Majority rule for profiles of arbitrary length, with an emphasis on the consistency axiom

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.

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