[EM] Approval strategy from rankings

MIKE OSSIPOFF nkklrp at hotmail.com
Mon Jan 5 05:17:02 PST 2004

>In Approval, everyone should vote strategically. When there's no 
>about other voters' preferences or voting plans, people should vote for the
>above-mean candidates. But if the voter doesn't have ratings, but only a
>ranking of the candidates, then, as I said, s/he should vote for the best
>half of the candidates. That means that, for a group having a particular
>sincere ranking, half will vote for the middle candidate, as I was saying 
>a previous posting.

I don't think I agree with this, although it coincidentally works out in
the three-candidate case.

The main reason is that, while we have no information about the voters'
utilities for each candidate, the voters themselves surely would.

They don't. That's the assumption. All I said was that, if a voter doesn't 
have opinions about rating the candidate, but only has a ranking of the 
candidates, then that voter should vote for the best half of the candidates.

That's because, in the absense of other information or assumptions, the best 
guess is to assume that the candidates' utility varies uniformly in the 
ranking. So the best half of the candidates are the above-mean candidates.

You continued:

rather than the having the voters estimate utility based on rankings, it
should be the experimenter estimating the distribution of voters.  We
could base our estimates on a linear or gaussian distribution (or its
inverse), for example.  If just looking for a middle-of-the road
estimate in the absence of other information, a linear distribution
seems the most reasonable.

I reply:

Sure, one could make it more realistic by having the voters rate the 
candidates according to some sort of distribution.

More realistic looking, more work, but not really necessary. Sometimes Borda 
is recommended for individual decisionmaking, where each consideration by 
which alternatives are being compared is treated as a "voter" with a Borda 
ballot. One does that because it's easier to rank the alternatives by each 
consideration than to rate them by each consideration. Of course it would be 
good to rate (weight) the consisderations, but if one isn't inclined to rate 
the alternatives by the considerations, one likkewise might not be inclined 
to rate the considerations either. So it isn't out of the question to assume 
that these particular voters only judge the candidates by ranking, and not 
by rating.

Sure, I'd try to rate them, but in an one actual decision situation, I 
preferred to just rank the alternatives by each consideration and use Borda, 
instead of CR. I knew the merit order of the alternatives by each 
consideration, but didn't want to try to guess ratings. Voters could have a 
similar feeling about candidates.

You continued:

Thus in a four-way race, for a block of voters with identical preference
orders, I would assume that 1/3 approve of three candidates, 1/3 approve
two candidates, and the final 1/3 bullet vote. I believe this would give
results identical to Borda.

Sincere rankings:

50: ABCD
50: DCBA
50: BCDA

Approval votes inferred as described above:

50: ABC
50: DC
50: B

Winners: B & C

Borda scores:

A: 3
B: 6
C: 5

B is the unique winner.

It isn't guaranteed to match Borda, because it depends on which 1/3 of the 
voters you have voting for which number of candidates. You can get different 
results by assigning different roles to the different thirds of the voters.

You continued:

The other approach would have all voters approving exactly two
candidates, which doesn't seem correct to me.

I reply:

If everyone voted according to their ranking, rather than rating the 
candidates, then they'd all vote for 2 candidates. Maybe not likely (if 
voters would be inclined to rate the candidates), but not incorrect.

Mike Ossipoff

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