Kidney exchange in dynamic sparse heterogenous pools

The need for kidney exchange arises when a healthy person wishes to donate a kidney but is incompatible with her intended recipient. Two main factors determine compatibility of a donor with a patient: blood-type compatibility and tissue-type compatibility. Two or more incompatible pairs can form a cyclic exchange so that each patient can receive a kidney from a compatible donor. In addition, an exchange can be initiated by a non-directed donor (an altruistic donor who does not designate a particular intended patient), and in this case, a chain of exchanges need not form a closed cycle.Current exchange pools are of moderate size and have a dynamic flavor as pairs enroll over time. Further, they contain many highly sensitized patients, i.e., patients that are very unlikely to be tissue-type compatible with a blood-type compatible donor. One major decision clearinghouses are facing is how often to search for allocations (a set of disjoint exchanges). On one hand, waiting for more pairs to arrive before finding allocations will increase the number of matched pairs, especially with highly sensitized patients, and on the other hand, waiting is costly. This paper studies this intrinsic tradeoff between the waiting time, and the number of pairs matched under a myopic, "current-like", matching algorithm called Chunk Matching (CM) that accumulates a given number of incompatible pairs, or a chunk, before searching for an allocation in the pool that consists of easy and hard to match patients.We perform sensitivity analysis on the chunk size given different types of allocations; first, we first study the performance of CM when it searches for allocations limited to cycles of length 2 and show that if the waiting period between two subsequent match runs is a sub-linear function of the problem size (or the entire time horizon), CM matches approximately the same number of pairs as the online scenario (matching each time a new incompatible pair joins the pool) does. Waiting, however, a linear fraction between every two runs will result in matching linearly more pairs compared to the online scenario. We then analyze CM when cycles of length both 2 and 3 are allowed. We show that for some regimes, sub-linear waiting will result in a linear addition of matches comparing to the online scenario. Finally, we study the efficiency of dynamic matching with chains (chains are initiated by a non-directed donor), and show that in the online scenario, adding one non-directed donor will increase linearly the number of matches that CM will find over the number of matches it will find without a chain.Our results may be of independent interest to the literature on dynamic matching in random graphs. Kidney exchange serves well as an example for which we have distributional information on the underlying graphs, thus we can exploit this information to make analysis and prediction far more accurate than the worst-case analysis can do. We believe our average-case analysis can have implications beyond the kidney exchange and can be applied to other dynamic allocation problems with such distributional information. Further, from a theoretical perspective, our paper initiates a novel direction in matching over time, deviating from the online scenarios.While this paper focuses on kidney exchange, there are many dynamic markets for barter exchange for which our findings apply. There is a growing number of websites that accommodate a marketplace for exchange of goods (often more than 2 goods), e.g. ReadItSwapIt.com and Swap.com. In these markets, the demand for goods, cycle lengths and waiting times play a significant role in efficiency.