Much of the research on elections focuses on selecting a single winner or having a tied set of winners to be thinned to a single winner by some tie-breaking procedure. There is also substantial research interest in, and a real-world need for, electing fixed-size committees, and this paper is in that stream of work.

In particular, this paper is inspired by the existing notions of minisum and minimax committee elections, defined over approval ballots (that is, ballots in which each candidate is approved or disapproved). Minisum declares a committee of the right size to be a winning committee if the sum over all votes of the Hamming distance between the vote and the committee, with each of those viewed in the natural way as a 0-1 vector, is minimized. Minimax is the same, except one wants to minimize the maximum over all votes of the Hamming distance between the vote and the committee.

This paper extends this approach to voting types other than approval ballots. In particular, approval ballots are, in effect, range voting with two choices, and this paper studies range voting with three choices (trichotomous voting). It also studies the cases of voting by complete linear orders and of voting by incomplete linear orders. In each of these settings, for minisum and minimax committee elections, the paper suggests a distance measure to be used.

This paper is a short one, and it is preparing the field for further research in this direction. Some of its choices are provocative; for example, for the case of complete linear orders, it suggests the use of a ranksum distance. That choice is, arguably of necessity, taking ordinal position as specifying proportionally the degree of preference; doing that is considered a cardinal sin by some. Such charged issues will make whatever future discussions and research follow from the definitions and questions that this valuable paper presents all the more interesting.