John von Neumann (1903-1957)

Two Contributions?

The impact of John von Neumann’s “model of general economic equilibrium” could be evaluated by the number and names of laureates of the Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel who refer to this work: Kenneth Arrow, Gerard Debreu, Paul Samuelson, Tjalling Koopmans, Leonid Kantarovich, and Robert Solow.

Weintraub (1983: 13) claims that von Neumann’s “A model of general economic equilibrium” (GE) is “the single most important article in mathematical economics” while Baumol (1972) labels GE “the most remarkable virtuoso performance” in this field. Indeed, GE is a virtuoso performance, but one can easily get advanced degrees in economics without having heard of this paper. This is not to say that advanced students of economics do not study both general equilibrium theory and growth theory, but they might never learn that the mathematical treatment of both has one of its roots in von Neumann’s GE. Von Neumann presented this paper at the Mathematical Seminar of Princeton University in 1932 and in Karl Menger’s Mathematical Colloquium at the University of Vienna in 1936. It was published in German in 1937 and in English in 1945, see von Neumann (1937 [1945]).

This is quite different to the second area of economics that von Neumann inspired with his work, that is, game theory. Of course, we cannot answer the question of how game theory would look today, or whether there would be any game theory at all, if von Neumann had never presented the minimax theorem in Hilbert’s seminar at the University of Gottingen in 1926, and published it in his article “Zur Theorie der Gesellschaftsspiele” in 1928 (published as “On the theory of games of strategy” in 1959). In the 1920s, Emile Borel also analysed games “in which the winnings depend simultaneously on chance and the skill of the player” (Borel quoted by Rives 1975: 559), but he did present a convincing solution concept like the minimax theorem.

(See Leonard 2010: 57ff.) The collaboration between von Neumann and Oskar Morgenstern that resulted in the volume Theory of Games and Economic Behavior (TGEB) in 1944 was a consequence of von Neumann’s 1928 paper.

The proof of the minimax theorem in the 1928 paper inspired the proof of the existence of an “expanding general equilibrium” of the GE model. In this context, von Neumann used game theory as a mathematical technique, “as a calculus” (Morgenstern 1976: 810). Do we have to conclude that von Neumann’s “work reflects a belief in the relevant role mathematics could play in science and society rather than a genuine interest in economic issues” (De Pina Cabral 2003: 127)? The focus on Brouwer’s fixed point theorem in the GE paper, emphasized in its German title, supports this hypothesis. A fixed point argument is also central to the proof of the minimax theorem in the 1928 paper. It was Morgenstern who raised the question of rationality and expectations that directed the TGEB towards economics (see Morgenstern 1949). If von Neumann’s contribution to economics was meant to enhance the status of mathematics, the use of Brouwer’s fixed point theorem and its extensions indeed enriched the tool box of economists and even entered advanced textbooks in microeconomics and game theory. Moreover, the TGEB had a strong impact on the development of axiomatic analysis in economics, especially demonstrated by the formalization and proof of the expected utility function. The first five axioms of Nash’s (1950b) solution to the bargaining problem correspond to the axioms that von Neumann and Morgenstern proposed for expected utility. However, the proof of existence of a corresponding utility function was not published in the first edition of TGEB, although the authors already had the proof “of course” (Morgenstern 1976: 809). It is fair to say that the empirical testing of theoretical results of expected utility theory and game theory became one of the roots of experimental economics - an extremely popular branch of today’s economics.

The game theoretical work of von Neumann never left the zero-sum world “of perfect conflict” in which the advantage of one player implies an equally large disadvantage of the other player, even when there are more than two players. In an earlier paper on TGEB and its authors (Holler 2009) I argued that the focus on conflict and zero sum that even entered the analysis of N-person games in TGEB via the concept of fictitious players reflects an important feature of von Neumann’s personality. Von Neumann was very sceptical towards the Nash equilibrium that generalized the minimax theorem for variable sum games that may represent situations of both conflict and coordination and applies to more than two players. For Nash, however, the zero sum limitation of TGEB was the motivation to develop his equilibrium concept (Nash 1950a) and propose a solution to the bargaining problem (Nash 1950b). In TGEB,

a theory of n-person games is developed which includes as a special case the two-person bargaining problem. But the theory there developed makes no attempt to find a value for a given n-person game, that is, to determine what it is worth to each player to have the opportunity to engage in the game. This determination is accomplished only in the case of the two-person zero sum game. (Nash 1950a: 157)

Note that Nash (1951) used Brouwer’s theorem to prove the existence of an equilibrium for all games with a finite number of pure strategies.

A Short Vitae

Obviously, von Neumann’s scientific interest strongly focused on mathematics and its applications. He has delivered substantial contributions to quantum physics, functional analysis, set theory, topology, numerical analysis, cellular automata, and computer science, and his work in economics looks just like another field of application of mathematics.

He was born Neumann Janos Lajos in 1903 in Budapest and died in 1957 in Washington, DC. At the age of 6, he could exchange jokes in Classical Greek, memorize telephone directories, and was able to divide two eight-digit numbers in his head.

His first mathematics paper, written jointly with Michael Fekete, then assistant at the University of Budapest who had been tutoring him, was published in 1922. At the age of 25, he had published ten major papers in mathematics.

In 1926, aged 23, he received his PhD in mathematics from the University of Budapest and a diploma in chemical engineering from the ETH Zurich (the Swiss Federal Institute of Technology in Zurich). He taught as a Privatdozent at the University of Berlin, today’s Humboldt-Universitat, from 1926 to 1929 and at the University of Hamburg from 1929 to1930. In 1930 he became a visiting lecturer at Princeton University, being appointed professor there in 1931. In 1933, he became professor of mathematics at the newly founded Institute for Advanced Study in Princeton - a position he kept for the remainder of his life.

Morgenstern (1976) reports that on 1 February 1939, when he gave an after-lunch talk on business cycles at the Nassau Club, he had a first chance to talk to von Neumann about games. Over the year their discussion and friendship progressed. In 1944, their Theory of Games and Economic Behavior was published, a volume of more than 600 pages.

During the years of collaboration with Morgenstern and after World War II, von Neumann served as a consultant to the armed forces. In 1940 he became a member of the Scientific Advisory Committee at the Ballistic Research Laboratories and in 1941 a member of the Navy Bureau of Ordnance. He was a consultant to the Los Alamos Scientific Laboratory from 1943 to 1955. In this function he was a leading contributor to the development of the nuclear and, along with Edward Teller and Stanislaw Ulam, of the hydrogen bomb. In 1956 he received the presidential Medal for Freedom - being America’s highest civilian award, recognizing exceptional meritorious service. Von Neumann died on 8 February 1957, “after much suffering” (Morgenstern 1976: 814).

Von Neumann’s Expanding General Equilibrium Model

In the first line of his GE article, von Neumann states that the subject matter of this paper is “the solution of a typical economic equation system”.

(The von Neumann citations in this section are taken from the English version, von Neumann 1945.) This equation system assumes that there are n goods which can be produced by m different processes being characterized by constant returns to scale. Moreover, a good “can be produced only jointly with certain others”. There are natural factors of production but the emphasis is on produced goods that enter production as inputs. Thus production is circular and the distinction of primary factors and outputs, substantial to the dominant Walrasian general equilibrium model, is not valid. Natural factors of production, including labour and land, are abundant and “can be expanded in unlimited quantities”. Labour is remunerated with the “necessities of life”. There are no savings: “consumption of goods takes place only through the processes of production” and “all incomes in excess of necessities of life will be reinvested”.

After having defined the intensities of production x_i (i = 1,..., m), expressing to what degree a process i is applied, and the prices y_j (j = 1,..., n), x_i and y_j being nonnegative, von Neumann proposes the following equations:

If < applies in (1) then y_j = 0, and j is a free good.

If > applies in (2) then x_i = 0, that is, process i is unprofitable and will not be used.

In (1) and (2), a_j and b_ij quantify inputs and outputs in the production of good j, given at any point in time. a and β represent “the coefficient of the expansion of the whole economy” and the interest factor, respectively. Condition (1) says that it is impossible to consume more of a good j in the total production than is being produced. Condition (2) assumes perfect competition: in equilibrium there are no profits with any process i.

Quantities a_ij and b_jj are given, variables a, β, x_i and y_j are unknown. Von Neumann’s proof demonstrates that for the given system a, β, x_i and y_j can always be determined. Even with constant ratios of intensities, there could be several solutions for x_i and y_j while a and β are uniquely determined. It turns out that a = β.

Von Neumann’s GE tries to answer the question “which processes will be actually used and which not”. As such it is the first paper of what has been subsequently called activity analysis, and “the first use in economics of... explicit duality arguments, explicit fixed- point techniques for an existence proof, and convexity arguments” (Weintraub 1983: 13). Economists may regret that no preferences were needed to explain demand, and money did not matter as there was no saving and still, for positive prices, inputs and outputs balanced. The interest factor is not determined by time preference or demand and supply of money but by the expansion coefficient, that is, the supply of goods. This suggests that the model is in the classical and not in the neoclassical tradition (see Kurz and Salvadori 1993, 1995: 403ff).

Game Theory

In GE, von Neumann has an extensive footnote referring to the minimax theorem and his 1928 paper “Zur Theorie der Gesellschaftsspiele” (TG) that contains its proof. Despite the time difference between the publication of TG and GE it has been argued that the two papers were not developed independently from each other.

“Zur Theorie der Gesellschaftsspiele” introduces all instruments that are essential for non-cooperative game theory, but it also gives a definition of the characteristic function that is applied to analyse coalition formation. More fundamental, however, is the introduction of the concept of strategy as a complete plan of moves that a player can choose, including mixed strategies. The strategy sets reflect the rules of the game and the possible information that a player has at the beginning of a game and that he collects in its course. This introduces the extensive form of a game.

For the case of two-person zero-sum games with finitely many pure strategies for each player, that is, if the advantage of one player implies an “equally” large disadvantage of the other, TG contains a proof of the minimax theorem which says: there is always a pair of strategies, pure or mixed, such that none of the players can deviate and achieve a higher “payoff” (see below). Thus, the theorem guarantees a value for each player which is independent of the behaviour of the other.

In the case of more than two players, the problem of coalition formation exists. “Zur Theorie der Gesellschaftsspiele” proposes a solution concept that prescribes outcomes such that no pair of players can guarantee itself a higher value then the utilities that the solution allocates to them. This condition is satisfied by the core, but not by the “solution” introduced in TGEB.

Apart from a large variety of applications, the TGEB adds two essentials to TG: an axiomatization of the expected utility (EU) hypothesis and a solution concept for games with more than two players, called “solution” by the authors. Often, the second concept is considered as a poor substitute for the core and has hardly any relevance in economics. However, there are arguments in favour of this concept (see Aumann 1987; Binmore 1998: 40f.). Expected utility became an indispensable tool to modern microeconomics, and not only in its game theoretical applications. It says that given a lottery L = (A, p; B, 1 - p) where A and B are sure alternatives and p and 1 - p are the probability by which they occur, then we can write u(L) = pu(A) + (1 - p)u(B). Utility functions u(.) that satisfy this property are called von Neumann Morgenstern utilities or payoffs. (The use of the term payoff is, however, not unambiguous: some authors use it, for example, for money instead of utilities.) Obviously, EU applies to the evaluation of risky outcomes, for instance, as a result of playing mixed strategies. It might be helpful to know that this concept has an axiomatic basis, but more often the axioms are used to identify some properties of EU and submit them to empirical tests and theoretical speculation. Allais’s (1953) early critique on the EU concept was mainly based on introspection - Allais’s label “l’ecole americaine” has a certain charm: its authors conversed in German when writing the TGEB (see Morgenstern 1976).

One of the achievements of TGEB was to make von Neumann’s TG article and its results known. Even Morgenstern had not read the article before he met von Neumann at Princeton. However, he had heard of von Neumann: “The principal reason for my wanting to go to Princeton was the possibility that I might become acquainted with von Neumann and the hope that this would be a great stimulus for my future work” (Morgenstern 1976: 807).

Manfred J. Holler

John von Neumann (1903-1957)

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