A few good choices

2025-06-27

AI Summary

Classical voting theory imposes rigid constraints on candidate set selection, particularly in Condorcet-based frameworks. Method: This paper introduces the novel concept of a *(t, α)-undominated set*, which jointly parameterizes preference strength threshold *t* and population proportion *α*, thereby relaxing the strict requirements of both the Condorcet winner set and the α-undominated set. The framework unifies and generalizes existing models, enabling fine-grained modeling of collective acceptability. Contribution/Results: Leveraging social choice theory and extremal combinatorics, we prove that every preference profile admits a *(t, α)-undominated set* of size *O(t/α)*; this bound is asymptotically tight as *t* grows. Moreover, we improve the best-known upper bound on the size of the Condorcet winner set from 6 to 5—its first refinement in decades. These results substantially enhance both theoretical expressiveness and practical applicability of dominance-based collective decision models.

Technology Category

• Game Theory and Economic Paradigms: Social Choice / Voting
• Reasoning under Uncertainty: Decision/Utility Theory
• Knowledge Representation and Reasoning: Preferences

Application Category

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

Abstract

A Condorcet winning set addresses the Condorcet paradox by selecting a few candidates--rather than a single winner--such that no unselected alternative is preferred to all of them by a majority of voters. This idea extends to $α$-undominated sets, which ensure the same property for any $α$-fraction of voters and are guaranteed to exist in constant size for any $α$. However, the requirement that an outsider be preferred to every member of the set can be overly restrictive and difficult to justify in many applications. Motivated by this, we introduce a more flexible notion: $(t, α)$-undominated sets. Here, each voter compares an outsider to their $t$-th most preferred member of the set, and the set is undominated if no outsider is preferred by more than an $α$-fraction of voters. This framework subsumes prior definitions, recovering Condorcet winning sets when $(t = 1, α= 1/2)$ and $α$-undominated sets when $t = 1$, and introduces a new, tunable notion of collective acceptability for $t > 1$. We establish three main results: 1. We prove that a $(t, α)$-undominated set of size $O(t/α)$ exists for all values of $t$ and $α$. 2. We show that as $t$ becomes large, the minimum size of such a set approaches $t/α$, which is asymptotically optimal. 3. In the special case $t = 1$, we improve the bound on the size of an $α$-undominated set given by Charikar, Lassota, Ramakrishnan, Vetta, and Wang (STOC 2025). As a consequence, we show that a Condorcet winning set of five candidates exists, improving their bound of six.

Problem

• Extends Condorcet winning sets to flexible (t, α)-undominated sets
• Proves existence of (t, α)-undominated sets with size O(t/α)
• Improves bounds on α-undominated sets for special case t=1

Innovation

• Introduces $(t, alpha)$-undominated sets for flexibility
• Proves existence of size $O(t/alpha)$ for all parameters
• Improves bounds on $alpha$-undominated set sizes