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Aggregating Information and Preferences with Bounded-Size Deviations

Qishen Han, Grant Schoenebeck, Biaoshuai Tao, Lirong Xia

TL;DR

This work studies binary majority voting with two antagonistic groups whose preferences depend on an unobservable ground truth. It adopts the ex-ante Bayesian $k$-strong equilibrium framework to account for capacitated coalitional deviations and provides a complete characterization of when equilibria supporting the informed majority arise, via a closed-form threshold curve $\xi^*(\alpha)$. The curve is piecewise and can have four segments (Flat, Steep, Non-Linear, Tail), reflecting how majority share $\alpha$ and coalition capacity interact with signal structure; a symmetric case reduces to fewer segments. The results extend the Condorcet Jury framework to settings with bounded coordination and incomplete information, offering a nuanced understanding of strategic voting and facilitating more refined predictions and design insights in settings with conflicting agents and partial information.

Abstract

We investigate a voting scenario with two groups of agents whose preferences depend on a ground truth that cannot be directly observed. The majority's preferences align with the ground truth, while the minorities disagree. Focusing on strategic behavior, we analyze situations where agents can form coalitions up to a certain capacity and adopt the concept of ex-ante Bayesian $k$-strong equilibrium, in which no group of at most $k$ agents has an incentive to deviate. Our analysis provides a complete characterization of the region where equilibria exist and yield the majority-preferred outcome when the ground truth is common knowledge. This region is defined by two key parameters: the size of the majority group and the maximum coalition capacity. When agents cannot coordinate beyond a certain threshold determined by these parameters, a stable outcome supporting the informed majority emerges. The boundary of this region exhibits several distinct segments, notably including a surprising non-linear relationship between majority size and deviation capacity. Our results reveal the complexity of the strategic behaviors in this type of voting game, which in turn demonstrate the capability of the ex-ante Bayesian $k$-strong equilibrium to provide a more detailed analysis.

Aggregating Information and Preferences with Bounded-Size Deviations

TL;DR

This work studies binary majority voting with two antagonistic groups whose preferences depend on an unobservable ground truth. It adopts the ex-ante Bayesian -strong equilibrium framework to account for capacitated coalitional deviations and provides a complete characterization of when equilibria supporting the informed majority arise, via a closed-form threshold curve . The curve is piecewise and can have four segments (Flat, Steep, Non-Linear, Tail), reflecting how majority share and coalition capacity interact with signal structure; a symmetric case reduces to fewer segments. The results extend the Condorcet Jury framework to settings with bounded coordination and incomplete information, offering a nuanced understanding of strategic voting and facilitating more refined predictions and design insights in settings with conflicting agents and partial information.

Abstract

We investigate a voting scenario with two groups of agents whose preferences depend on a ground truth that cannot be directly observed. The majority's preferences align with the ground truth, while the minorities disagree. Focusing on strategic behavior, we analyze situations where agents can form coalitions up to a certain capacity and adopt the concept of ex-ante Bayesian -strong equilibrium, in which no group of at most agents has an incentive to deviate. Our analysis provides a complete characterization of the region where equilibria exist and yield the majority-preferred outcome when the ground truth is common knowledge. This region is defined by two key parameters: the size of the majority group and the maximum coalition capacity. When agents cannot coordinate beyond a certain threshold determined by these parameters, a stable outcome supporting the informed majority emerges. The boundary of this region exhibits several distinct segments, notably including a surprising non-linear relationship between majority size and deviation capacity. Our results reveal the complexity of the strategic behaviors in this type of voting game, which in turn demonstrate the capability of the ex-ante Bayesian -strong equilibrium to provide a more detailed analysis.
Paper Structure (34 sections, 51 theorems, 33 equations, 2 figures, 1 table)

This paper contains 34 sections, 51 theorems, 33 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Our Contribution
  3. Related Works
  4. Preliminaries
  5. Non-dominated strategies
  6. Proof Sketch of Lemma \ref{['lem:dominated']}.
  7. Sequence of Environments.
  8. Group Size Robustness of Equilibria in Antagonistic Voting
  9. Explicit Form of the Threshold Curve.
  10. Shape Analysis on the Threshold Curve.
  11. Proof Sketch
  12. Segment 1: $\theta \ge \alpha \ge \frac{1}{2P_{\ell L}}$.
  13. Segment 2, 3, and 4, $\frac{1}{2P_{\ell L}} \ge \alpha > \frac{1}{2}$.
  14. Preliminaries
  15. Transition to maximization problem.
  16. ...and 19 more sections

Key Result

Theorem \ref{thm:thresholdk}

Let $\xi^*(\alpha)$ be the threshold curve. For any $\xi < \xi^*(\alpha)$, there exists an $\varepsilon$-ex-ante Bayesian $\xi n$-strong equilibrium such that (1) no agents play weakly dominated strategies, (2) the informed majority decision is reached with probability converging to 1, and (3) $\var

Figures (2)

  • Figure 1: The threshold curve $\xi^*(\alpha)$. The segments are proved in our Theorem \ref{['thm:thresholdk']}. The existence of (unlimited) strong equilibrium is from deng2024aggregation.
  • Figure 2: The shape of $\xi^*$ under different signal distribution. We set $\xi^* = 1$ when $\alpha > \theta$, indicating the existence of a strong equilibrium.

Theorems & Definitions (57)

  • Example 1: Condorcet Jury Theorem
  • Example 2: Voters with conflicting preferences and incomplete information
  • Theorem \ref{thm:thresholdk}: Threshold $\xi^*$ (informal)
  • Proposition 1
  • Example 3: Information Structure and Agent Preferences
  • Definition 1
  • Lemma 1
  • Theorem \ref{thm:thresholdk}
  • Theorem \ref{thm:thresholdk}: Threshold $\xi^*$
  • Proposition 2
  • ...and 47 more