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3

First of all, there is a lot of information in this related question: Max Min of function less than Min max of function. That said, the source of your problem is a confusion about which choices are available to each player when it is their turn. Consider the left-hand side of your first example: writing this in matrix form, each player gets to choose ...


1

By Theorem 3.1 and the proof's IPS being one-way, for all real numbers $\epsilon$, if ​ $0<\epsilon$ ​ then the promise problem Input: ​ a game as described in my question must output YES if : ​ ​ ​ the game's value for Player 1 is greater than 1-$\hspace{.025 in}\epsilon$ must output ​ NO ​ if : ​ ​ ​ the game's value for Player 1 is less than ...



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