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

2025-11-10

AI Summary

Existing models for continuous-time, finite-state mean-field games lack a rigorous framework integrating individual state dynamics with population-level strategy interactions. Method: We introduce the concept of Mixed Stationary Nash Equilibrium (MSNE), rigorously establishing its equivalence to fixed points of the mean-field evolution equation. By incorporating state-transition-rate–dependent feedback from the population distribution, we derive local asymptotic stability criteria for MSNE and prove its existence in finite-state settings. Furthermore, we construct a convergence theory for mean-field approximations. Contribution/Results: This work provides the first unified analytical framework for discounted dynamic mean-field games that explicitly incorporates state evolution mechanisms while ensuring evolutionary stability. It addresses a key limitation of classical evolutionary game theory—its inability to model state-dependent strategic interactions—and delivers an interpretable, verifiable theoretical foundation for dynamic collective decision-making applications, including traffic scheduling and epidemic control.

Technology Category

• Multiagent Systems: Mechanism Design
• Game Theory and Economic Paradigms: Equilibrium
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Web Mining and Content Analysis: Models for Web evolution
• Responsible Web: Machine-in-the-loop, human agency and autonomy

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.

Problem

• Develops evolutionary framework for dynamic games with state transitions
• Proposes Mixed Stationary Nash Equilibrium for evolutionary interpretation
• Establishes connection between equilibrium and mean field model

Innovation

• Evolutionary framework for dynamic games with state evolution
• Mean field approximation with finite-population guarantees
• Mixed Stationary Nash Equilibrium with evolutionary interpretation