These computations could be refined to produce more exact figures, but a clear conclusion emerges: even if charitable motives are discounted, people still have a substantial personal interest in maintaining a system in which organ transplants are available.

NOTATION

The notations used in the equations presented in this appendix are defined as follows:

N

=

total relevant population,

w

=

annual number of additions to the waiting list for a solid-organ transplant,

y

=

number of years in which a person might be on a waiting list for an organ,

g

=

number of people in a person’s “close” social network, and

pg,y

=

probability that at least one of g specified people will be placed on a waiting list in a y-year period.

ASSUMPTIONS

For the simple base model presented here, assume that all people have the same probability of being placed on a waiting list for organ transplantation in any given year and that their fates are independent of each other. Also assume that the proportion of the population that is added to the waiting list in a given year and the age distribution of the population are constant over the years. These assumptions are not crucial to the qualitative results but greatly simplify the initial analysis and exposition and can be relaxed in a more general derivation later.

The probability that the average person will be placed on a waiting list for a solid organ in a given year (g = 1, y = 1) is a simple relative frequency

(1)

The probability that at least one of a specified group of g people will be placed on a waiting list in a given year is

(2)



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