[EM] A simple Approval PR method with no fractions

[Ross Hyman]

Here is a simple Approval PR method with no fractions.
I think Phragmen Approval is the best droop proportional representation system for approval ballots.  
But here is a version of PR approval that produces a proportional list like Phragmen, but it is simpler. It uses only integers.  No fractions.   Each stage is like a regular approval election except that ballots can have vote values equal to integers larger than 1.  Because of this restriction it is not Droop proportional but I think the differences are slight when the number of seats is small where strict proportionality is impossible anyway.  


Method:Ballots are approval style.  The vote value of each ballot is initially set to 1.
1. Calculate the total vote for each non-elected candidate by adding the vote value of each
ballot that approves of a candidate to each candidate’s total.  Do this similarly to the way one would for
majoritarian approval, using the same vote value for each approved candidate on a ballot regardless of how many candidates the ballot approves.  The only difference from regular approval is that ballots can have vote values larger than 1.
2. Elect the candidate with the most votes.
3. Reset the vote value of all ballots that approved of the elected candidate to 0.
4. Add 1 to the vote value of all ballots.
5. Return to step 1 until all candidates have been elected.


Example elect 105 a1 a2 a3….
3 b1 b2 b3 ….

2 c1 c2 c3 …

Below, the total vote for each party (A,B, or C) is shown at each stage.  The elected candidate at each stage is indicated by the total number candidates now elected from that party.


A              B              C5,1            3              2
5              6,1            4
10,2           3              6
5              6              8,1
10,3           9              2
5              12,2           4
10,4           3              6
5              6              8,2
10,5           9              2
5              12,3           4
 
For 10 seats this gives the desired result: 5 seats for A,  3 seats for B and 2 for C.  However if the number of seats were very large the method awards 50% to A, 25% to B and 25% to C but it should be 50%, 30%,20%.  But I think its simplicity makes it worthy of consideration when the seat number is small.