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:[EMAIL PROTECTED], 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