2025/06/25 by Boris Pittel, Pittel, Boris
Economics, Econometrics and Finance · #05A05 #05A15 #05A16 #05C05 #05C80 #05D40 #06B05 #60C05 #Combinatorics (math.CO) #FOS: Mathematics #Game Theory and Voting Systems
paper · pdf · doi:10.48550/arxiv.2506.20613
openalex publication_date 2025/06/25 · openalex created_date 2025/10/18 · openalex updated_date 2026/07/28
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→∞ and m≫ n4, then for each fixed ℓ the probability of a Condercet winner is at most of order n-ℓ + n2/m1/2, thus converges to zero.