[EM] Proportional Range Voting via Honest Approval PAV

[fsimmons at pcc.edu]

While writing the below it occurred to me that we could construct another Proportional Representation 
method based on ordinal ballots (ranked preferences) by the following technique:

(1)  Convert the ordinal rankings into cardinal ratings via the monotonic, clone free techinque that I outlined 
under the title "Borda Done Right" a few months back.

(2) Convert these ratings into approval style ballots via the honest approval strategy.

(3)  Apply PAV to the resulting ballot set.

Here's a brief review of the ballot conversion at the heart of Borda Done Right:

For each candidate C on the ballot in question let p(C) be the probability that a candidate ranked equal to or 
behind C would be elected in a random favorite election. Then treating these probabilities as scores, 
normalize them to a cardinal ratings scale of zero to one.

[Any other monotone, clone free lottery could be used in place of the random favorite lottery
 in the definition of p(C).]

I would be interested to see some simulations comparing this 
  
ordinal-->cardinal-->approval--->PAV

method of PR with other PR methods based on rankings.

Since all of the steps are monotone and clone proof, the composite method should share these properties 
to the extent that PAV does.

My Best to You All on this holiday weekend!

Forest

> -----------------------------------------------------------------
> Date: Sun, 25 Dec 2011 16:40:50 +0000 (GMT)
> From: fsimmons at pcc.edu
> To: election-methods at lists.electorama.com
> Subject: [EM] Proportional Range Voting via Honest Approval PAV
>> Now that we have a good definition of honest approval strategy, 
> we can automatically adapt methods 
> (like PAV) that are based on approval style ballots to cardinal 
> ratings style ballots.
> 
> Definition: Honest Approval Strategy: Approve your k top ranked 
> candidates, where k is the sum of 
> your cardinal ratings (on a scale of zero to one) rounded to the 
> nearest whole number.
> 
> So given a set of Range/Score/Grade ballots, first convert them 
> to cardinal ratings on a scale of zero to 
> one. Then use honest approval strategy to convert these ratings 
> ballots to approval style ballots. 
> Finally apply PAV (or whatever approval method you like) to 
> determine who gets elected.