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.” — VON NEUMANN AND MORGENSTERN, The Theory of Games and Economic Behavior, second edition, 1947
NASH 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
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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.
Nash’s theory assumes that both sides’ expectations about each other’s behavior are based on the intrinsic features of the bargaining situation itself. The essence of a situation that results in a deal is “two individuals who have the opportunity to collaborate for mutual benefit in more than one way.”15 How they will split the gain, he reasoned, reflects how much the deal is worth to each individual.
He started by asking the question, What reasonable conditions would any solution — any split — have to satisfy? He then posed four conditions and, using an ingenious mathematical argument, showed that, if his axioms held, a unique solution existed that maximized the product of the players’ utilities. In a sense, his contribution was not so much to “solve” the problem as to state it in a simple and precise way so as to show that unique solutions were possible.
The striking feature of Nash’s paper is not its difficulty, or its depth, or even its elegance and generality, but rather that it provides an answer to an important problem. Reading Nash’s paper today, one is struck most by its originality. The ideas seem to come out of the blue. There is some basis for this impression. Nash arrived at his essential idea — the notion that the bargain depended on a combination of the negotiators’ back-up alternatives and the potential benefits of striking a deal — as an undergraduate at Carnegie Tech before he came to Princeton, before he started attending Tucker’s game theory seminar, and before he had read von Neumann and Morgenstern’s book. It occurred to him while he was sitting in the only economics course he would ever attend.16
The course, on international trade, was taught by a clever and young Viennese émigré in his thirties named Bert Hoselitz. Hoselitz, who emphasized theory in his course, had degrees in law and economics, the latter from the University of Chicago.17 International agreements between governments and between monopolies had dominated trade, especially in commodities, between the wars, and Hoselitz was an expert on the subject of international cartels and trade.18 Nash took the course in his final semester, in the spring of 1948, simply to fulfill degree requirements.19 As always, though, the big, unsolved problem was the bait.
That problem concerned trade deals between countries with separate currencies, as he told Roger Myerson, a game theorist at Northwestern University, in 1996.20 One of Nash’s axioms, if applied in an international trade context, asserts that the outcome of the bargain shouldn’t change if one country revalued its currency. Once at Princeton, Nash would have quickly learned about von Neumann and Morgenstern’s theory and recognized that the arguments that he’d thought of in Hoselitz’s class had a much wider applicability.21 Very likely Nash sketched his ideas for a bargaining solution in Tucker’s seminar and was urged by Oskar Morgenstern — whom Nash invariably referred to as Oskar La Morgue — to write a paper.22
Legend, possibly encouraged by Nash himself, soon had it that he’d written the whole paper in Hoselitz’s class — much as Milnor solved the Borsuk problem in knot theory as a homework assignment — and that he had arrived at Princeton with the bargaining paper tucked into his briefcase.23 Nash has since corrected the record.24 But when the paper was published in 1950, in Econometrica, the leading journal of mathematical economics, Nash was careful to retain full credit for the ideas: “The author wishes to acknowledge the assistance of Professors von Neumann and Morgenstern who read the original form of the paper and gave helpful advice as to the presentation.”25 And in his Nobel autobiography, Nash makes it clear that it was his interest in the bargaining problem that brought him into contact with the game theory group at Princeton, not the other way around: “as a result of that exposure to economic ideas and problems I arrived at the idea that led to the paper The Bargaining Problem’ which was later published in Econometrica. And it was this idea which in turn, when I was a graduate student at Princeton, led to my interest in the game theory studies there.”26