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.
Computer scientists grew an interest in the subject of epistemic logic in general – and of common knowledge in particular – starting in the 1980s.
[1] There are numerous puzzles based upon the concept which have been extensively investigated by mathematicians such as John Conway.
[3] The philosopher Stephen Schiffer, in his 1972 book Meaning, independently developed a notion he called "mutual knowledge" (
The idea of common knowledge is often introduced by some variant of induction puzzles (e.g. Muddy children puzzle):[2] On an island, there are k people who have blue eyes, and the rest of the people have green eyes.
The problem: finding the eventual outcome, assuming all persons on the island are completely logical (every participant's knowledge obeys the axiom schemata for epistemic logic) and that this too is common knowledge.
The answer is that, on the kth dawn after the announcement, all the blue-eyed people will leave the island.
For k > 1, the outsider is only telling the island citizens what they already know: that there are blue-eyed people among them.
When the outsider's public announcement (a fact already known to all, unless k=1 then the one person with blue eyes would not know until the announcement) becomes common knowledge, the blue-eyed people on this island eventually deduce their status, and leave.
Intuitively, common knowledge is thought of as the fixed point of the "equation"
This syntactic characterization is given semantic content through so-called Kripke structures.
A Kripke structure is given by a set of states (or possible worlds) S, n accessibility relations
Alternatively (yet equivalently) common knowledge can be formalized using set theory (this was the path taken by the Nobel laureate Robert Aumann in his seminal 1976 paper).
That is, Ki(e) is the set of states where the agent will know that event e obtains.
It is a subset of e. Similar to the modal logic formulation above, an operator for the idea that "everyone knows can be defined as e".
The equivalence with the syntactic approach sketched above can easily be seen: consider an Aumann structure as the one just defined.
We can define a correspondent Kripke structure by taking the same space S, accessibility relations
, and a valuation function such that it yields value true to the primitive proposition p in all and only the states s such that
is the event of the Aumann structure corresponding to the primitive proposition p. It is not difficult to see that the common knowledge accessibility function
defined in the previous section corresponds to the finest common coarsening of the partitions
Common knowledge was used by David Lewis in his pioneering game-theoretical account of convention.
Robert Aumann introduced a set theoretical formulation of common knowledge (theoretically equivalent to the one given above) and proved the so-called agreement theorem through which: if two agents have common prior probability over a certain event, and the posterior probabilities are common knowledge, then such posterior probabilities are equal.
A result based on the agreement theorem and proven by Milgrom shows that, given certain conditions on market efficiency and information, speculative trade is impossible.
For several years it has been thought that the assumption of common knowledge of rationality for the players in the game was fundamental.
It turns out (Aumann and Brandenburger 1995) that, in two-player games, common knowledge of rationality is not needed as an epistemic condition for Nash equilibrium strategies.
Computer scientists use languages incorporating epistemic logics (and common knowledge) to reason about distributed systems.
In his 2007 book, The Stuff of Thought: Language as a Window into Human Nature, Steven Pinker uses the notion of common knowledge to analyze the kind of indirect speech involved in innuendoes.
The comedy movie Hot Lead and Cold Feet has an example of a chain of logic that is collapsed by common knowledge.
We see the Kid realizing that his carefully constructed “edge” has collapsed into common knowledge.