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

中文翻译：

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Stackelberg 肾脏交换问题是 $\Sigma_2^p$-完全

我们介绍了 Stackelberg 肾脏交换问题。在这个问题中，代理人（例如医院或国家组织）可以控制许多不相容的患者-供体对，其患者需要移植。代理有机会加入旨在增加可实现的最大移植总数的合作努力。然而，个体代理只对最大化其控制下的一组患者中的移植数量感兴趣。然后，问题就变成了哪些患者接受协作。我们表明，只要我们允许交换最多涉及固定数量的 $K\ge 3$ 对，回答这个问题就是 $\Sigma_2^p$-complete。然而，当我们仅限于成对交换时，问题就可以在多项式时间内解决