We consider a collective choice problem in which the number of alternatives and the number of voters vary. Two fundamental axioms of consistency in such a setting, reinforcement and composition-consistency, are incompatible. We first observe that the latter implies four conditions each of which can be formulated as a consistency axiom on its own right. We find that two of these conditions are compatible with reinforcement. In fact, one of these, called composition-consistency with respect to non-clone winners, turns out to characterize a class of scoring rules which contains the Plurality rule. When combined with a requirement of monotonicity, composition-consistency with respect to non-clone winners uniquely characterizes the Plurality rule. A second implication of composition-consistency leads to a class of scoring rules that always select a Plurality winner when combined with monotonicity.

中文翻译：

---

评分规则的一致性：重新研究组合一致性

我们考虑一个集体选择问题，其中备选方案的数量和选民的数量各不相同。在这样的环境中，强化和组合一致性这两个基本的一致性公理是不相容的。我们首先观察到后者包含四个条件，每个条件都可以单独表述为一致性公理。我们发现其中两个条件与强化相容。事实上，其中一个被称为非克隆获胜者的组合一致性，结果证明了一类包含 Plurality 规则的评分规则。当与单调性要求相结合时，非克隆获胜者的组成一致性是复数规则的独特特征。