# Characterizing Necessary Losers to Explain Tournaments Solutions

arXiv: https://arxiv.org/abs/2608.23446 · HTML: https://mlsanity.com/paper/2608.23446/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 7 of its 34 references that are in the map.
- First posted on arXiv 24 Aug 2026.

## Builds on (most cited first, 7 of 7)

- [Explainable AI is Dead, Long Live Explainable AI! Hypothesis-driven decision support](https://mlsanity.com/paper/2302.12389/) (2023)
- [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)
- [Abductive and Contrastive Explanations for Scoring Rules in Voting](https://mlsanity.com/paper/2408.12927/) (2024)
- [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)
- [Explaining Tournament Solutions with Minimal Supports](https://mlsanity.com/paper/2509.09312/) (2025)

## Abstract (arXiv)

We study the problem of formally explaining why a candidate was not selected by a given tournament rule, by identifying sub-tournaments in which the candidate loses independently of how the rest of the tournament is completed. We define destructive minimal supports as any minimal sub-tournament satisfying this property, which in formal explainable artificial intelligence corresponds to abductive explanations for the question "Why does the loser lose the tournament?". For six common tournament solutions (maximin, uncovered set and its weighted variant, top cycle, Copeland, and Borda) we provide characterizations of when a candidate is either a necessary loser or a possible winner, we determine the size of the smallest destructive minimal supports, complemented by polynomial-time algorithms for their computation except for the case of Borda and Copeland rules which we conjecture to also be polynomial.