The connection between Arrow theorem and Sperner lemma

TL;DR

This paper establishes a direct proof that Arrow's theorem is equivalent to Sperner's lemma, and explores other statements also equivalent to Arrow's theorem, deepening the understanding of their logical connections.

Contribution

It provides a direct proof of the equivalence between Arrow's theorem and Sperner's lemma and identifies additional statements equivalent to Arrow's theorem.

Findings

Arrow's theorem is directly equivalent to Sperner's lemma.

Several other statements are shown to be equivalent to Arrow's theorem.

The paper clarifies the logical relationships among key theorems in social choice and topology.

Abstract

It is well known that Sperner lemma is equivalent to Brouwer fixed-point theorem. Tanaka [12] proved that Brouwer theorem is equivalent to Arrow theorem, hence Arrow theorem is equivalent to Sperner lemma. In this paper we will prove this result directly. Moreover, we describe a number of other statements equivalent to Arrow theorem.