Payoff Dynamic Models and Evolutionary Dynamic Models: Feedback and Convergence to Equilibria

We consider that at every instant each member of a population, which we refer to as an agent, selects one strategy out of a finite set. The agents are nondescript, and their strategy choices are described by the so-called population state vector, whose entries are the portions of the population selecting each strategy. Likewise, each entry constituting the so-called payoff vector is the reward attributed to a strategy. We consider that a general finite-dimensional nonlinear dynamical system, denoted as payoff dynamical model (PDM), describes a mechanism that determines the payoff as a causal map of the population state. A bounded-rationality protocol, inspired primarily on evolutionary biology principles, governs how each agent revises its strategy repeatedly based on complete or partial knowledge of the population state and payoff. The population is protocol-homogeneous but is otherwise strategy-heterogeneous considering that the agents are allowed to select distinct strategies concurrently. A stochastic mechanism determines the instants when agents revise their strategies, but we consider that the population is large enough that, with high probability, the population state can be approximated with arbitrary accuracy uniformly over any finite horizon by a so-called (deterministic) mean population state. We propose an approach that takes advantage of passivity principles to obtain sufficient conditions determining, for a given protocol and PDM, when the mean population state is guaranteed to converge to a meaningful set of equilibria, which could be either an appropriately defined extension of Nash's for the PDM or a perturbed version of it. By generalizing and unifying previous work, our framework also provides a foundation for future work.