[EM] Bucklin and detecting the highest generalized median rank

[Forest Simmons]

Bucklin is sometimes described as the method which gives the win to the
candidate with the highest median rank, analogous to Borda as the method
that gives the win to the candidate with the highest average rank.

The usual version of Bucklin uses only the crude median, and then (in the
case of several candidates tied for the highest median rank) gives the win
to the candidate with the greatest number of ballots ranking him/her at or
above the (common highest) median rank, i.e. the highest value of M+A in
the notation used below.

Here's how to detect which of several candidates has the highest
generalized median rank when they all have the same ordinary median rank:

For each candidate calculate the quantity Q = (A-B)/(M+0^M) , where A is
the number of ballots above the median rank, B is the number below median
rank, and M is the number at the (common) median rank.

[The term 0^M merely serves to keep Q defined even when M=0, in which case
A=B, so Q=0.]

The candidate with the highest value of Q has the highest generalized
median rank (among the candidates having the same ordinary median rank).

This doesn't mean that the value Q is the generalized median rank; it
isn't.  But it does detect the generalized median rank order.

Notice that this order reverses when the ballot rankings are reversed, so
the generalized median order satisfies a strong reverse symmetry criterion
like Borda does.

To get a better picture of the generalized median, make a histogram of the
candidate's ballot ranks.  Cut the histogram area in half with a vertical
line.  That line marks the generalized median rank for the candidate.


Forest


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