# [EM] Modified Bucklin

Forest Simmons fsimmons at pcc.edu
Fri Jan 4 18:50:06 PST 2002

```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

```