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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