A Descending Price Auction for Matching Markets

TL;DR

This paper introduces a polynomial-time descending-price auction algorithm for finding the maximum market-clearing price vector in unit-demand matching markets, leveraging combinatorial structure without LPs.

Contribution

It presents a novel strongly polynomial algorithm that efficiently computes the maximum MCP using combinatorial methods and skewness-based selection, avoiding linear programming.

Findings

Algorithm runs in $O(m^4)$ time

Converges to maximum MCP in $O(m^2)$ steps

Uses skewness function for efficient item selection

Abstract

This work presents a descending-price-auction algorithm to obtain the maximum market-clearing price vector (MCP) in unit-demand matching markets with m items by exploiting the combinatorial structure. With a shrewd choice of goods for which the prices are reduced in each step, the algorithm only uses the combinatorial structure, which avoids solving LPs and enjoys a strongly polynomial runtime of $O(m^4)$. Critical to the algorithm is determining the set of under-demanded goods for which we reduce the prices simultaneously in each step of the algorithm. This we accomplish by choosing the subset of goods that maximize a skewness function, which makes the bipartite graph series converges to the combinatorial structure at the maximum MCP in $O(m^2)$ steps. A graph coloring algorithm is proposed to find the set of goods with the maximal skewness value that yields $O(m^4)$ complexity.