Margin of Victory for Weighted Tournament Solutions

TL;DR

This work extends the margin of victory (MoV) concept from unweighted to weighted tournaments and analyzes three weighted tournament solutions: Borda (BO), Split Cycle (SC), and weighted Uncovered Set (wUC). It provides polynomial-time algorithms for destructive MoV across all three rules, while constructive MoV is polynomial for BO and NP-complete for SC and wUC, with corresponding hardness proofs and reductions. The study establishes monotonicity, cover-consistency, and the absence of degree-consistency for these MoV refinements, and derives tight, rule-specific MoV bounds. Empirical experiments across multiple generation models quantify how MoV behaves in practice, revealing that MOV can effectively refine winner sets, especially for SC and wUC, and highlighting implications for robustness and manipulation under weight-based voting settings.

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.

Figures (23)

• Figure 1: Counterexamples for $\mathsf{wUC}\xspace\not\subseteq\mathsf{SC}\xspace$ on the left and $\mathsf{SC}\xspace\not\subseteq\mathsf{wUC}\xspace$ on the right.
• Figure 2: MoV values of all alternatives $x\in\{a,b,c,d\}$ of the $10$-weighted tournament $T$ for each tournament solution $\mathsf{BO}$, $\mathsf{SC}$ and $\mathsf{wUC}$ together with possible minimum reversal sets.
• Figure 3: Multiedge construction for edges $e_1=(u,w)$ and $e_2=(u,w)$, with costs $c(e_1)=+1$ and $c(e_2)=-1$, illustrated by green, resp. red colour, and capacities $u(e_1)=7$ and $u(e_2)=3$.
• Figure 4: A 10-weighted tournament $T$ with $\mathsf{BO}\xspace(T)=\{a\}$. On the right is $T^R$ with minimum wCRS $R$, $R(b,c)=-2$, $R(a,c)=+2$, for $c$ corresponding to the flow in \ref{['fig:IllustrationConstructionBordaExample_ReversalFlow']}.
• Figure 5: Illustration of $G_{c,17}$ for $T$ in \ref{['fig:IllustrationConstructionBorda1']}. s The edges $e_{\textcolor{red}{red}}$ are drawn in red, resp. $e_{\textcolor{ForestGreen}{green}}$ in green.

Theorems & Definitions (69)

• Example 1
• Definition 2.1
• Example 2
• Theorem 3.1
• proof
• proof
• Theorem 3.2
• proof
• Definition 3.3
• Example 3