[EM] Modified Bucklin

[Forest Simmons]

Here's a suggested hybrid method that completely orders the candidates
while respecting the order of the various Condorcet equivalence classes:

Seed Bubble Sort with the Modified Bucklin order described below.

[Bubble Sort recursively sorts the top m-1 seeded candidates, and then
percolates the m_th candidate as far up the list as possible by pairwise
comparisons with adjacent candidates.]

Forest

On Fri, 4 Jan 2002, Forest Simmons wrote:

> I want to make a (hopefully) final modification to my previous versions of
> Modified Bucklin. Here it is: 
> 
> The context is a single winner election with N candidates. Each ballot has
> (potentially) R distinguishable levels (counting truncations as the lowest
> level), some of which may go unused by some or all of the voters.
> 
> For each candidate C let L be the highest level at which candidate C has
> fewer than 1/N of the ballots showing that candidate below level L. 
>  
> [In some cases level L will be the lowest possible level, below which
> every candidate has zero showings, which certainly represents fewer than
> 1/N of the ballots.]
> 
> Also for candidate C let k be the difference in the number of ballots
> showing C above the level L and the number of ballots showing C below the
> level L.
> 
> So now each candidate C has an associated ordered pair of numbers (L,k).
> 
> Order the candidates according to the lexicographical order of their
> associated number pairs.
> 
> This means that candidate C' is higher than C in the ordering 
> 
>                   if and only if
> 
>               L' is greater than L, 
>                        OR 
>                 L'= L AND k' > k.
> 
> The highest candidate in this lexicographical order is the Modified
> Bucklin winner.
> 
> This method is summable.  A running sum of how many ballots each candidate
> receives in each level is possible via an N by R matrix, where N is the
> number of candidates and R is the number of possible levels.
> 
> Question.  Is this method consistent or even monotone?
> 
> Question.  Given an N by R matrix of this type computed from some set S of
> ballots, what is the smallest number of factions that a set S' (of
> Cardinal Ratings style ballots) can have while yielding the same N by R
> matrix of candidate level summaries? 
> 
> Forest
> 
>