The the replicator equation is usually defined for pure strategies. More specifically, the replicator eqn for $n$ strategies is given by: \begin{equation} \dot x_{i} = x_{i} \left( \sum_{j=1}^{n} a_{ij}x_{j} - \phi \right) \qquad i = 1, \ldots, n \end{equation} where $x_i$ is the frequency of the strategy $i$, $A=[a_{ij}]$ is the payoff matrix, and $\phi$ is the average fitness of the population. Basically, what the replicator equation is saying is that the per capita rate of increase of the strategy $i$ (i.e., $\dot x_i/ x_i$) is the difference between the fitness of $i$ (i.e., $\sum_{j=1}^{n} a_{ij}x_{j}$) and the average fitness $\phi$ of the whole population.

Question: Does anyone know of a formulation of the replicator eqn for mixed strategies?


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