The power of mediators: Price of anarchy and stability in Bayesian games with submodular social welfare

TL;DR

This paper analyzes how mediators influence social welfare in Bayesian games with submodular functions, providing bounds on the price of anarchy and stability across various equilibrium concepts.

Contribution

It introduces bounds on the price of anarchy and stability in Bayesian games with submodular social welfare, highlighting the strategy representability gap and differences among equilibrium classes.

Findings

Bounds on PoA and PoS for independent priors

Bounds on PoA and PoS for correlated priors

Identification of a fundamental gap across equilibrium classes

Abstract

This paper investigates the role of mediators in Bayesian games by examining their impact on social welfare through the price of anarchy (PoA) and price of stability (PoS). Mediators can communicate with players to guide them toward equilibria of varying quality, and different communication protocols lead to a variety of equilibrium concepts collectively known as Bayes (coarse) correlated equilibria. To analyze these equilibrium concepts, we consider a general class of Bayesian games with submodular social welfare, which naturally extends valid utility games and their variant, basic utility games. These frameworks, introduced by Vetta (2002), have been developed to analyze the social welfare guarantees of equilibria in games such as competitive facility location, influence maximization, and other resource allocation problems. We provide upper and lower bounds on the PoA and PoS for a broad class of Bayes (coarse) correlated equilibria. Central to our analysis is the strategy representability gap, which measures the multiplicative gap between the optimal social welfare achievable with and without knowledge of other players' types. For monotone submodular social welfare functions, we show that this gap is $1-1/\mathrm{e}$ for independent priors and $\Theta(1/\sqrt{n})$ for correlated priors, where $n$ is the number of players. These bounds directly lead to upper and lower bounds on the PoA and PoS for various equilibrium concepts, while we also derive improved bounds for specific concepts by developing smoothness arguments. Notably, we identify a fundamental gap in the PoA and PoS across different classes of Bayes correlated equilibria, highlighting essential distinctions among these concepts.