Fictitious play in an evolutionary environment

We consider a continuous time version of the ctitious play model in an evolutionary environment. We derive two forms of the continuous time limit, both of which are approximations to a determinate version of the ctitious play updating algorithm. The rst is in the form of a rst-order partial di erential equation ('continuity equation') which we solve explicitly. The dynamic for a distribution of strategies is also derived, which we show that can be written in a form similar to a positive de nite dynamic. The asymptotic solution (in the ultra long run) is discussed in detail 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 (because 2nd-order terms are no-longer neglected) is in the form of a di usion equation. This is considerably more di cult to analyse. However, we show that it leads to the same asymptotic limit as the 1st-order approximation.