Common Knowledge as a Coinductive Modality

I prove in Coq Aumann’s Theorem: In perfect information games, common knowledge of rationality implies backward induction equilibrium. The notion of common knowledge is formalized, using a coinductive definition, as a modality containing an infinite amount of information. 1. The Green-Eyed Amazons In the land of the Amazons a strange taboo was enforced.1 None of them was allowed to know the colour of her own eyes. Everybody could see the colour of the eyes of the others, but there were no mirrors or other reflective surfaces on which to check one’s own appearance. It was also strictly forbidden to ask somebody else to reveal the colour of one’s eyes. The ironic fact is that every one of them had green eyes. Each of them could see that all her companions were green-eyed, but she was in doubt about herself. Another peculiarity of the Amazons was that they were all proficient logicians: They could make the most complex reasoning without error. One day the goddess Artemis came to the village and announced: ‘I decided to give a golden bow to every one of you who has green eyes. To apply for the prize you will need to paint your front door green before midnight of any day. Then, the following morning, I will come to check whether your eyes are indeed green. If they are, you will receive the price. But be warned, there will be no tolerance for cheaters: If your eyes turn out to be of a different colour, I will strike you down with one of my arrows. I am quite sure that I didn’t come here in vain, because I can see that at least one of you has green eyes.’ My question to you is: Can you predict how the Amazons behaved. Did any of them paint her door green, and when did they do it? The solution is revealed at the end of the article. 1This is a variant of the cheating wives [5] (or husbands [6]) puzzle, also known as the muddy children puzzle [2]. REFLECTIONS ON TYPE THEORY, λ-CALCULUS, AND THE MIND Essays Dedicated to Henk Barendregt on the Occasion of his 60th Birthday Copyright c © 2007 by Venanzio Capretta