The Psychology of the Stimulus in a Game of Numbers

An intriguing mathematical-psychological game sheds a little light on the size of the stimulus package recently passed by Congress. In particular it illustrates the truism that passing legislation calls for an understanding of the interplay between politics and economics.

First, note that most economists believe that the $800 billion ($0.8 trillion) stimulus is too small to do the job. They say the package needs to be much greater to close the $2.9 trillion gap between what the Congressional Budget Office forecasts the economy is capable of producing over the next three years and what it's likely to produce without the stimulus.

Further argument and arm-twisting might reduce the gap further, but several iterations probably would be required to get something sufficiently effective. Recognizing political realities is essential, however. If the stimulus package presented had been significantly bigger, it likely would have failed.

The 80 Percent Game

Now for the game which I sometimes present to my classes.

Consider a situation in which people in a large group are each asked to independently choose a number (not necessarily a whole number) between 0 and 100. They are further directed to pick the number that they think will be closest to 80 percent of the average number chosen by the group.

The one who comes closest will receive $10,000 for his efforts. (This latter offer I don't make to my students.) Stop for a bit and think what number you would pick.

Some in the group might reason that if people choose numbers at random, the average number chosen is likely to be 50 and so these people would guess 40, which is 80 percent of 50. Others, looking around at the thoughtful looks on people's faces, might anticipate that many of them will reason in the same way and guess 40 too, and so they would guess 32 (or perhaps a bit higher), which is 80 percent of 40.

Still others, seeing the intense concentration of other smart people in the group, might anticipate that they will guess 32 (or so) for the same reason and so they would guess 25.6 (or a bit higher), which is 80 percent of 32.

If the group continues to play this game, they will gradually learn to engage in ever more iterations of this meta-reasoning about others' reasoning until eventually they all reach the optimal response, which is 0. Since they all want to choose a number equal to 80 percent of the average, the only way they can all do this is by choosing 0, the only number equal to 80 percent of itself.

Choosing 0 leads to the so-called Nash equilibrium of this game. It results when individuals modify their actions until they can no longer benefit from changing them given what the others' actions are.

Being Right Mathematically vs. Being Right Politically

What makes this problem interesting is that anyone bright enough to cut to the heart of the problem and guess 0 right away is almost certain to be wrong (that is, almost certain not to win the $10,000 prize), since different individuals will engage in different degrees of meta-reasoning about others' reasoning.

Some, to increase their chances, will choose numbers a little above or a little below the natural guesses of 40 or 32 or 25.6 or 20.48. There will be some random guesses as well and some guesses of 50 or more. Unless the group is very unusual, few will guess 0 initially. Not everyone reasons so trenchantly.

If someone plays this game only once or twice, guessing the average of all the guesses is as much a matter of reading the others' intelligence and psychology as it is of following an idea to its logical conclusion.

By the same token and to return to the stimulus, understanding the constraints on legislators is often as important as assessing the legislation under consideration. And it's likely to be more difficult as well.

Economics is nowhere near as clear-cut as mathematics and the above analogy is no doubt a bit strained, but it is nevertheless suggestive. Insisting on being right mathematically (that is, choosing 0 as your number) or right economically (enacting a much larger stimulus bill) is not always a smart thing to do.

It's certainly not the same thing as being right psychologically (gauging others' number choices) or being right politically (assessing others' partisan commitments).

Some variant of this idea also helps answer the question perennially asked of academics and others: If you're so smart, why aren't you rich? The inverse question posed to financiers and Wall Street types remains unanswered: If you're so dumb, why are you rich?

John Allen Paulos, a professor of mathematics at Temple University, is the author of the best-sellers "Innumeracy" and "A Mathematician Reads the Newspaper," as well as "Irreligion: A Mathematician Explains Why The Arguments for God Just Don't Add Up." His "Who's Counting?" column on appears the first weekend of every month.