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Lecture 11

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• There are more than 90,000 patients on the waiting
list for cadaver kidneys in the U.S. today, and over
11,000 in Australia
• In 2011 33,581 patients were added to the kidney waiting list, and 28,625 patients were removed from the list.
• In 2011 there were 11,043 transplants of cadaver kidneys performed in the U.S.
• In the same year, 4,697 patients died while on the waiting list (and 2,466 others were removed from the list as “Too Sick to Transplant”.
• In 2011 there were also 5,771 transplants of kidneys from living donors in the US.
• Sometimes donors are incompatible with their intended recipient.
• This opens the possibility of exchange .

Donor 1
Blood type A
Recipient 1
Blood type B
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Donor 2
Blood type B
Recipient 2
Blood type A
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A classic economic problem:
Coincidence of wants
(Money and the Mechanism of Exchange, Jevons 1876)
Chapter 1: "The first difficulty in barter is to find two persons whose disposable possessions mutually suit each other's wants. …to allow of an act of barter, there must be a
double coincidence, which will rarely happen. ... the owner of a house may find it unsuitable, and may have his eye upon another house exactly fitted to his needs. But even if the owner of this second house wishes to part with it at all, it is exceedingly unlikely that he will exactly reciprocate the feelings of the first owner, and wish to barter houses.
Sellers and purchasers can only be made to fit by the use of some commodity... which all are willing to receive for a
time, so that what is obtained by sale in one case, may be used in purchase in another. This common commodity is called a medium, of exchange..."

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42 U.S.C. 274e

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Australia: HUMAN TISSUE ACT--1983
• 32. Trading in tissue prohibited
• (1) A person must not enter into, or offer to enter into, a contract or arrangement under which any person agrees, for valuable
consideration, whether given or to be given to any such person or to any other person:(a) to the sale or supply of tissue from any such person’s body or from the body of any other person, whether before or after that person’s death or the death of that other person

•
Public Law 110-144, 110th Congress, Dec. 21, 2007
• Section 301 of the National Organ Transplant
Act (42 U.S.C. 274e) is amended-- (1) in subsection (a), by adding at the end the following:
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Pair 1
Donor 1 Recipient 1
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Donor 3 Recipient 3
Pair 3
Donor 2 Recipient 2
Pair 2

Pair 2
Pair 1
Pair 4
Pair 3
Pair 5
Non-directed donor
Pair 6
Pair 7
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Suppose exchanges involving more than two pairs are impractical?
• Our New England surgical colleagues have (as a first approximation) 0-1 (feasible/infeasible) preferences over kidneys.
– (see also Bogomolnaia and Moulin (2004) for the case of two sided matching with 0-1 prefs)
• 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.

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• Let (V,E) be the graph whose vertices are incompatible patient-donor pairs, with mutually compatible pairs connected by edges.
• A matching M is a collection of edges such that no vertex is covered more than once.
• Let S ={S} 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.

Factors determining transplant opportunity
O
• Blood compatibility
A B
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AB
So type O patients are at a disadvantage in finding compatible kidneys—they 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)
 High sensitivity patients (80 < PRA < 100)

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A. Patient ABO Blood Type
O
A
B
AB
B. Patient Gender
Female
Male
C. Unrelated Living Donors
Spouse
Other
D. PRA Distribution
Low PRA
Medium PRA
High PRA
Frequency
48.14%
33.73%
14.28%
3.85%
Frequency
40.90%
59.10%
Frequency
48.97%
51.03%
Frequency
70.19%
20.00%
9.81%

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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 two related questions:
What would efficient matches look like in an “ideal” large world?
What is the efficiency loss from requiring the outcome to be individually rational for hospitals?

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 edges—the M edges are distributed randomly between the nodes
Erdos-Renyi : For any p(n) ¸ (1+ ² )(ln n)/n almost every large graph
G(n,p(n)) has a perfect matching, i.e. as n !1
the probability that a perfect matching exists converges to 1.
“Giant connected component”
Similar lemma for a random bipartite graph G(n,n,p).
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• The large graph model with constant p (for each kind of patient-donor pair) predicts that only short chains are useful.
• But we now see long chains in practice.
• They could be inefficient—i.e. competing with short cycles for the same transplants.
• But this isn’t the the case when we examine the data.

Why?
Very many very highly sensitized patients
Previous simulations: sample a patient and donor from the general population, discard if compatible (simple live transplant), keep if incompatible. This yields 13% High PRA.
The much higher observed percentage of high PRA patients means compatibility graphs will be sparse
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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
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Jellyfish structure of the compatibility graph: highly connected low sensitized pairs, sparse hi-sensitized pairs
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• We’ll consider random graphs with two kinds of nodes
(patient-donor pairs): Low sensitized and high sensitized
• L nodes will have a constant probability of an incoming edge
(compatible kidney)
• H nodes will have a probability that decreases with the size of the graph (e.g. in a simple case we’ll keep the number of compatible kidneys constant, pH = c/n)
• In the H subgraph, we’ll observe trees but almost no short cycles
• A non-directed donor can be modeled as a donor with a patient to whom anyone can donate—this allows nondirected donor chains to be analyzed as cycles
• (We also consider the effect of different assumptions about
23 how the number of non-directed donors grows…)

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H
Cycles and paths in random dense-sparse graphs
• n nodes. Each node is L w.p. À and H w.p. 1À
• incoming edges to L are drawn w.p.
• incoming edges to L are drawn w.p.
L

Cycles and paths in random sparse (sub)graphs (v=0, only highly sensitized patients)
The number of cycles of length O(1) is
Theorem.
O(1) . (a)
But when p
H is a large constant there is cycle with length O(n) (b)
Proof (a):
H
26 To be logistically feasible, a long cycle must be a chain, i.e. contain a NDD

Cycles and paths in random sparse graphs (v=0)
Theorem.
(a) The number of cycles of length O(1) is O(1) .
(b) But when p
H is a large constant there is path with length O(n)
Since cycles need to be short (as they need to be conducted simultaneously) but chains can be long (as they can be initiated by an altruistic donor,) the value of a non-directed donor is very large!
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H

Case v>0 (some low sensitized, easy to match patients . Why increasing cycle size helps
Theorem. Let C k cycles · cycles · be the largest number of transplants achievable with k . Let D k be the largest number of transplants achievable with k plus one non-directed donor . Then for every constant k there exists
½>0
Furthermore , C k and D k cover almost all L nodes.
L
H
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Case v>0 . Why increasing cycle size helps
Increasing cycle lengths significantly increases transplants.
Highly sensitized patients are the principal beneficiaries.
Low sensitized pairs of all blood types are overdemanded: it ’s easy to start a cycle from L to H since there are many H, and easy to end it back in L since most blood type compatible donors will do …
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Long chains benefit highly sensitized patients (without harming low-sensitized patients)

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NKR non-directed donor chain:2012. 60 lives, 30 kidneys: the practical implications are clear

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• We are seeing some hospitals withhold internal matches, and contribute only hard-tomatch 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”.

b
1 b
2 a
2 a
3 a
1 a
4 b
3

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• No strategyproof deterministic mechanism can give ½ the efficiency a
4 b
1 a
2 a
3 b
2 a
1 b
3

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• No strategyproof randomized mechanism can give 7/8 the efficiency a
4 b
1 a
2 a
3 b
2 a
1 b
3

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Individual rationality and efficiency: an impossibility theorem with a (discouraging) worst-case 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.

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a
3 a
1 a
2 c b d

No. of
Hospitals 2 4 6 8 10 12 14 16 18 20 22
IR,k=3 6.8 18.37 35.42 49.3 63.68 81.43 97.82 109.01 121.81 144.09 160.74
Efficient, k=3 6.89 18.67 35.97 49.75 64.34 81.83 98.07 109.41 122.1 144.35 161.07
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But the cost of not having IR could be very high if it causes centralized matching to break down

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But current mechanisms aren’t IR for hospitals
• Current mechanisms: Choose (~randomly) an efficient allocation.
Proposition : Withholding internal exchanges can (often) be strictly better off (non negligible) for a hospital regardless of the number of hospitals that participate.
A-O
O-A
And hospitals can withhold individual overdemanded pairs

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• Infinite horizon
• In each timestep, a hospital samples its patients from some known distribution
• Then there exists a truthful mechanism with efficiency 1 – o(1)

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• Initially the hospital has zero credits
• In the beginning of the round, if the hospital has zero credits, each patients enters the match with probability 1 – 1/k 1/6
• For each positive credit, the hospital increases this probability by 1/k 2/3 and the credit is gone
• For each negative credit, the hospital decreases this probability by 1/k 2/3 and the credit is gone. The probability is always > ½

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• For each patient over k, the hospital gets 1 credit
• For each patient below k, the hospital looses 1 credit
• These credits only affect the next rounds

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• Hiding a patient can give an additive advantage, but causes a multiplicative loss
• Number of credit doesn’t matter – you always care about the future
• Can work for every distribution of patients

What is the tradeoff between waiting and number of matches?
Dynamic matching in dense graphs (Unver,
ReStud,2010).

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Simulation results using 2 year data from NKR*
Matches
550
500
450
400
350
300
2-ways
3-ways
2-ways & chain
3-ways & chain
1 5 10 20 32 64 100 260 520 1041
Waiting period between match runs
In order to gain in current pools, we need to wait probably “too” long
*On average 1 pair every 2 days arrived over the two years

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Simulation results using 2 year data from NKR*
Matches – high PRA
230
210
190
170
150
130
110
90
1 5 10 20 32 64 100 260 520 1041
In order to gain in current pools, we need to wait probably “too” long
2-ways
3-ways
2-ways & chain
3-ways & chain
*On average 1 pair every 2 days arrived over the two years

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295
290
285
280
275
270
265
260
255
250
Simulation results using 2 year data from NKR*
Matches
1D 1W 2W 1M 3M 6M 1Y
Waiting Time
240
220
200
180
160
140
120
100
1D 1W 2W 1M 3M 6M 1Y
In order to gain in current pools, we need to wait probably “too” long
*On average 1 pair every 2 days arrived over the two years


Online:
A H-node forms an edge with each node u of U with probability ξ/n.
 A L-node forms an edge with each
Arriving pair and n is large enough (even with just 2way cycles)
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Either a sparse finite horizon model or an infinite horizon model and analyze steady state

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H
L
Dynamic matching in dense-sparse graphs n nodes. Each node is L w.p. q< 1/2 and H w.p. 1-q • incoming edges to L are drawn w.p. • incoming edges to L are drawn w.p. •

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H
L
Dynamic matching in dense-sparse graphs n nodes. Each node is L w.p. q< 1/2 and H w.p. 1q • incoming edges to L are drawn w.p. • incoming edges to L are drawn w.p. •
At each time step 1,2,…, n, one node arrives.

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 (PRA).
 PRA determines the likelihood that a patient cannot receive a kidney from a blood-type compatible donor.
 PRA < 79: Low sensitivity patients (L-patients).
 80 < PRA < 100: High sensitivity patients (H-patients).

 Distribution of High PRA patients in the pool is different from the population PRA.
𝑝 2

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At time steps Δ, 2Δ, …, n:
Find maximum matching in H-L; remove the matched nodes.
Find maximum matching in L-L; remove the matched nodes.

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Chunk matching finds a maximum matching at time steps Δ, 2Δ, …, n.
M(Δ) - expected number of matched pairs at time n
.
Theorem (Ashlagi, Jalliet and Manshadi): When matching only 2way cycles:
1. If Δ = o(n),
M(Δ) = M(1) + o(n)
2. Δ = αn, then
M(Δ) = M(1) + f(q,p)n for strictly increasing f()>0.

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When matching 2 and 3-way cycles:
1. If Δ = 𝜔(𝑛) M(Δ) = M(1) + f(q,p) (n)
(formally this is still a conjecture  )

57 𝑝
𝐻
= ξ 𝑛 −1+𝜀 :
M(Δ) = M(1) + o(n)
1. If Δ ≤ 𝑛
Theorem:
1−2𝜀
< 1/ 𝑝
𝐻
,
2. If Δ = α/ 𝑝
𝐻
M(Δ) = M(1) + f(q)n for strictly increasing f()>0.
Need to wait less time to gain…
If the graph is dense (large) – no need to wait at all…

 Special structure: Sparse H-L and dense L-L.
 ( PRA).
 PRA determines the likelihood that a patient cannot receive a kidney from a blood-type compatible donor.
 PRA < 79: Low sensitivity patients (L-patients).
 80 < PRA < 100: High sensitivity patients (H-patients).
 Most blood-type compatible pairs that join the pool have H-patients.
 Distribution of High PRA patients in the pool is different from the population PRA.
58 𝑝 2
 Compare the number of H-L matchings.

In H-L graph, Δ = o(n):
 No edge in the residual graph.
arrived chunk
 Tissue-type compatibility: Percentage Reactive Antibodies (PRA).
 PRA determines the likelihood that a patient cannot receive a kidney from a blood-type compatible donor.
 PRA < 79: Low sensitivity patients (L-patients).
 80 < PRA < 100: High sensitivity patients (H-patients).
residual graph
Most blood-type compatible pairs that join the pool have H-patients.
 Distribution of High PRA patients in the pool is different from the population PRA.
 Decision of online and chunk matching are the same on depthone trees. M(Δ) = M(1) + o(n).
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In H-L graph, Δ = αn:
 Find f(α)n augmenting paths to the matching obtained by online.
 Given M the matching of the online scheme: h1 h2
60 l1 l2
 Chunk matching would choose
(l1,h1) and ( l2,h2 ).
M(Δ) = M(1) + f(α)n,

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M(Δ) - expected number of matches using only 2-ways
MC(Δ) - expected number of using 2-ways and allowing an unbounded chain
.
Theorem (Ashlagi, Jalliet and Manshadi):
MC(1) = M(1) + f(q)n