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Yvain comments on Game Theory As A Dark Art - Less Wrong
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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.