Cumulative benefit games: Achieving cooperation when players discount the future

Experimental studies with captive animals show strong preferences for immed iate 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, dubbe d cumulative games, that can, in principle, promote cooperative action even when there is strong temporal discounting. In the simplest type of cumulat ive game a pair of players does not receive benefits at the end of each pla y, as in a conventional repeated game, but must complete a sequence of game s before collecting the accumulated benefits. In a preliminary analysis pit ting tit-for-tat against all-D, I show that accumulation can promote a cond itionally cooperative strategy even when there is strong temporal discounti ng. However, the delays created by accumulation de-value the pairwise inter action, so although the relative value of cooperation increases, the total value of the interaction decreases. I investigate accumulation further by s imulating the evolution of a broader class of strategies. These simulation studies show that accumulation, and small discounting rates (high future va lue) can both promote cooperative action. The limitations of these results are discussed. (C) 2000 Academic Press.