Mrs. Rasmusen and I were talking in the car about the problem of what to do at a 4-way stop when you arrive at almost the same time as another car to your right, but just a bit earlier. It’s a good practical example of the importance of common knowledge, the difference between both people knowing a fact and both people knowing that both people know that both people know…
If it were common knowledge that I got there first, I would start again and cross the intersection first. But suppose I’m not sure that the other driver saw that I arrived first. He might then cross first, thinking he had the right of way, and I would want to cross second in that case.
It could easily happen that both of us know that I arrived first, but since we don’t know that each other knows, we hold back– and nobody crosses.
After we both see that neither of us is crossing, we realize that something is wrong. I’d have to think harder than I have time for to analyze the resulting game, but we can rule out certain possibilities such as “The other driver believes that he arrived first and that it is common knowledge that he arrived first.”