# Explaining Tournament Solutions with Minimal Supports

arXiv: https://arxiv.org/abs/2509.09312 · HTML: https://mlsanity.com/paper/2509.09312/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 38 references that are in the map.
- First posted on arXiv 11 Sep 2025.

## Builds on (most cited first, 12 of 12)

- [Explanation in Artificial Intelligence: Insights from the Social Sciences](https://mlsanity.com/paper/1706.07269/) (2017)
- [A Symbolic Approach to Explaining Bayesian Network Classifiers](https://mlsanity.com/paper/1805.03364/) (2018)
- [On The Reasons Behind Decisions](https://mlsanity.com/paper/2002.09284/) (2020)
- [Logic-Based Explainability in Machine Learning](https://mlsanity.com/paper/2211.00541/) (2022)
- [On Relating 'Why?' and 'Why Not?' Explanations](https://mlsanity.com/paper/2012.11067/) (2020)
- [Enabling the Digital Democratic Revival: A Research Program for Digital Democracy](https://mlsanity.com/paper/2401.16863/) (2024)
- [Justifying Social-Choice Mechanism Outcome for Improving Participant Satisfaction](https://mlsanity.com/paper/2205.15863/) (2022)
- [Axiomatic Characterisations of Sample-based Explainers](https://mlsanity.com/paper/2408.04903/) (2024)
- [Abductive and Contrastive Explanations for Scoring Rules in Voting](https://mlsanity.com/paper/2408.12927/) (2024)
- [Computational Complexity Characterization of Protecting Elections from Bribery](https://mlsanity.com/paper/2007.02533/) (2020)
- [Margin of Victory in Tournaments: Structural and Experimental Results](https://mlsanity.com/paper/2012.02657/) (2020)
- [Margin of Victory for Weighted Tournament Solutions](https://mlsanity.com/paper/2408.06873/) (2024)

## Built on by (most cited first, 2 of 2)

- [Soft Tournament Equilibrium: Differentiable Set-Valued Inference for Non-Transitive Pairwise Comparisons](https://mlsanity.com/paper/2604.04328/) (2026)
- [Characterizing Necessary Losers to Explain Tournaments Solutions](https://mlsanity.com/paper/2608.23446/) (2026)

## Abstract (arXiv)

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.