Asymptotic Value of Monitoring Structures in Stochastic Games

TL;DR

This paper investigates how enhanced public monitoring in stochastic games enlarges the set of equilibrium payoffs, introducing a new comparison method called weighted garbling to analyze the impact of information improvements.

Contribution

It introduces weighted garbling as a novel way to compare monitoring structures and proves the monotonicity of equilibrium payoff sets with respect to this order.

Findings

Limit PPE payoff set expands with more informative monitoring.

Monotonicity of equilibrium payoffs holds for symmetric stochastic games.

Weighted garbling helps compare payoff sets across different state laws.

Abstract

This paper studies how improved monitoring affects the limit equilibrium payoff set for stochastic games with imperfect public monitoring. We introduce a simple generalization of Blackwell garbling called weighted garbling in order to compare different monitoring structures for this class of games. Our main result is the monotonicity of the limit perfect public equilibrium (PPE) payoff set with respect to this information order. We show that the limit PPE payoff set expands when the monitoring structure gets more informative with respect to the weighted garbling order. We also show that a similar monotonicity holds for strongly symmetric equilibrium for symmetric stochastic games. Finally, we show that our weighted garbling order is useful to compare the limit PPE payoff set for different state transition laws and monitoring structures when the limit feasible payoff set is the same.