[EM] Thiele’s method for Range Ballots

[Ross Hyman]

Thiele’s method for Range Ballots


Thiele’s proportional representation method is another way, besides Phragme’n’s method, of producing a house monotonic proportional list.  The method for approval ballots reduces the weight of a ballot according to the number of already approved elected candidates on the ballot. The weight factor is 1/(n+1) for D’Hondt. Thiele’s method can be extended to range ballots using the transformation of range to approval ballots I posted about last week.


For example, for the range ballotCandidate a: 0.9
Candidate b: 0.7
Candidate c: 0.4
Candidate d: 0.3
In which candidate a and candidate b have already been elected, the D’Hondt weight of the ballot is:
w = (1-0.9)*(1-0.7)*1 + [0.9*(1-0.7) + (1-0.9)*.7]*(1/2) + 0.9*0.7*(1/3) 
  = 0.03 + 0.17 + .21 = 0.41


The amount that this ballot contributes to electing candidate c is v_c*w = 0.4*0.41 =  0.164.
And the amount that this ballot contributes to electing candidate d is v_d *w =0.3*0.41 = 0.123.