CVD page about plurality winners

[Forest Simmons]

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.

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 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
>