Just as the STV system of proportional representation uses the front end
of the Borda Count (ballots with the candidates ranked), so can the front
end of Approval Voting (ballots marked with approved candidates) be used
to achieve another system of proportional representation. 

Think in terms of layers.  If you are bundling up for extreme cold, the
more layers the better, but the first one or two layers count the most; it
doesn't do much good to add a third layer of sweaters for the torso if the
legs are still uncovered. An extra pair of mittens is good, but that first
pair of ear muffs is more urgently needed.

The layers of protection for various body parts are analogous to layers of
representation for the various parts of the body politic. (And just as
long johns cover arms, legs, and torso, so some candidates may represent
more than one segment of the electorate.)

Various multiwinner systems of voting can be specified by assigning
various weights to the various layers.

It turns out that for proportional representation, the nth layer must
receive 100/n percent of the weight of the first layer.

For example, in any multiwinner election having six or more positions to
be filled, the sixth layer should count only one sixth as much as the
first layer.

In general we have n positions to be filled from among N candidates. Let C
be one of the N!/n!/(N-n)! possible combinations (coalitions) of n
candidates, and let B be one of the approval ballots cast in the election.
Suppose that ballot B approves exactly k of the members of the coalition
C.  Then this ballot contributes support for this coalition in the amount
1 + 1/2 +...+ 1/k .

The coalition with the highest total wins the election (unless side
conditions are not met).

A side condition might be, for example, that no one approved by fewer than
1/(n+1) of the voters could be elected. This reduces to requiring majority
approval in a single winner race. This condition is easily incorporated by
eliminating such candidates before examining the various possible
coalitions. 

Another side condition might be that the candidate with the greatest
over-all approval should be in the winning coalition. (This is not
automatic.) This side condition would be appropriate if the winning
coalition is to be the entire government, and one member of the coalition
is to be the moderator or president. 

This condition can be incorporated by automatically including him in all
of the coalitions to be evaluated. In other words, every admissible
coalition is made up of the most popular candidate and n-1 other
candidates.

If there are three positions of preeminence, say a triumvirate, the method
can be used to choose the coalition of three first, and then another
application of the method in which every admissible coalition contains the
winning three and n-3 other candidates.

If the triumvirate is to have a distinguished member, three successive
applications of the method to the voted ballots would do the trick; the
first round picks the prez by restricting all coalitions to one member.
The second round restricts coalitions to three members, one of which is
the prez.  The third round considers only coalitions containing the
triumvirate.

Note that the same ballots (without alteration) are used in all three
rounds.

Well, that pretty well describes the voting procedure.

If you want the gory details of some examples and a proof of the
proportionality property, I'll post another message next time. 

I hope you like it.

Forest

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