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 conclude 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 issue—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 utility. 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.)