Non-Zero-Sum Game as Zero-Sum Game

Theorem
Let $G$ be a non-zero-sum game for $n$ players.

Then $G$ can be modelled as a zero-sum game for $n + 1$ players.

Proof
At each outcome, the total payoff of $G$ will be an amount which will (for at least one outcome) not be zero

Let an $n + 1$th player be introduced to $G$ who has one move:


 * $(1): \quad$ Select any player $m$.


 * $(2): \quad$ If the total payoff of $G$ is $+k$, receive in payment $k$ from player $m$.


 * $(3): \quad$ If the total payoff of $G$ is $-k$, pay $k$ to player $m$.

This new game is the same as $G$ but with an extra (dummy) player, and is now zero-sum.