by John Allen Paulos
The stock market, social media, award contests, product reviews, beauty contests, social media, fashion styles, job applications, award contests, product reviews, and even elections, don’t seem to belong in the same crowded sentence. What do they have in common? Before I get there, a couple of abstract analogues to pave the way.
The great economist John Maynard Keynes noted the similarity between deciders, evaluators, reviewers, and judges of all sorts and readers in newspaper beauty contests, which were very popular in his day. The stated 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 large 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”.
Whether in politics, business, or everyday life, how such group judgments about group judgments develop is unclear, but various mathematical tools ranging from network theory to recursion are useful. A simple game I’ve written about elsewhere and have often asked my classes to play is also relevant. In the game, people in some designated 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 read 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 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.)
The problem of guessing 80 percent of the average guess is a bit like Keynes’s description of the readers’ task. What makes it tricky is that different individuals will engage in different degrees of meta‑reasoning about others’ reasoning. So if a group plays this game only once, trying to guess what 80% the average of all the guesses is much more a matter of assessing the others’ psychologies and logical abilities than it is of following an idea to its natural conclusion.
Returning to the first paragraph, observe that there are many examples of this general phenomenon in everyday life. Social media in general is one since it seems quite plausible that the postings and tweets that people share are often those that they judge others in the feed will find most politically pertinent, funny, or intriguing and the others, of course, are doing the same thing. The payoff isn’t $1,000 but rather the coin of the realm, virality, or simply a small increase in popularity.
Of course the stock market and politics naturally belong in this discussion of recursive reckoning. To state the obvious, for most investors the quality of a stock (its fundamentals or how it aligns with their interests and basic values) is at most secondary to an assessment of its popularity among other investors who are similarly guided by their assessments of others who in turn ….
As for politicians, most of them have their fingers in the air to quickly sense changes in public and political sentiment on an issue that they can then lemming-like “wholeheartedly” espouse, especially if their authoritarian leader directs them to do so. Such politicians and many other people as well remind me of the Groucho Marx description of them, “Those are my principles. If you don’t like them, I have others.”
Fashion broadly conceived provides another instance. So many people often wear torn jeans, say, or what they assume their peers will judge to be fashionable even if the fashion is not something that flatters them or something they might personally prefer. The same might be said about tatoos.
A similar story suggests a reason why online dating profiles, photos, and bios are so frequently less than sincere or bonafide, but rather are written in order to appeal to people that the writers they think other people are trying to appeal to.
Those judging art and movie competitions as well as book prizes can also succumb to a desire to vote the way they assume others will vote and not vote their authentic preferences. This is especially true with ranked voting or simply with public voting. I really like X, but both Y and Z are more likely to win, so I’ll go with one of them. College admission statements and job applications are other domains with such a tale.
Obviously and happily, however, the attempt to choose the “prettiest” simply because one wishes to agree with what one imagines the general consensus will be doesn’t always work. A small example: A friend of mine who owns a small successful business and needed a replacement for a departing employee ran an ad for such a person. All except three of the dozens of responders wrote that the requirements briefly mentioned in the ad were exactly the ones that they prized and possessed and no doubt assumed other applicants did as well. Furthermore the grammar of their responses was perfect and there were no misspellings or solecisms, but, and it’s a big but, their writing was totally banal. My friend judged that ChatGPT wrote almost all of the applications, and she gave the job to one of the few responders whose writing and comments showed more verve, authenticity, and independence.
There are countless other examples, but this isn’t the place for a boring political screed. Readers can no doubt construct their own.
In any case, the underappreciated bottom line is that it’s difficult, important, and dizzying for people to engage in recursive reckoning. Sometimes, however, it’s more difficult, more important, and clarifying not to. Doing and saying what you truly like is often righteous reckoning.
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John Allen Paulos is an emeritus Professor of Mathematics at Temple University and the author of Innumeracy and A Mathematician Reads the Newspaper. These and his other books are described and available here.
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