Objections to Saari's suggestion for solving Arrow's problem in social choice
 

Thomas Cool
JEL: A00
Report-no: TC2000-03-03
March 15 2000
 
 

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. 

  • 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 - and thus don't provide proper clarification. 
I am afraid that Saari's paper creates a new illusion in the off-direction.

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

Saari: Arrow posed a deep problem, that took 50 years to solve, and it needed deep mathematical insights in symmetry' to solve it.

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'.

(2) Nature or morals, 1 alternative or many, assuming or proving Saari poses that symmetry is 'natural'. This would relieve human beings from moral questions. He thus proposes 1 alternative to Arrow's "Axiom of Independence of Irrelevant Alternatives" (AIIA). Arrow says that AIIA is reasonable and morally desirable. Saari says that symmetry is reasonable and morally desirable. Now, how are you going to decide that ?  
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.
 

Comment: I prove that AIIA is neither reasonable nor morally desirable.
(3) Does less than claimed
What Saari essentially proves is only: Symmetry -> Borda. (Note: I take this on face value - have not strictly checked it.)

It is up to human discourse on morality whether we want symmetry (Borda) or something else, depending upon location, time and purpose. Sometimes people don't vote explicitly, but assign 'wisdom' or 'experience' to selected individuals, and accept their decisions. These are situations that could be morally acceptable but they are not covered by Saari's 'natural' symmetry.

And symmetry does not help either in the case of a deadlock, for example.


Conclusion


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

  1. Useful links
  2. My email to The Economist, Seppo Honkapohka (EER) and Donald Saari March 6 2000
  3. My email to Saari March 7 1999
====================================

References and links:

  • Cool (1999), "The Economics Pack User Guide - 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 90-802263-2-7, Samuel van Houten Genootschap. The PDF is at http://thomascool.eu/Papers/Drgtpe/Index.html 
  • The article in The Economist March 4 2000. 

  • "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,
Dear The Economist,
Dear professor Saari,

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]
>To: d_saari @ math.nwu.edu
>From: Thomas Cool <cool @ dataweb.nl>
>Subject: Connecting and resolving Sen's and Arrow's theorems
>Cc: 00ATTabarrok @ BSU.EDU
>
>Dear professor Saari,
>
>I came across your papers on the internet: "Connecting and resolving Sen's and Arrow's theorems" and "The symmetry and complexity of elections".
>
>As I enjoyed your papers, I am sure that you will enjoy mine:
>"The solution to Arrow's difficulty in social welfare", at
>http://econwpa.wustl.edu/eprints/get/papers/9707/9707001.abs
>
>Our approaches seem to overlap for, say, 90%, but there is a 
>crucial difference of the remainder 10%. I hope that you don't mind 
>me saying that my approach is the better one: there is no point in 
>losing this clarity.
>
>I don't think that the real problem is that IIA doesn't discriminate between rational and irrational voters. Or, I think that you cannot argue that it is not surprising that unsophisticated procedures do not lead to sophisticated results. On the contrary, it has been the strength of the Arrow argument that 
>very weak assumptions already generated a contradiction.
>
>My argument is, on the other hand, that IIA blocks the flow of information that is required for group decision making. This blocking would be 
>whether voters are rational or irrational, and my argument thus 
>differs from yours. (Though of course the conventional axioms presume rational voters anyway.)
>
>I hope that you will find the time to study my paper,
>and look forward to your reaction. Indeed, I find your 
>reasoning of an encouraging quality and level of sophistication,
>and it may well be that you are one of the first who will 
>recognise the validity of my argument.
>
>Many regards,
>Thomas Cool
>http://thomascool.eu [updated after 2000]
>
>PS. I send a copy to professor A. Tabarrok, who I happened 
>to contact earlier today on Mathematica, and whose page provided
>the link to your page. (Some of my Mathematica programs contain 
>reasonable constitutions.)
>
>PPS. Could you, by the way, check on the PDF file 
>on "Connecting and resolving ..." ? The copy that I downloaded 
>drops uppercase letters at curious places. Though I can follow 
>your argument, also from the other paper of which the PDF was okay,
>I'd rather have a fully specified copy. I would appreciate it if 
>you could verify whether there indeed is a problem with this PDF.
>
>