- Email: [email protected]
John Nash's Contribution to Economics
GAMES AND ECONOMIC BEHAVIOR ARTICLE NO.
14, 287–295 (1996)
0053
REVIEWS AND COMMENTS
With the intent of stimulating discussion, this section is reserved for book reviews, comments, and letters; your input is welcome. By nature, this material may be subjective, reflecting the opinions of the authors; your responses are therefore encouraged.
John Nash’s Contribution to Economics
1. INTRODUCTION For many years, there have been conferences where John Nash should have been honored as a leader of our field, and his presence has been much missed. I can imagine no higher privilege than to offer words in his honor today.1 The story of Nash’s contribution to economics is one of the most remarkable in the history of the field, beginning with an amazing paper that he started as an undergraduate in his first economics course. In graduate school, concerned that his Nobel-winning work might not be accepted as a doctoral dissertation, Nash also studied the mathematics of manifolds, which led to his solution of a classical embeddability problem. Nash’s mathematical talents could rival the genius of von Neumann like few others in the history of economics. But what I want to emphasize here above all is the sheer importance of Nash’s contribution as a turning point in the history of economics. In a room full of Nobel laureates, there is no one with a greater claim to have launched such a fundamental transformation in the scope of the field. In a series of four short papers published between 1950 and 1953, John Nash laid the foundations for game theory as we now know it. Two of these papers introduced the analysis of bargaining without transferable utility, and alone they 1
This paper was written for presentation at a luncheon in honor of the 1994 Nobel laureates, at the American Economics Association annual meetings in San Fransisco on January 6, 1996. Since 1994, several important articles in appreciation of Nash’s work have been published; see Leonard (1994), Milnor (1995), Nasar (1994), Nash (1994), Rubinstein (1995), and van Damme and Weibull (1995). Like others who have sought to better understand the work of John Nash, I am deeply indebted to Harold Kuhn for his help and advice. 287 0899-8256/96 $18.00 Copyright © 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
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would have made Nash one of the great leaders of game theory. But the other two papers, which develop Nash’s concept of equilibrium for general noncooperative games, must be ranked among the greatest watershed breakthroughs in the history of social science. We should recall others who contributed to the uphill climb to this watershed, and we should observe that the road down from the top was not straight or smooth. But while the publication of von Neumann and Morgenstern’s book attracted much more attention, it was Nash’s major paper on noncooperative equilibrium that truly marked the beginning of a new era in economics. Why is Nash equilibrium so important? Questions of institutional reform will always generate a practical demand for theories of social institutions. Of course, human behavior is wonderfully complex, and we cannot expect any social theory to attain an accuracy like Newton’s theory of planetary motion. Many alternative theories should be explored. But there are good reasons to expect Nash equilibrium analysis to be one of the most valuable methods of applied theory in all areas of economics and social science. In any area of applied social theory, we must begin with a model. To be able to handle normative questions, there must be some concept of human welfare in our model. If individuals are not motivated to maximize their own welfare, then any loss of welfare can be blamed on individual behavior, rather than on the structure of social institutions. So an argument for reform of institutions, rather than re-education of individuals, is most persuasive when it is based on a model that assumes rational maximizing behavior of individuals. Nash equilibrium is the logical formulation of this assumption: that each member of society will act, within his (or her) domain of control, to maximize welfare as he (or she) evaluates it, given the predicted behavior of others. Given that the concept of Nash equilibrium can be so fundamental to the analysis of any institution, it may seem surprising that classical social philosophers left the general formulation of equilibrium analysis to this generation. To unravel this mystery, we should review the history to see why the idea of Nash equilibrium was not so easy to see. Of course, given the fundamental importance of Nash equilibrium, its implicit application can be found in classical social philosophy. Reformulating ideas of Hobbes and Machiavelli into rigorous game-theoretic models can be an interesting and rewarding exercise. But the first clear application of Nash equilibrium in a mathematical model comes in the work of Augustin Cournot.
2. COURNOT AND VON NEUMANN Cournot’s (1838) book on economic theory is dazzling even today. He constructed a theory of oligopolistic firms that includes monopolists and perfect competitors as limiting extremes. He developed game models of oligopolistic competition, which he analyzed by the methodology of Nash equilibrium.
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But of course he was writing more than a century before Nash. Should Cournot get credit for the equilibrium concept? Should we speak rather of “Cournot equilibrium,” or “Cournot–Nash equilibrium”? I want to argue that this terminology would be wrong. We may speak of Cournot as the founder of oligopoly theory, but to give him credit for the fundamental solution concept of noncooperative game theory would be to confuse one application of a methodology with its general formulation. This distinction is one which Cournot would have appreciated. He wrote a short book on mathematical economics, but he wrote at greater length on the philosophy of science and the foundations of our knowledge. If he had recognized that noncooperative game theory can provide a general unifying structure for analyzing all kinds of social institutions, he would have wanted to write about it more than anyone else in his generation. But he did not see it. Cournot did not develop the conceptual distinction between the formulation of his specific game models and the general methodology used to analyze them.2 Indeed, far from finding a general analytical methodology in Cournot, readers from Bertrand (1883) to Fellner (1949) found specific models of oligopoly which had some interesting applied predictions, but which seemed to make some invalid assumptions. In particular, once Cournot has shown that the optimal output of firm 2 depends on the output of firm 1, it may seem irrational for the manager of firm 1 to assume that 2’s output would remain fixed if he changed 1’s output. Until this critique could be answered, Cournot’s methodology did not look like a compelling general theory of rational behavior. The answer came as von Neumann’s first great contribution to game theory. Von Neumann began his 1928 paper on the minimax theorem by formulating a general model of extensive games, in which players move sequentially over time with imperfect information about each others’ previous moves. Because players may get some information about other players’ previous moves, we cannot assume that players’ moves are independent in such extensive games. But von Neumann then defined a strategy for each player to be a complete plan that specifies a move for the player, at each stage where he is active, as a function of his information at that stage. A rational player can choose his strategy before the game begins, with no loss of generality, because a strategy lets him specify a different move for every situation in which he might find himself during the game. But ”before the game begins” means before any consequences of other players’ decisions can be observed. So each player must choose his 2 Cournot first analyzed competition among firms that compete to sell the same consumer good, and then he analyzed a second model of producers of complementary inputs for a manufactured good. In the analysis of the latter model, Cournot wrote “if we apply to the mutual relations of producers the same method of reasoning which served for analyzing the effects of competition. . . ” These words are the closest to an articulation of a general game-theoretic methodology that I have found in Cournot’s book.
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strategy without being informed of the other players’ strategy choices. Thus, von Neumann argued, there is no loss of generality in studying games where players make their strategic decisions independently. This insight is what allows us today to accept Cournot’s basic assumption that competitors make their decisions independently. Perhaps firm 2 can base its production next year on firm 1’s production this year; but that just means that firm 2 has a larger strategy space than Cournot admitted. At the level of strategic planning, we can still assume that firm 2 chooses its strategy independently of firm 1’s strategy choice. Thus, von Neumann argued that virtually any competitive game can be modeled by a mathematical game with the following simple structure: There is a set of players, each player has a set of strategies, each player has a payoff function from the Cartesian product of these strategy sets into the real numbers, and each player must choose his strategy independently of the other players. This structure is von Neumann’s normal form for representing general extensive games. Von Neumann did not consistently apply the principle of strategic independence, however. In his analysis of games with more than two players, von Neumann assumed that players would not simply choose their strategies independently, but would coordinate their strategies in coalitions. Furthermore, by computing max–min values for each coalition, von Neumann implicitly assumed that each coalition must be prepared to have its strategy announced first, allowing the other players to modify their decisions in response. Before Nash, however, no one seems to have noticed that these assumptions were inconsistent with von Neumann’s own argument for strategic independence of the players in the normal form. Von Neumann (1928) also added two restrictions to his normal form that severely limited its claim to be a general model of social interaction for all the social sciences. He assumed that payoff is transferable and that all games are zero-sum. To see why he added these seemingly unnecessary restrictions, we must recall his second great contribution to game theory: the minimax theorem. Von Neumann (1928) recognized that randomized strategies had to be admitted to prove the existence of minimax solutions for two-person zero-sum games. To analyze games with randomization, however, we need a theory of how players make decisions under uncertainty. Von Neumann used the traditional assumption that, when there is uncertainty, each player wants to maximize the expected value of his payoff. But he was uncomfortable with this assumption. In 1928 and again in his 1944 book with Morgenstern, he tried to justify this assumption by identifying all payoffs with monetary transfer payments, which led him to the restriction that payoff is transferable and all games are zero-sum. The fact that the zero-sum restriction also gave him the two-person minimax theorem was probably what committed him intellectually to these restrictions, but the discussion in von Neumann and Morgenstern (1944, Section 2.1.1) suggests that the initial motivation was to defer the problem of measuring utilities.
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In 1947 (in their book’s second edition), von Neumann and Morgenstern published their third great contribution to game theory: the axiomatic derivation of expected-utility maximization from a substitution argument. This new justification for measurable utility should have prompted them to drop their zero-sum transferable-payoff restrictions and to re-examine the foundations of their theory. But they did not. This task was left to John Nash, a young graduate student who saw almost immediately that the whole structure of game theory needed to be broken up and put back together right.
3. NASH’S RECONSTRUCTION OF GAME THEORY Nash’s first great contribution was his 1950 article on bargaining (Nash, 1950a), which began in 1948 with an idea that he had in his undergraduate economics course. This paper was the first work in game theory without transferable utility. By a beautiful axiomatic derivation, Nash introduced a bargaining solution that was virtually unanticipated in the literature.3 Then, on November 16, 1949, the Proceedings of the National Academy of Sciences received from Nash a short note, which was published the next year (1950b). In this two-page note, Nash gave the general definition of equilibrium for normal-form games, and he neatly sketched a fixed-point argument to prove that equilibria in randomized strategies must exist for any finite normal-form game. In his subsequent 1951 paper (which was based on his Princeton doctoral dissertation), Nash gave a fuller development of the idea of equilibrium. This 1951 paper includes versions of some 2×2 games like the Prisoners’ Dilemma, which have filled the basic game-theory literature ever since. But most importantly, Nash (1951) argued that his equilibrium concept, together with von Neumann’s normal form, gives us a complete general methodology for analyzing all games.4 Referring to the other ”cooperative” theories of von Neumann and Morgenstern, Nash wrote: This writer has developed a “dynamical” approach to the study of cooperative games based on reduction to non-cooperative form. One proceeds by constructing a model of the larger pre-play negotiation so that the steps of negotiation become moves in a larger non-cooperative game . . . describing the total situation.
3
Nash’s (1950a) axiomatic bargaining theory builds on the insight that individuals’ utility scales can be defined up to separate increasing linear transformations, but this result follows only from von Neumann and Morgenstern’s 1947 derivation of utility. Thus, Nash’s bargaining solution could not have been appreciated before 1947. It is remarkable that Nash found this solution so quickly thereafter. 4 Nash (1951) noted that the assumption of transferable utility can be dropped without loss of generality, because possibilities for transfer can be put into the moves of the game itself, and he dropped the zero-sum restriction from von Neumann’s normal form.
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Thus Nash applied the normalization argument to show that any other theory of games should be reducible to equilibrium analysis. With this step, Nash carried social science over the watershed into a new world where a unified analytical structure can be found for studying all situations of conflict and cooperation. Von Neumann’s normal form is our general model for all games, and Nash’s equilibrium is our general solution concept. In his 1953 paper, Nash offered an application of his program for reducing cooperative game theory to noncooperative equilibrium analysis. He modeled the two-person bargaining process by a simple game of simultaneous demands. This game has an infinite number of Nash equilibria, but Nash gave an ingenious perturbational argument for selecting a unique stable equilibrium which coincides with the bargaining solution that he previously derived axiomatically.
4. SUBSEQUENT DEVELOPMENT OF NONCOOPERATIVE GAME THEORY The impact of Nash’s reconstruction of game theory spread slowly. At first, more attention was focused on the cooperative analysis that von Neumann favored.5 Later, as more people realized the importance of Nash’s program, it became apparent that there were a number of technical problems that needed further study before noncooperative game theory could meet its promise as a general analytical methodology for applied work. John Harsanyi (1967–1968) questioned the argument for the normal form in situations where players have different information at the beginning of the game, and he developed a more versatile Bayesian-game model for such situations. Further challenging von Neumann’s argument for the normal form, Selten (1965, 1975) argued that we need to study Nash’s equilibria directly in the extensive form, to look for equilibria that satisfy some form of sequentially rationality. The fact that some equilibria might be less stable or less rational than others was noted by Nash (1951, Example 6). To exclude such unstable equilibria, we need some refinement of the Nash equilibrium concept, and the work of Selten (1975) launched a major research effort on this refinement problem. Papers in this literature have principally analyzed dominated strategies and perturbed games, ideas which Nash (1951, 1953) explored first. Schelling’s (1960) concept of the focal-point effect addressed the crucial question of how to interpret a multiplicity of equilibria in a game. Harsanyi (1973) showed how to realistically interpret the randomized strategies that von Neumann and Nash needed for existence. Powerful methods for analyzing communication 5 The brief reference to Nash’s work in the preface to the 1953 edition of von Neumann and Morgenstern’s book is particularly disappointing.
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in games were introduced by Aumann (1974). All of this work goes beyond Nash, but much of it is in directions in which he pointed.
5. THE IMPACT OF NASH’S GENERAL THEORY So today we model games in normal form, in Bayesian form, and in extensive form. We solve games by computing Nash equilibria, Bayesian equilibria, sequential equilibria, or correlated equilibria. The theory of noncooperative games that Nash founded has developed into a practical calculus of incentives that can help us to better understand the problems of conflict and cooperation in virtually any social, political, or economic institution. But why is it important to have a unifying analytical methodology of such broad scope? Practical research that affects real policy decisions is done by scholars who have long studied the specific institutions in question, not by mathematicians whose principal expertise is in differential topology. Can a general methodology developed by a mathematician really affect applied work? All researchers need a methodology to give a framework to their inquiry and debate. Our methodologies enable us to see connections that may be obscure to the untrained layman. But we also are aware that our expertise is diminished beyond the scope of our methodology, and we learn to stay within its boundaries. Before Nash, price theory was the one broad analytical methodology available to economics. The power of price-theoretic analysis enabled economists to serve as highly valued guides in practical policy making, to a degree that has not been approached by scholars in any other area of social science. But even within the traditional scope of economics, price theory has serious limits. Bargaining situations where individuals have different information do not fit easily into standard price-theoretic terms. The internal organization of a firm is largely beyond the scope of price theory. In the great debates about socialism, price-theoretic models have not been very useful for probing the defects of a nonprice command economy. Institutions for the enforcement of property rights are a crucial factor in the performance of economic markets, but such enforcement is set aside as a primitive assumption in price theory. Price theory prepares economists to advise the government, but not to study the government. Noncooperative game theory has liberated economists from these methodological restrictions, and the scope of applied economic analysis has grown to include all these topics. Game-theoretic models of moral hazard and adverse selection have spawned the new economics of information and organization. Methodological limitations no longer deter us from recognizing the essential interconnections between economic, social, and political institutions in economic development.
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Cournot (1838, Section 5) wrote: From a standpoint of mere etymology, whatever appertains to the organization of society belongs to the field of Political Economy; but it has become customary to use this last term in a sense much more restricted . . . being occupied principally with the material wants of mankind.
Today, the original scope of political economy has been restored to economic theorists, because of the general methodology that John Nash introduced.
REFERENCES Aumann, R. J. (1974). “Subjectivity and Correlation in Randomized Strategies,” J. Math. Econ. 1, 67–96. Bertrand, J. (1883). “Review of Walras’s ‘Theorie mathematique de la richesse sociale,’ and Cournot’s ’Recherches sur les principes mathematiques de la theorie des richesses,’” Journal des Savants 499–508. Translation by J. W. Friedman in Cournot Oligopoly (A. F. Daughety, Ed.), pp. 73–81. Cambridge: Cambridge Univ. Press (1988). Cournot, A. (1838). Recherches sur les Principes Mathematiques de la Theorie de la Richesse. Paris: Hachette. English translation by N. T. Bacon, Researches into the Mathematical Principles of the Theory of Wealth, New York: MacMillan (1927). Fellner, W. (1949). Competition Among the Few. New York: Knopf. Harsanyi, J. C. (1967–1968). “Games with Incomplete Information Played by ‘Bayesian’ Players,” Management Sci. 14, 159–182, 320–334, 486–502. Harsanyi, J. C. (1973). “Games with Randomly Disturbed Payoffs: A New Rationale for Mixed-Strategy Equilibria,” Int. J. Game Theory 2, 1–23. Leonard, R. J. (1994). “Reading Cournot, Reading Nash: The Creation and Stabilisation of the Nash Equilibrium,” Econ. J. 104, 492–511. Milnor, J. (1995). “A Nobel Prize for John Nash,” Math. Intelligencer 17(3), 11–17. Nasar, S. (1994). “The Lost Years of a Nobel Laureate,” New York Times, November 13, Section 3. Nash, J. F. (1950a). “The Bargaining Problem,” Econometrica 18, 155–162. Nash, J. F. (1950b). “Equilibrium Points in n-Person Games,” Proc. Nat. Acad. Sci. U.S.A. 36, 48–49. Nash, J. F. (1951). “Noncooperative Games,” Ann. Math. 54, 289–295. Nash, J. F. (1953). “Two-Person Cooperative Games,” Econometrica 21, 128–140. Nash, J. F. (1994). “Nobel Seminar: The Work of John Nash in Game Theory” (with contributions by H. W. Kuhn, J. C. Harsanyi, R. Selten, J. W. Weibull, E. van Damme, and P. Hammerstein), Les Prix Nobel 1994, 274–310. Rubinstein, A. (1995). “John Nash: The Master of Economic Modeling,” Scand. J. Econ. 97, 9–13. Schelling, T. (1960). Strategy of Conflict. Cambridge, MA: Harvard Univ. Press. Selten, R. (1965). “Spieltheoretische Behandlung eines Oligopolmodells mit Nachfragetragheit,” Z. ges. Staatswiss. 121, 301–329, 667–689. Selten, R. (1975). “Reexamination of the Perfectness Concept for Equilibrium Points in Extensive Games,” Int. J. Game Theory 4, 25–55. van Damme, E., and Weibull, J. W. (1995). “Equilibrium in Strategic Interaction: The Contributions of John C. Harsanyi, John F. Nash, and Reinhard Selten,” Scand. J. Econ. 97, 15–40.
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von Neumann, J. (1928). “Zur Theories der Gesellschaftsspiele,” Math. Ann. 100, 295–320. English translation by S. Bergmann in Contributions to the Theory of Games IV (R. D. Luce and A. W. Tucker, Eds.), pp. 13–42, Princeton, NJ: Princeton Univ. Press (1959). von Neumann, J., and Morgenstern, O. (1944). Theory of Games and Economic Behavior. Princeton, NJ: Princeton Univ. Press (2nd ed., 1947; 3rd ed., 1953).
Roger B. Myerson∗ MEDS Department J. L. Kellogg Graduate School of Management Northwestern University Evanston, Illinois 60208 ∗ Fax:
(847) 467-1220. E-mail: [email protected].
14, 287–295 (1996)
0053
REVIEWS AND COMMENTS
With the intent of stimulating discussion, this section is reserved for book reviews, comments, and letters; your input is welcome. By nature, this material may be subjective, reflecting the opinions of the authors; your responses are therefore encouraged.
John Nash’s Contribution to Economics
1. INTRODUCTION For many years, there have been conferences where John Nash should have been honored as a leader of our field, and his presence has been much missed. I can imagine no higher privilege than to offer words in his honor today.1 The story of Nash’s contribution to economics is one of the most remarkable in the history of the field, beginning with an amazing paper that he started as an undergraduate in his first economics course. In graduate school, concerned that his Nobel-winning work might not be accepted as a doctoral dissertation, Nash also studied the mathematics of manifolds, which led to his solution of a classical embeddability problem. Nash’s mathematical talents could rival the genius of von Neumann like few others in the history of economics. But what I want to emphasize here above all is the sheer importance of Nash’s contribution as a turning point in the history of economics. In a room full of Nobel laureates, there is no one with a greater claim to have launched such a fundamental transformation in the scope of the field. In a series of four short papers published between 1950 and 1953, John Nash laid the foundations for game theory as we now know it. Two of these papers introduced the analysis of bargaining without transferable utility, and alone they 1
This paper was written for presentation at a luncheon in honor of the 1994 Nobel laureates, at the American Economics Association annual meetings in San Fransisco on January 6, 1996. Since 1994, several important articles in appreciation of Nash’s work have been published; see Leonard (1994), Milnor (1995), Nasar (1994), Nash (1994), Rubinstein (1995), and van Damme and Weibull (1995). Like others who have sought to better understand the work of John Nash, I am deeply indebted to Harold Kuhn for his help and advice. 287 0899-8256/96 $18.00 Copyright © 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
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would have made Nash one of the great leaders of game theory. But the other two papers, which develop Nash’s concept of equilibrium for general noncooperative games, must be ranked among the greatest watershed breakthroughs in the history of social science. We should recall others who contributed to the uphill climb to this watershed, and we should observe that the road down from the top was not straight or smooth. But while the publication of von Neumann and Morgenstern’s book attracted much more attention, it was Nash’s major paper on noncooperative equilibrium that truly marked the beginning of a new era in economics. Why is Nash equilibrium so important? Questions of institutional reform will always generate a practical demand for theories of social institutions. Of course, human behavior is wonderfully complex, and we cannot expect any social theory to attain an accuracy like Newton’s theory of planetary motion. Many alternative theories should be explored. But there are good reasons to expect Nash equilibrium analysis to be one of the most valuable methods of applied theory in all areas of economics and social science. In any area of applied social theory, we must begin with a model. To be able to handle normative questions, there must be some concept of human welfare in our model. If individuals are not motivated to maximize their own welfare, then any loss of welfare can be blamed on individual behavior, rather than on the structure of social institutions. So an argument for reform of institutions, rather than re-education of individuals, is most persuasive when it is based on a model that assumes rational maximizing behavior of individuals. Nash equilibrium is the logical formulation of this assumption: that each member of society will act, within his (or her) domain of control, to maximize welfare as he (or she) evaluates it, given the predicted behavior of others. Given that the concept of Nash equilibrium can be so fundamental to the analysis of any institution, it may seem surprising that classical social philosophers left the general formulation of equilibrium analysis to this generation. To unravel this mystery, we should review the history to see why the idea of Nash equilibrium was not so easy to see. Of course, given the fundamental importance of Nash equilibrium, its implicit application can be found in classical social philosophy. Reformulating ideas of Hobbes and Machiavelli into rigorous game-theoretic models can be an interesting and rewarding exercise. But the first clear application of Nash equilibrium in a mathematical model comes in the work of Augustin Cournot.
2. COURNOT AND VON NEUMANN Cournot’s (1838) book on economic theory is dazzling even today. He constructed a theory of oligopolistic firms that includes monopolists and perfect competitors as limiting extremes. He developed game models of oligopolistic competition, which he analyzed by the methodology of Nash equilibrium.
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But of course he was writing more than a century before Nash. Should Cournot get credit for the equilibrium concept? Should we speak rather of “Cournot equilibrium,” or “Cournot–Nash equilibrium”? I want to argue that this terminology would be wrong. We may speak of Cournot as the founder of oligopoly theory, but to give him credit for the fundamental solution concept of noncooperative game theory would be to confuse one application of a methodology with its general formulation. This distinction is one which Cournot would have appreciated. He wrote a short book on mathematical economics, but he wrote at greater length on the philosophy of science and the foundations of our knowledge. If he had recognized that noncooperative game theory can provide a general unifying structure for analyzing all kinds of social institutions, he would have wanted to write about it more than anyone else in his generation. But he did not see it. Cournot did not develop the conceptual distinction between the formulation of his specific game models and the general methodology used to analyze them.2 Indeed, far from finding a general analytical methodology in Cournot, readers from Bertrand (1883) to Fellner (1949) found specific models of oligopoly which had some interesting applied predictions, but which seemed to make some invalid assumptions. In particular, once Cournot has shown that the optimal output of firm 2 depends on the output of firm 1, it may seem irrational for the manager of firm 1 to assume that 2’s output would remain fixed if he changed 1’s output. Until this critique could be answered, Cournot’s methodology did not look like a compelling general theory of rational behavior. The answer came as von Neumann’s first great contribution to game theory. Von Neumann began his 1928 paper on the minimax theorem by formulating a general model of extensive games, in which players move sequentially over time with imperfect information about each others’ previous moves. Because players may get some information about other players’ previous moves, we cannot assume that players’ moves are independent in such extensive games. But von Neumann then defined a strategy for each player to be a complete plan that specifies a move for the player, at each stage where he is active, as a function of his information at that stage. A rational player can choose his strategy before the game begins, with no loss of generality, because a strategy lets him specify a different move for every situation in which he might find himself during the game. But ”before the game begins” means before any consequences of other players’ decisions can be observed. So each player must choose his 2 Cournot first analyzed competition among firms that compete to sell the same consumer good, and then he analyzed a second model of producers of complementary inputs for a manufactured good. In the analysis of the latter model, Cournot wrote “if we apply to the mutual relations of producers the same method of reasoning which served for analyzing the effects of competition. . . ” These words are the closest to an articulation of a general game-theoretic methodology that I have found in Cournot’s book.
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strategy without being informed of the other players’ strategy choices. Thus, von Neumann argued, there is no loss of generality in studying games where players make their strategic decisions independently. This insight is what allows us today to accept Cournot’s basic assumption that competitors make their decisions independently. Perhaps firm 2 can base its production next year on firm 1’s production this year; but that just means that firm 2 has a larger strategy space than Cournot admitted. At the level of strategic planning, we can still assume that firm 2 chooses its strategy independently of firm 1’s strategy choice. Thus, von Neumann argued that virtually any competitive game can be modeled by a mathematical game with the following simple structure: There is a set of players, each player has a set of strategies, each player has a payoff function from the Cartesian product of these strategy sets into the real numbers, and each player must choose his strategy independently of the other players. This structure is von Neumann’s normal form for representing general extensive games. Von Neumann did not consistently apply the principle of strategic independence, however. In his analysis of games with more than two players, von Neumann assumed that players would not simply choose their strategies independently, but would coordinate their strategies in coalitions. Furthermore, by computing max–min values for each coalition, von Neumann implicitly assumed that each coalition must be prepared to have its strategy announced first, allowing the other players to modify their decisions in response. Before Nash, however, no one seems to have noticed that these assumptions were inconsistent with von Neumann’s own argument for strategic independence of the players in the normal form. Von Neumann (1928) also added two restrictions to his normal form that severely limited its claim to be a general model of social interaction for all the social sciences. He assumed that payoff is transferable and that all games are zero-sum. To see why he added these seemingly unnecessary restrictions, we must recall his second great contribution to game theory: the minimax theorem. Von Neumann (1928) recognized that randomized strategies had to be admitted to prove the existence of minimax solutions for two-person zero-sum games. To analyze games with randomization, however, we need a theory of how players make decisions under uncertainty. Von Neumann used the traditional assumption that, when there is uncertainty, each player wants to maximize the expected value of his payoff. But he was uncomfortable with this assumption. In 1928 and again in his 1944 book with Morgenstern, he tried to justify this assumption by identifying all payoffs with monetary transfer payments, which led him to the restriction that payoff is transferable and all games are zero-sum. The fact that the zero-sum restriction also gave him the two-person minimax theorem was probably what committed him intellectually to these restrictions, but the discussion in von Neumann and Morgenstern (1944, Section 2.1.1) suggests that the initial motivation was to defer the problem of measuring utilities.
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In 1947 (in their book’s second edition), von Neumann and Morgenstern published their third great contribution to game theory: the axiomatic derivation of expected-utility maximization from a substitution argument. This new justification for measurable utility should have prompted them to drop their zero-sum transferable-payoff restrictions and to re-examine the foundations of their theory. But they did not. This task was left to John Nash, a young graduate student who saw almost immediately that the whole structure of game theory needed to be broken up and put back together right.
3. NASH’S RECONSTRUCTION OF GAME THEORY Nash’s first great contribution was his 1950 article on bargaining (Nash, 1950a), which began in 1948 with an idea that he had in his undergraduate economics course. This paper was the first work in game theory without transferable utility. By a beautiful axiomatic derivation, Nash introduced a bargaining solution that was virtually unanticipated in the literature.3 Then, on November 16, 1949, the Proceedings of the National Academy of Sciences received from Nash a short note, which was published the next year (1950b). In this two-page note, Nash gave the general definition of equilibrium for normal-form games, and he neatly sketched a fixed-point argument to prove that equilibria in randomized strategies must exist for any finite normal-form game. In his subsequent 1951 paper (which was based on his Princeton doctoral dissertation), Nash gave a fuller development of the idea of equilibrium. This 1951 paper includes versions of some 2×2 games like the Prisoners’ Dilemma, which have filled the basic game-theory literature ever since. But most importantly, Nash (1951) argued that his equilibrium concept, together with von Neumann’s normal form, gives us a complete general methodology for analyzing all games.4 Referring to the other ”cooperative” theories of von Neumann and Morgenstern, Nash wrote: This writer has developed a “dynamical” approach to the study of cooperative games based on reduction to non-cooperative form. One proceeds by constructing a model of the larger pre-play negotiation so that the steps of negotiation become moves in a larger non-cooperative game . . . describing the total situation.
3
Nash’s (1950a) axiomatic bargaining theory builds on the insight that individuals’ utility scales can be defined up to separate increasing linear transformations, but this result follows only from von Neumann and Morgenstern’s 1947 derivation of utility. Thus, Nash’s bargaining solution could not have been appreciated before 1947. It is remarkable that Nash found this solution so quickly thereafter. 4 Nash (1951) noted that the assumption of transferable utility can be dropped without loss of generality, because possibilities for transfer can be put into the moves of the game itself, and he dropped the zero-sum restriction from von Neumann’s normal form.
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Thus Nash applied the normalization argument to show that any other theory of games should be reducible to equilibrium analysis. With this step, Nash carried social science over the watershed into a new world where a unified analytical structure can be found for studying all situations of conflict and cooperation. Von Neumann’s normal form is our general model for all games, and Nash’s equilibrium is our general solution concept. In his 1953 paper, Nash offered an application of his program for reducing cooperative game theory to noncooperative equilibrium analysis. He modeled the two-person bargaining process by a simple game of simultaneous demands. This game has an infinite number of Nash equilibria, but Nash gave an ingenious perturbational argument for selecting a unique stable equilibrium which coincides with the bargaining solution that he previously derived axiomatically.
4. SUBSEQUENT DEVELOPMENT OF NONCOOPERATIVE GAME THEORY The impact of Nash’s reconstruction of game theory spread slowly. At first, more attention was focused on the cooperative analysis that von Neumann favored.5 Later, as more people realized the importance of Nash’s program, it became apparent that there were a number of technical problems that needed further study before noncooperative game theory could meet its promise as a general analytical methodology for applied work. John Harsanyi (1967–1968) questioned the argument for the normal form in situations where players have different information at the beginning of the game, and he developed a more versatile Bayesian-game model for such situations. Further challenging von Neumann’s argument for the normal form, Selten (1965, 1975) argued that we need to study Nash’s equilibria directly in the extensive form, to look for equilibria that satisfy some form of sequentially rationality. The fact that some equilibria might be less stable or less rational than others was noted by Nash (1951, Example 6). To exclude such unstable equilibria, we need some refinement of the Nash equilibrium concept, and the work of Selten (1975) launched a major research effort on this refinement problem. Papers in this literature have principally analyzed dominated strategies and perturbed games, ideas which Nash (1951, 1953) explored first. Schelling’s (1960) concept of the focal-point effect addressed the crucial question of how to interpret a multiplicity of equilibria in a game. Harsanyi (1973) showed how to realistically interpret the randomized strategies that von Neumann and Nash needed for existence. Powerful methods for analyzing communication 5 The brief reference to Nash’s work in the preface to the 1953 edition of von Neumann and Morgenstern’s book is particularly disappointing.
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in games were introduced by Aumann (1974). All of this work goes beyond Nash, but much of it is in directions in which he pointed.
5. THE IMPACT OF NASH’S GENERAL THEORY So today we model games in normal form, in Bayesian form, and in extensive form. We solve games by computing Nash equilibria, Bayesian equilibria, sequential equilibria, or correlated equilibria. The theory of noncooperative games that Nash founded has developed into a practical calculus of incentives that can help us to better understand the problems of conflict and cooperation in virtually any social, political, or economic institution. But why is it important to have a unifying analytical methodology of such broad scope? Practical research that affects real policy decisions is done by scholars who have long studied the specific institutions in question, not by mathematicians whose principal expertise is in differential topology. Can a general methodology developed by a mathematician really affect applied work? All researchers need a methodology to give a framework to their inquiry and debate. Our methodologies enable us to see connections that may be obscure to the untrained layman. But we also are aware that our expertise is diminished beyond the scope of our methodology, and we learn to stay within its boundaries. Before Nash, price theory was the one broad analytical methodology available to economics. The power of price-theoretic analysis enabled economists to serve as highly valued guides in practical policy making, to a degree that has not been approached by scholars in any other area of social science. But even within the traditional scope of economics, price theory has serious limits. Bargaining situations where individuals have different information do not fit easily into standard price-theoretic terms. The internal organization of a firm is largely beyond the scope of price theory. In the great debates about socialism, price-theoretic models have not been very useful for probing the defects of a nonprice command economy. Institutions for the enforcement of property rights are a crucial factor in the performance of economic markets, but such enforcement is set aside as a primitive assumption in price theory. Price theory prepares economists to advise the government, but not to study the government. Noncooperative game theory has liberated economists from these methodological restrictions, and the scope of applied economic analysis has grown to include all these topics. Game-theoretic models of moral hazard and adverse selection have spawned the new economics of information and organization. Methodological limitations no longer deter us from recognizing the essential interconnections between economic, social, and political institutions in economic development.
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Cournot (1838, Section 5) wrote: From a standpoint of mere etymology, whatever appertains to the organization of society belongs to the field of Political Economy; but it has become customary to use this last term in a sense much more restricted . . . being occupied principally with the material wants of mankind.
Today, the original scope of political economy has been restored to economic theorists, because of the general methodology that John Nash introduced.
REFERENCES Aumann, R. J. (1974). “Subjectivity and Correlation in Randomized Strategies,” J. Math. Econ. 1, 67–96. Bertrand, J. (1883). “Review of Walras’s ‘Theorie mathematique de la richesse sociale,’ and Cournot’s ’Recherches sur les principes mathematiques de la theorie des richesses,’” Journal des Savants 499–508. Translation by J. W. Friedman in Cournot Oligopoly (A. F. Daughety, Ed.), pp. 73–81. Cambridge: Cambridge Univ. Press (1988). Cournot, A. (1838). Recherches sur les Principes Mathematiques de la Theorie de la Richesse. Paris: Hachette. English translation by N. T. Bacon, Researches into the Mathematical Principles of the Theory of Wealth, New York: MacMillan (1927). Fellner, W. (1949). Competition Among the Few. New York: Knopf. Harsanyi, J. C. (1967–1968). “Games with Incomplete Information Played by ‘Bayesian’ Players,” Management Sci. 14, 159–182, 320–334, 486–502. Harsanyi, J. C. (1973). “Games with Randomly Disturbed Payoffs: A New Rationale for Mixed-Strategy Equilibria,” Int. J. Game Theory 2, 1–23. Leonard, R. J. (1994). “Reading Cournot, Reading Nash: The Creation and Stabilisation of the Nash Equilibrium,” Econ. J. 104, 492–511. Milnor, J. (1995). “A Nobel Prize for John Nash,” Math. Intelligencer 17(3), 11–17. Nasar, S. (1994). “The Lost Years of a Nobel Laureate,” New York Times, November 13, Section 3. Nash, J. F. (1950a). “The Bargaining Problem,” Econometrica 18, 155–162. Nash, J. F. (1950b). “Equilibrium Points in n-Person Games,” Proc. Nat. Acad. Sci. U.S.A. 36, 48–49. Nash, J. F. (1951). “Noncooperative Games,” Ann. Math. 54, 289–295. Nash, J. F. (1953). “Two-Person Cooperative Games,” Econometrica 21, 128–140. Nash, J. F. (1994). “Nobel Seminar: The Work of John Nash in Game Theory” (with contributions by H. W. Kuhn, J. C. Harsanyi, R. Selten, J. W. Weibull, E. van Damme, and P. Hammerstein), Les Prix Nobel 1994, 274–310. Rubinstein, A. (1995). “John Nash: The Master of Economic Modeling,” Scand. J. Econ. 97, 9–13. Schelling, T. (1960). Strategy of Conflict. Cambridge, MA: Harvard Univ. Press. Selten, R. (1965). “Spieltheoretische Behandlung eines Oligopolmodells mit Nachfragetragheit,” Z. ges. Staatswiss. 121, 301–329, 667–689. Selten, R. (1975). “Reexamination of the Perfectness Concept for Equilibrium Points in Extensive Games,” Int. J. Game Theory 4, 25–55. van Damme, E., and Weibull, J. W. (1995). “Equilibrium in Strategic Interaction: The Contributions of John C. Harsanyi, John F. Nash, and Reinhard Selten,” Scand. J. Econ. 97, 15–40.
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von Neumann, J. (1928). “Zur Theories der Gesellschaftsspiele,” Math. Ann. 100, 295–320. English translation by S. Bergmann in Contributions to the Theory of Games IV (R. D. Luce and A. W. Tucker, Eds.), pp. 13–42, Princeton, NJ: Princeton Univ. Press (1959). von Neumann, J., and Morgenstern, O. (1944). Theory of Games and Economic Behavior. Princeton, NJ: Princeton Univ. Press (2nd ed., 1947; 3rd ed., 1953).
Roger B. Myerson∗ MEDS Department J. L. Kellogg Graduate School of Management Northwestern University Evanston, Illinois 60208 ∗ Fax:
(847) 467-1220. E-mail: [email protected].