[EM] Approval's bad-example

[MIKE OSSIPOFF]

Jobst--

You wrote:

Yesterday I wondered whether under Approval Voting there
would always be some equilibrium of the following kind:
All voters specify "sincere" approvals in the sense that when they prefer X 
to Y
they do not approve of Y without approving of X; and no group of voters can 
improve
their result by changing their specified approvals to some different
but still "sincere" (!) set of approvals.

I hoped that such weak kinds of equilibria might exist always.

I reply:

This is a quick reply., using a simpler example. James posted a 
co-operation/defection exmple in which co-operation wasn't a Nash 
equilibriuim.

Actual preferences:

40: C
31: ABC
29: BAC

Say the voters don't know that A is any more popular than B.

If the A and B votes vote for A and B, then C loses. But say the A voters 
co-operate by voting for A and B, but the B voters defect by voting only for 
B. B wins. The B voters have succeeded in taking advantage of the 
co-operation of the A voters.

That's the worst bad-example for Approval, and it's been mentioned by 
anti-Approval authors.

Of course it's a problem, but I've told some reasons why I don't consider it 
to be a serious problem.

If the B voters take victory in that way, it isn't a majorilty rule 
violation. The example suggests some reason to believe that A & B are 
similar, being a mutual majority set, in which case it won't matter so much 
which one wins.

If I were an A voter, and there were a possibility that the B voters would 
defect, I'd likely vote for both anyway, because if the B voters want B to 
win that badly, then maybe it's more important to them than it is to me 
which of {A,B} wins.

But the problem can be avoided by discussion, negotiation, threats, etc. 
James reasonably said that those solutions just evade the problem, because 
the method itself isnt solving the problem. Nevertheless, if there's any 
reason to believe that A is more popular than B, or more of a compromise, a 
CW (maybe the C voters prefer A to B), or more ethical or honest, etc., then 
the A voters could publicize that they're voting only for A, because they 
know that the B voters are inclined to defect, and because of principle, 
since A is more honest, more popular, the CW, etc.

Anyway, if the B voters do that, and the A voters aren't content to let them 
get away with it, then, if the defection is resented as cheating, the A 
voters won't help the B voters' candidate(s) again, and the B voters have 
hurt their cause.

For all these reasons, and probably a few more, I don't consider that 
co-operation/defection problem to be a serious problem.

With Approval, and with wv Condorcet, if there's a CW, then there's always 
at least one Nash equilibrium in which the CW wins and no one reverses a 
preference.

That can't be said for Pluirality, IRV, or margins Condorcet, with which 
there are situations with a CW where the only Nash equilibria in which the 
CW wins have people reversing a preference to protect the CW's win.

Mike Ossipoff

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