Limit value of dynamic zero-sum games with vanishing stage duration

TL;DR

This paper investigates the limiting behavior of dynamic zero-sum games with vanishing stage durations, establishing the existence of the limit value as the partition mesh approaches zero across various game types.

Contribution

It proves the existence of the limit value for dynamic zero-sum games with continuous-time Markov controls as the stage duration shrinks, extending to stochastic and symmetric no-information cases.

Findings

Limit value exists as partition mesh approaches zero.

Results apply to stochastic and symmetric no-information games.

Analysis reduces to a deterministic differential game.

Abstract

We consider two person zero-sum games where the players control, at discrete times {tn} induced by a partition $\Pi$ of R + , a continuous time Markov state process. We prove that the limit of the values v$\Pi$ exist as the mesh of $\Pi$ goes to 0. The analysis covers the cases of : 1) stochastic games (where both players know the state) 2) symmetric no information. The proof is by reduction to a deterministic differential game.