John von Neumann’s contribution to economic science

Maria Joao Cardoso De Pina Cabral

Introduction

Often described as a genius, John von Neumann made significant contributions in a wide range of fields. His work in mathematical physics won him praise as the primary intellectual influence responsible for the emergence of game theory, digital computing, and cellular automata. Despite the mathematical nature of his scientific interest, von Neumann also made important contributions to economics. His influence on the study of economics has received universal acclaim from prominent economists. Despite disagreeing with von Neumann on crucial questions, Paul Samuelson, a Nobel Laureate in Economics, observed: “the incomparable Johnny von Neumann. He darted briefly into our domain, and it has never been the same since.” (1) Richard Stone, another Nobel Laureate in Economics, avers that von Neumann and Oskar Morgenstern’s The Theory of Games and Economic Behavior (1944) is ” … the most important textbook since [John Maynard] Keynes’ General Theory.” (2) E. Roy Weintraub, current President of the History of Economics Society, described von Neumann’s “A Model of General Economic Equilibrium” as ” … the greatest paper in mathematical economics that was ever written.” (3) Richard Goodwin, a specialist in the study of economic dynamics, echoed Weintraub’s praise for von Neumann’s work, characterizing “A Model of General Economic Equilibrium,” as ” … one of the great seminal works of the century…. ” (4) This study seeks to validate such admiration for von Neumann’s influence on economic science by analyzing his two major contributions to the study of economics, “The Expanding Economic Model” and The Theory of Games and Economic Behavior.

John von Neumann: The Brilliant Young Mathematician

Janos (5) von Neumann was born on December 28, 1903 into a wealthy Jewish banking family (6) in Budapest, Hungary. He benefited from an elitist education in that ‘booming’ city. (7) At the age of ten, he was recognized a child prodigy. (8) By the age of seventeen, von Neumann enrolled at the University of Budapest to study mathematics. Between 1921 and 1923 he went to Budapest only to take exams while he attended the University of Berlin to take courses in physics, including statistical mechanics taught by Albert Einstein. (9) In Berlin, he associated with other distinguished Hungarian emigres, including Eugene Wigner, Leo Szilard, and Dennis Gabor, all prominent members of the “Hungarian Phenomenon” which later exerted an enormous impact on American physics and mathematics. (10) At the same time, von Neumann met the mathematician David Hilbert, a major influence on much of his work, at Gottingen. (11) He then attended the Swiss Federal Institute of Technology at Zurich, where he received a degree in Chemical Engineering in 1925. One year later, von Neumann earned a Ph.D. in mathematics with the highest honors from the University of Budapest. (12) After teaching mathematics at Princeton University for a semester in 1930, von Neumann moved permanently to the United States in 1933. He taught near Princeton, as a Professor of Mathematics at the newly formed Institute for Advanced Study (IAS) created by Abraham Flexner alongside such notable scientists and mathematicians as Einstein, Enrico Fermi, Hermann Weyl, and Wigner. (13)

Von Neumann proved to be a brilliant young mathematician. Between 1922 and 1927, he produced eighteen major mathematical articles. Much of his early research can best be described as a response to Hilbert’s mathematical program. (14) Sometimes called the “formalist” school of mathematics, (15) this approach was concerned with both the foundations of mathematics and axiomatization of mathematical physics. The attempt to establish various areas of mathematics on a secure axiomatic basis stood at the core of Hilbert’s program, and von Neumann’s projects were included in it. (16) Both men were interested in demonstrating how mathematics could become a widely applicable tool, even in fields that were, until then, not easy to formalize mathematically. (17) In explaining von Neumann’s contributions to economics in this manner, his work reflects a belief in the relevant role mathematics could play in science and society rather than a genuine interest in economic issues.

Von Neumann’s Contributions to Economics

Von Neumann made two major contributions to economic science. The first, the General Economic Equilibrium Model, is often referred to as von Neumann’s ‘Expanding Economic Model’ (EEM). (18) The second, The Theory of Games and Economic Behavior (TGEB), von Neumann co-authored with Oskar Morgenstern. In both works, von Neumann applied mathematical theory to economics. (19) He first expressed an interest in studying economics during the 1920s to Nicholas Kaldor and at a seminar in Berlin. His first written works on theoretical economics appeared in the early 1930s. (20) In 1932, while traveling regularly between Berlin and America, von Neumann presented an economic model at a Princeton mathematics seminar. The text, delivered in German, was originally entitled “On Certain Equations of Economics and A Generalization of Brower’s Fixed-Point Theorem (translation).” The paper would be published in 1937 as part of the proceedings of Karl Menger’s (21) Vienna Colloquium. (22) Two years later, von Neumann sent a copy of his paper to Kaldor, then chair of the editorial committee of the Review of Economic Studies, who believed that it deserved a wider audience. After a delay caused by World War II, von Neumann’s work was published in October 1945, entitled “A Model of General Economic Equilibrium.” (23)

Von Neumann’s growth model works with n goods and m production processes with constant returns to scale. The real wage rate corresponds to the necessities of life and all income in excess is reinvested. The real wage rate is given and profits have a residual nature. In this context, von Neumann determines how the process operates, rates of interest and prices, and the growth rate of the economic system. (24) Accordingly, the growth rate is determined endogenously. In addition to proving that a general equilibrium solution was possible, von Neumann’s main achievement was the resulting harmony between the model assumptions and the different aspects of the solution. Specifically, he demonstrated that the rate of interest is the same throughout the economy and all outputs expand at the same rate. (25)

One of the novelties of von Neumann’s model was removing the distinction between primary factors and outputs. This means that there are no ‘original’ factors as labor in traditional theory. ‘Labor’ is now a factor of production because workers need to consume commodities to produce other commodities. In essence, then, the model was particularly concerned with the circular aspects of the production process. (26)

The use of mathematical tools proved crucial in achieving the equilibrium solution. In fact, von Neumann’s use of Brower’s fixed theorem point helped him prove the existence of a dynamic equilibrium rate of growth. The resolution of the economic problem is made in such a way that all goods are produced at the lowest possible cost in the highest possible quantity. (27) According to von Neumann’s model, maximum growth implied the existence of a dynamic equilibrium, namely, the existence of a saddle point of a function relating the input and output matrices. (28)

Initially, von Neumann’s EEM produced much bewilderment among economists. A stranger to their profession had achieved something very important in the study of economics. Not surprisingly, several objections were raised, some the result of misunderstandings. One criticism leveled against von Neumann’s findings charged that he was advocating a slave economy. Other critics questioned von Neumann’s assumption that activities ended when they do not earn the market profit rate. Apart from these misinterpretations, the most serious limitation of the EEM was its non-monetary character. In particular, it is not possible to determine the effects of central bankers’ actions in over-expanding or over-restricting money. (29)

These criticisms notwithstanding, von Neumann’s EEM had an enormous impact on economic science. First, it enlarged economists’ mathematical toolbox with many complicated new instruments such as convex set theory and mathematical programming. Second, it would allow for a better future understanding of some of the differences between economic planning and free-market effects. Third, it helped in the development of dynamic models of economic growth. In so doing, von Neumann’s EEM would influence the work of at least six Nobel Laureates in Economics: Kenneth Arrow, Gerard Debreu, Paul Samuelson, T. C. Koopmans, A. Kantarovich, and Robert Solow. (30)

While the development of von Neumann’s model is known, its antecedents remain a source of debate. Von Neumann’s paper failed to acknowledge the related work of Karl Schlesinger or Abraham Wald. Moreover, it appears that Gustav Cassel’s Theoretische Sozialokonomie (Theory of Social Economy) (1918), better known as the Walras-Cassel model, (31) served as the starting point for von Neumann’s analysis. There also appears to be a link between von Neumann’s work and that of Viennese economists in the 1930s. First, the story of the paper itself suggests this affiliation. Von Neumann’s paper was naturally viewed as part of the general equilibrium group associated with the Menger Colloquium. Since colloquium members dealt with the problem of the existence of an equilibrium solution to the ‘Walras-Cassel model’, von Neumann was, plausibly, concerned with the same issue. Second, von Neumann’s EEM shares several important features emphasized in the work of the Viennese economists: the use of inequalities rather than equations; the complementary slackness conditions of free disposal and zero price for goods in excess supply; and, an emphasis on long-run equilibrium without profits. Furthermore, some of the mathematical tools generalized by von Neumann in his growth model became essential in neoclassical economics. This is particularly true for Brouwer’s fixed theorem point. Hence, a ‘family resemblance’ between the analyses appears evident (32)

This ‘affiliation’ of von Neumann’s growth model has been the dominant view in economic literature. Heinz D. Kurz and Neri Salvadori argue that von Neumann’s model emerged from the classical tradition of economic thought. (33) This, they claim, is evident from von Neumann’s reliance on the concept of ‘the circular nature of the production process’; (34) the notion of a uniformly expanding economy in which the rate of expansion is endogenously determined; (35) the duality of the monetary variables and the technical variables; (36) and, the way in which the rule of free goods was applied to primary factors of production and to products. (37) Kurz and Salvadori add that von Neumann’s paper is related to Robert Remark, a colleague at the Berlin Institute of Mathematics who studied the problem of the conditions under which positive solutions of systems of linear equations are obtained. According to Kurz and Salvadori, von Neumann was particularly concerned with refuting Remark’s assertion that the ‘normal price mechanism’ in a capitalist economy is inefficient. (38)

Although the ‘classical interpretation’ of von Neumann’s growth model has represented a dissident perspective from the dominant ‘neoclassical’ view, arguments in its favor emerged quickly. The first to point out the classical features of the model was David Gawen Champernowne, a Keynesian economist, whose writing on the subject appeared in the same review that published the English version of von Neumann’s paper. (39) Since then, Nicholas Kaldor, Michio Morishima, and Richard Goodwin, to name a few economists, have stressed the ‘classical nature’ of some features of von Neumann’s model. (40)

The Theory of Games and Economic Behavior (TGEB)

The source of von Neumann’s interest in games remains unknown. It appears that his interest in the relationship between set theory and parlor games led him to develop the minimax theorem. (41) This theorem, first presented in a paper delivered to the Gottingen Mathematical Society in December 1926, was overshadowed by von Neumann’s contemporaneous work. The paper was published two years later in Mathematische Annalen (Mathematical Annals). (42)

The minimax theorem, the first mathematical theorem of game theory, was demonstrated by von Neumann independent of any economic considerations. (43) Von Neumann wrote the paper for an audience of mathematicians with the hope of rendering abstract games amenable to mathematical treatment. The main focus of the paper is a long and difficult existence proof based on functional calculus and topology for all two-person, zero-sum games with a finite number of strategies. (44) Here, von Neumann proved that it is possible to work out the best strategy that would maximize potential gains or minimize potential losses. (45)

During the 1930s, von Neumann continued to show an occasional interest in the mathematics of games (46) and knew that the minimax theorem was relevant to economic theory as noted in his EEM paper. The significance of the first general proof of the minimax theorem shows that the mathematical origin of game theory is quite separate from economic thought. A little more than a decade later, game theory re-emerged in association with economy as a privileged field of interpretation in The Theory of Games and Economic Behavior (1944) co-authored by von Neumann and Oskar Morgenstern.

Collaboration between Von Neumann and Morgenstern

Born in Silesia, Germany, in 1902, (47) Morgenstern obtained his Ph.D. at the University of Vienna in 1925. He began teaching economics at the same university in 1928, and three years later succeeded Friedrich Hayek as director of Vienna’s Institute for Business Cycle Research. In 1938, when Hitler’s Germany annexed Austria (Anschluss), Morgenstern, then visiting the United States, decided to remain in America (48) and became a permanent faculty member at Princeton. (49)

In the early stages of his career, Morgenstern was a typical fourth generation Austrian economist. (50) His work focused on business cycles and methodological critique. He was particularly interested in such issues as the relationship of time and foresight to general equilibrium theory. Morgenstern’s interest in logic led to a presentation of the Holmes/Moriarty problem (51) at Karl Menger’s Mathematical Colloquium. There, Eduard Cech, a mathematician best known for his work on typology and the development of functional analysis, told Morgenstern that von Neumann had already formalized the issue in 1928. (52)

According to Morgenstern, he first discussed games and experiments with von Neumann on February 1, 1939 at the Nassau Club. (53) These conversations later included Einstein, Neils Bohr, and Weyl. After April 1940, their relationship intensified. The two met for several long discussions and soon became very close friends.

In the fall of 1940, von Neumann wrote his first paper on games, “Theory of Games I, General Foundations,” quickly followed by a second paper, delivered in January 1941, entitled, “Theory of Games II, Decomposition Theory.” (54) In both papers, von Neuman tried to synthesize his work on game theory. (55) Although these working papers do not list an author, the mathematical style and the fact that Morgenstern placed the papers in a folder which included other works written by von Neumann indicates that he was the author. Then, on May 17, 1941, von Neumann asked Morgenstern to write a paper on ‘maxims of behavior,’ a frequent topic of interest throughout their discussions. Although never published, this paper, entitled “Quantitative Implications of Maxims of Behavior” apparently had a direct impact on von Neumann’s decision to collaborate with Morgenstern. Initially this collaboration was expected to produce a fifty-page manuscript for submission to the Journal of Political Economy. It would later be expanded into a pamphlet (approximately 100 pages), then a short book, and finally a book that exceeded 600 pages. (56)

The issue of von Neumann-Morgenstern’s collaboration has been a subject of much discussion. The fact that Morgenstern wrote about this collaboration in his diary helps in determining the contribution of each author. In the first place, the asymmetry between von Neumann and Morgenstern in their collaboration is obvious. (57) Their names are not listed in alphabetical order on the cover of the book. (58) Additionally, the preface states that “the theory had been developed by one of them since 1928.” (59) Yet, while it is quite clear that von Neumann was indispensable for the progress of the work, (60) Morgenstern’s contribution proved crucial. He brought to the theory of games the other stream of work recognized, in retrospect, as analysis of games, developed by Antoine August Cournot’s economic contribution on duopoly, and the work of Eugene Bohm-Bawerk. (61)

The Theory of Games and Economic Behavior (TGEB) represents the emergence of game theory as a distinct recognized discipline. Although economics was a central concern for the authors, the scope of the book extended far beyond economics, reaching political science and sociology. In the first chapter, which provides the economic context for game theory, the authors advanced axiomatization of the measurable utility, one of the most important contributions of the book to the study of economics. Using Bernoulli’s analysis (62) as their starting point, they established a system of axioms for a numerical utility and achieved the ‘von Neumann-Morgenstern utility function.’ This represented an essential advance in general demand theory, particularly under risk and uncertainty situations. (63)

Another major contribution of TGEB is the concept of static economic equilibrium. Although the application of this concept is model-dependent, it does not require any particular ‘rules of the game.’ Thus, von Neumann and Morgenstern’s equilibrium solution, contrary to previous treatments such as the general competitive equilibrium of Marie-Espirit Leon Walrus, Vilfredo Pareto, and Irving Fisher, does not depend on perfect competition, or, even, on market contexts, which limited interaction. (64)

Von Neumann and Morgenstern’s solution depends on the concept of ‘dominance.’ This means that players rule out strategies that will definitely be disadvantageous to them. The application of the concept of ‘dominance’ depends on the players’ objectives and the ‘rules of the game’ played. This definition of solution applies to problems of individual optimization, cooperative games, and games of politics. (65) In this context, a solution is not a single imputation but a set of imputations, each set being stable in that none of its member imputations dominates any other, and every imputation outside the set is dominated by at least one imputation inside. (66)

Von Neumann and Morgenstern stated explicitly that their concept of solution was neither optimum nor, in general, exclusive. It has been observed that, unlike many current game theorists, von Neumann and Morgenstern were attracted rather than disturbed by a multiplicity of equilibrium. (67) The theory does not predict which solution will be observed or which imputation within any solution will be obtained. (68) A solution may be correlated with a standard of behavior, customs, or institutions governing social organization at a particular time. Hence, unlike other approaches, there is a vast range of indeterminism. Von Neumann and Morgenstern were conscious of this break from conventional economics. The theory of games was intended to represent a departure from the Hicks-Samuelson variant of neoclassical economics. The main concern was no longer the ordinary maximum or minimum problems, but rather conceptually different situations.

Perhaps the most influential contribution of TGEB to economic science is its systematization of game theory as a branch of choice theory. The first step in the development of the theory of games involved the construction of a formal, mathematical description of a game. Von Neumann and Morgenstern were the first to describe games as a class, delimit the information structure of a game, draw a game tree, and define a solution to a game. Thus, whereas earlier authors (i.e., Cournot) had analyzed problems that would later become identified as part of game theory, von Neumann and Morgenstern established game theory as a distinct and autonomous field. (69)

TGEB also reflects the earlier influence of Hilbert’s mathematical program on von Neumann. Like those developments in quantum physics that von Neumann focused on during the late 1920s, the correspondence between the theory and the world in the theory of games had been abandoned. Thus, it was no longer just a theory of economic interdependence decision; it became part of a general shift in science to ‘formalism.’ In short, it represented the rejection of determinism, continuity, and calculus in favor of indeterminism, probability, and discontinuous changes of state.

Despite Morgenstern’s contribution, TGEB is primarily viewed as a study of formal reasoning provided by von Neumann. Apart from a few young mathematicians and “mathematically-inclined” economists, most economists relied on review articles by Leonid Hurwicz, Herbert A. Simon, Richard Stone, and Abraham Wald to understand TGEB. (70) In general, these review articles expressed enthusiasm for and familiarized most economists with game theory concepts such as pure and mixed strategies, randomization, solution to a game, and the minimax theorem. Still, economists balked at accepting von Neumann and Morgenstern’s work because of their aversion to mathematics and failure to read a long and technical book. This response to TGEB by mainstream economists also represented a negative reaction to von Neumann and Morgenstern’s critical view of eminent works of more conventional economic theory (i.e., John R. Hicks Value and Capital (1939)).

The main response to TGEB came from the academic community of applied mathematicians at Princeton which proved especially receptive to the importance of statistical decision theory in von Neumann and Morgenstern’s work. This interest in game theory at Princeton’s mathematics department was shared by strategists at the RAND Corporation and Office of Naval Research but did not reach the economics profession in the late 1940s or 1950s. (71) Still, the mathematical work in this study became the starting point for a generation of economists who later applied game theory to various fields such as industrial organization, microeconomics theory, macroeconomics policy coordination, and international trade negotiations.

Although economists initially neglected TGEB, in the long run it exerted an enormous influence on the discipline as is evident in the Nobel lectures delivered by Arrow and Debreu. As these economists have noted, TGEB freed economics from the limits of differential calculus. The influences of the new axiomatization of economic decision were spread among the different areas of economics. Thus, von Neumann and Morgenstern’s game theory pushed the study of economics into new directions such as the linking of the core and competitive general equilibrium in the late 1950s and 1960s, as well as Reinhard Selten’s important sequence of refinements of the Nash equilibrium for analyzing dynamic strategic interaction. TGEB is also responsible for the increased emphasis on models of learning and constrained rationality as well as laboratory experimentation. (72) More recently, von Neumann and Morgenstern’s study influenced Robert E. Lucas, Nancy L. Stokely, and Edward C. Prescott’s Recursive Methods in Economic Dynamics (1989), which, like TGEB, relies on topology and mathematical theory. (73)

The influence exerted by von Neumann and Morgenstern’s book has not been limited to economic science. Actually, the new field of game theory that emerged with the publication of TGEB impacted not only economics but science in general. This is due to the universal and versatile nature of the concepts involved. The intellectual interest in the “game” comes from different areas of knowledge besides economics, notably psychology, strategy, operational investigation, and politics. Moreover, the “game” involves strategic behavior and interaction among decisions, common problems studied in the social sciences. Thus, beyond traditional war problems, this basic structure of rational behavior can be identified everywhere in social relationships. It is thus not surprising that game theory influenced various social sciences. Recently, the utility of game theory was extended to the study of phenomena where interaction exists but whose nature is not rational. Such areas of interest include disease propagation, ecosystems evolution, and meteorological movements. (74) It thus appears that the influence of von Neumann and Morgenstern’s work on science is not yet complete.

Conclusion

Von Neumann was a brilliant mathematician whose contributions to other sciences stem from his belief that impartial rules could be found behind human interaction. Accordingly, his work proved crucial in converting mathematics into a key tool to social theory. This was particularly evident in the study of economics. After von Neumann, economics remained intrinsically linked with mathematics. Moreover, this connection produced a new mathematical theory that has provided a better understanding of the nature of economics. Consequently, von Neumann’s work implied a “profound and extensive transformation of economic theory.” (75)

Von Neumann’s work also had a major impact on society at large. After 1939, he became the ‘fundamental force’ in the deliberate organization of science to war in the United States. He provided extensive military consulting and participated in projects whose results would become crucial in ending the war. During his involvement in the Manhattan Project at Los Alamos, New Mexico, for example, he studied the direction of blast waves in the atomic bomb design. After the war, von Neumann’s work on game theory emerged as a central influence upon the application of scientific methods to conflict. Indeed, President Harry S. Truman recognized this by appointing von Neumann to the United States Atomic Energy Commission.

Von Neumann’s contribution to economics represents a minor part of his legacy. Aside from economics and game theory, he pursued a variety of interests, ranging from work on mathematical physics to foundational research on digital computing and cellular automata. Near the end of his life, von Neumann worked not only on computer design but was deeply concerned with a theory of automata. At the time of his death (February 8, 1957), he was regarded as the “main brain” behind the modern digital computer and numerical meteorology.

ENDNOTES

(1) Quoted in Norman Macrae, John von Neumann: The Scientific Genius Who Pioneered the Modern Computer, Game Theory, Nuclear Deterrence, and Much More (New York: Pantheon Books, 1999), 266.

(2) Ibid., 247.

(3) Ibid., 248.

(4) Ibid., 249.

(5) The equivalent in Hungarian for John.

(6) The family name Neumann became von Neumann when John’s father, Max, was ennobled by Emperor Franz Joseph in 1913 “for his contribution to the economic development of Hungary.” See Robert J. Leonard, “Creating a Context for Game Theory,” History of Political Economy 24(1992):39.

(7) This environment produced a distinguished generation. Five of the six Hungarian Nobel Prize winners were Jews born between 1875 and 1905. Macrae, John von Neumann, 32.

(8) By the time he left secondary school, von Neumann won the nationwide ‘Eotvos Prize’ for ‘excellence in mathematics and scientific reasoning.’ Leonard, “Creating a Context for Game Theory,” 40.

(9) Robert J. Leonard, “From Parlor Games to Social Science: von Neumann, Morgenstern and the Creation of Game Theory, 1928-1944,” Journal of Economic Literature 33 (June 1995):732.

(10) Leonard, “Creating a Context for Game Theory,” 40.

(11) At Gottingen, von Neumann worked closely with Hilbert, who became a major influence on von Neumann’s work. Actually, most of von Neumann’s projects are best understood as responses to Hilbert’s program to axiomatize all of mathematics. See Urs Rellsatb, “New Insights into the Collaboration between John von Neumann and Oskar Morgenstern,” in John von Neumann and Modern Economics, eds. Mohammed Dore, Sukhamoy Chakravarty, and Richard Goodwin (Oxford: Clarendon Press, 1989), 117.

(12) Robert W. Dimand and Mary Ann Dimand, The History of Game Theory, vol. 1, From the Beginning to 1945 (London: Routledge, 1996), 129.

(13) Von Neumann was the youngest member of the faculty at IAS.

(14) Actually, von Neumann’s earlier work focused on topics of interest to Hilbert such as set theory, operator theory, and the mathematical foundations of quantum mechanics.

(15) Philip Mirowsky, “What were von Neumann and Morgenstern Trying to Accomplish?” History of Political Economy 24 (1992):117.

(16) Ibid.

(17) Leonard, “From Parlor Games to Social Science,” 732.

(18) Macrae, John von Neumann, 253.

(19) Christian Schmidt, “Game Theory and Economics: An Historical Survey,” Revue d’Economie Politique 100 (September-October 1990):605.

(20) Leonard, “Creating a Context for Game Theory,” 50.

(21) Menger, a professor of mathematics and the son of the famous economist, was one of the three founders of the neoclassical school.

(22) It seems likely that von Neumann just sent the text to the colloquium’s organizer. Macrae, John von Neumann, 248.

(23) John von Neumann, “A Model of General Economic Equalibrium,” Review of Economic Studies XIII (October 1945):1-9.

(24) Heinz D. Kurz and Neri Salvadori, “Von Neumann Growth Model and the Classical Tradition,” European Journal of the History of Economic Thought 1 (Autumn 1993):134.

(25) Nicholas Kaldor, “John von Neumann: A Personal Recollection,” in John von Neumann and Modern Economics, eds. Dore, Chakravarty, and Goodwin, 303.

(26) Ibid.

(27) Macrae, John von Neumann, 253. Fixed-point theorems are useful for establishing the existence of solutions to a system of non-linear equations. In economics, fixed point theorems, such as the Brower Fixed Theorem, are often used to guarantee the existence of equilibrium in a wide variety of models of the economy. See Jerry Green and Walter P. Heller, “Mathematical Analysis and Convexity with Applications to Economics,” in Handbook of Mathematical Economics, vol. 1, eds. Kenneth Arrow and Michael Intrilligator (New York: North-Holland, 1981):49-50.

(28) Leonard, “Creating a Context for Game Theory,” 50.

(29) Macrae, John von Neumann, 253-55.

(30) Ibid., 248.

(31) Leonard, “Creating a Context for Game Theory,” 50. Walras’ theory refers to the general equalibrium theory developed by the Frenchman Leon Walrus in his Elements of Pure Economics (1874). In the “Walrasian” approach, the fundamental tool of analysis was a system of simultaneous market demand and supply equations. In this context, Walrus’ theory was concerned with grand themes such as the existence of an equilibrium solution, the stability of that equalibrium, the incorporation of capital and growth, and the introduction of money.

(32) Leonard, “From Parlor Games to Social Science,” 736.

(33) Kurz and Salvadori, “Von Neumann Growth Model and the Classical Tradition,” 138.

(34) Ibid.

(35) Ibid., 151.

(36) Ibid., 140.

(37) In support of their position, Kurz and Salvadori argue that there are difficulties in conciliating neoclassical analysis with von Neumann’s growth model. They stress that in von Neumann’s model: The initial endowment of the economy is not present; the growth rate is determined endogenuously; the direct marginalist connection between prices and quantities is not assumed in contrast to Walras’ formula; there is an asymmetry in the theory of distribution that is characteristic of classical analysis; and, it takes into account both circulating and fixed capital. Ibid., 135-37.

(38) Ibid., 151.

(39) It was Kaldor who asked David Champernowne to write an explanatory paper to appear alongside in the Review of Economic Studies. Kaldor, “John von Neumann,” in Dore, Chakravarty, and Goodwin, eds., 305.

(40) Kurz and Salvadori referred to Nichaolas Kaldor, “Capital Accumulation and Economic Growth,” in The Theory of Capital, eds., Freidrich August Lutz and Douglas Hague (London: Macmillan, 1961); Michio Morishima, Marx’s Economics: A Dual Theory of Value and Growth (New York: Cambridge University Press, 1973); and, Richard Goodwin, “Swinging Along the Turnpike with von Neumann and Sraffa,” Cambridge Journal of Economics 10 (September 1986):203-10.

(41) Leonard, “From Parlor Games to Social Science,” 732.

(42) Dimand and Dimand, The History of Game Theory, vol. I, From the Beginning to 1945, 129.

(43) Schmidt, “Game Theory and Economics,”591.

(44) Leonard, “From Parlor Games to Social Science,” 734.

(45) Macrae, John von Neumann, 256.

(46) In 1939, he mentioned having unpublished material on playing poker. See Leonard, “Creating a Context for Game Theory,”50.

(47) When he was twelve, he moved to Vienna with his family.

(48) He was dismissed by the Nazis.

(49) Until his retirement in 1970, when he moved to New York University. He remained there until his death in 1976.

(50) Mirowski, “What Were von Neumann and Morgenstern Trying to Accomplish?” 128.

(51) The Holmes-Moriarty game, which is included in TGEB, can be described as follows: Sherlock Holmes, the famous London detective, is being pursued by his nemesis, the evil genius, James Moriarty. Holmes leaves London’s Victoria Station on a train to Dover. The train has one stopover at Canterbury. Holmes saw Moriarty at the train station intending to leave on the same train. From Moriarty’s behavior, Holmes is convinced that Moriarty knows that he will be on the train to Dover. Once aboard the train, each of the protagonists has two alternatives: to get off at Canterbury or to continue to Dover. If they choose different stations, Holmes wins since he gets away with his life, otherwise Moriarty wins. The problem is how should Holmes play the game? The solution to the Holmes-Moriarty game is an infinite regress of prediction, reaction, and revised prediction from which there is no escape (you know that I know that you know that I know …).

(52) Morgenstern noted that one of the principle reasons he chose Princeton was the possibility of being acquainted with von Neumann. See Oskar Morgenstern, “The Collaboration Between Oskar Morgenstern and John von Neumann on the Theory of Games,” Journal of Economic Literature 14 (1976):807.

(53) According to Morgenstern, he and von Neumann could not remember the date when they first met. See Ibid.

(54) Ibid., 803.

(55) Leonard, “Creating a Context for Game Theory,” 54.

(56) Dimand and Dimand, The History of Game Theory, vol. I, From the Beginning to 1945, 146.

(57) This fact was not hidden, and Morgenstern himself wrote that he was mainly just a spectator to the reactions that occurred in von Neumann’s mind. See Rellstab, “New Insights into the Collaboration Between John von Neumann and Oskar Morgenstern,”78.

(58) Morgenstern wrote,”… Johnny wanted to have our names listed alphabetically, I absolutely refused to entertain this proposal, and after some struggle he gave in.” Morgenstern, “The Collaboration of Oskar Morgenstern and John von Neumann on the Theory of Games,” 813.

(59) Quoted in Rellstab, “New Insights Into the Collaboration Between John von Neumann and Oskar Morgenstern,” 77.

(60) In his diaries, Morgenstern remarks that several times von Neumann wrote as much as seventeen pages overnight and calculated examples. Quoted in Ibid., 78.

(61) Eugene von Bohm was one of the leading members of the first generation of the Austrian School of economics. His work on free markets influenced subsequent generations of economists, including Morgenstern. His work was cited five times in von Neumann and Morgenstern’s The Theory of Games and Economic Behavior (Princeton, NJ: Princeton University Press, 1944), more often than anyone else except the mathematician George David Birkhoff. See Dimand and Dimand, The History of Game Theory, vol. I, From the Beginning to 1945, 144.

(62) Bernoulli’s analysis, which von Neumann and Morgenstern picked up, refers to the expected utility hypothesis. This stems from Daniel Bernoulli’s (1738) solution to the famous St. Petersburg paradox posed in 1713 by his cousin, Nicholas Bernoulli. The paradox challenges the old idea that people value random ventures according to its expected return. The paradox posed the following situation: “a fair coin will be tossed until a head appears; if the first head appears on the nth toss, then the payoff is 2n ducats. How much should one pay to play this game? The paradox is that the expected return is infinite.”

(63) Joao Cesar das Neves, “Cinquentenario de ‘Theory of Games and Economic Behavior,” Economia 18 (1994): 159.

(64) Dimand and Dimand, The History of Game Theory, vol. I, From the Beginning to 1945, 148.

(65) Ibid.

(66) Leonard, “From Parlor Games to Social Science,” 755.

(67) Dimand and Dimand, The History of Game Theory, vol. I, From the Beginning to 1945, 149.

(68) Leonard, “From Parlor Games to Social Science,” 755.

(69) There is a divergent opinion concerning The Theory of Games and Economic Behavior. According to this view, the origins of game theory can be found in the selected work of Antoine August Cournot, Recherches sue les Principes Mathematiques de la Richesse (Paris: Calmann Levy, 1838), Francis Ysidro Edgeworth, Mathematical Physics (New York: A.M. Kelley Publishers, 1881), and Frederik Zeuthen, Problems of Monopoly and Economic Warfare (London: Routledge, 1930), not in von Neumann and Morgenstern’s book.

(70) Leonid Hurwicz, “Theory of Economic Behavior,” American Economic Review 35:5 (December 1945):909-23; Herbert A. Simon, “Review of The Theory of Games and Economic Behavior by John von Neumann and Oskar Morgenstern,” American Journal of Sociology 50 (May 1945):558-60; Richard Stone, “The Theory of Games,” Economic Journal 58 (1948):185-201; Abraham Ward, “Review of The Theory of Games and Economic Behavior by John von Neumann and Oskar Morgenstern,” Economic Statistics 39 (1947):47-52.

(71) Dimand and Dimand, The History of Game Theory, vol. I, From the Beginning to 1945, 155.

(72) Leonard, “From Parlor Games to Social Science,” 757.

(73) Robert E. Lucas, Nancy L. Stokely, and Edward Prescott, Recursive Methods in Economic Dynamics (Cambridge, MA: Harvard University Press, 1989), 588.

(74) Cesar das Neves, “Cinquentenario de ‘Theory of Games and Economic Behavior,'” 162.

(75) Gerard Debreu, “Mathematical Theory in the Economic Mode,” Nobel Memorial Lecture, December 8, 1983.

MARIA JOAO CARDOSO DE PINA CABRAL is an Associate Professor in the Economics Department at Instituto Superior de Contabiliade e Administracao de Coimbra in Portugal.

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