# [EM] Multi-Winner Approval Strategy

MIKE OSSIPOFF nkklrp at hotmail.com
Mon Feb 2 01:14:02 PST 2004

```Disclaimer: Multiwinner strategy is completely new to me, and so I can't
guarantee that I haven't missed something. But it seems to me that, till
someone devises something better, the 1-winner Approval strategies are a
good way to vote in multiwinner Approval.

Yesterday I was suggesting, for 0-info elections with multi-winner Approval,
the 1-winner 0-info strategy of voting for the above-mean candidates.

That's a good strategy, even for multi-winner. With several winners, Pij
could mean the probability that, if there are 2 candidates between whom your
ballot can make a break a tie for 8th place, those 2 candidates are i & j.

If i & j are in a tie for, say, 3rd place, and you make i win that tie,
surely j will win 4th place, or at least win one of the 8 seats. So 8th
place is the one that makes a difference between being on the committee and
not being on the committee. Whether it's strictly true or not, it seems a
reasonable assumption that a candidate who loses a close count for, say, 7th
place will still be the 8th place winner.

So Pij(Ui-Uj) is the probability that your ballot could make a difference
between i & j, multiplied by their utility difference. Weber's formula seems
just as valid for that multi-winner situation.

The problem is that, on the actual committee, the utility of a
committee-composition isn't just the sum of its members' utilities, because,
for instance, with a 9-member committee, it makes a big difference whether
your faction has 4 members on the committee or 5 members on the committee.

Other than that, the above-mean strategy seems pretty good. Unless someone
devises a strategy especially for multi-winner Approval, that deals with
committee majorities, then the ordinary above-mean strategy is good enough
for multi-winner Approval with 0-info.

As with 1-winner, the Pij are all equal, because it's 0-info, and so they
can all be dropped from the formula. The algebra is as with 1-winner, and so
you reach the same conclusion: Vote for the above-mean candidates.

What about when it isn't 0-info? I suggested the better-than-expectation
strategy. Of course it too has the disadvantage that it doesn't consider how
much difference one more winner can make when it gives your faction a
majority.

Also, an expectation is a complete outcome, and those are more complicated
for multi-winner, where an outcome is a combination of 8 candidates. It
might not make much sense to talk about a "value of the election" that's
different from your expectation, but still the situation might be simplified
and partly saved by defining a "simplified approximate value of the
election" (SAVE).

A candidate's contribution to the SAVE would be his utility multiplied by
his probability of being one of the winners. How likely he is to be a
winnner, times how good it would be for him to be one of the winners. So the
SAVE would have the same form as your expectation in a 1-winner election.

If a candidate's Pi is the probability that, if there are 2 candidates
between whom you can make or break a tie for 8th place, that candidate is
one of those 2, then things have the form that they have with 1 winner, and
that suggests that if we make the reasonable, though rough, approximation
that the Pi are proportional to the Wi (win probabilities), and the other
approximations that go with better-than-expectation, then I'd expect that we
should get the same conclusion: Vote for the candidates who are better than
your expectation for the election--each candidate who is so good that you'd
rather have him/her in charge than hold the election.

Because of these arguments, and because of yesterday's arguments about
slates, then, it seems that, till someone finds somethng  better, the
1-winner Approval strategies would be good for multiwinner Approval. Also,
those strategies are plausible just on the face of it too: Voting for the
above mean, or voting for the candidates who seem better than what you can
expect from the election seems natural and plausible.

Even when you know that you don't have as good a strategy as you'd like, the
best strategies that you have are still the thing to use.

Lastly, of course, it's also true that often, maybe usually, in Approval
elections, you just know how you want to vote. In those instances, you don't
need to make estimates about mean-merit or election-expectation. You just
vote as you want to. In fact, some people have defined utility in terms of
choices like that.

Mike Ossipoff

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