On relationships between substitutes conditions

In the matching with contract literature, three well-known conditions (from stronger to weaker): substitutes, unilateral substitutes (US), and bilateral substitutes (BS) have proven very critical both in theory and practice. This paper aims to deepen our understanding of them by separately axiomatizing the gap between BS and the other two. We first introduce a new "doctor separability" condition (DS) and show that BS, DS, and irrelevance of rejected contracts (IRC) are equivalent to US and IRC. Due to Hatfield and Kojima (2010) and Ayg�ün and S�önmez (2012), we know that US, "Pareto separability" (PS), and IRC are the same as substitutes and IRC. This along with our result implies that BS, DS, PS, and IRC are equivalent to substitutes and IRC. All of these results are given without IRC whenever hospitals have preferences.