[EM] How to make summable any method based on range ballots.

[fsimmons at pcc.edu]

How to make summable any method based on range ballots.

If no range ballot rates more than one alternative at the top range value, then replace each ballot with the 
average of all of the ballots that have the same favorite.

Otherwise, first split each ballot into n ballots, one for each equal top alternative on the ballot, and 
assign a weight of 1/n to each such ballot for use in the averaging.

This can be done at the local precinct level as long as the weights are conveyed to the next level along 
with the (weighted) average ballots for the alternatives.

After all of the ballots are combined this way there will be a range ballot for each alternative that was 
rated top on some ballot, and each such ballot will have a weight assigned that indicates the total 
support for the respective alternative, as well as the number of original ballots (or fractions thereof) from 
which this ballot was averaged.

These ballots can be used in any method based on range ballots, including Range itself, where the 
result will be identical to that obtained by counting the range ballots in the usual way, i.e. the alternative 
with the highest average range score will win.

But now I want to specialize to a new way of using these summary ballots:

Note that the number of “factions” will be less than or equal to the number of candidates, and the range 
values on the weighted average range ballots will in general be irregularly spaced.

This is the ideal set up for an elimination method based on collapsing weakest preferences when they 
are too weak for making a reasonable elimination decision.

We take advantage of the fact that as follows:

When there are k levels, we eliminate every alternative that is beaten pairwise by a margin ratio greater 
than (k-1)/1.

[An alternative Y is beaten pairwise by X with a margin ratio of  x/y iff the total weight of the ballots on 
which X is rated above Y divided by the total weight of the ballots on which Y is rated above X is x/y.]

This ratio (k-1)/1 is chosen because it makes cycling impossible, and therefore leaves at least one 
alternative in play after the eliminations.

After the eliminations based on k levels, we decrement k by collapsing the weakest preference on each 
ballot, and make the same kind of eliminations based of the new value of k.  

[When we collapse a preference we shrink the gap to zero without changing the size of any other gap.   
Equivalently, one could convert all of the range ballots to ranked ballots with the preferences ranked in 
strength before proceeding with this method.]

We rationalize that it is OK to make eliminations on the basis of a smaller k because 

(1)	the ballot preferences are stronger and hence more reliable, and
(2)	with the smaller number of levels it is impossible to eliminate all of the alternatives on the 
basis of the smaller margin ratio.

When k reaches the value of 2, the method automatically elects the approval winner among the 
remaining alternatives, provided we think of the approval cutoff as the largest rating gap on each ballot.

Comments?