Data is a commodity. Recent research has considered the mathematical problem of setting prices for different queries over data. Ideal pricing functions need to be flexible --defined for arbitrary queries (select-project-join, aggregate, random sample, and noisy privacy-preserving queries). They should be fine-grained -- a consumer should not be required to buy the entire database to get answers to simple "low-information" queries (such as selecting only a few tuples or aggregating over only one attribute). Similarly, a consumer may not want to pay a large amount of money, only to discover that the database is empty. Finally, pricing functions should satisfy consistency conditions such as being "arbitrage-free" -- consumers should not be able to circumvent the pricing function by deducing the answer to an expensive query from a few cheap queries.

Previously proposed pricing functions satisfy some of these criteria (i.e. they are defined for restricted subclasses of queries and/or use relaxed conditions for avoiding arbitrage). In this paper, we study arbitrage-free pricing functions defined for arbitrary queries. We propose new necessary conditions for avoiding arbitrage and provide new arbitrage-free pricing functions. We also prove several negative results related to the tension between flexible pricing and avoiding arbitrage, and show how this tension often results in unreasonable prices.