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Yvain comments on Game Theory As A Dark Art - Less Wrong
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Considering the reasoning that ends in "everyone is kicked off the board," wouldn't they all talk about it for a few minutes and then reject the proposal 4-1 (or maybe 3-2)?
This seems much like the Prisoners' Dilemma. Yes, you can avoid it easily if you can talk beforehand and trust everyone to go through with their precommitments. If you can't talk or you don't trust what they say, then it's much harder to avoid. After all, if the first two directors cooperated with the plan by voting no, then the second two directors would have a very high incentive to defect and vote yes.
In practice people are usually able to solve these for much the same reasons they can usually solve prisoners' dilemmas - things like altruism and reputational penalties.
What occurred to me when I read it is "Why is this guy allowed to propose a motion which changes its actions based on how many people voted in favor of, or against, it?" While it's likely the company's bylaws don't specifically prohibit it, I'm not sure what a lawyer would make of it, and even if it worked, I don't think these sort of meta-motions would remain viable for long. I suspect the other members of the board would either sign a contract with each other, (gaining their own certainty of precommitment,) or refuse to acknowledge it on the grounds that it isn't serious.
Even without a precommitment etc., there isn't direct incentive to be the first or second "yes" vote, only the third. If you had two shills on the board, it's a much stronger scenario.
But since there's such a strong incentive to be the third, if you are the second-most-senior-director and know that all the directors are strawmen-rational-actors, you can be pretty confident that if you vote yes, the most-senior-director will also vote yes.
Of course, it all gets into careful opponent analysis then, which makes the whole exercise quite fuzzy and into "well, Tom really hates the new guy, so he'll probably vote no because he's ornery" territory. All the directors are basing their decisions on the decisions of each other, since there is no reward for acting alone. Again, a second confederate in the beginning makes all the difference.