[EM] Phragme’n method for range ballots

[Ross Hyman]

Phragme’n method for range ballots.
Sometimes proportional budgeting elections use range style ballots.  Here is how to do a Phragme’n proportional representation election for a priority spending list using range ballots.  I have posted this generalization before on this list (the search function is not working so I can’t refer to it) but it received no feedback, perhaps because I had not adequately explained the transformation from range to approval ballots which I have now posted separately. 


The method is the same as for approval ballots with transformed value of V_m,n and V_n, They are explicitly transformed in the method below.


The method:
Spending items are elected sequentially.  The first elected spending item is spending item 1.  The next is spending item 2, etc.  The first item on the priority list is the one that minimizes


s_1 = C_1/V_1 

and the nth item on the spending priority list is the one that minimizes 

s_n = (C_n + sum_m V_m,n * s_m)/V_n

C_n is the monetary cost of the nth spending item. (All C_n = 1 for candidate elections)

sum_m is a sum over the elected spending items 1 to n-1.

V_m,n = sum_i r_i,m * r_i,n * product_j (1 – r_i,j) is the total weight of ballots that approve spending items m and  n, and not spending items between m and n.

sum_i is the sum over all range ballots.


r_i,j is the score given to spending item j on the ith range ballot.  It takes values on a continuum from 0 to 1.

product_j is over all spending items between m and n, from m+1 to n-1.


V_n = sum_i r_i,n is the sum of the range votes for spending item n.