Allocating multiple units by sealed-bid auctions

This paper studies the allocation and rent distribution between buyers and the seller in multiple units sealed-bid auctions. We restrict attention to the complete information case. A simple diagrammatic exposition of equilibria in the first-price and uniform-price auctions is provided and their equilibria are compared one to another and to the second-price (or Vickrey) auction. We show that the truthful equilibrium in the first-price auction is revenue equivalent to the second-price auction. Both these auctions are discriminatory auctions. We also show that the revenue raised in the truthful equilibrium of the first-price auction is less than that raised in the truthful equilibirum of the uniform-price auction. The approach is also related to a number of themes found more broadly in economics such as the convergence to competition.