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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)=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. • 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 $60 registrant $30 $15 Optimal x=R/H 9 10 8 To do list • Non-uniform HLA distribution • Numbers for races • And More…