The Arrow Impossibility Theorem (Kenneth J. Arrow Lecture Series)
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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.
impossibility theorem. Still, there is an important sense in which this conclusion is too pessimistic: It presumes that, in order to satisfy an axiom, a voting rule must conform to that axiom regardless of what voters’ preferences turn out to be.8 In practice, however, some preferences may be highly unlikely. One reason for this may be ideology. As Black (1948) notes, in many elections the typical voter’s attitudes toward the leading candidates will be governed largely by how far away they are
extensions. 2. THE MODEL Our model in most respects falls within a standard social choice framework. Let X be the set of social alternatives (including alternatives that may turn out to be infeasible), which, in a political context, is the set of candidates. For technical convenience, we take X to be finite with cardinality m ≥ 3. The possibility of individual indifference often makes technical arguments in the social choice literature a great deal messier (see, e.g., Sen and Pattanaik
Conversely, on the domain we have for the profile R°, as in the Introduction, in which the distribution of rankings is as follows: Proportion of voters 0.47 0.49 0.04 Ranking x y w y z x z x y w w z By Lemma 3, FM works well on ′ defined by equation (25). Moreover, by Lemmas 1 and 2, FRO and FP do not work well ′. Hence, we have an example of why Theorem 2 applies to plurality rule and rank-order voting. In the Introduction we mentioned May’s
scenario (see footnote 2), for example, it might be considered proper to give more weight to citizens with low incomes. 4. Neutrality is hard to quarrel with in the setting of political elections. But if instead the “candidates” are, say, various amendments to a nation’s constitution, then one might want to give special treatment to the status quo—namely, to no change—and so ensure that constitutional change occurs only with overwhelming support. 5. It is called this after the
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