Games,Theory and Applications

This is a Dover reprint of a text, first published in 1986, covering various topics in game theory at a fairly sophisticated undergraduate level. The official prerequisites for understanding the text, according to the preface, are “an ability to differentiate and integrate simple functions, the idea of maxima and minima of functions, familiarity with matrix notation (though not the properties of matrices) and summation signs, and a basic knowledge of probability.” To these, however, I would add, as an “unofficial” but very real prerequisite, a degree of mathematical maturity commensurate with having taken several upper-level courses. This may defeat to some extent the author’s intent of making the book available “to as wide a numerate audience as possible”, but I think that the book is sufficiently concise that a person without such maturity will find it rather hard going at times. No prior knowledge of linear programming is assumed and the subject is not used in any detailed way in the text, though occasional references to it are made.

The text opens with a short chapter introducing the basic terminology of game theory, giving a first look at some examples, and providing a short (less than two pages long) history of the subject. The next three chapters discuss what I think of as the core material for an introductory course; they cover, in order, two-person zero-sum games (extensive and normal form; pure and mixed strategies; saddle points; a statement, without proof. of the minimax theorem; domination; solutions of some games of specific size), two-person non-zero-sum games (differences between these and zero-sum games; equilibria; a statement and sketch of a proof. assuming the Brouwer fixed point theorem, of Nash’s theorem on equilibria; cooperative games and bargaining theory), and n-person games (imputation; the core; stable sets; Shapley value). As an illustration of the conciseness of the author’s writing style, note that the text covers all this material in about 80 pages; by contrast, these three topics comprise most of Straffin’s book Game Theory and Strategy, and he spends about three times as many pages on it. Another particularly nice (and more leisurely) source for the two-person theory is Part III (“Conflict”) of the text A Mathematical Look at Politics by Robinson and Ullman. 

Thomas’ chapter 5 discusses applications to economics, specifically market games of various sizes, and duopoly and oligopoly theory. (These are situations where the market is controlled by two, or more than two, firms, and the issue is how they should set their prices.) Straffin’s text contains a section on the duopoly problem, but does not address the subject in the kind of depth that Thomas does.

The next four chapters of the text discuss more advanced aspects of game theory, specifically: metagames (some games, when played in practice, result in outcomes different than those predicted by the theory, because strategies selected are based to some extent on what the player thinks he or she is going to do); multi-stage games (essentially, these are games where the outcome requires playing the game again); evolutionary games (game theory applied to long-term behavior of animal species); and auctions (where the players are the bidders and the strategic choice involves determining how much to bid). These four chapters are largely independent of each other and of chapter 5, so a person who has read the first four chapters can proceed, according to his or her particular interests, to any of chapters 5 through 9. I found these chapters to be rather demanding reading, with the chapter on auctions probably the most accessible.

The final, non-mathematical, chapter of the text is on gaming, a term which might mean different things to different people; here, it is used to refer to “the playing of a situation involving two or more decision-makers which can be modeled as a game.” The author discusses the questions of why one should do gaming, and how to model a gaming situation.

Each chapter (except the last one) ends with a selection of exercises, some calling for computations and some for proofs. Solutions to what seems like all of the exercises in the book appear in a thirty-page section at the end of the text. There is also an extensive (but now, not surprisingly, quite dated) bibliography.

Bottom line: somebody who just wants to learn the basic theory would likely, I think, be better served by Straffin’s text, but a more knowledgeable reader who wants to explore some of the more sophisticated aspects of the theory might find this book useful.

Mark Hunacek (mhunacek@iastate.edu) teaches mathematics at Iowa State University.