Value Bounds and Best Response Violations in Discriminatory Share Auctions

This paper analyzes a discriminatory share auction in which bidders submit nonincreasing step functions with a bounded number of steps, the type space consists of private non-increasing marginal valuation functions, and the number of participants is random. I show that the interim utility can be written as a simple functional of the distribution of the allocated quantity. This allows me to derive equilibrium existence and to give a characterization of the equilibrium bid schedules in terms of the individual bidders' optimality conditions. The characterization facilitates the formulation of bounds on the estimates of marginal valuations between the submitted quantity points and permits a simple estimator of the fraction of best response violations among the submitted bids. Proofs of concept for the bounds and the estimator are given by using a novel data set from meat import quota auctions in Switzerland.