On likelihood of a Condorcet winner for uniformly random and independent voter preferences

Abstract

We study a mathematical model of voting contest with $m$ voters and $n$ candidates, with each voter ranking the candidates in order of preference, without ties. A Condorcet winner is a candidate who gets more than $m/2$ votes in pairwise contest with every other candidate. An ``impartial culture'' setting is the case when each voter chooses his/her candidate preference list uniformly at random from all $n!$ preferences, and does it independently of all other voters. For impartial culture case, Robert May and Lisa Sauermann showed that when $m=2k-1$ is fixed ($k=2$ and $k>2$ respectively), and $n$ grows indefinitely, the probability of a Condorcet winner is small, of order $n^{-(k-1)/k}$. We show if $m, n\to\infty$ and $m\gg n^4$, then for each fixed $\ell$ the probability of a Condercet winner is at most of order $n^{-\ell} + n^2/m^{1/2}$, thus converges to zero.