<br><h3> Chapter One </h3> <b>Kenneth Arrow on Social Choice Theory</b> <p> <p> Kenneth Arrow, Amartya Sen, and Kotaro Suzumura <p> <p> Contents <p> Part I: An Editorial Note 3 Part II: An Interview with Kenneth J. Arrow 4 References 23 Part III: The Classification of Social Choice Propositions 24 <p> <p> <b>Part I: An Editorial Note <p> Amartya Sen and Kotaro Suzumura</b> <p> Kenneth Arrow founded the modern form of social choice theory in a path-breaking contribution at the middle of the twentieth century. We—the editors of the <i>Handbook of Social Choice and Welfare</i> other than Arrow—want to begin this final volume by noting the continuing need to read Arrow's decisive contribution in his epoch-making book, <i>Social Choice and Individual Values</i>, which started off the contemporary round of research on social choice theory. (Kenneth J. Arrow, <i>Social Choice and Individual Values</i>, New York: John Wiley & Sons, 1st edition, 1951, 2nd edition, 1963.) <p> We are also including in this chapter an interview that Kenneth Arrow gave to Professor Jerry Kelly some years ago, which was published in <i>Social Choice and Welfare</i> in 1987. This presents Arrow's thinking on the subject as it developed since his own pioneering contribution. <p> Finally, this chapter also includes some new observations by Arrow, "The Classification of Social Choice Propositions," dealing particularly with the distinction between normative and descriptive statements in social choice theory. These notes, which Arrow has written for this volume at a very difficult time for him, reflect, inevitably in a highly compressed form, some recent thoughts of the founder of the discipline on an important methodological issue in social choice theory. <p> This is also an occasion for us—the editors other than Arrow-to acknowledge the huge benefit that the subject of social choice theory has received through the active contributions of the founder of this modern discipline over the last sixty years. We also would like to say how privileged we have been to work with Arrow in editing this two-volume <i>Handbook of Social Choice and Welfare</i>. <p> April 2010 <p> <p> <b>Part II: An Interview with Kenneth J. Arrow <p> Kenneth Arrow and J. S. Kelly</b> <p> The following is an edited transcript of an interview conducted on March 4, 1986, with Professor Arrow while he was visiting Syracuse University to deliver the Frank W. Abrams Lecture Series to be published as <i>The Uncertain Future and Present Action</i> by Syracuse University Press. This interview was to elaborate on his description, presented in Volume 1 of his <i>Collected Papers</i> (Harvard University Press, 1983) of the origins of his work in collective choice theory. <p> <p> <i>JK. You started off the story in the collected papers with remarks about studying relational logic while you were in Townsend-Harris High School in New York City. <p> KA.</i> Not in high school in the sense of [being] in my high school courses, but during this period I was an omnivorous reader and got into all sorts of things. One of them was Bertrand Russell's <i>Introduction to Mathematical Philosophy</i>, and it made a tremendous impression on me. It was the idea of logic that was in there. I don't really recall, for example, if there was a formal definition of a relation as a set of ordered pairs, but I learned the ideas of mathematical logic and its applications to mathematics in Russell's book. It seems to me that I also read one or two other logic books around that time. <p> <p> <i>JK. Later, when you went to City College of New York as a mathematics major, you encountered more mathematical logic. <p> KA.</i> Yes, but again the logic study was on my own, there were no courses in it. I don't really remember exactly what I read. I remember once taking out the <i>Principia Mathematica</i>, but of course it's not the sort of thing one really reads from. I was looking up some theorems in it and things like that. I really am not prepared to tell you what I read, but at some point things like the idea of defining rational numbers by ordered pairs and equivalence classes by ordered pairs was something I got to know. I was fascinated by this and used to aggravate my professors by writing out proofs in very strictly logical form, avoiding words as much as possible and things of that kind. <p> <p> <i>JK. You did take a formal course with Tarski in the Philosophy Department; how did you happen to take that course? <p> KA.</i> Yes. Well, I <i>knew</i> that Alfred Tarski was a great and famous logician, and there he was in my last term in school, and obviously I was going to take a course with Alfred Tarski. It turned out he had two courses. One was a kind of introductory course and I felt I knew more than <i>that</i>. The other course he gave was in the calculus of relations. To say it was in the <i>calculus</i> of relations meant that he gave an axiomatic treatment of relations, although he motivated it of course by motivating the axioms. You never had <i>xRy</i>; you only had <i>R</i> and <i>S</i> and <b>T</b>. You see, he never mentioned <i>individuals</i> in the formal theory. He had an axiomatic theory like an axiomatic treatment of set theory. Relations have some special aspects, in particular the idea of relative product, <i>RS</i>. If there is a <i>z</i> such that <i>xRz</i> and <i>zSy</i>, then <i>xRSy</i>. The relative square, <i>R</i><sup>2</sup> = <i>RR</i>, is especially interesting; if the relative square is included in <i>R</i> you have transitivity. <p> So it was a fascinating thing, although it was really very elementary, really very easy. The concepts were not very subtle compared with the deep things he was working on like the truth principle. <p> <p> <i>JK. At this point you were involved in translating some of Tarski's work. <p> KA.</i> He wrote a textbook called <i>Introduction to Logic</i> (Tarski 1941), which is one of the modern treatments, modern as of 1940. It had been published in German, may even have been orginally published in Polish. I didn't translate it. What happened was he had a translator and I read the proofs. I was just finishing college and he asked me to read the proofs for him. He didn't know any English, you see. This was the interesting thing. He came to this country in September 1939, for some kind of congress or conference and was trapped here by the outbreak of the war. He knew Polish, he knew German, but he didn't know any English, so he spent the Fall term learning some English so he could teach us in the Spring. At first we couldn't understand a word he was saying, but after about a week or so we began to catch on and we realized it wasn't <i>his</i> rate of progress it was <i>our</i> rate of progress that was relevant. His stresses were all wrong. He was aware of this and therefore felt he couldn't proofread in English. It's rather interesting as a coincidence that the translator was a German philosopher named Olaf Helmer, and Helmer comes back into my story eight years later. <p> It's interesting ... Tarski, although his English was weak, had a very good sense of language, and he kept on asking me, "Is that really good English?" Not in the sense of being grammatically correct, but, well for example, Helmer was very fond of using the word "tantamount," and Tarski got the feeling that somehow it's not a word used very often. Actually his instincts for language were extremely good. I suppose that was connected with his general work on formalizations and metalanguages. Anyway, I was just a proofreader. <p> <p> <i>JK. You write that as a graduate student at Columbia you spent time, as an exercise, translating consumer theory in the logic of relations and orderings. What got you started on that and what did you get out of it? <p> KA.</i> I went to Columbia because ... well there were several problems. One was that we were extremely poor and the question of going anywhere depended on resources. Columbia had the great advantage, of course, that I could live at home, which wasn't true anywhere else. I didn't get any financial support for my first year, none at all. <p> But another of the things I had learned on my own at college was mathematical statistics, and I really had become fascinated with it. There was a course in statistics [at City College New York]; the teacher, a man by the name of Robinson, had no <i>real</i> knowledge of it I would say, basically—I won't even say he had a good reading list—but he did list one book, J. F. Kenney (1939), if I remember correctly, which happened to have an excellent bibliography. It was not one of those cookbooks in statistics but actually did have some attempts at mathematics. Kenney had references to R. A. Fisher and gave you enough to get you interested. So I started reading Fisher, and one of the first things was trying to work out his derivation of the distribution of the correlation coefficient under the null hypothesis, which was an integration in <i>n</i>-dimensional space. In Fisher it was done by intuition. I mean it's rigorous if you're sufficiently sophisticated; to me it was gibberish. But I knew enough multivariate calculus to be able to translate it into rigorous form, at least a form that I understood, and then I could see that he really was right. But I couldn't see it the way he wrote it. Then I suppose because of my logical background what was really important was reading the Neyman–Pearson papers, which were then new and written in rather obscure places, but they were available in the [CCNY] library. From Fisher alone, I think I would have been hopelessly confused about the logic of statistical tests, although Fisher was great on deriving distributions. <p> So, I knew I wanted to study mathematical statistics, which however was not a field, not a Department at Columbia. It was spread out in other Departments. I knew that Hotelling was one of the major figures, but he was in the Economics Department. I rather naively thought I would study mathematics and then would take the statistics from Hotelling. I had no interest in Economics. <p> I was in the Mathematics Department, taking courses like Functions of a Real Variable, but I was going to take courses from Hotelling. In the first term he happened to give a course in Mathematical Economics. So out of curiosity I took this and got completely transformed. <p> The course to an extent revolved around Hotelling's own papers. But, as it happens, they were kind of central. He gave a rigorous derivation of supply and demand. There was one paper on the theory of the firm, one on the theory of the consumer (Hotelling 1932, 1935). And he gave a rigorous derivation of demand functions in the consumer theory paper and derived the Slutsky equations. I think he knew about Slutsky's work, though I'm not sure he actually referred to Slutsky. So, anyway, this was one of the best papers around at the time. It's now a staple of our literature but then really was novel. One of the things, he was a very, <i>very</i> strong ordinalist, emphasized that all these results were invariant under monotone transformations, which was not a normal practice in economics at that time. Of course, all those who were coming of age, like Paul Samuelson, would jump to that position; it was the normal position of the avant garde. <p> Well, the idea was that it was an <i>ordering</i>. It was clear that what they were saying was "<i>x</i> is better than <i>y</i>" and that this is a transitive relationship. And I recognized that there were certain continuity axioms that had to be added to that. I was already familiar with that because there were certain similar things in the foundations of probability theory. In fact I think I worked that out for myself. I was playing around once in college trying to work out an axiom system for probability theory, that was work on an Honors paper or something, and I ran across a set of axioms by Karl Popper. Research methods were pretty primitive; I looked through the Union catalog and there was a reference to an article in <i>Mind</i> by Popper (1938). I realized that his axiom system really couldn't explain certain things that we take for granted, like the fact that cumulative distributions have a one-sided continuity property. So I realized that you need some kind of extra continuity axiom, and I sort of invented countable additivity all by myself. Later, of course, I found that Kolmogoroff and others had done this, but I could see there had to be an axiom. <p> So I was kind of familiar from having worked it out there that you needed these continuity axioms in order to close your preference theory system. It was easy to provide, and I suppose others were doing the same. I could also see that while it was clarifying for me, it was hardly a contribution to knowledge, because all I was doing was translating to a language that I knew. At least it got me thinking; whenever I saw a <i>U</i> for a utility function I translated to a preference ordering. <p> In fact one thing that struck me as an interesting problem—this is digressing a bit, but not entirely—why should there be a utility function representing an ordering? Hotelling had never really asked that question. Although he emphasized that the indifference map was the primitive, and the utility function only represented it, he didn't really ask, "Why should you have a representation in terms of numbers?" I was really thinking about this problem when I happened to run across some papers by Herman Wold (1943, 1944) who gave what he called a "Synthesis" in some papers in <i>Skandinavisk Aktuarietidskrift</i>, which gave a long treatment of demand analysis that did have essentially an axiomatic point of view. There he said you've got to prove there is a utility function representation. He was the first person I know to realize, in print, that this was a problem. He gave an answer, extremely weak because he needed strong assumptions. <p> Anyway, then I switched to Economics from Mathematics. I had gone to Hotelling asking for a letter of recommendation for a fellowship in the Mathematics Department and he said, "Well, I'm sure I don't have any influence in the Mathematics Department, but if you should enroll in Economics, I've found in the past they are willing to give one of my students a fellowship." I was bought. <p> Incidentically, I impressed him on about the second day of the class because he was fascinated by Edgeworth's taxation paradox; in fact his paper on the theory of the firm was called "Edgeworth's Taxation Paradox and the Nature of Supply and Demand Theory" (Hotelling 1932). Consider a case where there are first-class and third-class railroad tickets as in the English system. It turns out that if you impose a tax on one ticket then, with suitable demand functions, you could lower the price of both commodities. At the time there was a lot of excitement about that; the public finance people were pooh-poohing it, saying, "How can this be?" It had to do with the nature of interrelated demand curves and that was the big thing Hotelling stressed, that demand functions depended on <i>n</i> variables, not one variable. But he said he was puzzled by the fact that he had never been able to produce an example of Edgeworth's paradox with linear demand functions. So I sat down and wrote out the conditions for linear demand functions to yield the paradox; these conditions were certain inequalities on the coefficients and the inequalities were inconsistent. So I came in the next day and showed it to him. Really it was just a few lines, but from that point on he was really impressed with me. It was an extremely easy calculation, but thinking in inequality terms was not common. Little pieces were quite easy to prove, but you couldn't do it in the mechanical fashion that you were doing with, say, solving simultaneous equations or maximizations. <p> <i>(Continues...)</i> <p> <p> <!-- copyright notice --> <br></pre> <blockquote><hr noshade size='1'><font size='-2'> Excerpted from <b>Handbook of Social Choice and Welfare</b> Copyright © 2011 by Elsevier BV. Excerpted by permission of North-Holland. All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.<br>Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site.