The author describes this book as “an investigation of encounters between logic and games.” Thus, the author considers both logic in games and logic as games. Logic in games addresses the issue of describing things like game strategies in logic terms, whereas logic as games concerns itself with viewing questions such as satisfiability as a game between a verifier and a falsifier. The book is divided into six parts. The author suggests that readers can view the whole as two books, with Parts 1 through 3 concerning the use of logical systems to analyze the structure of games and Parts 4 through 6 covering (more or less) logic games.

The book begins with an introduction, “Exploring the Realm of Logic in Games,” which can be read both before reading the rest of the book and at the conclusion of the book as a review. Most chapters begin with a brief introduction and end with a conclusion; often, there is also a section on literature and directions for further exploration. The author requires of readers a familiarity with modal logic, as various forms of modal logic are used throughout the book. It is hard to suggest that a long book be made even longer, but an appendix addressing notation and some brief description of modal logic would provide a helpful reference for the reader. This is especially true as some notation, such as the use of # as a modifier meaning “realization,” is introduced only briefly.

Readers familiar with descriptions of games in terms of a payoff matrix will find the presentation to be quite a bit different from what they are used to. The presentation is most often in terms of a game tree. This allows the author to introduce incomplete information as the inability of a player to distinguish between certain states, and to show this on the game tree using dotted lines between the indistinguishable states.

Most of the book deals with finite games, although there is some discussion of infinite games in Part 1, where evolution is considered as an infinite game, and again in Part 5. Part 1, “Game Logics and Process Structure,” provides a logical view for such things as player preferences, strategies, and models for games. Games with incomplete information and infinite games are also described. Centipede games (so called because their tree resembles a centipede) provide an example where backward induction argues for strategy where the final outcome is not as good for either player as another strategy, thus requiring some collusion between the players to reach the more favorable outcome--a kind of prisoner’s dilemma.

Part 2, “Logical Dynamics and Theory of Play,” discusses, among other things, the way in which a player’s beliefs can be updated as the play proceeds. The muddy children puzzle--where two of three children have mud on their faces and the father says, “At least one of you is dirty,” and asks, “Does anyone know if he is dirty?”--provides a nice example here.

Part 3, “Players’ Powers and Strategic Games,” addresses issues such as the ability of a player to force play into a particular subset of states. Matrix representation of games makes its appearance here.

Part 4, “Logic Games,” considers formula evaluation and model comparison as games. The author suggests that any logical task can be “gamified.” A chapter on argumentation and dialog considers these as games with a proponent defending a claim against an opponent who grants initial concessions.

Part 5, “Operations on Games,” initiates a study of an algebra of games, which permits the composition of games in various ways, including disjunction, conjunction, and negation, as well as a parallel construction obtained by interleaving sequences of play from the component games.

Part 6, “Comparisons and Merges,” puts together ideas from earlier in the book to look at hybrids of some of the games discussed. An example of such a hybrid is an independence-friendly (IF) game, in which a player must choose some value without knowing the values of variables in some set. This can be used to model linguistic ambiguity. Sabotage games, discussed here, apply to questions of reachability under adverse circumstances. A computer science (CS) example of a sabotage game is that of a system reacting to changes in its environment.

The book emphasizes the logic view. While there are several examples of games that recur in the text, it would have been helpful if there were more examples, particularly when new concepts are introduced. While this would have made a long book even longer, it would have made the book easier to read. Readers will need to be comfortable with modal logic and logical reasoning, and those from the computing field will sometimes need to supply the connections to CS. Specific questions of computation are only addressed on the level of asserting that a particular system is decidable. There are many ideas that could be fruitful, and the author makes an effort to suggest avenues for further exploration.