Rationing at the Margin: Existence of Partially Revealing Equilibria in Large Uniform-Price Auctions

In large common-value uniform-price auctions with finitely many states and signals, partially revealing prices make bidders tie at the clearing price. Existence of equilibrium then has two components: price feasibility, governed by restrictions shared by all rules that ration only tied bids, and rationing implementation by the auction’s own rule. Both can fail robustly. With two signals and three states, an equilibrium generically exists if and only if the informationally intermediate state does not have the lowest value. With three signals, there is an open set of economies on which the pro-rata auction has no equilibrium but rationing by announced signals, an endogenous type-dependent sharing rule, restores one: there, an anonymous rule cannot divide the surplus of a tie between two marginal types with different information. With two signals, no alternative rule that rations only tied bids restores an equilibrium that is absent under pro rata. Under pro rata, existence at a candidate price function is a linear system plus one product condition per pooling price, and reduces to linear programming when pools are thin.