Probabilistic Copeland Tournaments Sam Saarinen University of Kentucky Lexington, KY samuel.saarinen@uky.edu Judy Goldsmith University of Kentucky Lexington, KY goldsmit@cs.uky.edu Craig Tovey Georgia Tech Atlanta, GA ctovey@isye.gatech.edu INTRODUCTION We consider a probabilistic model of round-robin tournaments, or equivalently, Copeland voting, where candidates are the vot- ers. 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 prob- ability 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 particu- lar 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 probabilis- tic 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, al- though they did not introduce probabilities [5]. Instead, they con- sidered 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 in- complete ordered ballots. In their interpretation, the ranked candi- dates are preferred to all candidates not mentioned, and all comple- tions 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 rep- resented 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, 1 each distinct pair of which will play one game. Team i defeats team j with probability Ti,j , and game outcomes are jointly indepen- dent. 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, sim- ple 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 1 It is more common in AI to refer to “agents" or “candidates". We stick to “teams" for consistency. Appears in: Proceedings of the 14th International Conference on Autonomous Agents and Multiagent Systems (AAMAS 2015), Bor- dini, Elkind, Weiss, Yolum (eds.), May, 4–8, 2015, Istanbul, Turkey. Copyright c  2015, International Foundation for Autonomous Agents and Multiagent Systems (www.ifaamas.org). All rights reserved. Figure 1: All four Eulerian orientations of a graph weight wij has the same meaning as Tij . Team i defeats team j with probability wij and loses to team j with complementary prob- ability 1 − wij . This graph representation employs only one edge between each distinct pair of vertices i and j . Definition: Let t be a distinguished agent of a PCT represented by matrix T . The Probabilistic Tournament Problem (PTP) is the problem of computing the probability, denoted PW (T,t), that t wins at least as many games as any other agent. If the PCT is rep- resented by graph G, the probability is denoted PW (G, t). Definition: Let T represent a PCT with distinguished agent t. The Unique-winner Probabilistic Tournament Problem (UPTP) is the problem of computing the probability, denoted PUW (T,t), that t wins strictly more games than any other agent. If the PCT is rep- resented by graph G, the probability is denoted PUW (G, t). The PTP was shown to be in #P [8], and a similar argument shows that the UPTP is in #P. However, no other previous hardness results are known. Work by Aziz, et al. has explored probabilistic knockout tournaments [1], but their results are not applicable to a PCT. COMPLEXITY RESULTS THEOREM 1. The PTP is #P-complete, even if all probabilities are in {0, 1 2 , 1}. Given the previous result, which places the PTP in the complex- ity class #P, it suffices to show that the PTP is #P-hard. We prove the hardness of the PTP by a reduction from counting the number of Eulerian orientations of an Eulerian graph. An Eulerian graph is an undirected graph all of whose nodes have even degree. An Eulerian orientation of an Eulerian graph is a choice of orientation for each edge such that every vertex’s indegree equals its outdegree. (See Figure 1.) The problem of counting the number of Eulerian orien- tations of an Eulerian graph has been shown to be #P-complete by a reduction from counting perfect matchings [9]. Assume we are given an Eulerian graph H =(U, F ). We will now construct a tournament graph G =(V,E) with U ⊂ V ; F ⊂ 1851