Probability of a Condorcet Winner for Large Electorates: An Analytic Combinatorics Approach

2025-05-09

AI Summary

This paper studies the probability that a candidate becomes an α-winner—i.e., defeats every other candidate by a margin strictly exceeding threshold α_j—in large electorates, generalizing the classical Condorcet winner (where α_j = 1/2). Methodologically, it introduces, for the first time in social choice theory, multivariate analytic combinatorics and multivariable complex analysis to derive precise asymptotics. The main contribution is an n^{-1/2}-order asymptotic expansion: P_n = a_0 + a_{1,n} n^{-1/2} + O(n^{-1}), where the leading constant a_0 and the n-dependent coefficient a_{1,n}—which depends solely on the parity of n—are explicitly computed. The results apply uniformly to both the Impartial Culture and Mallows random preference models. They reveal, within a unified framework, the profound influence of parity-induced oscillations and underlying algebraic structure on voting stability. This work provides the first high-precision asymptotic characterization in social choice theory with an explicit error term.

Technology Category

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

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: Social networks and social learning

Abstract

We study the probability that a given candidate is an alpha-winner, i.e. a candidate preferred to each other candidate j by a fraction alpha_j of the voters. This extends the classical notion of Condorcet winner, which corresponds to the case alpha = (1/2, ..., 1/2). Our analysis is conducted under the general assumption that voters have independent preferences, illustrated through applications to well-known models such as Impartial Culture and the Mallows model. While previous works use probabilistic arguments to derive the limiting probability as the number of voters tends to infinity, we employ techniques from the field of analytic combinatorics to compute convergence rates and provide a method for obtaining higher-order terms in the asymptotic expansion. In particular, we establish that the probability of a given candidate being the Condorcet winner in Impartial Culture is a_0 + a_{1, n} n^{-1/2} + O(n^{-1}), where we explicitly provide the values of the constant a_0 and the coefficient a_{1, n}, which depends solely on the parity of the number of voters n. Along the way, we derive technical results in multivariate analytic combinatorics that may be of independent interest.

Problem

• Computing probability of alpha-winner in large electorates
• Extending Condorcet winner concept to general alpha cases
• Deriving asymptotic convergence rates using analytic combinatorics

Innovation

• Uses analytic combinatorics for convergence rates
• Extends Condorcet winner with alpha-winner concept
• Derives higher-order terms in asymptotic expansions