Common knowledge means everyone knows a fact at every order of knowing. A three-state example separates shared knowledge from coordination and public announcements.
Common knowledge
Common knowledge of a fact means that everyone in a specified group knows it, everyone knows that everyone knows it, and this continues through every finite order of knowing. In a trading model, knowing that a shared execution service is ready can be insufficient: a desk may also need to reason about what another desk knows before acting.
The group matters. A fact can be common knowledge between two desks without being common knowledge across the market. The recursive definition is introduced in MIT's common-knowledge lecture.
Everyone knows, but does everyone know that?
Consider an original fictional model with two desks, A and B, and three possible states. Readiness is an event, E={r1,r2}; in state d the service is not ready. Each desk's information set determines which states it can distinguish:
Desk
Its possible information cells
A
One cell contains r1 and r2; the other contains only d.
B
One cell contains only r1; the other contains r2 and d.
Assume the actual state is r1. A knows the service is ready because both states A considers possible are in E. B knows it too because B identifies r1 exactly. This is mutual knowledge: everyone knows the fact.
Yet A considers r2 possible. In that state, B cannot distinguish a ready service from a service in state d. Thus A does not know that B knows readiness. The second order already fails, so readiness is not common knowledge at r1.
Three states form an A-indistinguishability link from r1 to r2 and a B-indistinguishability link from r2 to d. At actual state r1 both desks know readiness, but A considers r2 possible, where B cannot rule out the not-ready state d.
Original fictional information model. Blue shading marks the ready event; solid and dashed links show which states A and B respectively cannot distinguish. The two-link path exposes a failure of higher-order knowledge.
The distinction concerns information, without needing probabilities or irrational behaviour. A is correctly uncertain about B's knowledge. B can know the true state while A remains uncertain about what B knows.
The recursive test
In the partition model, write Pi(ω) for the information cell observed by person i at state ω. The knowledge operator is
Ki(E)={ω:Pi(ω)⊆E}.
It collects states where every possibility left open to i satisfies E. This is knowledge of a true event, not merely a high-confidence belief. For the group, define
M(E)=i⋂Ki(E),M1(E)=M(E),Mn+1(E)=M(Mn(E)).
Then the common-knowledge event is
C(E)=n=1⋂∞Mn(E).
These definitions follow the knowledge-operator formulation in Sobel's notes, page 1. In our model, M(E)={r1} but M2(E)=∅. No state survives every order. Checking a few successful orders would not, in general, establish the infinite requirement.
A public announcement changes the model
Suppose a truthful announcement of readiness is heard and understood by both desks, and the fact that this observation is shared is itself common knowledge. In the ideal partition model, both restrict attention to E: state d is removed from every cell. All remaining possibilities now satisfy readiness, so readiness becomes common knowledge. This is an application of the public-event characterisation in Border, §§5–6.
The observation assumptions do the work. Sending the same notification separately to two terminals does not establish them if each desk is uncertain whether the other received or understood it. A “public” data source can be widely accessible without satisfying this stronger model of public observation. More acknowledgements can establish additional orders; an arbitrary finite exchange alone is not a general proof of common knowledge.
Knowing readiness does not choose an action
Even grant common knowledge that the service is ready. Consider this simultaneous, one-round game, with its payoffs also common knowledge. Both desks choose Execute or Wait. Entries are (A utility, B utility) in stipulated utility units, rather than measured trading profits.
A chooses
B: Execute
B: Wait
Execute
(1, 1)
(−1, 0)
Wait
(0, −1)
(0, 0)
There are two pure Nash equilibria. At (Execute, Execute), switching alone to Wait lowers utility from 1 to 0. At (Wait, Wait), switching alone to Execute lowers it from 0 to −1. Both waiting remains stable against unilateral deviations, although both desks prefer joint execution.
For either desk, if the probability it assigns to the other executing is q, executing has expected utilityq−(1−q)=2q−1; waiting gives 0. Execution is strictly better only when q>1/2. Knowing the service's condition does not itself supply that belief about the other's action.
This separates two modelling tasks: specify who knows what, then check the incentives and any mechanism for selecting among equilibria. Common knowledge of a fact can change strategic reasoning while leaving multiple mutually consistent actions available.
Kim C. Border, Notes on the Structure of Common Knowledge, revision 6 April 2016, §§3, 5–6: partition-based knowledge, public events and the reachability characterisation.
Joel Sobel, Econ 208, Problem Set 3, Fall 2007, page 1 and Problems 0–5: knowledge operators, iterated mutual knowledge and common knowledge.
