Sequential Auctions with Decreasing Reserve Prices

We study sequential sealed bid auctions with decreasing reserve prices when there are two identical objects for sale and unit-demand bidders (existing literature has dealt with the case of weakly increasing reserve prices). Under decreasing reserve prices bidders may have an incentive not to bid in the first auction, and no equilibrium exists with a strictly increasing stage one bidding function. However, we find that an equilibrium always exists, and its shape depends on the distance between the two reserve prices. The equilibrium exhibits some pooling at the stage one auction, which disappears in the limit as the number of bidders tends to infinity. We also show revenue equivalence between first-price and second-price sequential auctions under decreasing reserve prices. Finally, our results allow us to shed some light on an optimal order problem (increasing versus decreasing exogenous reserve prices) for selling the two objects.