The consequences of switching strategies in a two-player iterated survival game

We consider two-player iterated survival games in which players are able to switch from a more cooperative behavior to a less cooperative one at some step of an n-step game. Payoffs are survival probabilities and lone individuals have to finish the game on their own. We explore the potential of these games to support cooperation, focusing on the case in which each single step is a Prisoner's Dilemma. We find that incentives for or against cooperation depend on the number of defections at the end of the game, as opposed to the number of steps in the game. Broadly, cooperation is supported when the survival prospects of lone individuals are relatively bleak. Specifically, we find three critical values or cutoffs for the loner survival probability which, in concert with other survival parameters, determine the incentives for or against cooperation. One cutoff determines the existence of an optimal number of defections against a fully cooperative partner, one determines whether additional defections eventually become disfavored as the number of defections by the partner increases, and one determines whether additional cooperations eventually become favored as the number of defections by the partner increases. We obtain expressions for these switch-points and for optimal numbers of defections against partners with various strategies. These typically involve small numbers of defections even in very long games. We show that potentially long stretches of equilibria may exist, in which there is no incentive to defect more or cooperate more. We describe how individuals find equilibria in best-response walks among n-step strategies.