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I'm using a textbook called Playing for Real, and I would like some help in understanding a proof in the text. Before that, here are the relevant sections.

Definition of strictly competitive game

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Relevant Theorem

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Definition of Value of a game

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Extra assumption

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Proof that I want to understand enter image description here

My question is about the sentence let $W_v$ be the smallest set into which player 1 can force the outcome. This would mean that the player can only force the outcome into sets that look like $\{u_j, u_{j+1} .... u_k \}$.

Say we have 5 outcomes, $u_1, u_2, u_3, u_4, u_5$. Why can't it be that player 1 can force the outcome into be in the set $\{u_3, u_5\}$?

Or why couldn't we have that player 1 can force the outcome to be in the set $\{u_3\}$?

And so on and so forth

Thanks for your help!

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  • $\begingroup$ It probably should be that $W_v$ is the smallest set in $\{W_{u_i} : 1\leq i \leq k\}$ into which player I can force the outcome, not just the smallest set. $\endgroup$ – Brian Moehring Jan 18 '20 at 4:08
  • $\begingroup$ @BrianMoehring, I think that would make sense. Thanks! $\endgroup$ – Phil Jan 18 '20 at 16:14

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