Repeated Games with Tail-Measurable Payoffs

TL;DR

This paper investigates multiplayer Blackwell games with tail-measurable payoffs, demonstrating the existence of approximate equilibria through four different proof methods, extending classical two-player game results.

Contribution

It provides four novel proofs that multiplayer Blackwell games with finite players, actions, and tail-measurable payoffs admit epsilon-equilibria.

Findings

Blackwell games with finite players and actions have epsilon-equilibria.

Multiple proof techniques confirm the existence of approximate equilibria.

The results extend classical two-player game theory to multiplayer settings.

Abstract

We study multiplayer Blackwell games, which are repeated games where the payoff of each player is a bounded and Borel-measurable function of the infinite stream of actions played by the players during the game. These games are an extension of the two-player perfect-information games studied by David Gale and Frank Stewart (1953). Recently, various new ideas have been discovered to study Blackwell games. In this paper, we give an overview of these ideas by proving, in four different ways, that Blackwell games with a finite number of players, finite action sets, and tail-measurable payoffs admit an $\varepsilon$-equilibrium, for all $\varepsilon>0$.