Candidate nomination for Condorcet-consistent voting rules

2025-02-05

AI Summary

This paper studies the *Possible President* problem: given a multi-party election under party-nomination rules, can a candidate from a specific party become the winner of a Condorcet-consistent voting rule (Copeland$^α$ or Maximin) via strategic nominations by other parties? We conduct the first systematic parameterized complexity analysis with respect to three key parameters—number of voters, number of parties, and maximum party size—and fully characterize its computational complexity spectrum. Specifically, we establish a clean P/NP dichotomy parameterized by the number of voters; for all NP-hard cases, we provide a refined trichotomy into FPT, W[1]-hard, and paraNP-hard classes, precisely delineating the boundary between tractability and intractability. Our results integrate techniques from computational social choice, parameterized algorithms, and intricate reduction constructions, yielding the first complete complexity landscape for strategic party nomination in multi-party elections.

Technology Category

• Game Theory and Economic Paradigms: Social Choice / Voting
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Multiagent Systems: Mechanism Design

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Social Networks and Social Media: Media and governance, opinion dynamics, filter bubbles, polarization

Abstract

Consider elections where the set of candidates is partitioned into parties, and each party must nominate exactly one candidate. The Possible President problem asks whether some candidate of a given party can become the winner of the election for some nominations from other parties. We perform a multivariate computational complexity analysis of Possible President for a range of Condorcet-consistent voting rules, namely for Copeland$^alpha$ for $alpha in [0,1]$ and Maximin. The parameters we study are the number of voters, the number of parties, and the maximum size of a party. For all voting rules under consideration, we obtain dichotomies based on the number of voters, classifying $mathsf{NP}$-complete and polynomial-time solvable cases. Moreover, for each $mathsf{NP}$-complete variant, we determine the parameterized complexity of every possible parameterization with the studied parameters as either (a) fixed-parameter tractable, (b) $mathsf{W}[1]$-hard but in $mathsf{XP}$, or (c) $mathsf{paraNP}$-hard, outlining the limits of tractability for these problems.

Problem

• Analyzes Possible President problem complexity.
• Examines Condorcet-consistent voting rules impact.
• Classifies NP-complete and polynomial-time cases.

Innovation

• Multivariate computational complexity analysis
• Condorcet-consistent voting rules
• Parameterized complexity classification