Hi, my name is Liang Zhou Koh.
I am a graduate student in philosophy at the University of Toronto,
and today I want to talk to you about common knowledge.
One distinctive aspect of human mind reading
is our ability to entertain and reason about
higher-order knowledge and belief that agents
may have about the mental states of others.
In other words, our ability to have knowledge and belief about what other
people know and believe, not only about the world, but also about other minds.
In this video, we will be exploring an important social phenomenon
that is closely tied to this advanced mindreading ability – common knowledge.
We sometimes call something that is widely known by everyone
or almost everyone in our community common knowledge.
“Paris is the capital of France” or “water is H2O”
are some examples of this kind of common knowledge.
Common knowledge, however, is also a technical notion that is widely used in
philosophy, economic theory, theoretical computer science and linguistics.
According to a classical definition: some agents
commonly know some fact if they all know it,
they all know that they all know it, they all know that
they all know that they all know it, and so on, infinitely.
Why should we be interested in this phenomenon?
Common knowledge appears to play an important role
in underpinning social coordination.
One classic case for thinking about this function of common knowledge
is the coordinated attack problem.
Imagine two armies camped on separate hilltops,
preparing to attack a fortified city the next day.
They can win the battle only if they attack the city simultaneously
and will suffer defeat if they fail to coordinate their attacks.
Unfortunately, the battlefield is shrouded in a thick fog
and the only way that the generals commanding
those armies can communicate is via messengers.
Moreover, since it is possible for their messengers to be captured by the enemy,
there is no guarantee that their messages will reach the other general.
Let us consider what it would take for the generals to coordinate their attack.
Suppose General Anne sends the message “attack at dawn”
and General Bill receives it.
However, since Anne does not know if Bill received the message, it would not be
rational for her to risk a catastrophic defeat
and command her army to attack at dawn.
Knowing this, it would also not be rational for Bill to attack.
Suppose Bill tries to improve their situation by sending a
message back to Anne, confirming that he has received her message.
Since Bill does not know if his reply reached Anne,
he does not know if Anne now knows
that he received her initial message, it would not be rational for him to attack.
Therefore, even if Anne does receive Bill’s reply,
since she knows that Bill does not know that
she has received it and would not attack,
it would not be rational for her to attack either.
Suppose Anne sends another message to let Bill know that she received his message …
The same line of reasoning should lead us to conclude
that doing so would still not allow them to rationally coordinate.
It would appear that no matter how many messages they sent,
it would not be rational for the generals to attack.
Let us now put things in terms of what the generals know.
Consider what happens after Anne sent the initial
message to attack and Bill received it.
Since there is a chance for her message to be intercepted,
Anne does not know whether Bill received it.
Suppose, after waiting for some time,
she receives Bill’s reply. Anne would now know that Bill received her message.
However, since Bill does not know whether his reply
reached Anne, he does not know that Anne now knows that.
Suppose Anne sends another message to acknowledge that she received Bill’s reply.
Once Bill receives this, both Anne and Bill would now know
that they both know that Bill received the initial message.
However, since Anne does not know that her latest message got through, she would
not know that Bill now knows that she knows that he received her initial message.
We may extend this line of reasoning indefinitely.
Recall that, by definition, for Anne and Bill to
commonly know that Bill received the message to attack,
they both need to know that the other person knows
that Bill received the message, and so on, infinitely.
In this scenario, however, the unreliable channel of
communication makes it so that there will always be
one level of knowledge in this infinite series that the
person who last sends a message will fail to attain.
Thus, the generals will not attain common knowledge that Bill
received the message to attack no matter how many messages they send.
This absence of common knowledge may explain why
rational coordination is impossible in this case.
But it also seems wrong to say that the generals would not be able to
rationally coordinate their attack no matter how many messages they send.
Intuitively, it seems rational for the generals to commit
to the attack after they have exchanged a certain number of messages.
Suppose the generals managed to successfully exchange
5 messages between them.
Imagine how it would be for Bill to then reason as follows:
Anne does not know that I know that she knows
that I know that she knows that I received
her message to attack, so she will not attack at dawn,
therefore I should also not attack.
When put in this way, Bill’s reasoning does not seem so compelling.
The generals’ decision not to attack would seem even more bizarre
when we go from 5 messages to, say, 27 messages.
This conflict between our intuition
and the line of reasoning that we’ve just sketched
is at the heart of the paradox of the coordinated attack problem.
In the next video, we will look at some responses to this paradox.