[EM] number of possible ranked ballots given N candidates

[rob brown]

On 12/14/05, Paul Kislanko <kislanko at airmail.net> wrote:
>
> Just a thought for your endeavor. A way to save "more" information than
> the pairwise matrix but less than saving counts of each ballot configuration
> is to save an NxN matrix with column headings being count of #1, #2, ... #N
> ranks and rows being the Alternatives. Use the rule:
>
> If equals are found, assign the next alternative rank (r+1) where where r
> is the rank the previous alternative listed would've had if there were no
> equals. I.e. ranks are assigned 1,1,3 for A=B>C.
>
> This takes no more space than the pairwise matrix, but contains much more
> information. See http://football.kislanko.com/2005/bucklincomps.html for
> an example with 119 alternatives and any number of voters.
>
> Our messages overlapped, each trying to find a way to store more data
without storing all ballots. :)

Your suggestion seems like something you'd want to store in addtion to the
matrix, no?  It seems like it would be hard to perform the sort of
calculations that the matrix works well for, i.e. picking a candidate in a
way that doesn't encourage strategic voting and doesn't punish (or reward)
candidates for having other candidates that appeal to the same constituency.

BTW, if it was me I would handle ties differently, for two candidates tied
for first, I would probably want to store half a point for first place and
half a point for second place for each candidate.  In other words treat 2
A=B>C ballots as being functionally identical to 1 A>B>C ballot and 1 B>A>C
ballot.  But that's just me.

-rob
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