EXAMPLE 11 Organ Transplant Policies and Arrow’s Impossibility Theorem

Finding an equitable procedure for determining a rank ordering of patients in need of an organ transplant is complicated: there are several criteria that should be considered in arriving at such a “priority ranking.” Three such criteria are, for example, (1) the length of time that a patient has been waiting, (2) the probability of success as measured by the numbers of antigens that the patient and donor have matched, and (3) the fraction of the population unsuitable as donors for this potential recipient due to the presence of certain antibodies. A further discussion of these issues occurs in Section 13.3.

Each of the three criteria gives us a ranking (with ties) of the patients according to the more appropriate recipient of the next available organ, according to that particular criterion. Although these rankings are often determined by measurements, the use of different scales for different criteria muddies the water sufficiently so that you might want to work simply with the rankings derived from the measurements, as opposed to working directly with the measurements themselves. This is the context in which we will frame the problem.

So what does the search for a procedure to rank-order potential recipients of an organ have to do with voting? In a sense, everything, if looked at the right way. We can think of each criterion as a “voter” and each potential recipient as an “alternative.” The procedure that we seek is what social choice theorists call a social welfare function. It differs from a social choice procedure in that the result of an election is not a single winner or a group tied for the win, but a listing of the alternatives—the priority ranking, in our organ-transplant situation.

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For a moment, let’s return to the particular task of seeking a priority ranking of the potential recipients of an organ based on how they are ranked according to each of several criteria, like the three we mentioned earlier. What “reasonable” properties might we expect any such procedure to satisfy? Consider the following:

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  1. If one potential recipient is ranked above another potential recipient with respect to every single criterion, then we should expect to be ranked above in the priority ranking.
  2. If potential recipient is ranked above potential recipient in the priority ranking, and there are subsequent changes in how potential recipients are ranked with respect to one or more of the criteria, then potential recipient should not be ranked above potential recipient in the priority ranking unless has moved from being below to being above with respect to at least one criterion.
  3. No single criterion should dominate, in the sense that one potential recipient ’s ranking above another potential recipient ’s ranking, with respect to that criterion, guarantees that will be ranked above in the priority ranking.

If we accept these as being required of any “reasonable” procedure, then we have a striking (and highly non-obvious) fact to report: Our task of finding a reasonable procedure is impossible! In fact, this is precisely the statement of Arrow’s impossibility theorem in the context of social welfare functions: There is no social welfare function (for three or more alternatives) that satisfies Pareto (our first condition above), IIA (our second condition above), and nondictatorship (our third condition above).