 The example in Section 5.1 hints that mutual knowledge is not the only weakening of common knowledge that is relevant to coordination. Brandenburger and Dekel (1987) and Monderer and Samet (1989) explore another option, which is to weaken the properties of the \(\mathbf{K}^{*}_N\) operator. Monderer and Samet motivate this approach by noting that even if a mutual knowledge hierarchy stops at a certain level, agents might still have higher level mutual beliefs about the proposition in question. So they replace the knowledge operator \(\mathbf{K}_i\) with a belief operator \(\mathbf{B}^p_i\):