Field and Lab Convergence in Poisson LUPI Games

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 fixed 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 field data from a Swedish gambling company, and laboratory data from a scaled-down version of the field game. The field 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 can account for the basic discrepancies between the equilibrium prediction and the data.