Replicator equation for mixed strategies?

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?

• When you allow for mixed strategies, you essentially have an infinite strategy set. There is a branch of evolutionary dynamics concerned with infinite strategy spaces. I know of this paper, for example: link.springer.com/article/10.1007/PL00004092 Nov 18 '16 at 0:38