The Arrow Impossibility Theorem (Kenneth J. Arrow Lecture Series) by Maskin Eric Sen Amartya (2014-07-22) Hardcover

Kenneth J. Arrow Lecture Series

by Amartya Sen

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.

Genres: Economics, Philosophy, Politics

Unknown Binding

First published June 3, 2014

About the author

Amartya Kumar Sen is an Indian economist who was awarded the 1998 Nobel Prize in Economic Sciences for his contributions to welfare economics and social choice theory, and for his interest in the problems of society’s poorest members.

Sen was best known for his work on the causes of famine, which led to the development of practical solutions for preventing or limiting the effects of real or perceived shortages of food. He is currently the Thomas W. Lamont University Professor and Professor of Economics and Philosophy at Harvard University. He is also a senior fellow at the Harvard Society of Fellows and a Fellow of Trinity College, Cambridge, where he previously served as Master from the years 1998 to 2004. He is the first Asian and the first Indian academic to head an Oxbridge college.

Amartya Sen's books have been translated into more than thirty languages. He is a trustee of Economists for Peace and Security. In 2006, Time magazine listed him under "60 years of Asian Heroes" and in 2010 included him in their "100 most influential persons in the world".

Reviews

[BlackOxford · Rating 4 out of 5]

Why Nothing Works Right

The Nobel Prize-winning economist. Kenneth Arrow, died this past week. His famous Impossibility Theorem is, like many profound ideas, more talked about than understood. Sen's book is both a tribute to the man and an introduction this very disconcerting idea.

The Arrow Impossibility Theorem is arrived at through some tricky maths. But its conclusion is easy to state: In any group of people who have to reach a decision together but who have even slightly different preferences about where they want to end up, the decision they will reach is that which they all can accept but which none of them wants.

It applies to any group - electorates, markets, corporate boards, local councils, ad hoc committees, working parties, family meetings - wherever a choice among alternative courses of action is required. It applies even when, in fact especially when, members of the group are respectful and concerned about each other and the continued unity of the group. It is the default condition of all human deliberations.

The Theorem has been popularised as, among other things, the Abilene Paradox (https://www.goodreads.com/book/show/6... ) which is frequently mentioned but rarely explored by management consultants and experts in decision-theory. The full impact of its meaning has never really been appreciated either in the social sciences disciplines where it should be of paramount concern, or among the general public who deserve to know the inevitable consequences of decision-making in areas as diverse as democratic politics and corporate strategy.

The truth is that none of these decisions have any solid claim to representative rationality. The Theorem is not some quaint paradox that is of marginal importance to mathematicians. It is the WMD of collective choice, the granddaddy poisoner of political wells, the hydrogen bomb targetted on our pretensions of social rationality. The implications are profound, perhaps too profound to deal with. So the Theorem is largely ignored.

But we may have reached a point in both political and corporate life in which the Arrow Theorem can no longer be ignored. It is obvious, for example, that the national politics in the United States has cracked under just the kind of pressure Arrow predicted. Its principle electoral effect of dissatisfaction is growing. Frustration with democracy is intensifying. Acceptance of results wears increasingly thin as decisions disadvantage voters disproportionately.

The psychological effects are just as important. Continuous compromise promotes the sentiment of dictatorship, the strong man who can cut through messy democratic politics as the only real solution to the problem. Dictatorship can at least promise the satisfaction of some segment of society. Others may be tremendously disadvantaged but that's merely the price necessary to pay for coherence of action. It is to a sort of dictatorial sump that democratic attitudes deteriorate in a possibly inevitable progression.

In short, the Arrow Impossibility Theorem points to a fatal flaw in every form of social governance upon which we rely in civic and corporate life. It is a formulation of our social original sin. We are born into it without a choice. It sits in our somewhat smug society waiting for a chance to demonstrate its power. To date, no one has devised a convincing way to even mitigate its effects much less neutralise them. The least we can expect as the paradox bites deeper into social life is more vituperative party politics and more of Donald Trump.

Postscript: For a discussion of one promising approach to the Arrow problem see https://www.goodreads.com/review/show...

[Athan Tolis · Rating 4 out of 5]

Let’s say all the boys in class decide to rank their favorite 20 cars.

So every boy writes down on a piece of paper his own ranking of those 20 cars and then we go about listing how we would like to work it all out.

0. Suppose we want our ranking to be transitive. So if I know that in the final ranking the Mercedes is above the Honda and the Honda is above the Fiat, then the Mercedes must be above the Fiat.

1. Suppose that no matter how many boys there are in class, no matter how many of the (more than 20 factorial, to accommodate draws, missing cars etc.) permutations are submitted, you want to want to come up with a voting scheme that will deliver one, definitive, ranking. (a property of the scheme known us “Unrestricted Domain”)

2. Suppose that in this definitive ranking the Ferrari beats the Mercedes. And then suppose we take the Chevrolet out of the list. If then we repeat the vote without the Chevrolet but with all the other 19 cars, the Ferrari had better still beat the Mercedes, or else our voting scheme is unsatisfactory. (a property known as “Independence of Irrelevant Alternatives”)

3. Suppose that in every single boy’s ranking the Daihatsu gets beaten by the Lamborghini. It had better also get beaten in the final ranking. (a property known as the “Pareto Principle”)

4. Suppose you don’t want there to be a boy who always prevails over the rest if he likes one car more than another (this is the “Non-Dictatorship Principle”)

The Arrow Impossibility Theorem says you’re out of luck. You can’t get all of the above.

I guess everybody who’s been to elementary school already knows this, but Kenneth J Arrow gave mathematical proof. The proof’s rather easy to follow and I close this review with my version of it. The book is dedicated to an exposition of the Theorem and its ramifications.

The result is not a big surprise, obviously, but it is the cornerstone of a beautiful theory. Armed with this result, other economists and philosophers have over the years looked at a number of “voting rules” such as the Anglo-Saxon “first past the post,” the French runoff system, the plurality voting rule, ranking the candidates etc. and worked out when they will yield satisfactory results.

This monograph of a book is written by some of the most prominent such theoreticians, including Amartya Sen, Eric Maskin and Partha Dasgupta, with short contributions from Joseph Stiglitz and Kenneth Arrow himself, all beautiful in their own way, though I must say I was confused by the introduction by Prasanta Pattanaik.

Also, there is a full paper here that derives some very significant results concerning when “rank-order” voting “works well” (i.e. satisfies conditions such as the ones I describe above), when “plurality rule” voting “works well,” when majority rule is decisive (answer: when there’s no “Concordet triplet” such that x>y>z for fewer than half the voters, y>z>x for fewer than half the voters and z>x>y for a third set of fewer than half the voters) and finally all this work yields the extremely powerful result that if, given a set of preferences, you can come up with some rule that “works well,” then so will majority rule (and that therefore we have not wasted 200 years of democracy using this rule)

That said, mathematical symbols are used when words would have fully sufficed. The complex math symbols are never, ever “pushed” in the proof. A Lebesgue integral is defined for no reason. (No use is ever made of measure theory anywhere past this definition) The author never says anything along the lines of “we recognize that this set is a group and apply Theorem X from group theory.” It’s 100% math for the sake of math, and I found that annoying, especially since the book is riddled with errata.

For example, on p. 112 there are extra brackets around the main expression that don’t belong; on p. 119 (and again on p. 120 and again on p.p. 143, 144) xRy and xRz and yRxRz and zRxRy are written with the second x’s and y’s as subscripts when the notation for “x dominates y under R” has been defined as xRy; on p. 123 we are assured that for some t
So here’s me cribbing Amartya Sen and having a go at proving the main result, always in the context of boys ranking cars:

Lemma 1: The stickler. Suppose there is a stickler, who is allowed to impose his will on all the other boys on just one pair of cars. Suppose little Johnny can get away with making everybody rank the Ford Pinto higher than the Aston Martin. It can be shown that this automatically makes him the dictator.

Suppose everybody (Johnny included) likes the Mustang more than the Pinto. And suppose everybody (Johnny included) likes the Aston Martin more than the Jaguar. Then, because Johnny likes the Pinto more than the Aston and because just for this one pair Johnny can impose his will on the entire class, the Mustang must (like for Johnny, to whom transitivity also applies) rank higher than the Jaguar in everybody’s ranking. Not only that, but we said it should make no difference if we take a car or two out. So then we can take both the Pinto and the Aston out of the reckoning and everybody should (like Johnny) still like the Mustang more than the Jaguar by the independence of irrelevant alternatives. And since everybody ranks the Mustang higher than the Jaguar, then by the Pareto principle the Mustang will rank higher than the Jaguar in the final ranking. And it’s all because of Johnny the stickler, who for all we know could be the only boy who likes the Mustang above the Pinto and the Pinto above the Aston Martin and the Aston above the Jaguar. In math speak, if there’s an individual who is locally decisive (i.e. over one pair), he is globally decisive.

Lemma 2: Contraction of decisive sets.

Suppose the football team runs the dictatorship. (i.e. it is decisive)

Suppose the offense team all like the Ferrari more than the Lamborghini and the offense team also all like the Ferrari more than the Maserati, but the offense team don’t have any preference between the Maserati and the Lamborghini.

Suppose the defence team all like the Ferrari more than the Lamborgini and the defense team also like the Maserati more than the Lamborghini, but the defense team don’t have a preference between the Ferrari and the Maserati.

Clearly, the whole football team likes the Ferrari more than the Lamborghini, so we all know which way they two rank.

Now suppose that in some (non-football-playing) boys’ lists the Maserati appears above the Ferrari or equal to the Ferrari. Because the football team as a whole ranks the Ferrari above the Lamborghini and the football team as a whole are the dictators, it means that in these boys’ lists the Maserati will have to appear above the Lamborghini too, because the football team ranks the Ferrari above the Lamborghini, so this means there are boys who had to obey the defense team’s orders with respect to ranking the Maserati and the Lamborghini. They could not have been obeying the offense team as the offense team could not care. So if there are boys who list the Maserati above the Ferrari and if the football team are dictators, then it was the defense team were decisive between the Maserati and the Lamborghini, which makes the globally decisive (by Lemma 1) while the offense team had nothing to do with this and don’t belong to the dictatorship.

To avoid this possibility, say there isn’t a single boy outside the football team who ranks the Maserati above the Ferrari. But we know the offense team rank it below. So the Maserati ranks below the Ferrari, and that could not have been the work of the defense team, since they don’t care. And thus the offense team are decisive between the Ferrari and the Maserati, so by Lemma 1 they are globally decisive (they are the dictators) and the defense team is not.

So we have shown that, unless they coincide in every single ranking, dictatorships are divisible and we have established the second lemma, “contraction of decisive sets.”

Armed with these proofs, we’re home and dry. By the Pareto Principle, if there’s a pair of cars where everybody agrees what the ranking should be, that had better be the final ranking.

In other words, the whole class put together gets to be “the stickler” on any car pair where there is unanimity. By lemma 1, this means the whole class is decisive. The whole class is the dictatorship. By lemma 2, a subset of the class must be the dictatorship. Keep repeating until you’re left with one dictator. QED

[Rob · Rating 3 out of 5]

This is just a few lectures and papers published elsewhere compiled together, but both Sen and Maskin are good writers, so it provides a very useful introduction to Arrow's Impossibility Theorem. I can't imagine why anyone would want such a thing, except for fairly academic reasons. In order to read the proof, it would be helpful to be at least somewhat familiar with how preferences work, so best for students in the second half of their undergraduate degree. Alternatively, if taking the proof as given, it would of use to any student who has completed first year micro. The technical details of the proof are not particularly illuminating - none of the intermediate steps really mean much, so you can skip it.

[Mhamed Limam · Rating 4 out of 5]

I was truly astonished by the insights presented in this Theorem. It is remarkable how a set of seemingly fair axioms within the framework of a social welfare function (such as the Pareto principle, Independence of irrelevant alternatives, and unrestricted domains) can ultimately lead to the emergence of a dictatorship. Arrow's impossibility theorem holds immense implications, even for someone like myself who is not an economist. It extends far beyond academia and economy and has direct relevance to our day-to-day lives.
For instance, after reading this book , I couldn't care less about other people's opinions of me. Mathematically speaking, there is simply no social function that can simultaneously satisfy all their individual preferences. Many of these preferences are inconsistent, illogical, and can even give rise to impossibilities (such as the Condorcet paradox). Therefore, why bother caring about them hahahahaha ?

[Willis · Rating 3 out of 5]

A bit more academic and mathematical than I would have liked - it really is the compilation of some lectures on the topic. I can't believe I'd never heard of this theorem previously which has some interesting applications to politics. In essence, one key takeaway from Arrow's theorem is that you can't devise any voting mechanism that will work to give a consistent overall result on the composite of user preferences. People's preferences are too complicated.

[Diego · Rating 4 out of 5]

Una breve pero muy interesante discusión entre Amartya Sen, Eric Maskin y Kenneth Arrow sobre el origen, presente y futuro de los campos de la elección social, la economía de bienestar y la agenda de investigación iniciada por Arrow en Social Choice and Individual Values.

[Carter · Rating 3 out of 5]

The first of this series of books, is written, almost like a lecture, in part by Amarta Sen, the Nobel Prize winning Economist. The topic is the famous impossibility theorem. I found the pace a bit too leisurely. The topic is easier to cover in some notes, or a short tutorial paper.