On values of repeated games with signals

TL;DR

This paper investigates the existence of various value concepts in two-player zero-sum repeated games with evolving states and signals, establishing conditions under which these values exist or fail to exist.

Contribution

It demonstrates the non-existence of limsup and uniform values in general, and proves the existence of the value under specific signaling conditions and game structures.

Findings

Limsup and uniform values may not exist in general.

Existence of the value when players observe a public signal including actions.

Existence of the $ ext{sup}$ value and uniform value in recursive games with nonnegative payoffs.

Abstract

We study the existence of different notions of value in two-person zero-sum repeated games where the state evolves and players receive signals. We provide some examples showing that the limsup value (and the uniform value) may not exist in general. Then we show the existence of the value for any Borel payoff function if the players observe a public signal including the actions played. We also prove two other positive results without assumptions on the signaling structure: the existence of the $\sup$ value in any game and the existence of the uniform value in recursive games with nonnegative payoffs.