CVD page about plurality winners (correction)

[Forest Simmons]

The part about several candidates having the same highest 1/N quantile
rating needs correction.  See below.

On Thu, 27 Dec 2001, Forest Simmons wrote:

> This observation of Borda reported in Steve Barney's posting below
> suggests that perhaps Bucklin's method should be modified to use an
> (N-1)/N super majority instead of 1/2 . 
> 
> Bucklin's method gives the win to the candidate with the highest median
> rank, i.e. the candidate with the highest (100/2) percentile rank.
> 
> Bucklin, modified to take into account Borda's observation, would give the
> win to the candidate with the highest (100/N) percentile rank.
> 
> In the case of N=4, this would give the win to the candidate with the
> highest quartile rank.  In other words, the candidate with the highest
> rank R such that a quarter or fewer of all of the voters ranked it below
> R, would be the winner.  If several candidates share the same quartile
> rank R, then the one with the fewest votes at that level (hence, most
> above that level) is winner.
> 

This should say that if several candidates share the same quartile rank R,
then (among them) the one with the most votes above that level is winner. 

> Note that if there are no truncations, then at least one candidate will
> have a (100/N) percentile rank above the lowest rank. For example it is
> impossible for all four candidates in a four way race to have more than
> 25 percent of their votes at any one rank (in particular, the bottom
> rank) when no voter can rank more than one candidate at that rank.
> 
> When there are truncations (and when generalizing the method to
> unconstrained CR) it may be possible that every candidate has more than
> (100/N) percent at the bottom rank or rate.  
> 
> In that case, all candidates share the bottom rank as the highest rank R
> below which fewer than 100/N percent of the voters ranked them. So,
> according the decision rule, the candidate with the fewest bottom level
> votes wins.
> 
> Has this modification of Bucklin ever been advocated by anybody?
> 
> It seems to me that the biggest drawback would be a strategic incentive
> for a voter to insincerely rank her less preferred of the two (perceived)
> front runners below her sincere least preferred candidate. This problem
> does not exist in the case of unconstrained CR ballots nor in the case of
> ranked ballots when truncations are allowed.
> 
> Here is a more precise description of the method adapted to CR ballots,
> constrained or unconstrained, in an N candidate race: 
> 
> For each candidate C let R(C) be the maximum rating R such that candidate
> C shows below level R on fewer than 100/N percent of the ballots. 
> 
> Let M be the maximum of R(C) as C varies over the candidates.
> 
> Let S be the set of candidates attaining this maximum, i.e.
> 
>       S = { C :  R(C)=M }.
> 
> Let C' be the member of S with the fewest number of ballot ratings at
> level M.
> 

This should say that among the members of S, C' is the candidate with the
greatest number of ballot ratings above level M.

> This candidate C' wins the election.
> 
> [end of description of method]
> 
> Forest
> 
> On Sun, 23 Dec 2001, Steve Barney wrote:
> 
> > This CVD site is interesting, but Jean Charles de Borda argued that we
> > cannot be sure the plurality winner is the most preferred candidate, on
> > the whole, unless he/she receives more than 1 - 1/N of the votes, where
> > 'N' is the number of candidates (see Borda's article in Iain McLean and
> > Arnold B Urken; eds.;  _Classics of Social Choice_; Ann Arbor;
> > University of Michigan Press; 1995). In a 3 candidate contest, for
> > example, that means that the plurality winner must get a super-majority
> > of more than 2/3 of the votes before we can be sure. The only time a
> > mere simple majority (more than half) is enough is when there are only 2
> > candidates (according to Borda's formula). 
> > 
> > Steve Barney
> > 
> 
>