Second-Price Auctions with Power-Related Distributions

We analyze a class of parametric second-price auction models where asymmetry is modeled by allowing bidders to take different numbers of draws from the same distribution. We compute the closed-form distribution of price and construct likelihood and method-of-moments estimators to recover the underlying value distribution from observed prices. We derive a Herfindahl-like formula that predicts merger effects and find that merger effects depend on the shares of the merging bidders, the variance, and the "shape" of the distribution. We generalize the model by allowing bidders to mix over power-related distributions. The dominant strategy equilibrium implies that an auction among bidders who mix over distributions can be expressed as a mixture of auctions. This implies that an auction among bidders with potentially correlated values can be expressed as a mixture over independent power-related auctions