On the equivalence of the Arrow impossibility theorem and the Brouwer fixed point theorem

摘要

We will show that in the case where there are two individuals and three alternatives (or under the assumption of free-triple property) the Arrow impossibility theorem [K.J. Arrow, Social Choice and Individual Values, second ed., Yale University Press, 1963] for social welfare functions that there exists no social welfare function which satisfies transitivity, Pareto principle, independence of irrelevant alternatives, and has no dictator is equivalent to the Brouwer fixed point theorem on a 2-dimensional ball (circle). Our study is an application of ideas by Chichilnisky [Economics Letters 3 (1979) 347–351] to a discrete social choice problem, and also it is in line with the work by Baryshnikov [Advances in Applied Mathematics 14 (1993) 404–415]. But tools and techniques of algebraic topology which we will use are more elementary than those in Baryshnikov [Advances in Applied Mathematics 14 (1993) 404–415].