An Algorithm for the Assignment Game Beyond Additive Valuations

TL;DR

This paper introduces an efficient algorithm for computing competitive equilibria in assignment games that incorporate both imperfectly transferable utility and gross substitutes valuations, extending previous work to more complex market models.

Contribution

It presents the first efficient algorithm for assignment games with combined imperfectly transferable utility and gross substitutes valuations, using augmenting path and matroid intersection techniques.

Findings

The algorithm efficiently computes competitive equilibria in the new setting.

Computing competitive equilibrium becomes NP-hard in a generalized model.

The approach extends existing algorithms for special cases to more complex valuation settings.

Abstract

The assignment game, introduced by Shapley and Shubik (1971), is a classic model for two-sided matching markets between buyers and sellers. In the original assignment game, it is assumed that payments lead to transferable utility and that buyers have unit-demand valuations for the items being sold. Two important and mostly independent lines of work have studied more general settings with imperfectly transferable utility and gross substitutes valuations. Multiple efficient algorithms have been proposed for computing a competitive equilibrium, the standard solution concept in assignment games, in these two settings. Our main result is an efficient algorithm for computing competitive equilibria in a setting with both imperfectly transferable utility and gross substitutes valuations. Our algorithm combines augmenting path techniques from maximum matching and algorithms for matroid intersection. We also show that, in a mild generalization of our model, computing a competitive equilibrium is NP-hard.