As multi-hospital kidney exchange has grown, the set of players has grown from patients and surgeons to include hospitals. Hospitals can choose to enroll only their hard-to-match patient-donor pairs, while conducting easily-arranged exchanges internally. This behavior has already been observed. We show that as the population of hospitals and patients grows the cost of making it individually rational for hospitals to participate fully becomes low in almost every large exchange pool (although the worst-case cost is very high), while the cost of failing to guarantee individual rationality is high, in lost transplants. We identify a mechanism that gives hospitals incentives to reveal all patient-donor pairs. We observe that if such a mechanism were to be implemented and hospitals enrolled all their pairs, the resulting patient pools would allow efficient matchings that could be implemented with two and three way exchanges.