Against the Gods: The Remarkable Story of Risk

by Peter L. Bernstein

  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.

  The example becomes more realistic when we translate it into a technique for measuring the utility-the degree of satisfaction-of pos sessing $1 compared to the utility of possessing a second dollar, for a total of $2. This man's favored outcome must now be $2, which takes the place of milk in the above example; no money takes the place of tea, the least favored outcome, and $1 becomes the middle choice and takes the place of coffee.

  Once again we ask our subject to choose between a sure thing and a gamble. But in this case the choice is between $1 versus a gamble that pays either $2 or nothing. We set the odds in the gamble at a 50% chance of $2 and a 50% chance of nothing, giving it a mathematical expectancy of $1. If the man declares that he is indifferent between the $1 certain and the gamble, then he is neutral on the subject of risk at this low level of the gamble. According to the formula proposed by von Neumann and Morgenstern, the probability on the favorite possibilityin this case the $2 outcome-defines how much the subject prefers $1 over zero compared with how much he prefers $2 over zero. Here 50% means that his preference for $1 over zero is half as great as his preference for $2 over zero. Under these circumstances, the utility of $2 is double the utility of $1.

  The response might well differ with other people or under other circumstances. Let us see what happens when we increase the amount of money involved and change the probabilities in the gamble. Assume now that this man is indifferent between $100 certain and a gamble with a 67% probability of paying $200 and a 33% probability of coming up zero. The mathematical expectancy of this gamble is $133; in other words, the man's preference for the certain outcome-$100-is now larger than it was when only a couple of dollars were involved. The 67% probability on $200 means that his preference for $100 over zero is two-thirds as great as his preference for $200 over zero: the utility of the first $100 is larger than the utility of the second $100. The utility of the larger sum diminishes as the amount of money at risk increases from single digits to triple digits.

  If all this sounds familiar, it is. The reasoning here is precisely the same as in the calculation of the "certainty equivalent," which we derived from Bernoulli's fundamental principle that the utility of increases in wealth will be inversely related to the amount of wealth already possessed (page 105). This is the essence of risk aversion-that is, how far we are willing to go in making decisions that may provoke others to make decisions that will have adverse consequences for us. This line of analysis puts von Neumann and Morgenstern strictly in the classical mode of rationality, for rational people always understand their preferences clearly, apply them consistently, and lay them out in the fashion described here.

  Alan Blinder, a long-time member of the Princeton economics faculty, co-author of a popular economics textbook, and Vice Chairman of the Federal Reserve Board from 1994 to 1996, has provided an interesting example of game theory.17 The example appeared in a paper published in 1982. The subject was whether coordination is possible, or even desirable, between monetary policy, which involves the control of short-term interest rates and the supply of money, and fiscal policy, which involves the balance between federal government spending and tax revenue.

  The players in this game are the monetary authorities of the Federal Reserve System and the politicians who determine the mix between government spending and tax revenues. The Federal Reserve authorities perceive control of inflation as their primary responsibility, which makes them favor economic contraction over economic expansion. They serve long terms-fourteen years for members of the Board, and until retirement age for presidents of the Federal Reserve Banks-so they can act with a good deal of independence from political pressures. The politicians, on the other hand, have to run regularly for election, which leads them to favor economic expansion over economic contraction.

  The object of the game is for one player to force the other to make the unpleasant decisions. The Fed would prefer to have tax revenues exceed spending rather than to have the government suffer a budget deficit. A budget surplus would tend to hold inflation in check, thereby protecting the members of the Fed from being seen as the bad guys. The politicians, who worry about being elected, would prefer the Fed to keep interest rates low and the money supply ample. That policy would stimulate business activity and employment and would relieve Congress and the President of the need to incur a budget deficit. Neither side wants to do what the other side wants to do.

  Blinder sets up a matrix that shows the preferences of each side in regard to each of three decisions by the other: contract, do nothing, or expand. The numbers above the diagonal in each square represent the order of preference of the members of the Fed; the numbers below the diagonals represent the order of preference of the politicians.

  Blinder's payoff matrix. (Adapted from Alan S. Blinder, 1982, "Issues in the Coordination of Monetary and Fiscal Policies," in Monetary Policy Issues in the 1980s, Kansas City, Missouri: Federal Reserve Bank of Kansas City, pp. 3-34.)

  The highest-ranked preferences of the Fed (1, 2, and 3) appear in the upper left-hand corner of the matrix, where at least one side is contractionary while the other is either supportive or does nothing to rock the boat. The members of the Fed clearly prefer to have the politicians do their job for them. The three highest-ranked preferences of the politicians appear in the lower right-hand corner, where at least one side is expansionary while the other is either supportive or does nothing to rock the boat. The poli
ticians clearly prefer to have the Fed adopt expansionary policies and for the politicians to do nothing. The lowest-ranked preferences of the politicians appear in the left-hand column, while the lowest-ranked preferences of the Fed appear in the bottom row. This is hardly a situation in which much accommodation is likely.

  How will the game end? Assuming that the relationship between the Fed and the politicians is such that collaboration and coordination are impossible, the game will end in the lower left-hand corner where monetary policy is contractionary and fiscal policy is expansionary. This is precisely the outcome that emerged in the early Reagan years, when Blinder wrote this paper.

  Why this outcome and no other? First, both sides are displaying their character here-an austere Fed and generous politicians. Under our assumption that the Fed cannot persuade the politicians to run a budget surplus and that the politicians cannot persuade the Fed to lower interest rates, neither side has any desire to alter its preferences nor can either dare to be simply neutral.

  Look upward and to the right from those two 7s. Note that there is no number below the diagonal (the politicians' preference) looking upward on the left-hand vertical that is lower than 7; there is no number above the diagonal (the Fed's preference) looking horizontally to the right that is lower than 7. As long as the Fed is contractionary and the politicians are expansionary, both sides are making the best of a bad bargain.

  That is not the case in the upper right-hand corner, where the Fed's monetary policy is less tight and a budget surplus emerges. Looking left horizontally and above the diagonals, we note that both the choices rank higher than 4: the Fed would rather do nothing or even be contractionary as compared to contributing to a business expansion that might end up in an inflationary situation. The opposite view would prevail among the politicians. Looking downward vertically, we find that both the choices rank higher than 4: the politicians would rather do nothing or run a deficit than follow a policy that cost them their jobs if their constituents lose their jobs as a result.