The Simplest Unlimited-Player Game of Skill

Abstract

The Least Unique Positive Integer game (LUPI) is among the simplest games that can be played by any number of players, N > 2, and has a nontrivial strategic component. In LUPI, players try to pick the smallest positive integer nobody else picks. Despite its simplicity, the game was not widely known until fairly recently. It was actually offered as a state run lottery game in Sweden in 2007, but players collaborated, and the game was quickly stopped. LUPI has also been proposed as the basis for a reverse auction system, and here it is proposed as a “party game”. The Nash equilibrium for the game has been previously worked out in the case where the numbers of players that make each choice are independent Poisson random variables, an assumption that can often be justified when N is large. Here we summarize previous work and derive a number of interesting new results on Nash equilibria when N is small, and when N is large using the Poisson assumption. We also investigate whether the Nash equilibrium strategies for LUPI games are evolutionary stable strategies. Finally, we look at cheating strategies for LUPI and devise ways to make it harder to cheat.