[EM] [OT] Kenneth Arrow theory... anyone?

[Sampo Syreeni]

On 2003-11-21, David GLAUDE uttered:

>[[Do you know that a multi-cultural society cannot be democratic? The
>Nobel Prize Kenneth Arrow mathematically showed, in 1952, that there was
>no possible democracy via a voting system (theorem of impossibility),
>except if the voters share the same culture and close values (Nobel Prize
>Amartya SEN)]]

I agree with Alex. This is a typical, vulgar misrepresentation of Arrow.
But there is also a seed of truth in it.

Arrow talks about whether individual linear rankings can be fit into a
collective linear ranking over a broad range of conditions and shows that
this cannot be achieved without breaking some simple, intuitive rules. In
this sense, if people are permitted to disagree broadly enough
(multiculturalism), there's no coherent way to define the "will of the
people" which doesn't devalue or misrepresent some people's preferences.
But if people indeed think alike about most issues, their preference
orderings will be very similar and the likelihood that there will be
voting cycles decreases dramatically. In this limited case quite a number
of social choice functions will probably define the will of the people in
a manner which most people would consider sensible.

That is, the problem with dictatorship isn't that it's inherently a bad
voting method. It's just that everybody has to agree with the dictator in
order for it to work. Is this democracy, then? That sorta depends on the
viewpoint.

>33% find A > B > C
>33% find B > C > A
>33% find C > A > B
>Then we have a (basic) problem.
>The theorem would be related to that???

This is Condorcet's paradox, also called the problem of cyclic majorities.
It's connected to Arrow's theorem, but Arrow is considerably more
sophisticated than the simple, isolated problem we're seeing here.

>* Do you know of any other extremist party using that argument and making
>reference to Kenneth Arrow?

Not in a systematic manner, no. But among the libertarians I know, similar
arguments are often used to oppose naive democracy and to argue that
collective choice isn't an all-powerful decision-making mechanism.

>* I remember reading that there are no perfect voting system and that
>given some realistic assumption on the goal and choosing a voting method
>it is possible to create a set of ballot that will give "unexpected" or
>"unsatisfying" result... is it true and related to the statement above?

Sort of, but this issue is far broader than Arrow's theorem. The criteria
used to evaluate voting systems include Arrow's, but certainly aren't
limited to them. Different voting systems have different weaknesses and no
voting system satisfies all the different criteria we'd like them to,
simultaneously.

You are probably referring to the fact that there are no general,
strategy-free voting methods. This is called the Gibbard-Satterthwaite
theorem. Essentially it says that all voting methods satisfying a couple
of intuitive conditions can be manipulated by voting insincerely. In other
words, there are no well-behaved voting systems where the best way for an
individual to vote is to always tell the truth. In a sense this means that
voting sincerely can always lead to weird outcomes, at least when others
vote strategically.

>* If that "mathematical proof" turn valid, would there be some assumption
>that can be proven wrong or discuss enough to say that it does not apply
>to the real world.

Arrow's reasoning is solid, but applying it to the real world is a tricky
business. If we look at the characterization above, it places rather
brutal constraints on what can be called a democracy -- it's sort of the
same as claiming that there can be no market economy because no market can
be perfect.
-- 
Sampo Syreeni, aka decoy - mailto:decoy at iki.fi, tel:+358-50-5756111
student/math+cs/helsinki university, http://www.iki.fi/~decoy/front
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