Attainable results in committee elections

Abstract

The committee election problem is to choose from a finite set S of candidates a nonempty subset T of committee members as the consequence of an election in which each voter expresses a preference for a candidate in S. Solutions of this problem can be modelled by functions which map each partition of 1 (i.e., normalized vote tallies of candidates who have been ordered canonically by tally) into a nonempty subset of positive integers (i.e., sizes of committees). To solve this problem, we recently described a parameterized voting scheme, the ratio-of-sums or ras p consensus rule, in which p controls the degree to which votes must be concentrated in elected committees. It is desirable to identify the attainable results of such rules so as to understand their properties and to facilitate their comparison. For all p, we characterize the attainable ras p results in the general case where the partition's parts are real, and in the special case where p as well as its parts are rational.