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…