Voting Profiles Admitting All Candidates as Knockout Winners
Bernard De Baets, Emilio De Santis
TL;DR
The paper proves that in knockout tournaments with 2^n candidates (n≥3), there exist explicitly constructible majority graphs and small voting profiles that permit any candidate to win given an appropriate bracket. It provides a recursive G_n construction built from a base G_3, along with a Bracket design pi^(n;j) ensuring each candidate j can win; it then delivers a concrete voting profile R_n (and tilde{R}_n) realizing G_n with size linear in n (4n−3). The work further analyzes a Poisson-distributed random number of voters, showing that with sufficiently large λ_n the resulting random majority graph matches G_n with high probability, via Chernoff bounds. Overall, it demonstrates that manipulation via bracket design is feasible with compact, explicitly constructible profiles, and extends the results to a random-voter setting with high-probability guarantees.
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.