The New Palgrave Dictionary of Economics

2018 Edition
| Editors: Macmillan Publishers Ltd

von Neumann, John (1903–1957)

  • Gerald L. Thompson
Reference work entry
DOI: https://doi.org/10.1057/978-1-349-95189-5_1406

Abstract

Jansci (John) von Neumann was born to Max and Margaret Neumann on 28 December 1903 in Budapest, Hungary. He showed an early talent for mental calculation, reading and languages. In 1914, at the age of ten, he entered the Lutheran Gymnasium for boys. Although his great intellectual (especially mathematical) abilities were recognized early, he never skipped a grade and instead stayed with his peers. An early teacher, Laslo Ratz, recommended that he be given advanced mathematics tutoring, and a young mathematician Michael Fekete was employed for this purpose. One of the results of these lessons was von Neumann’s first mathematical publication (joint with Fekete) when he was 18.

JEL Classifications

B31 

His Life

Jansci (John) von Neumann was born to Max and Margaret Neumann on 28 December 1903 in Budapest, Hungary. He showed an early talent for mental calculation, reading and languages. In 1914, at the age of ten, he entered the Lutheran Gymnasium for boys. Although his great intellectual (especially mathematical) abilities were recognized early, he never skipped a grade and instead stayed with his peers. An early teacher, Laslo Ratz, recommended that he be given advanced mathematics tutoring, and a young mathematician Michael Fekete was employed for this purpose. One of the results of these lessons was von Neumann’s first mathematical publication (joint with Fekete) when he was 18.

Besides his native Hungarian, Jansci (or Johnny, as he was universally known in his later life) spoke German with his parents and a nurse and learned Latin and Greek as well as French and English in school. In 1921 he enrolled in mathematics at the University of Budapest but promptly left for Berlin, where he studied with Erhard Schmidt. Each semester he returned to Budapest to take examinations without ever having attended classes. While in Berlin he frequently took a three-hour train trip to Göttingen, where he spent considerable time talking to David Hilbert, who was then the most outstanding mathematician of Germany. One of Hilbert’s main goals at that time was the axiomatization of all of mathematics so that it could be mechanized and solved in a routine manner. This interested Johnny and led to his famous 1928 paper on the axiomatization of set theory. The goal of Hilbert was later shown to be impossible by Kurt Gödel’s work, based on an axiom system similar to von Neumann’s, which resulted in a theorem, published in 1930, to the effect that every axiomatic system sufficiently rich to contain the positive integers must necessarily contain undecidable propositions.

After leaving Berlin in 1923 at the age of 20, von Neumann studied at the Eidgenossische Technische Hochschüle in Zurich, Switzerland, while continuing to maintain his enrolment at the University of Budapest. In Zurich he came into contact with the famous German mathematician, Hermann Weyl, and also the equally famous Hungarian mathematician, George Polya. He obtained a degree in Chemical Engineering from the Hochschüle in Zurich in 1925, and completed his doctorate in mathematics from the University of Budapest in 1926. In 1927 he became a privatdozent at the University of Berlin and in 1929 transferred to the same position at the University of Hamburg. His first trip to America was in 1930 to visit as a lecturer at Princeton University, which turned into a visiting professorship, and in 1931 a professorship. In 1933 he was invited to join the Institute for Advanced Study in Princeton as a professor, the youngest permanent member of that institution, at which Albert Einstein was also a permanent professor. Von Neumann held this position until he took a leave of absence in 1954 to become a member of the Atomic Energy Commission.

Von Neumann was married in 1930 to Marietta Kovesi, and his daughter Marina (who became a vice-president of General Motors) was born in 1935. The marriage ended in a divorce in 1937. Johnny’s second marriage in 1938 was to Klara Dan, whom he met on a trip to Hungary. They maintained a very hospitable home in Princeton and entertained, on an almost weekly basis, numerous local and visiting scientists. Klara later became one of the first programmers of mathematical problems for electronic computers, during the time that von Neumann was its principal designer.

In 1938 Oskar Morgenstern came to Princeton University. His previous work had included books and papers on economic forecasting and competition. He had heard of von Neumann’s 1928 paper on the theory of games and was eager to talk to him about connections between game theory and economics. In 1940 they started work on a joint paper which grew into their monumental book, Theory of Games and Economic Behavior published in 1944. Their collaboration is described in Morgenstern (1976).

Von Neumann became heavily involved in defence-related consulting activities for the United States and Britain during World War II. In 1944 he became a consultant to the group developing the first electronic computer, the ENIAC, at the University of Pennsylvania. Here he was associated with John Eckert, John Mauchly, Arthur Burks and Herman Goldstine. These five were instrumental in making the logical design decisions for the computer, for example, that it be a binary machine, that it have only a limited set of instructions that are performed by the hardware, and most important of all, that it run an internally stored program. It was acknowledged by the others in the group that the most important design ideas came from von Neumann. The best account of these years is Goldstine (1972). After the war von Neumann and Goldstine worked at the Institute of Advanced study where they developed (with others) the JONIAC computer, a successor to the ENIAC, which used principles some of which are still being used in current computer designs.

In 1943 von Neumann became a consultant to the Manhattan Project which was developing the atomic bomb in Los Alamos, New Mexico. This work is still classified but it is known that Johnny performed superbly as a mathematician, an applied physicist, and an expert in computations. His work continued after the war on the hydrogen bomb, with Edward Teller and others. Because of this work he received a presidential appointment to the Atomic Energy Commission in 1955. He took leave from the Institute for Advanced Study and moved to Washington. In the summer of 1955 he fell and hurt his left shoulder. Examination of that injury led to a diagnosis of bone cancer which was already very advanced. He continued to work very hard at his AEC job, and prepared the Silliman lectures (von Neumann 1958), but was unable to deliver them. He died on 8 February 1957 at the age of 53 in the Walter Reed Hospital, Washington, DC.

The Theory of Games

Without question one of von Neumann’s most original contributions was the theory of games, with which it is possible to formulate and solve complex situations involving psychological, economic, strategic and mathematical questions. Before his great paper on this subject in 1928 there had been only a handful of predecessors: a paper by Zermelo in 1912 on the solution in pure strategies of chess; and three short notes by the famous French mathematician E. Borel. Borel had formulated some simple symmetric two-person games in these notes, but was not able to provide a method of solution for the general case, and in fact conjectured that there was no solution concept applicable to the general case. For a commentary on the priorities involved in these two men’s work see the notes by Maurice Frechet, translations (by L.J. Savage) of the three Borel papers, and a commentary by von Neumann, all of which appeared with von Neumann (1953a).

The three main results of von Neumann’s 1928 paper were: the formulation of a restricted version of the extensive form of a game in which each player either knows nothing or everything about previous moves of other players; the proof of the minimax theorem for two-person zero-sum games; and the definition of the characteristic function for and the solution of three-person zero-sum games in normal form. Von Neumann also carried out an extensive study of simplified versions of poker during this time, but they were not published until later.

The extensive form of a game is the definition of a game by stating its rules, that is, listing all of the possible legal moves that a player can make for each possible situation he can find himself in during a play of the game. A pure strategy in a game is a much more complicated idea – a listing of a complete set of decisions for each possible situation in which the player can find himself. A complete enumeration of all possible strategies shows that the number of such strategies is equal to the product of the number of legal moves for each situation, which implies that there is an astronomical number of possible strategies for any non-trivial game such as chess. Most of these are bad, and would never be used by a skilful player, but they must be considered to find its solution. The normalized form of a game is obtained by replacing the definition of a game as a statement of its rules, as is done in its extensive form, by a listing of all of the possible pure strategies for each player. To complete the normalized form of the game, imagine that each player has made a choice of one of his pure strategies. When pitted against another a unique (expected) outcome of the game will result. For the moment we will imagine that the outcome of the game is monetary, and therefore each player gets a ‘payoff’ at the end of the game which is actually money. (Later we will replace money by ‘utility’.) If the sum of the payments to all players is zero the game is said to be zero-sum; otherwise it is a non-zero-sum game.

The normalized form of a game is also called a matrix game, and any real m × n matrix can be considered a two-person zero-sum game. The row player has m pure strategies, i = 1, …, m, and the column player has n pure strategies, j = 1, …, n. If the row player chooses i and the column player chooses j then the payoff a(i, j) is exchanged between them, where a(i, j) > 0 means that the row player receives a(i, j) from the column player, while a negative payoff means that the column player receives the absolute value of that amount from the row player.

The importance of the careful analysis of the extensive and normalized forms of a game is that it separates out the concept of strategy and psychology in any discussion of a game. As an example, in poker bidding high when having a weak hand is commonly called ‘bluffing’, and considered an aggressive form of play. As a result of this formulation, and the solution of simplified versions of the game von Neumann showed that in order to play poker ‘optimally’ it is necessary to bluff part of the time, i.e., it is a required part of the strategy of any good poker player. A similar analysis for simplified bridge shows that a required part of an optimal bridge strategy is to signal, via the way one discards low cards in a suit, whether the player holds higher cards in that suit.

The analysis of special kinds of games shows that some of them can be solved by using pure strategies. This class includes the games of ‘perfect information’ such as the board games of chess and checkers. However, even such a simple game as matching pennies shows that an additional strategic concept is needed, namely, that of a ‘mixed strategy’. This concept appeared first in the context of symmetric two-person games in Borel’s 1921 paper. Briefly, a mixed strategy for either player is a finite probability function on his set of pure strategies. For matching pennies the common strategy of flipping the penny to choose whether to play heads or tails is a mixed strategy that chooses both alternatives with equal probability (1/2), and is, in fact, an optimal strategy for that game.

We now discuss the way that von Neumann made precise the definition of a solution to a matrix game. Let A be an arbitrary m × n matrix with real number entries. Let x be an m-component row vector, and let f be an m-component column vector all of whose components are ones. Then x is a mixed strategy vector for the row player in the matrix game A if it satisfies: xf = 1 and x ≥ 0. Similarly, let y be an n-component column vector, and let e be an n-component row vector of all whose components are ones. Then y is a mixed strategy vector for the column player in the matrix game A if it satisfies: ey = 1 and y ≥ 0. Mixed strategy vectors are also called probability vectors because they have non-negative components that sum to one, and hence could be used to make a random choice of a pure strategy by spinning a pointer, choosing a random number, etc. To complete the definition of the solution to a game, we need a real number v, called the value of the game. The solution to the matrix game A is now a triple, a mixed strategy x for the row player, a mixed strategy y for the column player, and a value v for the game: these quantities must solve the following pair of (vector) inequalities:
$$ xA\ge ve\;\mathrm{and}\; Ay\le vf. $$

Because these are linear inequalities, one might suspect (and would be correct) that the optimal x, y and v can be found by using a linear programming code and a computer.

However, in the 1920s it was not clear that such a solution existed. In fact, Borel conjectured that it did not. The most decisive result of von Neumann’s 1928 paper was to establish, using an argument involving a fixed point theorem, his famous minimax theorem to the effect that for an arbitrary real matrix A there exists a real number v and probability vectors x and y such that
$$ \underset{x}{Maximum}\;\underset{y}{Maximum}\kern0.36em xAy=\underset{y}{Maximum}\;\underset{x}{Maximum}\kern0.36em xAy $$

This theorem became the keystone not only for the theory of two-person matrix games, but also for n-persons games via the characteristic function (to be discussed later).

We now discuss the major differences between von Neumann and Morgenstern (1944) and von Neumann’s 1928 paper. The information available to each player was assumed, in the 1928 paper, to be the following: when required to move, each player knows either everything about the previous moves of his opponents (as in chess), or nothing (as in matching pennies). By using information trees, and partitioning the nodes of such trees into information sets, in 1944 this concept was extended to games in which players have only partial information about previous moves when they are required to make a move. This was a difficult but major extension, which has not been substantially improved upon since its exposition in the 1944 treatise.

A second major change in the basic theory of games was in the treatment of payoff functions. In the 1928 paper payoffs were treated as if they were monetary, and it was implicitly assumed that money was regarded as equally important by each of the players. In order to take into account the well-known objections, such as those of Daniel Bernoulli, to the assumption that a dollar is equally important to a poor man as a rich man, a monetary outcome to a player was replaced by the utility of the outcome. Although Bernoulli had suggested that the utility of x dollars should be the natural logarithm of x, so that the addition of a dollar to a rich man’s fortune would be valued less than the addition of a dollar to a poor man’s fortune, this specific utility concept was never universally accepted by economists. So utility remained a fuzzy, intuitive concept. Von Neumann and Morgenstern made the absolutely decisive step of axiomatizing utility theory, making it unambiguous and they can properly be said to have started the modern theory of utility, not only for game theory, but for all of economics and the social sciences.

Almost two-thirds of the 1944 treatise consists of the theory of n-person constant-sum games, of which only a small part, the three person zero-sum case, appears in the 1928 paper. When n > 2, there are opportunities for cooperation and collusion as well as competition among the players, so that there arises the problem of finding a way to evaluate numerically the position of each player in the game. In 1928 von Neumann handled this problem for the zero-sum case by introducing the idea of the characteristic function of a game defined as follows: For each coalition, that is, subset S of players, let v(S) be the minimax value that S is assured in a zero-sum two-person game played between S and its complementary set of players.

To describe the possible division of the total gain available among the players the concept of an imputation, which is a vector (x(1),…, x(n)) where x(i) represents the amount the player i obtains, was introduced. For a coalition C in a constant-sum game v(C) is the minimum amount that the coalition C should be willing to accept in any imputation, since by playing alone against all the other players, C can achieve that amount for itself. Except for this restriction there is no other constraint on the possible imputations that can become part of a solution. An imputation vector x is said to dominate imputation vector y if there exists a coalition C such that (1) x(i) ≥ y(i) for all i in C, and (2) the sum of x(i) for i in C does not exceed v(C). The idea is that that the coalition C can ‘enforce’ the imputation x by simply threatening to ‘go it alone’, since it can do no worse by itself.

One might think, or hope, that a single imputation could be taken as the definition of a solution to an n-person constant-sum game. However, a more complicated concept is needed. By a von Neumann–Morgenstern solution to an n-person game is meant a set S of imputations such that (1) if x and y are two imputations in S then neither dominates the other; and (2) if z is an imputation not in S, then there exists an imputation x in S that dominates z. Von Neumann and Morgenstern were unable (for good reasons, see below) to prove that every n-person game had a solution, even though they were able to solve every specific game they considered, frequently finding a huge number of solutions.

At the very end of the 1944 book there appears a chapter of about 80 pages on general non-zero-sum games. These were formally reduced to the zero-sum case by the technique of introducing a fictitious player, who was entirely neutral in terms of the game’s strategic play, but who either consumed any excess, or supplied any deficiency so that the resulting n + 1 person game was zero-sum. This artifice helped but did not suffice for a completely adequate treatment of the non-zero-sum case. This is unfortunate because such games are the most likely to be found useful in practice.

About 25 years after the treatise appeared, William Lucas (1969) provided as a counter-example, a general sum game that did not have a von Neumann–Morgenstern solution. Other solution concepts have been considered since, such as the Shapley value, and the core of a game.

One of the most interesting non zero-sum games considered in that chapter was the so-called market game. The first example of a market game (though it was not called that) was the famous horse auction of Böhm-Bawerk, published in 1881. The horses had identical characteristics, each of 10 buyers had a maximum price he was willing to bid, and each of 8 sellers had a minimum price he was willing to accept. Böhm-Bawerk’s solution was to find the ‘marginal pairs’ of prices, which turned out to be included in the von Neumann–Morgenstern solution to this kind of game. Later work on this problem was done by Shapley and Shubik (1972) and Thompson (1980, 1981).

The Expanding Economy Model

Another of von Neumann’s original contribution to economics was von Neumann (1937), which contained an expanding economy model unlike any other economic model that preceded it. When von Neumann gave a seminar to the Princeton economics department in 1932 on the model, which was stated in terms of linear inequalities not equations, and whose existence proof depended upon a fixed point theorem more sophisticated than any published in the mathematics literature of the time, it is little wonder that he made no impression on that group. He repeated his talk on the subject at Karl Menger’s mathematical seminar in Vienna in 1936, and published his paper in German in 1937 in the seminar proceedings. The paper became more widely known after it was translated into English and published in The Review of Economic Studies in 1945 together with a commentary by Champernowne.

Von Neumann’s model consists of a closed production economy in which there are m processes and n goods. In order to describe it we use the vectors e and f previously defined together with the following notation:
  • x is the m × 1 intensity vector with xf = 1 and x ≥ 0.

  • y is the 1 × price vector with ey = 1 and y ≥ 0.

  • α = 1 + a/100 is the expansion factor, where a is the expansion rate.

  • β = 1 + b/100 is the interest factor, where b is the interest rate. The model satisfies the following axioms:
    • Axiom 1. xBαxA or x(BαA) ≥ 0.

    • Axiom 2. ByβAy or x(BβA)y ≤ 0.

    • Axiom 3. x(BαA)y = 0.

    • Axiom 4. x(BβA)y = 0.

    • Axiom 5. xBy > 0.

Axiom 1 makes the model closed, i.e., the inputs for a given period are the outputs of the previous. Axiom 2 makes the interest rate be such that the economy is profitless. Axiom 3 requires that overproduced goods be free. Axiom 4 forces inefficient processes not to be used. And Axiom 5 requires the total value of all goods produced to be positive.

In order to demonstrate that for any pair of nonnegative matrices A and B, solutions consisting of vectors x and y and numbers α and β exist, an additional assumption was needed:

Assumption V.  A + B > 0.

This assumption means that every process requires as an input or produces as an output some amount, no matter how small, of every good. With this assumption, and the assumption that natural resources needed for expansion were available in unlimited quantities, von Neumann showed that necessarily α = β, that is, that the expansion and interest factors were equal. In his paper, von Neumann proved a sophisticated fixed point theorem and used it to prove the existence theorem for the EEM.

D.G. Champernowne (1945) provided the first acknowledgement that the economics profession had seen the article, and also provided its first criticisms. We mention three:
  1. (1)

    Assumption V which requires that every process must have positive inputs or outputs of every other good was economically unrealistic.

     
  2. (2)

    The fact that the model has no consumption, so that labour could receive only subsistence amounts of goods as necessary inputs for production processes, also seems unrealistic.

     
  3. (3)

    The consequence of Axiom 3 that overproduced good should be free is too unrealistic.

     
Criticisms 1 and 2 were removed by Kemeny et al. (1956), who replace Assumption V by:
  • Assumption KMT-1. Every row of A has at least one positive entry.

  • Assumption KMT-2. Every column of B has at least one positive entry. The interpretation of KMT-1 is that every process must use at least one good as an input. And the interpretation of KMT-2 is that every good must be produced by some process. With these assumptions they were able to show that there were a finite number of possible expansion factors for which intensity and price vectors existed satisfying the axioms. They also showed how consumption could be added into the model, which responded to criticism 2.

An alternative way of handling these criticisms appears in Gale (1956).

In Morgenstern and Thompson (1969, 1976), the third criticism above was answered by generalizing the model to become an ‘open economy’. In such an economy the price of an overproduced good cannot fall below its export price, and it cannot rise above its import price. Generalizations of the open model have been made by Los (1974) and Moeschlin (1974).

Von Neumann’s Influence on Economics

Although von Neumann has only three publications that can directly be called contributions to economics, namely, his 1928 paper on the theory of games, his 1937 paper (translated in 1945) on the expanding economy model and his 1944 treatise (with Morgenstern) on the theory of games, he had enormous influence on the subject. The small number of contributions is deceptive because each one consists of several different topics, each being important. We discuss these separately.

The expanding economy model, von Neumann (1937) consisted of two parts: the first input–output equilibrium model that permits expansion; and second the fixed point theorem. The linear input–output model is a precursor of the Leontief model, of linear programming as developed by Kantorovich and Dantzig, and of Koopman’s activity analysis. This paper, together with A. Wald (1935) raised the level of mathematical sophistication used in economics enormously. Many current younger economists are high-powered applied mathematicians, in part, because of the stimulus of von Neumann’s work.

The theory of games, von Neumann (1928) and von Neumann and Morgenstern (1944), was an enormous contribution consisting of several different parts: (1) the axiomatic theory of utility; (2) the careful treatment of the extensive form of games; (3) the minimax theorem; (4) the concept of a solution to a constant-sum n-person game; (5) the foundations of non-zero-sum games; (6) market games. Each of these topics could have been broken into a series of papers, had von Neumann taken the time to do so. And he could have forged a brilliant career in economics by publishing them. However, he found that making an exposition of the results that he had worked out in notes or in his head was less interesting to him than investigating still other new ideas.

Von Neumann’s indirect contributions, such as the theory of duality in linear programming, computational methods for matrix games and linear programming, combinatorial solution methods for the assignment problem, the logical design of electronic computers, contributions to statistical theory, etc. are equally, important to the future of economics. Each of his contributions, direct or indirect, was monumental and decisive. We should be grateful that he was able to do so much in his short life. His influence will persist for decades and even centuries in economics.

Selected Works

  • 1928. Zur Theorie der Gesellschaftsspiele. Mathematische Annalen 100: 295–320.

  • 1937. Über ein ökonomisches Gleichungssystem und eine Verallgemeinerung des Brouwerschen Fixpunktsatzes. In Ergebnisse eines mathematische Kolloquiums, vol. 8, ed. Karl Menger. Trans. as ‘A model of general equilibrium’, Review of Economic Studies 13 (1945–6): 1–9.

  • 1944. (With O. Morgenstern.) Theory of games and economic behavior. Princeton: Princeton University Press. 2nd ed, 1947; 3rd ed, 1953.

  • 1947. Discussion of a maximum problem. Unpublished working paper, Princeton, November, 9 pp.

  • 1948. A numerical method for determining the value and the best strategies of a zero-sum two-person game with large numbers of strategies. Mimeographed, May, 23 pp.

  • 1953a. Communications on the Borel notes. Econometrica 21: 124–125.

  • 1953b. (With G.W. Brown.) Solutions of games by differential equations. In Contributions to the theory of games, Annals of mathematics studies no. 28, vol. 1, ed. H.W. Kuhn and A.W. Tucker. Princeton: Princeton University Press.

  • 1953c. (With D.B. Gillies and J.P. Mayberry.) Two variants of poker. In Contributions to the theory of games, Annals of mathematics studies no. 28, vol. 1, ed. H.W. Kuhn and A.W. Tucker. Princeton: Princeton University Press.

  • 1954. A numerical method to determine optimum strategy. Naval Research Logistics Quarterly 1: 109–115.

  • 1958. The computer and the brain. New Haven: Yale University Press.

  • 1963. Collected works, vols. I–VI. New York: Macmillan.

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© Macmillan Publishers Ltd. 2018

Authors and Affiliations

  • Gerald L. Thompson
    • 1
  1. 1.