Margin of Victory in Tournaments: Structural and Experimental Results

Abstract

Tournament solutions are standard tools for identifying winners based on pairwise comparisons between competing alternatives. The recently studied notion of margin of victory (MoV) offers a general method for refining the winner set of any given tournament solution, thereby increasing the discriminative power of the solution. In this paper, we reveal a number of structural insights on the MoV by investigating fundamental properties such as monotonicity and consistency with respect to the covering relation. Furthermore, we provide experimental evidence on the extent to which the MoV notion refines winner sets in tournaments generated according to various stochastic models.

Margin of Victory in Tournaments: Structural and Experimental Results

“…In particular, even though the problem is computationally hard for MoV k-kings for any constant k ≥ 4 (Brill et al 2020b), there exists an efficient heuristic that correctly computes the MoV value in most cases. In the full version (Brill et al 2020a), we give an example showing that the heuristic is not always correct. More precisely, for any positive integer , we construct a tournament such that {MoV TC (x, T ) | x ∈ V (T )} contains the values 1, 2, .…”

“…For two representative stochastic models (uniform random and urn), Figure 2 depicts the average size of the set of alternatives with maximum MoV value, and Figure 3 shows the average number of unique MoV values. Results on the other four models (impartial culture, Mallows, and two variants of Condorcet noise) can be found in the full version (Brill et al 2020a).…”

“…In the full version (Brill et al 2020a), we show that neither monotonicity nor transfer-monotonicity can be dropped from the condition of Lemma 4. This also means that neither of the two properties implies the other.…”

“…All omitted proofs can be found in the full version of this paper (Brill, Schmidt-Kraepelin, and Suksompong 2020a).…”

“…We used six stochastic models to generate preferences: the uniform random model (which was used in Section 3.3), two variants of the Condorcet noise model (with and without voters), the impartial culture model, the Pólya-Eggenberger urn model, and the Mallows model. Detailed descriptions of these models can be found in the full version of our paper (Brill et al 2020a).…”