The Arrow Impossibility Theorem

Eric Maskin, Amartya Sen, Kenneth Arrow, Partha Dasgupta, Prasanta Pattanaik, Joseph E. Stiglitz

• Publisher: Columbia University Press
• Year: 2014
• ISBN:
  9780231526869
• Topic: Volkswirtschaftslehre
• Subtopic: Wirtschaftstheorie

Kenneth J. Arrow's pathbreaking "impossibility theorem" was a watershed innovation in the history of welfare economics, voting theory, and collective choice, demonstrating that there is no voting rule that satisfies the four desirable axioms of decisiveness, consensus, nondictatorship, and independence.

In this book Eric Maskin and Amartya Sen explore the implications of Arrow's theorem. Sen considers its ongoing utility, exploring the theorem's value and limitations in relation to recent research on social reasoning, and Maskin discusses how to design a voting rule that gets us closer to the ideal—given the impossibility of achieving the ideal. The volume also contains a contextual introduction by social choice scholar Prasanta K. Pattanaik and commentaries from Joseph E. Stiglitz and Kenneth J. Arrow himself, as well as essays by Maskin, Dasgupta, and Sen outlining the mathematical proof and framework behind their assertions.

PART I

THE LECTURES

OPENING REMARKS

JOSEPH E. STIGLITZ

The Kenneth J. Arrow lectures are given in honor of Kenneth Arrow, who was one of Columbia’s most distinguished graduates. The topic of the second annual Arrow lecture, on which this book is based, was Ken’s thesis Social Choice and Individual Values. For anyone doing a PhD, it would be no bad thing to aspire to the standards this work has set.

The fact that Ken’s PhD thesis remains an icon more than a half century after its writing shows just how much it changed the way we think about the whole problem of social choice. That someone could even formulate the question his thesis poses reveals its author’s novel cast of mind. I find that it is still inspiring when I read it today.

The speakers for this lecture—Amartya Sen and Eric Maskin—were particularly suitable for the occasion because of their enormous contributions to the theory of social choice, elaborating on some of the ideas and aspects of the field that Ken opened up more than fifty years ago.

Amartya Sen has been so generously frequent and welcome a visitor at Columbia that I feel we can almost claim him as one of our own. I have known Amartya for more than forty years—we met in England when I was a graduate student in the late sixties—so his presence at the lecture was a particular pleasure for me. He is now the Thomas Lamont University Professor and Professor of Economics and Philosophy at Harvard University, and until recently he was the Master of Trinity College in Cambridge.

His research has ranged over a number of fields in economics, philosophy, and decision theory, and he has made particularly important contributions to the theory of social choice, which is our broad theme today. In 1998 he received the Nobel Memorial Prize in Economics for his contribution to welfare economics, and his work on social choice was notably mentioned in the citation of the prize.

Amartya and I worked together on the Commission on the Measurement of Economic Performance in Social Progress (2009), which was set up by French president Nicholas Sarkozy to translate some of Amartya’s ideas into the measurements of a country’s economic performance. An earlier related work, the Human Development Index of the United Nations Development Program, has become a standard metric, especially in the context of evaluating the performance in developing countries. Amartya played a central role in creating and shaping that index.

Amartya’s recent book The Idea of Justice is an important work that critically takes on a range of thinkers from Adam Smith to Rawls who have written on this central subject in philosophy, politics, and economics. Some of the ideas surface in his contribution to this book.

The second speaker, Eric Maskin, was the Albert Hirschman Professor at the Institute for Advanced Study at Princeton at the time of the lecture and is now the Adams University Professor at Harvard University. He is particularly well known for his work on mechanism design, including his work on how to design institutions for achieving particular social or economic goals. In recognition of his fundamental contributions, he shared the 2007 Nobel Memorial Prize in Economics.

Eric, like Amartya, has worked in many different fields. Indeed, his work has had a deep influence in almost every area of economics. It was a real pleasure to welcome him to Columbia.

ARROW AND THE IMPOSSIBILITY THEOREM¹

AMARTYA SEN

I

It was wonderful for me to have the opportunity to pay tribute to Kenneth Arrow, who is not only one of the greatest economists of our time but also one of the finest thinkers of our era. That itself made the occasion of the second annual Arrow lecture very special for me, but on top of that, it was marvelous to have the company of Eric Maskin, with whom I used to teach a most enjoyable joint course on social choice theory at Harvard, until he deserted us for the Institute for Advanced Study at Princeton.² And it was very pleasing for me to have Joe Stiglitz as the participating chair of the meeting (having known Joe for many years, I can assure you that there was no danger of Joe being an aloof chair) and to know that Akeel Bilgrami’s intellectual vision was behind the planning of this event. I was in admirable company at the lecture and want to begin this piece by expressing my appreciation of that, but most especially by thanking Ken Arrow himself, for making us all think in new lines, and personally for me, for being such a major influence on my own intellectual life.

I shall be particularly concerned in this essay with Arrow’s pathbreaking “impossibility theorem,” for which Arrow managed to find, in line with his sunny temperament, a rather cheerful name: “General Possibility Theorem.”³ This result, and with it the formulation of the demands of mathematical social choice theory, were real watersheds in the history of welfare economics as well as of voting theory and collective choice.

The informational foundation of modern social choice theory relates to the basic democratic conviction that social judgments and public decisions must depend, in some transparent way, on individual preferences, broadly understood. (I have investigated the implications of this perspective in social choice theory in my paper “The Informational Basis of Social Choice,” which is reprinted in part 2 of this book.)⁴ The emergence of this democratic instinct relates closely to the ideas and events that surrounded the European Enlightenment. Even though the pursuit of democratic social arrangements drew also on various earlier sources and inspirations, it received a definitive delineation and massive public acknowledgment only during the Enlightenment, particularly—but not exclusively—during the second half of the eighteenth century, which also saw the French Revolution and American independence.

What can be called “preferences” of persons can, of course, be variously interpreted in different democratic exercises, and the differences are well illustrated by the contrasts between (1) focusing on votes or ballots (explored in the classic works of Borda and Condorcet), (2) concentrating on the interests of individuals (explored, in different ways, in the pioneering writings of David Hume, Jeremy Bentham, and John Stuart Mill), and (3) drawing on the diverse judgments and moral sentiments of individuals about societies and collectivities (explored by Adam Smith and Immanuel Kant, among many others over the centuries). These contrasts, between alternative interpretations of preferences, can be very important for some purposes. I shall visit that territory before long. However, for the moment I shall use the generic term preference to cover all these different interpretations of individual concerns that could be invoked, in one way or another, to serve as the informational bases of public decisions and of social judgments.⁵

In contemporary social choice theory, pioneered by Kenneth Arrow, this democratic value is absolutely central, and the discipline has continued to be loyal to this basic informational presumption. For example, when an axiomatic structure yields the existence of a dictator as a joint implication of chosen axioms that seemed plausible enough (on this more presently), this is immediately understood as something of a major embarrassment for that set of axioms, rather than being taken to be just fine on the ground that it is a logical corollary of axioms that have been already accepted and endorsed. We cannot begin to understand the intellectual challenge involved in Arrow’s impossibility theorem without coming to grips with the focus on informational inclusiveness that goes with a democratic commitment, which is deeply offended by a dictatorial procedure. This is so, even when the dictatorial result is entailed by axiomatic requirements that seem reasonable, taking each axiom on its own.

So let me begin by discussing what Arrow’s impossibility theorem asserts and how it is established. The theorem has the reputation of being “formidable,” which is a good description of its deeply surprising nature as well as of its vast reach, but the air of distanced respect is not particularly helpful in encouraging people to try to understand how the result emerges. It is, however, important for people interested in political science, in welfare economics, or in public policy to understand the analytical foundations of Arrow’s far-reaching result, and there is no reason it should be seen as a very difficult result to comprehend and appreciate. A closer understanding is also relevant for seeing what its implications are and what alleged implications, often attributed to it, may be misleading.

The proof of the Arrow theorem I shall present follows Arrow’s own line of reasoning, but through some emendations that make it agreeably short and rather easy to follow. However, it is a completely elementary proof, using nothing other than basic logic, like Arrow’s own.⁶ The important issue here is not just the shortness of getting to the Arrow theorem but the ease with which it can be followed by anyone without any technical reasoning or any particular knowledge of mathematics or advanced mathematical logic. So I have spelled out fully the reasoning behind each step (perhaps too elaborately for some who are very familiar with this type of logical reasoning).

II

The basic engagement of social choice with which Arrow was concerned involved evaluating and choosing from the set of available social states (x, y,…), with each x, y, etc., describing what is happening to the individuals and the society in the respective states of affairs. Arrow was concerned with arriving at an aggregate “social ranking” R defined over the set of potentially available social states x, y, etc. With his democratic commitment, the basis of the social ranking R is taken to be the collection of individual rankings {R_i}, with any R_i standing for person i’s preference ranking over the alternative social states open for social choice. It is this functional relation that Kenneth Arrow calls the “social welfare function.” Given any set of individual preferences, the social welfare function determines a particular aggregate social ranking R.

That there could be problems of consistency in voting rules was demonstrated by the Marquis de Condorcet in the eighteenth century....