Deferred Acceptance with Compensation Chains by Piotr Dworzak

 Here's an interesting look at deferred acceptance algorithms, published online early in Operations Research 

Deferred Acceptance with Compensation Chains  by Piotr Dworczak 

Published Online:18 Feb 2021https://doi.org/10.1287/opre.2020.2042

Abstract: I introduce a class of algorithms called deferred acceptance with compensation chains (DACC). DACC algorithms generalize the Gale–Shapley algorithm by allowing both sides of the market to make offers. The main result is a characterization of the set of stable matchings: a matching is stable if and only if it is the outcome of a DACC algorithm. The proof of convergence of DACC algorithms uses a novel technique based on a construction of a potential function.