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

[Alex Small]

Is this "single-peakedness" the same as saying all voters fall on a 1D
ideological spectrum?

e.g. if all voters and candidates fit on the left-right spectrum, then all
voters will have one of these preferences:

Left>Middle>Right
Right>Middle>Left
Middle>Left>Right
Middle>Right>Left

But if issue space is more complicated, e.g. the "middle" guy scores low
with a lot of voters on some issue that doesn't fall on a left-right
spectrum (maybe something like intelligence or experience) then you could
also get Right>Left>Middle or Left>Right>Middle (e.g. voters who think
"middle" is an idiot and would rather have a smart guy that they
completely disagree with than an idiot that they sometimes agree with).

Or is single-peakedness more complicated than that?


Joseph Malkevitch said:
> 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
>> ----
>> Election-methods mailing list - see http://electorama.com/em for list
>> info
>
> --
> Joseph Malkevitch    |
> Mathematics Dept.    |
> York College(CUNY)   |
> Jamaica, NY 11451
>
> Phone: 718-262-2551
> Web page:
> http://www.york.cuny.edu/~malk
>
>
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