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Thomas Cool
Introduction
I assume that you know about Kenneth Arrow's theorem on social choice, and that you know about my solution to it, in Cool (1997), or even better in Cool (2000) and supportive Cool (1999). Donald Saari also claims that he has found a solution - see http://www.math.nwu.edu/~d_saari/. The Economist March 4th 2000, p97, gives a popular description of Saari's result - see the appendix. I think that Saari is right for 90% but still dangerously off-track. There is sufficient reason to become quite frustrated by events.
Please note that I have the impression that Saari gives nice work in general, and that he should be recommended for his critical approach to the conventional views on Arrow's Theorem. Please note that my reaction does not come from 'priority' considerations. I have no doubt about it that my analysis simply stands. My worry is on content, i.e. the policy errors and the failing academia. Suppose, indeed, that you
think that Saari is right: then you would agree with me that there have
been serious policy errors ! Then you should worry as I do ! And then you
should acknowlegde that OK, he was right on a point. And then you should
read my work ! Will you please do that ?
Discussion
There are the following objections to Saari's approach (as summarised by The Economist). (1) Deep or common sense Comment: Arrow did not pose a deep problem, his view should have been killed from the start. Economists and other theorists alike should be deeply ashamed - for dropping common sense in the face of some math. It is erroneous again to think that you need 'symmetry'. A person in the street might say: "Well I like symmetry - a nice short word - and I never heard about 'independence of irrelevant alternatives' and it sounds like something really bad". It may be that Saari suddenly gets very popular for such frivolous reasons. But as scientists we should
be wary about such reactions, and provide proper schemes for reasoning.
What Saari essentially proves is only: Symmetry -> Borda. (Note: I take this on face value - have not strictly checked it.)
I think that Saari provides
some useful insights in the matter. He might well be excused for thinking
that he has found the solution. But he should also study my analysis -
and then might be the first to see that this is the real solution.
Appendices
References and links:
"The mathematics of voting; Democratic symmetry" The Economist, March 4th
2000, p97:
(...) In a paper just published in the journal Economic Theory, Donald Saari, a mathematician at Northwestern University in Evanston, Illinois, claims to have got to the root of the problem. It is, he says, all to do with symmetry (technically, with something called the wreath product of symmetry groups). Essentially, says Dr Saari, voting paradoxes arise when the system fails to respect natural cancellations of votes. In a two-candidate contest, for example, nobody would deny that the candidate with the most first-preference votes should win. One way to explain this is that votes of the form AB (ie, candidate A is preferred to candidate B) should cancel out votes of the form BA. if this leaves a surplus of A then A wins. These cancellations are a form of reflectional symmetry. But votes in a three-candidate election should cancel out, too (....) This is a form of rotational symmetry, since the three votes form a rotating cycle. Taking these two symmetries into account, it is possible to characterise all paradoxes for a three-candidate election under any voting procedure. Dr Saari's results can also be generalised for elections with more than three candidates using more complicated, but closely related symmetries. It is thus possible to evaluate the "fairness" of different voting systems. (...) The fairest voting system,
says Dr Saari, would respect both symmetries, (....)
============================== Email March 6 2000:
Dear professor Honkapohja,
I noticed The Economist reporting on Saari's results - see the enclosed text. I think that Saari is 90% correct - but still 10% off. There is a better analysis available. There is sufficient reason to become quite frustrated by events. Serious errors are being made in real life politics, based upon improper understanding of the issue. The academia make their errors with regards to logic. I am afraid that Saari's paper creates a new illusion in the off-direction. I kindly ask you to not regard the matter as one on 'priority', since I have no doubt about it that my analysis simply stands. My position on this is on content: the policy errors and the failing academia. Let me invite to really study my analysis, and then come back to me. Best regards, Thomas Cool http://thomascool.eu [updated after 2000] See my email (dated 1993 but this must be 1999 since I reset my clock for the risk of the millennium bug). Unfortunately Saari did not react. My paper on this (see the email) was rejected with rather dumb reasons by the European Economic Review (Honkapohja). My analysis is included in The Economics Pack - applications of Mathematica, ISBN 90-804774-1-9, the JEL reference number is JEL 0820 (Journal of Economic Literature, volume 37, no. 3, September 1999). The paper itself is included as a chapter in my "Definition and reality in the General theory of Political economy", First Edition, March 5 2000, ISBN 90-802263-1-8, Samuel van Houten Genootschap. The PDF is at http://thomascool.eu/Papers/Drgtpe/Index.html
======================= >Date: Sun, 07 Mar 1993 19:04:55
+0100 [Note: 1993 -> 1999]
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