Repeated prisoner's dilemma with errors and option to leave

We study a population of players who are coupled up to play a repeated prisoner's dilemma, with the possibility of making errors in the implementation of their strategies. The partnerships may be broken due to external factors (exogenous separation) or because one of the partners decides to leave (endogenous separation). Players who separate and go solo are randomly matched with another single player and keep on playing the game with their new partner.

Strategies in this setting map the histories of play within the partnership to one of the following actions: Cooperate, Defect or leave. We focus on memory-1 strategies, which are strategies in which players condition their actions on the outcome of their last interaction with their partner. Some examples are Tit-For-Tat, Win–Stay–Lose–Shift, Grim and AllD. With the option to leave, there are 162 memory-1 strategies, since one must decide on their initial action in a new partnership (Cooperate or Defect) and what to do (Cooperate, Defect or leave) after each of the four possible outcomes in the game. Consequently, there are 2 x 3⁴ = 162 strategies.

We consider an evolutionary setting where players occasionally revise their strategies. When revising them, players preferentially switch to strategies that are performing well, and they may also try new ones (experimentation).

We study the levels of cooperation and the success of each strategy, and we compare models with and without the option to leave. With low probabilities of an exogenous breakup, cooperation is significantly greater and more robust with the option to leave than without it. Furthermore, classical strategies that support cooperation in the repeated prisoner's dilemma without the option to leave do not fare well when players are allowed to leave their partner: with moderate probabilities of error and experimentation, classical strategies—which do not include leaving—are basically wiped out in the population.