There was no answer in https://physics.stackexchange.com/questions/600494/is-there-a-bell-inequality-for-2-times-2-and-1-times1-times1-times1-configur. Hence posting in mathoverflow on the possibility it might belong to mathoverflow as it involves non-trivial inequalities.
Bell inequality violation in two party setting has survived experimental verification.
Instead of two players Alice and Bob let us include two additional players Chihiro and Denzel.
In the original game Alice and Bob are independent players and let us denote the situation $1\times 1$.
If Alice, Bob, Chihiro and Denzel interact independently let us denote the situation $1\times1\times1\times1$.
If Alice and Bob collude let us denote the situation $2\times1\times1$.
If Alice and Bob collude and Chihiro and Denzel collude let us denote the situation $2\times2$.
If Alice, Bob and Chihiro collude let us denote the situation $3\times1$.
- Are the Bell inequality violations for these cases quantified somewhere?
- Is there experimental results somewhere (it would be out of scope for MO)?