We study misaligned prices for logically related contracts in prediction markets. First, we uncover persistent arbitrage opportunities for risk-neutral investors between identical contracts on different exchanges. Examining the impact of several thousand dollars of transactions on the exchanges themselves in a randomized field trial, we document that price support extends well beyond what is seen in the published order book and that arbitrage opportunities are significantly larger than purely observational measurements indicate. Second, we demonstrate misalignment among identical and logically related contracts listed on the same exchange that cluster around moments of high information flow, when related contracts systemically shut down or fail to respond efficiently. Third, we document bounded rationality in prediction markets; examples include: consistent asymmetry between buying and selling, leaving the average return for selling higher than for buying; and persistent price lags between exchanges. Despite these signs of departure from theoretical optimality, the markets studied function well on balance, considering the sometimes complex and subtle relationships among contracts. Yet, we detail how to improve prediction markets by moving the burden of finding and fixing logical contradictions into the exchange and providing flexible trading interfaces, both of which free traders to focus on providing meaningful information in the form they find most natural.