On the Cardinality of the Smith Set Under Impartial Culture with Many Alternatives

2026-10-01

AI Summary

This study investigates the asymptotic probability distribution of the cardinality of the Smith set under the impartial culture assumption for multiple alternatives. Employing a random preference model with pairwise majority comparisons and asymptotic probabilistic methods, we systematically analyze the decay behavior of the Smith set size for a fixed number of voters. The research precisely characterizes the decay orders across different cardinality regimes under constant, polynomial, and exponential scenarios. Furthermore, it resolves a related conjecture proposed by Liu et al. and reveals the specific rate at which the probability of the Smith set encompassing all alternatives converges to one. These findings refine the theoretical boundaries within this domain.

Abstract

For some integer $\ell\geq 2$, consider $m=2\ell-1$ voters, each of which has a preference ranking over $n$ alternatives. The Smith set is the smallest nonempty set of alternatives each of which defeats every alternative outside the set in a pairwise majority comparison. Under the standard benchmark impartial culture where each voter uniformly and independently chooses a random preference ranking over all alternatives, we study the asymptotic probability that the Smith set has cardinality $s$ for fixed $\ell$ and $n\to \infty$. First, we prove that the probability that the Smith set has constant cardinality $s$ goes to zero at a rate of $\Theta_{\ell,s} (n^{-s(\ell-1)/\ell})$. Next, if $\min\{s,n-s\}\to\infty$, we show that this probability decays superpolynomially. Further, when $s$ and $n-s$ are both $\Theta(n)$, we prove that the probability decays exponentially and determine the exact exponential base. Finally, we prove that the probability that the Smith set contains all alternatives approaches one at a rate of $\Theta_\ell(n^{-(\ell-1)/\ell})$. These theorems resolve the conjectures in [Liu et al, Ann Stat 2026] under impartial culture.

Problem

• Smith set
• impartial culture
• cardinality
• asymptotic probability
• social choice

Innovation

• Smith set
• Impartial culture
• Asymptotic probability
• Cardinality
• Social choice theory