[EM] Summability (Efficient Parallelizability)

[Forest W Simmons]

For practical purposes any method based on rankings or range style 
ballots, can be closely approximated by a summable version. Since 
approval cutoffs can be incorporated into rankings and ratings, methods 
that require approval cutoffs can also be efficiently accomodated.

It's based on the idea that if you agree with me that A is best, B 
second best, and C third best, then it is likely that your ratings of 
the remaining candidates will be highly positively correlated to mine.

If there are N candidates we need an array with M=N*(N-1)*(N-2) rows 
and N+1 columns.

Each row number stands for one of the permutations of N candidates 
taken three at a time.

If your ballot agrees in the order of the top three candidates with the 
ith permutation, then your rating or ranking (as the case may be) of 
the jth candidate is added into the ith row and jth column of the 
array.  And a one is added into column (N+1) of that row.

Each row of the summed array is a summary of the votes of one of the 
basic M factions, together with the number of voters in that faction.  

The first three numbers in the row are the ones in which the members of 
that faction are in precise agreement.  The remaining numbers (divided 
by the faction size) represent average rankings or ratings of the other 
candidates by members of the faction.

Since the dimension of the Array is N*(N-1)*(N-2) by (N+1) it grows as 
the fourth power of N.  For 100 candidates the array would have only 
one hundred million entries, which is tiny by today's standards of data 
sets.

It is practically infinitesimal compared to the 100 factorial possible 
permutations of the candidates.

Forest