# Nonlocal games: The CSCH game

The problem that I am having is that I don't see how they come up with the probabilities mentioned on Slide #7 here:

https://www.ipam.ucla.edu/publications/opws1/opws1_9367.pdf

first I don't understand what $a_{r}(s), b_{r}(t)$ mean. On slide #6 they are defined as $a_{r}:S \rightarrow A$ and $b_{r}:T \rightarrow B$ and am missing what the subscript $_{r}$ is supposed to dictate.

Now I see that $s \in S \text{ and } t \in T$ where $S \text{ and } T$ are the set of questions for player 1 and 2 respectively. So is, for example $b_{r}:T \rightarrow B$ the probability of answering in such a way that combined with player $S$'s question and answers that they end up winning? If however this was the case, what does slide #7's saying that $a_{r}(s) = 0 \text{ and } b_{r}(s) = 0$ mean? note the $b_{r}(s)$ instead of $b_{r}(t)$ on the slide. Is this a mistake? Either way I don't understand what they are trying to say. Lastly leading up to the question, why:

$$p = max \sum_{s,t}π(s, t)V (a_{r}(s), b_{r}(t)|s, t) = \frac{3}{4} ?$$

so, how did they get $\frac{3}{4}$ in the game $s \cdotp t = a \oplus b$? Perhaps someone could show the numerical example?

Thanks much,

Brian

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