Against the Gods: The Remarkable Story of Risk (46 page)

Von Neumann first presented his theory of games of strategy in a
paper that he delivered in 1926, at the age of 23, to the Mathematical
Society at the University of Gottingen; the paper appeared in print two
years later. Robert Leonard of the University of Quebec, a leading historian of game theory, has surmised that this paper was not so much the
product of a "detached moment of inspiration" as an effort by von Neumann to focus his restless fancy on a subject that had been attracting the attention of German and Hungarian mathematicians for some
time. Apparently the stimulus for the work was primarily mathematical,
with little or nothing to do with decision-making as such.

Although the subject matter of the paper appears to be trivial at first
glance, it is highly complex and mathematical. The subject is a rational
strategy for playing a childhood game called match-penny. Each of two
players turns up a coin at the same moment as the other. If both coins
are heads or if both are tails, player A wins. If different sides come up,
player B wins. When I was a boy, we played a variation of this game in
which my opponent and I took turns shouting either "Odds!" or
"Evens!" as, at an agreed call, we opened our fists to show either one
finger or two.

According to von Neumann, the trick in playing match-penny
against "an at least moderately intelligent opponent" lies not in trying
to guess the intentions of the opponent so much as in not revealing
your own intentions. Certain defeat results from any strategy whose
aim is to win rather than to avoid losing. (Note that dealing with the
possibility of losing appears here for the first time as an integral part of
risk management.) So you should, play heads and tails in random fashion, simulating a machine that would systematically reveal each side of
the coin with a probability of 50%. You cannot expect to win by
employing this strategy, but neither can you expect to lose.

If you try to win by showing heads six times out of every ten plays,
your opponent will catch on to your game plan and will win an easy
victory. She will play tails six times out of every ten plays if she wins
when the pennies fail to match; she will play heads six times out of
every ten plays if she wins when the pennies do match.

So the only rational decision for both players is to show heads and
tails in random fashion. Then, over the long run, the pennies will
match half the time and will fail to match half the time. Fun for a little
while, but then boring.

The mathematical contribution von Neumann made with this
demonstration was the proof that this was the only outcome that could
emerge from rational decision-making by the two players. It is not the
laws of probability that decree the 50-50 payoff in this game. Rather,
it is the players themselves who cause that result. Von Neumann's paper
is explicit about this point:

... [E]ven if the rules of the game do not contain any elements of
"hazard" (i.e., no draws from urns) ... dependence on ... the statistical element is such an intrinsic part of the game itself (if not of the
world) that there is no need to introduce it artificially.4

The attention von Neumann's paper attracted suggests that he had
something of mathematical importance to convey. It was only later
that he realized that more than mathematics was involved in the theory
of games.

In 1938, while he was at the Institute for Advanced Study socializing with Einstein and his friends, von Neumann met the German-born
economist Oskar Morgenstern. Morgenstern became an instant acolyte.
He took to game theory immediately and told von Neumann he wanted
to write an article about it. Though Morgenstern's capability in mathematics was evidently not up to the task, he persuaded von Neumann to
collaborate with him on a paper, a collaboration that extended into the
war years. The results of their joint efforts was Theory of Games and
Economic Behavior, the classic work in both game theory and its application to decision-making in economics and business. They completed the
650 pages of their book in 1944, but the wartime paper shortage made
Princeton University Press hesitant to publish it. At last a member of
the Rockefeller family personally subsidized the publication of the
book in 1953.

The economic subject matter was not entirely new to von
Neumann. He had had some interest in economics earlier, when he
was trying to see how far he could go in using mathematics to develop
a model of economic growth. Always the physicist as well as the mathematician, his primary focus was on the notion of equilibrium. "As
[economics] deals throughout with quantities," he wrote, "it must be a
mathematical science in matter if not in language ... a close analogy to
the science of statical mechanics."

Morgenstern was born in Germany in 1902 but grew up and was
educated in Vienna. By 1931, he had attained sufficient distinction as an
economist to succeed Friedrich von Hayek as director of the prestigious
Viennese Institute for Business Cycle Research. Though he was a
Christian with a touch of anti-Semitism, he left for the United States in 1938, following the German invasion of Austria, and soon found a position on the economics faculty at Princeton.5

Morgenstern did not believe that economics could be used for predicting business activity. Consumers, business managers, and policymakers, he argued, all take such predictions into consideration and alter their decisions and actions accordingly. This response causes the forecasters to change their forecast, prompting the public to react once again. Morgenstern compared this constant feedback to the game played by Sherlock Holmes and Dr. Moriarty in their attempts to outguess each other. Hence, statistical methods in economics are useless except for descriptive purposes, "but the diehards don't seem to be aware of this."6

Morgenstern was impatient with the assumption of perfect foresight that dominated nineteenth-century economic theory. No one, he insisted, can know what everybody else is going to do at any given moment: "Unlimited foresight and economic equilibrium are thus irreconcilable with each other."' This conclusion drew high praise from Frank Knight and an offer by Knight to translate this paper by Morgenstern from German into English.

Morgenstern appears to have been short on charm. Nobel Laureate Paul Samuelson, the author of the long-run best-selling textbook in economics, once described him as "Napoleonic.... [A]lways invoking the authority of some physical scientists or other."*'
Another contemporary recalls that the Princeton economics department `just hated Oskar."9 Morgenstern himself complained about the lack of attention his beloved masterpiece received from others. After visiting Harvard in 1945 he noted "none of them" had any interest in game theory.10 He reported in 1947 that a fellow economist named Ropke said that game theory "was Viennese coffeehouse gossip."t
When he visited a group of distinguished economists in Rotterdam in 1950, he discovered that they "wanted to know nothing about [game theory] because it disturbs them."

Although an enthusiast for the uses of mathematics in economic
analysis-he despised Keynes's nonrigorous treatment of expectations
and described The General Theory as "simply horrible"-Morgenstern
complained constantly about his problems with the advanced material
into which von Neumann had lured him.11 Throughout their collaboration Morgenstern held von Neumann in awe. "He is a mysterious
man," Morgenstern wrote on one occasion. "The moment he touches
something scientific, he is totally enthusiastic, clear, alive, then he sinks,
dreams, talks superficially in a strange mixture.... One is presented
with the incomprehensible."

The combination of the cool mathematics of game theory and the
tensions of economics seemed a natural fit for a mathematician with an
enthusiasm for economics and an economist with an enthusiasm for
mathematics. But the stimulus to combine the two arose in part from
a shared sense that, to use Morgenstern's words, the application of
mathematics to economics was "in a lamentable condition."12

An imperial motivation was also there-the aspiration to make
mathematics the triumphant master in the analysis of society as well as
in the analysis of the natural sciences. While that approach would be
welcomed by many social scientists today, it was probably the main
source of the resistance that game theory encountered when it was first
broadly introduced in the late 1940s. Keynes ruled the academic roost
at the time, and he rejected any sort of mathematical description of
human behavior.

The Theory of Games and Economic Behavior loses no time in advocating the use of the mathematics in economic decision-making. Von
Neumann and Morgenstern dismiss as "utterly mistaken" the view that
the human and psychological elements of economics stand in the way of
mathematical analysis. Recalling the lack of mathematical treatment in
physics before the sixteenth century or in chemistry and biology before
the eighteenth century, they claim that the outlook for mathematical
applications in those fields "at these early periods can hardly have been
better than that in economics-mutatis mutandis-at present."13

Von Neumann and Morgenstern reject the objection that their
rigidly mathematical procedures and their emphasis on numerical quan tities are unrealistic simply because "the common individual ... conducts his economic activities in a sphere of considerable haziness."14
After all, people respond hazily to light and heat, too:

[I]n order to build a science of physics, these phenomena [heat and
light] had to be measured. And subsequently, the individual has come
to use the results of such measurements-directly or indirectly-even
in his everyday life. The same may obtain in economics at a future
date. Once a fuller understanding of human behavior has been
achieved with the aid of a theory that makes use of [measurement],
the life of the individual may be materially affected. It is, therefore,
not an unnecessary digression to study these problems.15

The analysis in The Theory of Games and Economic Behavior begins
with the simple case of an individual who faces a choice between two
alternatives, as in the choice between heads and tails in match-penny. But
this time von Neumann and Morgenstern go more deeply into the
nature of the decision, with the individual making a choice between two
combinations of events instead of between two single possibilities.

They take as an example a man who prefers coffee to tea and tea to
milk." We ask him this question: "Do you prefer a cup of coffee to a
glass that has a 50-50 chance of being filled with tea or milk?" He
prefers the cup of coffee.

What happens when we reorder the preferences but ask the same
question? This time the man prefers milk over both coffee and tea but
still prefers coffee to tea. Now the decision between coffee for certain
and a 50-50 chance of getting tea or milk has become less obvious than
it was the first time, because now the uncertain outcome contains
something he really likes (milk) as well as something he could just as
well do without (tea). By varying the probabilities of finding tea or milk
and by asking at what point the man is indifferent between the coffee
for certain and the 50-50 gamble, we can develop a quantitative estimate-a hard number-to measure by how much he prefers milk to
coffee and coffee to tea.