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

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.