Veto theorems with expansion consistency conditions and without the weak Pareto principle
type: Item
instance of: scholarly article
title: Veto theorems with expansion consistency conditions and without the weak Pareto principle
zbMATH Open document ID: 0669.90008
DOI: 10.1016/0165-4896(88)90030-3
author: Ruvin Gekker
published in: Mathematical Social Sciences
publication date: 1988-01-01T00:00:00Z
review text: Let X and N be two finite sets, respectively the set of social alternatives (\(| X| \geq 3)\) and the set of individuals (\(| N| \geq 2)\). To any individual \(i\in N\) is associated a binary preference relation \(R_ i\) on X which is reflexive complete and transitive (with asymmetric part \(P_ i)\). Given any profile of individual preferences \(u=(R_ i)_{i\in N}\) and any non-empty subset S of X a social choice function C gives one (and only one) non-empty subset \(C_ u(S)\) of S. A basic social choice preference R on X (reflexive and complete) is derived from C by letting: \(x,y\in X\); \(xRy\to x\in C_ u(\{x,y\})\) (with P the asymmetric part of R). In order for the social choice function to be selective over larger sets than pairs a non-null condition (NN) is imposed: for any profile \(u=(R_ i)_{i\in N}\) and any \(S\subset X\), \(| S| >2\to C_ u(S)\neq S.\) The familiar independence of irrelevant alternatives condition (I.I.A) is also required: for any two profiles \(u=(R_ i)_{i\in N}\) and \(v=(R_ i')_{i\in N}\) such that \((xR_ iy\to xR_ i'y\) and \(yR_ ix\to yR_ i'x)\) for all \(i\in N\), we have \(C_ v(\{x,y\})=C_ u(\{x,y\})\). Consider two more conditions: (1) weak non imposition (WNI): for any pair \(x,y\in X\) of alternatives there is a profile \(u=(R_ i)_{i\in N}\) such that \(x\in C_ u(\{x,y\});\) (2) strong dominance (SD): for every \(S\subset X\) and every profile \(u=(R_ i)_{i\in N}\) for any \(x,y\in S\), \(x\in C_ u(\{x,y\})\to (y\in C(S)\to x\in C(S)).\) The first is weaker than the standard (weak) Pareto optimality property (requiring \(y\in C_ u(S)\) for any profile such that \(xP_ 1y\) for any \(i\in N)\). The second condition introduces a choice-consistency requirement. The central result of the paper is that for a social choice function satisfyng NN, IIA, WNI and SD there exists a vector, namely an individual \(i\in N\) such that for any \(x,y\in X\) and any profile \(u=(R_ i)_{i\in N}\), \(xP_ iy\) implies \(x\in C_ v(\{x,y\})\). The conclu…
Mathematics Subject Classification ID: 91B14
zbMATH DE Number: 4097069
zbMATH Keywords: social choice function
zbMATH Keywords: independence of irrelevant alternatives
zbMATH Keywords: weak non imposition
zbMATH Keywords: strong dominance
zbMATH Keywords: vector
zbMATH Keywords: quasi-dictator
zbMATH Keywords: Possibility results
MaRDI profile type: Publication
full work available at URL: 0165-4896(88)90030-3
OpenAlex ID: W1977248806
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description: scientific article; zbMATH DE number 4097069