= CHSH game
{c}
{title2=$x\oplus y=ab$}
Two separated players receive uniformly random input <bits> $a,b$ and output <bits> $x,y$ without communicating after receiving their inputs. They win when $x\oplus y=ab$, with <exclusive or> $\oplus$. For deterministic local response functions, the four desired equalities have inconsistent parity, so at most three can hold. Shared randomness only averages such strategies, proving classical winning <probability> at most $3/4$. In terms of correlations $E_{ab}=\mathbb E[(-1)^{x+y}\mid a,b]$, the winning <probability> is $1/2+(E_{00}+E_{01}+E_{10}-E_{11})/8$, directly connecting the game to the <CHSH inequality>.
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