Mutual knowledge (logic)

Mutual knowledge is a fundamental concept about information in game theory, (epistemic) logic, and epistemology. An event is mutual knowledge if all agents know that the event occurred.^[1]: 73 However, mutual knowledge by itself implies nothing about what agents know about other agents' knowledge: i.e. it is possible that an event is mutual knowledge but that each agent is unaware that the other agents know it has occurred.^[2] Common knowledge is a related but stronger notion; any event that is common knowledge is also mutual knowledge.

The philosopher Stephen Schiffer, in his book Meaning, developed a notion he called "mutual knowledge" which functions quite similarly to David K. Lewis's "common knowledge".^[3]

Communications (verbal or non-verbal) can turn mutual knowledge into common knowledge. For example, in the Muddy Children Puzzle with two children (Alice and Bob, ${\displaystyle G=\{a,b\}}$), if they both have muddy face (viz. ${\displaystyle M_{a}\land M_{b}}$), both of them know that there is at least one muddy face. Written formally, let ${\displaystyle p=[\exists x\!\in \!G(M_{x})]}$, and then we have ${\displaystyle K_{a}p\land K_{b}p}$. However, neither of them know that the other child knows (${\displaystyle (\neg K_{a}K_{b}p)\land (\neg K_{b}K_{a}p)}$), which makes ${\displaystyle p=[\exists x\!\in \!G(M_{x})]}$ mutual knowledge. Now suppose if Alice tells Bob that she knows ${\displaystyle p}$ (so that ${\displaystyle K_{a}p}$ becomes common knowledge, i.e. ${\displaystyle C_{G}K_{a}p}$), and then Bob tells Alice that he knows ${\displaystyle p}$ as well (so that ${\displaystyle K_{b}p}$ becomes common knowledge, i.e. ${\displaystyle C_{G}K_{b}p}$), this will turn ${\displaystyle p}$ into common knowledge (${\displaystyle C_{G}E_{G}p\Rightarrow C_{G}p}$), which is equivalent to the effect of a public announcement "there is at least one muddy face".