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Individual Strategy and Social Structure

A more interesting case, and one relevant to a study of the reproduction of norms of cooperation, is that of a population in which several competing strategies are present at any given time. What we want to know is whether the strategy frequencies that exist at a time are stable, or if there is a tendency for one strategy to become dominant over time. If we continue to rely on the ESS solution concept, we see a classic example in the hawk-dove game. If we assume that there is no uncorrelated asymmetry between the players, then the mixed Nash equilibrium is the ESS. If we further assume that there is no structure to how agents interact with each other, this can be interpreted in two ways: either each player randomizes her strategy in each round of play, or we have a stable polymorphism in the population, in which the proportion of each strategy in the population corresponds to the frequency with which each strategy would be played in a randomizing approach. So, in those cases where we can assume that players randomly encounter each other, whenever there is a mixed solution ESS we can expect to find polymorphic populations.

If we wish to avoid the interpretive challenge of a mixed solution ESS, there is an alternative analytic solution concept that we can employ: the evolutionarily stable state. An evolutionarily stable state is a distribution of (one or more) strategies that is robust against perturbations, whether they are exogenous shocks or mutant invasions, provided the perturbations are not overly large. Evolutionarily stable states are solutions to a replicator dynamic. Since evolutionarily stable states are naturally able to describe polymorphic or monomorphic populations, there is no difficulty with introducing population-oriented interpretations of mixed strategies. This is particularly important when random matching does not occur, as under those conditions, the mixed strategy can no longer be thought of as a description of population polymorphism.