[EM] electing a variable number of seats

[Andy Jennings]

Another way to elect a variable number of seats is to use Monroe's method.

Here's the overview for a fixed number of winners, N.  Every voter gives
every candidate a score on a numeric scale.  Then we find the optimal way to
choose the N winners AND divide up the voters equally and "assign" them to
the winners.  The quality metric is the sum of how each voter feels about
the one candidate to which he was assigned (that is, the numeric score they
gave to the candidate they ended up being assigned to).

(If you really think the voters should rank, not grade, all of the
candidates, then you can pull the grades from the rankings a la Borda.)

Finding the optimal such "assignment" can be done with integer programming.
 Worst cast, it is NP-hard (but not in the number of voters, just in the
number of candidates and the number of winners), but I have done some
simulations and in many cases the problems are solved pretty quickly.

So you can do this multiple times, for N=5,6,7,8,9, and compare the "quality
metric" to see which choice of N is best.  It seems that higher N would have
a natural advantage.  Maybe this is acceptable, or maybe we want to use an
"adjusted quality metric" that penalizes the different N values with
different multiplicative or additive constants.  In fact, any set of
monotone adjustment functions could be used, as long as they were chosen
before the election.

Andy Jennings
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