Discussion of Aleskerov’s Paper

To give some background to this paper it is worth briefly reviewing some of the Arrovian results relating rationality principles of a social choice function, σ, to the nature of the ‘decisiveness structure’ Dσ defined by σ. (A coalition M is in Dσ if the coalition can get its way under σ when its members agree.) Assuming the usual Pareto and universal domain conditions, the theory focused on constraints on Dσ sufficient to restrict the image, Im(σ), of σ in an appropriate way. For example, suppose that one wishes σ to have as image the class of acyclic relations (with domain the acyclic profiles). Then a necessary condition is that Dσ be a prefilter: essentially this means the collegium K(Dσ) = ∩ {M: M ∈ Dσ} is non-empty (Brown, 1973). On the other hand if σ is to have as image the strict partial orders, then the collegium must be decisive, that is an oligarchy. If σ is to have as image the weak orders then Dσ must be a fixed filter, which in the case of a finite society means there is a dictator (Kirman and Sondermann, 1972; Arrow, 1951). Note these need not be sufficient conditions: however if σ is simple, defined entirely by Dσ, then these restrictions on Dσ are also sufficient to constrain the image as required.