Note that \(y\) is not strictly dominated for Charles. It is easy to find a probability measure \(p\in\Delta(S_A\times S_B)\) such that \(y\) is a best response to \(p\). Suppose that \(p(u,l)=p(d,r)=\frac{1}{2}\). Then, \(EU(x,p)=EU(z,p)=1.5\) while \(EU(y,p)=2\). However, there is no probability measure \(p\in \Delta(S_A\times S_B)\) such that \(y\) is a best response to \(p\) and \(p(u,l)=p(u)\cdot p(l)\) (i.e., Charles believes that Ann and Bob’s choices are independent). To see this, suppose that \(a\) is the probability assigned to \(u\) and \(b\) is the probability assigned to \(l\). Then, we have: