[EM] Multi-Winner Approval Strategy

[MIKE OSSIPOFF]

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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