Uniform price auctions admit a continuum of “collusive seeming” equilibria due to bidders’ market power. In this paper, I modify the uniform price auction rules in allowing the seller to discount the offer price and to ration bidders after having observed the bids as usually happens in financial markets. Within a framework where bidders cannot exactly anticipate the seller’s strategy, I show that many of these equilibria are eliminated. Another interesting contribution is that the vulnerability of “collusive seeming” equilibria to strategic rationing is increased when bidders have a private cost of acquisition of the shares. Moreover, when the seller is unconstrained in the rationing scheme she can use, the unique surviving equilibrium price is the true value of shares. This may explain why auctioneers in financial auctions usually set the offer price below the market clearing price without giving explicit rules. This allows them to reduce bidders’ ability to manipulate the uniform price auction.