[EM] Approximating Non-Summable Methods with Summable Methods

[fsimmons at pcc.edu]

Recently Kathy Dopp initiated a thread rerquesting the best summable, monotone, PR method ideas.

In general non-summable methods can be approximated by summable methods if range ballots are used 
in the following way:

At each precinct, for each candidate C take a weighted average of the ballots that rate candidate C at 
the top where the weight of a ballot is one over the number of candidates rated at the top on that ballot.  
Then the weighted averages (together with the total weights for each average) are passed along to the 
central location where they can be averaged together using their respective weights, to get one weighted 
ballot for each candidate.  The method you had in mind can be applied to these ballots.

In particular, if you have a monotone PR method based on Range Ballots, you can apply that method.  If 
you don't have a monotone PR method based on Range Ballots, you can go to Warren Smith and get 
one; he collects them.

Of course, after the averaging, you can convert the range ballots into ordinal rankings, and then use your 
favorite method based on rankings if you don't like range ballots.

Note that if the original ratings are zero/one only, i.e. approval ballots, then the method still works.

In summary, you can get a summable version of your favorite method.  And if you want, no matter what 
kind of ballots you method uses, you can get the required information via approval ballots.