[EM] Another Grand Compromise

[Forest Simmons]

In this "Grand Compromise" voters can choose which kind of ballot they 
want.  The ballot styles to choose from are ordinal rankings,and a variety 
of cardinal ratings style ballots including range ballots, grade ballots 
(whether A to F or A to Z), 0 to 10 olympic, 0 to seven psychological, 
Yes/No approval, etc.

1. After the marked ballots are collected they are first tallied pairwise, 
i.e. the pairwise information available on each ballot is summed to a 
grand pairwise matrix M.

2. Using the pairwise win/loss information from M, a Condorcet Lottery 
winner is determined as described in Jobst's recent post on the "Condorcet 
Lottery" method.  [There is always a CW among "lotteries," which are 
winning probability distributions for the candidates.]

3. Use the winning probability distribution from step 2 to determine 
approval cutoffs on the original ballots:

3a. On all of the CR ballots (including approval ballots, which are just 
CR ballots with resolution 2) put the cutoff at the expected CR based on 
the probabilities from step 2. If this weighted mean CR value is the 
rating of some candidate (or the common rating of some set of candidates) 
on the ballot, then convert that rating to a fraction of approval, and 
assign each candidate at that level that fraction of approval. [This won't 
change which candidates are approved on an approval ballot.]

3b.  On each ordinal rankings ballots use the probabilities from step 2 as 
weights. If the total weight of candidates below a given rank is as great 
as the total weight of candidates above that rank, then approve the 
candidates at that rank, otherwise not.

In any case put the cutoff below the top rank (or rating) and above the 
bottom possible rank (or rating). Truncated candidates are not approved.

4. The approval winner (based on the cutoffs determined in steps 3a and 
3b) is the Grand Compromise Winner.

Forest