Explaining Tournament Solutions with Minimal Supports

2025-09-11

AI Summary

This paper addresses the problem of winner explainability in tournaments: given a candidate and a voting rule, it seeks a minimal, verifiable supporting subtournament—i.e., a smallest subset of pairwise comparisons whose outcomes guarantee the candidate’s victory under any completion of the remaining matches. It introduces the notion of “minimum support set” into tournament theory as a concise, formal abductive explanation. The framework covers major tournament rules—including top-cycle, uncovered set, Copeland, Borda, maximin, and weighted uncovered set. Polynomial-time algorithms are devised for all rules except the weighted uncovered set, which is shown to be NP-complete; tight bounds on the size of minimum support sets are also established. Experiments demonstrate that the method produces compact, certifiable, and intuitively interpretable explanations. This work advances both the theoretical foundations and practical deployment of explainable AI in pairwise comparison settings.

Technology Category

• Machine Learning: Learning Preferences or Rankings
• Game Theory and Economic Paradigms: Social Choice / Voting
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

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 but 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.

Problem

• Explaining tournament winners with minimal sub-tournaments
• Identifying necessary winners under various tournament rules
• Computing certified explanations for tournament solutions

Innovation

• Minimal sub-tournaments as certified explanations
• Polynomial algorithms compute minimal supports
• NP-complete for weighted uncovered set