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MAX and MIN apply the minimax algorithm each time before making their move, searching to some predefined depth. Imagine, however, that the MAX player applies the minimax algorithm only once, at the beginning of the game, and then uses the obtained minimax values for all his future moves. In what case would MAX player implemented in this way surely maximize his minimum gain?

A There is no guarantee that such a player would maximize his minimum gain

B If he has reached all final states of the game when calculating the minimax-value

C If MAX’s heuristic is more informed than MIN’s heuristics

D If he uses a heuristic that is identical to that of MIN player

Can someone explain why the answer is A and not B?

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It's not too clear from your question what you mean, but I guess this is not even originally a question coming from you, so I'll suppose you cannot give more information

In any case, if the result of your minimax algorithm is a path of the game tree, aka a Nash Equilibrium, then the problem is that there might be more than one Nash Equilibrium. If the result of the minimax is a whole strategy, then it implies that you have found a SubGame Nash Equilibrium, and by definition you will maximize your minimum gain

This is assuming you are dealing with a perfect information game

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