# Game Theory - Extensive Zero-Sum Game Property Proof

How I might I go about to prove (or disprove, but I believe that this is true) the following: We call a 2-player extensive game $\Gamma$ a zero-sum game if the sum of the 2 payoffs for an terminal history $h$ is $0$. Suppose that $\Gamma$ is a zero-sum extensive game with finitely many terminal histories. Is there a value $v \in \mathbb{R}$ such that the payoff of player 1 is $v$ for every subgame perfect equilibrium? Argue that this is true, or give a counter example.

I think that this should be true, but I am not quite sure how to approach the proof for my assignment. I tried playing with a few examples, but cannot seem to derive a counter example, which also leads me to believe that it's true. Furthermore, I tried a proof by induction, but that did not quite work, as I was unsure how to use the fact that $\Gamma$ was zero-sum in the inductive-step.

Help would be appreciated. Thanks!

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Maybe include some definitions for these terms? – Benjamin Dickman Nov 7 '12 at 8:56