In drawing analogies between economics and physics, von Neumann and Morgenstern talked a lot about the theory of heat (or, as it is more pretentiously known, thermodynamics). They pointed out, for instance, that measuring heat precisely did not lead to a theory of heat; physicists needed the theory first, in order to understand how to measure heat in an unambiguous way. In a similar way, game theory needed to be developed first to give economists the tools they needed to measure economic variables properly.
The example of the theory of heat played another crucial role—in articulating a basic issue within game theory itself. At the outset, von Neumann and Morgenstern made it clear that they did not want to venture into the philosophical quagmire of defining all the nuances of utility. For them, to develop game theory for use in economics, it was enough to equate utility with money. For the businessman, money (as in profits) is a logical measure of utility; for consumers, income (minus expenses) is a good measure of util- ity, or you could think of the utility of an object as the price you were willing to pay for it. And money can be used as a currency for translating what anyone wants into more specific objects or events or experiences or whatever. So equating utility with money is a convenient simplifying assumption, allowing the theory to focus
on the strategic aspects of how to achieve what you want, without worrying about the complications involved in defining what you want.
However, there remained an important aspect of utility that von Neumann and Morgenstern had to address. Was it even pos- sible, in the first place, to define utility in a numerical way, to make it susceptible to a mathematical theory? (Bernoulli had proposed a way to calculate utility, but he had not tried to prove that the concept could be a basis for making rational choices in a consistent way.) Money (which obviously is numerical) could really be a good stand-in for the more complex concept of utility only if utility can really be represented by a numerical concept. And so they had to show that it was possible to define utility in a mathematically rig- orous way. That meant identifying axioms from which the notion of utility could be deduced and measured quantitatively.
As it turned out, utility could be quantified in a way not unlike the approach physicists used to construct a scientifically rigorous definition of temperature. After all, primitive notions of utility and temperature are similar. Utility, or preference, can be thought of as just a rank ordering. If you prefer A to B, and B to C, you surely prefer A to C. But it is not so obvious that you can ascribe a num- ber to how much you prefer A to B, or B to C. It was once much the same with heat—you could say that something felt warmer or cooler than something else, but not necessarily how much, cer- tainly not in a precise way—before the development of the theory of heat. But nowadays the absolute temperature scale, based on the laws of thermodynamics, gives temperature an exact quantita- tive meaning. And von Neumann and Morgenstern showed how you could similarly convert rank orderings into numerically pre- cise measures of utility.
You can get the essence of the method from playing a modi- fied version of Let’s Make a Deal. (For the youngsters among you, that was a famous TV game show, in which host Monty Hall of- fered contestants a chance to trade their prizes for possibly more valuable prizes, at the risk of getting a clunker.) Suppose Monty offers you three choices: a BMW convertible, a top-of-the-line big-
screen plasma TV, or a used tricycle. Let’s say you want the BMW most of all, and that you’d prefer the TV to the tricycle. So it’s a simple matter to rank the relative utility of the three products. But here comes the deal. Your choice is to get either the plasma TV, OR a 50-50 chance of getting the BMW. That is, the TV is behind Door Number 1, and the BMW is behind either Door Number 2 or Door Number 3. The other door conceals the tricycle.
Now you really have to think. If you choose Door Number 1—the plasma TV—you must value it at more than 50 percent as much as the BMW. But suppose the game is more complicated, with more doors, and the odds change to a 60 percent chance of the BMW, or 70 percent. At some point you will be likely to opt for the chance to get the BMW, and at that point, you could con- clude that the utilities are numerically equal—you value the plasma TV at, say, 75 percent as much as the BMW (plus 25 percent of the tricycle, to be technically precise). Consequently, to give utility a numerical value, you just have to arbitrarily assign some number to one choice, and then you can compare other choices to that one using the probabilistic version of Let’s Make a Deal.
So far so good. But there remains the problem of operating in a social economy where your personal utility is not the only is- sue—you have to anticipate the choices of others. And in a small- scale Gilligan’s Island economy, pure strategic choices can be subverted by things like coalitions among some of the players. Again, the theory of heat offers hope.
Temperature is a measure of how fast molecules are moving. In principle, it’s not too hard to describe the velocity of a single molecule, just as you could easily calculate Robinson Crusoe’s util- ity. But you’d have a hard time with Gilligan’s Island, just as it becomes virtually impossible to keep track of all the speeds of a relatively few number of interacting molecules. But if you have a trillion trillion molecules or so, the interactions tend to average out, and using the theory of heat you can make precise predictions about temperature. (The math behind this is, of course, statistical mechanics, which will become even more central to the game theory story in later chapters.)
As von Neumann and Morgenstern pointed out, “very great numbers are often easier to handle than those of medium size.”21 That was exactly the point made by Asimov’s psychohistorians: Even though you can’t track each individual molecule, you can predict the aggregate behavior of vast numbers, precisely what taking the temperature of a gas is all about. You can measure a value related to the average velocity of all the molecules, which reflects the way the individual molecules interact. Why not do the same for people? It worked for Hari Seldon. And it might work for a sufficiently large economy. “When the number of participants becomes really great,” von Neumann and Morgenstern wrote, “some hope emerges that the influence of every particular partici- pant will become negligible.”22
With the basis for utility established at the outset, von Neumann and Morgenstern could proceed simply by taking money to be utility’s measure. The bulk of their book was then devoted to the issue of finding the best strategy to make the most money.
At this point, it’s important to clarify what they meant by the concept of strategy. A strategy in game theory is a very specific course of action, not a general approach to the game. It’s not like tennis, for instance, where your strategy might be “play aggres- sively” or “play safe shots.” A game theory strategy is a defined set of choices to make for every possible circumstance that might arise. In tennis, your strategy might be to “never rush the net when your opponent serves; serve and volley whenever you are even or ahead in a game; always stay back when behind in a game.” And you’d have other rules for all the other situations.
There’s one additional essential point about strategy in game theory—the distinction between “pure” strategies and “mixed” strategies. In tennis, you might rush the net after every serve (a pure strategy) or you might rush the net after one out of every three serves, staying back at the baseline two times out of three (a mixed strategy). Mixed strategies often turn out to be essential for making game theory work.
In any event, the question isn’t whether there is always a good
rules for strategic behavior that covers all eventualities. And in fact, there is—for two-person zero-sum games. You can find the best strategy using the minimax theorem that von Neumann published in 1928. His proof of that theorem was notoriously complicated. But its essence can be boiled down into something fairly easy to remember: When playing poker, sometimes you need to bluff.