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

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.

Figures (4)

• Figure 1: The timing of the game, the designer first commits to a mechanism $\hat{c}:A \rightarrow \Delta C$, the agent chooses a contingent strategy $\hat{a}:S\times\Omega\rightarrow \Delta A$. The payoff for players in each state $s\in S$ is then determined by the realization of $\omega\sim\mathbb{P}$, $s$$m\sim \hat{a}(s)$ and $c\sim\hat{c}(a)$.
• Figure 2: The information that can be communicated with invariant preferences is limited by the measure of types $\omega$ whose indifference curves are multi-tangent to the same convex set $\Gamma$. Even if this holds for a specific type (left), it will fail for very nearby types (right).
• Figure 3: The dashed line indicates the feasible payoffs of our public goods game. The red line describes the continuation payoff set in our Atonement equilibrium with two players, where a deviation results in a proportional loss to one's own continuation utility, and a proportional gain to the other's.
• Figure 4: Inferential DAGs relating past private ($s_{i,\le t}$) and public ($s_{\le t}^p$) signals to strategies ($a_{i,>t}$), square nodes are payoff-relevant inferences. In PPE public signals form a coordination device and private information is irrelevant; in BFE agents condition on signals that do not carry payoff-relevant information; in BBE signals are informative about others' past actions ($a_{j,\le t}$) and, through that, their future strategy.

Theorems & Definitions (17)

• Proposition 1
• proof
• Example 1: Communication with Transparent Preferences and Receiver Commitment
• Corollary 1
• Definition 1
• Theorem 1
• proof
• Proposition 2
• Definition 2: Monitoring Terminology
• Example 2: Public Goods Game