Dictatorial domains
The well known Gibbard Satterthwaite (GS) theorem says that the only social choice function that is Pareto optimal and strategyproof is the dictatorial one, if there are at least 3 alternatives. It is also well known that if the domain of the social choice functions is restricted to, for instance, single peaked preferences, also non dictatorial choice functions may exist. The problem addressed in this paper is: how far can the universal domain of all possible profiles be restricted while still retaining the impossibility result of dictatorship. The main result of the paper is: if the domain \(D\) of strict orderings on \(A\) is linked (and there are at least 3 alternatives), then it is dictatorial, i.e., for all social choice functions \(f: D^{N} \rightarrow A\) satisfying unanimity, if \(f\) is strategyproof, then \(f\) is dictatorial. Linked domains are described as follows. Two alternatives \(a\) and \(b\) are said to be connected if there exists a strict ordering in \(D\) where \(a\) is ranked first and \(b\) second and another one where \(b\) is first and \(a\) is second. A domain \(D\) is linked if one can arrange all the alternatives in a sequence satisfying the following property: the second alternative is connected to the first and every alternative after the second is connected to at least two others before it in the sequence. The assumption that the domain is linked is significantly weaker than the assumption that the domain is universal. Hence, this theorem generalizes the GS theorem. The proof requires about 5 pages. The authors demonstrate the existence of a linked domain which has exactly \(4m - 6\) orderings, where \(m\) is the number of alternatives, while the universal domain is of size \(m!\). The authors apply their main result to a model of voting under constraints, due to Barbera, Sonnenschein and Zhou, intended to represent a situation where members are being elected to a club. A domain \(D\) has the unique seconds property if there exists a pair of alternatives \(a\), \(b\), such that for any preference ordering \(P\) in \(D\) with \(a\) ranked first in \(P\) it is the case that \(b\) is ranked second in \(P\). The domain of single peaked preferences satisfies this property. The authors show that if a domain satisfies the unique seconds property, then it is non-dictatorial. Next they conclude that for \(m = 3\), the domain \(D\) is dictatorial if and only if \(D\) is the set of all strict orderings on \(A\). Dictatorial domains in the case of \(m = 4\) are characterized in another paper.
scientific article; zbMATH DE number 1983470