8 - Other game dynamics

Summary

Many of the properties of the replicator equation are valid for other game dynamics which may serve, for instance, to model imitation processes. We describe a large class of game dynamics that eliminate pure strategies which are iteratively strictly dominated, and discuss some instances of the best-reply dynamics motivated by fictitious play. We show that no reasonable dynamics can converge to equilibrium for all games.

Imitation dynamics

The replicator dynamics mimics the effect of natural selection (although it blissfully disregards the complexities of sexual reproduction). In the context of games played in human societies, however, the spreading of successful strategies is more likely to occur through imitation than through inheritance. How should we model this imitation processes?

Let us start with symmetric games defined by an n × n payoff matrix A, and assume that the pure strategies R₁ to R_n are adopted by a (large) population of players with frequencies x_i(t) at time t, so that the state is given, at any instant, by a point x ∈ S_n. Strategy R_i then earns (Ax)_i = ∑ a_ijx_j a s expected payoff, and the average payoff in the population is given by x · Ax. We shall suppose that occasionally a player is picked out of the population and afforded the opportunity to change his strategy. He samples another player at random and adopts his strategy with a certain probability.