Subject: Re: [HM] Topics in the history of Social Choice
From: Thomas Bartlow (thomas.bartlow@villanova.edu)
Date: Thu Dec 30 1999 - 11:05:22 EST
Kenneth J. Arrow discovered the (im)possibility theorem in the theory of
social choice in the summer of 1948 while working for the Rand Corporation.
He tells the story in "The Origins of the Impossibility Theorem," History of
Mathematical Programming, edited by J.K. Lenstra, A.H.G.R. Kan, A. Schrijver,
North-Holland, 1991, pp. 1-4. In this paper he recalls the year as 1949 but
in the preface to the 1951 book he says it was 1948 and a preliminary version
of the theorem was read at the December 1948 meeting of the Econometric
Society ("A difficulty in the Concept of Social Welfare," Journal of Political
Economy 58 (1950), 328-346, reprinted in Landmarks in Political Economy, The
University of Chicago Press, 453-480). The basic discovery was developed into
his Ph.D. thesis which appeared in 1951 as the book Social Choice and
Individual Values. Here is part of what Arrow wrote in 1991.
In the course of trying to write a dissertation (subsequently abandoned),
I was concerned with the fact that firms in the modern world typically
had many owners (shareholders). ... The natural assumption was that one
investment policy would be chosen over another if a majority of the
stockholders preferred the first to the second. My training [in
mathematics] led me to check whether this relation was transitive; there
was no difficulty in constructing a counterexample. ... The next year I
was idly thinking about electoral politics for some reason. I started with
a model in which candidates (or parties) arranged on a Left-Right scale,
so that supporters of one party would prefer a party nearer to them to
one further in the same direction. ... It was quickly apparent that under
these conditions, majority voting would indeed define an ordering. However,
about a month after these reflections, I found the identical idea in a
paper by Duncan Black in the Journal of Political Economy
...
I spent the summer... at the Rand Corporation. ... There was a philosopher
on the staff, named Olaf Helmer. ... He was troubled about the application
of game theory when the players were interpreted as nations. The
interpretation of utility or preference for an individual was clear
enough, but what was meant by that for a collectivity of individuals? I
assured him that economists had thought about the problem in connection
with the choice of economic policies and that the appropriate formalism
had been developed by Abram Bergson in a paper in 1938. ... He asked me
to write up an exposition. I started to do so and immediately realized
the problem was the same I had already encountered. I knew already that
majority voting would not aggregate to a social ordering but assumed there
must be alternatives. A few days of trying them made me suspect there was
an impossibility result, and I found one very shortly.
I believe the recognition that majority voting defines a collective order
when alternatives can be arranged on a scale and individual preferences are
determined by choosing an optimal position on this scale is the only overlap
between Arrow's ideas and Black's work.
Arrow has also reflected on his work on social choice in an autobiographical
essay in Lives of the Laureates, vol. 2, ed. by William Brett and Roger W.
Spencer, MIT Press, 1990, and in J.S. Kelly, An Interview with Kenneth J.
Arrow, Social Choice and Welfare 4 (1987), 43-62.
Moshe' Machover wrote:
> I would like to draw the attention of historians of mathematics to two
> topics in the history of social choice (which, roughly speaking, is
> mathematics applied to political science and collective decision-making).
>
> I feel that in the known story of both these topics there are some gaps,
> and there is much scope for further research by competent historians of
> mathematics (which excludes me ... ).
>
> 1. The (re-)invention of social choice.
>
> The story as it is known at present is *roughly*--NB: I am not a
> historian--as follows.
>
> In 1951, when K J Arrow published his celebrated book *Social Choice and
> Individual Values* (containing his famous theorem and related work, for
> which he was awarded the Nobel prize for economics), he had little
> awareness that the subject had been dealt with before.
>
> However, Duncan Black, another modern pioneer of social choice (who
> independently discovered some of the same results as Arrow), unearthed
> earlier work on this topic, including (inter alia) contributions by
> Condorcet, Laplace, and Dodgson (aka Lewis Carroll, author of the Alice
> books). See Black, *The Theory of Committees and Elections*, Cambridge UP
> 1958 (republished 1987 by Kluwer).
>
> Later, Iain McLean and others discovered earlier writings on the subject,
> including a brief remark by Pliny the Younger and detailed discussions by
> Ramon Lull (end of 13th century) and Niclolaus Cusanus (15th Century).
> BTW, Lull and N Cusanus are connected also by both being Hermeticians.
>
> On these matters see McLean and Urken, *Classics of Social Choice*,
> Michigan UP 1995 and the extensive bibliography therein. (See also
> Fesenthal and Machover, "After two centuries, should Condorcet's voting
> procedure be implemented?", Behavioral Science 37:250--274, 1992, in the
> Appendix of which we dispute McLean's interpretation of Lull's
> prescription.)
>
> I feel there are still big gaps in this story. Also, even the texts that
> have been re-discovered may benefit from the attention of a competent
> historian *of mathematics*. (McLean, the leader in the field, is a
> professional historian, but is far from being a mathematician and is not
> really a historian of mathematics.)
>
> 2. The history of indices of voting power.
>
> The story, as far as it is known to me, is as follows.
>
> Until fairly recently is was widely believed, and often stated in papers on
> social choice, that the study of voting power (measuring the power of each
> voter in a decision-making body, whose decision rule is not necessarily
> symmetric) was pioneered by Shapley and Shubik in their 1954 paper, "A
> method for evaluating the distribution of power in a committee system",
> American Political Science Review 48:787--792, in which they applied the
> Shapley value (proposed by Shapley in 1953 for arbitrary cooperative games)
> to simple games. Their index is known as the Shapley-Shubik index
>
> In 1965 J F Banzhaf, in his paper "Weighted voting doesn't work", Rutgers
> Law Review 19:317--343 proposed an alternative index, known as the Banzhaf
> index, which he studied further in subsequent papers.
>
> However, it transpired that Lionel Penrose had pioneered the subject in
> 1946, in 'The elementary statistics of majority voting", J of the Royal
> Statistical Society 109:53--57. The index he proposed is effectively the
> same as Banzhaf's; and in fact much of the latter's work was a rediscovery
> of Penrose's results.
>
> Regarding all this, see Felsenthal and Machover, *The Measurement of Voting
> Power*, Edward Elgar, 1998. See also Peter Morriss, *Power--A Philosophical
> Analysis*, Manchester UP 1987. (Morriss, who work is not in the mainstream
> of social choice, was one of the very few writers who were aware of
> Penrose's work.)
>
> Again, I feel that there are gaps in this story. L Penrose was a well-known
> mathematical statistician. Why was his contribution not noticed by
> social-choice theorists (mainly in the US)? this and other questions
> suggest themselves.
>
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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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