This article is really a chapter from the 1994 Handbook of Game Theory on the implications of common knowledge. As has been discussed on this site, common knowledge of rationality (I know that you know that I know that you know…that we are rational) is enormously powerful. Aumann famously described common knowledge of rationality (CKR) in terms of information partitions for each agent. With common priors, CKR allows you to derive a number of fantastic results, such that agents cannot agree to disagree, even if they use non-Bayesian decision rules, that agents cannot speculate, and that agents will not bet (a famous “no-trade theorem”). Writing knowledge in terms of information partitions also allows for some spectacular proofs, such as the following due originally to Kreps and mentioned by Geanakoplos:
Let agents have (perhaps different) beliefs about the state of the world in the future, and let each be given an endowment which is ex-ante Pareto optimal. Then the prices p and bundles equal to the endowments are a rational expectations equilibrium; that is, knowing the prices, and hence learning something about the demand of other players and their beliefs about the future, will not induce an agent to trade away from her endowment. The proof here is simple. First note that by property of partitions, more information cannot make you worse off as an optimizer. With no information, the agent can get utility equal to that from consuming his endowment. So by adding information (from prices), you can do no worse than the utility from the endowment. But since that is true for all players, and since the endowment was Pareto, no one is willing to trade with the agent.
This article also has extensive discussion of two major sticking points regarding common knowledge: the Harsanyi doctrine (that common knowledge of rationality can be derived from regular rationality, with some minor assumptions), and the possibility of knowledge without information partitions (roughly, what happens if people make mistakes). Of course, this article is old, but it nonetheless contains all of the major examples and proofs that form that modern core of knowledge theory.
http://cowles.econ.yale.edu/~gean/art/p0882.pdf (Kudos to John for including final versions of all of his papers going back to the early 1980s on his website)