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.Osborne, Martin J., and Ariel Rubinstein. ''A Course in Game Theory''. Cambridge, MA: MIT, 1994. Print. 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".Stephen Schiffer, ''Meaning'', 2nd edition, Oxford University Press, 1988. The first edition was published by OUP in 1972. Also, David Lewis, ''Convention'', Cambridge, MA: Harvard University Press, 1969. For a discussion of both Lewis's and Schiffer's notions, see Russell Dale,
The Theory of Meaning
' (1996).

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

See also

* Elephant in the room
* The Emperor's New Clothes
* Distributed knowledge

External links

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References

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Game theory
Concepts in epistemology
Epistemic logic