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

TL;DR

The paper develops an evolutionary framework for continuous-time, finite-state mean field games with discounted payoffs, introducing a Mixed Stationary Nash Equilibrium (MSNE) that accommodates heterogeneity in deterministic policies across agents. It proves a mean field approximation with existence of MSNE via Kakutani fixed points, and establishes that MSNE correspond to rest points of the proposed mean field evolutionary dynamics under common revision protocols, with pairwise comparison revisions yielding equivalence in both directions. It further shows that non-MSNE rest points are generically unstable and that strict MSNE are locally asymptotically stable under several revision classes, supported by an illustrative MAC example. Overall, the work highlights qualitative differences between discounted-payoff and average-payoff settings and provides a foundation for future finite-population approximations and broader stability analyses in discounted mean field games.

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.

Figures (1)

• Figure 1: Trajectories of evolutionary dynamics in the illustrative MAC game.

Theorems & Definitions (38)

• Definition 1: Imitative
• Definition 2: Excess payoff
• Definition 3: Pairwise comparison
• Theorem 1
• proof
• Definition 4: BSNE
• Definition 5: MSNE
• Theorem 2
• proof
• Theorem 3