This paper considers voting rules for two alternatives, viz. the simple majority rule of K.O. May, the majority rule with bias of Fishburn and others, and the majority rule by difference of votes of Goodin and List, and of Llamazares. These all have been characterized axiomatically for profiles of fixed length, that is, for a fixed population. The aim of this paper is to study analogs of these results in the situation where various populations are considered and disjoint populations can be combined into one population. The effect of this shift of focus is that now the domain of the rule consists of all finite nonempty sequences of votes. Young (1974) introduced the axiom of consistency, by which two populations can be combined into one as long as they agree on the output of the voting rule. We use a version of this axiom as given by Roberts (1991). Our paper can be seen as making a strong case for this simple, and natural axiom.

In the simple majority rule as well as the majority rule by difference of votes, the outcome of a tie is undecided, that is, both alternatives are in the output. In the majority rule with bias, ties are broken. In our case of profiles of variable lengths there are reasons to distinguish between strong bias (a tie is always broken) and weak bias (in certain cases a tie is not yet broken). The case of weak bias fits nicely in the context of a new characterization of simple majority rule that follows from previous results of the authors.

本文考虑了两种替代方案的投票规则。KO May的简单多数票制，Fishburn等人偏见的多数票制以及Goodin和List和Llamazares票数的不同决定了多数票制。对于固定长度的轮廓，即对于固定的人口，这些都是公理化的特征。本文的目的是在考虑各种人群并且不相交的人群可以组合为一个人群的情况下研究这些结果的类似物。焦点转移的结果是，规则的域现在由所有有限的非空票序列组成。Young（1974）引入了一致性公理，只要他们同意投票规则的输出，就可以将两个总体合并为一个。我们使用罗伯茨（1991）给出的该公理的一个版本。

在简单多数规则以及通过票数差的多数规则中，平局的结果不确定，也就是说，两种选择都在输出中。在有偏见的多数统治下，纽带破裂。在我们的可变长度轮廓情况下，有理由区分强偏差（始终打断领带）和弱偏差（在某些情况下尚未打结）。从作者先前的结果得出的对简单多数规则的新描述中，弱偏置的情况非常合适。