A Beautiful Mind (18 page)

The Theory of Games and Economic Behavior
was in every way a revolutionary book. In line with Morgenstern’s agenda, the book was “a blistering attack” on the prevailing paradigm in economics and the Olympian Keynesian perspective, in which individual incentives and individual behavior were often subsumed, as well as an attempt to ground the theory in individual psychology. It was also an effort to reform social theory by applying mathematics as the language of scientific logic, in particular set theory and combinatorial methods. The authors wrapped the new theory in the mantle of past scientific revolutions, implicitly comparing their treatise to Newton’s
Principia
and the effort to put economics on a rigorous mathematical footing to Newton’s mathematization, using his invention of the calculus, of physics.
²⁸
One reviewer, Leo Hurwicz, wrote, “Ten more such books and the future of economics is assured.”
²⁹

The essence of von Neumann and Morgenstern’s message was that economics was a hopelessly unscientific discipline whose leading members were busily peddling solutions to pressing problems of the day — such as stabilizing employment — without the benefit of any scientific basis for their proposals.
³⁰
The fact that much of economic theory had been dressed up in the language of calculus struck them as “exaggerated” and a failure.
³¹
This was not, they said, because of the “human element” or because of poor measurement of economic variables.
³²
Rather, they claimed, “Economic problems are not formulated clearly and are often stated in such vague terms as to make mathematical treatment
a priori
appear hopeless because it is quite uncertain what the problems really are.”
³³

Instead of pretending that they had the expertise to solve urgent social problems, economists should devote themselves to “the gradual development of a theory.”
³⁴
The authors argued that a new theory of games was “the proper instrument with which to develop a theory of economic behavior.”
³⁵
The authors claimed that “the typical problems of economic behavior become strictly identical with the mathematical notions of suitable games of strategy.”
³⁶
Under the heading “necessary limitations of the objectives,” von Neumann and Morgenstern admitted that their efforts to apply the new theory to economic problems had led them to “results that are already fairly well known,” but defended themselves by
contending that exact proofs for many well-known economic propositions had been lacking.
³⁷

Before they have been given the respective proofs, theory simply does not exist as a scientific theory. The movements of the planets were known long before their courses had been calculated and explained by Newton’s theory… .

We believe that it is necessary to know as much as possible about the behavior of the individual and about the simplest forms of exchange. This standpoint was actually adopted with remarkable success by the founders of the marginal utility school, but nevertheless it is not generally accepted. Economists frequently point to much larger, more burning questions and brush everything aside which prevents them from making statements about them. The experience of more advanced sciences, for example, physics, indicates this impatience merely delays progress, including the treatment of the burning questions.

When the book appeared in 1944, von Neumann’s reputation was at its peak. It got the kind of public attention — including a breathless front-page story in
The New York Times
— that no other densely mathematical work had ever received, with the exception of Einstein’s papers on the special and general theories of relativity.
³⁸
Within two or three years, a dozen reviews appeared by top mathematicians and economists.
³⁹

The timing, as Morgenstern had sensed, was perfect. The war had unleashed a search for systematic attacks on all sorts of problems in a wide variety of fields, especially economics, previously thought to be institutional and historical in character. Quite apart from the new theory of games, a major transformation was under way — led by Samuelson’s
Foundations of Economic Theory
— making economic theory more rigorous through the use of calculus and advanced statistical methods.
⁴⁰
Von Neumann was critical of these efforts, but they surely prepared the ground for the reception of game theory.
⁴¹

Economists were actually somewhat standoffish, at least compared to mathematicians, but Morgenstern’s antagonism to the economics profession no doubt contributed to that reaction. Samuelson later complained to Leonard, the historian, that although Morgenstern made “great claims, he himself lacked the mathematical wherewithal to substantiate them. Moreover [Morgenstern] had the irksome habit of always invoking the authority of some physical scientist or another.”
⁴²
In Princeton, Jacob Viner, the chairman of the economics department, heaped scorn on the unpopular Morgenstern by saying that if game theory couldn’t even solve a game like chess, what good was it, since economics was far more complicated than chess?
⁴³

It must have become obvious to Nash fairly early on that “the bible,” as
The Theory of Games and Economic Behavior
was known to students, though mathematically innovative, contained no fundamental new theorems beyond von Neumann’s stunning min-max theorem.
⁴⁴
He reasoned that von Neumann had
succeeded neither in solving a major outstanding problem in economics using the new theory nor in making any major advance in the theory itself.
⁴⁵
Not a single one of its applications to economics did more than restate problems that economists had already grappled with.
⁴⁶
More important, the best-developed part of the theory — which took up one-third of the book — concerned zero-sum two-person games, which, because they are games of total conflict, appeared to have little applicability in social science.
⁴⁷
Von Neumann’s theory of games of more than two players, another large chunk of the book, was incomplete.
⁴⁸
He couldn’t prove that a solution existed for all such games.
⁴⁹
The last eighty pages of
The Theory of Games and Economic Behavior
dealt with non-zero-sum games, but von Neumann’s theory reduced such games formally to zero-sum games by introducing a fictitious player who consumes the excess or makes up the deficit.
⁵⁰
As one commentator was later to write, “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.”
⁵¹

To an ambitious young mathematician like Nash, the gaps and flaws in von Neumann’s theory were as alluring as the puzzling absence of ether through which light waves were supposed to travel was to the young Einstein. Nash immediately began thinking about the problem that von Neumann and Morgenstern described as
the
most important test of the new theory.

9
The Bargaining Problem
Princeton, Spring 1949

We hope however to obtain a real understanding of the problem of exchange by studying it from an altogether different angle; that is, from the perspective of a “game of strategy.”

— V
ON
N
EUMANN AND
M
ORGENSTERN,
The Theory of Games and Economic Behavior,
second edition, 1947

N
ASH WROTE HIS FIRST PAPER,
one of the great classics of modern economics, during his second term at Princeton.
¹
“The Bargaining Problem” is a remarkably down-to-earth work for a mathematician, especially a young mathematician. Yet no one but a brilliant mathematician could have conceived the idea. In the paper, Nash, whose economics training consisted of a single undergraduate course taken at Carnegie, adopted “an altogether different angle” on one of the oldest problems in economics and proposed a completely surprising solution.
²
By so doing, he showed that behavior that economists had long considered part of human psychology, and therefore beyond the reach of economic reasoning, was, in fact, amenable to systematic analysis.

The idea of exchange, the basis of economics, is nearly as old as man, and deal-making has been the stuff of legend since the Levantine kings and the pharaohs traded gold and chariots for weapons and slaves.
³
Despite the rise of the great impersonal capitalist marketplace, with its millions of buyers and sellers who never meet face-to-face, the one-on-one bargain — involving wealthy individuals, powerful governments, labor unions, or giant corporations — dominates the headlines. But two centuries after the publication of Adam Smith’s
The Wealth of Nations,
there were still no principles of economics that could tell one how the parties to a potential bargain would interact, or how they would split up the pie.
⁴

The economist who first posed the problem of the bargain was a reclusive Oxford don, Francis Ysidro Edgeworth, in 1881.
⁵
Edgeworth and several of his Victorian contemporaries were the first to abandon the historical and philosophical tradition of Smith, Ricardo, and Marx and to attempt to replace it with the mathematical
tradition of physics, writes Robert Heilbroner in
The Worldly Philosophers.
⁶

Edgeworth was not fascinated with economics because it justified or explained or condemned the world, or because it opened new vistas, bright or gloomy, into the future. This odd soul was fascinated by economics because economics dealt with
quantities
and because anything that dealt with quantities could be translated into
mathematics.
⁷

Edgeworth thought of people as so many profit-and-loss calculators and recognized that the world of perfect competition had “certain properties peculiarly favorable to mathematical calculation; namely a certain indefinite multiplicity and dividedness, analogous to that infinity and infinitesimality which facilitate so large a portion of Mathematical Physics … (consider the theory of Atoms, and all applications of the Differential Calculus).”
⁸

The weak link in his creation, as Edgeworth was uncomfortably aware, was that people simply did not behave in a purely competitive fashion. Rather, they did not behave this way all the time. True, they acted on their own. But, equally often, they collaborated, cooperated, struck deals, evidently also out of self-interest. They joined trade unions, they formed governments, they established large enterprises and cartels. His mathematical models captured the results of competition, but the consequences of cooperation proved elusive.
⁹

Is it peace or war? asks the lover of “Maud” of economic competition. It is both, pax or pact between contractors during contract, war, when some of the contractors without consent of others contract.

The first principle of Economics is that every agent is actuated only by self-interest. The workings of this principle may be viewed under two aspects, according as the agent acts without, or with, the consent of others affected by his actions. In a wide sense, the first species of action may be called war; the second contract.

Obviously, parties to a bargain were acting on the expectation that cooperation would yield more than acting alone. Somehow, the parties reached an agreement to share the pie. How they would split it depended on bargaining power, but on that score economic theory had nothing to say and there was no way of finding one solution in the haystack of possible solutions that met this rather broad criterion. Edgeworth admitted defeat: “The general answer is —(a) Contract without competition is indeterminate.”
¹⁰

Over the next century, a half-dozen great economists, including the Englishmen John Hicks and Alfred Marshall and the Dane F. Zeuthen, took up Edgeworth’s problem, but they, too, ended up throwing up their hands.
¹¹
Von
Neumann and Morgenstern suggested that the answer lay in reformulating the problem as a game of strategy, but they themselves did not succeed in solving it.
¹²

Nash took a completely novel approach to the problem of predicting how two rational bargainers will interact. Instead of defining a solution directly, he started by writing down a set of reasonable conditions that any plausible solution would have to satisfy and then looked at where they took him.

This is called the axiomatic approach — a method that had swept mathematics in the 1920s, was used by von Neumann in his book on quantum theory and his papers on set theory, and was in its heyday at Princeton in the late 1940s.
¹³
Nash’s paper is one of the first to apply the axiomatic method to a problem in the social sciences.
¹⁴

Recall that Edgeworth had called the problem of the bargain “indeterminate.” In other words, if all one knew about the bargainers were their preferences, one couldn’t predict how they would interact or how they would divide the pie. The reason for the indeterminacy would have been obvious to Nash. There wasn’t enough information so one had to make additional assumptions.