Fictitious play in an evolutionary environment

We consider continuous time versions of the fictitious play updating algorithm in an evolutionary environment. We derive two forms of continuous-time limit, both defining approximations to this algorithm. The first has the form of a first-order partial differential equation, which we solve explicitly. The dynamic for a distribution of strategies is also derived, which we show can be written in a form similar to a positive definite dynamic. The asymptotic solution (in the ultra long run) is discussed for 2-player, 2-strategy co-ordination and anti-coordination games, and we show convergence to Nash equilibrium in both cases. The second, and better, approximation is in the form of a diffusion equation. This is considerably more difficult to analyze. However, we derive a formal solution and show that it leads to the same asymptotic limit for the distribution of strategies as the 1st-order approximation for 2-player, 2-strategy anti-coordination games.