On Collective Rationality and a Generalized Impossibility Theorem.

The article shows that the conflict involving collective rationality goes much deeper than one might suppose on the basis of the impossibility theorems that focus on binary social choices. In effect, it is proved that generalized versions of the principal conditions used by researchers A. Mas-Colell and H. Sonnenschein are incompatible. The results are based on social choices from subsets of in alternatives rather than on social choices from subsets of two alternatives. It will be assumed throughout that the number n of voters is finite. To weaken one of the typical structural conditions slightly we shall assume later that all voters have linear preference orders on X, and, given this, adopt a universal domain condition which says that all n-tuples of linear preference orders on X are admissible or potential voter preference profiles. The article builds up to binary-choice impossibility theorems with special emphasis on notions of collective rationality. The collective-rationality idea along with other conditions to apply are generalized to m-alternative subsets of X, and states the new impossibility theorem. Proofs are given with conclusions in the final section.