Properties of scoring auctions.

In addition, we have shown that several other candidate procedures for buying differentiated products, including some, such as the menu auction and the beauty contest, that also combine competition with the flexibility of deciding on all the dimensions of the product, are dominated by scoring auctions. These results suggest that scoring auctions provide a useful mechanism (they are simple straightforward procedures) for buying differentiated products.

We conclude with a few remarks on potential venues for further research.

Suppliers' uncertainty about their costs at the time of bidding. In our model, it was immaterial whether bidders were committed to their offer or to the scores that their offer generated. Suppose now that the cost of attribute Q is given by c(Q, [theta], [tau]), where only [theta] is known to the supplier at the time of bidding and [tau] is known before the contract is executed. Define k([theta]) = [E.sub.[tau]] [[max.sub.Q] {[phi](Q) - c(Q, [theta], [tau])}]. Our equilibrium characterization results go through with this redefined pseudotype. The only difference is that the delivered quality level now generically differs from the offered quality level because the delivered quality will solve [max.sub.Q]{[phi](Q) - c(Q, [theta], [tau])} for the realization of [tau]. This provides a rationale for making the scores, rather than the actual offers, binding. Thus, low-cost realizations generate higher levels of quality and higher prices for the supplier, whereas negative cost shocks generate lower qualities and lower prices. (20) This added flexibility is another advantage of the scoring auction relative to the other procedures that set the quality to be delivered at the contracting stage.

Noncontractible quality dimensions. An essential assumption for all our results is that quality is contractible. When some dimensions of the good are contractible and others not, contracting can generate perverse incentives, as Holmstrom and Milgrom (1991) have shown. At the same time, it seems desirable to generalize the analysis of procurement mechanisms to such environments (see Che, forthcoming for a discussion of possible mechanisms).

Implications for empirical work. Even in the presence of symmetric suppliers, scoring auctions present interesting auction design questions (e.g., how can the buyer manipulate the scoring rule to his advantage?). However, scoring auctions present two difficulties from the point of view of identification: the identification of the functional form for the costs and the identification of the distribution of private information. One consequence of our sufficient statistics result is that the distribution of types will generally be nonidentified on the basis of auction data, even when the functional form for the costs is known. Indeed, the observed information (the scores) is one-dimensional, whereas the information to be inferred is multidimensional. This suggests two possible solutions. When the (p, Q) offers rather than the scores are binding, the observed data are again multidimensional. (21) Another possibility is to look at auction data where changes in the scoring rule can be exploited. In any case, our article provides a theoretical basis from which investigation of identification is feasible.

Appendix

* This Appendix contains proofs of Theorems 3 and 4 and a sketch of the proof of Theorem 5.

Proof of Theorem 3, part (i). We discretize the price grid. Let [DELTA] be the minimum price (and therefore profit and utility) increment. We proceed in two steps, comparing first the menu auction and then the beauty contest to the scoring auction.

Step 1: Menu auction. The following is an equilibrium. In round 1, each bidder submits a schedule that generates at most zero utility for the buyer, that is, {(p, Q); Q [member of] [R.sup.M], p - c(Q, [theta]) = constant and [max.sub.t], [max.sub.(p.Q)] {v(Q, t) - p} = 0}. Let [[pi].sub.t] be the profit level corresponding to the period t schedule for a given supplier. At round t, this supplier submits schedule {(p, Q); Q [member of] [R.sup.M], p - c(Q, [theta]) = [[pi].sub.t-i] - [DELTA]} as long as [[pi].sub.t - 1] - [DELTA] [greater than or equal to] 0 if his offer was not selected in round t - 1. (22) This is an equilibrium (we can adapt the arguments in Bikhchandani, Haile, and Riley, 2002 to argue that the outcome of this strategy is the unique equilibrium outcome in symmetric strategies). Each supplier participates as long as a positive profit can be made, otherwise they exit. The selected level of attributes, [Q.sup.*], satisfies [Q.sup.*] = arg [max.sub.Q]{v(Q, t) - c(Q, [theta])} for the realization of t. The final price satisfies p = v([Q.sup.*], t) - [max.sub.Q]{v(Q, t) - c(Q, [[??])} (modulo the increment), where [??] refers to the cost function of the second- best supplier. This is the outcome of the scoring auction.

Step 2: Beauty contest. The following is an equilibrium. In round 1, each bidder submits a bid in the schedule that generates at most zero utility for the buyer, that is, {(p, Q); Q [member of] [R.sup.M], p - c(Q, [theta]) = constant and [max.sub.t], [max.sub.(p.Q)] {v(Q, t) - p} = 0}. Let [[pi].sub.t], be the profit level corresponding to the bid in period t for a given supplier. At round t, if this supplier was not the winner in round t - 1, he submits any bid in schedule {(p, Q); Q [member of] [R.sup.M], p - c(Q, [theta]) = [[pi].sub.t-i]} that he has not submitted in the past. If no unsubmitted bid remains in this schedule, the supplier submits a bid in {(p, Q);Q [member of] [R.sup.M], p- c(Q, [theta]) = [[pi].sub.t-1] - [DELTA]} as long as [[pi].sub.t-1] - [DELTA] [greater than or equal to] 0. The process continues until no further bid is received. As before, the equilibrium strategies yield the unique equilibrium outcome. The winner in the beauty contest is the same as in the menu auction. However, the buyer might not be equally well off as in the menu auction, because here he cannot choose the (p, Q) pair that maximizes his utility. However, as [DELTA] goes to zero, the winning (p, Q) must maximize v(Q, t) - c(Q, [theta]). Otherwise, the winning bidder could have won with a higher level of profit. This is ruled out by his bidding behavior. Thus, the buyer is equally well off.

Proof of Theorem 3, part (ii). We proceed in two steps, comparing first the menu auction and then the beauty contest to the scoring auction.

Step 1: Menu auction. An argument mirroring the argument for the standard second-price auction establishes that {(p, Q); p = c(Q, [theta]), Q [member of] [R.sup.M]} is a dominant strategy equilibrium. Equivalence with the scoring auction follows from the fact that suppliers submit bids such that p = c(Q, [theta]) in the dominant strategy equilibrium of the scoring auction, with Q = arg max{v(Q, t) - c(Q, [theta])}. This is also the bid selected by the buyer in the winning schedule. The best second offers in both auctions are also identical.

Step 2: Beauty contest. In equilibrium, suppliers submit an equilibrium bid ([p.sup.*], [Q.sup.*]) in the schedule {(p, Q);p = c(Q, [theta]), Q [member of] [R.sup.M]}. Consider any alternative bid (p, Q) such that p - c(Q, [theta]) > 0. The expected profit generated by this bid is equal to

Pr ((p, Q) generates the highest score) (Ep - c(Q, [theta])),

where Ep is the expected resulting price determined by the second-best offer. The deviation ([??], c(Q, [theta])), where [??] = c(Q, [theta]) dominates. The expected profit it generates is equal to

Pr ((p, Q) generates the highest score)(Ep - c(Q, [theta]))+

Pr (([??], Q) generates the highest score but (p, Q) does not)(E[??] - c(Q, [theta])),

where E[[??] is the expected price given that ([??], Q) generates the highest score but (p, Q) does not. Clearly, E[??] - c(Q, [theta]) > 0. ([p, Q) such that p - c(Q, [theta]) < 0 is similarly dominated. From the buyer's point of view, the utility from ([p.sup.*], [Q.sup.*]) [member of] {(p, c(Q, [theta])); Q [member of] [R.sup.M], p = c(Q, [theta])} is lower than max{v(Q, t) - p; p = c(Q, [theta]), Q [member of] [R.sup.M]} with strictly positive probability. Q.E.D.

Proof of Theorem 4. If the menu auction equilibrium involves pooling of offers, then inefficiency is immediate. Thus, we direct attention to equilibria where full separation occurs (a full menu is offered by each supplier).

Consider the optimization problem faced by a supplier of type [theta]. Let U(t, [theta]) denote the utility received by a buyer of type t from a supplier of type [theta] in the equilibrium of the menu auction. The buyer's incentive compatibility constraint can be rewritten as

d/dt U(t, [theta]) = [v.sub.t](Q(t, [theta]), t) (A1)

(with second-order condition [[nabla].sub.t]Q(t, [theta]) [greater than or equal to] 0). Let Pr (U, t) denote the equilibrium probability that an offer generating a level of utility U for a buyer of type t wins. Ignoring the second-order condition, the supplier's problem can be written as (23)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

subject to (A1)

This is a standard optimal control problem, with U being the state variable and Q being the control variables. The Hamiltonian is given by

H(Q, [lambda], U, t) = (v(Q(t, [theta]), t) - c(Q(t, [theta]), [theta]) - U(t, [theta]))Pr(U(t, [theta]), t)h(t) + [lambda](t, [theta])[v.sub.t](Q(t, [theta]), t).

If a solution exists, it must satisfy the following first-order conditions (where we have dropped the arguments for simplicity):

([v.sub.Q](Q, t) - [c.sub.Q](Q, [theta]))Pr(U, t)h(t)+[lambda](t, [theta])[v.sub.Qt](Q, t) = 0 (A2)

-(v - c- U) 3/dUPr(U, t)h(t)+Pr(U, t)h(t) = [[lambda]'] (A3)