The Arrow Impossibility Theorem by Amartya Sen, Eric Maskin

By Amartya Sen, Eric Maskin

Kenneth Arrow's pathbreaking "impossibility theorem" was once a watershed within the historical past of welfare economics, balloting idea, and collective selection, demonstrating that there's no vote casting rule that satisfies the 4 fascinating axioms of decisiveness, consensus, nondictatorship, and independence. during this booklet, Amartya Sen and Eric Maskin discover the results of Arrow's theorem. Sen considers its ongoing application, exploring the theorem's price and boundaries with regards to fresh learn on social reasoning, whereas Maskin discusses how one can layout a vote casting rule that will get us in the direction of the right — provided that attaining definitely the right is very unlikely. the quantity additionally features a contextual advent by means of social selection pupil Prasanta ok. Pattanaik and commentaries from Joseph E. Stiglitz and Kenneth Arrow himself, in addition to essays through Sen and Maskin outlining the mathematical evidence and framework in the back of their assertions.

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In Arrow’s own framework, decisiveness is formulated as transitivity, the requirement that if, given voters’ rankings, candidate X is elected over candidate Y and Y is elected over Z, then X should be elected over Z. ) 6. There are two closely related conditions that go under the name “independence of irrelevant alternatives”: Arrow’s axiom and the condition formulated by J. Nash (“The Bargaining Problem,” Econometrica 18(2), 1950: 155–162). Here I am using the Nash formulation, because it is somewhat more convenient for my purposes.

Majority rule, by contrast, is easily seen to satisfy independence: if candidate X beats each other candidate by a majority, it continues to do so if one of those other candidates is dropped from the ballot. Unfortunately, majority rule violates our first axiom, decisiveness—it doesn’t always produce a clear-cut winner (this is a problem that Condorcet himself discussed). To see what can go wrong, consider an election with three candidates X, Y, and Z, and an electorate in which 35 percent of the population rank X first, Y second, and Z third; 33 percent rank Y first, Z second, and X third; and 32 percent rank Z first, X second, and Y third (see table 3).

You may remember that in that election everything came down to Florida. If George W. Bush carried the state, he would become president and the same for Al Gore. Now, Florida—like most other states7— uses plurality rule to determine the winner. In the event, Bush got somewhat fewer than six hundred more votes than Gore. Although this was an extraordinarily slim margin in view of the nearly 6 million votes cast, it gave Bush a plurality (and thus the presidency). And, leaving aside the accuracy of the totals themselves (hanging chads and the like), we might reasonably ask whether there was anything wrong with this outcome.

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