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

University of California, Santa Barbara

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? About 9 million different types Probability that two random people match –Both US whites : 1/11,000 –Both Afro-American: 1/100,000 –Both Asian-American: 1/30,000 –Afro and Caucasian : 1/110,000 In contrast to blood transfusions.

Distribution of type size is very nonuniform About half the white population are in groups smaller than 1/100,000 of population. About 20 per cent are in groups smaller than 1/1,000,000 of population.

Rarer half: Magnified view

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. U.S. registry has about 6 million World registry about 10 million

Costs Cost of tests and maintaining records about $140 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 –Somewhat less traumatic for donor –Somewhat more risky for recipient

Social benefits from an additional donor: Behind the Veil of Ignorance Every person in society faces some small probability of needing a life-saving transplant. Adding a donor increases the probability of a match for every person. We numerically calculate the effect of an extra registrant on survival probability for each individual and value this increment at a “value of statistical life”.

Probability of having no match Let p i be fraction of the population that is of HLA type i. Probability that a person in i has no match in the registry is (1-p i ) R. Probability that a randomly selected person has no match in the registry is Sum i p i (1-p i ) R

Gain from extra registrant Calculate the derivative with respect to R of the probability of match. That is the derivative of Sum i p i (1-p i ) R

Value of a New Registrant With 3 million in registry, an extra white registrant increases the probability that a given white person finds a match by about 1.8 x What is this worth?

Increase in expected matches In a single year, about 6,000 people seek matches. Thus a new registrant increases expected number of matches by about 6 x x 10 3 =6 x But a new registrant remains in registry for several years. Suppose 10 years. Then over 10 years, expected number of matches found increases by 6 x

Little things add up Recipient of transplant gains 1/3 in probability of healthy life. Thus expected number of “lives saved” by an extra registrant is about 1/3x6x10 -4 = 2x With “Value of life of $5 x 10 6, the value of an additional registrant is about $1000.

Benefit cost for white US registrants Estimated Cost of new registrant $140 Estimated benefit $1000

Differences by racial group in US Race% of Pop# in registry Prob of no match White603 mil.09 Black12500 k.30 Asian k.20

Dollar values of added registrant Race of New Registrant Expected value in lives saved by race WhiteBlackAsian White$1187$588$477 Black$27$510$21 Asian$18$17$351 Others$58$101$50 Total$1291$1219$900

What is going on? A new white donor is much more likely to be just a duplicate, yet new white donors are at least as valuable as new blacks or Asians. All lives saved are valued equally. Difference is in number of people seeking transplants.

Who is seeking transplants? % of pop% of seekers White 6986 Black 124 Asian other 156

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. Probability that you are called on if there are k registrants of your type is 1/k.

Conditional probability of being pivotal if called upon. Probability White-American.06 Afro-American.30 Asian-American.18

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.