[EM] Re: Multiwinner Approval Strategy

[MIKE OSSIPOFF]

With 0-info it isn't as difficult to show that the 1-winner strategy is 
good, in this case the above-mean strategy.

For 0-info, the slate approach isn't needed. The skates was  just one way to 
try to show that the better-than-expectation strategy is reasonable  for 
elections that are not 0-info.

If Pij is the probability that, if there are 2 candidates between whom my 
vote for one and not the other can make or break a tie involving which will 
beat the other out of being on the committee, then i & j will be those 
candidates--then, just as in the 1-winner case, because it's 0-info, all the 
Pij are the same and can be dropped.

Then the algebra is the same as in the 1-winner case, and leads to the 
above-mean strategy of 0-info Approval

The only problem is that, as you mentioned, the actual outcomes in the 
multilwinner are whole committees, and the utilitly of a whole committee 
isn't just the sum of the utilities of its members, because, for instance, 
if your faction has 5 of the 9 seats it has a majority, which is very 
different from if it has only 4 of the 9 seats. But taking that into account 
seems like it would be complicated. And if we ignore that problem, then the 
above-mean strategy for 0-info is sound.

Perfect or not, the best strategy available is the one to use, and, at least 
so far, for 0-info multiwinner Approval, the above-mean strategy is the best 
strategy available.

Mike Ossipoff

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