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The Arrow Impossibility Theorem (Kenneth J. Arrow Lectures Series) Hardcover – 15 Aug 2014


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Product details

  • Hardcover: 160 pages
  • Publisher: Columbia University Press (15 Aug. 2014)
  • Language: English
  • ISBN-10: 0231153287
  • ISBN-13: 978-0231153287
  • Product Dimensions: 21.1 x 14.5 x 1.8 cm
  • Average Customer Review: 4.0 out of 5 stars  See all reviews (1 customer review)
  • Amazon Bestsellers Rank: 386,611 in Books (See Top 100 in Books)
  • See Complete Table of Contents

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Without hyperbole, no postwar intellectual of the first rank has done more good for more people--above all, many of the world's poorest--than Amartya Sen. -- Boyd Tonkin The Independent What is Arrow's impossibility theorem? Why is it true? What are its implications for democratic decision making? Is its nihilism justified? These are the kinds of questions addressed in Maskin and Sen's masterful Arrow lectures. These lectures and the accompanying essays provide an accessible introduction to Kenneth J. Arrow's theorem for the neophyte and much food for thought for the cognoscente. -- John A. Weymark, Vanderbilt University How vital it is to understand the ideas behind Kenneth J. Arrow's impossibility theorem if we want to design reasonably fair ways of coming to consensus decisions that take equitable account of individual preferences. This book is a marvelous introduction to the theorem, a keystone in the theory of social choice. We are treated to a discussion of that theory--its origin, background, and the challenges it points to--by some of its great architects. -- Barry Mazur, Harvard University, author of Imagining Numbers The pioneers of social choice theory give us lively, enjoyable, and stimulating lectures and exchanges of ideas. Their views, more than sixty years after the publication of Kenneth J. Arrow's theorem, are of paramount interest to anyone aware of the difficulties of collective decisions. -- Marc Fleurbaey, Princeton University

About the Author

Eric Maskin is the Adams University Professor at Harvard University. He received the 2007 Nobel Memorial Prize in Economics (with L. Hurwicz and R. Myerson) for laying the foundations of mechanism design theory. He has also contributed to game theory, contract theory, social choice theory, political economy, and other areas of economics. Amartya Sen is the Thomas W. Lamont University Professor and Professor of Economics and Philosophy at Harvard University. In 1998 he was awarded the Nobel Memorial Prize in Economic Sciences, and in 1999 he was awarded the Bharat Ratna, India's highest civilian award. He is also a senior fellow at the Harvard Society of Fellows; distinguished fellow of All Souls College, Oxford; and a Fellow of Trinity College, Cambridge. His books have been translated into more than thirty languages.

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By Athan on 11 Dec. 2014
Format: Hardcover
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.
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3 of 3 people found the following review helpful
Short and to the point 11 Dec. 2014
By Athan - Published on Amazon.com
Format: Hardcover
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 of 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 and z'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<t expression (5) must hold, when clearly one of the two is the Greek tau etc. etc. Equally, important steps are left to the reader. I struggled with understanding why at the top of p. 121 Limited Favoritism necessarily meant there could only be one alternative that is top-ranked. I've a couple math degrees and I did not enjoy these proofs, I liked Amartya Sen's style more. That is the bottom line.

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
1 of 3 people found the following review helpful
Arrow and Jacques. A good team 21 Oct. 2014
By James H. Dobbins - Published on Amazon.com
Format: Kindle Edition Verified Purchase
This is a book that should be read and used in conjunction with the works of Elliott Jacques.
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