Increasable Supply and “Collusive-Seeming Equilibria” in the Uniform-Price Auction

When supply is fixed in advance, the uniform-price share auction is vulnerable to “collusive-seeming equilibria” in which the realized price is less than a common knowledge lower bound on all bidder’s per share value vi. Our main result is that no collusive-seeming equilibria exist in a variation of the uniform-price auction in which the auctioneer sets supply so as to maximize profits after receiving bidders’ announced demand schedules. Lack of precommitment to quantity levels, especially the ability to increase supply as a function of the bidding, is key. For example, consider the special case in which there is a common knowledge common value v (with n bidders): when the auctioneer precommits to a supply curve S (p) = a + bp , Nash equilibria exist in which any price p ∈ [ nbv−a b(n+1) , v ] can be realized; when the auctioneer is only able to increase supply from a prespecified minimum Ŝ, no collusive-seeming equilibria exist if bidders are restricted to submit differentiable demand schedules and to bid at prices no lower than c′ ( Ŝ ) , the auctioneer’s marginal cost; but when the auctioneer is only able to decrease supply from a prespecified maximum Ŝ, Nash equilibria exist in linear strategies in which any price p ∈ [ n−1 n v + c′(Ŝ) n , v ] can be realized. ∗I thank Susan Athey, Kerry Back, Hung-Ken Chien, Paul Klemperer, Eric Maskin, Paul Milgrom, Erik Stuart, Robert Wilson, participants at FERC and Stanford GSB seminars and especially Jeremy Bulow for helpful comments. I gratefully acknowledge the Federal Trade Commission for its support while this work was completed. Wilson (1979), Back and Zender (1993) and others have emphasized a weakness of the uniform-price auction. This auction often possesses equilibria in which very low prices are realized. For example, in a common values setting in which each bidder possesses private information about the per share value v, Wilson showed that a Bayesian Nash equilibrium exists in which the realized price is always exactly v 2 – no matter how many bidders participate! Wilson also provided the first example of what we shall call a “collusive-seeming equilibrium” of the uniform-price auction. When v is common knowledge and there are n bidders and S shares, Wilson showed that it is a Nash equilibrium for each bidder i to announce demand Di (p) = S n + S v 2 n (n− 1) (v 2 − p ) whereupon each bidder receives S n at price v 2 . Back and Zender expanded upon this example, observing that any price p < v can be realized in a Nash equilibrium. In this common knowledge common value environment, a collusive-seeming equilibrium is one in which the realized price p < v. More generally, when each bidder i receives private information si and has per share valuation vi (s1, ..., sn), 1 a collusive-seeming equilibrium is one in which the realized price is less than a lower bound on all bidders’ per share value with positive probability, i.e. in which Pr ( p (s1, ..., sn) < min i inf s1,...,sn vi (s1, ..., sn) ≡ v ) > 0. In any collusive-seeming equilibrium, each bidder announces demand that is (much) more inelastic than his true demand. This steepens each bidder’s residual supply curve. The best option, then, is to accept a smaller quantity at a lower Throughout this paper, we assume that all bidders have linear valuations for shares. That is to say, bidder i is willing to pay vi (q; s1, ..., sn) ≡ qvi (s1, ..., sn) for q shares. This assumption is not necessary to our analysis, all of which goes through if one redefines a collusive-seeming equilibrium as one in which the realized price is less than a lower bound on all bidders’ marginal value for shares, i.e. in which Pr ( p < min i inf q,s1,...,sn ∂ [vi (q; s1, ..., sn)] ∂q ≡ v ) > 0.