What risk should a selfish partner take in order to save the life of a non-relative, selfish friend? A stochastic game approach to the prisoner’s dilemma

A model of cooperation versus defection in a sequence of games is analyzed under the assumptions that the rules of the games are randomly changed from one encounter to another, that the decisions are to be made at each situation anew, according to the specific rules of the specific local game, and that the outcome of each such game affects the ability of a player to participate and, thus, cooperate (if in its own interest) in the next game. Players wish to maximise their total payment at the entire supergame. Under plausible assumptions it is shown that all Evolutionary Stable Strategy (ESS) of the supergame determine cooperation over a non-degenerate range of situations determining encounters of the P.D. type - A moderately altruistic cooperation is selected because it enables the survival of a potential partner for future cooperation. The model also explains the evolution of gratitude, rather than the assumption thereof, and predicts a qualitative difference between partnership altruism and kin altruism.