Some Further Properties of the Cumulative Offer Process

We investigate extension/resource monotonicity and respecting improvements properties of the cumulative offer process (COP). Extension monotonicity says that no doctor is to be better off whenever others start preferring more contracts to being unmatched. Resource monotonicity, on the other hand, requires that no doctor becomes worse off whenever hospitals start hiring more doctors. The COP becomes extension and resource monotonic whenever contracts are unilateral substitutes (US) satisfying an irrelevance of rejected contracts condition (IRC). This result implies that the COP is population monotonic under US and the IRC. We then turn to the respecting improvements property, which states that no doctor should be harmed if some of his contracts become more popular. With an additional law of aggregate demand condition, we show that the COP respects improvements.