From: John Conover <john@email.johncon.com>

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 logic." 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. John 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, john@email.johncon.com, http://www.johncon.com/

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