Kidney exchange 2000-2015: Algorithms and Incentives Al Roth Stanford Simons Institute for Theoretical Computer Science Berkeley Nov 17, 2015 Some mile posts of kidney exchange in the U.S. 2000: First U.S. exchange, in Rhode Island Hospital 2004: formation of New England Program for Kidney Exchange, Ohio consortium 2005: implemented algorithm for pairwise exchange Began to do short chains, and longer (3-way) cycles
Other kidney exchange networks formed (including federally sponsored pilot program in 2010) 2007: Norwood Act makes kidney exchange legal 2007: first long (non-simultaneous) chain. 2009: non-simultaneous chains become standard 2015--? Kidney exchange clearinghouse design Roth, Alvin E., Tayfun Snmez, and M. Utku nver, Kidney Exchange, Quarterly Journal of Economics, 119, 2, May, 2004, 457-488. ________started talking to docs________ ____ Pairwise Kidney Exchange, Journal of Economic Theory, 125, 2, 2005, 151188. ___ A Kidney Exchange Clearinghouse in New England, American Economic Review, Papers and Proceedings, 95,2, May, 2005, 376-380. _____ Efficient Kidney Exchange: Coincidence of Wants in Markets with Compatibility-Based Preferences, American Economic Review, June 2007, 97,
3, June 2007, 828-851 ___multi-hospital exchanges become commonhospitals become players in a new kidney game________ Ashlagi, Itai, David Gamarnik and Alvin E. Roth, The Need for (long) Chains in Kidney Exchange, May 2012 Ashlagi, Itai and Alvin E. Roth, New challenges in multi-hospital kidney exchange, American Economic Review Papers and Proceedings, 102,3, May 2012, 354-59. Ashlagi, Itai and Alvin E. Roth "Free Riding and Participation in Large Scale, Multihospital Kidney Exchange,Theoretical Economics 9(2014),81763. Anderson, Ross, Itai Ashlagi, David Gamarnik and Alvin E. Roth, Finding long chains in kidney exchange using the traveling salesmen problem, Proceedings of the National Academy of Sciences of the United States of America (PNAS), January 20, 2015 | vol. 112 | no. 3 | 663668. 3 And in the medical literature Saidman, Susan L., Alvin E. Roth, Tayfun Snmez, M. Utku nver, and Francis L. Delmonico,
Increasing the Opportunity of Live Kidney Donation By Matching for Two and Three Way Exchanges, Transplantation, 81, 5, March 15, 2006, 773-782. Roth, Alvin E., Tayfun Snmez, M. Utku nver, Francis L. Delmonico, and Susan L. Saidman, Utilizing List Exchange and Undirected Donation through Chain Paired Kidney Donations, American Journal of Transplantation, 6, 11, November 2006, 2694-2705. Rees, Michael A., Jonathan E. Kopke, Ronald P. Pelletier, Dorry L. Segev, Matthew E. Rutter, Alfredo J. Fabrega, Jeffrey Rogers, Oleh G. Pankewycz, Janet Hiller, Alvin E. Roth, Tuomas Sandholm, Utku nver, and Robert A. Montgomery, A Non-Simultaneous Extended Altruistic Donor Chain, New England Journal of Medicine , 360;11, March 12, 2009, 1096-1101. Ashlagi, Itai, Duncan S. Gilchrist, Alvin E. Roth, and Michael A. Rees, Nonsimultaneous Chains and Dominos in Kidney Paired Donation Revisited, American Journal of Transplantation, 11, 5, May 2011, 984-994 Ashlagi, Itai, Duncan S. Gilchrist, Alvin E. Roth, and Michael A. Rees, NEAD Chains in Transplantation, American Journal of Transplantation, December 2011; 11: 27802781. Rees, Michael A., Mark A. Schnitzler, Edward Zavala, James A. Cutler, Alvin E. Roth, F.
Dennis Irwin, Stephen W. Crawford,and Alan B. Leichtman, Call to Develop a Standard Acquisition Charge Model for Kidney Paired Donation, American Journal of Transplantation, 2012, 12, 6 (June), 1392-1397. Roth, Alvin E., Transplantation: One Economists Perspective, Transplantation, February 2015, Volume 99 - Issue 2 - p 261264. 4 Today: New Developments and Frontiers Fumo, D.E., V. Kapoor, L.J. Reece, S.M. Stepkowski,J.E. Kopke, S.E. Rees, C. Smith, A.E. Roth, A.B. Leichtman, M.A. Rees, Improving matching strategies in kidney paired donation: the 7-year evolution of a web based virtual matching system, American Journal of Transplantation, October 2015, 15(10), 2646-2654 . Afshin Nikzad, Mohammad Akbarpour,, Alvin E.
Roth and Michael A. Rees, Financing Transplant Costs of the Poor: Global Kidney Exchange, in preparation 5 Pre-kidney history: some abstract theory Shapley, Lloyd and Herbert Scarf (1974), On Cores and Indivisibility, Journal of Mathematical Economics, 1, 23-37. Roth, Alvin E. and Andrew Postlewaite (1977), Weak Versus Strong Domination in a Market with Indivisible Goods, Journal of Mathematical Economics, 4, 131137. Roth, Alvin E. (1982), Incentive Compatibility in a Market with Indivisible Goods, Economics Letters, 9, 127-132.
6 House allocation Shapley & Scarf [1974] housing market model: n agents each endowed with an indivisible good, a house. Each agent has preferences over all the houses and there is no money, trade is feasible only in houses. Gales top trading cycles (TTC) algorithm: Each agent points to her most preferred house (and each house points to its owner). There is at least one cycle in the resulting directed graph (a cycle may consist of an agent pointing to her own house.) In each such cycle, the corresponding trades are carried out and these agents are removed from the market together with their assignments. The process continues (with each agent pointing to her most preferred house that remains on the market) until no agents and houses remain.
7 Theorem (Shapley and Scarf 74): the allocation x produced by the top trading cycle algorithm is in the core (no set of agents can all do better than to participate) Proof : Let the cycles, in order of removal, be C1, C2,Ck. (In case two cycles are removed at the same period, the order is arbitrary.) Suppose agent a is in C1. Then xa is as first choice, so a cant do better at any other allocation y, so no y dominates x via a coalition containing a. Induction: suppose we have shown that no agent from cycles C1, Cj can be in a dominating coalition. Then if a is in cycle Cj+1, the only way that agent a can be better off at some y than at x is if, at y, a receives one of the houses that originally belonged to an agent b in one of C1,Cj . But, by the inductive hypothesis, y cannot
dominate x via a coalition that contains b 8 Theorem (Shapley and Scarf 74): the allocation x produced by the top trading cycle algorithm is in the core (no set of agents can all do better than to participate) When preferences are strict, Gales TTC algorithm yields the unique allocation in the core (Roth and Postlewaite 1977). 9 Theorem (Roth 1982): if the top trading cycle procedure is used, it is a dominant strategy for every agent to state his true preferences.
10 Theorem (Roth 1982): if the top trading cycle procedure is used, it is a dominant strategy for every agent to state his true preferences. Sketch of proof: Cycles and chains i 11 The cycles leave the system (regardless of where i points), but is choice set (the chains pointing to i) remains, and can only grow
i 12 Allocating dormitory rooms Abdulkadirolu & Snmez [1999] studied the housing allocation problems on college campuses, which are in some respects similar: A set of houses (rooms) must be allocated to a set of students. Some of the students are existing tenants each of whom already occupies a room and the rest of the students are newcomers. In addition to occupied rooms, there are vacant rooms. Existing tenants are not only entitled to keep their current houses but also apply for other houses.
13 Transplantssome background Transplantation is the best treatment for a number of diseases For kidneys both deceased and live donation is possible For most organs, deceased donation is the only possibility But theres a big shortage of organs compared to the need 14 Kidneys More than 100,000 people on the waiting list for deceased-donor kidneys in the U.S.
The wait can be many years, and many die while waiting. (In 2012, 4,543 died, and 2,668 became too sick to transplant) Transplantable organs can come from both deceased donors and living donors. In 2013 there were 5,732 transplants from living donors Now including more than 10% from kidney exchange Sometimes donors are incompatible with their intended recipient. This opens the possibility of exchange. 15 Simple two-pair kidney exchange
Donor 1 Blood type A Donor 2 Blood type B Recipient 1 Blood type B Recipient 2 Blood type A 16
Modeling kidney exchange, when it was new Players: patients, donors, surgeons Incentives: Need to make it safe for them to reveal relevant medical information Blood type is the big determinant of compatibility between donors and patients The pool of incompatible patient-donor pairs looks like the general pool of patients with incompatible donors Efficient exchange can be achieved in large enough markets (but not infinitely large) with exchanges and chains of small sizes 17 Roth, Alvin E., Tayfun Snmez, and M. Utku nver, Kidney
Exchange, Quarterly Journal of Economics, 119, 2, May, 2004, 457488. Top trading cycles and chains Top Trading Cycles and Chains The mechanism we propose relies on an algorithm consisting of several rounds. In each round each patient ti points either towards a kidney in Ki {ki} or towards w, andki} or towards w, and each kidney ki points to its paired recipient ti. A cycle is an ordered list of kidneys and patients (k1, t1, k2 , t2, . . . , km, tm) such that
kidney k1 points to patient t1, patient t1 points to kidney k2 kidney km points to patient tm, and patient tm points to kidney k1.. Note that no two cycles can intersect. 19 There can be w-chains as well as cycles A w-chain is an ordered list of kidneys and patients (k1,t1, k2,t2, km,tm) such that
kidney k1 points to patient t1, patient t1 points to kidney k2 kidney km points to patient tm, and patient tm points to w. Unlike cycles, w-chains can intersect, so a kidney or patient can be part of several w-chains, so an algorithm will have choices to make. w 20 Lemma
Consider a graph in which both the patient and the kidney of each pair are distinct nodes, as is the waitlist option w. Suppose each patient points either towards a kidney or w, and each kidney points to its paired recipient. Then either there exists a cycle or each pair is at the end of a w-chain. 21 Kidney Exchange For incentive reasons, all surgeries in an exchange are conducted simultaneously, so a 2-way exchange involves 4 simultaneous surgeries 22
Suppose exchanges involving more than two pairs are impractical? Roth, Alvin E., Tayfun Sonmez , and M. Utku Unver, "Pairwise Kidney Exchange ," Journal of Economic Theory, 125, 2, December 2005, 151-188. Our New England surgical colleagues had (as a first approximation) 0-1 (feasible/infeasible) preferences over kidneys. Initially, exchanges were restricted to pairs. This involves a substantial welfare loss compared to the unconstrained case But it allows us to tap into some elegant graph theory for constrained efficient and incentive compatible mechanisms. 23
Pairwise matchings and matroids Let (V,E) be the graph whose vertices are incompatible patient-donor pairs, with mutually compatible pairs connected by (undirected) edges. A matching M is a collection of edges such that no vertex is covered more than once. Let S ={ki} or towards w, andS} be the collection of subsets of V such that, for any S in S, there is a matching M that covers the vertices in S Then (V, S) is a matroid: If S is in S, so is any subset of S. If S and S are in S, and |S|>|S|, then there is a point in S that can be added to S to get a set in S.
24 Pairwise matching with 0-1 preferences Theorems: All maximal matchings match the same number of couples. If patients have priorities, then a greedy priority algorithm produces the efficient (maximal) matching with highest priorities. Any priority matching mechanism makes it a dominant strategy for all couples to accept all feasible kidneys reveal all available donors So, there are efficient, incentive compatible
mechanisms Hatfield 2005: these results extend to a wide variety of possible constraints (not just pairwise) 25 Gallai-Edmonds Decomposition 26 Who are the over-demanded and underdemanded pairs? We can ask this even for the more general problem of efficient exchange (i.e. once we surmounted the constraint of just 2-way exchanges). By 2006 we had successfully made the case for at least 3 way exchanges
Saidman, Roth, Snmez, nver, and Delmonico, Increasing the Opportunity of Live Kidney Donation By Matching for Two and Three Way Exchanges, Transplantation, 81, 5, March 15, 2006, 773-782. Roth, Sonmez, Unver Efficient Kidney Exchange: Coincidence of Wants in Markets with Compatibility-Based Preferences, American Economic Review, June 2007, 97, 3, June 2007, 828-851 Factors determining transplant opportunity O Blood compatibility A B
AB So type O patients are at a disadvantage in finding compatible kidneysthey can only receive O kidneys. And type O donors will be in short supply. Tissue type compatibility. Percentage reactive antibodies (PRA) Low sensitivity patients (PRA < 79): vast majority of patients High sensitivity patients (80 < PRA < 100): about 10% of general population, somewhat higher for those incompatible with a donor The presence of antibodies to donor HLAs--a positive crossmatch-significantly increases the likelihood of graft rejection by the 28 recipient HLA (class I and II)
http://www.biomedcentral.com/1471-2105/11/S11/S10/figure/F1?highres=y Compatibility graphs: An arrow points from pair i to pair j if the kidney from donor i is compatible with the patient in pair j. Pair 1 Pair 3 Pair 4 Pair 5 Pair 2
Pair 6 30 Random Compatibility Graphs n hospitals, each of a size c>0 D(n) - random compatibility graph: 1. n pairs/nodes are randomized compatible pairs are disregarded 2. Edges (crossmatches) are randomized Random graphs will allow us to ask : What would efficient matches look like in an ideal large world? 31
(Large) Random Graphs G(n,p) n nodes and each two nodes have a non directed edge with probability p Closely related model: G(n,M): n nodes and M edgesthe M edges are distributed randomly between the nodes Erdos-Renyi: For any p(n)(1+)(ln n)/n almost every large graph G(n,p(n)) has a perfect matching, i.e. as n goes to the probability that a perfect matching exists converges to 1. A natural case for kidneys is p(n) = p, a constant (maybe different for different kinds of patients), hence always above the threshold. Giant connected component Similar lemma for a random bipartite graph G(n,n,p). Can extend also for r-partite graphs, directed graphs 32 Ideally Efficient Allocations: if we were seeing all the patients in sufficiently large markets
2014 Patient-donor pairs by blood type. Over-demanded (shaded) patient-donor pairs are all matched. 33 Compatibility graph: cycles and chains Pair 1
Pair 3 Pair 4 Pair 5 Non-directed donor Pair 6 Pair 2 Waiting list patient Pair 7 34 Chains initiated by non-directed
(altruistic) donors Non-directed donation before kidney exchange was introduced Nondirected donor Wait list 35 Chains initiated by non-directed (altruistic) donors Non-directed donation before kidney exchange was introduced
Non-directed donation after kidney exchange was introduced Nondirected donor Wait list Nondirected donor R1 D1
R2 D2 Wait list 36 A chain in 2007 3-way chain March 22, 2007 BOSTON -- A rare six-way surgical transplant was a success in Boston.
There are only 6 people in this chain. Simultaneity congestion: 3 transplants + 3 nephrectomies = 6 operating rooms, 6 surgical teams 37 Chains in an efficient large dense pool A-A O-O AB-B B-B
AB-A AB-O A-O B-A ABAB B-O VA-B B-AB A-AB O-A
O-B Waiting list patient It looks like a non-directed donor can increase the match size by at most 3 A-B O-AB Non-directed donor blood type O 38
How about when hospitals become players? We are seeing some hospitals withhold internal matches, and contribute only hard-to-match pairs to a centralized clearinghouse. Mike Rees (APD director) writes us: As you predicted, competing matches at home centers is becoming a real problem. Unless it is mandated, I'm not sure we will be able to create a national system. I think we need to model this concept to convince people of the value of playing together. 39 Individual rationality and efficiency: an impossibility theorem with a (discouraging) worstcase bound
For every k> 3, there exists a compatibility graph such that no k-maximum allocation which is also individually rational matches more than 1/(k-1) of the number of nodes matched by a k-efficient allocation. 40 Proof (for k=3) a3 a1 e a2
c b d 41 There are incentives for Transplant Centers not to fully participate even when there are only 2-way exchanges The exchange A1-A2 results in two transplantations, but the exchanges A1-B and A2-C results in four. (And you can see why, if Pairs A1 and A2 are at the same transplant center, it might be good for them to nevertheless be submitted to a regional match) 42
PRA distribution in historical data 16% 14% 40% 35% 30% Percentage 25% 20% 15% 10% Percentage
12% 10% 8% NKR APD 6% 4% 2% 0% 95-96 96-97 97-98
98-99 99-100 PRA Range NKR APD 5% 0% PRA Range PRA probability for a patient to fail a tissue-type test with a
random donor 43 Graph induced by pairs with A patients and A donors. 38 pairs (30 high PRA). Dashed edges are parts of cycles. No cycle contains only high PRA patients. Only one cycle includes a high PRA patient 44 Cycles and paths in random dense-sparse graphs n nodes. Each node is L w.p. v and H w.p. v incoming edges to L are drawn w.p. incoming edges to H are drawn w.p. c/n:
L L nodes have many incoming arrows H 45 Case v>0 (some low sensitized, easy to match patients. Why increasing cycle size helps Theorem. Let Ck be the largest number of transplants achievable with cycles k. Let Dk be the largest number of transplants achievable with cycles k plus one non-directed donor. Then for every constant k there exists >0
Furthermore, Ck and Dk cover almost all L nodes. L H 46 Can simultaneity be relaxed in Non-directed donor chains? Cost-benefit analysis: If something goes wrong in subsequent transplants and the whole ND-chain cannot be completed, the worst outcome will be no donated kidney being sent to the waitlist and the ND donation would entirely benefit the
KPD [kidney exchange] pool. (Roth, Sonmez, Unver, Delmonico, and Saidman) AJT 2006, p 2704). 47 Simultaneous cycles and Nonsimultaneous extended altruistic donor (NEAD) chains D1 D2 R1 R2 Conventional cycle: why always simultaneous?
On day 1 donor D2 gives a kidney to recipient R1, and on day 2 donor D1 is supposed to give a kidney to recipient R2 48 Simultaneous cycles and Nonsimultaneous extended altruistic donor (NEAD) chains D1 D2 R1 R2 Conventional cycle
NDD D1 D2 R1 R2 Non-simultaneous chain Since NEAD chains can be non-simultaneous, they can be long Roth, Alvin E., Tayfun Snmez, M. Utku nver, Francis L. Delmonico, and Susan L. Saidman, Utilizing List Exchange and Undirected Donation through Chain
Paired Kidney Donations, American Journal of Transplantation, 2006 49 50 The First NEAD Chain (Rees, APD) July 2007 AZ 1 July 2007 OH
2 Sept 2007 OH 3 Sept 2007 OH 4 Feb 2008 MD 5
Feb 2008 MD 6 Feb 2008 MD 7 Feb 2008 NC 8
O A A B A A A AB
A O O A A B A A
A A Recipient PRA 62 0 23 0 100
78 64 3 100 46 Recipient Ethnicity Cauc Cauc
Cauc Cauc Cauc Hisp Cauc Cauc Cauc AA
Relationship Husband Wife Mother Daughter Sister Brother Wife Husband Father Daughter
Husband Wife Friend Friend Brother Brother Daughter Mother MI O
Daughter Mother # March March 2008 2008 MD OH 9 10 AB * A
* This recipient required desensitization to Blood Group (AHG Titer of 1/8). # This recipient required desensitization to HLA DSA by T and B cell flow cytometry. 51 52 Chains are important for hard to match pairs Why are chains essential? As kidney exchange became common and transplant centers gained experience they began withholding their easy to match pairs and transplanting them internally. This means that the flow of new patients to kidney exchange networks contain many who are hard to match
So chains become important: many pairs with few compatible kidneys can only be reached through chains. 53 Feb 2012, NKR: a NDD chain of length 60 (30 transplants) 54 Computational notes Finding maximal 2-way exchanges is a computationally easy problem. Finding maximal 2- and 3-way exchanges is computationally complex. There are many more chains than cycles
But so far there hadnt been a problem solving the necessary integer programs. 55 Matching algorithms have had to get more powerful to find optimal matchings involving cycles and chains in sparse graphs Integer Programming based algorithm for finding optimal cycle and chain based exchanges. Formulation I: MAX weighted # transplants Max Pair gives only if receives s.t. No cycles with length >b Cycle bound constraint is added only iteratively to speed up the computation
The running time of the algorithm is below 20 min for most instances (most are solved within seconds) Anderson, Ross, Itai Ashlagi, David Gamarnik and Alvin E. Roth, Finding long chains in kidney exchange using the traveling salesmen problem, Proceedings of the National Academy of Sciences of the United States of America (PNAS), January 20, 2015 | 663668. 56 Market design isnt just analytical Ive briefly mentioned some of the analytical models that have played a big role: Game theory Top trading cycles and chainsthe core Strategic behaviordominant strategy incentive compatibility Deterministic graph theory
Random graph theory Integer programming But market design is also economic engineering at an operational level 57 Changes in kidney exchange over time Operational level: How to manage long (non-simultaneous) chains How to increase acceptance rates of offers? Why do transplant centers withhold easy to match pairs? What makes kidney exchange among multiple hospitals hard? Why (else) are chains of such practical usefulness?
What are (some of) the obstacles that remain? 58 Most offers are rejected An offer is a particular cycle or chain segment an edge in the compatibility graph In 2007, only about 15% of offers resulted in transplants Today its closer to 50% Whats going on? X 59 Transplant centers declined donors for medical reasons
Transplant centers were consistently rejecting potential donors for reasons that could have been stated in advance But surgeons didnt/couldnt accept/reject kidneys in advance A threshold language was introduced for prerejection But its hard to specify in advance which kidneys are unacceptable, when the language is limited to thresholds in each dimensione.g. age, BMI, blood pressure 60 Reducing predictable rejections Now show transplant centers multiple combinations of donor-recipient offers prior to
making formal offers. Its easier to accept or reject from a relatively small set of possible kidneys 61 Remaining problems Center incentives Frequent flier accounting Finances Standard acquisition charge Broader pictureenlarging the pool International? 62 Encouraging transplant centers to participate
fully in kidney exchange Make it safe for hospitals and their patients to enroll easy to match pairs in kidney exchange (and not just hard to match pairs) Guarantees to hospitals that they and their patients wont lose if they enroll all pairs in exchange Frequent flier programs? 63 Frequent flier accounting: over and underdemanded patient-donor pairs Over-demanded Under-demandedblood type O patients
64 Growth of kidney exchange in the US Kidney exchange transplants per year 589 528 430 228 2 4
6 19 34 27 74 446 281 111
2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 5-fold increase since 2007, majority in chains 65 Kidney exchange outside the U.S. Friday, July 24, 2015 Kidney exchange in Turkey (1st exchanges there) April 10, 2015 A first non-directed donor kidney exchange chain in Italy March 30, 2015 A first kidney exchange in Argentina March 5, 2015 First kidney exchange in Poland Friday, November 7, 2014 Kidney exchange in Spain: now more than 100 transplants June 7, 2014 Kidney exchange in France December 19, 2013 Kidney exchange in Vienna August 19, 2013 Ten kidney exchange transplants on World Kidney Day in Ahmedabad, India July 28, 2013 First Kidney Exchange in Portugal: July 23, 2013 Kidney exchange chain in India
June 6, 2013 Kidney exchange between Jewish and Arab families in Israel December 26, 2012 Kidney exchange in Canada December 1, 2012 Kidney exchange in India June 1, 2012 Mike Rees and Greece: an intercontinental kidney exchange March 27, 2012 Kidney exchange in Britain February 5, 2012 Kidney exchange in Australia, 2011 April 29, 2011 First kidney exchange in Spain December 8, 2010 National kidney exchange in Canada August 3, 2010 Kidney Exchange in South Korea Tuesday, August 3, 2010 Kidney Exchange in South Korea Friday, July 30, 2010 Kidney transplantation advice from the Netherlands March 9, 2010 Kidney exchange news from Britain (1st 3-way there) January 27, 2010 The Australian paired Kidney eXchange (AKX) goes live June 25, 2009 Kidney exchange in Canada (1st exchange there) February 27, 2009 Kidney Exchange in Australia (in Western Australia)
66 Kidneys Transplanted per million population U.S.A. Nigeria, Bangladesh, Vietnam (Global Observatory on Donation & Transplantation) Global kidney exchange: a possibility of mutual aid United States Two-way exchange
Developing World Transplants unavailable 69 First global kidney exchange, with a pair from the PhilippinesJanuary 2015, Alliance for Paired Donation (Rees et al.) Nondirected donor Jose Kristine
R2 D2 Jose Mamaril received a kidney from a non-directed American donor in Georgia. His wife, Kristine, donated one of her kidneys to an American recipient in Minnesota, whose donor continued the chain by donating to a patient in Wait Seattle. list THE BLADE/JETTA FRASER 70 Kidney Disease costs Medicare $35 Billion per year
Dialysis costs Medicare approx $90,000/year Kidney transplantation costs approx $33,000/year In 5 years U.S. taxpayers save >$275,000 per kidney transplant More than enough to pay for surgery and post-operative care for a foreign pair. 71 Kidney Disease costs Commercial Insurance Companies > $14 Billion per year Dialysis cost $800,000 / 33 months Kidney transplantation costs $300,000 / 33 months Insurance Companies can save up to $500,000
per kidney transplant 72 Two Proposals and Three Planners To exploit gains from trade, we introduce two proposals: Global kidney exchange Non-directed donors proposal And analyze them from the perspectives of three planners: Medicare State department Private insurance 73 Planners Objectives
Medicare Maximize domestic transplants subject to budget constraint. Break ties in favor of policies with lower cost. State department Maximizes domestic transplants subject to budget constraint. Break ties in favor of policies with more international transplants. Profit-maximizing insurance corporation Minimize costs 74 Global Kidney Exchange The GKE proposal is self-financing.
Back of the envelope calculation: cost of hemodialysis $90, 000 per year average time under dialysis 5 years cost of transplant $120, 000 per surgery (plus $20,000 in maintenance therapy costs per patient per year) But in steady state, waiting time decreases. So dialysis costs will go downhow long will GKE remain self financing? 75 GKE: A (Stylized) Dynamic Model Domestic pairs arrive with rate m. International pairs arrive with rate m All pairs are ex ante biologically compatible with probability p
International pairs are financially incompatible All pairs perish with (normalized) rate 1 GKE(): the above model with policy parameter 76 GKE: Greedy Matching Policy Whenever a domestic pair arrives: If agent has any domestic matches, match her to one of them uniformly at random If agent has any international matches (but no domestic), match her to one of them uniformly at random Whenever an international pair arrives: If agent has any matches, match her to one
of them uniformly at random 77 GKE: Illustration (should become a dynamic animation) International pairs Domestic pairs 78 Large Market, Highly Sensitized Patients we model a large market regime, with highly sensitized domestic patients: p (prob of compatibility) is small and m (domestic arrival rate) is large. Formally:
m . p = (1) (mp, the average degree of a node gets arbitrarily large as # nodes grows m . p = o(n) (mp grows slower than # nodes: the percentage of other nodes connected to each node goes to zerothe graph remains sparse) 79 Costs Cd : cost of dialysis per patient per year Cs : cost of surgery (and maintenance therapy) per patient W() ) : Average waiting time of all agents in GKE() ) W(0) : Average waiting time under status quo
C() ): Total cost = Cd W() ) + Cs . (# of all matched patients) Let () ) = Cd . W() )/ Cs dialysis cost per patient / surgery cost 80 GKE Is Self-Financing Theorem: If > ln(2), then there exists a constant ) >0 such that GKE() ) is selffinancing. ln 2 = .693 < 1 81 Intuition Some domestic pairs immediately find a match Some other do not find a match upon arrival.
They increase the average waiting cost International pairs get matched to those the latter type of domestic pairs So even if the average dialysis cost is less than the surgery costs, GKE can still be self-financing because it matches domestic patients with higher-than-average dialysis costs. 82 Planners Choices Private insurers choice Medicare and State Departments choice
83 The Non-Directed Donor Proposal limitation of the GKE proposal: The number of international transplants is bounded by the size of the domestic exchange pool (presently on the order of 1,000 pairs) The NDD proposal can increase the number of international transplants by more than an order of magnitude (100,000 patients on U.S. deceased-donor waiting list) 84 The Non-Directed Donor Proposal Patient: Dana Donor: Erik
NDD: Ben Each international patient is accompanied by a donor and a non-directed donor 85 NDD: A Dynamic Model Domestic patients have arrival rate m. Deceased kidneys become available with rate md International patients (together with donors) arrive with rate mi Policy maker accepts international patients with rate (0 ) mi) is the policy parameter (controlled by the policy maker) When a deceased or international donation is accepted A match is found; number of domestic waiting patients decreases by one
All waiting patients perish with (normalized) rate 1 NDD(): the above policy with policy parameter 86 Matching Policy Medicare and State department An arbitrary policy such as First-ComeFirst-Served Insurance Corporation Last-Come-First-Served (U.S. private insurers only pay for first 33 months) LCFS policy minimizes the waiting cost 87
Planners Decision Theorem B. Average length of the domestic queue in NDD() is ) is md (m + ) ), unless the average queue length is very small. When the queue becomes too short, (i.e. when md (m + ) ) becomes very small), md (m + ) ) is not a sharp estimate of the average queue length But in this case, we have almost solved the problem For our purpose, it is safe to assume md (m + ) ) is a sharp estimate for the average queue lengthi.e. of average waiting time 88 The medical logistics may not be the hard part 89
Financial flows Savings: Medicarecomplex legislative/bureaucratic Private insurers (33 months) Costs: Surgeriestransplant centers Post surgical treatment in home countries Infrastructure development in home countries USAID?--Same Federal budget, but no change needed in Medicare Allow insurance companies to nominate patients? 90 Jose and Kristine: Safely home $50,000 escrow fund for post-surgical care
91 Why do we have laws against simply buying and selling kidneys? I sure dont know the answer to this one, but I think its a subject that social scientists need to study It isnt just about body parts 92