Solution
Two-Thirds of the Average
Show the problem again
Everyone in a large group simultaneously picks a number in [0, 100]. The winner is whoever is closest to two-thirds of the group average. If all players are perfectly rational and this is common knowledge, what number does everyone pick in the unique Nash equilibrium?
Worked solution
The answer is 0. No rational player picks above 66.67, since 2/3 of the average can never exceed that. But if everyone's choice is capped at 66.67, no one should pick above 44.4, and so on: iterated elimination of dominated strategies drives all choices down to 0. At 0, no one can profitably deviate, making it the unique Nash equilibrium. (Real humans average around 20–35, which is the interviewer's favorite follow-up.)
Source: The beauty-contest game, from Rosemarie Nagel's experimental study (1995), after Keynes's beauty-contest metaphor (1936). Statement written for AxiomIQ.