The blocking lemma and group incentive compatibility for matching with contracts

• We show that the blocking lemma for matchings with contracts holds.
• The preexisting blocking lemmas for matchings without contracts are special cases of our result.
• As an immediate consequence of the blocking lemma, we show that the doctor-optimal stable mechanism is group strategy-proof for doctors.

This paper considers a general class of two-sided many-to-one matching markets, so-called matching markets with contracts. We study the blocking lemma and group incentive compatibility for this class of matching markets. We first show that the blocking lemma for matching with contracts holds if hospitals’ choice functions satisfy substitutes and the law of aggregate demand. The blocking lemma for one-to-one matching (Gale and Sotomayor, 1985) and that for many-to-one matching (Martínez et al., 2010) are special cases of this result. Then, as an immediate consequence of the blocking lemma, we show that the doctor-optimal stable mechanism is group strategy-proof for doctors if hospitals’ choice functions satisfy substitutes and the law of aggregate demand. Hatfield and Kojima (2009) originally obtain this result by skillfully using the strategy-proofness of the doctor-optimal stable mechanism. In this paper we provide a different proof for the group incentive compatibility by applying the blocking lemma.