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

[Joseph Malkevitch]

If one can order the alternatives being voted on (candidates) on a linear
scale so that all of the alternatives are "single peaked" (using ordinal
ranking ballots) then if there are an odd number of voters the Condorcet
method will always choose a winner. (This result is due to Duncan Black.)
Being single peaked is a very strong condition. In fact if there are n
alternatives and rankings are done without ties then at most 2 to the nth
power can be single peaked while there might be as many as n! rankings. One
can think of the existence of such a scale for the candidates as being a sign
of "homogeneous" values shared by the voters on the alternatives being ranked.

Best wishes,

Joe Malekvitch

Sampo Syreeni wrote:

> 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
> openpgp: 050985C2/025E D175 ABE5 027C 9494 EEB0 E090 8BA9 0509 85C2
> ----
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--
Joseph Malkevitch    |
Mathematics Dept.    |
York College(CUNY)   |
Jamaica, NY 11451

Phone: 718-262-2551
Web page:
http://www.york.cuny.edu/~malk