There are a number of important questions associated with statistical experiments: when does one yield more information than another; how can we measure the difference in information; how fast does information accumulate by repeating the experiment? The means of answering such questions has emerged from the work of Wald, Blackwell, LeCam and others and is based on the ideas of risk and deficiency. The present work, which is devoted to the various methods of comparing statistical experiments, is essentially self-contained, requiring only some background in measure theory and functional analysis. Chapters introducing statistical experiments and the necessary convex analysis begin the book, and are followed by others on game theory, decision theory and vector lattices. The latter are a natural framework for studying statistical problems, as they help to avoid purely technical questions. The notion of deficiency, which measures the difference in information between two experiments, is then introduced. The relation between it and other concepts, such as sufficiency, randomization, distance, ordering, equivalence, completeness and convergence are explored. The last three chapters are devoted to applications of the theory to linear models, local comparison of differentiable experiments, majorization, and discussions of topics complementary to the main text. This is a comprehensive treatment of the subject and will be an essential reference for mathematical statisticians.