Properties of scoring auctions.

Theorem 1 ensures that there is no loss of generality in concentrating on pseudotypes when deriving the equilibrium in the scoring auction, even if the scoring rule does not correspond to the buyer's true preference. (Note that Theorem 1 does not rule out equilibria where different types submit different (p, Q) bids--but given that they yield the same score and the same probability of winning at equilibrium, they are payoff irrelevant.)

Most theoretical analyses of scoring auctions have implicitly or explicitly taken advantage of pseudotypes to derive an equilibrium in these auctions (Che, 1993; Bushnell and Oren, 1994, 1995). Theorem 1 suggests that doing so does not discard any other equilibria of interest. Although this might not be totally surprising when types are one-dimensional, this result is not trivial for environments where types are multidimensional. This property is a consequence of the combination of the quasilinear scoring rule, the single dimensionality of the allocation decision, and the independence of types across bidders. We cannot reduce the strategic environment to one-dimensional private information if any of these conditions does not hold. As argued in Section 1, the quasilinearity of the scoring rule is necessary to be able to summarize suppliers' preferences over contracts by a single number. As noted after Lemma 1, independence was needed to make suppliers' beliefs independent of their types. Neither condition is necessary to use pseudotypes to derive an equilibrium in the one-dimensional model (for example, Branco, 1997 extends Che's model to correlated private information).

The next result makes the relationship between scoring auctions and standard one-object auctions even more explicit.

Corollary 1. The equilibrium in quasilinear scoring auctions with independent types inherits the properties of the equilibrium in the related single-object auction where (i) bidders are risk neutral, (ii) their (private) valuations for the object correspond to the pseudotype k in the original scoring auction and are distributed accordingly, (iii) the highest bidder wins, and (iv) the payment rule is determined as in the scoring auction, with bidders' scores being replaced by bidders' bids.

Corollary 1 has practical implications for the derivation of the equilibrium in scoring auctions. It suggests the following simple algorithm for deriving equilibria in scoring auctions: (1) given the scoring rule, derive the distribution of pseudotypes, [G.sub.i](k), (2) solve for the equilibrium in the related IPV auction where valuations are distributed according to [G.sub.i](k), [b.sub.i](k), and (3) the equilibrium bid in the scoring auction is any (p, Q) such that S(p, Q) = [b.sub.i](k). (The actual (p, Q) delivered are easy to derive given [b.sub.i](k) and the solution to equation (1).)

4. Expected utility equivalence across auction formats

In this section, we extend the revenue equivalence theorem (Myerson, 1981; Riley and Samuelson, 1981) to multi-attribute environments. Che (1993) proved the utility equivalence between the first- and second-score scoring auction when types are one- dimensional and the scoring rule corresponds to the buyer's true preference. Theorem 2 shows that this result extends to multidimensional private information and scoring rules that do not correspond to the buyer's true preference.

Theorem 2 (Expected utility equivalence). Any two scoring auctions that:

(i) use the same quasilinear scoring rule,

(ii) use the same allocation rule [x.sub.i]([k.sub.i], [k.sub.-i]), i = 1, ..., N, and

(iii) yield the same expected payoff for the lowest pseudotype [[k.bar].sub.i] i = 1, ..., N,.

generate the same expected utility for the buyer

Proof. Because the buyer's utility is quasilinear, his expected utility from a given auction is

([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

where ESS([k.sub.i]) is the expected social surplus generated by awarding the contract to bidder i with pseudotype [k.sub.i].

By Theorem 1, we can focus on equilibria which are only functions of pseudotypes. Incentive compatibility implies that [U.sub.i]([k.sub.i]) is almost everywhere differentiable and that d/d[k.sub.i] [U.sub.i]([k.sub.i]) = [x.sub.i]([k.sub.i]), where [x.sub.i]([k.sub.i]) is a well-defined function almost everywhere by Lemma 2. Hence, (ii) and (iii) imp y that [U.sub.i](k) is the same across both auctions.

Next, fix [k.sub.i] and let ([p.sup.*] ([[theta].sub.i], [s.sub.i]), [Q.sub.*] ([[theta].sub.i], [s.sub.i])) be the realized contract of supplier i with type [[theta].sub.i] [member of] [[theta].sub.i]([k.sub.i]), when the score to satisfy is [s.sub.i]. Because the scoring rule is quasilinear, [Q.sub.*] ([[theta].sub.i], [s.sub.i]) is only a function of the scoring rule and [[theta].sub.i], and not of [s.sub.i] (cf (1)). Hence,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is independent of [s.sub.i] and equal across the two auctions given (i). The claim follows. Q.E.D.

Three points are worth noting concerning this result. First, the assumption that the scoring rule is quasilinear is key. Without it, suppliers' choice of product characteristics (p, Q) would depend on the form of the resulting obligation, that is, the auction format.

Second, the proof of Theorem 2 relies on the fact that any equilibrium is essentially pure as a function of pseudotypes (i.e., [x.sub.i] are functions). Without this property, expected utility equivalence between two auctions that yield the same distribution of allocations as a function of pseudotypes would only hold when the scoring rule corresponds to the true valuation (cf the argument before Lemma 2). In that case, ESS([k.sub.i]) = [k.sub.i] and the result holds trivially.

Third, Theorem 2 implies the standard equivalence between the first-score auction, the second-score auction, and the Dutch and English auctions when bidders are symmetric. However, the symmetry requirement is with respect to the distribution of pseudotypes and not the distribution of types. In particular, some bidders can (stochastically) be stronger for one attribute and others for another attribute, yet, when it comes to pseudotypes, they can be symmetric.

5. Comparison with alternatives

In this section, we consider three alternatives to scoring auctions under three different auction formats, and for simplicity we focus on the case where suppliers are ex ante symmetric. We show that, except for the first-price menu auction, these alternatives generate equal or lower expected utility for the buyer than a scoring auction that uses the true preference of the buyer. Thus, a fortiori, a scoring auction with an optimally chosen scoring rule dominates these alternatives. We next describe these procedures in detail and discuss some of their properties.

Menu auctions. (12) In the menu auction, the buyer does not reveal his type. Instead, suppliers are asked to submit (p, Q) schedules. The buyer selects the offer that generates the highest level of utility. This alternative comes in three versions. In the "ascending" version (A), the auction takes place over several rounds. In each round, the buyer selects the supplier whose schedule generates the greatest utility. In the next round, the other suppliers are invited to submit new schedules and the process continues until no further offers are made. The winner is the supplier who offers the best schedule in the last round. The resulting contract is the (p, Q) in his schedule that the buyer prefers. In the "first-price" version (FP), the winner is the supplier offering the (p, Q) contract that generates the highest utility to the buyer and this is the resulting contract. Finally, in the "second-price" version (SP), the winner is the supplier offering the (p, Q) contract that generates the highest utility to the buyer and the resulting contract is ([??], Q), where [??] is adjusted so that ([??], Q) generates the same score as the best offer by the losers. (13)

Menu auctions introduce an interesting new twist: suppliers must now account for the fact that the buyer selects the offer he prefers in the submitted menus. Let (Q(t, [theta]), p[(t, [theta])).sub.t[member of]T] denote the menu submitted by a supplier of type [theta], with the indexing such that (Q(t, [theta]), p(t, [theta])) is the offer preferred by the buyer with taste t. Incentive compatibility for the buyer requires that

v(Q(t, [theta]), t) - p(t, [theta]) [greater than or equal to] v(Q([??],[theta]), t) - p([??],[theta]) [for all]t, [??] [member of] T, (8)

that is, using standard arguments and the fact that [v.sub.Qt], > 0 ([v.sub.Qt], ensures that Q is monotonic and thus a.e. differentiable),

[v.sub.Q](Q(t, [theta]), t)Q,(t, [theta]) = p,(t, [theta]) for all [theta] and all t at which Q is differentiable. (9)

Lemma 3. Consider any incentive-compatible menu (Q(t, [theta]), [p(t, [theta])).sub.l[member of]T]. This menu induces efficient production for all t, [theta] if and only if (i) it corresponds to an ex post iso-profit curve of supplier with type [theta] and (ii) Q(t, [theta]) is a.e. differentiable with Q,(t, [theta]) > 0 a.e.

Proof. Efficient production requires

[v.sub.Q](Q(t, [theta]), t) = [c.sub.Q](Q(t, [theta]), [theta]) for all t, [theta]. (10)

Condition (ii) follows directly from the requirement of efficiency together with the assumption that [v.sub.Qt], > 0. Suppliers' ex post iso-profit curves are described by the locus of offers such that p(t, [theta]) - c(Q(t, [theta]), [theta]) is constant, that is,