Non-Zero-Sum Game as Zero-Sum Game
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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 $m$.
This new game is the same as $G$ but with an extra (dummy) player, and is now zero-sum.
$\blacksquare$
Sources
- 1956: Steven Vajda: The Theory of Games and Linear Programming ... (previous) ... (next): Chapter $\text{I}$: An Outline of the Theory of Games: $2$