Refining Tournament Solutions via Margin of Victory

TL;DR

This paper introduces a framework for refining tournament solutions by measuring the margin of victory, which quantifies the robustness of alternatives against pairwise comparison reversals, enhancing discriminative power.

Contribution

It proposes the concept of margin of victory for tournament solutions, analyzes its computational complexity, and provides bounds, improving the ability to distinguish between winners and non-winners.

Findings

MoV measures robustness of alternatives in tournament solutions.

Complexity results for computing MoV for common solutions.

Bounds on MoV for winners and non-winners.

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.