Evolutionary Analysis of Continuous-time Finite-state Mean Field Games with Discounted Payoffs

TL;DR

This paper develops an evolutionary framework for continuous-time mean field games with finite states and discounted rewards, introducing a new equilibrium concept and establishing its stability and approximation guarantees.

Contribution

It introduces a mean field approximation for dynamic games with state evolution and proposes the Mixed Stationary Nash Equilibrium, linking it to evolutionary stability.

Findings

Established approximation guarantees for the mean field model.

Characterized the equivalence between MSNE and evolutionary rest points.

Provided conditions for the evolutionary stability of MSNE.

Abstract

We consider a class of continuous-time dynamic games involving a large number of players. Each player selects actions from a finite set and evolves through a finite set of states. State transitions occur stochastically and depend on the player's chosen action. A player's single-stage reward depends on their state, action, and the population-wide distribution of states and actions, capturing aggregate effects such as congestion in traffic networks. Each player seeks to maximize a discounted infinite-horizon reward. Existing evolutionary game-theoretic approaches introduce a model for the way individual players update their decisions in static environments without individual state dynamics. In contrast, this work develops an evolutionary framework for dynamic games with explicit state evolution, which is necessary to model many applications. We introduce a mean field approximation of the finite-population game and establish approximation guarantees. Since state-of-the-art solution concepts for dynamic games lack an evolutionary interpretation, we propose a new concept - the Mixed Stationary Nash Equilibrium (MSNE) - which admits one. We characterize an equivalence between MSNE and the rest points of the proposed mean field evolutionary model and we give conditions for the evolutionary stability of MSNE.