Social Choice and Game Theory: Recent Results with a Topological Approach

Abstract

This chapter presents a summary of recent results obtained in game and social choice theories, and highlights the application and the development of tools in algebraic topology. The purpose is expository: no attempt is made to provide complete proofs, for which references are given, nor to review the previous work in this area, which covers a significant subset of the economic literature.The use of topological tools has a long tradition in economic analysis, which goes back to the work of Von Neumann on balanced economic growth of 1937 and 1945. He proved a generalization of Brouwer's fixed point theorem that was the basis of Kakutani's theorem. In game theory and in general equilibrium market analysis, fixed point methods are the topological methods most frequently utilized to show existence of solutions. As a matter of fact, fixed point theorems are the largest part of applications of topology to economics as a whole. Instead of topological methods, social choice theory has traditionally been formulated in a combinatorial fashion, following the first formal works of Arrow (1951) and Black (1948) in this area.However, we shall now show that many problems of game and social choice theories, when properly formulated, exhibit an intrinsic topological structure that may be fruitfully examined with algebraic topology tools that go beyond fixed point theorems, such as homotopy and cohomology theories. This allows us to tap a wealth of existing topological techniques, as well as to develop new ones, to resolve problems in social choice and game theories.This chapter studies certain social choice paradoxes and their resolution, the relation of a fixed point theorem with the social choice paradox, the equivalence of the Pareto condition and the existence of a dictator, majority rules, aggregation in large economies, and the aggregation of Von Neumann-Morgenstern utilities for choices under uncertainty.Within game theory, we summarize results on the manipulation of games, the existence of Nash equilibrium of certain non-convex games and the fairness of these games, and the existence and characterization of strategy-proof games, a problem that appears in the literature on economies with public goods.