Got it, sort of. Once you have 3 people, then each person has a conjecture about the actions of the other two people. This means that your distribution might not be the product of the marginals over your distributions over the actions of each opponent, so you might be maximizing expected utility wrt your actual beliefs, but not wrt the product of the marginals—and the marginals are what are supposed to form the Nash equilibrium. Common knowledge and common priors mean stop this by forcing your conjecture over the different players to be independent. I still have a hard time explaining in words why this has to be true, but at least I understand the proof.
Got it, sort of. Once you have 3 people, then each person has a conjecture about the actions of the other two people. This means that your distribution might not be the product of the marginals over your distributions over the actions of each opponent, so you might be maximizing expected utility wrt your actual beliefs, but not wrt the product of the marginals—and the marginals are what are supposed to form the Nash equilibrium. Common knowledge and common priors mean stop this by forcing your conjecture over the different players to be independent. I still have a hard time explaining in words why this has to be true, but at least I understand the proof.