Modelling Reciprocal Altruism

Biologists rely extensively on the iterated Prisoner's Dilemma game to model reciprocal altruism. After examining the informal conditions necessary for reciprocal altruism, I argue that formal games besides the standard iterated Prisoner's Dilemma meet these conditions. One alternate representation, the modified^Prisoner's Dilemma game, removes a standard but unnecessary condition; the other game is what I call a Cook's Dilemma. We should explore these new models of reciprocal altruism because they predict different stability characteristics for various strategies; for instance, I show that strategies such as Tit-for-Tat have different stability dynamics in these alternate models. 1 The altruism puzzle and the standard model 2 Informal conditions for reciprocal altruism 3 Criticism of Axelrod's justification of the anti-exploitation condition 4 A menu of formal models of reciprocal altruism 5 Modelling reciprocal altruism in guppies, baboons, and bats 5.1 Modelling simultaneous cooperation in guppies 5.2 Nonsimultaneous cooperation in baboons 5.3 Modelling reciprocal altruism with the Cook's Dilemma 6 Characteristics of the alternate models 7 Conclusions Appendix Resistance to invasion results for TFT and ALT 1 The altruism puzzle and the standard model Scientists use mathematical models to make vague ideas and claims more precise. Besides having the virtue of making such assertions easily testable, the precision of mathematical models often yields surprising theoretical results that could not be predicted a priori by pre-mathematical intuition. But care must also be taken in formalizing our attempts to describe the world—scientists face many choices in deciding how to make their models and theories rigorous; one hopes to avoid the relatively unimportant details and preserve what is needed for a model to have predictive and explanatory success. Theoretical biologists are no exception, and face these sorts of decisions. One of the puzzles facing biologists is the pervasiveness of cooperative behaviour among organisms. Prima facie, the theory of evolution by © Oxford University Press 1996 at T he U nirsity of B rtish C olom ia Lrary on July 6, 2010 http://bjpordjournals.org D ow nladed fom 534 Christopher Stephens natural selection implies that helping behaviour should not exist because organisms that do not help should do better than helpers by reaping the rewards of the help without incurring the costs. But, of course, organisms frequently do help one another. Why is this? William Hamilton [1964] developed inclusive fitness theory (kin selection theory) to provide a partial explanation for the prevalence of cooperation. Helping behaviour among close relatives is selected for because close relatives share a large percentage of genes. But kin selection theory cannot provide a complete explanation of cooperative behaviour because helping often occurs between organisms that are not close relatives (Trivers [1971]; Wilkinson [1988]). Biologists appeal to reciprocal altruism to explain cooperation that cannot be accounted for by kin selection. Roughly, reciprocal altruism evolves because organisms do better by accepting the immediate costs of helping another organism in order to reap the comparatively greater benefits of receiving help at a later time. In order to develop this idea precisely, biologists rely extensively on the two-person iterated Prisoner's Dilemma game (Trivers [1971]; Axelrod and Hamilton [1981]; Maynard Smith [1982]; Axelrod [1984]; Dugatkin [1988]; Nowak and Sigmund [1993, 1994]). In a Prisoner's Dilemma game, each player has two possible behaviours: defect or cooperate. The game may be represented in the following way: Player 2 Cooperate Defect Player Cooperate W, W X, Y