A Beautiful Mind

Where many of the more remarkable mathematicians of the period are withdrawn and awkward, John von Neumann is seen as somewhat dashing and attractive. In fact, he is “universally considered the most cosmopolitan, multifaceted, and intelligent mathematician” (79) of the twentieth century.

Von Neumann’s career would include numerous roles including “physicist, economist, weapons expert, and computer visionary” (79). His powers of recall, mental arithmetic, and general knowledge, along with his sometimes cold, blunt manner, lead to people joking that he is “really an extraterrestrial who [has] learned how to imitate a human perfectly” (80).

During the war, von Neumann co-wrote the highly significant text The Theory of Games and Economic Behavior and, amongst other things, came up with a method for detonating the A-bomb that is “credited with shortening the time needed to develop the bomb by as much as a year” (81). In 1948, he is a respected researcher at the Institute for Advanced Studies in Princeton.

In 1928, von Neumann wrote a paper about “the theory of games” (84) in which he suggested that studying games might provide insights into economics. In 1938, he teamed up with professor Oskar Morgenstern to expand on his theory in the groundbreaking text The Theory of Games and Economic Behavior.

Published in 1944, the book serves as “a blistering attack” (85) on accepted economic theory, seeking to reintegrate individual behavior into analysis of economics and “reform social theory by applying mathematics as the language of scientific logic” (85).

Thanks in part to von Neumann’s prestige, the book catches the public’s attention, receiving “a breathless front-page story in The New York Times” (86). Many of the students at Princeton begin referring to it as “the bible” (86). Economists are generally skeptical, however, suggesting that the book fails to follow through on its bold claims.

Although he recognizes that it is “mathematically innovative” (86), Nash is also critical of the book’s merits. He notices that a large section of the theory “appear[s] to have little applicability in social science” (87) because it is focused on “games of total conflict” (87) which do not often occur in real-life economic scenarios. He also realizes that the discussion of games with more than two players does not “prove that a solution exist[s] for all such games” (87) and that the analysis of non-zero-sum games—games where each player’s gains and losses are not balanced exactly by the other players’ gains and losses—is similarly incomplete. He quickly begins thinking about solutions to these flaws.  

In his second term at Princeton, Nash writes his first academic paper, titled The Bargaining Problem. Despite very little training in economics, he manages to offer a remarkably original and significant insight, revealing that behavior usually considered simply part of individual psychology and therefore “beyond the reach of economic reasoning” is actually “amenable to systematic analysis” (88).

The “idea of exchange” or the “one-to-one bargain” (88) is central to economics, but no theory has yet been able to provide real insight into how people engaged in exchange will behave or how a bargain will be struck. The central problem here is that people do not “behave in a purely competitive fashion” (89) all of the time. Instead, while still acting “out of self-interest” (89), they often cooperate and collaborate with others. Unlike pure competition, there is no mathematical model to explain or predict this behavior or its results.

Nash takes an original approach to this problem, asking, “What reasonable conditions would any solution–any split–have to satisfy?” (90). In answer, he proposes “four conditions” and employs “an ingenious mathematical argument” (90) to demonstrate that “unique solutions [are] possible” (90) in an economic theory of bargains.

Importantly, the reason Nash’s theory is so original, and therefore so insightful, is because he first had the idea for it long before he was exposed to von Neumann’s work or the high-level mathematics of Princeton. Instead, it had first “occurred to him while he was sitting in the only economics course he would ever attend” (90), an undergraduate course at Carnegie.  

Nash is thinking more about the theory of games in the summer of 1949. However, he must spend the summer preparing for, and then successfully passing, his general examination. This is “the effective end of Nash’s years as a student” (92).

His generals out of the way, Nash returns to game theory and approaches von Neumann with the beginnings of a new theory. However, Nash has barely begun to outline “the proof he had in mind for an equilibrium in games of more than two players” (94) before von Neumann cuts him off and dismisses his proposition as “trivial” (94).

The conflict between Nash and von Neumann on this matter is not surprising as they each approach game theory from different personal perspectives. Von Neumann has a background in collaborative research and group discussions. As a result, he sees “people as social beings who [are] always communicating” (94). Nash, on the other hand, sees “people as out of touch with one another and acting on their own” (94).

Not everyone is as dismissive as von Neumann. Game theory analyst David Gale is encouraging, recognizing that the theory could be applied so widely that it could even be “generalized to disarmament” (95) and later commenting, “The mathematics was so beautiful” (95).

Nash’s theory highlights interdependence and the fact that “the outcome of a game for one player depends on what all the other players choose to do” (97). In games where moves are made simultaneously, such as poker, this requires players to try to second-guess the other players in order to calculate what their own best strategy should be, while the other players are doing the same thing.

Nash responds to this by proposing that all games within a “very broad class” (97) have at least one point of equilibrium, defined as a situation in which each player is employing the best available strategy or making the best available decision, taking into account the decisions of the other players, which they are making based on the same evaluation of their fellow players.

Not even Nash immediately recognizes the true significance of his theory, which will go on to become “one of the basic paradigms in social sciences and biology” (98) and will eventually lead to Nash receiving a Nobel Prize.