[EM] Transformation from range to approval ballots

[Toby Pereira]

This is a method that I have considered in the past, but I decided it wasn't the best method. For your example:
Candidate a: 0.9
Candidate b: 0.7
Candidate c: 0.4
I would convert to approvals as follows:
0.4: Approves a, b, c0.3: Approves b, c0.2: Approves a0.1: Approves none
I would define it so that each "fraction" of a voter that approves a candidate with a score of s will also approve all candidates with a score of s or above.
This way is simpler, retains within-voter Pareto dominance, and is scale invariant - that is to say that if all scores are multiplied by a constant, then it would not affect how the approvals are spread across the candidates so any election result would be the same.
Toby

      From: Ross Hyman <rahyman at sbcglobal.net>
 To: "election-methods at lists.electorama.com" <election-methods at lists.electorama.com> 
 Sent: Sunday, 11 February 2018, 15:01
 Subject: [EM] Transformation from range to approval ballots
   
Transformation from range to approval ballots.
Ranges, r_a, span from 0 to 1.  Each range ballot is transformed into many approval ballots, each with its own weight.  Its weight is a product of the r_a’s  for a ballot that approves candidate a, and (1-r_a) for a candidate that does not approve candidate a.


Example:  Three candidates. A range ballot gives them the following scores
Candidate a: 0.9
Candidate b: 0.7
Candidate c: 0.4


This is transformed into a set of approval ballots of every type with the weights:


don’t approve a, don’t approve b, don’t approve c: (1-0.9)*(1-0.7)*(1-0.4) = 0.018
don’t approve a, don’t approve b, approve c: (1-0.9)*(1-0.7)*0.4 = 0.012
don’t approve a, approve b, don’t approve c: (1-.09)*0.7*(1-0.4) = 0.042
don’t approve a, approve b, approve c: (1-0.9)*0.7*0.4 = 0.028 
approve a, don’t approve b, don’t approve c: 0.9*(1-0.7)*(1-0.4) = 0.162
approve a, don’t approve b, approve c: 0.9*(1-0.7)*0.4 = 0.108
approve a, approve b, don’t approve c: 0.9*0.7*(1-0.4) = 0.378
approve a, approve b, approve c: 0.9*0.7*0.4 = 0.252


The total is 1, as required.
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