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

Joseph Malkevitch joeyc at cunyvm.cuny.edu
Fri Nov 21 12:58:18 PST 2003

```I did not explain what I wanted very clearly in my haste. Single-peakedness is
a property of a collection of ballots with respect to an ordering of the
alternatives. (one plots the height of the alternative on the ballot against
the linear ordering getting a line or broken line segments what are single
peaked). For each ballot, preference goes down with respect to that ballots
top choice, with respect to the given linear ordering. One fixes the ordering
and then checks that all the ballots are single peaked respect to it.

So yes, for 3 alternatives there are 6 potential ballots (no ties). At most 4
of these 6 can be single peaked with respect to a particular ordering of the
alternatives. Furthermore, there is an ordering for which the ballots you list
are single peaked.

With 4 alternatives, if voters have 17 or more of the 24 possible types of
ballots then the ballots they have can not be single peaked.

There are some attempts in the literature to broaden Black's theorem but I do
not remember the details.

My only point was that if one knew in advance that there was limited enough
diversity of opinion on the alternatives there might be methods such as
Condorcet's, not always guaranteed to produce a winner, which would produce a
winner. Arrow's Theorem assumes that all possible ballots may be presented, so
in this environment, Condorcet's method will not always produce a winner.

Cheers,

Joe

Alex Small wrote:

> 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
> >
> >
> > ----
> > Election-methods mailing list - see http://electorama.com/em for list
> > info
>
> ----
> 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

```