Blackwell Equilibrium in Repeated Games

TL;DR

This paper explores the application of Blackwell optimality to repeated games, establishing conditions under which equilibrium strategies are sequentially rational across different monitoring structures.

Contribution

It extends Blackwell optimality concepts to repeated games, analyzing equilibrium existence and structure under perfect, public, and private monitoring scenarios.

Findings

Under perfect monitoring, a folk theorem applies with minmax payoffs.

Without public randomization, perfect public equilibria are often pure or stage-game Nash.

In private monitoring, stage-game Nash equilibria are played in every round in certain games.

Abstract

We apply Blackwell optimality to repeated games. An equilibrium whose strategy profile is sequentially rational for all high enough discount factors simultaneously is a Blackwell (subgame-perfect, perfect public, etc.) equilibrium. The bite of this requirement depends on the monitoring structure. Under perfect monitoring, a ``folk'' theorem holds relative to an appropriate notion of minmax. Under imperfect public monitoring, absent a public randomization device, any perfect public equilibrium generically involves pure action profiles or stage-game Nash equilibria only. Under private conditionally independent monitoring, in a class of games that includes the prisoner's dilemma, the stage-game Nash equilibrium is played in every round.