Explaining Tournament Solutions with Minimal Supports

TL;DR

This work identifies minimal supports, minimal sub-tournaments in which the candidate is guaranteed to win regardless of how the rest of the tournament is completed, and shows how minimal supports can serve to produce compact, certified, and intuitive explanations for tournament solutions.

Abstract

Tournaments are widely used models to represent pairwise dominance between candidates, alternatives, or teams. We study the problem of providing certified explanations for why a candidate appears among the winners under various tournament rules. To this end, we identify minimal supports, minimal sub-tournaments in which the candidate is guaranteed to win regardless of how the rest of the tournament is completed (that is, the candidate is a necessary winner of the sub-tournament). This notion corresponds to an abductive explanation for the question,"Why does the winner win the tournament?", a central concept in formal explainable AI. We focus on common tournament solutions: the top cycle, the uncovered set, the Copeland rule, the Borda rule, the maximin rule, and the weighted uncovered set. For each rule we determine the size of the smallest minimal supports, and we present polynomial-time algorithms to compute them for all solutions except for the weighted uncovered set, for which the problem is NP-complete. Finally, we show how minimal supports can serve to produce compact, certified, and intuitive explanations for tournament solutions.

Figures (5)

• Figure 1: A tournament and a 5-weighted tournament.
• Figure 2: MSs for $a\in\mathop{\mathrm{UC}}\nolimits(G)$ from Figure \ref{['fig:exunw']}.
• Figure 3: MSs for $a\in\mathop{\mathrm{MM}}\nolimits(G_w)$ from Figure \ref{['fig:exw']}.
• Figure 4: From a tournament (a) and a weighted tournament (e), we compute a smallest minimal support for $a\in\mathop{\mathrm{UC}}\nolimits(G)$ (b) and for $a\in\mathop{\mathrm{MM}}\nolimits(G)$ (f). We identify and visualize their underlying structure as a $a$-rooted tree (c) or by mean of the out-going edges in the neighborhood of the winning candidate and the in-going edges in those of the losing candidates (g). Finally, we produce textual explanations based on these structures (d) and (h). Examples for the remaining tournament solutions are available in Section \ref{['sec:app_expl']} of the Appendix.
• Figure :

Theorems & Definitions (90)

• Example 1
• Definition 1
• Definition 2
• Example 2
• Example 3
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• Example 4
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• Example 5
• Definition 5