The Arrow Impossibility Theorem

I am very grateful to the two lecturers; it was indeed an honor to have Eric Maskin and Amartya Sen speak about my now quite old impossibility theorem. I cannot imagine two better discussants. Let me first turn to Eric’s presentation. He presented an extraordinarily interesting theorem about the situations in which the impossibility theorem fails. In other words, he imposes restrictions on the sets of individual preferences and finds that under his conditions majority voting will work. Of course he added the condition of anonymity to the ones I impose in order to achieve that result. It is an extraordinary simple equation, and he puts great emphasis on the role of majority voting in the sense of Condorcet. That is to say, we consider all the pairwise comparisons and pick the candidate who beats everyone individually.

However, there is of course one condition that does seem to be essential to any kind of social choice rule—namely, that it give a result. That is to say, in the usual terminology, that it be decisive. I do not yet quite understand how Eric’s results can help us in the case where his conditions fail. Something has to happen if majority voting is intransitive or, in other words, where the restrictive set of preferences is insufficient to overcome the impossibility theorem. This is a pretty key issue. So I leave that as an open question for Eric to analyze further.

I will now turn to Amartya’s insights and beautiful exposition of my theorem. I fully agree with his comments, but I have two remarks. First, he notes that I called the result a possibility theorem and attributes this to my sunny disposition. The facts are a little different. I have always regarded myself rather as a gloomy realist—but perhaps I am wrong. Instead, someone else, Tjalling Koopmans,² insisted on using the word possibility. He was upset by the term impossibility. Now, I cannot say that Tjalling had an extraordinarily sunny disposition, either. He was not necessarily a lively or cheerful person, nor was he really an optimist. But he did dislike the feeling that things could not happen or change. And given that the dissertation was originally posed as a Cowles Commission monograph, I felt that to please Tjalling, I would call it a “possibility theorem.” It was not, however, my idea at all.³

Another issue concerns Amartya’s discussion of the informational basis of preference in relation to interpersonal comparisons. He notes that in my original formulation the orderings of individuals were considered separately without any interpersonal comparisons. And Amartya is correct that I did not address the question of conversation and dialogue in the formation of preferences, the meaning of changing your opinion when talking to someone else. That said, the contention that the impossibility theorem ignores interpersonal comparisons slightly misinterprets my intent. While I did not allow for the interpersonal comparison of utilities, this did not mean that interpersonal comparisons were completely excluded. Instead, the individual preferences that form the basis of the social welfare function involve such comparisons. Individual preferences are about society. Of course the individual might give more weight to themselves than others, but the pairwise comparisons are rankings of different social orderings. Thus in constructing the impossibility theorem, a particular type of interpersonal comparison was present.

The organizers of the lecture and this volume have asked me to address a few other items. One concerns what aspects of social choice theory I would be interested in pursuing today. From a technical point of view, I would like to see more research on the condition that I find, in a way, to be the most problematic: the independence of irrelevant alternatives. In other words, if this condition is relaxed is there a decision mechanism that will satisfy the other restrictions? I might, for instance, put in a few candidates on the ballot who were not actually available. They could serve, in a way, to give some measurement to the others. Now I do not exactly know how to do this in a way that is going to be consistent. But by dropping the independence of irrelevant alternatives condition, you could then employ a Borda count, the method proposed by Balinski and Laraki,⁴ or perhaps some other alternative. Regardless, what emerges is a consistent ranking, which satisfies the other conditions, not based on cardinal utilities but instead on the rankings. Of course if a candidate drops out, different results emerge. Whether this is a good decision mechanism or not, I am not prepared to say. Regardless, it is an interesting area for further exploration.

Related to the relaxation of the condition of irrelevant alternatives are efforts to develop an interpersonal scale. If you look at the questionnaires that economists use to assess, for instance, the happiness of people, they will ask a respondent, “How happy are you?” The person then ranks their happiness on a scale. Now a rock bottom, hard-boiled economist might say to the individual, “You had a happiness of one, and now it’s three? What do you mean?” Such an approach is somewhat equivalent to the work of the psychologist Stanley Smith Stevens whose research had people make comparisons that were equivalent to asking, “Is this light brighter than this sound is loud?”⁵ Instead of saying, “I don’t know what you are talking about,” people respond to these questions and find them meaningful. And if people find them meaningful, then I have to say that in my point of view they must be so. The responses of individuals permit, moreover, a systematic representation. Thus the work of economists, such as Daniel Kahneman and Alan Krueger and their “Day Reconstruction Method,”⁶ is more or less equivalent to the development of an interpersonal scale and overcoming the independence of irrelevant alternatives. While I do not have a mathematical theorem capturing the findings of these studies, the formulation of some other kind of condition based on this research that replaces the independence of irrelevant alternatives would be an interesting avenue to explore.

Another and somewhat related area, and one that has important relevance to climate change issues, is the comparison of utility streams over time and what special properties we want associated with such comparisons. Some people try to argue for complete symmetry or that people a few hundred years from now are equal to individuals today—in other words, the principle of universalizability. Such a stance leads, however, to a paradox of sorts if you go out to infinity. If we impose the condition, for example, that if one stream is better in every period than another, or at least as good in every period and better in one, it should be better. But if we are all alike, and I change, would the second person or the individual next to them, and so on down to the person in a hundred years also change so that you could switch them around? It should make no difference, yet it does. Under the condition of universalizablity, if future individuals are going to be better off than we are, then our willingness to sacrifice on their behalf is certainly reduced.⁷ Such paradoxes stress the central importance of the axioms that we utilize when thinking about utility streams over time and issues of intergenerational equity in relation to climate change.

Related to the previous matter is the issue of constructability. When you are dealing with infinite dimensional elements, can you really compute the results? Some things are simply quite extremely difficult to compute. They’re not constructible in the sense that there is no finite process that will enable an individual to carry out the calculation. This applies to a lot of problems, not just those that are social in nature, such as climate change, but also to individual as well as social choice problems. To put it more simply, you could say, “You choose the best of that heap.” But then how one exactly does that can be quite complicated if not impossible in a finite length of time.

While the points just made are focused on future research paths related to the impossibility theorem, the organizers asked me to speak about the various inspirations for my dissertation. Many of these sources have been discussed at length elsewhere, but I do have a related anecdote to end with that speaks to the, shall we say, unexpected sources of ideas. I cannot say that my interest in elections was extraordinary and that it steered me toward the impossibility theorem. Yet I do distinctly recall an incident when I was eleven years old. It revolved around the 1932 Democratic Convention. In those days the conventions were a big deal. There were a lot of candidates, and at the time only a handful of states had primaries. Instead the local political machines would designate their delegates, and the presidential candidate was selected at the convention (that year the New York delegation was predominantly for Alfred Smith and not for Franklin Delano Roosevelt). The convention really decided things, and so, like most people, I listened. I remember turning on the radio and getting my sister, who was seven years old. I made a big chart with all the candidates on the left and all the states along the top or something to that effect. A lot of the states had their own local candidate who would be run for several ballots, just to get the name nationally known. During the voting every state reported and their selections were called out, such as, “Alabama casts 24 votes for Alfred Smith” and so and so. I then dictated to my sister what to write down on the chart during all the rounds of voting. To this day she teases me that given her assistance in this endeavor that she was the one who started me on my social choice career!

2. Tjalling Charles Koopmans (1910–1985): MA in physics and mathematics, University of Utrecht, 1933; PhD in mathematical statistics, University of Leiden, 1936; lecturer, Netherlands School of Economics, 1936–1938; economist, League of Nations, Geneva, 1938–1940; research associate, Princeton, 1940–1941; Penn Mutual Life Insurance Co., 1941–1942; statistician, Combined Shipping Adjustment Board, 1942–1944; Cowles Commission, 1944–1945; professor, University of Chicago, 1946–1955; professor, Yale, 1955–1985.

3. For a more lengthy discussion of the origins of the impossibility theorem, please see the following two sources: Kenneth Arrow, Amartya Sen, and Kotaro Suzumura, “Kenneth Arrow on Social Choice Theory,” in Handbook of Social Choice and Welfare, vol. 2 (Elsevier BV, 2011), pp. 3–27; and Kenneth Arrow, “The Origins of the Impossibility Theorem,” in History of Mathematical Programming, eds. Jan Karel Lenstra, Alexander H. G. Rinnooy Kan, and Alexander Schrijver (Amsterdam: Elsevier Science, 1991), pp. 1–5; also reprinted in this volume.