Two-Thirds of the Average
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?
Related problems & prerequisites
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Source: The beauty-contest game, from Rosemarie Nagel's experimental study (1995), after Keynes's beauty-contest metaphor (1936). Statement written for AxiomIQ.