  We introduce the Stackelberg kidney exchange problem. In this problem, an
agent (e.g. a hospital or a national organization) has control over a number of
incompatible patient-donor pairs whose patients are in need of a transplant.
The agent has the opportunity to join a collaborative effort which aims to
increase the maximum total number of transplants that can be realized. However,
the individual agent is only interested in maximizing the number of transplants
within the set of patients under its control. Then, the question becomes which
patients to submit to the collaborative effort. We show that, whenever we allow
exchanges involving at most a fixed number $K \ge 3$ pairs, answering this
question is $\Sigma_2^p$-complete. However, when we restrict ourselves to
pairwise exchanges only, the problem becomes solvable in polynomial time
