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I am given a payoff matrix $A$ of the second player. So $a_{ij}$ denotes an outcome of the second player if the appropriate strategies were chosen. Thus, we can solve another game with matrix $-1A$ - the payoff matrix of the first player and obtain the answer for the original problem. Am I right?

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From the other player's point of view, the payoff matrix would be the negative transpose, i.e., $-A^T$.

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