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.
Cheap Talk 
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May 16, 2012 at 1:09 am
Lones Smith
Curiously, the social security system is based on “paying it backward”: Younger generation bequeathes to older generation. The younger generation that eventually sees the end of the passing game is worse off than if the system had never existed. But it is a subgame perfect equilibrium.
So suppose that the chain is not initiated by a good samaritan, but a selfish one. Call this pair #1. The sickly member 1-S wants a kidney, but like social security, we do not ask that his beloved 1-B donate one. Indeed, maybe he has no beloved. Or he won a lottery. Rather, we simply find someone 2-B who can compatibly donate to 1-S who has a sickly beloved 2-S in need of a kidney. Then we find someone, call him 3-B, who can compatibly donate to 2-S whose sickly beloved 3-S needs a kidney. And so on… At each stage, someone is donating to shoot their beloved to the top of the queue. But we have no need of a good samaritan.
What am I missing? Why not use this game? Economics usually does best when it relies on self-interest, and not benevolence. “Paying is forward” is noble, but “paying it forward” sounds like sounder economics.
May 16, 2012 at 6:54 am
Tayfun Sonmez
Jeff: Thanks for the publicity!
Lones: Thanks for the question. People are horrified with the possibility that, a pair might donate a kidney without eventually receiving one. A good samaritan donor assures that this never happens. Consider a pair (donor 1, patient 1). Suppose donor 1 donates a kidney in the hopes of getting patient 1 a kidney as you propose, and assume that shortly after the donation it becomes clear that patient 1 is too sick to receive one (or say he/she dies). Something like that will eventually happen. There will be quite a crisis that day potentially harming the program. There are also some patients who are very difficult to match because of antibodies against a very large fraction of the society. I have seen many patients where that fraction is more than 99 percent. These are called (in this case very) highly sensitized patients. If the donor of such a pair starts the sequence of donations, there is a good chance they may never receive one.
A chain that is started by the donor of a pair is essentially hoping to start a sequence of cycles. There is a 2000 consensus statement by the transplantation community explicitly asking all operations in a cycle start simultaneously (in the sense of all donors and patients receiving narcosis simultaneously). The main purpose of the consensus statement is supporting these exchanges, but it is explicitly states that no pair shall donate one w/o getting a kidney.
Of course none of the above arguments is an economics one; so
let me finally provide one: I actually don’t believe efficiency will be significantly improved, at least in a mature market, by encouraging pairs to donate ahead in the hopes of starting these cycles. In a relatively reasonably sized pool (say at least 100 pairs) one can get all efficient allocations with at most cycles of size four. This has to do with blood donation partial order structure. So donating a kidney before receiving one does not buy a pair too much. It might help with pairs receiving transplants several months earlier on average, but at the cost of several mistakes in the sense of failing to provide a kidney to a pair that has donated. I don’t think potential benefits are high enough to justify potentially discouraging pairs from kidney exchanges altogether.
May 16, 2012 at 7:13 am
lauradoval
Illustrated in Grey’s Anatomy : http://www.imdb.com/title/tt1263297/synopsis
May 16, 2012 at 7:47 am
Tayfun Sonmez
Lones: I realized that I failed to give one very important piece of information.
There is indeed precedence for a version of your proposal. When we started working on kidney exchange, there was a practice called “indirect exchange.” Indeed we proposed improved versions of indirect exchanges in our very first kidney exchange paper. Under an indirect exchange a pair donates a kidney to the deceased donor list and gets a priority upgrade in the list. So under an indirect exchange, a pair donates (a live) kidney before receiving a deceased donor kidney. About 20 or so of such exchanges are conducted in New England in the period 2000-2004; that’s before our involvement. We proposed improving this practice through chains very much along the lines you proposed. Even though indirect exchanges were conducted, the practice received lots of fire because of its distributional implications. Pairs who participate in such exchanges had mostly O blood type patients but typically A or B or AB blood type donors. So indirect exchanges were adversely affecting patients on the O-blood type deceased donor pool. At the time we argued that chains can be used to avoid that or minimize the adverse welfare impact, but given that the rest of our proposal was already providing lots of potential benefits, New England decided to drop indirect exchanges altogether.
I always thought that was a mistake… Since we proposed a number of improvements, the total benefits were more than enough to compensate any adverse welfare consequence of indirect exchanges to O-blood type patients w/o living donors. But doctors opted to choose the individual elements of our proposal each of which are Pareto improvements individually. An indirect chain increases aggregate efficiency but it is not Pareto improving by itself. However our total package was a Pareto improvement, and indirect exchange chains could have easily passed together with the other innovations. It didn’t happen. Once other innovations are accepted and became the norm, it is no longer possible to include indirect exchange chains in the system of course…
This above discussion also brings the following issue, a bit related to some of the points on my earlier post. Most pairs who will be willing to donate ahead will be those who have O patients with A or B or AB donors.
That is because they know they are at a disadvantage at the long side of the system. This will result in an unstable system since you are feeding lots of A,B,AB kidneys to the system in the hopes of eventually getting O kidneys. So the system will be compromised very fast. You will have too many O patients with priority whose loved ones donated non-O kidneys. What is possible with priority in deceased donor list is not possible here. Then again even with priority at deceased donor list the distribution of kidneys changed, which resulted in ending the practice.
There you go: Another economic argument.
May 16, 2012 at 1:46 pm
Antonio Nicolo
There are other possible ways of improving KPE program. One possibility is to expand kidney exchanges through the participation by compatible pairs. In order to give incentives to compatible pairs to join the program, a part from altruistic motivations, patients of a compatible pairs who join the program should have the chance of a better match. Donor age has been identified as the most significant factor influencing long-term outcomes of living donor kidney transplants and compatible pairs can be incentivized by matching them with donors who are younger than their compatible donors. This may be an effective way to reduce the unbalancedness in the number of 0-blood type patients in the program.
May 16, 2012 at 9:27 pm
Tayfun Sonmez
Antonio: You are absolutely right! I think inclusion of compatible pairs will be the most important breakthrough. I know you are working on this version of the problem; I am also working on a second project with Utku Unver to make a stronger case for inclusion of compatible pairs.