Collaborative Research: The Optimal Timing of Kidney Exchanges: A Markov Game Approach

这一合作研究项目的目的是为潜在的配对肾脏交换(PKE)提供更准确的评估。在给定任意大小的PKE的情况下，考虑何时(就患者健康而言)应该进行交换的问题。只有在交换中的每个患者都愿意的情况下，交换才是可能的。这个问题自然而然地被描述为一个无限水平的随机对策。分析这个随机博弈揭示了几个独特的特征(例如，交换概率完全表征了均衡，并且人们可以将注意力限制在只有一个患者随机的均衡上)。然后，我们可以证明该博弈的均衡与某一混合整数规划的可行点之间的一一对应关系。扩展包括对这个游戏的迭代方法，以及建模扩展，例如身体肾脏到达和无私捐赠者。这一目标将通过三种方式实现：1)通过比较质量调整后的预期寿命，它将比其他方法更重视患者和捐赠者匹配良好的交换；2)它将考虑动态的患者健康；3)它允许患者决定何时应该进行移植。鉴于肾脏的严重短缺和配对肾脏交换的潜力，这项研究可以导致患者和捐赠者之间更好的匹配。医学研究人员、研究生和本科生将从这项研究中受益，因为它融合了运筹学、医学和经济学的重要方面。

The objective of this collaborative research project is to provide much more accurate valuations of potential paired kidney exchanges (PKEs). Given an arbitrary-sized PKE, consider the question of when (in terms of patient health) an exchange should occur. An exchange is only possible if every patient in the exchange desires it. This problem is naturally formulated as an infinite-horizon stochastic game. Analyzing this stochastic game reveals several unique features (e.g. the exchange probabilities completely characterize the equilibria, and one can restrict attention to equilibria in which only one patient randomizes). One can then demonstrate a one-to-one correspondence between the equilibria of this game and the feasible points of a certain mixed-integer program. Extensions include iterative approaches to this game, as well as modeling extensions, such as cadaveric kidney arrivals and altruistic donors. This goal will be accomplished in three ways: 1) By comparing quality-adjusted life expectancies, it will value exchanges where the patients and donors are well matched more than others; 2) It will consider dynamic patient health; and 3) It allows patients to determine when the transplantations should occur.Given the severe shortage of kidneys and the potential of paired kidney exchanges, this research can lead to much better matching between patients and donors. Medical researchers, graduate students and undergraduate students will benefit from this research as it blends important aspects of operations research, medicine, and economics.