[EM] Amalgamation details

[fsimmons at pcc.edu]

To amalgamate factions so that there is at most one faction per candidate X (in the context of range 
style ballots) take a weighted average of all of the ballots that give X top rating, where each ballot has 
weight equal to one over the number of candidates rated equal top on that ballot.  The total weight of the 
resulting "faction rating vector" for candidate X is the sum of the weights that that were used for the 
weighted average.

Note that these faction rating vectors are efficiently summable.  A running sum (together with its weight) 
is kept for each candidate.  Any single ballot is incorporated by taking a weighted average of the running 
sum and the ballot, where the respective weights are those mentioned above.  For the running sum it is 
the running sum weight.  For the ballot it is zero if the candidate is not rated top, and 1/k if it is rated top 
with (k-1) other candidates..

To combine two running sums for the same candidate take a weighted average of the two using the 
running sum weights, and then add these weights together to get the combined running sum weight.

If you multiply each faction rating vector by its weight and add up all such products, you get the vector of 
range totals.

Of course Range as a method is summable more efficiently without amalgamating factions, but other 
non-summable methods, when willing to accept amalgamated factions, thereby become summable.

So, for example, we can make a summable form of Dodgson:

(1) Use ratings instead of rankings.

(2) amalgamate the factions.

(3) let each candidate (with help from advisors) propose a modification of the ballots that will created a 
Condorcet Winner.

(4) the CW that is created with the least total modification is the winner.  

Modifications are measured by how much they change the ratings on how many ballots.

For example if you change X's rating by .27 on 10 of the 537 ballots of one faction, and by .32 on 15 
ballots from another faction, then the total modification is  2.7 + 4.8 = 7.5

The reason for the competition is that otherwise the method would be NP-complete.