Common Knowledge and Nash Equilibrium
Where every man is Enemy to every man … the life of man is soli-
tary, poore, nasty, brutish, and short.
In the case of any person whose judgment is really deserving of
confidence, how has it become so? Because he has kept his mind
open to criticism of his opinions
John Stuart Mill
This chapter applies the modal logic of knowledge developed in §4.1 and §5.10 to explore sufficient conditions for a Nash equilibrium in two-player games (§8.1). We then expand the modal logic of knowledge to multiple agents and prove a remarkable theorem, due to Aumann (1976), that asserts that an event that is self-evident for each member of a group is common knowledge (§8.3).
This theorem is surprising because it appears to prove that individuals know the content of the minds of others with no explicit epistemological assumptions. We show in §8.4 that this theorem is the result of implicit epistemological assumptions involved in construction of the standard semantic model of common knowledge, and when more plausible assumptions are employed, the theorem is no longer true.
Aumann's famous agreement theorem is the subject of §8.7, where we show that the Aumann and Brandenburger (1995) theorem, which supplies sufficient conditions for rational agents to play a Nash equilibrium in multiplayer games, is essentially an agreement theorem. Because there is no principle of Bayesian rationality that gives us the commonality of beliefs on which agreement depends, our analysis entails the demise of methodological individualism, a theme explored in §8.8.
Suppose that rational agents know one another's conjectures (§4.1) in state ω, so that for all i and j ≠ i, ifand Sj ∊ Sj is player j's pure strategy in s-i, then Sj is a best response to his conjecture . We