Reducing finding a Nash Equilibrium for a 3-player Zero-Sum game to finding a Nash Equilibrium for a two-player zero sum games.

Question

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

Answer 1

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).