# An Axiomatic Theory of Tie-Breaking: Impossibility, Characterization, and Decomposition

arXiv: https://arxiv.org/abs/2605.22846 · HTML: https://mlsanity.com/paper/2605.22846/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 15 May 2026.

## Built on by (most cited first, 1 of 1)

- [Cross-View Correspondence Is a Measurement Intervention: Two-Sided Validation for Agent Evaluation and Credit Assignment](https://mlsanity.com/paper/2608.17713/) (2026)

## Abstract (arXiv)

We develop an abstract axiomatic theory of tie-breaking. A tie-breaking input consists of a finite set N of players, a weak order on N representing the standings to be refined, and an auxiliary information item drawn from a set on which the symmetric group Sym(N) acts. Within this minimal framework we prove three theorems. First, no tie-breaking rule producing a strict linear order can be anonymous, provided the input space contains even one intrinsically symmetric situation, a condition met in essentially every realistic application. Second, when we allow the rule to output a partition of N (rather than a strict ranking), there is a unique rule satisfying two natural axioms: it is the partition of N into orbits of the joint stabilizer of the input. Third, every reasonable strict tie-breaking rule decomposes uniquely as the canonical orbit partition followed by an arbitrary completion. The decomposition makes precise the informal observation that real tie-breaking systems are honest until forced to be arbitrary. The framework is broad enough to capture chess tournament tie-breakers, sports league regulations, voting tie-breakers, tie-breaking among symmetric players in cooperative games, and ranking by network centrality measures, all within a single uniform formalism.