We introduce a graph-theoretic generalization of classical Arrow-Debreu economics, in which an undirected graph specifies which consumers or economies are permitted to engage in direct trade, and the graph topology may give rise to local variations in the prices of commodities. Our main technical contributions are: (1) a general existence theorem for graphical equilibria, which require local markets to clear; (2) an improved algorithm for computing approximate equilibria in standard (non-graphical) economies, which generalizes the algorithm of Deng et al. [2002] to non-linear utility functions; (3) an algorithm for computing equilibria in the graphical setting, which runs in time polynomial in the number of consumers in the special but important case in which the graph is a tree (again permitting non-linear utility functions). We also highlight many interesting learning problems that arise in our model, and relate them to learning in standard game theory and economics, graphical games, and graphical models for probabilistic inference.

We introduce new online models for two important aspectsof modern financial markets: Volume Weighted Average Pricetrading and limit order books. We provide an extensivestudy of competitive algorithms in these models and relatethem to earlier online algorithms for stock trading.

We provide a natural learning process in which the joint frequency of (time-averaged) empirical play converges into the set of convex combinations of Nash equilibria. Furthermore, the actual distribution of players' actions is close to some (approximate) Nash equilibria on most rounds (on all but a vanishing fraction of the rounds). In this process, all players rationally choose their actions using a public prediction made by a deterministic, weakly calibrated algorithm. For this to be possible, we show that such a deterministic (weakly) calibrated learning algorithm exists.