The Ann and Bob example

This is a simple example of the way we humans make use of common knowledge every day. In [1], Clark and Marshall give the following example: Suppose Ann says to Bob: "Have you ever seen the movie showing at the Roxy tonight?" The movie that is actually showing that evening is Monkey Business (MB), but an earlier, incorrect announcement stated that the movie that was playing that evening was A Day At The Races (DR).
 
How does Ann know, that Bob knows which movie is playing that evening? In order for Bob to understand Ann it must be common knowledge that the movie that is playing that night is MB. The condition that they both know that MB is being played is not sufficient for this situation.
 
Suppose that Ann and Bob read the morning newspaper together. They both see that DR is playing that night at the Roxy. Later that day, they separately both read the correction that MB is playing that night. When Ann ask Bob the question about having seen the movie before, Bob is likely to think that Ann is talking about DR. So, it is neccessary for him to know that Ann knows that MB is playing, and vice versa. But even these two assumption do not suffice. Mutual knowledge is needed.
 
Mutual knowledge in this case means: Both know that MB, both know that both know that MB, etc. This is an infinite conjunction, and therefore not very usable statement. This is a problem we will encounter in the Byzantine Generals example as well.

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