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Foundations of game theory: Noncooperative games
Au~~omotica, Vol. 32, No. 9, pp. 1341-1342, 1996 Copyright 0 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved COX-10981% $15.00 + 0.00
Pergamon
Book Review
Foundations of Game Theory: Noncooperative Games* Nicolai N. Vorob’ev
should have liked more down-to-earth examples in order to elucidate the abstract concepts in the introductory chapter. To the novice, the beginning may be a bit rough. The author, however, once having adopted this philosophy, has done an excellent job: the book contains a wealth of material, is self-contained and is clearly written. The author is an authority in the field of game theory. The original version of the book was published in Russian in 1984. This means that some newer developments have not found a place in the book and most references are from the early 1980s or earlier (though I did spot at least one 1990 reference). In the preface to the English edition, the author regrets this omission, but gives some references where one can find more recent developments. That no connection to H, theory has been made is no surprise: that theory is both recent and is specifically related to dynamic game theory, which is not the main theme of this book. According to the preface again, the English edition differs slightly from the Russian original. A positive point is that some interesting results published so far in the Russian literature only, and not or not widely known in the western hemisphere, have become accessible for the first time (I think) to an English-reading public. As far as prerequisites are concerned, some mathematical sophistication is surely needed. More specifically, a general mathematical background in calculus, fixed-point theory, linear algebra and probability theory is required. No previous knowledge of game theory is required. It seems to me that the book can also be used as a textbook for (mathematically inclined) students, provided that the teacher gives some motivating and popularizing examples (especially in the beginning) and some background material. In contrast to many textbooks, this book does not have exercises at the end of its chapters. With respect to the audience that the author has in mind, he ‘did not intend to appeal to readers who are only concerned with pure mathematics as such, but rather addresses those who specialize in applications of mathematics in such fields as economics, psychology, social psychology, or law’ (from the Foreword). The book consists of a non-numbered Introduction and four numbered chapters devoted respectively to noncooperative games, finite noncooperative games, two-person zero-sum games and finally, matrix games. Occasionally the author refers back to previous chapters, but the chapters themselves can be read (rather) independently, depending on the reader’s objectives, interests and mathematical training. Each chapter (including the Introduction), ends with (historical) notes and a discussion of references. Some sections in chapters, not related to the main stream of the book, are printed in small type. Once in a while, a problem is stated and a complete analysis of its solution is given; this is particularly helpful. The Introduction itself is a full-grown chapter, consisting of 32 pages. It starts with the mathematization of informal ideas that explains such issues as informal and mathematial models. Another section deals with systematization of approaches, where it is explained that the methodological and metaphysical aspect is more essential for game theory than for other branches of mathematics, owing to the fact that game theory is oriented toward so&o-economic problems. Subsequently, the essential ingredients of a game, namely decision making, the conflict and the optimality of a decision are discussed. An example of going from the abstract to the concrete is found in Section 1.5. There a class %?of games is considered ‘that are not solvable under some
Reviewer: GEERT JAN OLSDER Delft University of Technology, Faculty of Technical Mathematics and Computer Science, Applied Analysis Group, P.O. Box 5031, 2600 GA Delft, The Netherlands. THEORY, the subject of the book under review, is still a relatively new branch of mathematics. It has applications in, among other things, economics, engineering and politics. Expressions with more or less the same contents as ‘game theory’ are ‘conflict analysis’ and ‘multiperson decision making’. What is essential is that two or more parties (which could be persons, firms, countries, etc.) are involved with partly or completely conflicting interests. Mathematically, such an interest is usually expressed as a cost function to be minimized or a profit function (a payoff) to be maximized. The cost functions of the various parties involved in the game are usually different. If only one party were present then the game would be degenerate and one would thus enter the field of optimization and/or (optimal) control theory. In spite of its youth (measured along the time scale of mathematical evolution), game theory has grown remarkably well, and various branches can already be distinguished. A common division of the existing theory is that between cooperative and noncooperative games. Loosely speaking, a game is noncooperative if each party pursues its own interests as well as possible. In cooperative games, on the other hand, the parties are willing to negotiate such that each party gets a ‘fair share of the cake’. Another division of games is one according to which they are static or dynamic. In static games the parties make in principle one decision, independently of each other, and then the outcome of the game is known. In dynamic games, on the other hand, time plays a role, and the parties make frequent decisions. During such a dynamic game, new information may become available (such as the previous decisions of the other parties) upon which more recent decisions can be based. The current book deals with noncooperative and mainly static games. Even if one restricts oneself to static noncooperative games, quite a few books on this subject have already appeared in the English language, especially on an introductory level. The book under review distinguishes itself clearly from most of the others in the sense that it fills a gap between the elementary introductions and the speciahsed monographs. Another striking feature of this book is that the build-up is from the general to the particular. It starts with general principles in the Introduction (mathematization of informal ideas, indifference relations, the process of cognition, metastrategies, etc.; even a quotation by Karl Marx on the interpretation of the world is used) and ends in the last chapter with matrix games (where topics like matix games and linear programming, graphical solution methods are dealt with-topics that one encounters earlier on in many other books). This philosophy from the general to the particular has been used consistently. In the early chapters results are given in a rather abstract setting, and become more concrete in the later chapters. Often independent proofs of such concrete results are given, so that one does not need to refer back to the earlier chapters. Imagining myself to be a novice to the theory of games, I did have some hesitations about the appropriateness of this philosophy. I GAME
* Foundations of Game Theory: Noncooperative Games, by Nicolai N. Vorob’ev. Birkhluser-Verlag, Base1 (1994). ISBN 3-7643-2378-7. ISBN 0-8116-2378-7. 1341
1342
Book Review
know another class X of games that are solvable in terms of their own optimality principle 4#‘. Subsequently, various mappings are introduced, such as f: %?+ X At the end of this discussion, which for me, as teacher, would be difficult to convey well to students, it turns out that the class Xof games might for instance be an extension of the original class % in the sense that in the original class only pure strategies are allowed, while in the extended version mixed strategies are also allowed. Thus one is down to earth again! Other important sections deal with the introduction of noncooperative games, optimality principles, noncooperative games with payoffs, and games of timing. Chapter 1 deals with ‘Noncooperative Games’ and consists of 100 pages. It starts with a thorough description of the components that define a game, i.e. the players’ set I, the strategies and the payoff functions. Notions such as finite game, coalition, topological game game, constant-sum (depending on the topological properies of the strategy sets), affinely strategically and equivalent games, Elhomomorphisms between two noncooperative games, and symmetric games, are introduced. Next, optimality principles are studied in detail. Dominance and stability with respect to subsets of players (coalitions) play an important role. Nash and Pareto solutions follow as special cases of more generally introduced principles. Since various optimality principles may lead to empty solution sets, realizability with respect to these principles (also in metastrategies and mixed extensions) is studied in the next sections. It is here that Kakutani’s and Brouwer’s fixed-point theorems are used to prove the existence of solutions, provided that precisely formulated convexity conditions are fulfilled. Metastrategies are strategies that may depend on other strategies (the Stackelberg solution principle and other hierarchically defined strategy dependences fall within this context, though ‘Stackelberg’ himself is not exclusively referred to here). The last section of this chapter deals with the intrinsic topology and intrinsic metric on spaces of strategies. Chapter 2, of 72 pages, deals with ‘Finite Noncooperative Games’. The fundamental concepts and various properties already introduced in Chapter 1 in a general setting are revisited here for the narrower class of finite games. Thus more specific results can be obtained. Specific sections are devoted to dyadic games (where the strategy set of each palyer consists of precisely two elements) and solution methods (in which the Lemke-Howson algorithm plays an important role). The last section deals with the possibility of uniformly reducing the theory of equilibrium in arbitrary finite noncooperative games to that in finite three-person games. Decreasing the number of players to three has to be paid for by a substantial increase in the number of strategies in the new game, but the reduction of players to three remains very interesting, both mathematically and philosophically. This fact, discovered and proved by V. S. Bubyalis, has not been widely known. Chapter 3, of 149 pages, is devoted to ‘Two-person Zero-sum Games’. Though zero-sum games are a special case of the nonzero-sum games studied in Chapter 2, the current chapter is not, strictly speaking, a further specialization of the previous one, since the strategy spaces are allowed to be infinite here. Thus the study of ‘games on the square’ (where each of the two players can choose a real number from the unit interval) belongs to this chapter. In the beginning the surprising, though very trivial, fact is discussed that each solution in a two-person zero-sum game is Pareto-optimal, which leads immediately to the conclusion that Pareto optimality is not a very suitable solution principle for such games. One encounters the notions of c-equilibria, saddle points, value, and various minimax theorems (where one distinguishes various kinds of slightly different ‘convexity’ definitions). It turns out that many games do not have (c-)saddle-point solutions, not even in the mixed sense. As a further extension, finitely additive strategies are introduced. For games on the square, the payoff function is often given by an integral over the square. Special sections are devoted to cases where the kernel of this integral has special properties (such as being an analytic function or being separable in the strategies). An example of a subsection that
goes into elegant mathematical subleties is the one on ‘rational games with singular solutions’. Such games are defined as having rational payoff functions, and form a class of games narrower than the class of analytic games. The chapter ends with a section on ‘games of timing’ and a special class of ‘games on the square’, where the kernel of the payoff function is usually discontinuous along the diagonal of the unit square. An elegant and not widely known theorem by E. B. Yanovskaya gives conditions under which optimal mixed strategies exist for such games. Neither upper- nor lower-semicontinuity are required here. Chapter 4, of 97 pages, deals with ‘Matrix Games’, which form a subclass of two-person zero-sum games (some subsections, however, deal with the nonzero-sum extension to b&matrix games). Various results proved earlier for this more general class of games are now proved independently of the previous chapters and often in different, simplified, ways. As already noted, many textbooks start with a chapter on matrix games, whereas in this book it is the dessert of the complete meal of game theory. The chapter starts with the basic concepts of existence of an equilibrium point in mixed strategies (based on the result on two alternatives). Then the ‘solution of matrix games of small format’ is considered, where graphical methods are central. Subsequently, the relation with linear programming is discussed, which gives rise to solution methods of games ‘of larger format’. Equilibrium solutions in terms of extreme strategies for both matrix and bimatrix games are discussed, as well as the so-called Shapley-Snow theorem. There is an interesting section on matrix games where ‘sub-matrix games’ are defined as restrictions to a submatrix of the original larger matrix. These submatrices are then replaced by the values of the corresponding subrpatrix games, thus giving rise to an (approximate) version of the original game, but of smaller dimension. Other approximate solution methods are discussed, such as by means of the ‘method of differential equations’. After Chapter 4, there are a list with references (many of which are Russian), a list of joint authors (of the references given), an index of notation (given per chapter) and finally a general index. In conclusion, this is a very well-written book and should be of special interest for readers with a mathematical background and taste. It covers the area of noncooperative, static games rather completely. It can be used as a text in a game theory course by choosing appropriate parts (treating the whole book would take far too much time for an average course). The philosophy of starting with general principles and gradually adding more and more conditions so as to arrive at more concrete and more down-to-earth results, has been followed consistently; it may be a bit rough in the beginning for the novice, who would like to get some more grip on the subject, by means of more examples for instance. No exercises are provided. Though I do not know (nor can I read) the original Russian version of the book, the English version seems to be an excellent translation. Sometimes, with other translated texts, one (at least the current reviewer) notes (too) clearly that the translator has had difficulties with finding the appropriate translation of technical words with specific meanings. This is not the case for the book under review (the only word in the current book where one might suspect an original version in another language is the word ‘situation’, referring to the Carthesian product of the individual strategy spaces). About the reuiewer
Geert Jan Olsder (born in 1944) is Chairman of the Department of Applied Analysis within the Faculty of Technical Mathematics and Informatics at Delft University of Technology. His research centers around differential games (with T. Baqar, he authored the book Noncooperative Dynamic Game Theory, Academic Press, New York, 1982; revised edition, 1995) and discrete event systems (with F. Baccelli, G. Cohen and J. P. Quadrat, he authored the book Synchronization and Linearity, Wiley, New York, 1992). He is (Associate) Editor of four scientific journals, and he has served IFAC in various functions.
Pergamon
Book Review
Foundations of Game Theory: Noncooperative Games* Nicolai N. Vorob’ev
should have liked more down-to-earth examples in order to elucidate the abstract concepts in the introductory chapter. To the novice, the beginning may be a bit rough. The author, however, once having adopted this philosophy, has done an excellent job: the book contains a wealth of material, is self-contained and is clearly written. The author is an authority in the field of game theory. The original version of the book was published in Russian in 1984. This means that some newer developments have not found a place in the book and most references are from the early 1980s or earlier (though I did spot at least one 1990 reference). In the preface to the English edition, the author regrets this omission, but gives some references where one can find more recent developments. That no connection to H, theory has been made is no surprise: that theory is both recent and is specifically related to dynamic game theory, which is not the main theme of this book. According to the preface again, the English edition differs slightly from the Russian original. A positive point is that some interesting results published so far in the Russian literature only, and not or not widely known in the western hemisphere, have become accessible for the first time (I think) to an English-reading public. As far as prerequisites are concerned, some mathematical sophistication is surely needed. More specifically, a general mathematical background in calculus, fixed-point theory, linear algebra and probability theory is required. No previous knowledge of game theory is required. It seems to me that the book can also be used as a textbook for (mathematically inclined) students, provided that the teacher gives some motivating and popularizing examples (especially in the beginning) and some background material. In contrast to many textbooks, this book does not have exercises at the end of its chapters. With respect to the audience that the author has in mind, he ‘did not intend to appeal to readers who are only concerned with pure mathematics as such, but rather addresses those who specialize in applications of mathematics in such fields as economics, psychology, social psychology, or law’ (from the Foreword). The book consists of a non-numbered Introduction and four numbered chapters devoted respectively to noncooperative games, finite noncooperative games, two-person zero-sum games and finally, matrix games. Occasionally the author refers back to previous chapters, but the chapters themselves can be read (rather) independently, depending on the reader’s objectives, interests and mathematical training. Each chapter (including the Introduction), ends with (historical) notes and a discussion of references. Some sections in chapters, not related to the main stream of the book, are printed in small type. Once in a while, a problem is stated and a complete analysis of its solution is given; this is particularly helpful. The Introduction itself is a full-grown chapter, consisting of 32 pages. It starts with the mathematization of informal ideas that explains such issues as informal and mathematial models. Another section deals with systematization of approaches, where it is explained that the methodological and metaphysical aspect is more essential for game theory than for other branches of mathematics, owing to the fact that game theory is oriented toward so&o-economic problems. Subsequently, the essential ingredients of a game, namely decision making, the conflict and the optimality of a decision are discussed. An example of going from the abstract to the concrete is found in Section 1.5. There a class %?of games is considered ‘that are not solvable under some
Reviewer: GEERT JAN OLSDER Delft University of Technology, Faculty of Technical Mathematics and Computer Science, Applied Analysis Group, P.O. Box 5031, 2600 GA Delft, The Netherlands. THEORY, the subject of the book under review, is still a relatively new branch of mathematics. It has applications in, among other things, economics, engineering and politics. Expressions with more or less the same contents as ‘game theory’ are ‘conflict analysis’ and ‘multiperson decision making’. What is essential is that two or more parties (which could be persons, firms, countries, etc.) are involved with partly or completely conflicting interests. Mathematically, such an interest is usually expressed as a cost function to be minimized or a profit function (a payoff) to be maximized. The cost functions of the various parties involved in the game are usually different. If only one party were present then the game would be degenerate and one would thus enter the field of optimization and/or (optimal) control theory. In spite of its youth (measured along the time scale of mathematical evolution), game theory has grown remarkably well, and various branches can already be distinguished. A common division of the existing theory is that between cooperative and noncooperative games. Loosely speaking, a game is noncooperative if each party pursues its own interests as well as possible. In cooperative games, on the other hand, the parties are willing to negotiate such that each party gets a ‘fair share of the cake’. Another division of games is one according to which they are static or dynamic. In static games the parties make in principle one decision, independently of each other, and then the outcome of the game is known. In dynamic games, on the other hand, time plays a role, and the parties make frequent decisions. During such a dynamic game, new information may become available (such as the previous decisions of the other parties) upon which more recent decisions can be based. The current book deals with noncooperative and mainly static games. Even if one restricts oneself to static noncooperative games, quite a few books on this subject have already appeared in the English language, especially on an introductory level. The book under review distinguishes itself clearly from most of the others in the sense that it fills a gap between the elementary introductions and the speciahsed monographs. Another striking feature of this book is that the build-up is from the general to the particular. It starts with general principles in the Introduction (mathematization of informal ideas, indifference relations, the process of cognition, metastrategies, etc.; even a quotation by Karl Marx on the interpretation of the world is used) and ends in the last chapter with matrix games (where topics like matix games and linear programming, graphical solution methods are dealt with-topics that one encounters earlier on in many other books). This philosophy from the general to the particular has been used consistently. In the early chapters results are given in a rather abstract setting, and become more concrete in the later chapters. Often independent proofs of such concrete results are given, so that one does not need to refer back to the earlier chapters. Imagining myself to be a novice to the theory of games, I did have some hesitations about the appropriateness of this philosophy. I GAME
* Foundations of Game Theory: Noncooperative Games, by Nicolai N. Vorob’ev. Birkhluser-Verlag, Base1 (1994). ISBN 3-7643-2378-7. ISBN 0-8116-2378-7. 1341
1342
Book Review
know another class X of games that are solvable in terms of their own optimality principle 4#‘. Subsequently, various mappings are introduced, such as f: %?+ X At the end of this discussion, which for me, as teacher, would be difficult to convey well to students, it turns out that the class Xof games might for instance be an extension of the original class % in the sense that in the original class only pure strategies are allowed, while in the extended version mixed strategies are also allowed. Thus one is down to earth again! Other important sections deal with the introduction of noncooperative games, optimality principles, noncooperative games with payoffs, and games of timing. Chapter 1 deals with ‘Noncooperative Games’ and consists of 100 pages. It starts with a thorough description of the components that define a game, i.e. the players’ set I, the strategies and the payoff functions. Notions such as finite game, coalition, topological game game, constant-sum (depending on the topological properies of the strategy sets), affinely strategically and equivalent games, Elhomomorphisms between two noncooperative games, and symmetric games, are introduced. Next, optimality principles are studied in detail. Dominance and stability with respect to subsets of players (coalitions) play an important role. Nash and Pareto solutions follow as special cases of more generally introduced principles. Since various optimality principles may lead to empty solution sets, realizability with respect to these principles (also in metastrategies and mixed extensions) is studied in the next sections. It is here that Kakutani’s and Brouwer’s fixed-point theorems are used to prove the existence of solutions, provided that precisely formulated convexity conditions are fulfilled. Metastrategies are strategies that may depend on other strategies (the Stackelberg solution principle and other hierarchically defined strategy dependences fall within this context, though ‘Stackelberg’ himself is not exclusively referred to here). The last section of this chapter deals with the intrinsic topology and intrinsic metric on spaces of strategies. Chapter 2, of 72 pages, deals with ‘Finite Noncooperative Games’. The fundamental concepts and various properties already introduced in Chapter 1 in a general setting are revisited here for the narrower class of finite games. Thus more specific results can be obtained. Specific sections are devoted to dyadic games (where the strategy set of each palyer consists of precisely two elements) and solution methods (in which the Lemke-Howson algorithm plays an important role). The last section deals with the possibility of uniformly reducing the theory of equilibrium in arbitrary finite noncooperative games to that in finite three-person games. Decreasing the number of players to three has to be paid for by a substantial increase in the number of strategies in the new game, but the reduction of players to three remains very interesting, both mathematically and philosophically. This fact, discovered and proved by V. S. Bubyalis, has not been widely known. Chapter 3, of 149 pages, is devoted to ‘Two-person Zero-sum Games’. Though zero-sum games are a special case of the nonzero-sum games studied in Chapter 2, the current chapter is not, strictly speaking, a further specialization of the previous one, since the strategy spaces are allowed to be infinite here. Thus the study of ‘games on the square’ (where each of the two players can choose a real number from the unit interval) belongs to this chapter. In the beginning the surprising, though very trivial, fact is discussed that each solution in a two-person zero-sum game is Pareto-optimal, which leads immediately to the conclusion that Pareto optimality is not a very suitable solution principle for such games. One encounters the notions of c-equilibria, saddle points, value, and various minimax theorems (where one distinguishes various kinds of slightly different ‘convexity’ definitions). It turns out that many games do not have (c-)saddle-point solutions, not even in the mixed sense. As a further extension, finitely additive strategies are introduced. For games on the square, the payoff function is often given by an integral over the square. Special sections are devoted to cases where the kernel of this integral has special properties (such as being an analytic function or being separable in the strategies). An example of a subsection that
goes into elegant mathematical subleties is the one on ‘rational games with singular solutions’. Such games are defined as having rational payoff functions, and form a class of games narrower than the class of analytic games. The chapter ends with a section on ‘games of timing’ and a special class of ‘games on the square’, where the kernel of the payoff function is usually discontinuous along the diagonal of the unit square. An elegant and not widely known theorem by E. B. Yanovskaya gives conditions under which optimal mixed strategies exist for such games. Neither upper- nor lower-semicontinuity are required here. Chapter 4, of 97 pages, deals with ‘Matrix Games’, which form a subclass of two-person zero-sum games (some subsections, however, deal with the nonzero-sum extension to b&matrix games). Various results proved earlier for this more general class of games are now proved independently of the previous chapters and often in different, simplified, ways. As already noted, many textbooks start with a chapter on matrix games, whereas in this book it is the dessert of the complete meal of game theory. The chapter starts with the basic concepts of existence of an equilibrium point in mixed strategies (based on the result on two alternatives). Then the ‘solution of matrix games of small format’ is considered, where graphical methods are central. Subsequently, the relation with linear programming is discussed, which gives rise to solution methods of games ‘of larger format’. Equilibrium solutions in terms of extreme strategies for both matrix and bimatrix games are discussed, as well as the so-called Shapley-Snow theorem. There is an interesting section on matrix games where ‘sub-matrix games’ are defined as restrictions to a submatrix of the original larger matrix. These submatrices are then replaced by the values of the corresponding subrpatrix games, thus giving rise to an (approximate) version of the original game, but of smaller dimension. Other approximate solution methods are discussed, such as by means of the ‘method of differential equations’. After Chapter 4, there are a list with references (many of which are Russian), a list of joint authors (of the references given), an index of notation (given per chapter) and finally a general index. In conclusion, this is a very well-written book and should be of special interest for readers with a mathematical background and taste. It covers the area of noncooperative, static games rather completely. It can be used as a text in a game theory course by choosing appropriate parts (treating the whole book would take far too much time for an average course). The philosophy of starting with general principles and gradually adding more and more conditions so as to arrive at more concrete and more down-to-earth results, has been followed consistently; it may be a bit rough in the beginning for the novice, who would like to get some more grip on the subject, by means of more examples for instance. No exercises are provided. Though I do not know (nor can I read) the original Russian version of the book, the English version seems to be an excellent translation. Sometimes, with other translated texts, one (at least the current reviewer) notes (too) clearly that the translator has had difficulties with finding the appropriate translation of technical words with specific meanings. This is not the case for the book under review (the only word in the current book where one might suspect an original version in another language is the word ‘situation’, referring to the Carthesian product of the individual strategy spaces). About the reuiewer
Geert Jan Olsder (born in 1944) is Chairman of the Department of Applied Analysis within the Faculty of Technical Mathematics and Informatics at Delft University of Technology. His research centers around differential games (with T. Baqar, he authored the book Noncooperative Dynamic Game Theory, Academic Press, New York, 1982; revised edition, 1995) and discrete event systems (with F. Baccelli, G. Cohen and J. P. Quadrat, he authored the book Synchronization and Linearity, Wiley, New York, 1992). He is (Associate) Editor of four scientific journals, and he has served IFAC in various functions.