Refining Tournament Solutions via Margin of Victory

Abstract

Tournament solutions are frequently used to select winners from a set of alternatives based on pairwise comparisons between alternatives. Prior work has shown that several common tournament solutions tend to select large winner sets and therefore have low discriminative power. In this paper, we propose a general framework for refining tournament solutions. In order to distinguish between winning alternatives, and also between non-winning ones, we introduce the notion of margin of victory (MoV) for tournament solutions. MoV is a robustness measure for individual alternatives: For winners, the MoV captures the distance from dropping out of the winner set, and for non-winners, the distance from entering the set. In each case, distance is measured in terms of which pairwise comparisons would have to be reversed in order to achieve the desired outcome. For common tournament solutions, including the top cycle, the uncovered set, and the Banks set, we determine the complexity of computing the MoV and provide worst-case bounds on the MoV for both winners and non-winners. Our results can also be viewed from the perspective of bribery and manipulation.

Margin of Victory in Tournaments: Structural and Experimental Results

“…Theorem 18 suggests that when tournaments are generated according to the uniform random model, MoV TC and MoV k-kings for k ≥ 4 can likely be computed by a simple formula based on the degrees of the alternatives. In particular, even though the problem is computationally hard for MoV k-kings for any constant k ≥ 4 (Brill et al 2020), there exists an efficient heuristic that correctly computes the MoV value in most cases. In Appendix A.2, we give an example showing that the heuristic is not always correct.…”

“…While we introduced the MoV concept for tournament solutions (Brill et al 2020), similar concepts have been applied to a large number of settings, perhaps most notably voting. In addition, various forms of bribery and manipulation have been considered for both elections and sports tournaments.…”

“…Since all alternatives in TC (T ) dominate all alternatives outside, it cannot be that x ∈ TC (T ) and y ∈ TC (T ). If x, y ∈ TC (T ), Brill et al (2020) showed that MoV TC (x, T ) = −1 = MoV TC (y, T ). If x ∈ TC (T ) and y ∈ TC (T ), then MoV TC (x) > 0 > MoV TC (y).…”

“…For instance, Fey (2008) showed that the top cycle, the uncovered set, and the Banks set are likely to include all alternatives in a large random tournament. To address this issue, we recently introduced the notion of margin of victory (MoV) for tournaments (Brill, Schmidt-Kraepelin, and Suksompong 2020). The MoV of a winner is defined as the minimum number of pairwise comparisons that need to be reversed in order for the winner to drop out of the winner set.…”

“…1 In addition to refining tournament solutions, the notion also has a natural interpretation in terms of bribery and manipulation: the MoV of an alternative reflects the cost of bribing voters or manipulating match outcomes so that the status of the alternative changes with respect to the winner set. For a number of common tournament solutions, we studied the complexity of computing the MoV and provided bounds on its values for both winners and non-winners (Brill et al 2020).…”