New proof shows fictitious play converges at super-exponential rate in stochastic games
First convergence analysis for fully coupled FBSDEs reveals geometric to super-exponential speed.
In a new paper on arXiv, researchers Adam Andersson, Kristoffer Andersson, and Per Ljung provide the first rigorous convergence analysis of the fictitious-play approximation procedure for fully coupled forward-backward stochastic differential equations (FBSDEs) arising in finite-player non-zero-sum stochastic differential games. Under a general set of assumptions, they prove that the iterative method converges at a geometric rate. Remarkably, when an additional structural condition holds, the convergence rate accelerates to super-exponential in a special class of games.
The results have immediate relevance to financial modeling—the authors validated their theory with a numerical experiment on a linear-quadratic interbank borrowing and lending problem, confirming geometric convergence. This work bridges a gap between game theory and numerical analysis, offering a provably efficient way to solve complex multi-agent decision problems where agents' strategies are coupled through both the state and the adjoint (costate) variables. The paper spans 36 pages and includes two figures.
- First convergence proof for fictitious play applied to fully coupled FBSDEs in finite-player stochastic games
- Geometric convergence under general assumptions; super-exponential rate in a restricted class
- Numerical validation on a linear-quadratic interbank borrowing and lending problem
Why It Matters
Provides a rigorous algorithmic guarantee for solving complex multi-agent economic and financial games efficiently.