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In 1997 the American behavioural economist Richard Thaler (b. 1945) ran an experiment in the Financial Times a Guessing Game which was a version of Keynes’ Beauty Contest.
Guess a number from zero to 100, with the goal of making your guess as close as possible to two-thirds of the average guess of all those participating in the contest. To help you think about this puzzle, suppose there are three players who guessed 20, 30 and 40 respectively. The average guess would be 30, two-thirds of which is 20, so the person who guessed 20 would win.
If you did not enter the contest, you might consider what your guess might have been.
Now that you have thought, consider what I will call a zero-level thinker. He says: “I don’t know. This seems like a maths problem. I will just pick a number at random.” Lots of people guessing a number between zero and 100 at random will produce an average guess of 50.
How about a first-level thinker? She says: “The rest of these players don’t like to think much, they will probably pick a number at random, averaging 50, so I should guess 33, two-thirds of 50.”
A second-level thinker will say: “Most players will be first-level thinkers and think that other players are a bit dim, so they will guess 33. Therefore I will guess 22.”
A third-level thinker: “Most players will discern how the game works and will figure that most people will guess 33. As a result they will guess 22, so I will guess 15.”
Of course, there is no convenient place to get off this train of thinking. Do you want to change your guess?
Here is another question: what is the Nash equilibrium for this scenario? Named for John Nash, the mathematician and subject of the film A Beautiful Mind who sadly was recently killed in a car crash, the Nash equilibrium in this game is a number that if everyone guessed it, no one would want to change their guess. The only Nash equilibrium in this game is zero.
To see why, suppose everyone guessed three. Then the average guess would be three and you would want to guess two-thirds of that, or two. But if everyone guessed two you would want to guess 1.33, and so forth. If, and only if, all participants guessed zero would no one want to change his or her guess.
Formally, this game is identical to Keynes’s beauty contest: you have to guess what other people are thinking that other people are thinking. In economics, the “number guessing game” is commonly referred to as the “beauty contest”.
Thanks to the FT, this is the second time I have run this experiment on a large scale [see panel]. In 1997 we offered two business-class tickets to North America. Now, in these days of austerity, entrants were offered what I have been assured is a posh travel bag. Personally, I am also throwing in an autographed copy of my recent book Misbehaving, on which this essay is based.
How have things changed? Well, one finding will comfort tradition-bound economists. When the prize was two business-class tickets we had 1,382 contestants. With only a travel bag on offer, entrants dropped to 583. Economic theory is redeemed!
Even with the smaller number of entrants, the results were nearly identical. In 1997 the average guess was 18.9, meaning the winning guess was 13. This time the average guess was 17.3, leading to a winning guess of 12. The distribution of guesses also looks like the one from 1997.
Many contestants were able to figure out the Nash equilibrium and guessed zero or one, thinking everyone else would be as clever as they were. A large number also guessed 22, showing second-level thinking. Just as last time, there was an assortment of pranksters who guessed 99 or 100, trying to skew the results.
Keynes’s beauty-contest analogy remains an apt description of what money managers do. Many investors call themselves “value managers”, meaning they try to buy stocks that are cheap. Others call themselves “growth managers”, meaning they try to buy stocks that will grow quickly. But of course no one is seeking to buy stocks that are expensive or stocks of companies that will shrink. So what these managers are really trying to do is buy stocks that will go up in value — or, in other words, stocks that they think other investors will later decide should be worth more.
Buying a stock that the market does not fully appreciate today is fine, as long as the rest of the market comes around to your point of view sooner rather than later. Remember another of Keynes’s famous lines: “In the long run we are all dead.” The typical long run for a portfolio manager is no more than a few years; often just a few months! So to beat the market a money manager has to have a theory about how other investors will change their minds. In other words, their approach has to be behavioural.
The Thaler challenge
Reading through the submissions was to meet a wonderful slice of FT readers: from the witty to the saboteurs, from accountants to students, writes Caroline Daniel. Many were pleased with their own logic; most were men.
Inevitably, there were readers who suggested 42 on the grounds that in The Hitchhiker’s Guide to the Galaxy “42 is the answer to the ultimate question of life”. Another reader joked: “33, the age of Jesus when crucified by the Romans.”
Some readers chose numbers for personal reasons: “65 means both financial freedom and also the freedom to do as I will.” Another picked 56 as it was “the number I was issued when I became a special constable” in the Met. One chose a number based on “how many kilometres I ran in the park while thinking about Keynes and his idea of burying piles of money there”.
There were instances of sabotage. One entrant confessed he chose 99 “in an evil attempt to render the statisticians’ attempts at guessing a correct answer useless”. A round-robin group all nominated 100, ominously declaring, “All of your base belong to us.”
A sorry few did not even bother to explain their logic. “The answer is 10, because it isn’t nine or 11.”
One of my favourite responses came from Benjamin Mueller of Keble College, Oxford. He noted that the winning answer in 1997 was 13. “Schooled as I am in neoclassical economics, I also assume that incentives matter. In 1997 the prize was two business-class tickets for a flight from London to New York. This makes it likely that the contest attracted a higher calibre of participants who thought harder about the puzzle than when the reward is a bag. The education effect is cancelled out by the diminished worth of prize. My guess, therefore, is 13.”
He came close to winning — as did many readers (and I should add that the Dom Reilly bag is a very splendid one). I want to thank all those who participated. Several people identified 12, the winning number, but Richard Thaler picked Anatoly Lebedev, executive director, commodities electronic trading, at Goldman Sachs for his logic. Lebedev added this excellent warning: “If the competition was checked by a computer, there would be a ‘hacker’ solution of submitting a billion times one same number from fake accounts and then calling two-thirds of the number from a real account.” Saboteurs, watch out!
