Strong Equilibria in Bayesian Games with Bounded Group Size

TL;DR

This paper addresses group strategic behavior in Bayesian games when coalitions are realistically bounded in size. It introduces ex-ante Bayesian $k$-strong equilibrium ($k$-EBSE) and Bayesian $k$-strong equilibrium ($k$-BSE) to model deviations by coalitions of size at most $k$, with $k$-EBSE relying on ex-ante utilities and $k$-BSE on interim utilities conditioned on every type; it further proves that $k$-EBSE implies $k$-BSE. Focusing on peer prediction mechanisms, the authors derive precise thresholds $k_E$ and $k_B$ (and their per-signal variants) that determine when truthful reporting remains an equilibrium under each concept, showing how these thresholds depend on the number of agents $n$, the prior, and the scoring rule. They provide proof sketches and intuition based on convexity arguments and averaging over deviators, and they illustrate the impact of using different scoring rules (e.g., Brier vs. log) on the robustness to collusion. Beyond peer prediction, the work discusses applications to voting with partial information and Private Blotto, arguing that bounded coalition concepts yield finer-grained insights into coordination and robustness against collusion in diverse Bayesian settings.

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.

Figures (1)

• Figure 1: The illustration of Lemma \ref{['lem:subspace_h']} and \ref{['lem:subspace_l']}. The X-axis and Y-axis denote $\bar{\beta_{\ell}}$ and $\bar{\beta_{h}}$ respectively. The two half-planes characterized by two lines cover the $[0,1]^2$ area, so there always exists agents with a certain signal that do not wish to deviate. Two lines are not necessarily located above/below point (1, 0).

Theorems & Definitions (31)

• Example 1: Group strategic behavior in peer prediction
• Example 2
• Example 3
• Definition 1: ex-ante Bayesian $k$-strong equilibrium
• Definition 2: Bayesian $k$-strong equilibrium
• Proposition 1
• Remark 1
• Example 4
• Example 5
• Theorem 1