# Soft Tournament Equilibrium: Differentiable Set-Valued Inference for Non-Transitive Pairwise Comparisons

arXiv: https://arxiv.org/abs/2604.04328 · HTML: https://mlsanity.com/paper/2604.04328/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 23 of its 88 references that are in the map.
- First posted on arXiv 6 Apr 2026.

## Builds on (most cited first, 23 of 23)

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- [ELSPR: Evaluator LLM Training Data Self-Purification on Non-Transitive Preferences via Tournament Graph Reconstruction](https://mlsanity.com/paper/2505.17691/) (2025)
- [Collecting, Classifying, Analyzing, and Using Real-World Elections](https://mlsanity.com/paper/2204.03589/) (2022)
- [Soft Condorcet Optimization for Ranking of General Agents](https://mlsanity.com/paper/2411.00119/) (2024)
- [Explaining Tournament Solutions with Minimal Supports](https://mlsanity.com/paper/2509.09312/) (2025)
- [On the Structure of Stable Tournament Solutions](https://mlsanity.com/paper/2004.01651/) (2020)
- [Characterizing the Top Cycle via Strategyproofness](https://mlsanity.com/paper/2108.04622/) (2021)

## Abstract (arXiv)

Soft Tournament Equilibrium (STE) is a differentiable layer for Top-Cycle (TC) and Uncovered-Set (UC) inference from reciprocal pair probabilities. Normalized log-sum-exp reachability and covering give smooth scores with approximation, perturbation, and margin-recovery bounds. We distinguish structural supervision, posterior uncertainty, and the final set decision through controlled synthetic studies and reconstruction of recorded ordinal profiles. Matched-epoch training improves F1 under a common structural readout, but a separate equal-budget soft-UC comparison does not demonstrate an advantage. A prospective equal-budget hard-UC study improves selective F1 at 24 alternatives from 0.6327/0.6292 for ordinary/relational native heads to 0.6676, while exact recovery remains only 1.16%. On 36 human profiles held out by source, a separate exploratory comparison favors Jeffreys posterior inference over learned independent-edge and mixture distributions in native expected-F1 decoding (0.8742 versus 0.8704/0.8366). Only three references are non-singleton selective cores. Completion certificates and full-depth TC diagnostics clarify additional identification, smoothing, and computational limits. The evidence supports conditional synthetic overlap gains, while preserving the soft-set null and the absence of a demonstrated learning advantage over strong count-based human-profile controls.