One%20Chance%20in%20a%20Million:%20An%20equilibrium%20Analysis%20of%20Bone%20Marrow%20Donation

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One Chance in a MillionAn equilibrium Analysis
of BoneMarrow Donation

• Ted Bergstrom, Rod Garratt, and Damian
  Sheehan-Connor

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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.

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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.

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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.

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Costs

• Cost of tests and maintaining records about
• 60 per registrant. Paid for by registry.
• Cost to donor.
• Bone marrowneedle into pelvis
• Under anesthesia
• Some pain in next few days.
• Alternate methodblood filtering
• Less traumatic for donor
• More risky for recipient

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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.

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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.

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Notation

• N populationthink 250,000,000
• R registrantsthink 5,000,000
• H HLA types--think 1,000,000
• xR/H average no of registrants in group
• nN/H HLA group sizeassume equal
• pR/N
• P(k,x) Probability that an HLA type has k
  registrants.

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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).

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Probability of being pivotal as a function of
xR/H
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
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Benevolence theory

• C Cost of donating
• B Value of being pivotal in saving someone
  elses life
• W Warm glow from donating without having been
  pivotal.
• Assume BgtCgtW.
• Person will donate if H(x)gt (C-V)/(B-V)

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Plausible numbers?

• Suppose V0
• If x5, then for registrants,
• C/Blt.034
• US registry has about 5 million donors or 2 of
  population.
• So the most generous 2 of population would need
  to have
• C/Blt 1/30.

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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.

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Optimal value of x
Marginal cost of registrant 60 30 15
Optimal xR/H 8 9 10
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To do list

• Non-uniform HLA distribution
• Numbers for races
• And More