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In population games, a large population of players, modeled as a continuum,is divided into subpopulations, and the fitness or payoff of each subpopulationdepends on the overall population composition. Evolutionary dynamics describehow the population composition changes in response to the fitness levels,resulting in a closed-loop feedback system. Recent work established aconnection between passivity theory and certain classes of population games,namely so-called "stable games". In particular, it was shown that a combinationof stable games and (an analogue of) passive evolutionary dynamics results instable convergence to Nash equilibrium. This paper considers the conversequestion of necessary conditions for evolutionary dynamics to exhibit stablebehaviors for all generalized stable games. Here, generalization refers to"higher order" games where the population payoffs may be a dynamic function ofthe population state. Using methods from robust control analysis, we show thatif an evolutionary dynamic does not satisfy a passivity property, then it ispossible to construct a generalized stable game that results in instability.The results are illustrated on selected evolutionary dynamics with particularattention to replicator dynamics, which are also shown to be lossless, aspecial class of passive systems.

Passivity Analysis of Higher Order Evolutionary Dynamics and Population Games