Correlated and product games: all equalizer conclusions

Let $S$ be a Bell experiment with exactly two parties and exactly two settings per party, each setting having an arbitrary positive finite number of outcomes. Let $\Omega=\{0,1\}^2$ be its setting tuples, $X_a$ its measured outcome tuples at setting $a$, and $\mathcal V$ its assignments $v_i(b)$ of an outcome to every setting of every party. Write $\Delta(E)$ for the probability distributions on a finite set $E$. Let $Q_a(x)$ be a real table whose rows are probability distributions and whose marginals agree on every subset of parties whenever the settings on that subset agree. Assume also that there exist positive local dimensions, a positive semidefinite joint matrix $\rho$ of trace $1$, and positive semidefinite local measurement matrices $M_{i,b,u}$ summing to the identity over $u$, such that $Q_a(x)=\operatorname{Re}\operatorname{tr}((M_{0,a_0,x_0}\otimes M_{1,a_1,x_1})\rho)$. Define $P_p(a,x)=\sum_{v:\,v_i(a_i)=x_i\ \forall i}p(v)$, and assume that no $p\in\Delta(\mathcal V)$ satisfies $P_p(a,x)=Q_a(x)$ for every $a,x$. For finite probability distributions $q,r$, let $D(q\Vert r)=\infty$ if some $q(z)>0=r(z)$, and otherwise let $D(q\Vert r)=\max\{0,\sum_{z:q(z)\ne0}q(z)\log_2(q(z)/r(z))\}$, as an extended nonnegative real. Define $D_s(p)=D((s(a)Q_a(x))_{a,x}\Vert(s(a)P_p(a,x))_{a,x})$, $F(s)=\inf_{p\in\Delta(\mathcal V)}D_s(p)$, and let $\mathcal A$ be the setting distributions of the form $s(a_0,a_1)=u_0(a_0)u_1(a_1)$, where $u_0,u_1\in\Delta(\{0,1\})$. For $A\subseteq\Delta(\Omega)$, put

and say that $(s,p)$ satisfies the saddle inequalities for $A$ when $s\in A$, $p\in\Delta(\mathcal V)$, and $D_t(p)\le D_s(p)\le D_s(r)$ for every $t\in A$ and $r\in\Delta(\mathcal V)$. Then there exists an extended nonnegative real $V$ with $0<V<\infty$ for which all the following hold: some $s\in\Delta(\Omega)$ and $p\in\Delta(\mathcal V)$ satisfy the saddle inequalities for $\Delta(\Omega)$ and $D_s(p)=V$;

the set $\mathcal M_{\Delta(\Omega)}$ is nonempty and equals $\mathcal M_{\mathcal A}$; every $p\in\mathcal M_{\Delta(\Omega)}$ satisfies $D_s(p)=V$ for every $s\in\Delta(\Omega)$; and a pair satisfying the saddle inequalities for $\mathcal A$ exists if and only if there exist $s\in\mathcal A$ and $p\in\Delta(\mathcal V)$ such that $D_s(p)=F(s)$ and, for some extended nonnegative constant $K$, $D_t(p)=K$ for every $t\in\Delta(\Omega)$. Moreover, for every $s\in\Delta(\Omega)$ and $p\in\Delta(\mathcal V)$, if $s\in\mathcal A$, $D_s(p)=F(s)$, and such a constant $K$ exists, then that particular pair $(s,p)$ satisfies the saddle inequalities for $\mathcal A$. Product distributions and other distributions with zero coordinates are included.