How to explain Tsirelson's inequality using extended probabilities?
Some people have tried explaining the Bell inequalities using extended probabilities.
For instance, a pair of entangled photons are created and sent off to Alice and Bob. Alice can set her polarizer to $0^\circ$ or $+30^\circ$. Bob can set his to $0^\circ$ or $-30^\circ$. If both polarizers are aligned, both outcomes always agree. If only one is rotated, 3/4 of the time, there's agreement. If both are, there's only agreement 1/4 of the time.
Extended probabilities. Assume each photon "secretly" has "actual" values for both polarization settings prior to measurement. WLOG, just consider the cases where the "hidden values" between the two Alice polarizations either (A)gree or (D)isagree. Ditto for Bob's.
Then, (A,A) prob 3/8 (A,D) prob 3/8 (D,A) prob 3/8 (D,D) prob -1/8
"explains" the violation of the Bell inequality.
This still leaves open the question why we can't have (A,A) prob 1/2 (A,D) prob 1/2 (D,A) prob 1/2 (D,D) prob -1/2 violating Tsirelson's bound.