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Reviewer: Irtaza Barlas

Simply put, this book is about prediction. The "learning" and "games" parts of the title are specific applications of the prediction problem. What is not simple is the generalization of the prediction problem the authors have put forward in a formal manner. As discussed throughout this book, the prediction problem has been approached by researchers from a variety of angles. Therefore, a statistical approach to prediction assumes that the prediction of a sequence of outcomes is a realization of a stationary stochastic process generating these outcomes. The authors abandon this assumption, and practically assume nothing about the process. This leads to the definition of effectiveness, and the formal goal of prediction, in an approach called "prediction of individual sequences." This style of sequential prediction finds its roots in the theory of zero-sum repeated games with fixed loss. The central idea behind this style of prediction is that, even though the predictor in question does not know anything about the process that is generating the outcome(s) of interest, there are certain experts available who serve as reference forecasters. The information obtained from these experts is available to the predictor before the next outcome is revealed, and it is left to make its own prediction. The responsibility of the predictor is then to minimize a cumulative loss. This minimization does not need to reach zero, but should be as good as the best expert. Chapter 1 of the book has a two-page review as a gentle start, but in my opinion this is too short and too gentle to be of any real use to a reader who is a newcomer to this subject. The authors introduce their general idea, and then move on to quickly define the problem formally in chapter 2. In particular, a weighted average forecaster is defined that serves as a benchmark throughout the text. Chapter 3 improves the performance boundaries under specific assumptions about the experts and decision space. The chapter discusses a myopic strategy that chooses the best expert in the overall sense, that is, one with the lowest cumulative loss over time. It also establishes the general conditions under which this strategy works fairly well. Another technique, termed the greedy forecaster, is also developed that turns out to perform as well as the weighted average forecaster in most scenarios. The results of this chapter are linked to the prediction work done in information theory, specifically to applications in compression and data encoding. In chapters 8 and 9, the authors look at the details of loss functions in this context. Chapter 11 introduces relationships with online learning schemes. An interesting idea of "prediction with side information" is developed. This information is apart from the past outcomes of the sequence. The problem is specialized to a binary problem in chapter 12, and is shown to be similar to that obtained by Rosenblatt [1] (as in perceptrons). Chapters 4 through 7 consider the problems in a game theoretic framework, in which the forecaster has limited information in some way. A model of randomized prediction is introduced and developed. In the absence of enough information for the predictor, the use of randomization is shown to achieve very good results, which would otherwise not be possible. The relationship between the prediction problem and a game theoretic framework is studied carefully, with repeated multiplayer games. Readers beware: the cute and colorful cover of the book is deceptive, in that it may seem like a friendly book, written as an introduction to the exotic topics of prediction, learning, and games. In fact, the book covers the basics very quickly, in a two-page "gentle introduction," and goes on to the not-so-mundane world of lemmas, theorems, corollaries, and their respective proofs. Since the book is supposed to be self-contained, the authors chose to place some of the proofs of the lemmas in the appendix. In my opinion, the "self-contained" claim can be easily challenged. Overall, the book can be used for serious graduate-level coursework on the topic. The authors provide detailed historical accounts that show their depth of understanding of many different fields. They are articulate in bringing these historically diverse fields under the umbrella of a combined context. They are also very careful in defining their assumptions and proofs.