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HE KNOWS THAT ALL THE OTHER PLAYERS ARE RATIONAL.

Agreed. 9 knows that 7 is rational and will give 9 as much money as is necessary to buy 9's vote, because otherwise 7 will die.

Are you saying it would be rational in any circumstance for 7 to refuse 9's offer and die? If not, then how is it irrational for 9 to extract as much money as possible based on the absolute certainty that 7 won't refuse?
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