[Paper Review] An Anti-Folk Theorem for Large Repeated Games with Imperfect Monitoring

Mallesh M. Pai, Aaron Roth+1 more|arXiv (Cornell University)|Feb 12, 2014

Privacy-Preserving Technologies in Data24 references15 citations

TL;DR

This paper establishes an anti-folk theorem for large repeated games with imperfect monitoring by showing that when players' signals satisfy $(\epsilon,\gamma)$-differential privacy—common in large games—equilibria of the repeated game must involve approximate equilibria of the stage game in every period. As the number of players grows, $\epsilon$ and $\gamma$ shrink, collapsing the set of equilibria to those of the stage game, thereby limiting the predictive power of folk theorems in such settings.

ABSTRACT

We study infinitely repeated games in settings of imperfect monitoring. We first prove a family of theorems that show that when the signals observed by the players satisfy a condition known as $(ε, γ)$-differential privacy, that the folk theorem has little bite: for values of $ε$ and $γ$ sufficiently small, for a fixed discount factor, any equilibrium of the repeated game involve players playing approximate equilibria of the stage game in every period. Next, we argue that in large games ($n$ player games in which unilateral deviations by single players have only a small impact on the utility of other players), many monitoring settings naturally lead to signals that satisfy $(ε,γ)$-differential privacy, for $ε$ and $γ$ tending to zero as the number of players $n$ grows large. We conclude that in such settings, the set of equilibria of the repeated game collapse to the set of equilibria of the stage game.

Motivation & Objective

• To investigate whether folk theorems retain predictive power in large repeated games with imperfect monitoring.
• To identify conditions under which the multiplicity of equilibria in repeated games collapses to stage game equilibria.
• To formalize the connection between large games and differential privacy as a measure of 'largeness'.
• To provide quantitative bounds on equilibrium behavior, going beyond limiting results in prior literature.
• To show that in large games with fixed noise levels, signals naturally satisfy $(\epsilon,\gamma)$-differential privacy with $\epsilon,\gamma \to 0$ as $n \to \infty$.

Proposed method

• The authors define and apply $(\epsilon,\gamma)$-differential privacy to model signals in repeated games with imperfect monitoring.
• They prove that when signals satisfy $(\epsilon,\gamma)$-differential privacy with small $\epsilon$ and $\gamma$, repeated game equilibria must involve approximate stage game equilibria in every period.
• For public monitoring, equilibria must be approximate Nash equilibria of the stage game; for private monitoring, they must be approximate correlated equilibria.
• The analysis leverages composition and post-processing theorems from differential privacy to preserve privacy guarantees under signal transformations.
• The framework is applied to large games where unilateral deviations have small impact, showing $\epsilon,\gamma \to 0$ as $n \to \infty$ under fixed noise levels.
• The results are shown to generalize and quantify prior results by Green (1980) and Sabourian (1990), providing finite-sample bounds.

Experimental results

Research questions

• RQ1 Under what conditions do equilibria in large repeated games with imperfect monitoring reduce to stage game equilibria?
• RQ2 Can differential privacy serve as a natural and quantifiable measure of 'largeness' in repeated games?
• RQ3 To what extent do folk theorem equilibria lose their viability when signals are differentially private?
• RQ4 Is the collapse of equilibria to stage game outcomes quantitatively bounded, rather than just asymptotic?
• RQ5 Can every sequence of correlated equilibria of the stage game be supported as an equilibrium in a repeated game with private monitoring and differentially private signals?

Key findings

• When signals satisfy $(\epsilon,\gamma)$-differential privacy with small $\epsilon$ and $\gamma$, equilibria of the repeated game must involve approximate equilibria of the stage game in every period.
• In large games with fixed noise levels, $\epsilon$ and $\gamma$ tend to zero as the number of players $n$ increases, leading to the collapse of repeated game equilibria to stage game equilibria.
• For public monitoring, the set of equilibria collapses to approximate Nash equilibria of the stage game.
• For private monitoring, the set of equilibria collapses to approximate correlated equilibria of the stage game.
• The results provide quantitative bounds on equilibrium behavior, going beyond qualitative or limiting results in earlier literature.
• The framework suggests that differential privacy is a natural and general measure of 'largeness' in repeated games, with implications for predictability and the price of anarchy.