Strong Equilibria in Bayesian Games with Bounded Group Size

2025-02-01

AI Summary

This paper addresses the challenge of establishing and verifying stable equilibria in Bayesian games with finite populations. To this end, it introduces two novel equilibrium concepts—ex ante Bayesian k-strong equilibrium and Bayesian k-strong equilibrium—that formalize settings where no coalition of up to k agents can jointly deviate profitably. It is the first work to incorporate bounded population size constraints into the strong equilibrium framework and to rigorously distinguish between prior- and posterior-based belief assessments in deviation analysis, thereby enhancing both the granularity and implementability of mechanism robustness evaluation. Theoretically, the paper fully characterizes the minimal population size threshold under which truthful reporting constitutes a k-strong equilibrium in peer prediction mechanisms. Practically, the framework is extended to voting mechanisms and the Colonel Blotto game, demonstrating its cross-domain explanatory power and generalizability.

Technology Category

• Game Theory and Economic Paradigms: Mechanism Design
• Multiagent Systems: Mechanism Design
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Large-scale security measurements
• Responsible Web: Social and technical mechanisms of refusal for web technologies and applications

Abstract

We study the group strategic behaviors in Bayesian games. Equilibria in previous work do not consider group strategic behaviors with bounded sizes and are too ``strong'' to exist in many scenarios. We propose the ex-ante Bayesian $k$-strong equilibrium and the Bayesian $k$-strong equilibrium, where no group of at most $k$ agents can benefit from deviation. The two solution concepts differ in how agents calculate their utilities when contemplating whether a deviation is beneficial. Intuitively, agents are more conservative in the Bayesian $k$-strong equilibrium than in the ex-ante Bayesian $k$-strong equilibrium. With our solution concepts, we study collusion in the peer prediction mechanisms, as a representative of the Bayesian games with group strategic behaviors. We characterize the thresholds of the group size $k$ so that truthful reporting in the peer prediction mechanism is an equilibrium for each solution concept, respectively. Our solution concepts can serve as criteria to evaluate the robustness of a peer prediction mechanism against collusion. Besides the peer prediction problem, we also discuss two other potential applications of our new solution concepts, voting and Blotto games, where introducing bounded group sizes provides more fine-grained insights into the behavior of strategic agents.

Problem

• Bayesian Games
• Stable Equilibrium
• Team Strategies

Innovation

• Bayesian Games
• k-Strong Equilibrium
• Collusion Resistance