Extremal shift rule for continuous-time zero-sum Markov games

TL;DR

This paper demonstrates that the Krasovskii--Subbotin extremal shift rule provides near-optimal strategies for a controlled continuous-time Markov chain with a finite number of types, approximating the limiting zero-sum differential game as particles grow large.

Contribution

It establishes the near-optimality of the extremal shift rule in the original Markov game, connecting finite particle systems to the limiting differential game.

Findings

Extremal shift rule is nearly optimal for large particle systems.

The limiting game is a zero-sum differential game.

The approach bridges finite Markov chains and continuous differential games.

Abstract

In the paper we consider the controlled continuous-time Markov chain describing the interacting particles system with the finite number of types. The system is controlled by two players with the opposite purposes. The limiting game as the number of particles tends to infinity is a zero-sum differential game. Krasovskii--Subbotin extremal shift provides the optimal strategy in the limiting game. The main result of the paper is the near optimality of the Krasovskii--Subbotin extremal shift rule for the original Markov game.