# Bonemarrow

```One Chance in a Million:
An equilibrium Analysis of Bone
Marrow Donation
Ted Bergstrom, Rod Garratt, and
Damian Sheehan-Connor
Background
• Bone marrow transplants dramatically
improve survival prospects of leukemia
patients.
• For transplants to work, donor must be of
same HLA type as recipient.
• Exact matches outside of family are
relatively rare.
How rare?
• At least 5 million possible types, not all
equally frequent.
• Probability that two randomly selected
people match is on order of 1/1,000,000.
• In sharp contrast to blood transfusions.
Bone marrow registry
• Volunteers are DNA typed and names
placed in a registry. A volunteer agrees to
donate stem cells if called upon when a
match is found.
• Matches are much more likely between
individuals of same ethnic background.
• Worldwide registry is maintained with
Costs
• Cost of tests and maintaining records about
\$60 per registrant. Paid for by registry.
• Cost to donor.
– Bone marrow—needle into pelvis
– Under anesthesia
– Some pain in next few days.
• Alternate method—blood filtering
– Less traumatic for donor
– More risky for recipient
Free rider problem for donors
• Suppose that a person would be willing to
register and donate if he new that this would
save someone who otherwise would not
find a match.
• But not willing to donate if he knew that
somebody else of the same type is in the
registry.
Nash equilibrium
• Need to calculate probability that a donor
will be pivotal, given that he is called upon
to donate.
• We do this with a simplified model.
Notation
•
•
•
•
•
•
•
N population—think 250,000,000
R registrants—think 5,000,000
H HLA types--think 1,000,000
x=R/H average no of registrants in group
n=N/H HLA group size—assume equal
p=R/N
P(k,x) Probability that an HLA type has k
registrants.
Distributions
• P(k,x)=xke-x/k!
(approximately Poisson).
Probability that you are pivotal given that you
are called on to donate
H(x)=Sumk P(k,x)/k =x/(ex-1).
Probability of being pivotal as a function of x=R/H
x
1
2
3
4
5
6
P(0)
.37 .14 .05 .02 .006 .0025 .0009 .00033
H(x) .58 .31 .16 .07 .034 .015
7
8
.0064 .00268
Benevolence theory
• C Cost of donating
• B Value of being pivotal in saving someone
else’s life
• W Warm glow from donating without
having been pivotal.
• Assume B>C>W.
• Person will donate if H(x)> (C-V)/(B-V)
Plausible numbers?
• Suppose V=0
• If x=5, then for registrants,
C/B<.034
US registry has about 5 million donors or 2% of
population.
So the most generous 2% of population would
need to have
C/B< 1/30.
Socially Optimal registry size
• Let N be the number of people who need
transplants and s be the probability that a
transplant saves a life.
year and s is about .4.
• Assume registrant remains in registry for 10 years.
• Expected number of lives saved by a new
registrant is 40,000 d/dx P(0,R/H) dx/dR.
• Value of statistical life, about \$5,000,000.
Optimal value of x
Marginal cost of \$60
registrant
\$30
\$15
Optimal x=R/H
9
10
8
To do list
• Non-uniform HLA distribution
• Numbers for races
• And More…
```
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