Convergence of fictitious play for fully coupled FBSDEs in finite-player stochastic differential games
2026-07-09 • Computer Science and Game Theory
Computer Science and Game Theory
AI summaryⓘ
The authors study how a method called fictitious play behaves when used to solve certain complex mathematical problems called coupled forward-backward stochastic differential equations (FBSDEs), which show up in games where multiple players make decisions that affect each other. They prove that this method converges steadily over time, and under some extra conditions, it converges even faster than usual in some special types of games. Their work is the first to analyze the convergence of fictitious play in this complicated setting. They also did a test case involving banks borrowing and lending money that supported their theoretical findings.
fictitious playforward-backward stochastic differential equationsFBSDEnon-zero-sum gamesstochastic differential gamesconvergence rategeometric convergencesuper-exponential convergencelinear-quadratic gameinterbank lending
Authors
Adam Andersson, Kristoffer Andersson, Per Ljung
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.