Probabilistic Copeland Tournaments

INTRODUCTION We consider a probabilistic model of round-robin tournaments, or equivalently, Copeland voting, where candidates are the voters. We assume that the outcomes of each game or pairwise vote are jointly independent. In particular, we do not assume that votes arise from voters’ ranked orderings of candidates. We can treat such games as pairwise preferences, without assuming any form of transitivity. We prove the #P-completeness of computing the probability of victory. As a consequence, it is #P-hard to manipulate a round-robin tournament by controlling the outcome of a subset of the games to raise the probability of winning above a particular threshhold. These results hold in the restricted case where all probabilities are zero, one half, or one. According to Faliszewski, et al. [4], the notion of probabilistic Copeland elections go back to a 1929 paper by Zermelo [15] and more recently to Levin and Nalebuff [7]. In 2005, Konczak and Lang looked at Copeland elections with incomplete ballots, although they did not introduce probabilities [5]. Instead, they considered possible and necessary winners (probabilities > 0 and 1, respectively). These notions have been well studied (e.g., [3, 6, 11, 13, 14]). Bachrach et al. [2] introduced a probabilistic interpretation of incomplete ordered ballots. In their interpretation, the ranked candidates are preferred to all candidates not mentioned, and all completions of the partial linear order are equally likely. This introduces correlations in the probabilities of individual pairings, which we do not assume. Bachrach et al. showed that computing the probability of a given candidate winning, in this setting, was #P-hard, using techniques that do not apply in the tournament setting. Definition: A Probabilistic Copeland Tournament (PCT) is represented by an n × n nonnegative matrix, T , where Ti,j + Tj,i = 1 ∀i, j. The n row and column indices of T represent n teams, each distinct pair of which will play one game. Team i defeats team j with probability Ti,j , and game outcomes are jointly independent. Therefore, the probability of a set of game outcomes equals the product of the probabilities of the individual game outcomes. Equivalently, a PCT is represented by a complete, directed, simple graph (V,E), where V is the set of teams, with edge weights we : e ∈ E such that 0 ≤ we ≤ 1 ∀e ∈ E. An edge (i, j) with