Friendship and the evolution of co-operation

Recent theoretical research on the repeated Prisoner's Dilemma game (PD) suggests that the evolution of co-operative behaviour depends on the rate at which social groups break apart and new groups form. Long-lasting social groups tend to favour the evolution of co-operation. It is therefore of interest to construct models in which individuals must choose whether to maintain their current social relationships. This is done in the present paper, and the analysis suggests that unco-operative individuals will seek to move quickly from one social relationship to another, while co-operative individuals will seek to maintain relationships with other co-operative individuals. As a result of their differing decisions about changing social relationships, co-operators tend to interact with other co-operators to a greater extent than do unco-operative individuals, and this difference is an important determinant of the results of the model. In contrast to the original analyses of the repeated Prisoner's Dilemma, the model presented here allows for the existence of stable equilibria with both co-operative and unco-operative individuals simultaneously present in the same population. It is proved that the position of these equilibria (and hence the frequency of co-operators) depends on the size of the various payoffs that define the Prisoner's Dilemma game. In addition, it is proved that co-operation can be maintained at a high frequency so long as co-operators are able to maintain their relationships with other co-operators over long periods of time. This is true even if unco-operative individuals are able to move from one relationship to another at a very high rate.