Equilibria in Multi-Unit Discriminatory Auctions

We extend Milgrom and Weber's affiliated valuations model to the multi-unit case. We show that the discriminatory auction has a unique equilibrium, that corresponds to Milgrom and Weber's first-price equilibrium in the 2-bidder, constant marginal valuations case. This unique equilibrium therefore leads to lower expected prices than the equilibrium of the English auction where the units are bundled together. Hence we show that in an auction of a single object where the object can be divided into k parts and a bidder's valuation for each part is the same, it is not possible to increase revenue by using a multi-unit discriminatory auction. With more than two bidders and constant marginal valuations we show that the first-price equilibrium is an equilibrium of the multi-unit discriminatory auction. Back and Zender show this in the common values case which is a special case of affiliated valuations. We also show that the first-price equilibrium does not hold with decreasing marginal valuations.