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Suppose,in a Gale-Stewart game, player I or player II's action space $A_i$ at an arbitary stage $i$ is fixed as $\{0,1\}$. $f:\{0,1\}^{\omega} \to \mathbb{R}\times\mathbb{R}$ is a payoff function that maps each outcome to a vector, in which, the first component is player I's payoff, and the second is player II's payoff.

Where can I read about the above general situation? It seems to me, the existence of equilibrium may relate to the cardinality of the partition induced by level sets, even each level set is required to open or closed.

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    $\begingroup$ Why do you introduce the partition $\mathcal{P}$. If you take the function $f$ to be defined on $\{0,1\}^\omega$ directly, you get a (coarser) partition into $f$-level-sets. $\endgroup$ – Michael Greinecker Apr 19 '13 at 7:26
  • $\begingroup$ @MichaelGreinecker: Thank you. $\endgroup$ – Metta World Peace Apr 19 '13 at 7:37

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