Properties of scoring auctions.

[c.sub.Q](Q(t, [theta]), [theta])[Q.sub.t](t, [theta]) = p,(t, [theta]) for all [theta] and all t at which Q is differentiable. (11)

Because Q, (t, [theta]) [not equal to] 0, conditions (10) and (11) are equivalent given (9) (continuity of the efficient production level in t ensures that (10) is satisfied at non differentiability points of Q). Q.E.D.

Beauty contests. In a beauty contest, the buyer does not reveal his type and the suppliers are asked to submit a single bid (p, Q). It potentially comes in three forms: the ascending format, and the first-price and the second-price auctions. These formats are self- explanatory given their description for the menu auction.

Price-only auctions. In price-only auctions, the buyer publishes a detailed request-for-quote that sets minimum levels for each attribute. All offers satisfying these conditions are evaluated on a price basis. Again, it comes in three guises: ascending, first-price and second- price.

We now compare the performance of these alternative procedures with the performance of a scoring auction that uses the true preference of the buyer as its scoring rule. Let [U.sup.k.sub.l] (t) be the expected utility of the buyer with taste t, in format k [member of] {A, FP, SP} and procedure l [member of ] {score, menu, beauty} (where "score" stands for a scoring auction of the type described in the previous sections with the scoring rule corresponding to the true preference of the buyer).

Theorem 3. For all t,

(i) [U.sup.A.sub.score](t) = [U.sup.A.sub.menu](t) = [U.sup.A.sub.beauty](t) as the bidding increment goes to zero,

(ii) [U.sup.SP.sub.score](t) = [U.sup.SP.sub.menu](t) = [U.sup.SP.sub.beauty](t).

Proof. See the Appendix.

For the ascending auction, we build the unique symmetric equilibrium for each procedure. The equivalence between all three procedures then stems from the direct comparison of the final allocations. For the second-price format, we show that submitting schedule S = {(p, Q) such that p = c(Q, [theta]), Q [member of] [R.sub.M]} is the unique dominant strategy equilibrium in the menu auction. The equivalence between the menu auction and the scoring auction with [phi](Q) = v(Q, t) follows directly. For the beauty contest, we argue that the equilibrium bid ([p.sup.*], [Q.sup.*]) must belong to S. Because there is a positive probability that ([p.sup.*], [Q.sup.*]) does not belong to [argmax.sub.(p,Q)[member of]S]{v(Q, t)--c(Q, [theta])} for the actual type-- unknown to the suppliers--of the buyer [U.sup.SP.sub.menu] > [U.sup.SP.sub.beauty] follows.

Theorem 3 understates the superiority of scoring auctions in two ways. First, scoring auctions are likely to dominate both procedures because they save on bidding costs for suppliers. In practice, the existence of bidding costs will limit the number of offers made in a menu auction. This favors the scoring auction. Likewise, the equivalence result for the beauty contest in the ascending format requires that suppliers submit a very high number of bids. Second, the comparison in Theorem 3 is with a scoring auction with scoring rule [phi]b(Q) = v(Q, t). As suggested by Che (1993) and Asker and Cantillon (2006), the buyer will in general be better off announcing [phi](Q) [not equal to] v(Q, t).

We next consider the first-price menu auction. We first show the following general result.

Theorem 4. Any equilibrium of the first-price menu auction is inefficient.

Proof. See the Appendix.

Theorem 4 follows from the following observations. If the equilibrium in the menu auction involves pooling (suppliers make the same offer to different buyer types), inefficiency is immediate. If, instead, full separation occurs at equilibrium, inefficiency arises from the tension between the requirements of incentive compatibility and those of profit maximization. For the purpose of profit maximization alone, suppliers are tempted to target different profit levels according to the buyer's type. The buyer's incentive compatibility constraint limits the ability of suppliers to do this. We argue that the buyer's incentive compatibility constraint binds generically in any separating equilibrium of the first-price menu auction and show that qualities are distorted as a result. (14)

The inefficiency of the first-price menu auction is not necessarily bad news for the buyer if it induces fiercer competition. To investigate this question further, we focus on the more restricted environment where private information is one-dimensional and the buyer can only have two types. (15,16) The following theorem suggests that buyers with different types are likely to rank the two procedures differently.

Theorem 5 (Adapted from Theorem 2 of Biglaiser and Mezzetti, 2000). (17) Suppose Q, [theta] [member of] R, t [member of] {[t.sub.L], [t.sub.H]} with [t.sub.L] < [t.sub.H]. Suppose [v.sub.Q](Q, [t.sub.H]) > [v.sub.Q](Q, [t.sub.L]) and [v.sub.QQ](Q, [t.sub.H]) [less than or equal to] [v.sub.QQ](Q, [t.sub.L]) for all Q and that [c.sub.Q[theta]]e > 0. Then, in the symmetric equilibrium of the menu auction, [U.sup.FP.sub.score]([t.sub.L]) [greater than or equal to] [U.sup.FP.sub.menu]([t.sub.L]) and [U.sup.FP.sub.score]([t.sub.H])[less than or equal to] [U.sup.FP.sub.menu]([t.sub.H]).

The proof of Theorem 5 closely follows that in Biglaiser and Mezzetti. A sketch is provided in the Appendix pointing out how to adapt their arguments.

At equilibrium, suppliers offer two contracts, one targeted at the low-type buyer, ([Q.sub.L], [p.sub.L), and the other targeted at the high-type buyer, ([Q.sub.H], [p.sub.H]). The inequalities in Theorem 5 are strict whenever one of the incentive compatibility constraints (8), evaluated at the equilibrium offers of the scoring auction, binds. Biglaiser and Mezzetti (2000) argue that this will be the case when [t.sub.L] and [t.sub.H] are sufficiently close. Intuitively, when [t.sub.L] and [t.sub.H] are sufficiently distinct, the two contracts offered by each supplier are sufficiently different that the low-type buyer is not tempted by the high-type contract and vice versa. In that case, the bidding equilibrium in the truthful scoring auction describes the equilibrium contracts for each type in the menu auction. (18) When the two buyer types are sufficiently close, the incentive compatibility constraints bind. Following the intuition from the single-principal single-agent case, the price and quality of the low-type buyer is distorted downward to satisfy the incentive compatibility constraint of the high-type buyer. However, competition means that the participation constraint of the buyer is now endogenous from the point of view of an individual supplier: it depends on the bids of the other suppliers. This increases the costs of distorting the low-type contract relative to the single- principal single-agent benchmark. As a result, suppliers also use the price offered to the high-type buyer to help ensure his incentive compatibility constraint is satisfied. Thus, the price offered to the high-type buyer decreases relative to the scoring auction, and the high-type buyer is better off. (19)

Biglaiser and Mezzetti point out that if the buyer knows his type prior to choosing a procedure, unraveling of the buyer's private information is likely: the low-type buyer chooses a scoring auction, leaving the high-type as the only type to choose the menu auction. Because he no longer has any private information, the menu and the scoring auction become equivalent again.

We now turn to the procedure where the buyer sets minimum quality standards and awards the contract on the basis of price only.

Theorem 6. Consider any standard auction format where the high bidder wins and its equivalent in the scoring auction. A buyer is always better off using a scoring auction with a scoring rule that corresponds to his true taste than imposing minimum quality standards/attribute levels and selecting the winner on the basis of price only.

Proof. Suppose the buyer sets minimum quality standards Q = [bar.Q] [member of] [R.sub.M]. Because costs are increasing, suppliers will set their quality levels at [bar.Q]. We are now back to a standard procurement auction with symmetric bidders and costs c([bar.Q], [[theta].sub.i]) [member of] R. Let [x.sup.(n:N)] denote the nth highest-order statistics from N draws of random variable x. From the revenue equivalence theorem, the expected utility of the buyer from this minimum quality standard auction is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

= expected utility from the truthful scoring auction by Theorem 2. Q.E.D.

6. Concluding remarks

Our article provides a systematic analysis of equilibrium behavior in scoring auctions when private information is multidimensional. We have characterized the set of equilibria in scoring auctions and have argued that a single number, the supplier's pseudotype, is sufficient to describe the equilibrium outcome in these auctions, when the scoring rule is quasilinear and types are independently distributed. In doing so, we have drawn on the equivalence between the reduced form of a scoring auction and that of a standard single-object IPV auction. We have also derived a new expected utility equivalence theorem for scoring auctions. Both results extend existing theories of scoring auctions.