On the asymptotic convergence to mixed equilibria in 2 × 2 asymmetric games

We analyse the stability properties of mixed equilibria in 2 × 2 asymmetric games under evolutionary dynamics. With the standard replicator dynamics these equilibria are stable but not asymptotically stable. We modified the replicator dynamics by introducing players of two types: myopics – as in the standard replicator dynamics – and best responders. The behaviour of the latter is described by a continuous time version of the best reply dynamics. Asymptotic convergence under the Modified Replicator Dynamics is proved by identifying a strictly decreasing Ljapunov function. We argue that the finding has important implications to justify the use of economic models with mixed strategy equilibria.