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This starts with an assumption that P knows that SB exists

Somehow I suspect this is nonsense, why does SB not have a strategy corresponding to not existing?

In TOM, if players have complete information about their opponent's preferences, they can look ahead and anticipate the consequences of their moves and thereby decide whether or not to move. They can do this using backwards induction.

You can't look ahead and anticipate SB's moves if SB does not exist ... .

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