Re: Arrows Impossibility Theorem

From: John Conover <>
Subject: Re: Arrows Impossibility Theorem
Date: 19 Dec 1998 23:47:19 GMT

Burkhard C. Schipper writes:
> The Arrows Impossibility Theorem's inherent Condorcet Paradox is as follows:

Hi Burkhard. Just for the historical perspective, Kenneth Arrow in
"History of Mathematical Programming", J.K. Lenstra and
A. H. G. Rinnooy Kan and A. Schrijver, CWI, Amsterdam, Holland, 1991,
ISBN 0 444 888187, pp. 1-4 entitled "The Origins of the Impossibility
Theorem" states that he can not claim originality in the discovery. It
was first developed by the French political philosopher and
probability theorist, Condorcet in 1785. He also cites the works by
Duncan Black in the "Journal of Political Economy", where it was
apparently "rediscovered," circa 1947.

The Impossibility Theorem is, indeed, a very uncomfortable paradox.

R. Duncan Luce and Howard Raiffa, "Games and Decisions," John Wiley &
Sons, New York, New York, 1957, ISBN 0-486-65943-7, pp. 374 addresses
the issue, and offers the opinion that the constraints in the proof
are too strong.

A non-technical implication/interpretation of the consequences of the
theorem can be found in "Archimedes' Revenge," Paul Hoffman, Fawcett
Crest, New York, New York, 1993, ISBN 0-449-21750-7, pp. 215, Chapter
    In 1951, Kenneth Arrow, ... [offered] ... a convincing
    demonstration that any conceivable democratic voting system can
    yield undemocratic results ...

    One year later, Paul Samuelson ... put it this way: "The search of
    the great minds of recorded history for the perfect democracy, it
    turns out, is the search for a chimera, for logical
    self-contradiction. New scholars all over the world-in
    mathematics, politics, philosophy, and economics-are trying to
    salvage what can be salvaged from Arrow's devastating discovery
    that is to mathematical politics what Kurt Godel's 1931
    impossibility-of-proving-consistency theorem is to mathematical

Hoffman's interpretation is interesting reading, whether one agrees
with it, or not, and contains several simple counter intuitive
examples of the consequences of the theorem taken from the real world.


BTW, I wonder, if the Impossibility Theorem were an iterated game, if
the outcome would have fractal dynamics? If so, a Nash equilibrium?
Or, how about path dependency? I mean, just because we can never have
a perfect social system, (or science thereof,) doesn't mean we can
just through up our hands in frustration. What if we try to make it
work, anyhow?


John Conover,,

Copyright © 1998 John Conover, All Rights Reserved.
Last modified: Fri Mar 26 18:53:34 PST 1999 $Id: 981219161210.22322.html,v 1.0 2001/11/17 23:05:50 conover Exp $
Valid HTML 4.0!