Generalizing deferred acceptance in many to one matching with contracts, by Hatfield, Kominers and Westkamp in RESTUD

 Stability, Strategy-Proofness, and Cumulative Offer Mechanisms, by John William Hatfield, Scott Duke Kominers, Alexander Westkamp, The Review of Economic Studies, Volume 88, Issue 3, May 2021, Pages 1457–1502, https://doi.org/10.1093/restud/rdaa052

Abstract: We characterize when a stable and strategy-proof mechanism is guaranteed to exist in the setting of many-to-one matching with contracts. We introduce three novel conditions—observable substitutability, observable size monotonicity, and non-manipulability via contractual terms—and show that when these conditions are satisfied, the cumulative offer mechanism is the unique mechanism that is stable and strategy-proof (for workers). Moreover, we show that our three conditions are, in a sense, necessary: if the choice function of some firm fails any of our three conditions, we can construct unit-demand choice functions for the other firms such that no stable and strategy-proof mechanism exists. Thus, our results provide a rationale for the ubiquity of cumulative offer mechanisms in practice.