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A new proof of Smithson's fixed point theorem for order preserving multifunctions

Haruki Kono, Mark Voorneveld

TL;DR

The paper addresses Smithson's fixed-point theorem for order-preserving multifunctions on a partially ordered set and the errors in its original proof. It presents a concise, direct proof using a Zorn's lemma-based construction of maximal isotone selections and derives a fixed point under the stated assumptions. It then exposes the flaw in Smithson's argument via a counterexample where the supremum of a maximal chain in a maximal subset need not lie in that subset. The results reinforce the reliability of monotone fixed-point arguments in economic theory, supporting monotone comparative statics in contexts like Nash equilibria with strategic complementarities and dynamic economies.

Abstract

Fixed point theorems are ubiquitous in economic research. Many studies cite Smithson (1971) ``Fixed points of order preserving multifunctions,'' yet the original proof contains errors. This note presents a new, concise proof and explains why Smithson's argument is invalid.

A new proof of Smithson's fixed point theorem for order preserving multifunctions

TL;DR

The paper addresses Smithson's fixed-point theorem for order-preserving multifunctions on a partially ordered set and the errors in its original proof. It presents a concise, direct proof using a Zorn's lemma-based construction of maximal isotone selections and derives a fixed point under the stated assumptions. It then exposes the flaw in Smithson's argument via a counterexample where the supremum of a maximal chain in a maximal subset need not lie in that subset. The results reinforce the reliability of monotone fixed-point arguments in economic theory, supporting monotone comparative statics in contexts like Nash equilibria with strategic complementarities and dynamic economies.

Abstract

Fixed point theorems are ubiquitous in economic research. Many studies cite Smithson (1971) ``Fixed points of order preserving multifunctions,'' yet the original proof contains errors. This note presents a new, concise proof and explains why Smithson's argument is invalid.
Paper Structure (3 sections, 1 theorem, 3 equations)

This paper contains 3 sections, 1 theorem, 3 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Smithson's fixed point theorem

Key Result

Theorem 1

Let $(X, \leq)$ be a partially ordered set and $F: X \rightrightarrows X$ a multifunction. Assume Then $F$ has a fixed point, i.e., $x^* \in F(x^*)$ for some $x^* \in X$.

Theorems & Definitions (2)

  • Theorem 1
  • proof