This is a fun Aumann paper that talks about what players have to believe to be in a Nash equilibrium. Here, instead of imagining agents randomizing, we’re instead imagining that the probabilities over actions live in the heads of the other agents: you might well know exactly what you’re going to do, as long as I don’t. It shows that in 2-player games, you can write down conditions that involve mutual knowledge but not common knowledge that imply that the players are at a Nash equilibrium: mutual knowledge of player’s conjectures about each other, players’ rationality, and players’ payoffs suffices. On the contrary, in 3-player games (or games with more players), you need common knowledge: common priors, and common knowledge of conjectures about other players.
The paper writes:
One might suppose that one needs stronger hypotheses in Theorem B [about 3-player games] than in Theorem A [about 2-player games] only because when n≥3, the conjectures of two players about a third one may disagree. But that is not so. One of the examples in Section 5 shows that even when the necessary agreement is assumed outright, conditions similar to those of Theorem A do not suffice for Nash equilibrium when n≥3.
This is pretty mysterious to me and I wish I understood it better. Probably it would help to read more carefully thru the proofs and examples.
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.
This is a fun Aumann paper that talks about what players have to believe to be in a Nash equilibrium. Here, instead of imagining agents randomizing, we’re instead imagining that the probabilities over actions live in the heads of the other agents: you might well know exactly what you’re going to do, as long as I don’t. It shows that in 2-player games, you can write down conditions that involve mutual knowledge but not common knowledge that imply that the players are at a Nash equilibrium: mutual knowledge of player’s conjectures about each other, players’ rationality, and players’ payoffs suffices. On the contrary, in 3-player games (or games with more players), you need common knowledge: common priors, and common knowledge of conjectures about other players.
The paper writes:
This is pretty mysterious to me and I wish I understood it better. Probably it would help to read more carefully thru the proofs and examples.
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.