In coordination games and speculative over-the-counter financial markets, solutions depend on higher-order average expectations: agents' expectations about what counterparties, on average, expect their counterparties to think, etc. We offer a unified analysis of these objects and their limits, for general information structures, priors, and networks of counterparty relationships. Our key device is an interaction structure combining the network and agents' beliefs, which we analyze using Markov methods. This device allows us to nest classical beauty contests and network games within one model and unify their results. Two applications illustrate the techniques: The first characterizes when slight optimism about counterparties' average expectations leads to contagion of optimism and extreme asset prices. The second describes the tyranny of the least-informed: agents coordinating on the prior expectations of the one with the worst private information, despite all having nearly common certainty, based on precise private signals, of the ex post optimal action.