Resolute refinements of social choice correspondences

Abstract

Many classical social choice correspondences are resolute only in the case of two alternatives and an odd number of individuals. Thus, in most cases, they admit several resolute refinements, each of them naturally interpreted as a tie-breaking rule, satisfying different properties. In this paper we look for classes of social choice correspondences which admit resolute refinements fulfilling suitable versions of anonymity and neutrality. In particular, supposing that individuals and alternatives have been exogenously partitioned into subcommittees and subclasses, we find out arithmetical conditions on the sizes of subcommittees and subclasses that are necessary and sufficient for making any social choice correspondence which is efficient, anonymous with respect to subcommittees, neutral with respect to subclasses and possibly immune to the reversal bias admit a resolute refinement sharing the same properties.

Breaking ties in collective decision-making

“…The concept of regular group works properly even to analyze consistency. Theorems 9 below is proved in "Appendix B" by widely extending the theory developed in Bubboloni and Gori (2016b) to the framework here considered.…”

“…The proof of the above result is formally the same as the one of Proposition 13 in Bubboloni and Gori (2016b) and so is omitted.…”

“…We recall a part of Proposition 24 in Bubboloni and Gori (2016b) which is interesting for our scope 20 : P U 2 = ∅ if and only if there exists (ϕ, ψ, ρ 0 ) ∈ U such that ψ is a conjugate of ρ 0 .…”

“…In the general case for the number of alternatives, some observations about different levels of anonymity and neutrality can be found in the paper by Kelly (1991). Bubboloni and Gori (2016b) consider, only for 1-sccs, weak versions of anonymity and neutrality, determined by partitions of individuals and alternatives. More recently, a wide literature about k-sccs which are representation-focused on some attributes of the alternatives (sex, age, profession, ethnicity) has been growing.…”

“…The techniques used in our paper are based on an algebraic approach focused on the theory of finite permutation groups and the notion of action of a group on a set. That approach was initially used by Egecioglu (2009) and Egecioglu and Giritligil (2013) for analyzing the set of preference profiles and later developed by Bubboloni and Gori (2014, 2016b to describe, in a unified framework, profiles and functions on them satisfying some symmetry properties. 9 We emphasize that, generalizing ideas and results in Gori (2015, 2016b) for managing the resolute refinements of spcs and k-scc is in no way a trivial adaptation of the results in there.…”