[Election-Methods] Two proportional subset methods

[Aaron Armitage]

This is my first message to the list, so I should introduce myself. I'm Aaron Armitage, and I'm interested in voting theory and a longtime lurker on the list. I have no special qualifications, but I hope the field still has room for for enthusiastic amateurs to contribute.
 
Now, on to the methods.
 
If we can treat the seats as equivalent, if, for example, we're willing to let the assignment of ministries be done internally by the cabinet, or if we're making committee assignments within American legislatures, we can elect them using top n plurality, where n is the number of seats open, provided that voters can change their votes. Preferably the votes would remain subject to change for the entire session. I think this would actually be more stable than closing them, because if there is an occasion for changing the membership and the political will to do it, it will happen anyway. By leaving the votes open, there can be more continuity. But even if the votes close after a certain point, there should be enough time for every actor to react to others' actions, say a week. Each faction will strategically distribute its votes to maximize its influence, and will eventually reach the Nash equilibrium, which is proportionality. Since the "game" is so
 simple and optimum strategy so easy to discover, this should happen fairly reliably.
 
The problem posed by Kristofer Munsterhjelm is more interesting. We want to make the overall cabinet relatively proportional, but we also want to be relatively majoritarian in electing particular ministries. I like ordinal ranking, so I'll suggest a method using those instead of cardinal ones. Every candidate is eligible for every seat; the voters provide separate ordinal rankings for each position. If the voters are MPs we could require full rankings, but this system should also be practical for large-scale elections and there will have to be some reasonable way of handling truncation.
 
Since no person can hold two positions and each ministry can be lead by only one person, we're trying to discover which combination of outcomes best satisfies the rankings provided. I suggest pairwise comparisons using a particular method to discover which of the two combinations being compared better matches the voters' rankings. Give each seat a "weight" of one and ignoring results that are the same in both combinations, each combination divides the "weight" of each seat among those voters who preferred that combination's winner over the other combination's winner for the same position.
 
If, for example, I'm  member of a 100 seat Parliament and I vote Adams > Baker > Clark for Prime Minister and Xavier > Ypres > Zumwalt for Minister of Silly Walks. Take the pairwise contest between Adams/Zumwalt and Baker/Ypres. 34 other members agree with me in preferring Adams to Baker, and 79 other members agree with me in preferring Ypres to Zumwalt.  Adams/Zumwalt gives me .0285714... and Baker/Ypres gives me .0125. If we're comparing Adams/Ypres to Baker/Zumwalt, then I take the scores for both of them because I favor both aspects of the outcome, so I get .0410714... for Adams/Ypres and nothing for Baker/Zumwalt.
 
Each pairwise comparison is settled in favor of the outcome which distributes support more evenly among the voters. The Condorcet winner, if there is one, will be the combination which takes office. Otherwise it will be the winner of your favorite completion method.
 
If we want to privilege the head of government (and possibly other positions we consider especially important, although the more we do this the less proportional the cabinet will be), make that one position elected on its own using a Condorcet method and then elect the rest of the cabinet using the method described.


      
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