Consistent social choice functions and systems of distinct representatives
type: Item
instance of: scholarly article
title: Consistent social choice functions and systems of distinct representatives
zbMATH Open document ID: 0573.90003
DOI: 10.1016/0165-4896(85)90004-6
published in: Mathematical Social Sciences
publication date: 1985-01-01T00:00:00Z
review text: According to \textit{B. Peleg} [Econometrica 46, 153-161 (1978; Zbl 0373.90002)] a social choice function is exactly and strongly consistent if and only if for any profile of preferences \((R^ 1,R^ 2,..,R^ n)\) there is a profile \((Q^ 1,Q^ 2,..,Q^ n)\) such that the latter profile gives the same social outcome as \((R^ 1,R^ 2,...,R^ n)\) and no coalition can obtain a social choice preferable to each of its members if the rest of the group maintains the \(Q^ i\) preferences. The profile \((Q^ 1,Q^ 2,...,Q^ n)\) is called an equilibrium point of the social choice function relative to \((R^ 1,R^ 2,...,R^ n).\) Peleg has proposed several social choice functions which are exactly and strongly consistent. They are based on a procedure of successive elimination of most disliked alternatives [see also \textit{B. Peleg}, ''Game theoretic analysis of voting in committees'', chapter 5 (1984; Zbl 0563.90002)]. The present paper modifies Peleg's functions and describes them in terms of ''systems of distinct representatives''. Let \(N=\{1,2,...,n\}\) denote a set of n individuals. Let X denote a set of alternatives. Let L(X) denote the set of linear orders on X. A coalition is a subset of N. Definition 1: A constrained social choice function (CSCF) is a pair consisting of a domain \(D\subset L(X)\) and a function F from \(D^ N=D\times D\times...\times D\) to X. Definition 2: Let \(n\geq m-1\) and \(| X| =m\), \(\beta\) : \(X\to N\) be such that \(\sum_{x\in X}\beta (x)=n+1\). Then a point y is \(\beta\)-feasible relative to \((R^ 1,R^ 2,..,R^ n)\subset (L(x))^ N\) if and only if there exists a sequence \(C_ 1\), \(C_ 2,...,C_{m-1}\) of coalitions and a sequence \(x_ 1\), \(x_ 2,...,x_{m-1}\) of distinct alternatives such that (1) \(| C_ i| =\beta (x_ i)\) for each i; (2) for each i, for all \(j\in C_ i\), for all \(z\in X\setminus \{x_ 1,x_ 2,...,x_{i-1}\}\), \(zR^ jx_ i\); (3) for all \(i\neq j\), \(C_ i\cap C_ j=\emptyset\); (4) for all i, \(y\neq x_ i\). Here \(| C_ i|\) deno…
Mathematics Subject Classification ID: 91B14
Mathematics Subject Classification ID: 91B10
Mathematics Subject Classification ID: 91B06
zbMATH DE Number: 3915955
zbMATH Keywords: theory of voting
zbMATH Keywords: decision-making in committees
zbMATH Keywords: consistent social choice functions
zbMATH Keywords: exact and strong consistency
zbMATH Keywords: equilibrium point
zbMATH Keywords: systems of distinct representatives
author: Ki Hang Kim
author: Fred Roush
reviewed by: Wulf Gaertner
MaRDI profile type: Publication
full work available at URL: 0165-4896(85)90004-6
OpenAlex ID: W2067132329
cites work: Q5578815
cites work: On Nicely Consistent Voting Systems
cites work: Consistent Voting Systems
cites work: Q3851094
description: scientific article; zbMATH DE number 3915955