Thursday, February 23, 2006

The Evolution of Co-operation: the Santos Model   posted by DavidB @ 2/23/2006 07:02:00 AM

The existence of cooperation is one of the major problems in human evolution. Among non-human animals, cooperation is rare except among individuals who are closely related. Among humans, in contrast, it is common. The problem is to explain this in view of the temptation to 'defect' from cooperation, obtaining its benefits without its costs. The problem is classically exemplified in the game of the Prisoner's Dilemma, where in any single play of the game it is always advantageous for an individual to defect, even though two players who cooperate will both do better than two who defect.

A variety of solutions to the problem have been suggested. They include:

- reciprocal altruism in repeated interactions (Trivers)

- group selection for benefits to the social group (D. S. Wilson, Boyd and Richerson, and others)

- cooperation enforced by punishment, including 'altruistic' punishment (see e.g. here)

- indirect fitness benefits to cooperators, such as sexual selection via the Handicap Principle.

An interesting alternative or addition to these solutions has been developed in recent work by F. C. Santos and colleagues. The main papers are available here. (NB some of these are in preprint form, so they may not correspond exactly to published versions). As far as I understand it (which may not be very far) the key feature of the Santos model is that societies have a heterogeneous structure, in such a way that some individuals have more social interactions than others. The results of simulations appear to show that for plausible values of the parameters cooperation can prevail even in the classical Prisoner's Dilemma game, where each pair of individuals meet at random and interact no more than once.

This appears paradoxical, because there is no doubt that if players interact at random, the average payoff per interaction is greater for defectors than cooperators, whatever their proportions in the population. If payoffs are measured in reproductive fitness one would therefore expect defectors to drive cooperators to extinction. The solution to the paradox is that the total payoff to each individual depends not only on his average payoff per interaction, but on his total number of interactions with all other individuals. Provided the average payoff is positive, a player with a lower average payoff, but a lot of interactions, may do better overall than one with a higher average payoff but fewer interactions. So if 'cooperative' individuals have more interactions than 'defectors', they may do better than defectors even if their interactions with cooperators and defectors are random in the sense that they are in line with their proportions in the population.

There may be a suspicion of some fallacy in this argument, and on reading the Santos papers I didn't understand how cooperators could have more interactions overall without also driving up the number of interactions (and the total payoff) for defectors. However, I worked through a few simple numerical examples to satisfy myself that the model can work. The explanation is that an increase in the total interactions of cooperators does increase the interactions of defectors, but not to the same extent as for cooperators. By analogy, suppose that males and females have both homosexual and heterosexual encounters. It would then be possible for males (or females) to increase their total number of both homosexual and heterosexual encounters, maintaining the same proportions of these as before, while the other sex increased only its heterosexual encounters. In the same way, if cooperators increase their number of interactions with both cooperators and defectors, they may increase their total number of interactions compared to defectors. The average payoff per interaction for cooperators is unchanged, while for defectors it increases (because a higher proportion of their interactions are with cooperators), but the total payoff to cooperators relative to defectors can still increase.

Of course, this shows only that the model is possible, not that it is realistic in practice. It is certainly realistic to suppose that different individuals have differing numbers of social interactions, but this does not explain why cooperators should have more interactions than defectors. Unless we suppose that the tendency to interact is somehow correlated with the tendency to cooperate, it would seem to be a matter of chance whether cooperation evolves.

But the problem disappears if we allow something equivalent to reputation to enter the model, because individuals with a reputation as cooperators will have more encounters than those with a reputation as defectors. I therefore suspect that the Santos model will be most useful in conjunction with models involving reputation and reciprocity, which arise mainly in humans and other social animals with advanced cognitive capacities.

Added: I should have emphasised that the ultimate outcome depends on the parameters. If the advantage per interaction of defection relative to cooperation is too large, the 'Santos Effect' will only slow down the elimination of cooperators, not prevent it. Cooperation will only prevail in the long run if the difference in payoffs is small-to-moderate.