Cooperation in Repeated Prisoner’s Dilemma with Outside Options

In many repeated interactions, repetition is not guaranteed but instead must be agreed upon. We formulate a model of voluntary repetition by introducing outside options to a repeated Prisoner’s Dilemma and investigate how the structure of outside options affects the sustainability of mutual cooperation. When the outside option is deterministic and greater than the value of mutual defection, the lower bound of the discount factors that sustain repeated cooperation is greater than the one for ordinary repeated Prisoner’s Dilemma, making cooperation more difficult. However, stochastic outside options with the same mean may reduce the lower bound of discount factors as compared to the deterministic case. This is possible when the stochasticity of the options increases the value of the cooperation phase more than the value of the punishment phase. Necessary ansufficient conditions for this positive effect are given under various option structures.