Sara Robinson explains How Real People Think in Strategic Games — which is nothing like mathematicians:
Once upon a time, Arianna, a Hollywood hostess, threw a large party for a diverse assemblage of guests. Bored with the usual rounds of “Hollywood Star Charades” and “Two Minutes with a Millionaire,” she initiated a novel game: Each guest was to choose a number from 0 to 100, write it on a piece of paper, and drop it in a passed basket. The goal, she explained, was to choose a number as close as possible to two thirds of the group average. The person or people who came closest would share a bottle of very expensive champagne.Minerva, the lone mathematician at the party, mulled over possible strategies. “Suppose the other participants’ choices are randomly distributed,” she reasoned. “The average of the other guesses would be about 50, so I should choose 33. . . . But wait, the other participants will realize this too, so their numbers would not be random—they’d average to 33. So I should choose 22. . . . But the others will see this too, so I should choose 14. . . .” As she quickly realized, this iterative process stops only when it reaches 0. In the end, assuming that the other guests would follow a similar line of reasoning, Minerva decided that the group average would be 0. Anticipating at least a taste of superb champagne, she confidently wrote down her guess and dropped it in the basket.
Minerva had happened on the unique “Nash equilibrium” for Arianna’s game, although, as a number theorist, she wasn’t familiar with the term. A Nash equilibrium is a collection of strategies, one for each player, such that even if all players know the others’ strategies, they have no incentive to change their own.
A little later, Arianna tabulated the guesses and, with fanfare, announced the group average: 30. Minerva was stunned. Warren, the lone economist in the group (who had, in fact, suggested the game to the hostess), won the champagne by guessing 31. He said his farewells and left the party with the champagne and a well-known actress on his arm.
What was the error in Minerva’s reasoning?
The error, of course, was thinking that everyone else would think rationally — and that they’d follow through in their analysis:
One focus of behavioral game theorists has been “dominance solvable games”—games that, like Arianna’s, can be solved by iterated reasoning. For mathematicians, the inductive leap from one or two iterations to the mathematical limit is an effortless one. But this is not the case for most people. From experimental studies, behavioral game theorists conclude that people tend to do only a few steps of iterated reasoning and then stop—either because the reasoning is too complicated for them or because they believe it’s too complicated for others.
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In 1995, Nagel had groups of German students, about fifteen in each group, try to guess two thirds of the group average within limits of 0 and 100. While the average guess for the groups was around 35, she observed that many students chose 33, two thirds of the midpoint of 50, or 22, two thirds of two thirds of the midpoint. Very few players chose 0.Three years later, Teck Ho, Colin Camerer, and Keith Wiegelt replicated the experiment, replacing two thirds with 0.7, 0.9, 1.1, and 1.3. With their data, the researchers precisely quantified the number of steps of iterated reasoning the subjects seemed to be doing. Most subjects, they concluded, were doing between 0 and 3 rounds.
In a fascinating series of unpublished experiments, Camerer, a professor of economics at the California Institute of Technology, also tried the game with several distinct subject pools. When the goal was to guess 70% of the average, some of the groups, including those made up of Caltech undergraduates, game theorists, and computer scientists, averaged under 20, while other groups, such as CEOs, 70-year-olds, and Pasadena City College students, averaged over 50.
Again using a statistical analysis of the guesses, Camerer inferred the average number of “steps of thinking” for each of the groups; the numbers range from 0 (for a small group of Pasadena City College students) to 3.8 (for the computer scientists).