Working Paper BETA #2023-05

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Title : Show your strength in the hammer-nail game: a Nim game with incomplete information

Author(s) : Gisèle Umbhauer

Abstract : We study the hammer-nail game, a game played in the French TV show “Fort Boyard”, by transforming this game into a Nim game with incomplete information. In this game, two players are in front of a nail slightly driven into a wooden support. Both have a hammer and in turn hit the nail. The winner is the first player able to fully drive the nail into the support. A player is of strength f if he is able, with one swing of the hammer, to drive the nail at most f millimeters into the support. We study the perfect Bayesian Nash equilibria of this game with incomplete information on the players’strength, and we also look at the equilibrium behavior when strength is combined with dexterity.

Key-words : Nim game, incomplete information, subgame perfect Nash equilibrium, perfect Bayesian equilibrium, Fort Boyard

JEL Classification : C72