Introduction to Game Theory: Common Knowledge

Common knowledge is a term used in game theory.

A normal person would probably define ‘common knowledge’ as something that is known by everyone playing a game.

Of course, a game theorist isn’t a normal person so they have a more complicated definition.

Something is common knowledge if:

– it is known by all the players; and

– all the players know that all the other players know it; and

– all the players know that all the other players know that all the other players know it.

And so on.

An example is the Two Generals Problem. In this situation two generals need to co-ordinate an attack on a town, they are only able to communicate by sending messengers between them.

The first general sends a message to the second saying that they will attack at 9am on the 19th July, but he doesn’t know whether the message has been received by the other general. Unless he is sure that the message has been received he will not attack as it is too dangerous to attack when there is a chance that the other general has not received the message.

The second general knows this so he sends a message back to the first general confirming that he has received the message and will attack at the agreed time. But now the second general cannot be sure his message has been received unless he receives another message back from the first.

When the first sends this next message he again cannot be sure that it has been received until he gets a message back from the second.

This loop goes on forever, the generals can never be sure that the time of the attack is common knowledge between them so they will not attack.

Today’s takeaway: Common knowledge is a bit more complicated than you might think!

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