The optimal size for a minority

We investigate a setting in which members of a population, bifurcated into a majority and a minority, transact with randomly matched partners. All members are uniformly altruistic, and each transaction can be carried out cooperatively or through a market mechanism, with cooperative transactions saving on transaction costs. Externalities are introduced, whereby cooperation by members of one group and the relative size of that group, affect the incentives to cooperate by members of the other group. Under these conditions, we determine the optimal size of the minority from the minority's perspective, and consider the conditions under which such a size might be attained. The model provides insights on social conflicts both between groups and within groups.