Separating equilibria in public auctions

We consider two private-value auctions where the prize in one is higher than the prize in the other. We show that a separating equilibrium exists in which bidders with a high valuation attend the auction with the higher prize while bidders with a low valuation attend the auction with the lower prize. In addition, we prove that a weak separating equilibrium exists where the strong bidders attend the high prize auction while the weak bidders randomize and may attend either auction, although with a higher probability of attending the low prize auction. In the set of auctions with separating equilibrium, we find the optimal minimum bids that maximize a seller's expected revenue.