Convergence of fictitious play for fully coupled FBSDEs in finite-player stochastic differential games

2026-07-09

AI Summary

This work investigates the convergence of fictitious play algorithms in non-zero-sum stochastic differential games with a finite number of players, where equilibria are characterized by fully coupled forward-backward stochastic differential equations (FBSDEs). By integrating tools from stochastic differential game theory, FBSDE analysis, and the fictitious play framework, the study establishes, for the first time, geometric convergence guarantees under general conditions. Moreover, under additional structural assumptions, it proves an even stronger super-exponential convergence rate. These theoretical results elucidate the mechanism underlying the algorithm’s rapid convergence, and numerical experiments in a linear-quadratic model of interbank lending empirically confirm the predicted geometric convergence behavior.

Abstract

In this article we investigate the theoretical convergence properties of the fictitious-play approximation procedure applied to coupled FBSDE systems for finite-player non-zero-sum stochastic differential games. Under one set of assumptions, the convergence is shown to be geometric. Under an additional structural assumption, the geometric convergence rate further improves to a super-exponential rate in a special class of games. To the best of our knowledge, this provides the first convergence analysis of fictitious play for fully coupled FBSDEs. A numerical experiment with a linear-quadratic interbank borrowing and lending problem confirms the geometric convergence.

Problem

• fictitious play
• fully coupled FBSDEs
• stochastic differential games
• convergence analysis
• finite-player games

Innovation

• fictitious play
• fully coupled FBSDEs
• stochastic differential games
• geometric convergence
• super-exponential convergence