Winning in the Limit: Average-Case Committee Selection with Many Candidates

2026-02-04

AI Summary

This work addresses the problem of selecting a fixed-size committee that can defeat every external candidate with a prescribed majority level, under random preferences of a large number of voters and candidates. Within the impartial culture model, the study establishes sharp thresholds for the existence of α-winning sets and α-dominating sets by combining probabilistic methods, duality arguments, and rounding techniques. Specifically, it proves that the threshold for α-winning sets is \(1 - 1/k\), while that for α-dominating sets is \(1/2 - 1/(2k)\). These results precisely characterize the boundary of feasibility for such committee selection problems and significantly improve upon the previously known upper bound on the infeasibility of α-dominating sets, thereby offering stronger theoretical guarantees.

Technology Category

• Game Theory and Economic Paradigms: Social Choice / Voting
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Knowledge Representation and Reasoning: Preferences

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments

Abstract

We study the committee selection problem in the canonical impartial culture model with a large number of voters and an even larger candidate set. Here, each voter independently reports a uniformly random preference order over the candidates. For a fixed committee size $k$, we ask when a committee can collectively beat every candidate outside the committee by a prescribed majority level $\alpha$. We focus on two natural notions of collective dominance, $\alpha$-winning and $\alpha$-dominating sets, and we identify sharp threshold phenomena for both of them using probabilistic methods, duality arguments, and rounding techniques. We first consider $\alpha$-winning sets. A set $S$ of $k$ candidates is $\alpha$-winning if, for every outside candidate $a \notin S$, at least an $\alpha$-fraction of voters rank some member of $S$ above $a$. We show a sharp threshold at \[ \alpha_{\mathrm{win}}^\star = 1 - \frac{1}{k}. \] Specifically, an $\alpha$-winning set of size $k$ exists with high probability when $\alpha<\alpha_{\mathrm{win}}^\star$, and is unlikely to exist when $\alpha>\alpha_{\mathrm{win}}^\star$. We then study the stronger notion of $\alpha$-dominating sets. A set $S$ of $k$ candidates is $\alpha$-dominating if, for every outside candidate $a \notin S$, there exists a single committee member $b \in S$ such that at least an $\alpha$-fraction of voters prefer $b$ to $a$. Here we establish an analogous sharp threshold at \[ \alpha_{\mathrm{dom}}^\star = \frac{1}{2} - \frac{1}{2k}. \] As a corollary, our analysis yields an impossibility result for $\alpha$-dominating sets: for every $k$ and every $\alpha>\alpha_{\mathrm{dom}}^\star = 1 / 2 - 1 / (2k)$, there exist preference profiles that admit no $\alpha$-dominating set of size $k$. This corollary improves the best previously known bounds for all $k \geq 2$.

Problem

• committee selection
• impartial culture
• majority threshold
• collective dominance
• probabilistic voting

Innovation

• committee selection
• sharp threshold
• impartial culture
• α-dominating set
• probabilistic method