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Economics 1944

Theory of Games and Economic Behavior

John von Neumann & Oskar Morgenstern

Rational choice under conflict and risk, made mathematical.

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In depth · the introduction

When your best move depends on what someone else is about to do, ordinary maths runs out — so two thinkers invented the maths of outguessing each other, and of choosing wisely when the outcome is a gamble.

The big idea

Most of economics until then imagined a lone person facing fixed prices, like shopping in a store where nothing you do changes the tags. But real life is full of situations where what's best for you depends on what others choose — bargaining, bidding, competing for customers, even bluffing at cards. Von Neumann and Morgenstern argued these are all the same kind of problem: a game of strategy. And they showed it could be done with rigorous mathematics, not just hand-waving.

Their first result is striking. In a strictly competitive two-player game — where one wins exactly what the other loses — there is always a provably best way to play, if you're willing to mix up your moves randomly so you can't be read. Their second result tackled risk: they built a clean way to turn your gut preferences between safe bets and gambles into actual numbers — a personal “utility” scale — so that the rational choice is simply the option with the highest expected utility.

How it came about

John von Neumann was one of the great mathematicians of the century, a prodigy who worked on everything from quantum mechanics to the first computers and the atomic bomb. As a young man in 1928 he had proved a theorem about two-player games. Oskar Morgenstern was an Austrian economist, exiled to Princeton, who was convinced economics needed firmer mathematical foundations and kept pressing the point.

The two met at Princeton and started a collaboration that grew far beyond what either expected — a planned pamphlet swelled into a 625-page book. Published in 1944, in the middle of the Second World War, it founded an entire field. Honesty compels a note on credit: the heavy mathematics is overwhelmingly von Neumann's; Morgenstern's gift was seeing, and arguing relentlessly, that these tools belonged at the heart of economics.

Why it mattered

Before this book, economists had no rigorous way to model people who are reasoning about each other. After it, “game theory” became a language spoken across economics, politics, biology, and computing. Just as importantly, the utility idea gave a precise meaning to risk aversion — why a guaranteed $40 can feel better than a coin-flip for $100 — which sits underneath how insurance is priced and how investments are weighed. Later thinkers found the limits of the theory, too, and those limits launched the field of behavioral economics.

A way to picture it

Think of rock-paper-scissors. If you always throw rock, a clever opponent will always throw paper and beat you every time. The only unbeatable plan is to randomize — one-third rock, one-third paper, one-third scissors — so no one can predict you. Von Neumann's minimax theorem says every strictly competitive game has exactly this kind of best, unpredictable strategy. And for risky choices, the utility curve is like the difference between hunger levels: the first slice of pizza thrills you, the eighth barely registers — so a sure meal can be worth more to you than a gamble for a feast.

An interactive decision panel: a utility-of-money curve on the left and a lottery wheel against a sure $40 on the right. Slide the gamble's winning chance and a caution slider, and the panel says whether you take the gamble or the sure money, and what guaranteed amount the gamble is worth to you.

Where it sits

A century and a half after Adam Smith described an economy of independent traders guided by an “invisible hand,” this book added the missing piece: what happens when people don't just respond to prices but actively scheme against each other. The thread runs on through John Nash, whose 1950 equilibrium extended the idea to games that aren't strictly win-lose, and reaches today's auction designers and the “adversarial” training that pits two neural networks against each other. It also begins the story that behavioral economics would later complicate — the discovery that real people don't always obey the tidy logic of expected utility.

The original document
Original source text

The program — mathematics for economics

John von Neumann & Oskar Morgenstern · Theory of Games and Economic Behavior · 1944 · Ch. 1, §1.2
It is not that there exists any fundamental reason why mathematics should not be used in economics. The arguments often heard that because of the human element, of the psychological factors etc., or because there is — allegedly — no measurement of important factors, mathematics will find no application, can all be dismissed as utterly mistaken.
The reason why mathematics has not been more successful in economics must, consequently, be found elsewhere. … The lack of real success is largely due to a combination of unfavorable circumstances, some of which can be removed gradually. To begin with, the economic problems were 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.

Why economics is a game of strategy

Ch. 1, §1.1 — Formulation of the Economic Problem
We hope to establish satisfactorily, after developing a few plausible schematizations, that the typical problems of economic behavior become strictly identical with the mathematical notions of suitable games of strategy.
It will be seen, however, that this theory of games of strategy is the proper instrument with which to develop a theory of economic behavior. One would misunderstand the intent of our discussions by interpreting them as merely pointing out an analogy between these two spheres. We hope to establish satisfactorily … that the typical problems of economic behavior become strictly identical with the mathematical notions of suitable games of strategy.

Measuring utility by a gamble (§3.3)

Ch. 1, §3.3 — The Notion of Utility
Let us for the moment accept the picture of an individual whose system of preferences is all-embracing and complete, i.e. who, for any two objects or rather for any two imagined events, possesses a clear intuition of preference.
If he now prefers A to the 50-50 combination of B and C, this provides a plausible base for the numerical estimate that his preference of A over B is in excess of his preference of C over A.
[ … ]
It is well known that thereby utilities — or rather differences of utilities — become numerically measurable. … We have practically defined a numerical scale for the quantity which we used to call ‘utility.’

Where and when it was signed

The two authors — a Hungarian-born mathematician at the Institute for Advanced Study and an Austrian economist at Princeton — finished the preface to the first edition together.
Princeton, N. J. · January 1943