Common knowledge is a special kind of
knowledgeKnowledge is defined by the Oxford English Dictionary as expertise, and skills acquired by a person through experience or education; the theoretical or practical understanding of a subject, what is known in a particular field or in total; facts and information or awareness or familiarity gained...
for a group of 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.
The
conceptThere are two prevailing theories in contemporary philosophy which attempt to explain the nature of concepts . The representational theory of mind proposes that concepts are mental representations, while the semantic theory of concepts holds that they are abstract objects...
was first introduced in the philosophical literature by
David Kellogg LewisDavid Kellogg Lewis was a 20th-century American philosopher. Lewis taught briefly at UCLA and then at Princeton from 1970 until his death. He is also closely associated with Australia, whose philosophical community he visited almost annually for more than thirty years...
in his study
Convention (1969). It has been first given a mathematical formulation in a
set-theoreticalThe modern study of set theory was initiated by Cantor and Dedekind in the 1870s. After the discovery of paradoxes in informal set theory, numerous axiom systems were proposed in the early twentieth century, of which the Zermelo–Fraenkel axioms, with the axiom of choice, are the best-known.The...
framework by
Robert AumannRobert John Aumann is an Israeli/American mathematician and a member of the United States National Academy of Sciences. He is a professor at the Center for the Study of Rationality in the Hebrew University of Jerusalem in Israel...
(1976).
Computer scientistsComputer science is the study of the theoretical foundations of information and computation, and of practical techniques for their implementation and application in computer systems. It is frequently described as the systematic study of algorithmic processes that create, describe and transform...
grew an interest in the subject of
epistemic logicEpistemic modal logic is a subfield of modal logic that is concerned with reasoning about knowledge. While epistemology has a long philosophical tradition dating back to Ancient Greece, epistemic logic is a much more recent development with applications in many fields, including philosophy,...
in general — and of common knowledge in particular — starting from the 1980s. There are numerous
puzzlesA logic puzzle is a puzzle deriving from the mathematics field of deduction.-History:This branch was produced by Charles Lutwidge Dodgson, who is better known under his pseudonym Lewis Carroll, the author of Alice's Adventures in Wonderland...
based upon the concept which have been extensively investigated by mathematicians such as
John ConwayJohn Conway may refer to:* John B. Conway, mathematician, functional analyst, George Washington University* John Horton Conway, mathematician at Princeton University. Popularly known for Conway's Game of Life...
.
Example
It is common to introduce the idea of common knowledge by some variant of the following puzzle: On an island, there are
k people who have blue eyes, and the rest of the people have green eyes. There is at least one blue-eyed person on the island (
k ≥ 1). If a person ever knows herself to have blue eyes, she must leave the island at dawn the next day. Each person can see every other person's eye color, there are no mirrors, and there is no discussion of eye color. At some point, an outsider comes to the island and makes the following public announcement, heard and understood by all people on the island: "at least one of you has blue eyes". The problem: Assuming all persons on the island are truthful and completely logical, what is the eventual outcome?
The answer is that, on the
k dawns after the announcement, all the blue-eyed people will leave the island.
This can be easily seen with an inductive argument. If
k = 1, the person will recognize that he or she has blue eyes (by seeing only green eyes in the others) and leave at the first dawn. If
k = 2, no one will leave at the first dawn. The two blue-eyed people, seeing only one person with blue eyes,
and that no one left on the 1st dawn, will leave on the second dawn. So on, it can be reasoned that no one will leave at the first
k-1 dawns if and only if there are at least
k blue-eyed people. Those with blue eyes, seeing
k-1 blue-eyed people among the others and knowing there must be at least
k, will reason that they have blue eyes and leave.
What's most interesting about this scenario is that, for
k > 1, the outsider is only telling the island citizens what they already know: that there are blue-eyed people among them. However, before this fact is announced, the fact is not
common knowledge.
For
k = 2, it is merely "first-order" knowledge. Each blue-eyed person knows that there is someone with blue eyes, but each blue eyed person does
not know that the other blue-eyed person has this same knowledge.
For
k > 2, it is "(
k − 1)th order" knowledge. After
k − 1 days, each blue-eyed person knows that a second blue-eyed person knows that a third blue-eyed person knows that.... (repeat for a total of
k − 1 levels) a
kth person has blue eyes, but no one knows that there is a "
kth" blue-eyed person with that knowledge, until the
kth day arrives. The notion of
common knowledge therefore has a palpable effect. Knowing that everyone knows does make a difference. When the outsider's public announcement (a fact already known to all) becomes common knowledge, the blue-eyed people on this island eventually deduce their status, and leave.
Modal logic (syntactic characterization)
Common knowledge can be given a logical definition in
multi-modal logicA modal logic is any system of formal logic that attempts to deal with modalities. Modals qualify the truth of a judgment. For example, if it is true that "John is happy," we might qualify this statement by saying that "John is very happy," in which case the term "very" would be a modality...
systems in which the modal operators are interpreted
epistemicallyEpistemic modal logic is a subfield of modal logic that is concerned with reasoning about knowledge. While epistemology has a long philosophical tradition dating back to Ancient Greece, epistemic logic is a much more recent development with applications in many fields, including philosophy,...
. At the propositional level, such systems are extensions of propositional logic. The extension consists of the introduction of a group
G of
agents, and of
n modal operators
Ki (with
i = 1, ...,
n) with the intended meaning that "agent
i knows." Thus
Ki (where is a formula of the calculus) is read "agent
i knows ." We can define an operator
EG with the intended meaning of "everyone in group
G knows" by defining it with the axiom
By abbreviating the expression with and defining , we could then define common knowledge with the axiom
with
There is however a complication. The languages of epistemic logic are usually
finitary, whereas the
axiomIn traditional logic, an axiom or postulate is a proposition that is not proved or demonstrated but considered to be either self-evident, or subject to necessary decision...
above defines common knowledge as an infinite conjunction of formulas, hence not a
well-formed formulaIn the formal languages used in mathematical logic and computer science, a well-formed formula or simply formula is an idea, abstraction or concept which is expressed using the symbols and formation rules of a particular formal language...
of the language. To overcome this difficulty, a
fixed-point definition of common knowledge can be given. Intuitively, common knowledge is thought of as the fixed point of the "equation" . In this way, it is possible to find a formula implying from which, in the limit, we can infer common knowledge of .
This
syntactic characterization is given semantic content through so-called
Kripke structures. A Kripke structure is given by (i) a set of states (or possible worlds)
S, (ii)
n accessibility relations , defined on , intuitively representing what states agent
i considers possible from any given state, and (iii) a valuation function assigning a truth value, in each state, to each primitive proposition in the language. The semantics for the knowledge operator is given by stipulating that is true at state
s iff is true at
all states
t such that . The semantics for the common knowledge operator, then, is given by taking, for each group of agents
G, the reflexive and transitive closure of the , for all agents
i in
G, call such a relation , and stipulating that is true at state
s iff is true at
all states
t such that .
Set theoretic (semantic characterization)
Alternatively (yet equivalently) common knowledge can be formalized using
set theoryThe modern study of set theory was initiated by Cantor and Dedekind in the 1870s. After the discovery of paradoxes in informal set theory, numerous axiom systems were proposed in the early twentieth century, of which the Zermelo–Fraenkel axioms, with the axiom of choice, are the best-known.The...
(this was the path taken by the Nobel laureate
Robert AumannRobert John Aumann is an Israeli/American mathematician and a member of the United States National Academy of Sciences. He is a professor at the Center for the Study of Rationality in the Hebrew University of Jerusalem in Israel...
in his seminal 1976 paper). We will start with a set of states
S. We can then define an event
E as a subset of the set of states
S. For each agent
i, define a
partitionIn mathematics, a partition of a set X is a division of X into non-overlapping and non-empty "parts" or "blocks" or "cells" that cover all of X...
on
S,
Pi. This partition represents the state of knowledge of an agent in a state. In state
s, agent
i knows that one of the states in
Pi(
s') obtains, but not which one.
We can now define a knowledge function K
in the following way:
That is K
i
(e
) is the set of states where the agent will know that event e
obtains.
Similar to the modal logic formulation above, we can define an operator for the idea that "everyone knows e
".
As with the modal operator, we will iterate the E
function, and . Using this we can then define a common knowledge function,
The equivalence with the synctactic 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 (i) the same space S
, (ii) accessibility relations that define the equivalence classes corresponding to the partitions , and (iii) a valuation function such that it yields value true
to the primitive proposition p
in all and only the states s
such that , where 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 for all , which is the finitary characterization of common knowledge also given by Aumann in the 1976 article.
Applications
Common knowledge was used by David Lewis in his pioneering game-theoretical account of convention. In this sense, common knowledge is a concept still central for linguists and philosophers of language (see Clark 1996) maintaining a Lewisian, conventionalist account of language.
Robert AumannRobert John Aumann is an Israeli/American mathematician and a member of the United States National Academy of Sciences. He is a professor at the Center for the Study of Rationality in the Hebrew University of Jerusalem in Israel...
introduced a set theoretical formulation of common knowledge (theoretically equivalent to the one given above) and proved the so-called "agreement theorem" through it: if two agents have common
prior probabilityIn Bayesian inference, a prior probability distribution, often called simply the prior, is a probability distribution representing knowledge or belief about an unknown quantity a priori, that is, before any data have been observed...
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.
The concept of common knowledge is central in
game theoryGame theory is a branch of applied mathematics that is used in the social sciences, most notably in economics, as well as in biology, engineering, political science, international relations, computer science, and philosophy...
. 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 2-player games, common knowledge of rationality is not needed as an epistemic condition for
Nash equilibriumIn game theory, Nash equilibrium is a solution concept of a game involving two or more players, in which each player is assumed to know the equilibrium strategies of the other players, and no player has anything to gain by changing only his or her own strategy unilaterally...
strategiesIn game theory, a player's strategy in a game is a complete plan of action for whatever situation might arise; this fully determines the player's behaviour...
.
Computer scientists use languages incorporating epistemic logics (and common knowledge) to reason about distributed systems. Such systems can be based on logics more complicated that simple propositional epistemic logic, see Wooldridge Reasoning about Artificial Agents
, 2000 (in which he uses a first-order logic incorporating epistemic and temporal operators) or van der Hoek et al. "Alternating Time Epistemic Logic".
In his 2007 book, The Stuff of Thought: Language as a Window into Human NatureThe Stuff of Thought: Language As a Window Into Human Nature is a New York Times best-selling book by Harvard experimental psychologist Steven Pinker published in 2007. It is his fifth book on the topics of language and cognitive science written for a general audience...
,
Steven PinkerSteven Arthur Pinker is a Canadian-American experimental psychologist, cognitive scientist, and author of popular science...
uses the notion of common knowledge (dubbing it mutual knowledge, as it is often done in the linguistics literature) to analyze the kind of indirect speech involved in innuendoes.
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