J nash contribution to economics


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J nash contribution to economics

  1. 1. GAMES AND ECONOMIC BEHAVIOR 14, 287–295 (1996)ARTICLE NO. 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 havebeen honored as a leader of our field, and his presence has been much missed. Ican 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 remarkablein the history of the field, beginning with an amazing paper that he started as anundergraduate in his first economics course. In graduate school, concerned thathis Nobel-winning work might not be accepted as a doctoral dissertation, Nashalso studied the mathematics of manifolds, which led to his solution of a classicalembeddability problem. Nash’s mathematical talents could rival the genius ofvon Neumann like few others in the history of economics. But what I wantto emphasize here above all is the sheer importance of Nash’s contribution as aturning point in the history of economics. In a room full of Nobel laureates, thereis no one with a greater claim to have launched such a fundamental transformationin the scope of the field. In a series of four short papers published between 1950 and 1953, John Nashlaid the foundations for game theory as we now know it. Two of these papersintroduced 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 theAmerican 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 toHarold 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.
  2. 2. 288 REVIEWS AND COMMENTSwould have made Nash one of the great leaders of game theory. But the other twopapers, which develop Nash’s concept of equilibrium for general noncooperativegames, must be ranked among the greatest watershed breakthroughs in the historyof social science. We should recall others who contributed to the uphill climb tothis watershed, and we should observe that the road down from the top was notstraight or smooth. But while the publication of von Neumann and Morgenstern’sbook attracted much more attention, it was Nash’s major paper on noncooperativeequilibrium that truly marked the beginning of a new era in economics. Why is Nash equilibrium so important? Questions of institutional reformwill always generate a practical demand for theories of social institutions. Ofcourse, human behavior is wonderfully complex, and we cannot expect anysocial theory to attain an accuracy like Newton’s theory of planetary motion.Many alternative theories should be explored. But there are good reasons toexpect Nash equilibrium analysis to be one of the most valuable methods ofapplied theory in all areas of economics and social science. In any area of applied social theory, we must begin with a model. To be ableto handle normative questions, there must be some concept of human welfarein 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 thestructure of social institutions. So an argument for reform of institutions, ratherthan re-education of individuals, is most persuasive when it is based on a modelthat assumes rational maximizing behavior of individuals. Nash equilibrium isthe logical formulation of this assumption: that each member of society willact, 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 anal-ysis of any institution, it may seem surprising that classical social philosophersleft the general formulation of equilibrium analysis to this generation. To un-ravel this mystery, we should review the history to see why the idea of Nashequilibrium was not so easy to see. Of course, given the fundamental importance of Nash equilibrium, its implicitapplication can be found in classical social philosophy. Reformulating ideas ofHobbes and Machiavelli into rigorous game-theoretic models can be an interest-ing and rewarding exercise. But the first clear application of Nash equilibriumin 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 con-structed a theory of oligopolistic firms that includes monopolists and perfectcompetitors as limiting extremes. He developed game models of oligopolisticcompetition, which he analyzed by the methodology of Nash equilibrium.
  3. 3. REVIEWS AND COMMENTS 289 But of course he was writing more than a century before Nash. Should Cournotget credit for the equilibrium concept? Should we speak rather of “Cournot equi-librium,” or “Cournot–Nash equilibrium”? I want to argue that this terminologywould be wrong. We may speak of Cournot as the founder of oligopoly theory,but to give him credit for the fundamental solution concept of noncooperativegame theory would be to confuse one application of a methodology with itsgeneral formulation. This distinction is one which Cournot would have appreciated. He wrote ashort book on mathematical economics, but he wrote at greater length on thephilosophy of science and the foundations of our knowledge. If he had recognizedthat noncooperative game theory can provide a general unifying structure foranalyzing all kinds of social institutions, he would have wanted to write about itmore than anyone else in his generation. But he did not see it. Cournot did notdevelop the conceptual distinction between the formulation of his specific gamemodels and the general methodology used to analyze them.2 Indeed, far from finding a general analytical methodology in Cournot, readersfrom Bertrand (1883) to Fellner (1949) found specific models of oligopoly whichhad some interesting applied predictions, but which seemed to make some invalidassumptions. In particular, once Cournot has shown that the optimal output offirm 2 depends on the output of firm 1, it may seem irrational for the managerof 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 likea 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 formulatinga general model of extensive games, in which players move sequentially overtime with imperfect information about each others’ previous moves. Becauseplayers may get some information about other players’ previous moves, wecannot assume that players’ moves are independent in such extensive games.But von Neumann then defined a strategy for each player to be a completeplan that specifies a move for the player, at each stage where he is active, as afunction of his information at that stage. A rational player can choose his strategybefore the game begins, with no loss of generality, because a strategy lets himspecify a different move for every situation in which he might find himselfduring the game. But ”before the game begins” means before any consequencesof 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, andthen he analyzed a second model of producers of complementary inputs for a manufactured good. Inthe analysis of the latter model, Cournot wrote “if we apply to the mutual relations of producers thesame method of reasoning which served for analyzing the effects of competition. . . ” These words arethe closest to an articulation of a general game-theoretic methodology that I have found in Cournot’sbook.
  4. 4. 290 REVIEWS AND COMMENTSstrategy without being informed of the other players’ strategy choices. Thus,von Neumann argued, there is no loss of generality in studying games whereplayers make their strategic decisions independently. This insight is what allows us today to accept Cournot’s basic assumptionthat competitors make their decisions independently. Perhaps firm 2 can base itsproduction next year on firm 1’s production this year; but that just means thatfirm 2 has a larger strategy space than Cournot admitted. At the level of strategicplanning, we can still assume that firm 2 chooses its strategy independently offirm 1’s strategy choice. Thus, von Neumann argued that virtually any competitive game can be mod-eled by a mathematical game with the following simple structure: There is a setof players, each player has a set of strategies, each player has a payoff functionfrom the Cartesian product of these strategy sets into the real numbers, and eachplayer must choose his strategy independently of the other players. This structureis von Neumann’s normal form for representing general extensive games. Von Neumann did not consistently apply the principle of strategic indepen-dence, however. In his analysis of games with more than two players, vonNeumann assumed that players would not simply choose their strategies in-dependently, but would coordinate their strategies in coalitions. Furthermore,by computing max–min values for each coalition, von Neumann implicitly as-sumed 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 inconsistentwith von Neumann’s own argument for strategic independence of the players inthe normal form. Von Neumann (1928) also added two restrictions to his normal form thatseverely limited its claim to be a general model of social interaction for all thesocial sciences. He assumed that payoff is transferable and that all games arezero-sum. To see why he added these seemingly unnecessary restrictions, wemust recall his second great contribution to game theory: the minimax theorem. Von Neumann (1928) recognized that randomized strategies had to be admittedto prove the existence of minimax solutions for two-person zero-sum games. Toanalyze games with randomization, however, we need a theory of how playersmake decisions under uncertainty. Von Neumann used the traditional assumptionthat, when there is uncertainty, each player wants to maximize the expectedvalue of his payoff. But he was uncomfortable with this assumption. In 1928and again in his 1944 book with Morgenstern, he tried to justify this assumptionby identifying all payoffs with monetary transfer payments, which led him tothe restriction that payoff is transferable and all games are zero-sum. The factthat the zero-sum restriction also gave him the two-person minimax theoremwas probably what committed him intellectually to these restrictions, but thediscussion in von Neumann and Morgenstern (1944, Section 2.1.1) suggests thatthe initial motivation was to defer the problem of measuring utilities.
  5. 5. REVIEWS AND COMMENTS 291 In 1947 (in their book’s second edition), von Neumann and Morgenstern pub-lished their third great contribution to game theory: the axiomatic derivation ofexpected-utility maximization from a substitution argument. This new justifica-tion for measurable utility should have prompted them to drop their zero-sumtransferable-payoff restrictions and to re-examine the foundations of their the-ory. But they did not. This task was left to John Nash, a young graduate studentwho saw almost immediately that the whole structure of game theory needed tobe 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 undergraduateeconomics course. This paper was the first work in game theory without trans-ferable utility. By a beautiful axiomatic derivation, Nash introduced a bargainingsolution that was virtually unanticipated in the literature.3 Then, on November 16, 1949, the Proceedings of the National Academy ofSciences received from Nash a short note, which was published the next year(1950b). In this two-page note, Nash gave the general definition of equilibriumfor normal-form games, and he neatly sketched a fixed-point argument to provethat equilibria in randomized strategies must exist for any finite normal-formgame. In his subsequent 1951 paper (which was based on his Princeton doctoral dis-sertation), Nash gave a fuller development of the idea of equilibrium. This 1951paper includes versions of some 2×2 games like the Prisoners’ Dilemma, whichhave filled the basic game-theory literature ever since. But most importantly,Nash (1951) argued that his equilibrium concept, together with von Neumann’snormal form, gives us a complete general methodology for analyzing all games.4Referring 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 scalescan be defined up to separate increasing linear transformations, but this result follows only from vonNeumann and Morgenstern’s 1947 derivation of utility. Thus, Nash’s bargaining solution could nothave 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 ofgenerality, because possibilities for transfer can be put into the moves of the game itself, and hedropped the zero-sum restriction from von Neumann’s normal form.
  6. 6. 292 REVIEWS AND COMMENTSThus Nash applied the normalization argument to show that any other theory ofgames should be reducible to equilibrium analysis. With this step, Nash carried social science over the watershed into a new worldwhere a unified analytical structure can be found for studying all situations ofconflict and cooperation. Von Neumann’s normal form is our general model forall games, and Nash’s equilibrium is our general solution concept. In his 1953 paper, Nash offered an application of his program for reducingcooperative game theory to noncooperative equilibrium analysis. He modeledthe two-person bargaining process by a simple game of simultaneous demands.This game has an infinite number of Nash equilibria, but Nash gave an inge-nious perturbational argument for selecting a unique stable equilibrium whichcoincides 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 Neumannfavored.5 Later, as more people realized the importance of Nash’s program,it became apparent that there were a number of technical problems that neededfurther study before noncooperative game theory could meet its promise as ageneral analytical methodology for applied work. John Harsanyi (1967–1968) questioned the argument for the normal form insituations 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 extensiveform, to look for equilibria that satisfy some form of sequentially rationality. The fact that some equilibria might be less stable or less rational than otherswas noted by Nash (1951, Example 6). To exclude such unstable equilibria, weneed some refinement of the Nash equilibrium concept, and the work of Selten(1975) launched a major research effort on this refinement problem. Papers in thisliterature 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 ques-tion of how to interpret a multiplicity of equilibria in a game. Harsanyi (1973)showed how to realistically interpret the randomized strategies that von Neumannand 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 Mor-genstern’s book is particularly disappointing.
  7. 7. REVIEWS AND COMMENTS 293in games were introduced by Aumann (1974). All of this work goes beyondNash, 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 extensiveform. We solve games by computing Nash equilibria, Bayesian equilibria, se-quential equilibria, or correlated equilibria. The theory of noncooperative gamesthat Nash founded has developed into a practical calculus of incentives that canhelp us to better understand the problems of conflict and cooperation in virtuallyany social, political, or economic institution. But why is it important to have a unifying analytical methodology of suchbroad scope? Practical research that affects real policy decisions is done byscholars who have long studied the specific institutions in question, not by math-ematicians whose principal expertise is in differential topology. Can a generalmethodology developed by a mathematician really affect applied work? All researchers need a methodology to give a framework to their inquiry anddebate. Our methodologies enable us to see connections that may be obscureto the untrained layman. But we also are aware that our expertise is diminishedbeyond the scope of our methodology, and we learn to stay within its boundaries. Before Nash, price theory was the one broad analytical methodology availableto economics. The power of price-theoretic analysis enabled economists to serveas highly valued guides in practical policy making, to a degree that has not beenapproached by scholars in any other area of social science. But even within the traditional scope of economics, price theory has seriouslimits. Bargaining situations where individuals have different information donot fit easily into standard price-theoretic terms. The internal organization ofa firm is largely beyond the scope of price theory. In the great debates aboutsocialism, price-theoretic models have not been very useful for probing thedefects of a nonprice command economy. Institutions for the enforcement ofproperty 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 thegovernment. Noncooperative game theory has liberated economists from these method-ological restrictions, and the scope of applied economic analysis has grown toinclude all these topics. Game-theoretic models of moral hazard and adverseselection have spawned the new economics of information and organization.Methodological limitations no longer deter us from recognizing the essential in-terconnections between economic, social, and political institutions in economicdevelopment.
  8. 8. 294 REVIEWS AND COMMENTS 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 economictheorists, because of the general methodology that John Nash introduced. REFERENCESAumann, 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.
  9. 9. REVIEWS AND COMMENTS 295von 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 DepartmentJ. L. Kellogg Graduate School of ManagementNorthwestern UniversityEvanston, Illinois 60208 ∗ Fax: (847) 467-1220. E-mail: myerson@casbah.acns.nwu.edu.