In the lowest unique positive integer (LUPI) game, players pick positive integers and the player who chose the lowest unique number (not chosen by anyone else) wins a …xed prize. We derive theoretical equilibrium predictions, assuming fully rational players with Poisson-distributed uncertainty about the number of players. We also derive predictions for boundedly rational players using quantal response equilibrium and a cognitive hierarchy of rationality steps with quantal responses. The theoretical predictions are tested using both …eld data from a Swedish gambling company, and laboratory data from a scaled-down version of the …eld game. The …eld and lab data show similar patterns: in early rounds, players choose very low and very high numbers too often, and avoid focal (“round”) numbers. However, there is some learning and a surprising degree of convergence toward equilibrium. The cognitive hierarchy model with quantal responses ca
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