Game theory

Population games model interactions among a large number of players, or agents, in which each agent’s payoff or fitness depends on its own strategy and the distribution of strategies of other agents. There has been extensive research in a variety of settings, ranging from societal to biological to engineered.

A central question in population games, as well as the related topic of learning in games  is understanding the long-run behavior of player strategies. In particular, under what conditions do population strategies converge to a solution concept such as Nash equilibrium? The outcome depends on both the underlying game and the particular evolutionary dynamics, and behaviors can range from convergence for classes of game/dynamics pairings to chaos in seemingly simple settings. Furthermore, a specific game can exhibit inherent obstacles to convergence for broad classes of evolutionary dynamics. Contrary to Nash equilibrium, there are relaxed solution concepts, such as coarse correlated equilibria, that are universally (i.e., for all games) induced by various forms of evolutionary dynamics.

Recent work established a connection between passivity theory and certain classes of population games, namely so-called “stable games”. In particular, it was shown that a combination of stable games and (an analog of) passive evolutionary dynamics results in stable convergence to Nash equilibrium.

We are considering the converse question of necessary conditions for evolutionary dynamics to exhibit stable behaviors for all generalized stable games. Here, generalization here refers to “higher-order” games where the population payoffs may be a dynamic function of the population state. Using methods from robust control analysis, we show that if an evolutionary dynamic does not satisfy a passivity property, then it is possible to construct a generalized stable game that results in instability.

We have illustrated on selected evolutionary dynamics with particular attention to replicator dynamics, which are also shown to be lossless, a special class of passive systems.