Nash and von Neumann may never have been able to see eye to eye on codifying the foundations of game theory. Perhaps Luce and Raiffa had this in mind when they published this work in 1957, giving it the apt subtitle “Introduction and Critical Survey”. It introduces the concepts of Nash solutions, von Neumann-Morgenstern theory and other key theories and applications, including those of “Social Choice and Individual Values” author Kenneth J. Arrow. These ideas are critically analyzed for applicability and weakness through detailed examples. The mathematics here requires only the most basic understanding of matrices, sequences, set theory, and simple linear programming. Any determined undergraduate or interested reader with a college mathematics background can follow Luce’s thoughtful application and criticism of game theory ideas.
The first chapters look deeply into the details and possibilities afforded by the classic examples of the Prisoner’s Dilemma and the World War II decision of Admiral Imamura to transport Japanese troops from the port of Rabaul in New Britain across the Bismarck Sea to New Guinea going either north of New Britain, where it was likely to be foggy, or south of New Britain, where the weather was likely to be clear. U.S. Admiral Kenney hoped to bomb the troop ships, but had to choose where to concentrate his search. This is modeled as a zero-sum game. By the time the reader is done with the first half-dozen chapters, she will have learned about zero-sum games, utility theory, the minimax theorem, equilibrium pairs, Nash’s bargaining problem, and Shapley Value.
Another half-dozen chapters follow. These explore n-person games and such added intricacies as side payments skewing a player’s utility, characteristic functions, stability theory, and defining a solution. Several examples underscore the application of the presented ideas, and the limits to their applicability. Weaknesses and limitations of theories are underlined in examples throughout the book. The closing chapters explore adding the dimension of uncertainty to decision making and the dynamics of group decision theory.
The final chapter on group decision I found very interesting in the light of the constant drumbeat of journal articles on mathematical analysis of alternate election systems offering fairness or transparency. I believe any reader interested such topics will find the “Group Decision Making” chapter and its topics of the social welfare function and Arrow’s Impossibility Theorem along with the Arrow Paradox to be illuminating, if not depressing.
Eight appendices augment this work making it truly comprehensive and useful for self-guided reading. They cover a probabilistic approach to utility, the minimax theorem, geometrical and LP approaches to zero-sum two-person games, solving methods, infinite games, and sequential compounding of two-person games. These appendices comprise nearly a fourth of the book and are a more direct path from the game form to the mathematical model.
This is an excellent nontechnical introduction to game theory. Reading this book will teach you a lot about game theory, but this reprint does not have any updated content. Cooperative games are hardly covered and their relation to projective geometry is not mentioned, for instance. Not only pure or applied mathematicians, but also social scientists, financial analysts, management theorists, etc. will benefit from the solid grounding this book provides on how to quantitatively model and assess the road to the bargain struck.
Tom Schulte is a lead systems engineer at Plex Systems in Michigan with a focus on statistical analysis and integration of heterogeneous systems through web services.