[EM] An opportunity cost-based approach for getting scores from social rankings

[Kristofer Munsterhjelm]

Here's a completely general way of getting scores from rankings. It 
probably won't work on all methods (a nonlinearity measure will show how 
well it fits), but it could be interesting even so:

In Plurality, what does it mean when one says that X has 49 votes and Y 
has 30? Among other things, it means that Y needs 49 - 30 = 19 
additional votes to tie X.

If we transport this to a ranked balloting system, we get a reasoning 
that Y is k points behind X if adding k copies of some kind of ballot 
that ranks Y first would make Y tie X. If the voting method has an 
abrupt change (from X > Y to Y > X), then, by median-esque reasoning, we 
could say that the tie happens at half the highest weight that still has 
X > Y plus half the lowest weight where Y > X.

By adding ballots that rank a lower candidate first until that lower 
candidate ties the next one up, we can get relative scores for pairs of 
candidates. If we do so for *all* candidates, we get a system of 
equation with n unknowns (number of candidates) and n-1 terms (number of 
relative scores). If we make an assumption for the nth term - e.g. that 
the sum of the scores should be the number of voters - we can then solve 
the set of equations and get scores.

For instance, if we put the 2009 Burlington data (restricted to 
Montroll, Kiss, and Wright alone) into Rob LeGrand's rbvote system at 
http://www.cs.wustl.edu/~legrand/rbvote/calc.html, then for Minmax 
(Simpson), we get that:

Montroll wins if we don't do anything (of course).
It takes 590 K>W>M votes for Kiss to tie Montroll (s_M - s_K = 590)
It takes 169 and a half W>M>K votes for Wright to tie Kiss. (s_K - s_W = 
169.5)
There are 8833 voters in all (s_M + s_K + s_W = 8833). There were 8980 
Burlington voters, but not all of them expressed preferences between M, 
W, and K.

Solving this, we get:

s_M = 20365 / 6 = 3394.16...
s_K = 16825 / 6 = 2804.16...
s_W =  7904 / 3 = 2634.66...

and the relative support is:

Montroll: 38.43%
Kiss:     31.75%
Wright:   29.83%