Voting Profiles Admitting All Candidates as Knockout Winners

TL;DR

This paper investigates voting profiles in knockout tournaments, showing that with a small number of voters, any candidate can be made a winner by choosing an initial order, and similar results hold probabilistically with many voters.

Contribution

It introduces a voting profile with a minimal number of voters that allows any candidate to win, and extends the results to probabilistic settings with large voter populations.

Findings

A voting profile with 4n - 3 voters ensures any candidate can win.

Results hold with high probability in large random voter settings.

The initial candidate order influences the tournament outcome.

Abstract

A set of $2^n$ candidates is presented to a commission. At every round, each member of this commission votes by pairwise comparison, and one-half of the candidates is deleted from the tournament, the remaining ones proceeding to the next round until the $n$-th round (the final one) in which the final winner is declared. The candidates are arranged on a board in a given order, which is maintained among the remaining candidates at all rounds. A study of the size of the commission is carried out in order to obtain the desired result of any candidate being a possible winner. For $2^n$ candidates with $n \geq 3$, we identify a voting profile with $4n -3$ voters such that any candidate could win simply by choosing a proper initial order of the candidates. Moreover, in the setting of a random number of voters, we obtain the same results, with high probability, when the expected number of voters is large.