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# One Chance in a Million: An equilibrium Analysis of Bone Marrow Donation Ted Bergstrom, Rod Garratt, and Damian Sheehan-Connor.

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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 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)=x k e -x /k! (approximately Poisson). Probability that you are pivotal given that you are called on to donate H(x)=Sum k P(k,x)/k =x/(e x -1).

x12345678 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<.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. 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/H8910

To do list Non-uniform HLA distribution Numbers for races And More…

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