Properties of scoring auctions.

with [lambda]([bar.t], [theta]) = [lambda]([t.bar], [theta]) = 0.

To complete the proof, we now argue that [lambda](t, [theta]) [not equal to] 0 for a positive measure of t, that is, the buyer's incentive constraint binds. Toward a contradiction, suppose it does not. Then the equilibrium in the menu auction corresponds to the equilibrium in the scoring auction as t varies. Denote by ([Q.sup.s](t, [theta]), [p.sup.s][(t, [theta])).sub.t[member of]T] the menu generated from the equilibrium offers submitted by a supplier of type [theta] in the scoring auction as t varies. From Corollary 1 and the known characterization of the equilibrium in the single-object IPV auction, the supplier's ex post profit in the scoring auction is given by

k(t, [theta]) - [E.sub.[??]][[k(t, [??]).sup.(1:N-1)] | [k(t, [??]).sup.(1:N-1)] < k(t, [theta])],

where k(t, [theta]) is the pseudotype that corresponds to type [theta] when the scoring rule is equal to v(Q, t) - p, and [k(t, [??]).sup.(1:N-1)] denotes the first-order statistics of N - 1 independent draws of pseudotypes. By the envelope theorem, [k.sub.t](t, [theta]) = [v.sub.t](Q(t, [theta]), t). It is independent of the distribution of [theta]. By contrast, the second term and its derivative with respect to t is a function of the distribution of [theta]. Thus the two terms do not cancel out and expost profits depend on t for a given [theta] (except if supplier [theta] wins with zero probability for all t). This implies--by Lemma 3--that the menu that corresponds to the equilibrium in the scoring auction is not incentive compatible, a contradiction.

When [lambda].(t, [theta]) [not equal to] 0, equation (A2) implies that qualities are distorted away from their efficient levels. Q.E.D. (24)

Sketch of proof of Theorem 5. Here we show how to adapt the arguments in Biglaiser and Mezzetti (2000) to fit our procurement framework. (25)

Let [U.sub.j]([theta]) be the utility of a type j buyer from the contract supplier [theta] offers to him. Let [[theta].sup.-1] (U.sub.j]) be the inverse of [U.sub.j]([theta]). Let Pr ([t.sub.L]) = p. The bidding problem faced by each supplier is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

subject to the following incentive compatibility constraints:

[IC.sub.H] : v([Q.sub.H], [t.sub.H]) - [p.sub.H] [greater than or equal to] v ([Q.sub.L], [t.sub.H])- [P.sub.L]

[IC.sub.L] : v([Q.sub.L], [t.sub.L] - [p.sub.L] [greater than or equal to] v ([Q.sub.H], [t.sub.L]) - [p.sub.H].

Assuming that [IC.sub.L] is slack (this is verified ex post) and rewriting [IC.sub.H] as follows,

[U.sub.H] [greater than or equal to] v([Q.sub.L], [t.sub.H])- v([Q.sub.L], [t.sub.L]) + [U.sub.L]

gives rise to the following Lagrangian for the bidders' optimization problem:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Of interest to us are the first-order conditions with respect to [U.sub.L] and [U.sub.H], which can be written as

[partial derivative][U.sub.L]/[partial derivative][theta] = [p(N - 1)g ([theta])/([lambda]([theta]) + p) (1 - G ([theta])) (v([Q.sub.H], [t.sub.H]) - c([Q.sub.H], [theta]) - [U.sub.H])] (A4)

[partial derivative][U.sub.H]/[partial derivative][theta] = [(1 - p)(N - 1)g([theta])/(1 - ([lambda](theta]) + p))(1 - G([theta])) (v([Q.sub.L], [t.sub.L] - c([Q.sub.L], [theta]) - [U.sub.L])]. (A5)

When [lambda]([theta]) = 0, [IC.sub.H] does not bind and the menu auction is equivalent to two independent scoring auctions. When [lambda]([theta]) > O, [IC.sub.H] does bind. In this case, [U.sub.H] increases and UL decreases. To see this, note that (A4) implies [U.sub.H] is increasing in [lambda]([theta]) and (A5) implies [U.sub.L] is decreasing in [lambda]([theta]). It follows that [U.sup.FP.sub.menu]([t.sub.L]) [less than or equal to] [U.sup.FP.sub.scoring] ([t.sub.L]) and [U.sup.FP.sub.menu]([t.sub.H] [greater than or equal to] [U.sup.FP.sub.scoring] ([t.sub.H]. Q.E.D.

This article supersedes an earlier paper distributed under the title "Equilibrium in Scoring Auctions." We thank Max Bazerman, Matthew Jackson, Luca Rigotti, Al Roth, Nicolas Sahuguet, Sasha Wolf, as well as audiences at Barcelona Auction Workshop 2005, Case Western Reserve, CORE, Decentralization Conference 2004, ECARES, Essex, Gerzeusee, Harvard/MIT, Inform 2003, London Business School, London School of Economics, New York University, Ohio State University, Rome Auction Conference 2004, University College London, and WZB Berlin for useful conversation and suggestions. The editor and two anonymous referees provided useful comments. Ioannis Ioannou provided excellent research assistance. The financial support of the Division of Research at Harvard Business School is gratefully acknowledged.

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(1) The road user cost is the (per-day) value of time lost owing to construction. By 2003, 38 states in the United States were using "A + B bidding" for large projects for which time is a critical factor. See, for instance, Arizona Department of Transport (2002) and Herbsman, Chen, and Epstein (1995).

(2) See Articles 55 and 56 of Directive 2004/17/EC and Articles 53 and 54 of Directive 2004/18/EC. If the authority does not resort to electronic auctions, it may publish a range of weightings for each criterion instead.

(3) http://europa,eu.int/comm/internal_market/publicprocurement/ introduction_en.htm.

(4) A variant of scoring auctions are auctions that involve the purchase of multiple items but where the buyer cannot commit, at the auction, to the quantity purchased. The scoring rule is used for allocating the contract, although the final contract depends on the realized quantities. This creates incentive problems we ignore (see Athey and Levin, 2001; Chao and Wilson, 2002; Ewerhart and Fieseler, 2003).

(5) A similar property (although through a much more subtle analogy to the standard IPV model) is exploited by Che and Gale (2006) to rank revenue in single-object auctions with multidimensional types and nonlinear payoffs.