common knowledge: one construct, many systems

Lewis introduced common knowledge in his 1969 Convention: A philosophical study to explain how conventions of language and behavior get sustained: a fact is common knowledge in a group when everyone knows it, everyone knows that everyone knows it, and so on through every level of the infinite hierarchy of mutual knowledge. It is stronger than merely everyone happening to know the fact, because each person must also be able to rely on the others' knowing, and their knowing that, and so forth. This recursive epistemic condition is what lets people coordinate on a shared convention, and it is the same construct that economics and computer science later took up.[1]

Unit

The depth of iterated “everyone knows that everyone knows” that holds

Recovery means

Not applicable

Measured by

Analyzed logically by whether the full hierarchy of mutual knowledge of a proposition obtains among the agents

Aumann's 1976 Agreeing to disagree gave common knowledge a set-theoretic formalization using agents' information partitions, and proved a striking consequence: if two agents share a common prior and their posterior probabilities of an event are common knowledge, those posteriors must be equal: rational agents cannot “agree to disagree.” Here the construct becomes a precise assumption in the foundations of game theory and the economics of information, underwriting results about equilibrium and belief. It is the identical recursive notion Lewis described, now stated in the language of events and knowledge operators and carrying analytical weight in models of rational interaction.[2]

Unit

Whether an event or posterior is common knowledge under the agents' partitions

Recovery means

Not applicable

Measured by

Formalized via the meet of agents' information partitions and the common-knowledge operator on events

Halpern and Moses's 1990 Knowledge and common knowledge in a distributed environment carried the same construct into computer science, modeling what each process in a distributed system knows about the global state. They showed that common knowledge is exactly the precondition for simultaneous coordinated action, and that over channels where messages can be lost it can never be attained: the classic “coordinated attack” impossibility. It is the same infinite epistemic ladder as in Lewis and Aumann, realized among communicating processes rather than people, and it explains why certain forms of agreement are unachievable in unreliable networks.[3]

Unit

Whether a fact is common knowledge across the processes of a system

Recovery means

Not applicable

Measured by

Reasoned about with epistemic logic over runs and system states, and the attainability results for common knowledge under given channel assumptions