Perfect Kidney Matching and Time
Link: https://scholar.rose-hulman.edu/cgi/viewcontent.cgi?article=1010&context=rhumj
Commentary: While learning about bipartite graphs and perfect matching, I began to wonder how perfect matching occurs in the real world outside of the example of auctions. Specifically, I wondered how perfect matching occurred when other quantifiers were used outside of money, such as time. As I looked through different real-life applications of perfect matching I found that in many cases kidney matching was done through perfect matching. This article describes the process of matching as given the donor is not a match for the recipient, the donor/recipient pair can be combined with other pairs to find some sequence of pairs such that some donor X can give to some receiver Y, and the original donor for receiver Y gives their kidney to X’s original receiver. A weighted bipartite graph can be created connecting each donor/recipient pair to each other.
This weighted graph can then be used to find an optimal matching using the Hungarian Algorithm, which uses the weighted graph to find the highest weighted pairings, where high-weighted pairings indicate high chances that both donor/recipient pairs can be satisfied. The article is also interesting in how it examines the moral dilemma which arises from the Hungarian Algorithm. Although it maximizes the satisfaction of the group, the pairings might not be equitable, and some patients will benefit much more than others. As such, it requires some sort of equalizer, which the article works to uncover. However, although this graph and algorithm account for different combinations of donors and receivers, and it does so through means other than money, it does not seem to take into account time. Rather, it claims that using the new process will lower wait times as a whole, despite certain recipients having specific time frames. This goes against what they wanted to fix with the new algorithm, and will ultimately lead to a system that is not entirely as equitable as they would like.
This article does a good job of explaining how current kidney donor systems work or might work in an ideal situation. Examining the possibilities of different donor/recipient pairs given a whole graph is truly interesting. However, I believe that a new perfect matching can be made, that uses the previous matchings as nodes on a graph, and uses time-sensitive patients as new potential donors in a new perfect matching using time as a quantifier. I wonder how this addition would impact our current system, and believe that it is something that should be considered.
Other resources: https://brilliant.org/wiki/hungarian-matching/