[EM] Transformation from range to approval ballots

[Ross Hyman]

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.