(Ab)using Indifference: Purification in Communication and Repeated Games

TL;DR

This paper demonstrates that in realistic settings with privacy, equilibria relying on private information conditioning are generally impossible, highlighting limitations of public equilibria and suggesting the need for belief-based approaches.

Contribution

It generalizes previous results by proving the impossibility of certain equilibria in games with privacy, and discusses implications for repeated games with noisy monitoring.

Findings

Equilibria conditioned on private, payoff-irrelevant information are impossible with privacy.

All equilibria without private randomization in certain repeated games are public equilibria.

Belief-based equilibria are necessary to understand repeated games with noisy monitoring.

Abstract

Recent papers in communication games construct equilibria by conditioning an agent's strategy on private, payoff-irrelevant information. I prove this is impossible in general games if there is any amount of realistic privacy in agents' preferences, generalizing previous results from cheap talk games to mediated cheap talk and communication with receiver commitment. Applying the result to repeated games with public+conditionally independent private monitoring, all equilibria without private randomization are perfect public equilibria, and non-trivial belief free equilibria are impossible. This result can be avoided if information is slightly correlated, or pay-off relevant. Due to undesirable properties of public perfect equilibria in some settings, I argue for further study of belief-based equilibria to understand equilibria of repeated games with noisy monitoring.