 The difficulty the agents face is trying to select an equilibrium to follow. For suppose that Robert hopes to coordinate with Liz on a particular equilibrium of the game, say \((s_2, s_2)\). Robert reasons as follows: “Since there are several strict equilibria we might follow, I should follow my end of \((s_2, s_2)\) if, and only if, I have sufficiently high expectations that Liz will follow her end of \((s_2, s_2)\). But I can only have sufficiently high expectations that Liz will follow \((s_2, s_2 )\) if she has sufficiently high expectations that I will follow \((s_2, s_2)\). For her to have such expectations, Liz must have sufficiently high (second-order) expectations that I have sufficiently high expectations that she will follow \((s_2, s_2)\), for if Liz doesn’t have these (second-order) expectations, then she will believe I don’t have sufficient reason to follow \((s_2, s_2)\) and may therefore deviate from \((s_2, s_2)\) herself. So I need to have sufficiently high (third-order) expectations that Liz has sufficiently high (second-order) expectations that I have sufficiently high expectations that she will follow \((s_2, s_2 )\), which involves her in fourth-order expectations regarding me, which involves me in fifth-order expectations regarding Liz, and so on.” What would suffice for Robert, and Liz, to have decisive reason to follow \((s_2, s_2)\) is that they each know that the other knows that … that the other will follow \((s_2, s_2)\) for any number of levels of knowledge, which is to say that between Liz and Robert it is common knowledge that they will follow \((s_2, s_2)\). If agents follow a strict equilibrium in a pure coordination game as a consequence of their having common knowledge of the game, their rationality and their intentions to follow this equilibrium, and no other, then the agents are said to be following a Lewis-convention (Lewis 1969).