Convergence of Fictitious Play for Fully Coupled FBSDEs in Infinite-Player Stochastic Differential Games

Jul 16, 2026

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This article studies theoretical convergence of the fictitious-play approximation for coupled FBSDE systems arising in finite-player, non-zero-sum stochastic differential games. Under one set of assumptions, the approximation converges geometrically. With an additional structural condition, the geometric rate improves to super-exponential convergence for a special class of games. The work is presented as the first convergence analysis of fictitious play for fully coupled FBSDEs, supported by a linear–quadratic interbank borrowing–lending numerical experiment.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages36

Reading Time1 h 31 min

Abstract

Introduction

Stochastic differential games and Nash equilibrium

Numerical approximation challenges

Related fictitious-play and deep-learning approaches

Contributions and convergence results

arXiv :2607 .08861v1 [math .OC] 9 Jul 2026 Convergence of fictitious play for fully coupled FBSDEs infinite-player stochastic differential games Adam Andersson∗ † Kristoffer Andersson‡ Per Ljung ∗§ July 13, 2026 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. 1 Introduction Stochastic differential games (SDG) provide a mathematical framework for modeling interacting agents or players whose actions or controls influence the evolution of a common stochastic system. In such problems, each player optimizes an individual objective while accounting for the strategies of the other players, leading to the famous Nash equilibrium solution concept. Non-zero-sum stochastic differential games arise naturally in multi-agent systems under uncertainty, where each agent optimizes an individual objective, with applications ranging from financial markets and energy systems to economics and engineering. Representative examples include interbank lending and systemic risk models [22], applications in economics such as dynamic oligopoly and resource management problems in environmental and engineering systems [15], multi-agent optimal execution problems with price impact [36], pursuit-evasion [8, 28, 34, 38] and missile guidance and control [3, 19, 33, 37, 39] . Similarly to stochastic control, stochastic differential games are either solved with the stochastic maximum principle [11, 17, 34], or by dynamic programming [11, 12, 19, 23, 31, 32, 33, 34, 36, 39, 38] . In this work, we focus on the latter approach. Approaching SDG via dynamic programming leads to a coupled system of Hamilton–Jacobi–Bellman (HJB) equations, the Nash PDE system, and toa coupled system of forward-backward stochastic differential equations (FBSDEs), the Nash FBSDE system. In regular settings, these two representations are equivalent, but the Nash FBSDE system is a more general representation for regimes with lower regularity. Only games with highly specific structure, such as the important and well-studied class of linear– quadratic games, admit analytical or semi-analytical solutions. Therefore, for many applications, numerical approximations are necessary. However, accurate, scalable, and practical numerical approximation of SDG remains an open problem. Finite difference methods have been studied for two-player zero-sum games [18] and for mean-field games [1, 2] . Grid-based finite difference and finite element methods for PDEs, however, suffer from the curse of dimensionality (COD) and are therefore limited to low-dimensional settings. To the best of our knowledge, direct approximation of the Nash PDE system for non-zero-sum stochastic differential games has not been studied in the literature, and would in any case be of limited interest because of the aforementioned scaling problem. A Markov ∗ Saab AB, Gothenburg, Sweden †Department of Mathematical Sciences, Chalmers University of Technology & University of Gothenburg ‡Department of Economics, University of Verona. Email: [kristofferherbert.andersson@univr.it](kristofferherbert.andersson@univr.it) § Department of Electrical Engineering, Chalmers University of Technology chain approximation framework for two-player zero-sum games was introduced in [31], and extended to finite-player non-zero-sum stochastic differential games in [32] . It was later extended to more general

FAQs

What convergence properties does the paper establish for fictitious play on coupled FBSDE systems?

The paper shows geometric convergence under one set of assumptions. With an added structural assumption, the convergence rate becomes super-exponential for a special class of games.

Why is the fully coupled FBSDE setting emphasized?

The authors aim to extend convergence analysis to fully coupled FBSDEs. Prior work and practical experience indicate that direct approaches may fail to converge when coupling is not handled appropriately, motivating robust deep FBSDE methods and new analysis.

How is the method supported empirically?

A numerical experiment on a linear–quadratic interbank borrowing and lending problem confirms the predicted geometric convergence.

Jul 16, 2026

Upload by Miles