Properties of scoring auctions.

(6) Until Section 5, in which we consider alternative mechanisms to the scoring auction, nothing is lost if t is assumed to be common knowledge. We introduce the notation here for completeness.

(7) Rezende (2004) studies a procurement model with fixed levels of nonmonetary attributes. In our model, the level of nonmonetary attributes is determined during the auction process.

(8) The requirement of quasilinearity of the scoring rule is only needed when private information is multidimensional. When private information is one-dimensional there is a one-to-one mapping between types and pseudotypes. The equivalence classes of types with the same preferences are thus singletons.

(9) Or, more generally, in the case of open formats, the strategies of the bidders.

(10) Note that when [[x.bar].sub.1](k) < [[x.bar].sub.1] (k), different types with the same pseudotype use different equilibrium strategies or they use the same mixed strategy.

(11) Without loss of generality, we focus on equilibria that involve optimal strategies for all realizations of types.

(12) Bichler and Kalagnanam (2003) use the expression "auctions with configurable offers" to describe such procedures.

(13) Note that commitment will again be essential here. The buyer must be able to convince suppliers that he will not manipulate the (unannounced) scoring rule in order to increase the value of the second-best offer.

(14) We find that there is no quality distortion at the top and at the bottom in the separating equilibrium of the first-price menu auction, a result that mirrors Rochet and Stole (2002). Rochet and Stole develop the intuition for this finding. They also reconcile it with the discrete type case in which there is distortion at the bottom (see Theorem 5).

(15) When the equilibrium in the menu auction is known to be a separating equilibrium, then it is easy to show, using the techniques developed in the proof of Theorem 4, that some buyer types prefer the menu auction whereas others prefer the scoring auction.

(16) Rochet and Stole (2002) note that pooling is a common feature of equilibrium in this class of models. How pooling affects the welfare of different buyers in the menu auction is unclear without a full characterization of the equilibrium.

(17) Biglaiser and Mezzetti consider a model in which multiple principals bid for the exclusive service of an agent. Each principal has private information about their valuation of the service, while the agent has private information about the disutility of providing the service. The first-price menu auction considered here is the procurement version of this model (with the agent being the buyer and the principals being the suppliers).

(18) Thus the outcome is efficient. This does not contradict Theorem 4 because Theorem 4 applies to the case where the buyer has a continuum of types.

(19) The quality offered to the high-type buyer remains at the first best.

(20) The U.S. highway procurement authorities use such a reward/penalty scheme.

(21) Bajari, Houghton, and Tadelis (2006) analyze data from highway paving procurement that have this feature. The approach they use involves a theoretical structure that employs aspects of the approach developed here.

(22) Note that it is a best response for the buyer to select the offer he truthfully prefers at each round. Note also that Lemma 3 ensures that the menu induces productive efficiency.

(23) If the solution to this problem violates the second-order condition, pooling must occur at equilibrium.

(24) [lambda](t, [theta]) must follow the law of motion (A3) together with the boundary conditions [lambda]([bar.t], [theta]) = [lambda]([t.bar], [theta]) = 0. In particular, this implies no distortion at the top and at the bottom.

(25) For details, the reader is referred to Biglaiser and Mezzetti (2000).

John Asker *

and

Estelle Cantillon **

* Stern School of Business and New York University; jasker@stern.nyu.edu.

** FNRS, Universite Libre de Bruxelles (ECARES), and CEPR; estelle.cantillon@ulb.ac.be.