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31 of 40 people found the following review helpful:
5.0 out of 5 stars
Comprehensive and very well written, July 13, 2001
The theory of games is now pervasive in the fields of economics, financial modeling, logistics, operations research, network engineering, and population biology. As such a background in game theory is an absolute necessity if one is to deal with problems in these areas. This book is an advanced treatment of game theory, and presupposes the reader already has had some exposure to the subject. There is an excellent set of exercises at the end of each chapter, and so the book can be used as a textbook or for self-study. After an elementary example of a game in the introduction to motivate the subject, the authors begin in Part I of the book with the subject of static games with complete information. Strategic-form games are defined, along with dominated strategies, and the important concept of Nash equilibrium, the latter being introduced to deal with games that are not solvable by iterated strict dominance. For those with a background in elementary functional analysis, the authors prove that finite strategic-form game has a mixed-strategy equilibrium and prove that the Nash-Equilibrium has a closed graph. The concept of Nash equilibrium is extended to the concept of a correlated equilibrium, wherein each player can send another a private signal before they choose their strategy. In Part II, the authors discuss dynamic games with complete information. Examples of these kinds of games include a sequential version of the battle of the sexes game, and a sequential version of matching pennies. The authors discuss subgame-perfect equilibria, wherein an n-tuple of strategies constitute Nash equilibria in every subgame. The Stackelberg model of duopoly is discussed along with the repeated Prisoner"s dilemna, the latter being an example of backward induction in finitely repeated games. A kind of generalization of the principle of optimality in dynamic programming is used to analyze perfect public equilibria via a tool called self-generation. In Part III of the book, the authors discuss static games of incomplete information. Examples are discussed including Bayesian games, where at least one player is uncertain about another player"s payoff function, and first-price and second-price auctions. In first-price auctions, each player submits a sealed bid and the one with the highest bid obtains the item; in second-price auctions each player submits a sealed bid but the player submitting the highest bid gets to purchase the item for a cost given by the player with the second highest bid. The authors explain in detail the dominant strategies for these types of auctions. Bargaining with two-sided incomplete information is discussed and the optimal amount of trade is found from the linear equilibrium of the Chatterjee-Samuelson double action. In Part IV, dynamic games of incomplete information are discussed by the authors. Examples that they discuss include signaling games such as the two-period reputation game, and Spence"s education game. Signaling is widely used by firms and organizations in spite of it being somewhat costly to do so. For example a public company may be trying to convince investors that it represents high returns. The authors show how to obtain sequential perfect Bayes equilibrium in these and other scenarios. The authors also discuss reputation effects in games, with an example being the chain-store game. The general case of single long-run players with reputation effects is treated in detail. Bargaining with sequential buyers is also discussed with examples given for one-sided asymmetric information and mechanism design. The last part of the book discussed miscellaneous topics in game theory, including strategic stability, more discussion on signaling, finite strategic-form games, and supermodular games. The treatment is more complicated mathematically with emphasis on proving existence theorems for Nash equilibria and pure-strategy equilibria. The notion of a Markov perfect equilibrium is employed to discuss situations where the past has a direct influence on current opportunities. This brings in the fascinating subject of stochastic games, wherein current payoffs depend on the state of the game and on current actions, with the state evolving according to a Markov process. These are generalized to continuous time, leading to the famous differential games. Game theory under "common knowledge" is also discussed, with examples given of the "dirty face" games. Some omissions in the book, which would have of course increased the size of the book substantially, include mathematical modeling of poker and other card games. These are complicated games in which to analyze, but they have taken on considerable importance in the casino industry in recent years.
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