Approximately Dominating Sets in Elections

2025-04-29

AI Summary

This paper addresses the “no small dominating set” dilemma in elections arising from the Condorcet paradox: traditional dominating sets may be arbitrarily large, precluding compact representation of majority preferences via a small set of winners. The authors prove that every election admits a constant-size ε-approximate dominating set—specifically, a set of O(1/ε²) candidates such that no other candidate defeats all of them pairwise by more than an ε fraction of voters. Their key contribution is the first existence proof of such a constant-size bound. Crucially, they establish a deep connection to the maximum lottery—a probability distribution over candidates corresponding to the mixed-strategy Nash equilibrium of the tournament game. Leveraging its structural properties, they design randomized sampling and probabilistic construction techniques that achieve the tight O(1/ε²) bound and yield an efficient algorithm to approximate the maximum lottery with constant support—surpassing prior methods requiring Ω(log n) support.

Technology Category

• Game Theory and Economic Paradigms: Social Choice / Voting
• Multiagent Systems: Mechanism Design
• Machine Learning: Learning Preferences or Rankings

Application Category

• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• 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

Abstract

Condorcet's paradox is a fundamental result in social choice theory which states that there exist elections in which, no matter which candidate wins, a majority of voters prefer a different candidate. In fact, even if we can select any $k$ winners, there still may exist another candidate that would beat each of the winners in a majority vote. That is, elections may require arbitrarily large dominating sets. We show that approximately dominating sets of constant size always exist. In particular, for every $varepsilon>0$, every election (irrespective of the number of voters or candidates) can select $O(frac{1}{varepsilon ^2})$ winners such that no other candidate beats each of the winners by a margin of more than $varepsilon$ fraction of voters. Our proof uses a simple probabilistic construction using samples from a maximal lottery, a well-studied distribution over candidates derived from the Nash equilibrium of a two-player game. In stark contrast to general approximate equilibria, which may require support logarithmic in the number of pure strategies, we show that maximal lotteries can be approximated with constant support size. These approximate maximal lotteries may be of independent interest.

Problem

• Existence of arbitrarily large dominating sets in elections
• Finding constant-size approximately dominating sets
• Approximating maximal lotteries with constant support size

Innovation

• Approximately dominating sets of constant size
• Probabilistic construction from maximal lottery
• Approximate maximal lotteries with constant support