Passivity Analysis of Replicator Dynamics and its Variations

In this paper, we focus on studying the passivity properties of different versions of replicator dynamics (RD). RD is an important class of evolutionary dynamics in evolutionary game theory. Evolutionary dynamics describe how the population composition changes in response to the fitness levels, resulting in a closed-loop feedback system. RD is a deterministic monotone non-linear dynamic that allows incorporation of the distribution of population types through a fitness function. Here, in this paper, we use a tools for control theory, in particular, the passivity theory, to study the stability of the RD when it is in action with evolutionary games. The passivity theory allows us to identify class of evolutionary games in which stability with RD is guaranteed. We show that several variations of the first order RD satisfy the standard loseless passivity property. In contrary, the second order RD do not satisfy the standard passivity property, however, it satisfies a similar dissipativity property known as negative imaginary property. The negative imaginary property of the second order RD allows us to identify the class of games that converge to a stable equilibrium with the second order RD.