From “Struggle” to “Balance”: Von Neumann’s Minimax Theorem of 1928
Born to a wealthy banking family of assimilated Hungarian Jews, John von Neumann (1903-1957) was a privately tutored mathematical prodigy. It is not insignificant for the present subject that, during his formative years, he was witness to not only the upheaval of the Great War but also, in Hungary, the 1919 Communist revolution of Bela Kun and its subsequent brutal suppression.
He watched too the growth of anti-Semitism, which would increasingly restrict the opportunities available to even well-integrated Jews such as himself. In the mid-1920s, he completed degrees in mathematics and chemical engineering at Budapest and Zurich, during which time he wrote several papers in the areas of axiomatic set theory and the consistency of mathematics, and in 1926 he became postdoctoral fellow at the University of Gottingen. Then a world centre in mathematics, Gottingen provided a vibrant intellectual setting, where he was able to work close to its leader David Hilbert and other mathematical luminaries such as Richard Courant and Hermann Weyl. During this period, he continued working on set theory and foundations and, in particular, the mathematical theory of quantum mechanics (see von Neumann 1932). While these works reveal features that would ultimately characterize von Neumann’s approach to economics, in particular the emphasis on axiomatic description,in the short-term it was not social science that drew his attention. His minor interest at Gottingen was, rather, the analysis of parlour games, activities then very much part of the culture.
The background to von Neumann’s first paper on games was twofold. First, there was the great prominence of chess in cultural life at the time, with the game capturing the imagination not only of the general public but also of writers, psychologists and mathematicians. (For psychological perspectives on chess during the period, see Binet 1894, and Djakow et al.
1927; or Nabokov’s chess novel The Defense 1930 [1964]). At the centre of the chess world stood mathematician Emanuel Lasker, who not only wrote about chess tactics, but also speculated on the possibility of a more general science of strategy and conflict. Secondly, quite unconnected to chess, and located in the French rather than German mathematical community, there was in the 1920s a series of papers on the theory of games by eminent French mathematician Emile Borel. Let us consider first Lasker and then Borel.With a 1902 doctorate in mathematics from Erlangen, and counting Hilbert and Max Noether among his mentors, Emanuel Lasker came from the same world of German mathematics into which von Neumann would later make his entry. As a student, turning to chess in order to pay his bills, Lasker took the world championship from Wilhelm Steinitz in 1897, and gradually attained mythical status by going on to hold the title for the next 24 years. Denied an academic post in mathematics, the rabbi’s son wrote prolifi- cally, providing counsel on chess strategy and also speculating about a possible science of “struggle”, applicable to the many realms in which strategic interaction was important. Most significant here was his 1907 pamphlet, Kampf, which presented an embryonic, discursive “game theory”, with discussion of “strategy”, the “economy principle” (achieving strategic effect with minimum effort), and “equilibrium and dominance”. Lasker felt that chess, although a complex game, with many moves available at any given moment, could, by virtue of the limited number of relevant moves, be inevitably made subject to scientific analysis. Similarly, he speculated that the struggles evident in economic and social life could also eventually be analysed scientifically, in terms of strategy and equilibrium, and he even emphasized the importance of focusing on expected payoffs, in order to allow for the riskiness of outcomes associated with different strategies.
Of the same generation as Lasker, and also a student of Hilbert, Ernest Zermelo was one of the first mathematicians to consider chess as a mathematical object.
His 1913 paper, “Uber eine Anwendung der Mengenlehre auf die Theorie des Schachspiels” (“On the application of set theory to the theory of the game of chess”), presented to the International Congress of Mathematicians at Cambridge, asked whether it was possible to characterize an arbitrary position in a game in mathematical terms, that is, what does it mean to be in a winning position, and is it possible to determine the number of moves necessary in order to secure victory? Zermelo supplied purely technical answers to these questions, in terms of the necessity and sufficiency of the emptiness and non-emptiness of certain strategic sets, and his paper gave rise to further discussions after World War I, involving Hungarian mathematicians Denes Konig and Lazslo Kalmar. Each of them produced a paper offering a refinement of Zermelo’s, and they cited discussions with their contemporary, von Neumann (see Konig 1927; Kalmar 1928-29).As for Emile Borel, the relevant papers were inspired, not by chess, but by his experience as a player of card games. Indeed, Borel followed Poincare in the view that chess, being confined to a chessboard that was 8 ? 8, was never generalizable to n ? n, and therefore was not a truly mathematical object. In a series of mathematical notes written throughout the 1920s, Borel presented in precise form the notion of strategy and the principle of random play (that is, the deliberate use of a mixed strategy), and investigated the range of two-person, zero-sum games in which the latter could be employed profitably (see Borel 1921, 1924, 1927). Having shown that an equilibrium was possible for games of three, five and seven strategies, he wondered if this would always be the case for increasingly “large” games. Not unlike Lasker, Borel also later speculated about the existence of a science of strategy, applicable to war, and economic and financial speculation (see Borel et al. 1938).
Presented at Gottingen in December 1926 and published two years later, von Neumann’s “Zur Theorie der Gesellschaftsspiele” (“On the Theory of Games of Strategy”) provides an answer to Borel’s question in the form of a theorem.
Given the subsequent history, the paper may be retrospectively viewed as the beginning of the “science” of strategy sought speculatively by the Frenchman and Lasker. Citing chess, baccarat, roulette and poker as examples, von Neumann considers the generic two-person, zero-sum game, which he defines by the strategies available to both players and their associated payoffs, and gives a tortuously difficult proof of the existence of a minimax equilibrium: a preferred way to play, possibly requiring the use of mixed strategies, that allows each player to minimize the amount ceded to the other. Such a game, says von Neumann, in the language of Lasker, is “well-balanced”, and it “makes no difference which of the two players is the better psychologist, the game is so insensitive that the result is always the same” (Von Neumann, 1928b: 23). (Further on, he promises to publish numerical examples of such two-person games as baccarat and a simplified poker, the “agreement of the results [of which] with the well-known rules of thumb of the games (for example, proof of the necessity to ‘bluff’ in poker) may be regarded as an empirical corroboration of the results of our theory”, ibid.: 42).Closing the paper with preliminary suggestions about how to extend the analysis to games of three players, he notes that the possibility of coalition-formation marks the reappearance in the game of something quite foreign to the two-person one, namely “struggle” - again, in the language of Lasker. He concludes with the suggestion that a similar approach could be taken with games of four and, ultimately, any number of players, the result of which would be a “satisfactory general theory” of all such games.
In May 1928, presumably when he became aware of Borel’s papers on the subject, von Neumann sent him a note, indicating that he had been independently working on the problem of existence of an equilibrium, and announcing the proof (see von Neumann 1928a). This Borel presented later to the Academie des Sciences.
In university courses given in 1936-37, and published in 1938, Borel commented on von Neumann’s theorem, insisting that it was unlikely to prove at all useful to the actual playing of games of strategy, where the presence of innovation and cycles in ways of playing made the practical application of mathematical calculation very difficult: “Perhaps [these remarks] will make clear... to those who would wish to turn games into an occupation how futile is the search for a perfect formula which is forever likely to elude us” (Borel et al. 1938: 117).After his 1928 paper, however, von Neumann went no further with the theory of games, essentially putting it aside for a decade. In 1930, he took a half-time appointment at Princeton’s Department of Mathematics, at the initiative of topologist Oswald Veblen, and, for two years, alternated between Berlin and Princeton, sharing the latter position with mathematical physicist and fellow Hungarian, Eugene Wigner. Gauging that his opportunities in Europe were limited, he then moved permanently to the US, becoming, along with Albert Einstein, one of the first members of the newly founded Institute for Advanced Study, which although located at Princeton was independent of the university. It was here that he would eventually be led back to game theory, when stimulated by both political developments and his eventual encounter with Oskar Morgenstern.