Cumulative benefit games: achieving cooperation when players discount the future

Experimental studies with captive animals show strong preferences for immediate reward. Several authors have argued that these tendencies to discount delayed reward may severely limit the Iterated Prisoner's Dilemma game as a model of animal cooperation. This paper explores a simple mechanism, dubbed cumulative games, that can, in principle, promote cooperative action even when there is strong temporal discounting. In the simplest type of cumulative game a pair of players does not receive benefits at the end of each play, as in a conventional repeated game, but must complete a sequence of games before collecting the accumulated benefits. In a preliminary analysis pitting tit-for-tat against all-D, I show that accumulation can promote a conditionally cooperative strategy even when there is strong temporal discounting. However, the delays created by accumulation de-value the pairwise interaction, so although the relative value of cooperation increases, the total value of the interaction decreases. I investigate accumulation further by simulating the evolution of a broader class of strategies. These simulation studies show that accumulation, and small discounting rates (high future value) can both promote cooperative action. The limitations of these results are discussed.