A leximin characterization of strategy-proof and non-resolute social choice procedures (Q1852663)

A social choice function associates a nonempty subset of the set of alternatives to a profile of individual linear orderings. Strategy-proofness is adapted to this set-function framework. It is shown that for a strategy-proof set-function, there is a positive integer \(k\) such that either (i) the selected subsets for all profiles have the same cardinality \(k\) and there is an individual \(i\) such that this selected subset of \(k\) alternatives coincide with his \(k\) highest ranked alternatives, or (ii) all subsets of cardinality 1 to \(k\) are chosen and there is a group of \(k\) individuals such that the selected subset is the union of the alternatives ranked first by these individuals. There does not exist a strategy-proof set-function such that the selected subsets are all of cardinality \(k^*\) with \(1< k^*< k\).