# Equilibrium strategy and population-size effects in lowest unique bid auctions

### Simone Pigolotti 1, 2, Sebastian Bernhardsson 1, 3, Jeppe Juul 1, Gorm Galster 1, Pierpaolo Vivo 4

#### Physical Review Letters 108 (2012) 088701

In lowest unique bid auctions, $N$ players bid for an item. The winner is whoever places the \emph{lowest} bid, provided that it is also unique. We use a grand canonical approach to derive an analytical expression for the equilibrium distribution of strategies. We then study the properties of the solution as a function of the mean number of players, and compare them with a large dataset of internet auctions. The theory agrees with the data with striking accuracy for small population size $N$, while for larger $N$ a qualitatively different distribution is observed. We interpret this result as the emergence of two different regimes, one in which adaptation is feasible and one in which it is not. Our results question the actual possibility of a large population to adapt and find the optimal strategy when participating in a collective game.

• 1. Niels Bohr Institute (NBI),
Niels Bohr Institute
• 2. Dept. de Fisica i Eng. Nuclear,
Universitat Politécnica de Catalunya
• 3. Swedish Defence Research Agency,
Swedish Defence Research Agency
• 4. Laboratoire de Physique Théorique et Modèles Statistiques (LPTMS),
CNRS : UMR8626 – Université Paris XI - Paris Sud