Social Choice and Resources Allocation : A Topological Perspective

Topological social choice is an area which has evolved rapidly in recent years. It involves the use of continuous or topological techniques to study what have come to be known as "Chichilnisky rules." The field has quickly moved from a separate and distinct area of social choice to one which is integrated into the mainstreams of both social choice theory and general equilibrium theory, and which has come to be a bridge between these previously disparate areas. One of the most striking results to emerge is Chichilnisky's result on the equivalence between Arrow's paradox, Chichilnisky's paradox, the existence of a competitive equilibrium in a general equilibrium model and the non-emptiness of the core: Arrow paradox <--> Chichilnisky paradox <--> Competitive equilibrium <--> Core. This shows that social choice is an integral part of the problem in economics and that at a deep level all forms of resource allocation are equivalent. The topological approach to social choice also leads to a set of results on incentive compatibility and manipulability. There is a result equivalent to the Gibbard-Satterthwaite result on manipulability: this result states that any social choice rule which is continuous and respects unanimity must admit a "strategic dictator", someone who has the ability to force whichever outcome he or she most wants, generally be misrepresentation (Chichilnisky, Chichilnisky and Heal, Koshevoy). There is also a very simple characterization of the conditions necessary and sufficient for non-constant manipulability of a continuous social choice rule as those which are locally constant or dictatorial (Chichilnisky and Heal). Finally, one can apply these techniques to the analysis of social choice with infinite populations and show that the rules which are possible in this case are limits of the preferences along sequences of voters which form ultrafilter (Chichilnisky and Heal)