A prediction market is a financial market designed to aggregate information. To facilitate trades, prediction markets are often operated by automated market makers. The market maker trades a set of securities with payoffs that depend on the outcome of a future event. For example, the market maker might offer a security that will pay off $1 if and only if a Democrat wins the 2012 US presidential election. A risk neutral trader who believes that the probability of a Democrat winning is p should be willing to purchase this security at any price below p, or sell it at any price above p. The current market price can then be viewed as the traders’ collective estimate of how likely it is that a Democrat will win the election. Market-based estimates have proved to be accurate in a variety of domains, including business, entertainment, and politics. However, when the number of outcomes is very large, it is generally infeasible to run a simple prediction market over the full outcome space. We propose a general framework for the design of securities markets over combinatorial or infinite state or outcome spaces. Our framework enables the design of computationally efficient markets tailored to an arbitrary, yet relatively small, space of securities with bounded payoff. We prove that any market satisfying a set of intuitive conditions must price securities via a convex cost function, which is constructed via conjugate duality. Rather than deal with an exponentially large or infinite outcome space directly, our framework only requires optimization over a convex hull. By reducing the problem of automated market making to convex optimization, where many efficient algorithms exist, we arrive at a range of new polynomial-time pricing mechanisms for various problems. Our framework also provides new insights into the relationship between market design and machine learning. In particular, we show that the tools that have been developed for online linear optimization are strikingly similar to those we have constructed for selecting pricing mechanisms. This is rather surprising, as the problem of learning in an online environment is semantically quite distinct from the problem of pricing securities in a prediction market: a learning algorithm receives losses and selects weights whereas a market maker manages trades and sets prices. We show that although the two frameworks have very different semantics, they have nearly identical syntax in a very strong sense.