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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