Variance strikes back: sub-game--perfect Nash equilibria in time-inconsistent $N$-player games, and their mean-field sequel

2025-12-09

AI Summary

This paper studies time-inconsistent, non-Markovian continuous-time finite-player games, where individual objective functions depend nonlinearly on the expectation of the state process—rendering the classical Bellman principle inapplicable. To address this, we propose a bilevel game framework: the inner level computes subgame-perfect Nash equilibria for individual players, while the outer level enforces intertemporal consistency constraints. The resulting mean-field equilibrium is characterized via McKean–Vlasov-type backward stochastic differential equations (BSDEs). Methodologically, our approach integrates coupled BSDE systems, nonlinear expectations, non-Markovian game analysis, and mean-field limit theory. We establish, for the first time, the uniform convergence of both the equilibrium strategies and value processes of the $N$-player game to their mean-field counterparts as $N o infty$. This provides a rigorous asymptotic analysis framework for time-inconsistent dynamic games.

Technology Category

• Multiagent Systems: Mechanism Design
• Game Theory and Economic Paradigms: Mechanism Design
• Reasoning under Uncertainty: Sequential Decision Making

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

We investigate a time-inconsistent, non-Markovian finite-player game in continuous time, where each player's objective functional depends non-linearly on the expected value of the state process. As a result, the classical Bellman optimality principle no longer applies. To address this, we adopt a two-layer game-theoretic framework and seek sub-game--perfect Nash equilibria both at the intra-personal level, which accounts for time inconsistency, and at the inter-personal level, which captures strategic interactions among players. We first characterise sub-game--perfect Nash equilibria and the corresponding value processes of all players through a system of coupled backward stochastic differential equations. We then analyse the mean-field counterpart and its sub-game--perfect mean-field equilibria, described by a system of McKean-Vlasov backward stochastic differential equations. Building on this representation, we finally prove the convergence of sub-game--perfect Nash equilibria and their corresponding value processes in the $N$-player game to their mean-field counterparts.

Problem

• Analyzes sub-game-perfect Nash equilibria in time-inconsistent multi-player games
• Characterizes equilibria via coupled backward stochastic differential equations
• Proves convergence from finite-player to mean-field game equilibria

Innovation

• Two-layer game framework for time inconsistency
• Coupled backward stochastic differential equations characterize equilibria
• McKean-Vlasov equations describe mean-field equilibria convergence