Classification theorem for smooth social choice on a manifold
The author studies a classification theorem for smooth social choice on a policy space, or manifold. By smooth social choice is meant a social preference relation, \(\sigma\) (u), obtained from a smooth preference profile, u, for the society, where \(\sigma\) (u) can be represented by piecewise differentiable trajectories in W. It is proved that the stability dimension, \(v^*(\sigma)\), of a general voting rule, \(\sigma\), without vetoers, is precisely v(\(\sigma)\)-2. In particular, in dimension at least v(\(\sigma)\)-1 it is possible to construct structurally stable cycles, in the sense that the cycles cannot be destroyed by small perturbations of preference. The second result of the paper is to obtain a lower bound on the instability dimension. In fact it is proved that if \(\sigma\) is an anonymous q rule (where any coalition of size q out of n players is decisive), and the dimension of the policy space is not greater than (n- 2)/(n-q) then the core may exist in a structurally stable fashion. It is then a simple corollary that: for such a rule the instability dimension \(w^*(\sigma)\) exceeds (n-2)/(n-q). These two results are particularly important in the classification of voting rules. The paper also reviews recent analyses of voting rules on a finite set of alternatives.
scientific article; zbMATH DE number 3878618
scientific article