Equilibrium payoff bounds from reputation effects are derived for repeated games with imperfect public monitoring in which a long-run player interacts frequently with a population of short-run ...players and the monitoring technology scales with the length of the period of interaction. The bounds depend on the monitoring technology through the flow of information, a measure of signal informativeness per unit of time based on relative entropy. Examples are shown where, under complete information, the set of equilibrium payoffs of the long-run player converges, as the period length tends to zero, to the set of static equilibrium payoffs, whereas when the game is perturbed by a small ex ante probability on commitment types, reputation effects remain powerful in the high-frequency limit.
A growing experimental literature studies the determinants of cooperation in infinitely repeated games, tests different predictions of the theory, and suggests an empirical solution to the problem of ...multiple equilibria. To provide a robust description of the literature’s findings, we gather and analyze a metadata set of experiments on infinitely repeated prisoner’s dilemma games. The experimental data show that cooperation is affected by infinite repetition and is more likely to arise when it can be supported in equilibrium. However, the fact that cooperation can be supported in equilibrium does not imply that most subjects will cooperate. High cooperation rates will emerge only when the parameters of the repeated game are such that cooperation is very robust to strategic uncertainty. We also review the results regarding the effect of imperfect monitoring, changing partners, and personal characteristics on cooperation and the strategies used to support it.
Experimental studies of infinitely repeated games typically consist of several indefinitely repeated games (“matches”) played in sequence with different partners each time, whereby match length, i.e. ...the number of stages of each game is randomly determined. Using a large meta data set on indefinitely repeated prisoner's dilemma games (Dal Bó and Fréchette, 2018) we demonstrate that the realized length of early matches has a substantial impact on cooperation rates in subsequent matches. We estimate simple learning models displaying the “power law of practice” and show that participants do learn from match length realization. We then study three cases from the literature where realized match length has a strong impact on treatment comparisons, both in terms of the size and the direction of the treatment effect. These results have important implications for our understanding of how people learn in infinitely repeated games as well as for experimental design.
Abstract
We show that the folk theorem holds generically for the repeated two-player game with private monitoring if the support of each player’s signal distribution is sufficiently large. Neither ...cheap talk communication nor public randomization is necessary.
More than half a century after the first experiment on the finitely repeated prisoner’s dilemma, evidence on whether cooperation decreases with experience—as suggested by backward induction—remains ...inconclusive. This article provides a meta-analysis of prior experimental research and reports the results of a new experiment to elucidate how cooperation varies with the environment in this canonical game. We describe forces that affect initial play (formation of cooperation) and unraveling (breakdown of cooperation). First, contrary to the backward induction prediction, the parameters of the repeated game have a significant effect on initial cooperation. We identify how these parameters impact the value of cooperation—as captured by the size of the basin of attraction of always defect—to account for an important part of this effect. Second, despite these initial differences, the evolution of behavior is consistent with the unraveling logic of backward induction for all parameter combinations. Importantly, despite the seemingly contradictory results across studies, this article establishes a systematic pattern of behavior: subjects converge to use threshold strategies that conditionally cooperate until a threshold round; conditional on establishing cooperation, the first defection round moves earlier with experience. Simulation results generated from a learning model estimated at the subject level provide insights into the long-term dynamics and the forces that slow down the unraveling of cooperation.
•We experimentally analyze behavior in repeated games when helping people in one's group hurts those in other groups.•Intergroup conflict aversion weakens intragroup cooperation as cooperation ...decreases with the degree of rivalry with the outgroup.•Cooperation increases with the degree of synergy with partners and previous cooperation by partners but not rivals.•To understand the motives and strategies that drive such behavior, we use a structural model to estimate group-specific social preferences and cooperative strategies.•It shows that boundedly rational participants cooperate with partners if they expect profits from mutual cooperation, but stop once partners defect, and avoid conflict because they are altruistic to rivals.
We experimentally analyze group-specific social preferences and dynamic strategies in finitely repeated games where helping partners hurts rivals. Cooperation with partners decreases when it inflicts losses on rivals, even when it is socially efficient. Cooperation decreases with rivalry, while it increases with synergy and previous cooperation by partners but not rivals. Our structural model of bounded rationality estimates preferences and strategies. It shows that participants anticipate future payoffs from partner cooperation and punish partner defection with unyielding defection, while further averting intergroup conflict due to altruism towards rivals. Intragroup cooperation and intergroup conflict weaken over time through these two channels.
We propose a new approach for running lab experiments on indefinitely repeated games with high continuation probability. This approach has two main advantages. First, it allows us to run multiple ...long repeated games per session. Second, it allows us to incorporate the strategy method with minimal restrictions on the set of pure strategies that can be implemented. This gives us insight into what happens in long repeated games and into the types of strategies that subjects construct. We report results obtained from the indefinitely repeated prisoner’s dilemma with a continuation probability of δ=.95. We find that during such long repeated prisoner’s dilemma games, cooperation drops from the first period of a supergame to the last period of a supergame. When analyzing strategies, we find that subjects rely on strategies similar to those found in the literature on shorter repeated games—specifically Tit-For-Tat, Grim Trigger, and Always Defect. However, we also identify features of strategies that depend on more than just the previous period that are responsible for the drop in cooperation within supergames, but that may be overlooked when using the common strategy frequency estimation approach.
Do people lie less in repeated interactions with the same partner than in a series of one-shot interactions with strangers? We find that under asymmetric information, senders lie substantially less ...if paired with the same receiver than when randomly re-matched with different receivers. However, the lying gap diminishes if the receiver is allowed to offer feedback to the sender.
•Subjects lie less in finitely repeated game than in one-shot game, without feedback.•The repetition effect greatly diminishes if feedback is present.•Repeated interaction and feedback acts as near substitutes to influence lying.
This paper compares behavior under four different implementations of infinitely repeated games in the laboratory: the standard random termination method proposed by Roth and Murnighan (J Math Psychol ...17:189–198,
1978
) and three other methods that de-couple the expected number of rounds and the discount factor. Two of these methods involve a fixed number of repetitions with payoff discounting, followed by random termination proposed by Sabater-Grande and Georgantzis (J Econ Behav Organ 48:37–50,
2002
) or followed by a coordination game proposed in (Andersson and Wengström in J Econ Behav Organ 81:207–219,
2012
; Cooper and Kuhn in Am Econ J Microecon 6:247–278,
2014a
). We also propose a new method—block random termination—in which subjects receive feedback about termination in blocks of rounds. We find that behavior is consistent with the presence of dynamic incentives only with methods using random termination, with the standard method generating the highest level of cooperation. Subject behavior in the other two methods display two features: a higher level of stability in cooperation rates and less dependence on past experience. Estimates of the strategies used by subjects reveal that across implementations, even when the discount rate is the same, if interactions are expected to be longer defection increases and the use of the Grim strategy decreases.
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. PUBLICATION ABSTRACT
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