# Game theory: Help with understanding the proof that any finite, strictly competitve game of perfect information without chance moves has a value

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

Relevant Theorem

Definition of Value of a game

Extra assumption

Proof that I want to understand

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

• 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. – Brian Moehring Jan 18 '20 at 4:08