On isotopic dictators and homological manipulators

We refine the analysis of Chichilnisky [Chichilnisky, G., The topological equivalence between the Pareto condition and the existence of dictator, J. Math. Econ. 9 (1982) 223-233] and Koshevoy [Koshevoy, G., Homotopy Properties od Pareto Aggregation Rules, 1993, preprint] of the topology of continuous Pareto aggregation rules. We take as the primary notion of the equivalence between aggregation rules the isotopy between them, that is a continuous family of rules satisfying the same axioms which the terminal rules do. We prove that the homotopic equivalences of Pareto rules to dictatorial ones known before can be made isotopies and give some generalizations to these results. We discuss also some homological properties of Pareto rules and their implications. The main result of the paper, however, is the construction of a Pareto rule which is not Pareto isotopic to a dictatorial rule (while homologically equivalent to it). This is the first example of existence results for the preference spaces with nontrivial topology (without domain restrictions).