A topological approach to the Arrow impossibility theorem when individual preferences are weak orders

Abstract

We will present a topological approach to the Arrow impossibility theorem of social choice theory that there exists no binary social choice rule (which we will call a social welfare function) which satisfies the conditions of transitivity, independence of irrelevant alternatives (IIA), Pareto principle and non-existence of dictator. Our research is in line with the studies of topological approaches to discrete social choice problems initiated by Baryshnikov [Y. Baryshnikov, Unifying impossibility theorems: a topological approach, 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 [Y. Baryshnikov, Unifying impossibility theorems: a topological approach, Advances in Applied Mathematics 14 (1993) 404–415]. Our main tools are homology groups of simplicial complexes. And we will consider the case where individual preferences are weak orders, that is, individuals may be indifferent about any pair of alternatives. This point is an extension of the analysis by Baryshnikov [Y. Baryshnikov, Unifying impossibility theorems: a topological approach, Advances in Applied Mathematics 14 (1993) 404–415].