Multi-Winner Approval Voting

[Eric Sanders]

Who should get the 3 seats? (and why?)

Multi-Winner Approval Voting

3 seats

4 candidates

3 voters

Voter 1: 

Approves of Candidate 1

Approves of Candidate 2

Approves of Candidate 3

Does Not Approve of Candidate 4

Voter 2:

Approves of Candidate 1

Approves of Candidate 2

Approves of Candidate 3

Does Not Approve of Candidate 4

Voter 3:

Does Not Approve of Candidate 1

Does Not Approve of Candidate 2

Does Not Approve of Candidate 3

Approves of Candidate 4

[Bruce Gilson]

First of all, with only 3 voters and 3 seats to fill, any proportional system should divide the seats identically to the voters. This means Candidate 4, with exactly one voter supporting him, gets a seat. The problem is that it's a three-way tie for Candidates 1 to 3. It seems that you need to draw lots to choose 2 of the three.

It is clear that Candidates 1, 2, and 3 form, as it were, a party

 

(even if not labeled as such) which has total support from 2 of the three voters. Candidate 4 has equally total support from the one remaining voter. So giving one seat to Candidate 4 and two to be divided among Candidates 1, 2, and 3 is as proportional as they come.

[Stephen Unger]

I fully agree with Bruce's analysis.

Steve
............

Stephen H. Unger
Professor Emeritus
Computer Science and Electrical Engineering
Columbia University
............

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[Dale Sheldon-Hess]

Also also agree, and using reweighted range voting gives the same answer.

(3-way tie for first seat among 1, 2, and 3 with two approvals vs. 4's
one; break tie by lots, the two who don't win are reweighted to have
one vote, the same as 4; break tie by lots; if 4 wins, final seat
decided by lot between remaining two candidates; if a candidate other
than 4 wins, final candidate out of 1,2,3 is reweighted to 2/3rds of a
vote, and 4 wins the final seat. Final result, no matter how the lots
are drawn: 4 and two candidates out of 1,2,3 are seated.)

Also: ugh, let's use letters instead of numbers for candidates; that
was confusing to type, reading it can't be better!