Strong Equilibria in Bayesian Games with Bounded Group Size

TL;DR

This paper introduces new equilibrium concepts for Bayesian games that account for bounded group sizes, analyzing their impact on collusion and strategic behavior in mechanisms like peer prediction, voting, and Blotto games.

Contribution

It proposes ex-ante Bayesian k-strong and Bayesian k-strong equilibria, providing a nuanced framework for group strategic behavior with bounded sizes in Bayesian games.

Findings

Characterized thresholds for group size k ensuring truthful reporting in peer prediction.

Demonstrated how the new equilibria serve as robustness criteria against collusion.

Applied the concepts to voting and Blotto games for deeper strategic insights.

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.