Two generals exchange a message and a confirmation (two rounds total). How many levels of nested knowledge do they now have about the attack plan?
ACommon knowledge — two rounds is sufficient for full coordination certainty
BThree levels: A knows, B knows A knows, A knows B knows A knows — but not common knowledge
COne level: each general knows the plan, and nothing more can be inferred
DInfinite levels, because each message implicitly contains all prior acknowledgments
After two rounds (message + confirmation): A knows (level 1), B knows A knows (level 2), A knows B knows A knows (level 3). Each round adds exactly one level. Common knowledge requires infinitely many nested levels — an infinite conjunction — which no finite exchange can achieve. This is the core insight of the coordinated attack problem: even if both generals know the plan and both know the other knows, the residual uncertainty at the next level is enough to rationally prevent commitment.
Question 2 Multiple Choice
Why does a public announcement (heard simultaneously by all parties with no private uncertainty) generate common knowledge, while a private message chain does not?
APublic announcements are legally binding in ways private messages are not
BWhen all parties simultaneously observe the same event, there is no residual uncertainty about who knows what — all levels of nesting collapse at once
CPrivate messages can be intercepted, destroying mutual knowledge
DPublic announcements repeat the information more times, increasing the probability that everyone heard it
In a public announcement, every agent simultaneously observes that every other agent is observing the same thing. There is no 'did they receive it?' uncertainty and no 'do they know I know?' uncertainty — all levels of the infinite iteration are satisfied at once. A private message always leaves uncertainty about whether it was received, which prevents the infinite nesting from closing. Ritual, ceremony, and publication work precisely because they engineer this simultaneous mutual witnessing.
Question 3 True / False
Common knowledge that p requires infinitely many nested levels: everyone knows p, everyone knows everyone knows p, and so on without end.
TTrue
FFalse
Answer: True
True. This is the formal definition. Letting E(p) mean 'everyone knows p,' common knowledge CK(p) = E(p) ∧ E(E(p)) ∧ E(E(E(p))) ∧ … — an infinite conjunction. This is not a philosophical idealization — it has real consequences. Even if you have level 1 through level 1,000,000 of mutual knowledge, the residual uncertainty at level 1,000,001 is logically sufficient to break coordination in iterated reasoning scenarios.
Question 4 True / False
If A and B both know p, then A and B have common knowledge that p.
TTrue
FFalse
Answer: False
False. Mutual knowledge — each agent knows p — is only the first level (E(p)). Common knowledge also requires E(E(p)): A knows that B knows p, and B knows that A knows p. Then E(E(E(p))), and so on infinitely. A and B can both know p without either knowing that the other knows, and this gap is practically significant: two people can both know a secret without having common knowledge of it, which is why coordination based on that secret remains fragile.
Question 5 Short Answer
Explain why the coordinated attack problem shows that no finite sequence of successful confirmations can achieve common knowledge, even if every message is received.
Think about your answer, then reveal below.
Model answer: Each round of messaging adds exactly one level of nested knowledge. After n rounds, the generals have n+1 levels: each knows the plan, each knows the other knows, ..., up to n+1 iterations. But common knowledge requires all infinite levels simultaneously. No matter how many rounds have been completed, there is always one more level of 'A knows B knows A knows...' that has not yet been established by a confirmation. At that level, one general cannot be certain the other general knows, so rational commitment is impossible — attacking alone (with no coordination) means certain defeat. The infinite requirement can never be closed by a finite process.
The deeper point: common knowledge is a fixed-point condition — CK(p) is defined as the state where CK(p) itself is already known to hold. No finite iteration of individual-knowledge claims reaches this fixed point. It requires a structural condition (simultaneous public observation) that creates the infinite nesting all at once, rather than building it level by level.