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.

翻译：我们发展了一种关于打破平局的抽象公理化理论。打破平局的输入包含：一个有限玩家集合N，一个表示待细化的排名状态的N上的弱序，以及一个从对称群Sym(N)作用其上的集合中抽取的辅助信息项。在这一最小化框架下，我们证明了三项定理。第一，若输入空间中存在哪怕一个内在对称情形（该条件在几乎所有实际应用中均满足），则任何产生严格线性序的打破平局规则都无法满足匿名性。第二，当我们允许规则输出N的一个划分（而非严格排序）时，存在唯一满足两个自然公理的规则：该规则即为将N划分为输入联合稳定化子轨道的划分。第三，任何合理的严格打破平局规则均可唯一分解为先进行规范轨道划分，再任意完成排序。这一分解精确刻画了一个非正式观察：现实的打破平局系统在被迫任意选择之前始终是诚实的。该框架足够广泛，能够统一涵盖国际象棋锦标赛打破平局规则、体育联赛条例、投票打破平局规则、合作博弈中对称玩家的打破平局以及网络中心性度量排名，所有情形均在同一统一形式体系下。