From time-average replicator to best-response dynamics, and back

When a game is played over time, the players can change their actions throughout the game. The choice of action can follow different learning mechanisms leading to different types of dynamics.    In this talk I shall look at the relation between the time-average of replicator dynamics (RD) and best-response dynamics (BRD), and its corresponding fictitious play (FP).  It is known that the time-average of RD converges to an invariant set under BRD but not whether given an invariant set BRD there always exists a corresponding RD orbit. Using results from the theory of dynamical systems and under some mild assumptions, I shall show that if there exists a hyperbolic invariant periodic orbit X for BRD then for each initial condition in X there exists a RD orbit whose time-average converges to the solution of FP through the given initial condition.   I shall then apply this result when knowledge of one type of dynamics exists and illustrate with an application to the Rock-Scissors-Paper game with two players.  The talk will start with the basics of evolutionary game theory so that anyone working on, or familiar with, dynamical systems should be able to understand it.