GAMES AND DECISION MAKING
By Charalambos D. Aliprantis and Subir K. Chakrabarti Oxford University Press, p.277 + xi pp., 2000
Aliprantis and Chakrabarti wrote this text while developing a course with content from economic theory as part of the National Science Foundation grant "Mathematics Throughout the Curriculum," awarded to Indiana University in 1995. After introducing optimization theory and decision theory, their book develops game theory. Their target audience is undergraduate and introductory graduate students in the social sciences, business, and "even mathematics". I used this text for an upper level undergraduate mathematics course in game theory for the Fall 2001 semester at Montclair State University.
Games and Decision Making presents a self-contained treatment of decision theory and game theory, requiring only minimal amounts of calculus and probability and statistics. The text is organized in seven chapters. Although the initial chapters build on one another, there is great flexibility in whether or not to cover, or the order in which to cover, the chapters devoted to applications. Chapter 1: "Choices" introduces optimization theory and decision theory. Chapter 2 covers strategic form games with both discrete (e.g., bimatrix form games) and continuous action spaces. The latter requires extensive use of optimization theory to determine equilibria, applying the Nash equilibrium test; this material is ideal for students to apply their knowledge of calculus and to interpret properties of derivatives. Chapter 3 is a short chapter that forms the graph-theoretic underpinning of the extensive form or sequential move games introduced in Chapter 4. The authors again build on the calculus by considering consequences of backward induction on sequential games with continuous action spaces, as well as the more standard use of backward induction on game trees. Even though Chapters 1-4 are essential material for the rest of the book, the foundation material in calculus and graph theory does not require too much inclass time. This leaves sufficient time in a course to cover Chapters 5-7: "Sequential Rationality", "Auctions", and "Bargaining", respectively.
Since game theory is often described as multi-player decision theory, it is appropriate that Aliprantis and Chakrabarti's text begins with decision theory. Many undergraduate mathematics texts on game theory cite only enough decision theory to motivate Von Neumann-Morgenstern utility functions. In Chapter 1 of Games and Decision Making, Sections 1-4 provide a terse review of differential calculus and Lagrange multipliers. Although this should be a review for mathematics students, Section 2 does provide an introduction to the language of optimization theory (e.g., feasible sets, objective functions, etc.). Section 5 is primarily a review of the requisite probability and statistics. Section 6 applies the material from the earlier sections to examine decision theory. The authors consider preferences over alternatives and lotteries, and include axioms necessary for a Von Neumann-Morgenstern utility function to represent the preferences. Risk aversion is explained nicely in terms of the concavity or convexity of the utility function. …