Hi, my name is Liang Zhou Koh.
I am a graduate student in philosophy at the University of Toronto,
and today I want to continue talking about common knowledge.
In the previous video, we introduced a paradox of
common knowledge – the coordinated attack problem.
A compelling line of reasoning seems to lead us
to the strange conclusion that no matter how
many messages they send, the generals would not
be able to rationally coordinate their attack.
How may we resolve this paradox?
In this video, we will explore two possible ways of answering this question.
As our discussion will show, both of these responses face some challenges.
First, we could try to stick to our original reasoning, and accept that it
is in fact strictly irrational for the generals to attack even after, say, 17
messages back and forth; maybe our feeling that the attack would be rational is
just an illusion, and no number of messages will ever make the attack rational.
One worry about this response is that many familiar cases of coordination
in everyday life are similar to the situation that the generals are in.
Think, for example, of how we set up meetings via email.
Like the generals, the person sending the email often does not
know whether the message they sent will be received or read.
Yet we often show up for these meetings and expect others to do
so as well without sending countless messages back and forth.
If we accept that it would be irrational for the generals to attack
even after they have exchanged a large number of messages,
should we accept that our actions in these everyday situations are irrational too?
If so, this might come at too high a cost.
Fortunately, it may be possible to find some key differences
between the generals’ situation and most everyday situations.
These differences may also suggest some ways of understanding
the relation between common knowledge and rational coordination.
For example, one distinctive feature of the generals’ situation
is that the risks that they run by attacking are very high.
The general who attacks faces the prospect of
catastrophic defeat if the other does not attack as well.
Perhaps common knowledge is only necessary
when coordination failure is especially costly.
Since this is not usually the case for everyday coordination,
we can hold that it is rational for us to show up for meetings
if we know that the others know the appointed time and place,
or even if we are just fairly confident that they do.
This response to the problem does leave us wondering
if the costs of coordination failure can be so high
that when our messaging system is not perfectly reliable,
no number of messages can make it rational for us to coordinate
After all, we need to factor in the potential benefits
that we will be giving up when we choose not to coordinate.
This puts some pressure on the idea that we should accept the conclusion that the
generals cannot rationally attack because common knowledge is out of their reach.
Another idea that has captured the attention of theorists is that the
coordinated attack problem seems to involve some unrealistic assumptions.
For example, we arrive at the conclusion that no number of messages makes the attack
rational by supposing that the generals can
extend the same reasoning to each new message.
This requires us to assume that the generals can calculate
what they know for each message sent and received.
Consider what this would involve after 17 messages have been exchanged.
The generals will have to reason that even though they both know that they
both know that they both know (… and, so on) that the plan is to attack
at dawn, they should not attack because they do not both know that,
or, in other words, because they lack a higher level of mutual knowledge.
It should be clear that we should not expect ordinary
agents to be able to go through such reasoning.
Common knowledge may be necessary for rational coordination
only for ideally rational agents who do not share our cognitive limitations.
A problem with this response is that it seems to
lead us to conclude that for ordinary agents
to rationally coordinate, we need to achieve a level of
mutual knowledge that matches our cognitive limits.
Yet this is at odds with what occurs in everyday coordination.
A message and a confirmation are usually all
it takes for us to successfully coordinate.
We will need to explain why sending a third message
is not required for rational coordination
since most of us do have the ability to understand how our
epistemic situation would be improved by doing so.
Therefore, a complete solution will not only have to explain why
this is so but also why the same may not be said for the generals.
An alternative response to the paradox that does not require
us to provide such an explanation might be worth exploring.
Maybe we should reject the conclusion that it would be irrational
for the generals to attack no matter how many messages are sent.
To do so, we would have to explain where we went wrong in our original reasoning.
One way to frame this question would be to ask: how many messages
would it take for the generals to rationally attack?
The worry here is that drawing the line at any
number of messages seems entirely arbitrary.
What is so special about, say, five messages that rational coordination
for the generals depends on 5, and not 4 or 6 messages being exchanged?
There is another puzzle here.
Suppose we say that it would be rational for the
generals to attack after they have exchanged 5 messages.
The problem with this is that the general sending the fifth message
has no way of knowing whether the message has been received,
and hence, no way of knowing whether the condition for rational coordination
has been met, without receiving another message.
But this leads us right back to the problem we started with.
We may avoid this problem only if we accept that it can be rational for the
generals to attack even if they do not know that it is rational to do so.
Maybe sometimes being rational isn’t the same as knowing that we are rational?
In this video, we discussed one of the most important paradoxes of common knowledge.
As our brief discussion indicates,
there is still no generally accepted solution to the paradox.
There is much to be discovered about exactly what common knowledge is,
how it is achieved and the role that it plays in our social lives.