Properties of scoring auctions.

Finally, a few papers consider alternatives to scoring auctions. Che (1993) provides a qualitative argument for why scoring auctions are better than price-only auctions with minimum quality standards in his one-dimensional framework. Bichler and Kalagnanam (2003) look at the "second-score" menu auction. They focus on the "winner determination problem" for a given set of offers received, not on equilibrium behavior. Menu auctions can be seen as a common agency problem where multiple principals (the suppliers) compete in offering menus of contracts to an agent (the buyer). From suppliers' perspective, menu auctions are also an example of a screening problem with random participation because a supplier's offer is accepted only if it is better than the competing offers the buyer received. We draw on these literatures when we study the first-price menu auction. We consider menu auctions, beauty contests, and price-only auctions with minimum quality standards, and systematically compare the outcome of equilibrium in these auctions with that in scoring auctions.

The rest of the article is organized as follows. Section 2 describes the model and introduces the notion of pseudotype. Section 3 proves that the pseudotypes are sufficient statistics in our environment, and establishes the correspondence between scoring auctions and regular IPV auctions. Our expected utility equivalence theorem is proved in Section 4. Section 5 compares the outcome of scoring auctions with that of menu auctions, beauty contests, and auctions with minimum quality standards. Section 6 concludes.

2. Model

Environment. We consider a buyer seeking to procure an indivisible good for which there are N potential suppliers. The good is characterized by its price, p, and M [greater than or equal to] 1 nonmonetary attributes, Q [member of] [R.sup.M.sub.+].

Preferences. The buyer values the good (p, Q) at v(Q, t) - p, where t [member of] [[t.bar], [bar.t]] indexes the buyer's taste for quality. (6) Supplier i's profit from selling good (p, Q) is given by p - c (Q, [[theta].sub.i]), where [[theta].sub.i] [member of] [R.sub.K], K [greater than or equal to] 1, is supplier i's type. We allow suppliers to be flexible with respect to the level of nonmonetary attributes they can supply. (7) We assume that v and c are twice continuously differentiable with [v.sub.Q], [c.sub.Q] > 0, v - c bounded, and [v.sub.QQ] - [c.sub.QQ] negative definite. In particular, this allows for costs that are independent across attributes and convex in individual attributes. We partially order the type space by assuming that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. When we analyze the first price menu auction, we will also impose [v.sub.Qt] > 0 and [v.sub.QQt], negative semidefinite.

Because social welfare is bounded and strictly concave in Q, the first-best level of nonmonetary attributes for each supplier, [Q.sup.FB]([[theta].sub.i]) = arg max{v(Q, t) - c(Q, [[theta].sub.i])}, is well defined and unique.

Information. Preferences are common knowledge among suppliers and the buyer, with the exception of suppliers' types, [[theta].sub.i] i = 1, ... N, and the buyer's taste parameter, t, which are privately observed. Types are independently distributed according to the continuous joint density function [f.sub.i](.) with support on a bounded and convex subset of [R.sup.K] with a nonempty interior, [[THETA].sub.i].Taste is distributed according to the continuous density h(.). These density functions are common knowledge.

[] Allocation mechanism. We now introduce the scoring auction. We start with two definitions:

A scoring rule is a function S : [R.sup.M+1.sub.+] [right arrow] + R : (p, Q) [right arrow] S(p, Q) that associates a score to any potential contract and represents a continuous preference relation over contract characteristics (p, Q). A scoring rule is quasilinear if it can be expressed as [phi](Q) - p or any monotonic increasing function thereof. We assume that the scoring rule is twice continuously differentiable and strictly increasing in Q, and that the resulting "apparent social welfare," [phi](Q) - c(Q, [theta]), is bounded and strictly concave in Q for all [theta]. For simplicity, we let [max.sub.Q]{[phi](Q) - c(Q, [theta])} [greater than or equal to] 0 for all [theta] to ensure that all suppliers participate in the auction at equilibrium.

A scoring auction is an allocation mechanism where suppliers submit bids of the type (p, Q) [member of] [R.sup.M+1.sub.+]. Bids are evaluated according to a scoring rule. The winner is the bidder with the highest score. The outcome of the scoring auction is a probability of winning the contract, [x.sub.i], a score to fulfill when the supplier wins the contract, [t.sup.W.sub.i], and a payment to the buyer in case he does not, [t.sup.L.sub.i]. A scoring auction is quasilinear when it uses a quasilinear scoring rule.

For example, in a first-score scoring auction, the winner must deliver a contract that generates the value of his winning score, that is, [t.sup.W.sub.i] = S([p.sub.i], [Q.sub.i]), [t.sup.L.sub.i] = 0. In an ascending scoring auction, the buyer progressively raises the required score to fulfill by any standing offer until all suppliers but one drop out. [t.sup.W.sub.i] is the value of that score and [t.sup.L.sub.i] = 0. In a second-score scoring auction, the winner must deliver a contract that generates a score equal to the score of the second-best offer received.

Note that when the scoring rule does not correspond to the buyer's preference--something which might be in his interest (Che, 1993; Asker and Cantillon, 2006)--commitment is essential. In public procurement, this might be easily done. The process must often abide by a strict set of rules and procedures, so that, in effect, the call for tender (and thus the scoring rule) is legally binding for the buyer. In private procurement, this might be harder, although, in principle, the buyer could sign a contract with the bidders before bidding takes place in which he commits to use the scoring rule. Such a contract could be enforced through an independent third-party audit. Repetition is an alternative mechanism.

We now proceed to the analysis of bidding behavior in the scoring auction. Consider supplier i with type [[theta].sub.i] who has won the contract with a score to fulfill [t.sup.W.sub.i]. Supplier i will choose characteristics (p, Q) that maximize his profit, that is,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Substituting for p into the objective function yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

An important feature of (1) is that the optimal Q is independent of [t.sup.W.sub.i]. Define

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

We shall call k([[theta].sub.i]) supplier i's pseudotype. It is the maximum level of apparent social surplus that supplier i can generate. Bidders' pseudotypes are well defined as soon as the scoring rule is given. The set of supplier i's possible pseudotypes is an interval in R. The density of pseudotypes inherits the smooth properties of [f.sub.i]. With this definition, supplier i's expected profit is given by

[x.sub.i] (k([[theta].sub.i]) - [t.sup.W.sub.i]) - (1 - [x.sub.i])[t.sup.W.sub.i]. (2)

In (2), supplier i's preference over contracts of the type ([x.sub.i], [t.sup.W.sub.i], [t.sup.L.sub.i]) is entirely captured by his pseudotype. Only quasilinear scoring rules have this property when private information is multidimensional. Indeed, consider a more general scoring rule S(p, Q). Assume that S is twice continuously differentiable, strictly increasing in Q and strictly decreasing in p. Bidder i's optimization problem becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (3)

Let [psi] (Q, [t.sup.W.sub.i]) be the price required to generate a score of [t.sup.W.sub.i] with nonmonetary attributes Q ([psi] is well defined because S is strictly decreasing in p and strictly increasing in Q; it is strictly increasing in Q and strictly decreasing in [t.sup.W.sub.i]). The objective function of bidder i becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and his expected payoff from contract ([x.sub.i], [t.sup.W.sub.i], [t.sup.L.sub.i]) is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Suppose we can organize types in equivalence classes such that all types in a given class share the same preferences over contracts. Concretely, suppose that types [[theta].sub.i] and [[??].sub.i] [not equal to] [[theta].sub.i] belong to such a class. It must be that

u([x.sub.i], [t.sup.W.sub.i], [t.sup.L.sub.i]; [[theta].sub.i]) = u([x.sub.i], [t.sup.W.sub.i], [t.sup.L.sub.i]; [[??].sub.i]) if and only if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for all pairs of contracts ([x.sub.i], [t.sup.W.sub.i], [t.sup.L.sub.i]), ([[??].sub.i], [[??].sup.W.sub.i], [[??].sup.L.sub.i]). (4)

Let Q([[theta].sub.i], [t.sup.W.sub.i]) = arg [max.sub.Q]{[PSI](Q, [t.sup.W.sub.i]) - c(Q, [[theta.sub.i])}. Condition (4) requires that [partial derivative]/[partial derivative][t.sup.W.sub.i][PSI] (Q([[theta].sub.i], [t.sup.W.sub.i]),[t.sup.W.sub.i]) = [partial derivative]/[partial derivative][t.sup.W.sub.i][PSI] (Q([??].sub.i], [t.sup.W.sub.i]), [t.sup.W.sub.i]). This equality will in general not be satisfied for [[??].sub.i] [not equal to] [[??].sub.i] unless is separable in Q and [t.sup.W.sub.i]). In turn, this requires that the scoring rule be quasilinear ([PSI] (Q, [t.sup.W.sub.i])) = [phi](Q) - [t.sup.W.sub.i]) for a quasilinear scoring rule). (8)