Convergence of deep fictitious play for stochastic differential games

Stochastic differential games have been used extensively to model agents' competitions in finance, for instance, in P2P lending platforms from the Fintech industry, the banking system for systemic risk, and insurance markets. The recently proposed machine learning algorithm, deep fictitious play, provides a novel and efficient tool for finding Markovian Nash equilibrium of large \begin{document}$N$\end{document}-player asymmetric stochastic differential games [J. Han and R. Hu, Mathematical and Scientific Machine Learning Conference, pages 221-245, PMLR, 2020]. By incorporating the idea of fictitious play, the algorithm decouples the game into \begin{document}$N$\end{document} sub-optimization problems, and identifies each player's optimal strategy with the deep backward stochastic differential equation (BSDE) method parallelly and repeatedly. In this paper, we prove the convergence of deep fictitious play (DFP) to the true Nash equilibrium. We can also show that the strategy based on DFP forms an \begin{document}$\epsilon$\end{document}-Nash equilibrium. We generalize the algorithm by proposing a new approach to decouple the games, and present numerical results of large population games showing the empirical convergence of the algorithm beyond the technical assumptions in the theorems.

Authors:
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Award ID(s):
Publication Date:
NSF-PAR ID:
10343586
Journal Name:
Frontiers of Mathematical Finance
Volume:
1
Issue:
2
Page Range or eLocation-ID:
287
ISSN:
2769-6715
2. We consider the well-known Lieb-Liniger (LL) model for \begin{document}$N$\end{document} bosons interacting pairwise on the line via the \begin{document}$\delta$\end{document} potential in the mean-field scaling regime. Assuming suitable asymptotic factorization of the initial wave functions and convergence of the microscopic energy per particle, we show that the time-dependent reduced density matrices of the system converge in trace norm to the pure states given by the solution to the one-dimensional cubic nonlinear Schrödinger equation (NLS) with an explict rate of convergence. In contrast to previous work [3] relying on the formalism of second quantization and coherent states and without an explicit rate, our proof is based on the counting method of Pickl [65,66,67] and Knowles and Pickl [44]. To overcome difficulties stemming from the singularity of the \begin{document}$\delta$\end{document} potential, we introduce a new short-range approximation argument that exploits the Hölder continuity of the \begin{document}$N$\end{document}-body wave function in a single particle variable. By further exploiting the \begin{document}$L^2$\end{document}-subcritical well-posedness theory for the 1D cubic NLS, we can prove mean-field convergence when the limiting solution to the NLS has finitemore »
3. For any finite horizon Sinai billiard map \begin{document}$T$\end{document} on the two-torus, we find \begin{document}$t_*>1$\end{document} such that for each \begin{document}$t\in (0,t_*)$\end{document} there exists a unique equilibrium state \begin{document}$\mu_t$\end{document} for \begin{document}$- t\log J^uT$\end{document}, and \begin{document}$\mu_t$\end{document} is \begin{document}$T$\end{document}-adapted. (In particular, the SRB measure is the unique equilibrium state for \begin{document}$- \log J^uT$\end{document}.) We show that \begin{document}$\mu_t$\end{document} is exponentially mixing for Hölder observables, and the pressure function \begin{document}$P(t) = \sup_\mu \{h_\mu -\int t\log J^uT d \mu\}$\end{document} is analytic on \begin{document}$(0,t_*)$\end{document}. In addition, \begin{document}$P(t)$\end{document} is strictly convex if and only if \begin{document}$\log J^uT$\end{document} is not \begin{document}$\mu_t$\end{document}-a.e. cohomologous to a constant, while, if there exist \begin{document}$t_a\ne t_b$\end{document} with \begin{document}$\mu_{t_a} = \mu_{t_b}$\end{document}, then \begin{document}$P(t)$\end{document} is affine on \begin{document}$(0,t_*)$\end{document}. An additional sparse recurrence condition gives \begin{document}$\lim_{t\downarrow 0} P(t) = P(0)$\end{document}.
4. In this note we study a new class of alignment models with self-propulsion and Rayleigh-type friction forces, which describes the collective behavior of agents with individual characteristic parameters. We describe the long time dynamics via a new method which allows us to reduce analysis from the multidimensional system to a simpler family of two-dimensional systems parametrized by a proper Grassmannian. With this method we demonstrate exponential alignment for a large (and sharp) class of initial velocity configurations confined to a sector of opening less than \begin{document}$\pi$\end{document}.
5. The purpose of this paper is to describe the feedback particle filter algorithm for problems where there are a large number (\begin{document}$M$\end{document}) of non-interacting agents (targets) with a large number (\begin{document}$M$\end{document}) of non-agent specific observations (measurements) that originate from these agents. In its basic form, the problem is characterized by data association uncertainty whereby the association between the observations and agents must be deduced in addition to the agent state. In this paper, the large-\begin{document}$M$\end{document} limit is interpreted as a problem of collective inference. This viewpoint is used to derive the equation for the empirical distribution of the hidden agent states. A feedback particle filter (FPF) algorithm for this problem is presented and illustrated via numerical simulations. Results are presented for the Euclidean and the finite state-space cases, both in continuous-time settings. The classical FPF algorithm is shown to be the special case (with \begin{document}$M = 1$\end{document}) of these more general results. The simulations help show that the algorithm well approximates the empirical distribution of the hidden states for large \begin{document}$M$\end{document}.