Editing Common knowledge (logic) (section)

{{short description|Statement that players know and also know that other players know (ad infinitum)}}
'''Common knowledge''' is a special kind of [[knowledge]] for a group of [[agent-based model|agents]]. There is ''common knowledge'' of ''p'' in a group of agents ''G'' when all the agents in ''G'' know ''p'', they all know that they know ''p'', they all know that they all know that they know ''p'', and so on ''[[ad infinitum]]''.<ref name=Osborne>Osborne, Martin J., and [[Ariel Rubinstein]]. ''A Course in Game Theory''. Cambridge, MA: MIT, 1994. Print.</ref> It can be denoted as <math>C_G p</math>.

The concept was first introduced in the philosophical literature by [[David Kellogg Lewis]] in his study ''Convention'' (1969).  The sociologist Morris Friedell defined common knowledge in a 1969 paper.<ref>Morris Friedell, "On the Structure of Shared Awareness," Behavioral Science 14 (1969): 28–39.</ref>  It was first given a mathematical formulation in a [[set theory|set-theoretical]] framework by [[Robert Aumann]] (1976). [[computer science|Computer scientists]] grew an interest in the subject of [[epistemic logic]] in general&nbsp;– and of common knowledge in particular&nbsp;– starting in the 1980s.{{ref|CSTexts}}  There are numerous [[Logic puzzle|puzzles]] based upon the concept which have been extensively investigated by mathematicians such as [[John Horton Conway|John Conway]].<ref>{{Cite book |title=Math Hysteria| author=Ian Stewart |year=2004 |publisher=OUP |chapter=I Know That You Know That... }}</ref>

The philosopher [[Stephen Schiffer]], in his 1972 book ''Meaning'', independently developed a notion he called "[[mutual knowledge]]" (<math>E_G p</math>) which functions quite similarly to Lewis's and Friedel's 1969 "common knowledge".<ref>Stephen Schiffer, ''Meaning'', 2nd edition, Oxford University Press, 1988.  The first edition was published by OUP in 1972.  For a discussion of both Lewis's and Schiffer's notions, see Russell Dale, ''[http://www.russelldale.com/dissertation/1996.RussellDale.TheTheoryOfMeaning.pdf The Theory of Meaning]'' (1996).</ref> If a trustworthy announcement is made in [[Dynamic epistemic logic#Public Events|public]], then it becomes common knowledge; However, if it is transmitted to each agent in private, it becomes mutual knowledge but not common knowledge. Even if the fact that "every agent in the group knows ''p''" (<math>E_G p</math>) is transmitted to each agent in private, it is still not common knowledge: <math>E_G E_G p \not \Rightarrow C_G p</math>. But, if any agent <math>a</math> publicly announces their knowledge of ''p'', then it becomes common knowledge that they know ''p'' (viz. <math>C_G K_a p</math>). If every agent publicly announces their knowledge of ''p'', ''p'' becomes common knowledge <math>C_G E_G p \Rightarrow C_G p</math>.