🤖 AI Summary
This study addresses the problem of fairly breaking ties in ranking procedures under natural axiomatic constraints. By leveraging algebraic structures—specifically group actions, stabilizer subgroups, and orbit partitions—the authors develop an axiomatic framework to formally model tie-breaking mechanisms. Their main contributions include proving the incompatibility between anonymity and strict ranking rules, establishing that orbit partitioning is the unique tie-breaking method satisfying the proposed natural axioms, and introducing a general decomposition theorem that, for the first time, unifies and characterizes the intrinsic structure of various real-world ranking rules within a single theoretical framework.
📝 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.