Mutual knowledge (logic)
# 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 $occurred.^{:73}$ 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. 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".

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

