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Matching markets are part of our daily lives, appearing on online platforms, school admissions and health systems. Their study attracts the interest of optimizers and game theorists. In this talk, we will focus on a particular matching market, the kidney exchange program (KEP), where combinatorial optimization and game theory play an important role.

A patient in need of a kidney transplant who has an incompatible donor can register on a KEP. The program seeks compatible donor exchanges between these patient-donor pairs to maximize patient benefit. KEPs resulting from the combination of patient-donor pools from different agents (which can be transplant centers, regions or countries) are currently being formed, requiring a game theoretical analysis. In this talk, we will recall literature on multi-player KEPs and see the extension of graph theoretical results to this game. Then, through computational experiments inspired by the Canadian KEP, we will investigate the social welfare generated by a non-cooperative multi-player KEP. These experiments will reveal the multiplicity of socially optimal solutions, leading to a new research question regarding individual patient fairness: Given multiple socially optimal matching plans, how should we select among them? We will end this talk with a proposition to tackle this question as well as an optimization methodology to efficiently implement it in practice.