13 Fair Division

13.4 13.3 Fair Division and Organ Transplant Policies

In 1984, the U.S. Congress passed the National Organ Transplant Act and established a unified transplant network known as the Organ Procurement and Transplantation Network (OPTN). One of the primary goals of the OPTN was to increase the equity in the national system of organ allocation.

Achieving an equitable system of organ allocation is complicated by factors other than demand exceeding supply. For example, should an available organ go to the patient who needs it the most or to the one for whom the likelihood of a successful transplant is greatest? Should both of these be taken into consideration, and if so, how? Questions such as these reveal the extent to which an equitable system of organ allocation is a challenging problem in fair division.

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To illustrate some of the issues (and paradoxes!) arising in the search for an equitable system for organ allocation, we’ll (roughly) follow Peyton Young’s synopsis of the fair division procedure for kidney allocation adopted by the OPTN in the late 1980s (see Suggested Readings on page 570).

There were three (main) criteria used in arriving at a final ranking of those needing a kidney, and each potential recipient was awarded points according to a fixed method that we now describe.

Criterion 1: Waiting time. A list of potential recipients was made according to how long they had been waiting for an organ. For each potential recipient, one calculates the fraction of people at or below the spot on the list he or she occupies and then awards that person a number of points equal to 10 times that fraction. So if there are five people on the list, the first (waiting the longest) gets points, the second gets points, the third gets points, and so on.
Criterion 2: Suitability. The donor and potential recipient each have six relevant antigens that are either matched or not matched; the likelihood of a successful transplant increases with more matches. Two points are awarded for each match.
Criterion 3: Disadvantage. Each person has antibodies that rule out a certain percentage of the population as being potential donors for that person. For some, only 10% are ruled out, whereas for others it may be as high as 90%. Those in the latter category are at a serious disadvantage compared with those in the former. Thus, potential recipients are awarded one point for each 10% of the population they are “sensitized against.”

To illustrate this allocation procedure, let’s assume we have five potential recipients—, , , , and —with the following characteristics:

Potential Recipient	Months Waiting	Antigens Matched	Percent Sensitized
	5	2	10
	4.5	2	20
	4	0	0
	2	3	60
	1	6	90

According to the procedure we described, points would be allocated as follows:

Months Waiting	Antigens Matched	Percent Sensitized	Total Points
10	4	1	15
8	4	2	14
6	0	0	6
4	6	6	16
2	12	9	23

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Thus, if one kidney became available, it would go to (with 23 points). Presumably, if two kidneys became available at the same time, would get one and (with 16 points) would get the other.

But now things get interesting. Peyton Young, being well versed in the paradoxes of voting theory, fair division, and apportionment (among other things), observed the following possibility: What if two kidneys become available, but one is delayed slightly? Presumably, gets the first one, and then we redo the chart with only , , , and . This yields the following:

Months Waiting	Antigens Matched	Percent Sensitized
5	2	10
4.5	2	20
4	0	0
2	3	60

According to the procedure that we described, points would be allocated as follows:

Months Waiting	Antigens Matched	Percent Sensitized	Total Points
10	4	1	15
7.5	4	2	13.5
5	0	0	5
2.5	6	6	14.5

Thus, (not ) now gets the second kidney, having 15 points to 14.5 for . This is an example of what is called the “priority paradox.” For more on this, consult Peyton Young’s book, listed in the Suggested Readings.