Kidney exchanges have saved many lives since economists Al Roth, Tayfun Sonmez, Utku Unver, and Atila Abdulkadiroglu first proposed them and then convinced doctors and hospitals to embrace them.

In paired kidney exchanges the transaction involves multiple pairs of patients. Each pair consists of a kidney patient who will receive a kidney, and a donor, typically a family member, who will give one. Each pair is incompatible: because of a blood-type or tissue-type mismatch the patient would reject the donor’s kidney.  The exchange works by creating a cycle of patients and donors who are compatible. For example, patient A’s wife donates her kidney to patient B whose husband donates his kidney back to patient A. Even longer cycles are possible.

As a rule all of  the transplantation operations in any paired exchange are carried out simultaneously and in the same hospital. This acts as a guarantee to each donor that they will give their kidney if and only if their loved one also receives one. If some donor along the cycle becomes ill or gets cold feet, the entire cycle is halted before it begins. Such a guarantee surely makes patients more willing to participate in the exchange but it also limits the size of the cycle since there is a limit to the number of surgeries that any one hospital can support.

Then there are the chain exchanges. Here, without any paired patient to receive a kidney in return, a good samaritan comes forth and offers to donate his kidney to any compatible stranger.  This good samaritan is going to save somebody’s life. And through the power of exchange, possibly many more than just one life. Because instead of just an arbitrary compatible recipient, the kidney can be given to a patient paired with a donor whose kidney is compatible with another patient paired with a donor whose kidney is compatible with…  That is, the good samaritan can activate a long chain of transplants that otherwise could not be completed by paired exchange because the chain of compatibility did not cycle back to its beginning.

The kidney exchange economists noticed a subtle difference between paired and chain exchanges. And based on their observation they convinced doctors to relax the rule on simultaneous surgeries in the case of chain exchanges. The ever-increasing record length chains of kidney transplants are only possible because of this.

Why were doctors willing to do sequential surgeries for chained exchanges while they insisted on simultaneity for paired exchanges? It’s not because they have any less concern that the chain would be broken before all patients receive their promised kidneys. It’s not because extending the size of a cycle is any less of a blessing than extending the length of a chain. The difference that the economists noticed can be boiled down to an esoteric concept known to mechanism designers as individual rationality.  

When a paired exchange cycle is broken because one surgery along the line is not carried out, one patient is necessarily made worse off than he would have been if the exchange had never happened.  Because that patient’s loved one has given her kidney and not only has the patient not received any kidney in return, but his donor no longer has a kidney to give. The patient has lost bargaining power in the kidney exchange market going forward.  The anticipation of this possibility would make patients and donors reluctant to participate in an exchange in the first instance.

By contrast, when the sequence of transplants in a chain is halted, every patient-donor pair who gave their kidney to the next patient downstream in the chain already received one from the previous upstream donor. Yes the patients at the end of the chain do not receive their promised kidneys but they are no worse off than if the chain had never been planned in the first place. Without any threat to individual rationality there is no reason not to extend the chain of surgeries as long as imaginable capitalizing on the original good samaritan’s altruism as much as compatibility allows.

Tayfun Sonmez is here at Northwestern giving a mini-course on market design, here are his lecture slides including a lecture on kidney exchange.