On Clearing Prices in Matching Markets: A Simple Characterization without Duality

TL;DR

This paper characterizes market-clearing prices in matching markets without relying on duality, revealing their convexity and their role in inducing maximum matchings.

Contribution

It provides a novel, duality-free characterization of market-clearing prices, highlighting their convex structure and connection to maximum matchings.

Findings

Market-clearing prices form a convex and closed set.

Any such prices induce all maximum matchings.

The approach bypasses traditional duality methods.

Abstract

Duality of linear programming is a standard approach to the classical weighted maximum matching problem. From an economic perspective, the dual variables can be regarded as prices of products and payoffs of buyers in a two-sided matching market. Traditional duality-based algorithms, e.g., Hungarian, essentially aims at finding a set of prices that clears the market. Under such market-clearing prices, a maximum matching is formed when buyers buy their most preferred products respectively. We study the property of market-clearing prices without the use of duality, showing that: (1) the space of market-clearing prices is convex and closed under element-wise maximum and minimum operations; (2) any market-clearing prices induce all maximum matchings.