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, [math]\displaystyle{ G=\{a,b\} }[/math]), if they both have muddy face (viz. [math]\displaystyle{ M_{a}\land M_{b} }[/math]), both of them know that there is at least one muddy face. Written formally, let [math]\displaystyle{ p=[\exists x\!\in\! G(M_{x})] }[/math], and then we have [math]\displaystyle{ K_{a}p\land K_{b}p }[/math]. However, neither of them know that the other child knows ([math]\displaystyle{ (\neg K_{a}K_{b}p)\land(\neg K_{b}K_{a}p) }[/math]), which makes [math]\displaystyle{ p=[\exists x\!\in\! G(M_{x})] }[/math] mutual knowledge. Now suppose if Alice tells Bob that she knows [math]\displaystyle{ p }[/math] (so that [math]\displaystyle{ K_{a}p }[/math] becomes common knowledge, i.e. [math]\displaystyle{ C_{G} K_{a}p }[/math]), and then Bob tell Alice that he knows [math]\displaystyle{ p }[/math] as well (so that [math]\displaystyle{ K_{b}p }[/math] becomes common knowledge, i.e. [math]\displaystyle{ C_{G} K_{b}p }[/math]), this will turn [math]\displaystyle{ p }[/math] into common knowledge ([math]\displaystyle{ C_{G}E_{G} p \Rightarrow C_{G} p }[/math]), 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. https://www.youtube.com/watch?v=5S1d3cNge24&t=46m02s.

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).

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[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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Mutual knowledge (logic)

Topic: Philosophy

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