Upload
manchu
View
24
Download
0
Embed Size (px)
DESCRIPTION
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. - PowerPoint PPT Presentation
Citation preview
One Chance in a Million:An equilibrium Analysis of Bone
Marrow DonationTed 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 about 10 million registrants.
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).
x 1 2 3 4 5 6 7 8
P(0) .37 .14 .05 .02 .006 .0025 .0009 .00033
H(x) .58 .31 .16 .07 .034 .015 .0064 .00268
Probability of being pivotal as a function of x=R/H
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<.034US 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.
• About 10,000 people in US had transplants last 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 registrant
$60 $30 $15
Optimal x=R/H 8 9 10
To do list
• Non-uniform HLA distribution
• Numbers for races
• And More…