On Discrete-Time Approximations to Infinite Horizon Differential Games

TL;DR

This work addresses approximating equilibria in infinite-horizon N-player differential games by a semi-Lagrangian discretization in time, then extends to a fully discrete (time and space) setting. It proves that discrete-time Nash equilibria converge to continuous-time equilibria and that discrete equilibria are $\epsilon$-Nash for the original game as the discretization parameters vanish, with error bounds $O(h)$ or $O(h+k)$ under suitable conditions. The approach relies on discrete Bellman equations, interpolation on the state space, and auxiliary discrete-time problems to bridge the discrete and continuous formulations. Numerical experiments on a two-player model validate convergence rates and illustrate practical computation of Markovian Nash equilibria in differential games.

Abstract

In this paper we study a discrete-time semidiscretization and a fully discretization (discrete-time, discrete-state) of an infinite time horizon noncooperative $N$-player differential game. We prove that as either the discretization time step or both time step and mesh size parameters approach zero the discrete value function approximates the value function of the differential game. Furthermore, the discrete Nash equilibrium is an $ε$-Nash equilibrium for the continuous-time differential game both in the discrete-time and fully discrete cases.

Figures (4)

• Figure 1: Model problem
• Figure 2: $W_1(\Psi_i^h,\phi_{-i}^h,p_{01},p_{02})-W_1(\phi_1^h,\phi_2^h,p_{01},p_{02})$ for $h=1/16, h=1/32, h=1/64$ and $h=1/128$
• Figure 3: $\max_{p_{01},p_{02}}\bigl\{W_1(\Psi_i^h,\phi_{-i}^h,p_{01},p_{02})-W_1(\phi_1^h,\phi_2^h,p_{01},p_{02})\bigr\}$ for $h=1/16, h=1/32, h=1/64$ and $h=1/128$.
• Figure 4: $\max_{p_1,p_2} | \phi_1(p_1,p_2)-\phi_{1}^h(p_1,p_2)|$ for $h=1/16, h=1/32, h=1/64, h=1/128$ and $h=1/256$

Theorems & Definitions (24)

• Definition 1
• Definition 2
• Theorem 1
• Definition 3
• Theorem 2
• Proposition 1
• Proof 1
• Proposition 2
• Proof 2
• Theorem 3