A Nonasymptotic Approach to Analyzing Kidney Exchange Graphs
Yichuan Ding, Dongdong Ge, Simai He, Christopher Thomas Ryan
Operations Research · 2018 · Volume 66 · Issue 4
Abstract
We propose a novel methodology to study kidney exchange. Using a random graph model of kidney exchange, we propose a nonasymptotic approach to quantifying the effectiveness of transplant chains in reducing the number of unmatched highly sensitized patients. Our approach is based on a two-phase random walk procedure where random walks are used to allocate chains, followed by allocation in cycles. The benefit of random walks is that they preserve the probabilistic structure of residual graphs, greatly facilitating analysis. Our approach allows us to analytically show the benefit of chains, as opposed to transplantation in cycles only, in nonasymptotic (medium-sized) graphs. We also derive useful analytical bounds that illustrate the performance of our proposed allocation procedure and more general kidney allocation procedures. Our results complement previous findings from analytical results in large (limit) graphs and empirical results based on data from fielded kidney exchanges demonstrating the benefits of chains. Moreover, our analysis sheds light on the relative importance of chains versus cycles in kidney allocation. In particular, our results show prioritizing chains over easy-to-transplant cycles, as opposed to prioritizing those cycles over chains, improves performance and provides analytical bounds on the associated benefits. A detailed simulation study numerically verifies our main results and provides additional insights. The online appendix is available at https://doi.org/10.1287/opre.2017.1717 .
Keywords
kidney exchange, random graph, long chains, pure death process, integer programming
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