort effects in their data. The mean intended defection period ranges from 5.5 in group I to 9.0 in group III, see their table 2.

[7] Rapoport, Guyer, Gordon (1976, p.234) do report that "the 'average pair' playing a long sequence of Prisoner's Dilemma eventually learns to cooperate," which is what the theory of repeated games predicts when combined with the selection principles of efficiency and symmetry. Of course, the view that there is more cooperation than game theory predicts persists to the current day, see for example Dawes/Thaler (1988).

[8] The fact that a subject's own money earnings may fail to induce the desired preferences is of interest in that it demonstrates that subjects are not purely money motivated, but may care about other things, like the money earnings of others (altruism) or about the experimenter having interesting results to report (experimenter induced bias). Andreoni (1993) investigates the more basic possibility that subjects are simply confused about how actions are related to outcomes in a public goods framework.

[9] Alger (1986) argues that most experiments don't report equilibrium behavior on the grounds that behavior is not "stable" over time.

[10] Axelrod (1984) also has some wonderful anecdotal evidence on the influence of prospective interaction on cooperation, including the observation that "a visiting professor is likely to receive poor treatment by other faculty members compared to the way these same people treat their regular colleagues (p.60)" and Caesar's explanation of why Pompey's allies stopped cooperating with him: "They regarded his [Pompey's] prospects as hopeless and acted according to the common rule by which a man's friends become his enemies in adversity. (p.59)"