Yes, Arrow's results concern voting. But voting is how things are decided in a democracy. The math is rock-solid. Here's a standard proof:
http://www.tulane.edu/~dnelson/COURSES/IntroPE/arrow.pdf
That, plus the impotrtance of the result, is why Arrow won the Nobel Prize, although Michigan didn't find the result interesting and (before Arrow won the Nobel) turned him down for tenure.
Arrow showed that you cannot have a voting system that satisfies the following conditions, four of which seem absolutely crucial to any system that calls itself democratic:
1) Nondictatorship (no one person's vote decides an issue (except as a tie breaker);
2) Monotonicity: No choice can be harmed in a vote by being ranked higher by an individual, or (stronger)
2a) Pareto optimality: if everyone prefers a certain choice, it will be the outcome society prefers
3)Universality: the voting system can in principle rank all choices that citizens might make relative to one another
4) Citizen Sovereignty. Every possible ranking of choices can be achieved from some set of individual votes.
5)Independence of irrelevant alternatives: You should not be able to affect the outcome of a vote by introducing another choice into it, unless the extra choice actually wins (The Nader problem!)
That is, in social choice-ese. the outcome of a choice among a limited set of options should be consiststent with the choice among all options, so that choices outset the subset (irrelevant ones) should not affect the choice within it. (This one is the least intuitive, but it seems like it should be true).
The theorem is that with two voters and three options, you cannot satisfy these conditions at the same time: not all choice sets are attainable without violating at least one of them.
There is a vast literature on getting out of the paradox because it is very trouble to democrats that no voting system can satisfy these criteria, all of which appear fundamental to democracy.
The problem isn't that no _single_ voting system can accurately represent the will of the majority -- you can't _combine_ voting systems to avoid the paradox.
The problem rather sugests that there is no such thing as "the will of the majority," if that is taken to be an aggregate of individual preferences. That of course is an interpretation, not a result. It's embraced by right wing, market-oriented public choice theorists like James Buchanan who want to take everything out of the public realm and make choices by buying and selling instead. So it is even more troubling for the left.
Nonetheless I suggest that nothing, not even a mathematical proof, would make us give up our faith in democracy. Even the public choice crowd does not advocate getting rid of democratic politics, just limiting its scope a lot.
jks
--- John Adams <jadams01 at sprynet.com> wrote:
> On Thursday, September 9, 2004, at 11:24 PM, andie
> nachgeborenen wrote:
>
> > By way of illustration, the economist Kennth Arrow
> has
> > a theorem -- a mathematical proof -- that four or
> five
> > absolutely fundamental and completely obvious
> > conditions for democracy are inconsistent.
>
> If I remember correctly--seems like David Duke was
> running for Senate
> that year, so it would've been '90 when we had the
> seminar--Arrow's
> work is specific to voting systems. This was given
> by the math
> department, though, so I can't swear whoever
> presented interpreted him
> correctly. (Didn't Arrow get a Nobel Prize in
> economics for this? There
> was some discussion during the seminar about whether
> the math of it was
> worthy--I seem to recall suggesting it was the
> cleverness of his
> application of the math that was being recognized,
> but then, that's the
> sort of argument I make around people more
> knowledgeable than myself.)
>
> > That doesn't make us think, oops, better go with
> > dictatorship. It makes us think there is something
> > funny about Arrow's proof, although no one has
> been
> > able to say just what.
>
> I recall the interpretation of his work as being a
> bit different--more
> that no single voting system can be guaranteed to
> accurately represent
> majority will given all possible election results.
> It's been a long
> time, though, and I was busy that fall, so I'm not
> shocked if I
> misremember.
>
> All the best,
>
> John A
>
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>
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