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

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  • Frank M. V. Feys

Abstract

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.

Suggested Citation

  • Frank M. V. Feys, 2026. "An Axiomatic Theory of Tie-Breaking: Impossibility, Characterization, and Decomposition," Papers 2605.22846, arXiv.org.
    • Handle: RePEc:arx:papers:2605.22846
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    References listed on IDEAS

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    1. Michele Gori, 2014. "Selecting anonymous, neutral and reversal symmetric minimal majority rules," Working Papers - Mathematical Economics 2014-04, Universita' degli Studi di Firenze, Dipartimento di Scienze per l'Economia e l'Impresa.
    2. Daniela Bubboloni & Michele Gori, 2021. "Breaking ties in collective decision-making," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 44(1), pages 411-457, June.
    3. Bubboloni, Daniela & Gori, Michele, 2016. "Resolute refinements of social choice correspondences," Mathematical Social Sciences, Elsevier, vol. 84(C), pages 37-49.
    4. László Csató, 2017. "On the ranking of a Swiss system chess team tournament," Annals of Operations Research, Springer, vol. 254(1), pages 17-36, July.
    5. Daniela Bubboloni & Michele Gori, 2014. "Anonymous and neutral majority rules," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 43(2), pages 377-401, August.
    6. Gibbard, Allan, 1977. "Manipulation of Schemes That Mix Voting with Chance," Econometrica, Econometric Society, vol. 45(3), pages 665-681, April.
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