Subject: [HM] Arrow's Theorem
From: Thomas Bartlow (thomas.bartlow@villanova.edu)
Date: Thu Jan 06 2000 - 10:46:34 EST
Moshe' Machover has directed the attention of this list to questions in
the history of the theory of social choice.
In a previous submission I said that Kenneth J. Arrow discovered his
impossibility theorem in 1948, published a preliminary version in 1950,
to be followed by the classic book of 1951, and that he has later
reflected in several places on the circumstances of the discovery. See
the following:
"A Difficulty in the Concept of Social Welfare, Journal of Political
Economy, 58(1950): 328-346, reprinted in Landmarks in Political Economy
edited by Earl J. Hamilton, Albert Rees, and Harry G. Johnson, The
University of Chicago Press and other places.
Social Choice and Individual Values, John Wiley and Sons, 1951. The
second edition of 1963 reprints the first edition with an added chapter
in which Arrow reviews the history of the topic both before and after
his discovery and offers a new formulation of the theorem. It has been
reissued by Yale University Press.
Collected Papers of Kenneth J. Arrow, Belknap Press of Harvard
University Press, 1983, vol. 1: 1-5
Kelly, J.S., An Interview with Kenneth J. Arrow, Social Choice and
Welfare 4 (1987): 43-62
Kenneth J. Arrow in Lives of the Laureates: Ten Nobel
Economists, edited by William Breit and Roger W. Spencer, Cambridge, MA:
The MIT Press, 1990
The Origins of the Impossibility Theorem, in History of Mathematical
Programming, ed. by J.K. Lenstra, A.H.G.R. Kan, and A. Schrijver,
North-Holland, 1991
The theorem can be paraphrased in this way: Suppose each individual in a
set of decision makers (voters) ranks the alternatives (candidates) in a
preference order. A social welfare function should assign to each vector
of individual orders, a single order which represents the collective
order of preference (the social choice). Such a function should satisfy
five conditions. The first requires that the domain of the social
welfare function be sufficiently large. Arrow's terms for the other
conditions were positive association of social and individual values,
independence of irrelevant alternatives, citizen sovereignty, and
non-dictatorship
In the years following the appearance of Arrow's book many authors
offered alternative versions of the theorem by modifying the conditions
and simplifying the proof. Here is a partial list.
Weldon, J. C., "On the Problem of Social Welfare Functions," Canadian
Journal of Economics and Political Science 18 (1952), 452--463
May, Kenneth O., "Intransitivity, Utility, and Aggregation of
Preferences," Econometrica 22 (1954), 1--13
Inada, K, "Alternative Incompatible Conditions for a Social Welfare
Function," Econometrica 23 (1955): 396-399
Blau, Julian H., "The Existence of a Social Welfare Function,"
Econometrica 25 (1957), 302--313. Here Blau shows that Arrow's theorem
is incorrect as stated because his domain condition is poorly
formulated.
Vickery, William, "Utility, Strategy, and Social Decision Rules," The
Quarterly Journal of Economics, vol. LXXIV (no. 4), Nov. 1960, 509-510
Riker, W. "Voting and Summation of Preferences: An Interpretive
Bibliographic Review of Selected Developments During the Last Decade,''
American Political Science Review 55 (Dec. 1961), 900--911
Fishburn, P. C., "Arrow's Impossibility Theorem: concise proof and
infinite voters," J. Economic Theory 2 (1970), 103-106
Blau, Julian H., "Arrow's Theorem with Weak Independence," Economica 38
(1971), 413-420
Blau, Julian, "A Direct Proof of Arrow's Theorem" Econometrica 40
(1972), 61--67
There has been some correspondence on this list about the involvement of
non-mathematicians in mathematical research. In this connection we may
note that none of this work appeared in mathematical journals and, among
the authors, only May and Fishburn may be considered to be
mathematicians.
May is well known to members of this list as a historian of mathematics
but his first professional work was in mathematical economics. Some may
be interested in my paper, "Kenneth O. May and the Theory of Social
Choice," Proceedings of the Canadian Society for the History and
Philosophy of Mathematics 9 (1996), 73-78.
I have an unpublished paper comparing the formulations of Arrow's
theorem in the works cited above. I will be glad to send it to any list
member who asks, either in Latex version as an e-mail attachment or a
paper copy by surface mail.
Bruce Buldt has asked about a possible connection with theorem of von
Neumann and Morgenstern. I know nothing about that and our library copy
of Theory of Games and Economic Behavior has gone missing so I cannot
check the reference which he gives.
Thomas L Bartlow
Assistant Professor
Department of Mathematical Sciences
Villanova University
800 Lancaster Avenue
Villanova PA 19085
fax: 610-519-6928
work: 610-519-7331
http://www66.homepage.villanova.edu/thomas.bartlow
Thomas.Bartlow@villanova.edu
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