One-Sided Matching Markets with Endowments: Equilibria and Algorithms

TL;DR

This paper studies the existence and computation of approximate equilibria in one-sided matching markets with endowments, introducing algorithms and showing limitations for certain utility functions.

Contribution

It introduces the $psilon$-approximate ADHZ model, proves equilibrium existence under linear utilities, and provides polynomial algorithms for specific utility cases.

Findings

Existence of equilibrium under linear utilities with desirable properties.

Polynomial-time algorithms for $psilon$-approximate equilibria in dichotomous and bi-valued utilities.

Counterexample showing non-existence of equilibrium in some cases.

Abstract

The Arrow-Debreu extension of the classic Hylland-Zeckhauser scheme for a one-sided matching market -- called ADHZ in this paper -- has natural applications but has instances which do not admit equilibria. By introducing approximation, we define the $\epsilon$-approximate ADHZ model, and we give the following results. * Existence of equilibrium under linear utility functions. We prove that the equilibrium satisfies Pareto optimality, approximate envy-freeness, and approximate weak core stability. * A combinatorial polynomial-time algorithm for an $\epsilon$-approximate ADHZ equilibrium for the case of dichotomous, and more generally bi-valued, utilities. * An instance of ADHZ, with dichotomous utilities and a strongly connected demand graph, which does not admit an equilibrium. Since computing an equilibrium for HZ is likely to be highly intractable and because of the difficulty of extending HZ to more general utility functions, Hosseini and Vazirani proposed (a rich collection of) Nash-bargaining-based matching market models. For the dichotomous-utilities case of their model linear Arrow-Debreu Nash bargaining one-sided matching market (1LAD), we give a combinatorial, strongly polynomial-time algorithm and show that it admits a rational convex program.