Recursive Regret Matching: A General Method for Solving Time-invariant Nonlinear Zero-sum Differential Games

TL;DR

This paper introduces Recursive Regret Matching, a novel approach for computing Nash equilibria in time-invariant nonlinear zero-sum differential games, capable of handling cases with or without saddle points.

Contribution

The method transforms differential games into a sequence of normal-form games solved via regret matching, improving real-time computation and avoiding saddle point existence analysis.

Findings

Successfully computes Nash equilibria in various game scenarios.

Handles cases where saddle points do not exist.

Demonstrates effectiveness through illustrative examples.

Abstract

In this paper, a new method is proposed to compute the rolling Nash equilibrium of the time-invariant nonlinear two-person zero-sum differential games. The idea is to discretize the time to transform a differential game into a sequential game with several steps, and by introducing state-value function, transform the sequential game into a recursion consisting of several normal-form games, finally, each normal-form game is solved with action abstraction and regret matching. To improve the real-time property of the proposed method, the state-value function can be kept in memory. This method can deal with the situations that the saddle point exists or does not exist, and the analysises of the existence of the saddle point can be avoided. If the saddle point does not exist, the mixed optimal control pair can be obtained. At the end of this paper, some examples are taken to illustrate the validity of the proposed method.