Replicator Equation: Applications Revisited

The replicator equation is a simple model of evolution that leads to stable form of Nash Equilibrium, Evolutionary Stable Strategy (ESS). No individuals get an incentive unilaterally deviating from the equilibrium. It has been studied in connection with Evolutionary Game Theory, a theory John Maynard Smith and George R. Price developed to predict biological reproductive success of populations. Replicator equation was originally developed for symmetric games, games whose payoff matrix is skew-symmetric and in general tells us, in a game, how individuals or populations change their strategy over time based on the payoff matrix of the game. Consider a large population of players where each player is assigned a strategy and players cannot choose their strategies which means rationality and consciousness don't enter the picture (players play based on a pre-assigned set of strategies). Evolutionary game theory assumes a scenario where a non-rational pairs of players, which play based on a pre-assigned set of strategies, repeatedly drawn from a large population plays a symmetric two-player game which drives the strategies with lower payoff to extinction. Since players interacts with another randomly chosen player in the population, a players expected payoff is determined by the payoff matrix and the proportion of each strategy in the population. The limiting behavior of the replicator dynamics (i.e., the evolutionary outcome) are the Evolutionary Stable Strategies, a NE with additional stability properties. Let x i (t) is the proportion of the population which plays strategy i ∈ N (set of strategies) at time t. The state of the population at any given instant is then given by x i (t) = [x 1 , x 2 , ..., x n ] ′ where ′ denotes transposition and n refers the size of available pure strategies, |N |. With A be the n × n payoff matrix (biologically measured as Darwinian fitness i.e reproductive success), the payoff for the i th-strategist, assuming the opponent is playing the j th strategy, is (a ij), the corresponding i th row and the j th column of A. If the population is in state x, the expected payoff earned by an the i th-strategist is (Ax) i while the mean payoff over the whole population is x ′ Ax. The growth rate of the population is then computed as the product of the current frequency of strategy with the own fitness relative to the average (the difference in the …