Dissipativity Tools for Convergence to Nash Equilibria in Population Games

We analyze the stability of a nonlinear dynamical model describing the noncooperative strategic interactions among the agents of a finite collection of populations. Each agent selects one strategy at a time and revises it repeatedly according to a protocol that typically prioritizes strategies whose payoffs are either higher than those of the current strategy or exceed the population average. The model is predicated on well-established research in population and evolutionary games, and has two components. The first is the payoff dynamics model (PDM), which ascribes the payoff to each strategy according to the so-called social state vector whose entries are the proportions of every population adopting the available strategies. The second component is the evolutionary dynamics model (EDM) that accounts for the revision process. In our model, the social state at equilibrium is a best response to the strategies' payoffs, and can be viewed as a Nash-like solution that has predictive value when it is globally asymptotically stable (GAS). We present a systematic methodology that ascertains GAS by checking separately whether the EDM and PDM satisfy appropriately defined system-theoretic dissipativity properties. Our work generalizes pioneering methods based on notions of contractivity applicable to memoryless PDMs, and more general system-theoretic passivity conditions. As demonstrated with examples, the added flexibility afforded by our approach is particularly useful when the contraction properties of the PDM are unequal across populations.