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

2025-06-03

AI Summary

This paper investigates the efficiency limits of (coarse) correlated equilibria mediated by a Bayesian coordinator in Bayesian submodular welfare maximization. Introducing the novel concept of “strategic expressibility gap,” it characterizes for the first time the fundamental efficiency disparity between independent and correlated priors: the price of anarchy (PoA) and price of stability (PoS) are tightly bounded by $1 - 1/e$ under independence, but degrade to $Theta(1/sqrt{n})$ under correlation. Leveraging submodular analysis, smoothness arguments, and Bayesian mechanism design, the paper derives tight PoA/PoS bounds for multiple classes of Bayes (coarse) correlated equilibria. It achieves the optimal $1 - 1/e$ approximation guarantee for monotone submodular welfare—significantly improving upon the efficiency loss inherent in non-mediated settings. These results provide the first theoretical performance guarantees for applications including facility location and influence maximization in Bayesian environments.

Technology Category

• Game Theory and Economic Paradigms: Mechanism Design
• Multiagent Systems: Mechanism Design
• Machine Learning: Bayesian Learning

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines

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.

Problem

• Analyzes mediators' impact on social welfare in Bayesian games
• Bounds price of anarchy and stability for submodular welfare games
• Measures strategy gap between optimal and player-known type outcomes

Innovation

• Mediators guide players to equilibria.
• Analyze submodular social welfare games.
• Bounds on price of anarchy/stability.