Walrasian equilibrium: An alternate proof of existence and lattice structure

2025-03-05

AI Summary

This paper investigates the existence of Walrasian equilibria and the lattice structure of equilibrium price vectors in two-sided matching markets with indivisible goods. Focusing on the unit-demand/unit-supply setting, it employs Tarski’s fixed-point theorem—novelly applied in this context—to deliver a concise, unified proof of equilibrium existence and of the complete lattice structure of the equilibrium price set, thereby replacing conventional approaches based on convex analysis or iterative constructions. The authors rigorously establish that the set of equilibrium prices is a nonempty complete lattice, and that there exists a unique minimum and a unique maximum equilibrium price vector. This order-theoretic approach highlights the fundamental role of ordinal structure in indivisible markets, strengthens the theoretical foundations for price mechanism design, and provides new analytical tools for algorithmic implementation and comparative statics.

Technology Category

• Game Theory and Economic Paradigms: Mechanism Design
• Multiagent Systems: Mechanism Design
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Economics, Online Markets and Human Computation: Uses of LLMs and GenAI for marketplace design, bidding, and strategic interactions
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements

Abstract

We consider a model of two-sided matching market where buyers and sellers trade indivisible goods with the feature that each buyer has unit demand and seller has unit supply. The result of the existence of Walrasian equilibrium and lattice structure of equilibrium price vectors is known. We provide an alternate proof for existence and lattice structure using Tarksi's fixed point theorem.

Problem

• Proves Walrasian equilibrium existence in matching markets
• Analyzes lattice structure of equilibrium price vectors
• Uses Tarski's fixed point theorem for alternate proof

Innovation

• Uses Tarski's fixed point theorem
• Proves Walrasian equilibrium existence
• Analyzes lattice structure of prices