Veto theorems with expansion consistency conditions and without the weak Pareto principle
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 conclusion is strengthened if a positive responsiveness (or monotonicity) condition replaces WNI: in which case there exists a quasi-dictator, i.e. a vector \(i\in N\) with the additional property that \(xP_ iy\) and \(xR_ jy\) for some \(j\neq i\) implies xPy. The paper also investigates consequences of requiring that no suboptimal alternative is socially preferenced to any optimal one (which is weaker than SD), or of dropping any Pareto optimality condition. ``Possibility results'' are shown to be expected only when the NN condition also is dropped. These conclusions actually turn out to be corollaries of a theorem by \textit{G. Bordes} [Rev. Econ. Stud. 43, 451-457 (1976; Zbl 0361.90003)].
scientific article; zbMATH DE number 4097069