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

TL;DR

This paper derives an analytical equilibrium strategy for lowest unique bid auctions, examines how population size affects bidding behavior, and compares theoretical predictions with real internet auction data.

Contribution

It introduces a grand canonical approach to analytically determine equilibrium distributions and explores the impact of population size on strategic adaptation in auctions.

Findings

Theory matches small N auction data accurately.

Large N auctions show different distribution patterns.

Two regimes of adaptation emerge depending on population size.

Abstract

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.