 The bounds in item 1 are optimal, in the sense that there exist probability functions \(P_L\) and \(P_U\) such that \(a_i \leq P_L(\gamma_i),\) \(P_U(\gamma_i)\leq b_i\) for \(1\leq i\leq n\), and \(L_{\Gamma,\phi}(a_1,\dots,a_n,b_1,\dots,b_n) = P_L(\phi)\) and \(P_U(\phi) = U_{\Gamma,\phi}(a_1,\dots,a_n,b_1,\dots,b_n)\).