  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.
