Multiplayer zero sum games

Question

In a course I'm taking, the professor mentioned that a zero sum games are only interesting for 2 players.
Can someone explain me that?

Answer 1

Take any $n$-player game. Add a $n+1$^th dummy player who has a single action available and a payoff function $u_{n+1}$ given by $u_{n+1}(a_1,\ldots,a_n,a_{n+1})=-\sum_{i=1}^n u_i(a_1,\ldots,a_n)$. This $n+1$-player game is a zero-sum game that is strategically equivalent to the original $n$-player game.

Answer 2

By definition of the zero-sum game, the losses of player p1 are entirely added the wining of player p2 and vice verse. If you introduce a third player, p3, and say if p1 losses 5 points, we can't move the 5 points to both players p2 and p3 simultaneously, otherwise the game will not be a zero-sum game. Also, we can't introduce a new distribution rule because the current methods will not then be valid.

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