How We Guess What Others Will Do

The stock market and beauty contests don't seem to belong in the same sentence. It's almost painful to think simultaneously about both WorldCom and Miss Universe. What do faceless corporations and captivating faces have in common?

One obvious answer is that companies and contestants are both frequently judged on superficial grounds. Another is that both companies and contestants sometimes try to hide unsavory aspects of their pasts — Enron's shredding of old documents, for example, or a model's lying about old nude pictures of herself.

John Maynard Keynes, arguably the greatest economist of the 20th century, pointed to a much deeper similarity, that between investors and judges. Specifically, he compared the position of investors in a market to that of reader/judges in newspaper beauty contests, which were very popular in his day. The apparent task of the readers was to pick the five prettiest out of, say, 100 contestants, but their real job was more complicated. The reason was that the newspaper rewarded them with prizes only if they picked the five contestants who received the most votes from the other readers.

That is, they had to pick the contestants that they thought were the most likely to be picked by the other readers, and the other readers had to try to do the same. They were not to give undue weight to their own taste. Instead they had to anticipate, in Keynes' words, "what average opinion expects the average opinion to be" (or, worse, anticipate what the average opinion expects the average opinion expects the average opinion to be).

A Simple Game

Countless situations — not just the stock market — require that we anticipate the actions of many other people, but our understanding of how we do this is hazy, to say the least. Whether in politics, war, business, or everyday life, how group consensus develops and sometimes suddenly changes remains intuitive and unformalized.

Various mathematical tools ranging from network theory to fractals are useful, but a simple game I discuss in my forthcoming book, A Mathematician Plays the Stock Market, is also relevant. In the game, people in a group are each asked to choose a number between 0 and 100. Furthermore, they're directed to pick the number that they think will be closest to 80 percent of the average number chosen by the group. The person who comes closest to this value will receive $1,000 for his or her efforts. Don't go on until you decide what number you would pick.

Some in the group might reason that the average number chosen is likely to be 50 and so these people would guess 40, which is 80 percent of this. Others might anticipate that people will guess 40 for this reason and so they would guess 32, which is 80 percent of 40. Still others might anticipate that people will guess 32 for this reason and so they would guess 25.6, 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 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.)

Stocks and Reasoning About Others' Reasoning

The problem of guessing 80 percent of the average guess is a bit like Keynes's description of the investors' task. What makes it tricky is that anyone insightful enough to cut to the heart of the problem and guess 0 right away is almost certain to be wrong, since different individuals will engage in different degrees of meta-reasoning about others' reasoning. (This may be sour grapes for my disastrous misadventure in the stock market.) 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.

If a group plays this game only once or twice, guessing the average of all the guesses is as much a matter of reading the others' psychologies as it is of following an idea to its logical conclusion. Gauging investors is often as important as gauging investments and is likely to be more difficult as well.

Finally, the recent SEC settlement to reform the investment industry reminds us of an unpleasant aspect of this market/contest analogy. It's not uncommon that some of the market's formerly prettiest contestants — WorldCom, Tyco, Adelphia, Global Crossing, and HealthSouth — turn out on closer inspection to have pimples and pustules all over their corporate faces.

Professor of mathematics at Temple University and adjunct professor of journalism at Columbia University, John Allen Paulos is the author of several best-selling books, including Innumeracy, and the forthcoming A Mathematician Plays the Stock Market. His Who’s Counting? column on appears the first weekend of every month.