Margin of Victory for Weighted Tournament Solutions

TL;DR

This paper extends the concept of margin of victory to weighted tournament solutions, analyzing computational complexity and properties for Borda's rule, the weighted Uncovered Set, and Split Cycle.

Contribution

It introduces a generalized MoV for weighted tournaments and studies its computational tractability and axiomatic properties for three specific rules.

Findings

MoV is tractable for some rules, intractable for others.

Provides bounds on the MoV values for the studied rules.

Generalizes properties from unweighted to weighted tournaments.

Abstract

Determining how close a winner of an election is to becoming a loser, or distinguishing between different possible winners of an election, are major problems in computational social choice. We tackle these problems for so-called weighted tournament solutions by generalizing the notion of margin of victory (MoV) for tournament solutions by Brill et. al to weighted tournament solutions. For these, the MoV of a winner (resp. loser) is the total weight that needs to be changed in the tournament to make them a loser (resp. winner). We study three weighted tournament solutions: Borda's rule, the weighted Uncovered Set, and Split Cycle. For all three rules, we determine whether the MoV for winners and non-winners is tractable and give upper and lower bounds on the possible values of the MoV. Further, we axiomatically study and generalize properties from the unweighted tournament setting to weighted tournaments.