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, $G=\{a,b\}$ ), 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 tells 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".

External links

Steven Pinker. "The Stuff of Thought: Language as a window into human nature". RSA.

References

^ Osborne, Martin J., and Ariel Rubinstein. A Course in Game Theory. Cambridge, MA: MIT, 1994. Print.
^ Peter Vanderschraaf, Giacomo Sillari (2007). Common Knowledge. Stanford Encyclopedia of Philosophy. Accessed 18 November 2011.
^ 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).

[Osborne-1] Osborne, Martin J., and Ariel Rubinstein. A Course in Game Theory. Cambridge, MA: MIT, 1994. Print.

[2] Peter Vanderschraaf, Giacomo Sillari (2007). Common Knowledge. Stanford Encyclopedia of Philosophy. Accessed 18 November 2011.

[3] 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).

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