On likelihood of a Condorcet winner for uniformly random and independent voter preferences
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| Format: | Preprint |
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2025
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| _version_ | 1866908627761102848 |
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| author | Pittel, Boris |
| author_facet | Pittel, Boris |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_20613 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On likelihood of a Condorcet winner for uniformly random and independent voter preferences Pittel, Boris Combinatorics 05A05, 05A15, 05A16, 05C05, 06B05, 05C80, 05D40, 60C05 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. |
| title | On likelihood of a Condorcet winner for uniformly random and independent voter preferences |
| topic | Combinatorics 05A05, 05A15, 05A16, 05C05, 06B05, 05C80, 05D40, 60C05 |
| url | https://arxiv.org/abs/2506.20613 |