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I am currently studying for an exam and got stuck on the following question:

We have seen that finding a Nash equilibrium in a two-player zerosum game is significantly easier than general two-player games. Now consider a three-player zero-sum game, that is, a game in which the rewards of the three players always sum to zero. Show that finding a Nash equilibrium in such games is at least as hard as that in general two-player games.

How can one reduce a 3-player zero sum game to a 2-player zero sum game?

Thanks a lot in advance

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  • $\begingroup$ The title of the math.stackexchange question you have written is totally different from what is actually in the exam question. One almost certainly cannot reduce a 3-player zeros-sum game efficiently to a 2-player zero-sum game; that is the whole point of the question. It actually asks you to reduce finding an equilibrium of any 2-player general-sum game to a 3-player zero-sum game. Hint: Start with an arbitrary 2-player general-sum game G and produce a 3-player zero-sum game G' from it such that you can efficiently recover an equilibrium of G from any equilibrium of G'. $\endgroup$ Mar 31, 2021 at 5:26

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Suppose you know an equilibrium strategy of the third player. Given this strategy, you need to find the corresponding equilibrium strategies of the first two players. This remaining problem is a general two-player game, as their payoffs might not sum to 0 (they due with the third, which is irrelevant at this point).

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