# Prisoners’ other Dilemma

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## Abstract

We introduce a measure for the *riskiness* of cooperation in the infinitely repeated discounted Prisoner’s Dilemma and use it to explore how players cooperate once cooperation is an equilibrium. Riskiness of a cooperative equilibrium is based on a pairwise comparison between this equilibrium and the uniquely safe all defect equilibrium. It is a strategic concept heuristically related to Harsanyi and Selten’s risk dominance. Riskiness 0 defines the same critical discount factor \(\delta ^{*}\) that was derived with an axiomatic approach for equilibrium selection in Blonski et al. (Am Econ J 3:164–192, 2011). Our theory predicts that the less risky cooperation is the more forgiving can parties afford to be if a deviator needs to be punished. Further, we provide sufficient conditions for cooperation equilibria to be *risk perfect*, i.e. not to be risky in any subgame, and we extend the theory to asymmetric settings.

### Keywords

Cooperation Repeated Prisoner’s Dilemma Equilibrium selection Forgiveness Perfection Strategic risk Strategic uncertainty Sucker’s payoff Collusion Coordination### JEL Classification

C72 C73 C92 L13 L14 M50## Notes

### Acknowledgments

We appreciate suggestions and remarks by an anonymous referee, Geir Asheim, Eric van Damme, Christian Ewerhart, David Levine, Klaus Ritzberger, Reinhard Selten and various seminar participants in Berkeley, Bonn, Frankfurt, Mannheim, Stanford, Stockholm, Tilburg, and Zürich. This Project has been supported by Grants from the DFG (BL418/4-1) for Blonski, and from the European Commission (EGEMTPS/RTN) for Spagnolo.

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