On 02/05/2013 06:50 PM, Peter Zbornik wrote:
The problem (after a slight simplification) is as follows: We want to elect five seats with any proportional ranking method (like Schulze proportional ranking, or Otten's top-down or similar), using the Hagenbach-Bischoff quota (http://en.wikipedia.org/wiki/Hagenbach-Bischoff_quota) under the following constraints: Constraint 1: One of the first two seats has to go to a man and the other seat has to go to a woman. Constraint 2: One of seat three, four and five has to go to a man and one of those seats has to go to a woman. Say the "default" proportional ranking method elects women to all five seats, and thus that we need to modify it in a good way in order to satisfy the constraints.
Oh, sorry. I didn't see the part about that you could use a proportional ranking method. In that case, the answer's simple. Pick the highest ranked council extension that doesn't violate the constraints.
E.g. for Schulze's proportional ranking method, say the candidates are W1, W2, W3, M1, M2, M3 (for Woman and Man respectively).
First round, you have a matrix with W1, W2, W3, M1, M2, and M3. Say the Schulze winner is M1. That's okay, M1 gets first place.
Second round, you have a matrix with {M1, W1}, {M1, W2}, {M1, W3}, {M1, M2}, and {M1, M3}. Determine the Schulze social ordering according to the Schulze proportional ordering weights (as defined in his paper). Remove {M1, M2} and {M1, M3} from the output social ordering since these aren't permitted. Say {M1, W1} wins.
Then you just continue like that. In essence, you're picking the best continuation of the ordering given what the constraints force you to do.
You could also just null out the defeat strengths in the proportional ordering matrix, but that would produce strategy incentives since Schulze doesn't satisfy IIA.
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