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Public Choice

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08-03-2018

Arrow, and unexpected consequences of his theorem

Author: Donald G. Saari

Published in: Public Choice | Issue 1-2/2019

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Abstract

A new way to interpret Arrow’s impossibility theorem leads to valued insights that extend beyond voting and social choice to address other mysteries ranging from the social sciences to even the “dark matter” puzzle of astronomy.

previous article Kenneth Arrow’s impossibility theorem stretching to other fields

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A proof of his theorem and of Theorem 1 (given below) using the geometry of an equilateral triangle is in Saari (1995, pp. 91–94); different proofs based on how a cube’s vertices are arranged are in Saari (2001, Sect. 8.4) and (2018, Chap. 6).

See my referenced papers for details. Much of this article is motivated by material in my book (Saari 2018).

Hazelrigg was the first to recognize the importance of Arrow’s theorem with respect to concerns from engineering. His first paper (Hazelrigg 1996) generated a continuing discussion in this area.

The comments about apportionments come from Chap. 5.4 in Saari (1995) and my lecture (Saari 2015).

Because the US house size is fixed at 435, this becomes an argument for using his method.

That is, a state does not lose a seat with an increase in house size.

Recall, the Borda Count tallies a N-candidate ballot by assigning a top ranked candidate \((N-1)\) points, a second ranked candidate \((N-2)\), points, …, a jth ranked candidate \((N-j)\) points, ….

Arrow, K. (1951). Social choice and individual values. New York, NY: Wiley (2nd edn. 1963).

Arrow, K., & Debreu, G. (1954). Existence of an equilibrium for a competitive economy. Econometrica, 22(3), 265–290.CrossRef

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Brown, J. (2009). Madoff report highlights SEC lapses in detecting fraud. PBS NewsHour.

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Greenberg, J. (1979). Consistent majority rule over compact sets of alternatives. Econometrica, 47, 627–636.CrossRef

Hazelrigg, G. (1996). The implications of Arrow’s Impossibility Theorem on approaches to optimal engineering design. Journal of Mechanical Design, 118(2), 161–164.CrossRef

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Saari, D. G. (1978). Methods of apportionment and the House of Representatives. The American Mathematical Monthly, 85, 792–802.CrossRef

Saari, D. G. (1995). Basic geometry of voting. New York, NY: Springer.CrossRef

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Saari, D. G. (2008). Disposing dictators: Demystifying voting paradoxes. New York, NY: Cambridge University Press.CrossRef

Saari, D. G. (2010). Aggregation and multilevel design for systems: Finding guidelines. Journal of Mechanical Design, 132, 081006-1–081006-9.CrossRef

Saari, D. G. (2014a). A new way to analyze paired comparison rules. Mathematics of Operations Research, 39, 647–655.CrossRef

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Saari, D. G. (2015). Confronting the modeling problem: Combining parts into the whole, Lecture: IMBS workshop on “Validation; What is it?” http://www.tents/events/conferencevideos.php. Accessed 23 June 2015.

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Saari, D. G. (2016b). Dynamics and the dark matter mystery, Invited 12/01/2016 article, SIAM News.

Saari, D. G. (2018). Mathematics motivated by the social and behavioral sciences. Philadelphia, PA: SIAM.CrossRef

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Title: Arrow, and unexpected consequences of his theorem
Author: Donald G. Saari
Publication date: 08-03-2018
Publisher: Springer US
Published in: Public Choice / Issue 1-2/2019
Print ISSN: 0048-5829
Electronic ISSN: 1573-7101
DOI: https://doi.org/10.1007/s11127-018-0531-7

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Other articles of this Issue 1-2/2019

Kenneth Arrow’s impossibility theorem stretching to other fields

Remembering Kenneth Arrow: discount rates

Introduction to a special issue in honor of Kenneth Arrow

Social welfare with net utilities

Reflections on Arrow’s theorem and voting rules

A defense of Arrow’s independence of irrelevant alternatives