Global COE Hi-Stat Discussion Paper Series 238

Research Unit for Statisticaland Empirical Analysis in Social Sciences (Hi-Stat)

Optimal Design of Scoring Auction with
Multidimensional Quality

Hi-Stat Discussion Paper

Takeshi Nishimura

July 2012

Hi-Stat
Institute of Economic Research
Hitotsubashi University
2-1 Naka, Kunitatchi Tokyo, 186-8601 Japan
http://gcoe.ier.hit-u.ac.jp

Optimal Design of Scoring Auction with
Multidimensional Quality
Takeshi Nishimura∗†
May 29, 2012

Abstract
This article studies the optimal design of scoring auction used in public procurement. In this auction, each supplier’s offer consists of both price and quality,
and a supplier whose offer achieves the highest score wins. The environment we
consider has the feature that quality is multi-dimensional. We show that there
exists a quasi-linear scoring rule which implements the optimal mechanism for thebuyer and is supermodular in quality if the virtual surplus is quasisupermodular
in quality. We further investigate how the buyer should classify quality attributes
when using a scoring rule which is additively separable in the attributes.

∗

Hitotsubashi University; ed091004@g.hit-u.ac.jp.
I am grateful to my advisor Akira Okada for his invaluable guidance and encouragement. I would
also like to thank Reiko Aoki, Gary Biglaiser, Makoto Hanazono, Hideshi Itoh, R. Vijay Krishna, Takashi
Kunimoto, Jun Nakabayashi, Ichiro Obara, Sergio Parreiras, Olivier Tercieux for their thoughtful comments and suggestions. This work was supported by Grant-in-Aid for Japan Society for the Promotionof Science (JSPS) Fellows.
†

1

1

Introduction

Auction rules of public procurement have changed from one-dimensional bidding to multidimensional bidding. In contrast with the former traditional rule in which each suppliersubmits only a price-bid, the latter auction rule requires suppliers to offer not only price
but also quality they promise to ensure in a project. For instance, in the EU, Article
53 of Directive 2004/18/EC specifies the “Most Economically Advantageous Tender”,
in which the procurement authorities award contracts based on various criteria such as
price, technical merit, aesthetic characteristics, delivery date, and so on. The design ofthe multi-dimensional auction is a matter of great concern to the procurement authorities,
reflecting the fact that public procurement accounts for about 16% of GDP in OECDmember countries (OECD, 2008).
The essential element of the multi-dimensional auction is a scoring rule. The rule,
which evaluates suppliers’ offers and gives them scores, should be carefully designed
because it considerably affects suppliers’ decisions what offers to make. There are many
examples of scoring rules adopted by state departments of transportation in the US:“A+B bidding” (Arizona, etc.), “weighted criteria” (Delaware, Idaho, Massachusetts,
Oregon, Utah, Virginia, etc.), “adjusted bid” (Arizona, Maine, Michigan, North Carolina,
South Carolina, South Dakota, etc.), and so on (see Molenaar and Yakowenko, 2007).
For instance, the rule of “weighted criteria” puts a weight on each of price and quality
attributes (e.g. delivery date, safety level) and evaluates each attribute individually, sothat a total score of each offer is a weighted sum of subscores and a supplier with thehighest total score wins a contract.
The previous studies have confirmed that the scoring auction performs very well fromthe buyer’s point of view. A common feature of the environments is that a buyer pro2

cures a single product (e.g. highway) differentiated by its quality from one of suppliers,
who have private information about cost parameters. In a seminal article, Che (1993)shows that a scoring auction with a properly designed scoring rule implements the buyer’s
optimal (i.e. utility-maximizing) mechanism characterized by Laffont and Tirole (1987),
McAfee and McMillan (1987), and Riordan and Sappington (1987). Branco (1997) extends this result to an environment where each supplier’s production cost has a commoncost component, so that his cost is correlated with the other suppliers’ costs.
In contrast with Che and Branco, who assume that both quality and each supplier’s
type are one-dimensional, Asker and Cantillon (2008) consider a fully general environment
where both elements are multi-dimensional. Although the main results of Asker andCantillon are the characterization of equilibrium bidding behavior and the expected utility
equivalence between some formats of scoring auction, they also show that the scoring
auction outperforms some other mechanisms from the buyer’s viewpoint. On the other
hand, they have not investigated whether a scoring auction can implement the optimalmechanism. The likely reason is that it is extremely difficult to characterize the optimalmechanism when the supplier has multi-dimensional private information. However, Asker
and Cantillon (2010) characterize the optimal mechanism in a specific environment where
quality is one-dimensional, each supplier’s type consists of two parameters (fixed costand marginal cost) and each parameter is a binary random variable. They show that thescoring auction yields a performance close to that of the optimal mechanism, taking a
numerical simulation approach.
All of the above studies focus on “quasi-linear” scoring rules under which a total scoreof each offer is given by a quality score minus price. A typical example of quasi-linear ruleis “weighted criteria”. In a recent study, Hanazono et al. (2011) consider “price-quality

3

ratio” scoring rules under which a total score of each offer is given by price divided by aquality score, and analyze the equilibrium bidding behavior.
In addition to these theoretical studies, there is some experimental evidence supporting the high performance of scoring auction compared to that of traditional price-only
auction (Bichler, 2000; Chen-Ritzo et al., 2005).
Although many scoring rules used in practice apply multiple quality criteria, there
are no theoretical studies investigating whether a scoring auction can implement the
optimal mechanism in an environment where quality is multi-dimensional. It is by nomeans trivial to answer the question and show what form of scoring rule succeeds inthe implementation, at least from the Revelation Principle (see, e.g., Myerson, 1981).
Moreover, with the model of multi-dimensional quality, it will be anticipated that theinteraction among quality attributes significantly affects the optimal form of scoring rule.
These observations motivate our current work. We generalize the model of Che (1993) byallowing for multi-dimensional quality. A scoring rule announced by the buyer induces
the following auction game. First, all suppliers’ (one-dimensional) cost parameters are
realized, and each supplier is privately informed about his own parameter respectively.
Second, each supplier simultaneously and independently offers both price and quality. A
supplier wins if the score of his offer is the highest among suppliers and higher than thepredetermined reserve score.1 The winner’s offer becomes a binding contract.
We derive two results. First, we show that there exists a quasi-linear scoring rule
which implements the optimal mechanism and is supermodular in quality if the so-called“virtual surplus” is quasisupermodular in quality. These properties of supermodularityand quasisupermodularity, which represent the concept of complementarity between qual1

The introduction of reserve score is also an extension of Che (1993).

4

ity attributes, are specific to the model of multi-dimensional quality (see Topkis, 1998 forthe definitions). In particular, our result implies that the scoring rule should evaluate all
quality attributes as a whole (not separately) to give a total score in a complementary
way. Second, we derive a necessary condition and a sufficient condition for the existenceof a quasi-linear scoring rule which implements the optimal mechanism and is additively
separable in some quality attributes. Our results imply that when the buyer establishes
some sets of subcriteria to use the additively separable rule, each pair of quality attributes
in the distinct sets should be complementary in terms of the production cost.
This article contributes to the literature on scoring auction in two ways. First, weconstruct the optimal scoring rule in a different manner from Che (1993) and Branco(1997). When quality is one-dimensional, the scoring rule constructed by them provides
suppliers with incentives to offer the appropriate quality level. On the other hand, ourexample shows that when quality is multi-dimensional, a natural extension of their scoring
rule may induce unsuitable quality offers from the buyer’s viewpoint. To resolve thisproblem, we construct the optimal scoring rule by applying the method of monotone
comparative statics (Topkis, 1998). The constructed rule satisfies supermodularity inquality to deter a supplier’s deviation from the desirable quality offer for the buyer.
Also, as a by-product of using the monotone comparative statics method, we require no
assumptions of concavity (or convexity) and differentiability in quality of the value andcost functions, and allow quality attributes to be discrete. Second, we provide a useful
guide to designing scoring rules which are additively separable in some quality attributes.
The result has important policy implications because additively separable scoring rules(e.g. “weighted criteria”) are widely adopted and it must be easier for procurement
authorities to administer those rules.

5The article is organized as follows. Section 2 presents the model which generalizes thatof Che (1993). Section 3 derives the equilibrium bidding strategy. Section 4 shows how
to design a scoring rule to implement the optimal mechanism. Section 5 investigates how
the buyer should classify quality attributes when using a scoring rule which is additively
separable in the attributes. Section 6 concludes. All proofs are in the Appendix.

2

The model

Consider a buyer who procures a single product from one of N suppliers. A (production)
contract between the buyer and a supplier i ∈ {1, ..., N } is denoted by (pi , qi ) ∈ R+ × Q,
under which the supplier i must deliver a product of quality qi = (qi1 , ..., qiM ) ∈ Q ⊂ RM
in exchange for price pi ∈ R+ .2 For each m ∈ {1, ..., M }, qim represents a level of quality(non-monetary) attribute. The supplier i’s cost parameter is given by θi ∈ [θ, θ̄] ⊂ R. The
suppliers’ types (θ1 , ..., θN ) are random variables which are independent across suppliers.
The cumulative distribution function of θi is given by F , with a density function f thatis strictly positive everywhere. Each supplier has private information about his realized
type respectively, but the prior probability distribution is common knowledge.
A supplier i of type θi earns profits p−c(q, θi ) from a contract (p, q), where c(q, θi ) > 0
is his production cost. The buyer’s utility from a contract (p, q) with a supplier of type
θi is a weighted sum of consumers’ surplus and profits, i.e. v(q) − p + α(p − c(q, θi )),where v(q) is the valuation for a product of quality q and α ∈ [0, 1] is a weight on profits.
We make the following assumptions.
mm
Assumption 1. Q = ×M
m=1 Q , where Q ⊂ R is a closed interval or a finite set.
2

In this article, bold letters denote some vectors: q ≥ q̂ means q m ≥ q̂ m for each m; q > q̂ meansq ≥ q̂ and q 6= q̂; q  q̂ means q m > q̂ m for each m.

6

Assumption 2. v is continuous in q.
Assumption 3. c is continuous in (q, θ) and strictly increasing in q. c is differentiable andstrictly increasing in θ. cθ ≡ ∂c/∂θ is continuous in (q, θ) and increasing in θ.
Assumption 4. c has strictly increasing differences in (q, θ), i.e. c(q 0 , θ)−c(q, θ) is strictlyincreasing in θ for each q 0 > q in Q. cθ has increasing differences in (q, θ), i.e. cθ (q 0 , θ) −cθ (q, θ) is increasing in θ for each q 0 > q in Q.
Assumption 5. Ff is increasing in θ.
We will apply the monotone comparative statics method. See Topkis (1998) for somenotions. Assumption 1 ensures that Q is a compact lattice. The notions of increasing
differences express the concept of complementarity between quality attributes and a cost
parameter. Assumptions 4 and 5 ensure that the “virtual surplus” defined later has
strictly decreasing differences in (q, θ).
There is an auction rule (mechanism) that is feasible for the buyer: a scoring auction.
We first define a scoring rule as S : R+ × Q → R. In a scoring auction, each supplieroffers both price and quality, and the scoring rule S assigns a score S(p, q) to each offer(p, q). With a reserve score which is normalized to zero, a supplier i wins only if his scoreis nonnegative and the highest among suppliers.3 We consider a first-score (sealed-bid)format, in which a winner i is awarded a binding contract (pi , qi ) he offered in the auction;
this format corresponds to the first-price format in the standard auction. We focus on aquasi-linear scoring rule, which takes a form of S(p, q) = s(q)−p. We also call the functions : Q → R a (quasi-linear) scoring rule. We assume that s is upper semicontinuous in q,
and s has a cost parameter θ̄s ∈ (θ, θ̄] which satisfies maxq∈Q [s(q) − c(q, θ̄s )] = 0.
3

We assume that if there is a nonnegative tie score, then each supplier achieving the highest scorewins with equal probability. All results hold for any other tie-breaking rule.

7

A scoring rule s publicly announced by the buyer induces the following auction game.
First, all suppliers’ types θ = (θ1 , ..., θN ) are realized, and each supplier is privately
informed about his own type respectively. Second, each supplier i simultaneously andindependently submits an offer (pi , qi ). Then, the game ends. If a supplier i of type θi who
offers (pi , qi ) such that s(qi )−pi = maxj [s(qj )−pj ] ≥ 0 wins, then he receives pi −c(qi , θi ),the other suppliers receive zero payoffs, and the buyer receives v(qi )−pi +α(pi −c(qi , θi )).
If no supplier wins, then the buyer and all suppliers receive zero payoffs.
In the next section, we explore the Bayesian Nash equilibrium of the auction game.
With a slight abuse of notation, we denote a supplier i’s bidding strategy by (pi , qi ) :[θ, θ̄] → R+ × Q. We assume that no supplier uses weakly dominated strategies.

3

Equilibrium bidding strategy

In this section, we derive the equilibrium bidding strategy. The following lemma characterizes a symmetric equilibrium, where all suppliers use the same bidding strategy.
Although our environment is substantially more general than Che (1993) and there areslight technical difficulties, we can apply his technique to prove the lemma.
Lemma 1. The auction game induced by a quasi-linear scoring rule s has a symmetricequilibrium in which (i) the bidding strategy (p∗ , q ∗ ) is such that for each θ ∈ [θ, θ̄s ],q ∗ (θ) ∈ arg max [s(q) − c(q, θ)]

(1)

q∈Q

∗

∗

Z θ̄s

p (θ) = c(q (θ), θ) +∗cθ (q (z), z)θ

1 − F (z)1 − F (θ)

N −1
dz,

(2)

and for each θ ∈ (θ̄s , θ̄], (p∗ (θ), q ∗ (θ)) is an arbitrary offer which satisfies s(q ∗ (θ)) −p∗ (θ) < 0, and (ii) a supplier of type θi wins only if θi = min{θ1 , ..., θN , θ̄s }. Moreover,for each θ ∈ [θ, θ̄s ], any offer (p0 , q 0 ) such that q 0 6∈ arg maxq∈Q [s(q) − c(q, θ)] is weakly
8dominated by (p, q ∗ (θ)) such that s(q ∗ (θ)) − p = s(q 0 ) − p0 .
It is important to note that in equilibrium any supplier who wins with positive probability chooses quality so as to maximize a quality score minus his production cost, asin (1). Actually, any offer which does not maximize s(q) − c(q, θ) is weakly dominatedby the quality offer q ∗ (θ) with some price offer. Given the optimal quality offer q ∗ (θ),
the optimal price offer p∗ (θ) is determined by (2). The price offer is greater than the
production cost due to the “shading behavior”.
Lemma 1 also implies that in equilibrium the most efficient supplier wins provided
that his type is lower than θ̄s . Let θ(N ) ≡ min{θ1 , ..., θN } be the lowest cost parameter
among suppliers, which is also a random variable. We denote the cumulative distributionfunction of θ(N ) by F(N ) (·) = 1 − (1 − F (·))N . Then, the buyer’s expected utility fromannouncing a quasi-linear scoring rule s isF(N ) (θ̄s )E v(q ∗ (θ(N ) )) − p∗ (θ(N ) ) + α[p∗ (θ(N ) ) − c(q ∗ (θ(N ) ), θ(N ) )] | θ(N ) ≤ θ̄sZ θ̄s 
F (θ)
∗
∗
∗
=
v(q (θ)) − c(q (θ), θ) − (1 − α)cθ (q (θ), θ)dF(N ) (θ),
f (θ)θ
where the equality follows from the substitution of p∗ (θ(N ) ) and the interchange of theorder of integration. We now define the virtual surplus as the function Φ ≡ v − c − (1 −(θ)α)cθ Ff . Its value Φ(q, θ) = v(q) − c(q, θ) − (1 − α)cθ (q, θ) Ff (θ)
times the density f (θ) is

the social surplus generated by trading a product of quality q between the buyer and a
supplier of type θ, minus the sum of information rents paid to the more efficient supplierthan θ. Using this virtual surplus, the buyer’s expected utility can be rewritten asZ θ̄sΦ(q ∗ (θ), θ)dF(N ) (θ).

θ

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4

Optimal scoring rule

In this section, we study the implementation problem. The analysis proceeds in two
steps. First, we characterize an optimal mechanism, following the standard mechanismdesign approach with the envelope theorem of Milgrom and Segal (2002). Second, we
find the condition under which a scoring auction implements the optimal mechanism,
and investigate how the scoring rule should be designed.
In a first step, we use the Revelation Principle to focus on “incentive compatible direct mechanisms”. A direct mechanism is an 3N -tuple of measurable functions(Pi , Qi , Xi )i∈{1,...,N } where (Pi , Qi , Xi ) : [θ, θ̄]N → R × Q × [0, 1]. For each profile of types
θ̂ = (θ̂1 , ..., θ̂N ) reported by suppliers, a payment schedule Pi (θ̂) specifies the expected
payment from the buyer to the supplier i, a quality schedule Qi (θ̂) specifies quality the
supplier i must ensure when delivering the product, and Xi (θ̂) withPN

i=1 Xi (θ̂) ≤ 1 spec-

ifies the trading probability between the buyer and the supplier i. A direct mechanism(Pi∗ , Q∗i , Xi∗ )i∈{1,...,N } is optimal (for the buyer) if it solves the following problem:
NX

max
ρ=(Pi ,Qi ,Xi )i∈{1,...,N }E [Xi (θ)v(Qi (θ)) − Pi (θ) + α[Pi (θ) − Xi (θ)c(Qi (θ), θi )]]i=1

s.t. Πρi (θi | θi ) ≥ Πρi (θˆi | θi ) for each θi , θˆi ∈ [θ, θ̄], i ∈ {1, ...N }Πρi (θi | θi ) ≥ 0

for each θi ∈ [θ, θ̄], i ∈ {1, ...N }

(3)
(4)

where Πρi (θˆi | θi ) ≡ Eθ−i [Pi (θˆi , θ−i ) − Xi (θˆi , θ −i )c(Qi (θˆi , θ −i ), θi )]. The first constraint is
the interim incentive compatibility (IC) constraint for each supplier, and the second oneis the interim individual rationality (IR) constraint for each supplier.
The next lemma characterizes an optimal mechanism. To explain the hypothesis of
the lemma and the related notions, we introduce some definitions. Denote q ∧ q̂ ≡(min{q 1 , q̂ 1 }, ..., min{q M , q̂ M }) and q ∨ q̂ ≡ (max{q 1 , q̂ 1 }, ..., max{q M , q̂ M }). Notice that10

q, q̂ ∈ Q implies q ∧ q̂, q ∨ q̂ ∈ Q because Q is a lattice. The virtual surplus Φ isquasisupermodular in quality if, for each q, q̂ ∈ Q and each θ ∈ [θ, θ̄], Φ(q ∧ q̂, θ) ≤ Φ(q, θ)
implies Φ(q̂, θ) ≤ Φ(q ∨ q̂, θ), and Φ(q ∧ q̂, θ) < Φ(q, θ) implies Φ(q̂, θ) < Φ(q ∨ q̂, θ).
The virtual surplus Φ is supermodular in quality if, for each q, q̂ ∈ Q and each θ ∈ [θ, θ̄],
Φ(q, θ) − Φ(q ∧ q̂, θ) ≤ Φ(q ∨ q̂, θ) − Φ(q̂, θ); we apply the same definition to a scoring rules. It is easy to show that if Φ is supermodular in quality, then Φ is quasisupermodular
in quality. Thus, quasisupermodularity is a weaker notion than supermodularity. Thesenotions express the concept of complementarity between non-monetary attributes.
Lemma 2. Suppose that the virtual surplus Φ is quasisupermodular in quality. Then, thefollowing direct mechanism (Pi∗ , Q∗i , Xi∗ )i∈{1,...,N } is optimal for the buyer:




1
if θi < min{θ1 , ..., θi−1 , θi+1 , ..., θN , θ̄∗ }




∗
1
Xi (θ) =if θi = min{θ1 , ..., θi−1 , θi+1 , ..., θN , θ̄∗ }]{j|θj =θi }






 0
if θi > min{θ1 , ..., θi−1 , θi+1 , ..., θN , θ̄∗ }Q∗i (θ) = Q∗ (θi ) ∈ arg max Φ(q, θi )
q∈Q
"
Z ∗

(6)

θ̄

Pi∗ (θ) = Xi∗ (θ)∗

c(Q (θi ), θi ) +(5)∗cθ (Q (z), z)θi

1 − F (z)1 − F (θi )

N −1

#
dz ,

(7)

where θ̄∗ ∈ [θ, θ̄] is a cost parameter such that Φ(Q∗ (θ), θ) ≥ 0 iff θ ∈ [θ, θ̄∗ ]. Moreover,Q∗ (θi ) is decreasing in θi .
The quasisupermodularity of the virtual surplus, which is trivially satisfied if quality
is one-dimensional, plays a key role in determining the property of the optimal quality
schedule Q∗i . The proof shows that a necessary and sufficient condition for the IC constraints (3) is given by the two conditions. With these conditions, we can rewrite the11buyer’s maximization problem as follows:max(Qi ,Xi )i∈{1,...,N }N
X

E [Xi (θ)Φ(Qi (θ), θi )]

i=1

s.t. For each θ, θ̂ ∈ [θ, θ̄] with θ < θ̂ and each i ∈ {1, ...N },Z θ̂Eθ−i [Xi (z, θ−i )cθ (Qi (z, θ−i ), z) − Xi (θ̂, θ−i )cθ (Qi (θ̂, θ−i ), z)]dz ≥ 0.

(8)

θThe condition (8), which we call the monotonicity condition, is automatically satisfied if
both Qi and Xi are decreasing in θi . Using the result of Milgrom and Shannon (1994),
the proof shows that the quasisupermodularity of the virtual surplus in quality together
with Assumptions 4 and 5 imply that a quality schedule Qi which maximizes the virtual surplus Φ(q, θi ) is decreasing in θi ; notice that this quality schedule Q∗ depends
only on a supplier’s type report because all suppliers are ex ante symmetric. The proof
also shows that the maximized virtual surplus Φ(Q∗ (θi ), θi ) is decreasing in θi . As a result, the quality schedule Q∗ and the trading probability Xi∗ defined in Lemma 2 satisfy
the monotonicity condition (8) and constitute the optimal mechanism with the payment
schedule Pi∗ . Thus, the more efficient a supplier is, the higher levels of all non-monetaryattributes he is required to achieve. On the other hand, without the quasisupermodularity, the optimal quality schedule Q∗ may be nonmonotonic in θi . This is because even if
Qm∗ is not decreasing in θi for some m, the monotonicity condition (8) can be satisfied
when cθ (Q(θi ), z) is decreasing in θi . The following example shows that this is the case.4
Example 1. Assume that M = 2, Q1 = Q2 = {1, 2}, v(q 1 , q 2 ) = 0 if q 1 = q 2 = 1 andv(q 1 , q 2 ) = 100 otherwise, c(q 1 , q 2 , θ) = 6θq 1 + (3θ + 9)q 2 − q 1 q 2 , θ is uniformly distributedon [1, 3], and α = 0. Then, one can show that Φ(q 1 , q 2 , θ) = v(q 1 , q 2 ) − (12θ − 6)q 1 −(6θ + 6)q 2 + q 1 q 2 , and Φ is not quasisupermodular in quality. Actually, Φ is submodular
4This can occur even in a single-agent screening model. See Laffont and Martimort (2002).

12

in quality, i.e. −Φ is supermodular in quality. The following levels of non-monetaryattributes maximizes Φ(q 1 , q 2 , θ) for each θ: (2, 1) if θ ∈ [1, 2)∗
1∗2∗
Q (θ) = (Q (θ), Q (θ)) = (1, 2) if θ ∈ [2, 3].
The quality schedule Q∗ together with Xi∗ defined in Lemma 2 and θ̄∗ = θ̄ = 3 satisfy themonotonicity condition (8) because cθ (Q∗ (θ), z) = 15 if θ ∈ [1, 2) and cθ (Q∗ (θ), z) = 12
if θ ∈ [2, 3], and thus cθ (Q∗ (θ), z) is decreasing in θ. Therefore, the optimal quality
schedule Q∗ is nonmonotonic in θ because Q1∗ is decreasing whereas Q2∗ is increasing.
In a second step, we discuss the implementation of the optimal mechanism via ascoring auction. We say that a (quasi-linear) scoring rule s implements the optimalmechanism if the auction game induced by s has a Bayesian Nash equilibrium whichyields the same outcome as (Pi∗ , Q∗ , Xi∗ )i∈{1,...,N } for each realization of θ. Lemmas 1
and 2 imply that the buyer’s goal is to construct a scoring rule s so that each supplier’s
equilibrium offer q ∗ (θ) is equal to Q∗ (θ) for each θ ∈ [θ, θ̄∗ ] and the inefficient suppliers(θ ∈ (θ̄∗ , θ̄]) are excluded by the reserve score. The next proposition demonstrates how
the scoring rule should be constructed under the same condition as Lemma 2.
Proposition 1. Suppose that the virtual surplus Φ is quasisupermodular in quality. Then,
there exists a quasi-linear scoring rule s∗ which implements the optimal mechanism andis supermodular in quality.
In the construction of the optimal scoring rule s∗ , we fully utilize the monotonicity ofthe optimal quality schedule Q∗ , which is guaranteed by the quasisupermodularity of thevirtual surplus in quality. Comparing (1) with (6), we see that if a weight α on profits
is equal to one so that the buyer does not care about information rents, then the scoring
rule s which is equal to her valuation v succeeds in the implementation. In general,13however, because α < 1, we must carefully devise the optimal scoring rule. Figure 1
shows how the rule s∗ is constructed. Suppose that quality is two-dimensional. In theleft figure, the solid curve represents the qualities which the optimal quality schedule Q∗
requires suppliers to ensure for each type θ ∈ [θ, θ̄∗ ]. Lemma 2 implies that this curve
is upward sloping due to the monotonicity of Q∗ . The rule s∗ is constructed so that ascore remains constant even if, starting from any point on the curve, the level of only
one attribute increases, and a score falls to zero if the level of at least one attribute islower than Qm∗ (θ̄∗ ). Thus, any supplier who wishes to win has no incentive to make aquality offer other than offers on the solid curve. In the right figure, the three solid curves
below the dotted curve represent the production costs for some cost parameters. These
production costs have the “single-crossing property” due to the monotonicity of Q∗ withthe assumption that the incremental cost is increasing in a cost parameter. Thus, there
exists a “lower envelope” of the cost curves shifted up, which is described by the dottedcurve and given by the following function of θ̂:
∗

Z θ̄∗

c(Q (θ̂), θ̂) +cθ (Q∗ (z), z)dz.

θ̂This is used as a score of the quality offer Q∗ (θ̂) with θ̂ ∈ [θ, θ̄∗ ]. The second term isa score added to the production cost c(Q∗ (θ̂), θ̂) to facilitate the separation of types.
Then, each supplier of type θ optimally makes the quality offer Q∗ (θ̂) = Q∗ (θ) so as tomaximize a quality score s∗ (Q∗ (θ̂)) minus his production cost c(Q∗ (θ̂), θ).
Proposition 1 states that the quasi-linear scoring rule s∗ , which is supermodular in
quality, implements the optimal mechanism. The supermodularity is derived from theLeontief-like shape of s∗ . With the quasisupermodularity of the virtual surplus in quality,
the buyer desires a more efficient supplier to achieve higher levels of all non-monetaryattributes, so that a scoring rule which gives a score in a complementary way works well.
14

On the other hand, without the quasisupermodularity, a supermodular scoring rule s may
fail in the implementation. This problem arises from the nonmonotonicity of the optimal
quality schedule, as shown in Example 1. The next example shows that this is the case.
Example 2. We make the same assumptions as Example 1. Suppose that a quasi-linear
scoring rule s which is supermodular in quality implements the optimal mechanism.
Because the optimal quality schedule satisfies Q∗ (θ) = (1, 2) for θ = 2, Lemma 1 implies
that s(2, 2) − s(1, 2) ≤ c(2, 2, θ = 2) − c(1, 2, θ = 2) = 10. Now, c(2, 1, θ) − c(1, 1, θ) =6θ − 1 > 10 for each θ > 11/6. Because s is supermodular in quality, s(2, 1) − s(1, 1) ≤s(2, 2) − s(1, 2). Hence, c(2, 1, θ) − c(1, 1, θ) > s(2, 1) − s(1, 1) for each θ > 11/6, and thus
a supplier of type θ ∈ (11/6, 2) never chooses Q∗ (θ) = (2, 1) in any equilibrium. This
contradicts the hypothesis that s implements the optimal mechanism.
We construct the scoring rule s∗ in a different way from Che (1993) (and Branco(1997)). We now show that a scoring rule a la Che (1993) which is naturally extended
to an environment where quality is multi-dimensional may fail in the implementation.
Suppose first that quality is one-dimensional and a continuous variable, and the optimal quality schedule Q∗ is strictly decreasing in θ. Then, a quasi-linear scoring rule s
constructed by Che isZ qs(q) = v(q) − (1 − α)0F ((Q∗ )−1 (y))∂ 2c(y, (Q∗ )−1 (y))
dy,∂q∂θf ((Q∗ )−1 (y))

where (Q∗ )−1 is the inverse function of Q∗ . The rule underrewards quality relative to thevaluation because the quality level which maximizes the social surplus v(q) − c(q, θ) is
excessive from the buyer’s viewpoint. With some differentiability assumptions and Inada
conditions, the first-order condition of the problem maxq [s(q) − c(q, θ)] is given bydv∂ 2c
F ((Q∗ )−1 (q)) ∂c(q) − (1 − α)(q, (Q∗ )−1 (q))− (q, θ) = 0,dq∂q∂θf ((Q∗ )−1 (q))∂q15which is satisfied if q = Q∗ (θ). Moreover, together with the assumptions in this article,
the assumption that v, −c and −cθ are concave in quality imply that s is also concave, as
shown by Che (1993). Thus, q = Q∗ (θ) is a global optimal solution to maxq [s(q)−c(q, θ)].
Suppose now that quality is multi-dimensional. Consider the following scoring rule:s(q) = v(q) − (1 − α)

M Z qmX
m=1

0

F (θm (y))
∂ 2c
−m∗ m
m
(y,
Q
(θ
(y)),
θ
(y))
dy,
∂q m ∂θf (θm (y))(9)

where θm is the inverse function of Qm∗ . With some differentiability assumptions andInada conditions, the first-order conditions of maxq [s(q) − c(q, θ)] are given by
m m
∂v
∂ 2c
∂c
m
−m∗ m mm m F (θ (q ))
(q)
−
(1
−
α)
(q
,
Q
(θ
(q
)),
θ
(q
))
− m (q, θ) = 0
m
m
m
m
∂q
∂q ∂θ
f (θ (q ))∂qfor each m, which are satisfied if q = Q∗ (θ). However, even if v, −c and −cθ are concavein quality, the scoring rule s defined by (9) may not be concave and cannot implement
the optimal mechanism. The next example shows that this is the case.
Example 3. Assume that M = 2, Q1 = Q2 = [0, 1], v(q) = q 1 + q 2 + 100, c(q, θ) =θ((q 1 )2 + (q 2 )2 ) + q 1 q 2 , θ is uniformly distributed on [1, 2], and α = 0. Note that because
v is linear in q and both −c and −cθ are strictly concave in q, Φ is strictly concave
in q. One can show that Φ(q, θ) = (q 1 + q 2 ) + 100 − (2θ − 1)((q 1 )2 + (q 2 )2 ) − q 1 q 2 ,Q∗ (θ) = (1/(4θ − 1), 1/(4θ − 1)), and θ̄∗ = θ̄ = 2. The scoring rule s defined by (9) iss(q) = v(q) −2 Z qm
X
m=1

0


2y

1
3
−
4y 4



1
3
dy = (q 1 + q 2 ) + ((q 1 )2 + (q 2 )2 ) + 100.
2
4Then, for some θ, s − c is not concave in q, so that the quality offer Q∗ (θ) = (1/(4θ −1), 1/(4θ − 1)) may not be a global optimal solution to maxq [s(q) − c(q, θ)]. Actually, forθ = 1, s(Q∗ (θ)) − c(Q∗ (θ), θ) = 100 + 1/6 < 100 + 1/4 = s(1, 0) − c(1, 0, θ).

16

5

Additively separable scoring rule

In this section, we investigate whether a quasi-linear scoring rule which is additively
separable in some quality attributes can implement the optimal mechanism.
The scoring rule s∗ constructed in Proposition 1 is supermodular in quality. Then, aquestion arises: Can an additively separable scoring rule such as “weighted criteria” mentioned in the Introduction implement the optimal mechanism? To answer this question,
we introduce a definition. We assume that the set of quality attributes {1, ..., M } is partitioned into two nonempty subsets M1 and M2 of criteria; our analysis can be extended tothe case of more than two subsets at the expense of notational complexity. A quasi-linear
scoring rule s is additively separable if the rule takes a form of s(q) = s1 (q 1 ) + s2 (q 2 ) for
each q = (q 1 , q 2 ) with q 1 ∈ ×m∈M1 Qm and q 2 ∈ ×m∈M2 Qm . In the next example, anadditively separable scoring rule can never implement the optimal mechanism.
Example 4. Assume that M = 2, Q1 = Q2 = {1, 2}, v(q 1 , q 2 ) = 9(q 1 + q 2 ) + 100,c(q 1 , q 2 , θ) = 3θ(q 1 + q 2 ) + (3 − θ)q 1 q 2 , θ is uniformly distributed on [1, 3], and α = 0.
Then, one can show that Φ(q 1 , q 2 , θ) = (12 − 6θ)(q 1 + q 2 ) − (4 − 2θ)q 1 q 2 + 100, and Φ isquasisupermodular in quality. The optimal quality schedule is given by (2, 2) if θ ∈ [1, 2)∗
1∗2∗
Q (θ) = (Q (θ), Q (θ)) = (1, 1) if θ ∈ [2, 3]
and θ̄∗ = θ̄ = 3. Suppose that a quasi-linear scoring rule s which is additively separable
implements the optimal mechanism. Because Q∗ (θ) = (1, 1) for θ = 2, Lemma 1 implies
that s(1, 2) − s(1, 1) ≤ c(1, 2, θ = 2) − c(1, 1, θ = 2) = 7. Now, c(2, 2, θ) − c(2, 1, θ) =6 + θ > 7 for each θ > 1. Hence, c(2, 2, θ) − c(2, 1, θ) > 7 ≥ s(1, 2) − s(1, 1) = [s1 (1) −s1 (1)] + [s2 (2) − s2 (1)] = [s1 (2) − s1 (2)] + [s2 (2) − s2 (1)] = s(2, 2) − s(2, 1) for each θ > 1,
and thus a supplier of type θ ∈ (1, 2) never chooses Q∗ (θ) = (2, 2) in any equilibrium.
17

This contradicts the hypothesis that s implements the optimal mechanism.
What is the cause of the failure? One can find the answer in the functional form of
the production cost. For each type except θ = 3, the incremental cost for each attributeis strictly increasing in the level of the other attribute. That is, the two quality attributes
are substitutable in terms of the production cost. Now, the set of types is partitioned
into two groups: From the buyer’s point of view, each type in the efficient group [1, 2)
should offer the greatest quality (2, 2), and each type in the inefficient group [2, 3] should
offer the least quality (1, 1). Then, given any additively separable rule which deters theinefficient suppliers’ deviations, an incentive for a supplier of type θ = 2 to deviate fromthe quality offer (1, 1) to (1, 2) is weaker than that for a supplier of type θ ∈ (1, 2) todeviate from the quality offer (2, 2) to (2, 1) because of the substitutability. This meansthat any additively separable rule causes deviations of some types in either group. As
one would expect, if the production cost has decreasing differences in (q 1 , q 2 ), then wecan construct an additively separable scoring rule which is immune to the deviation.
We can obtain the more general results although the underlying structure is the sameas Example 4. Here, the notion of increasing (decreasing) differences expresses the concept
of complementarity (substitutability) between some quality attributes in terms of the
production cost. The following proposition gives a necessary condition and a sufficientcondition for the existence of an additively separable scoring rule which implements the
optimal mechanism.
Proposition 2. Suppose that the virtual surplus Φ is quasisupermodular. Then: (i) If
c has increasing differences in (q 1 , q 2 ) and there exists (m, m0 ) ∈ M1 × M2 such that00c has strictly increasing differences in (q m , q m ), Qm∗ (θ−) > Qm∗ (θ+) and Qm ∗ (θ−) >0Qm ∗ (θ+) for some θ ∈ (θ, θ̄∗ ), then there is no additively separable scoring rule which18implements the optimal mechanism. (ii) If c has decreasing differences in (q 1 , q 2 ) and Qm
is finite for each m, then there is an additively separable scoring rule which implements
the optimal mechanism.
This proposition has several implications. First, the implementation possibility via
an additively separable scoring rule heavily depends on whether the cost function has
increasing differences or decreasing differences in quality attributes in the distinct setsof subcriteria, rather than the property of the value function. In particular, Example 4shows that an additively separable scoring rule cannot implement the optimal mechanism
even if the value function is additively separable in quality.
Second, the results of Proposition 2 provide a useful guide to designing additively
separable scoring rules. Consider an example of the highway construction. Supposethat the quality attributes represent delivery date (m = 1), durability of the highway(m = 2), maintenance service after delivery (m = 3), respectively. Moreover, we consider
the following plausible scenario: as the delivery date is earlier, it is more costly to increase
the durability level; as the durability level is higher, it is less costly to increase the levelof maintenance service; the incremental cost for the maintenance service is independentof the delivery date. That is, the production cost has strictly increasing differences in(q 1 , q 2 ), decreasing differences in (q 2 , q 3 ), and is additively separable in (q 1 , q 3 ). Thus,
the cost function has decreasing differences in (q 1 , q 2 ) with M1 = {1, 2} and M2 = {3}whereas the cost function has increasing differences in (q 1 , q 2 ) with M1 = {1} andM2 = {2, 3}. Proposition 2 then implies that the buyer should classify quality attributes
so that s(q) = s1 (q 1 , q 2 )+s2 (q 3 ) not s(q) = s1 (q 1 )+s2 (q 2 , q 3 ). When the buyer establishes
some sets of subcriteria to use the additively separable rule, each pair of quality attributes
in the distinct sets should be complementary in terms of the production cost.

19

6Concluding remarks

We have studied the optimal design of scoring auction in an environment where quality ismulti-dimensional. Our main result shows that if the virtual surplus is quasisupermodular
in quality, then there exists a quasi-linear scoring rule which implements the optimalmechanism and is supermodular in quality. Thus, when the virtual surplus exhibits akind of complementarity between quality attributes, a scoring rule which gives a quality
score in a complementary way works well. This in turn implies that the buyer shouldcarefully design scoring rules which are additively separable in some quality attributes.
One may wonder why each supplier should offer all quality attributes in a scoring
auction. Alternatively, the buyer can require each supplier to offer only one quality
attribute (with price). Then, with a scoring rule a la Che (1993), the most efficient
supplier achieves the highest score, and the winner’s type is revealed to the buyer. If
the levels of the remaining quality attributes are properly specified based on the winner’stype, then this mechanism can implement the optimal mechanism. The mechanism orprocedure, however, requires the buyer to prespecify the levels of the remaining quality
attributes for each cost parameter. In practice, it may be prohibitively costly for the buyer
to do so. For instance, the buyer may not be able to specify aesthetic and functional
characteristics of highway although suppliers with expertise can offer these characteristics.
Therefore, when there are at least two such quality attributes, this study is of significance.
Finally, we should point out a potential limitation of this analysis. In our model, amore efficient supplier (i.e. a supplier with less cost parameter) has a superior technology
of increasing the levels of all quality attributes. Thus, our model does not cover thecase in which one type of supplier has a superior technology of increasing the level of
one quality attribute to another type of supplier whereas the latter type has a superior
20technology of increasing the level of another quality attribute to the former type. To cover
the case, we need to allow each supplier’s type to be multi-dimensional. It is interesting
and challenging to study the optimal design of scoring auction in an environment where
both quality and a supplier’s type are multi-dimensional. This is left for future research.

Appendix
Proof of Lemma 1. (i) First, note that arg maxq∈Q [s(q) − c(q, θ)] is nonempty for eachθ because Q is compact and s − c is upper semicontinuous in q. (See, for example,
Kolmogorov and Fomin (1975).) We now show that in equilibrium a supplier of typeθ ∈ [θ, θ̄s ] never offers (p0 , q 0 ) such that q 0 6∈ arg maxq̂∈Q [s(q̂) − c(q̂, θ)]. Suppose, on thecontrary, that a supplier of type θ ∈ [θ, θ̄s ] makes such an offer (p0 , q 0 ). Consider anotheroffer (p, q) such that q ∈ arg maxq̂∈Q [s(q̂) − c(q̂, θ)] and s(q) − p = s(q 0 ) − p0 . The scoreof (p, q) is equal to that of (p0 , q 0 ), so that both offers yield the same winning probability
given the other suppliers’ strategies. The supplier’s expected profit from (p0 , q 0 ) is nothigher than his expected profit from (p, q) because[p0 − c(q 0 , θ)]P rob[win | S(p0 , q 0 )]≤ [p0 − c(q 0 , θ) + (s(q) − c(q, θ) − (s(q 0 ) − c(q 0 , θ)))]P rob[win | S(p0 , q 0 )]= [p − c(q, θ)]P rob[win | S(p, q)],where the inequality follows from the hypothesis that q 0 6∈ arg maxq̂∈Q [s(q̂) − c(q̂, θ)] 3
q, and the equality follows from the construction of (p, q). The inequality is strict if
P rob[win | S(p, q)] > 0, which occurs for some strategies of the other suppliers. This
contradicts the assumption that no supplier uses weakly dominated strategies. Thus, thelatter statement holds, and we can assume that a symmetric equilibrium bidding strategy21

(p, q ∗ ) satisfies q ∗ (θ) ∈ arg maxq∈Q [s(q) − c(q, θ)] for each θ ∈ [θ, θ̄s ].
Second, consider the following change of variable: k(θ) ≡ s(q ∗ (θ)) − c(q ∗ (θ), θ) for
each θ ∈ [θ, θ̄s ]. Because cθ is continuous and thus bounded on [θ, θ̄], it follows from theintegral form envelope theorem of Milgrom and Segal (2002) (see also Theorem 3.1 ofMilgrom (2004)) that k is absolutely continuous, and is given byZ θk(θ) = k(θ) −cθ (q ∗ (z), z)dz.

θFinally, we show that the bidding strategy (p∗ , q ∗ ) in the lemma constitutes a symmetric equilibrium. For each θ ∈ [θ, θ̄s ], the score s(q ∗ (θ)) − p∗ (θ) is given bys(q ∗ (θ)) − p∗ (θ) = s(q ∗ (θ)) − c(q ∗ (θ), θ) − [p∗ (θ) − c(q ∗ (θ), θ)]Z θ̄s1 − F(N −1) (z)= k(θ) −cθ (q ∗ (z), z)
dz.
1 − F(N −1) (θ)θ
Note that s(q ∗ (θ̄s )) − p∗ (θ̄s ) = s(q ∗ (θ̄s )) − c(q ∗ (θ̄s ), θ̄s ) = 0 by assumption. Because k iscontinuous in θ ∈ [θ, θ̄s ], so is s(q ∗ (θ)) − p∗ (θ). Moreover, the score s(q ∗ (θ)) − p∗ (θ) is
strictly decreasing in θ ∈ [θ, θ̄s ] because for each θ, θ0 ∈ [θ, θ̄s ] with θ < θ0 ,[s(q ∗ (θ)) − p∗ (θ)] − [s(q ∗ (θ0 )) − p∗ (θ0 )]
"Z s
#
Z θ̄s
θ̄
1
−
F
(z)
1
−
F
(z)
(N −1)(N −1)= (k(θ) − k(θ0 )) −cθ (q ∗ (z), z)dz −cθ (q ∗ (z), z)
dz
1 − F(N −1) (θ)
1 − F(N −1) (θ0 )
θ0
θ
Z θ0
1 − F(N −1) (z)0> (k(θ) − k(θ )) −cθ (q ∗ (z), z)
dz
1 − F(N −1) (θ)θ
Z θ00> (k(θ) − k(θ )) −cθ (q ∗ (z), z)dz
θ

Z θ0
=

[cθ (q ∗ (z), z) − cθ (q ∗ (z), z)] dz = 0.

θWhen the other suppliers follow the strategy (p∗ , q ∗ ), the expected profit of a supplier oftype θ ∈ [θ, θ̄s ] from offering (p, q ∗ (θ)) such that s(q ∗ (θ)) − p = s(q ∗ (θ̂)) − p∗ (θ̂) for some22θ̂ ∈ [θ, θ̄s ] can be written as[p − c(q ∗ (θ), θ)]P rob[win | S(p, q ∗ (θ))]= [s(q ∗ (θ)) − c(q ∗ (θ), θ) − s(q ∗ (θ̂)) + p∗ (θ̂)]P rob[win | S(p∗ (θ̂), q ∗ (θ̂))]= [k(θ) − s(q ∗ (θ̂)) + p∗ (θ̂)](1 − F(N −1) (θ̂))
Z θ̄s= (k(θ) − k(θ̂))(1 − F(N −1) (θ̂)) +cθ (q ∗ (z), z)(1 − F(N −1) (z))dz,
θ̂

where the second equality follows from the observation that the score S(p∗ (θ̂), q ∗ (θ̂)) =s(q ∗ (θ̂)) − p∗ (θ̂) is strictly decreasing in θ̂. The supplier cannot obtain a higher expected
profit by deviating from (p∗ (θ), q ∗ (θ)) to (p, q ∗ (θ)) such that s(q ∗ (θ)) − p = s(q ∗ (θ̂)) −p∗ (θ̂) because the difference between the expected profits is given byZ θ̂
− (k(θ) − k(θ̂))(1 − F(N −1) (θ̂)) +cθ (q ∗ (z), z)(1 − F(N −1) (z))dzθ=Z θ̂ hi−cθ (q (z), z)(1 − F(N −1) (θ̂)) + cθ (q (z), z)(1 − F(N −1) (z)) dz
∗

∗

θ

Z θ̂
=

cθ (q ∗ (z), z)(F(N −1) (θ̂) − F(N −1) (z))dz ≥ 0.

θIt is easy to show that the supplier cannot obtain a higher expected profit by deviatingfrom (p∗ (θ), q ∗ (θ)) to (p, q ∗ (θ)) such that s(q ∗ (θ)) − p 6∈ [0, s(q ∗ (θ)) − p∗ (θ)]. Also, a
supplier of type θ ∈ (θ̄s , θ̄] obtains a negative expected profit if he offers (p, q) suchthat s(q) − p ≥ 0 whereas he obtains zero profit by offering (p∗ (θ), q ∗ (θ)) such thats(q ∗ (θ)) − p∗ (θ) < 0. This completes the proof of part (i).
(ii) In the above equilibrium, the score s(q ∗ (θi )) − p∗ (θi ) is strictly decreasing inθi ∈ [θ, θ̄s ] with s(q ∗ (θ̄s )) − p∗ (θ̄s ) = 0, and is negative for each θi ∈ (θ̄s , θ̄]. Thus, a
supplier of type θi wins only if θi = min{θ1 , ..., θN , θ̄s }.
Proof of Lemma 2. (i) We show that a necessary and sufficient condition for the IC constraints (3) is given by the two conditions: envelope condition and monotonicity condition.
23

We say that a direct mechanism ρ satisfies the envelope condition if for each i and θ,Πρi (θ) = Πρi (θ̄) +Z θ̄Eθ−i [Xi (z, θ−i )cθ (Qi (z, θ−i ), z)]dz,
θ

where Πρi (θ) ≡ Πρi (θ | θ). We say that a direct mechanism ρ satisfies the monotonicity
condition if for each i, θ and θ̂ with θ̂ > θ,Z θ̂Eθ−i [Xi (z, θ−i )cθ (Qi (z, θ−i ), z) − Xi (θ̂, θ−i )cθ (Qi (θ̂, θ−i ), z)]dz ≥ 0.
θ

Because c has (strictly) increasing differences in (q, θ), it must hold that for each q, q 0 ∈ Qwith q ≤ q 0 and each z, z 0 ∈ [θ, θ̄] with z ≤ z 0 , c(q 0 , z) − c(q, z) ≤ c(q 0 , z 0 ) − c(q, z 0 ), andthus cθ (q 0 , z) ≥ cθ (q, z) > 0. Therefore, if both Xi (θi , θ−i ) and Qi (θi , θ−i ) are decreasingin θi for each θ−i , then the monotonicity condition is automatically satisfied.
First, we prove sufficiency. Suppose that a supplier i’s IC constraint is not satisfied.
Then, there exist θ and θ̂ such that Πρi (θ̂ | θ) > Πρi (θ). Hence, Eθ−i [Xi (θ̂, θ−i )c(Qi (θ̂, θ−i ), θ̂)−
Xi (θ̂, θ−i )c(Qi (θ̂, θ−i ), θ)] > Πρi (θ) − Πρi (θ̂) by definition of Πρi . Rewriting the left-handside as the definite integral and applying the envelope condition to the right-hand side,
we obtain
Z θ̂

Z θ̂
Eθ−i [Xi (θ̂, θ−i )cθ (Qi (θ̂, θ−i ), z)]dz >θEθ−i [Xi (z, θ−i )cθ (Qi (z, θ−i ), z)]dz.
θThis contradicts the monotonicity condition.
Next, we prove necessity. Using the integral form envelope theorem of Milgrom and Segal (2002), the IC constraints (3) imply that for each i and θ, Πρi (θ) = maxθ̂ Eθ−i [Pi (θ̂, θ−i )−Xi (θ̂, θ−i )c(Qi (θ̂, θ−i ), θ)] is given byΠρi (θ) = Πρi (θ̄) −

Z θ̄
θ

∂Πρi
(z | z)dz = Πρi (θ̄) +∂θZ θ̄Eθ−i [Xi (z, θ−i )cθ (Qi (z, θ−i ), z)]dz.
θ

We thus obtain the envelope condition. Also, the IC constraints (3) imply that for eachi, θ and θ̂, Eθ−i [Xi (θ̂, θ−i )c(Qi (θ̂, θ−i ), θ̂) − Xi (θ̂, θ−i )c(Qi (θ̂, θ−i ), θ)] ≤ Πρi (θ) − Πρi (θ̂).
24

Rewriting the left-hand side as the definite integral and applying the envelope condition
to the right-hand side, we obtain the monotonicity condition.
(ii) We solve the optimization problem. The IC constraints (3) imply that for eachi, Πρi (θ) is decreasing in θ because Πρi (θ) ≥ Πρi (θ0 | θ) ≥ Πρi (θ0 ) for each θ < θ0 ; thesecond inequality follows from the assumption that c is (strictly) increasing in θi . Hence,
the IR constraints (4) are replaced by Πρi (θ̄) ≥ 0 for each i. Using the result (i), theIC constraints (3) are replaced by the envelope and monotonicity conditions. By theenvelope condition and the interchange of the order of integration, E[Πρi (θi )] is given byZ θ̄Πρi (θi )f (θi )dθi = Πρi (θ̄) +Z θ̄ Z θ̄Eθ−i [Xi (z, θ−i )cθ (Qi (z, θ−i ), z)]dzf (θi )dθiθθ= Πρi (θ̄) +

θi

Z θ̄
Eθ−i [Xi (z, θ−i )cθ (Qi (z, θ−i ), z)]θF (z)f (z)dz.
f (z)Hence, the buyer’s objective function is rewritten asNX

E [Xi (θ)v(Qi (θ)) − Pi (θ) + αΠρi (θi )]

i=1
N
X

=

i=1
N
X

E [Xi (θ)[v(Qi (θ)) − c(Qi (θ), θi )] − (1 − α)Πρi (θi )]F (θi )
ρ=E Xi (θ) v(Qi (θ)) − c(Qi (θ), θi ) − (1 − α)cθ (Qi (θ), θi )− (1 − α)Πi (θ̄)
f (θi )
i=1
=

N
X





E Xi (θ)Φ(Qi (θ), θi ) − (1 − α)Πρi (θ̄) .

i=1

Note that arg maxq∈Q Φ(q, θi ) is nonempty for each θi because Q is compact and Φ iscontinuous in q. The above objective function is maximized when Πρi (θ̄) = 0, and Qi (θ)
and Xi (θ) are respectively given by Q∗ (θi ) and Xi∗ (θ) in the lemma. This is becauseQ∗ (θi ) maximizes Φ(q, θi ) and the maximized value Φ(Q∗ (θi ), θi ) is strictly decreasingin θi . The latter fact follows from Φ(Q∗ (θ), θ) ≥ Φ(Q∗ (θ0 ), θ) > Φ(Q∗ (θ0 ), θ0 ) for eachθ < θ0 ; the second inequality follows from the assumptions that c is strictly increasing in
θi , and both cθ and F/f are increasing in θi .
25

Finally, we show that the direct mechanism ρ∗ = (Pi∗ , Q∗i , Xi∗ )i∈{1,...,N } satisfies theignored monotonicity condition. Now, Φ is quasisupermodular in q by hypothesis, andhas strictly increasing differences in (q, −θi ) from Assumptions 4 and 5. It then follows
from Theorem 2.8.7 of Topkis (1998) that Q∗ (θi ) ≥ Q∗ (θi0 ) for each θi < θi0 . Also, Xi∗ isdecreasing in θi . These facts imply that ρ∗ satisfies the monotonicity condition.
Proof of Proposition 1. (i) We first show that there exists a quasi-linear scoring rule which
implements the optimal mechanism. From Lemmas 1 and 2, it suffices to show that
there exists an upper semicontinuous function s∗ : Q → R which satisfies Q∗ (θ) ∈arg maxq∈Q [s∗ (q) − c(q, θ)] for each θ ∈ [θ, θ̄∗ ] and s∗ (Q∗ (θ̄∗ )) − c(Q∗ (θ̄∗ ), θ̄∗ ) = 0.
Lemma 2 implies that Qm∗ is decreasing in θ. Hence, Qm∗ : [θ, θ̄] → R can have no
more than countably many points of discontinuity. Let {θ1 , θ2 , ..., θl , ...} be the discontinuous points of Q∗ in [θ, θ̄∗ ]. For each l with θl ∈ (θ, θ̄∗ ), there exist some m such thatQm∗ (θl +) ≡ limθ→θl + Qm∗ (θ) ≤ Qm∗ (θl ) ≤ limθ→θl − Qm∗ (θ) ≡ Qm∗ (θl −) with either orboth inequalities being strict.
First, we define a function σ : [θ, θ̄∗ ] → R: For each θ ∈ [θ, θ̄∗ ],
∗

Z θ̄∗

σ(θ) = c(Q (θ), θ) +cθ (Q∗ (z), z)dz.

θNote that if Q∗ (θ) = Q∗ (θ0 ), and thus Q∗ (θ) = Q∗ (θ00 ) for each θ00 ∈ [θ, θ0 ], then σ(θ) =σ(θ0 ). Using the function σ, we construct s∗ in the following way. (a) If q 6≥ Q∗ (θ̄∗ ),then let s∗ (q) = 0. (b) If there exists θ0 ∈ (θ, θ̄∗ ) \ {θ1 , θ2 , ..., θl , ...} or θ0 ∈ {θ, θ̄∗ } suchthat q ≥ Q∗ (θ0 ) and there exists no θ < θ0 which satisfies q ≥ Q∗ (θ) > Q∗ (θ0 ), then lets∗ (q) = σ(θ0 ).5 (c) If there exists l such that θl ∈ (θ, θ̄∗ ), q ≥ Q∗ (θl +) and q 6≥ Q∗ (θl ),then let s∗ (q) = σ(θl +). (d) If there exists l such that θl ∈ (θ, θ̄∗ ), q ≥ Q∗ (θl ) andFor some q, there may exist another θ(6= θ0 ) which satisfies the condition. However, because it thenfollows that Q∗ (θ) = Q∗ (θ0 ), the equality s∗ (q) = σ(θ) = σ(θ0 ) holds.
526q 6≥ Q∗ (θl −), then let s∗ (q) = σ(θl ). (e) If there exists l such that θl ∈ (θ, θ̄∗ ) and there
exists no θ < θl which satisfies q ≥ Q∗ (θ) > Q∗ (θl −), then let s∗ (q) = σ(θl −).
Second, we show that s∗ (Q∗ (θ̄∗ ))−c(Q∗ (θ̄∗ ), θ̄∗ ) = 0 and s∗ is upper semicontinuous on
Q. The former is trivial because s∗ (Q∗ (θ̄∗ )) = σ(θ̄∗ ) = c(Q∗ (θ̄∗ ), θ̄∗ ). By construction,
the function σ is continuous in [θ, θ̄∗ ] \ {θ1 , θ2 , ..., θl , ...}. Also, σ(θ) = σ(θ0 ) for eachθ, θ0 ∈ [θ, θ̄∗ ] with θ < θ0 if Q∗ (θ) = Q∗ (θ0 ) and thus Q∗ (θ) = Q∗ (θ00 ) = Q∗ (θ0 ) for eachθ00 ∈ [θ, θ0 ]. Hence, for each q ∈ Q and  > 0, there exists a neighborhood of q in whichs∗ (q 0 ) ≤ s∗ (q) + . This means that the function s∗ is upper semicontinuous on Q.
Finally, we show that Q∗ (θ) ∈ arg maxq∈Q [s∗ (q) − c(q, θ)]. By construction of s∗ ,q 6∈ {Q∗ (θ) | θ ∈ [θ, θ̄∗ ]} ∪ {Q∗ (θl +) | l = 1, 2, ...} ∪ {Q∗ (θl −) | l = 1, 2, ...} cannot
maximize s∗ (q) − c(q, θ) for each θ ∈ [θ, θ̄∗ ] because s∗ (Q∗ (θ)) − c(Q∗ (θ), θ) ≥ 0 whereass∗ (q) − c(q, θ) < 0 for each q 6≥ Q∗ (θ̄∗ ), and c is strictly increasing in q. Hence, it suffices
to show that for each θ, θ̂ ∈ [θ, θ̄∗ ], σ(θ) − c(Q∗ (θ), θ) ≥ σ(θ̂) − c(Q∗ (θ̂), θ), and for each l,σ(θ) − c(Q∗ (θ), θ) ≥ σ(θl +) − c(Q∗ (θl +), θ), σ(θ) − c(Q∗ (θ), θ) ≥ σ(θl −) − c(Q∗ (θl −), θ).
First, for each θ ∈ [θ, θ̄∗ ], θ̂ = θ maximizes σ(θ̂) − c(Q∗ (θ̂), θ) because for each θ̂ 6= θ,[σ(θ) − c(Q∗ (θ), θ)] − [σ(θ̂) − c(Q∗ (θ̂), θ)]
"
#
Z θ̄∗
Z θ̄∗
=
cθ (Q∗ (z), z)dz − c(Q∗ (θ̂), θ̂) +cθ (Q∗ (z), z)dz − c(Q∗ (θ̂), θ)
θ

Z θ̂
=

θ̂

[cθ (Q∗ (z), z) − cθ (Q∗ (θ̂), z)]dz ≥ 0.

θThe inequality holds because Lemma 2 implies that Q∗ (z) ≥ Q∗ (θ̂) for each z < θ̂,and the assumption that c has (strictly) increasing differences in (q, θ) implies thatcθ (Q∗ (z), z) ≥ cθ (Q∗ (θ̂), z). Second, for each θ ∈ [θ, θ̄∗ ] and l, σ(θ) − c(Q∗ (θ), θ) ≥27

σ(θl +) − c(Q∗ (θl +), θ) because[σ(θ) − c(Q∗ (θ), θ)] − [σ(θl +) − c(Q∗ (θl +), θ)]
"
#
Z θ̄∗
Z θ̄∗
∗
∗
∗
∗ l
=
cθ (Q (z), z)dz − lim c(Q (θ̂), θ̂) + limcθ (Q (z), z)dz − c(Q (θ +), θ)θ̂→θl +

θ

θ̂→θl +

"

Z θ̄∗

∗

∗

l

cθ (Q (z), z)dz − c(Q (θ +), θ ) +

=
=

∗

∗

l

cθ (Q (z), z)dz − c(Q (θ +), θ)
θl

θ

Z θl

#

Z θ̄∗

l

θ̂

[cθ (Q∗ (z), z) − cθ (Q∗ (θl +), z)]dz ≥ 0.

θThe first equality follows from the construction of σ, the second equality follows from thecontinuity of c in (q, θ), and the inequality holds because Lemma 2 and Assumption 3
imply that cθ (Q∗ (z), z) ≥ cθ (Q∗ (θl +), z) for each z ≤ θl . Similarly, we can show that
σ(θ) − c(Q∗ (θ), θ) ≥ σ(θl −) − c(Q∗ (θl −), θ).
(ii) We now show that the function s∗ is supermodular in quality. First, we show thats∗ is increasing in q. Note that σ is decreasing in θ because for each θ, θ0 with θ < θ0 ,
0

Z θ̄∗

∗

σ(θ) − σ(θ ) = c(Q (θ), θ) +

∗

∗

0

0

cθ (Q (z), z)dz − c(Q (θ ), θ ) −0cθ (Q∗ (z), z)dz

θ0

θ
∗

Z θ̄∗

∗

0

0

Z θ0

≥ c(Q (θ ), θ) − c(Q (θ ), θ ) +cθ (Q∗ (z), z)dz

θ

Z θ0
=

[cθ (Q∗ (z), z) − cθ (Q∗ (θ0 ), z)]dz ≥ 0.

θConsider any q, q 0 ∈ Q with q 0 > q. If q 6≥ Q∗ (θ̄∗ ), then s∗ (q 0 ) ≥ s∗ (q) = 0. If
q ≥ Q∗ (θ̄∗ ) and q 0 6≥ Q∗ (θ), then s∗ (q 0 ) ≥ s∗ (q) because there exist θ and θ0 with θ ≥ θ0such that s∗ (q) is equal to σ(θ−), σ(θ) or σ(θ+) and s∗ (q 0 ) is equal to σ(θ0 −), σ(θ0 ) orσ(θ0 +); note that s∗ (q) = σ(θ0 −) implies that s∗ (q 0 ) ≥ σ(θ0 −), and s∗ (q) = σ(θ0 ) implies
that s∗ (q 0 ) ≥ σ(θ0 ). If q 0 ≥ Q∗ (θ), then s∗ (q 0 ) = σ(θ) ≥ s∗ (q).
Second, we show that the function s∗ is supermodular in q. Fix any unorderedpair q and q̂ in Q, so that q m > q̂ m and q m̂ < q̂ m̂ for some m, m̂. Because s∗ isincreasing in q, s∗ (q ∨ q̂) ≥ s∗ (q̂) and s∗ (q) ≥ s∗ (q ∧ q̂). Thus, if s∗ (q) = s∗ (q ∧ q̂),28

then s∗ (q ∨ q̂) − s∗ (q̂) ≥ 0 = s∗ (q) − s∗ (q ∧ q̂). We claim that if s∗ (q) > s∗ (q ∧ q̂), thens∗ (q̂) = s∗ (q ∧ q̂). Suppose, on the contrary, that s∗ (q) > s∗ (q ∧ q̂) and s∗ (q̂) > s∗ (q ∧ q̂).
Then, by construction of s∗ , there exist m, m̂, θ, θ̂ such that q m ≥ Qm∗ (θ) > min{q m , q̂ m }with q m̂ ≥ Qm̂∗ (θ), and q̂ m̂ ≥ Qm̂∗ (θ̂) > min{q m̂ , q̂ m̂ } with q̂ m ≥ Qm∗ (θ̂). This contradicts
the result that Q∗ is increasing in θ because Qm∗ (θ) > Qm∗ (θ̂) and Qm̂∗ (θ) < Qm̂∗ (θ̂), sothat Q∗ (θ) and Q∗ (θ̂) are unordered. Thus, if s∗ (q) > s∗ (q ∧ q̂), then s∗ (q ∨ q̂) − s∗ (q̂) =s∗ (q ∨ q̂) − s∗ (q ∧ q̂) ≥ s∗ (q) − s∗ (q ∧ q̂). This completes the proof.
00

Proof of Proposition 2. (i) Suppose that Qm∗ (θ−) > Qm∗ (θ+) and Qm ∗ (θ−) > Qm ∗ (θ+)
with (m, m0 ) ∈ M1 × M2 and θ ∈ (θ, θ̄∗ ). Then, (Qm∗ (θ−))m∈M1 > (Qm∗ (θ+))m∈M1 and(Qm∗ (θ−))m∈M2 > (Qm∗ (θ+))m∈M2 . Because c has increasing differences in (q 1 , q 2 ) and0

strictly increasing differences in (q m , q m ), we obtain c(Q∨Q̂, θ)−c(Q̂, θ) > c(Q, θ)−c(Q∧Q̂, θ), where Q = ((Qm∗ (θ+))m∈M1 , (Qm∗ (θ−))m∈M2 ) and Q̂ = ((Qm∗ (θ−))m∈M1 , (Qm∗ (θ+))m∈M2 ),so that Q ∨ Q̂ = Q∗ (θ−) and Q ∧ Q̂ = Q∗ (θ+). Let  be a positive real number
such that 2 < [c(Q ∨ Q̂, θ) − c(Q̂, θ)] − [c(Q, θ) − c(Q ∧ Q̂, θ)]. Fix any θ0 , θ00 with
θ00 < θ < θ0 such that [c(Q ∧ Q̂, θ) − c(q ∧ q̂, θ0 )] − [c(Q, θ) − c(q, θ0 )] <  and [c(Q ∨Q̂, θ) − c(q ∨ q̂, θ00 )] − [c(Q̂, θ) − c(q̂, θ00 )] < , where q = ((Qm∗ (θ0 ))m∈M1 , (Qm∗ (θ00 ))m∈M2 )and q̂ = ((Qm∗ (θ00 ))m∈M1 , (Qm∗ (θ0 ))m∈M2 ), so that q ∨ q̂ = Q∗ (θ00 ) and q ∧ q̂ = Q∗ (θ0 ).
The existence of such θ0 and θ00 is guaranteed by the continuity of c in (q, θ).
Suppose that a scoring rule s implements the optimal mechanism. Then, it follows
from Lemma 1 that the inequality s(q) − c(q, θ0 ) ≤ s(q ∧ q̂) − c(q ∧ q̂, θ0 ) must hold29

because q ∧ q̂ = Q∗ (θ0 ) whereas q 6= Q∗ (θ0 ). Hence, we obtains(q) − s(q ∧ q̂) ≤ c(q, θ0 ) − c(q ∧ q̂, θ0 )< c(Q, θ) − c(Q ∧ Q̂, θ) + < c(Q ∨ Q̂, θ) − c(Q̂, θ) − < c(q ∨ q̂, θ00 ) − c(q̂, θ00 ).
Now, if the scoring rule s is additively separable, then s(q) − s(q ∧ q̂) = s2 (q 2 ) − s2 (q̂ 2 ) =s(q∨ q̂)−s(q̂) because q 1 = q 1 ∧ q̂ 1 = (Qm∗ (θ0 ))m∈M1 and q̂ 1 = q 1 ∨ q̂ 1 = (Qm∗ (θ00 ))m∈M1 .
Then, s(q ∨ q̂)−s(q̂) < c(q ∨ q̂, θ00 )−c(q̂, θ00 ), and thus a supplier of type θ00 never choosesq ∨ q̂ = Q∗ (θ00 ) from Lemma 1. Therefore, there is no additively separable scoring rule
which implements the optimal mechanism.
(ii) It follows from Lemma 1 that Qm∗ is decreasing in θ. Because Qm is finite,Qm∗ : [θ, θ̄] → R can have no more than finitely many points of discontinuity. Suppose,
without loss of generality, that the discontinuous points of Q∗ in [θ, θ̄∗ ] are indexed indecreasing order, i.e. θ1 > θ2 > ... > θL . Note that Q∗ (θl+1 +) = Q∗ (θ) = Q∗ (θl −) for
each θ ∈ (θl+1 , θl ), and Qm∗ (θl −) ≥ Qm∗ (θl +). Assume for simplicity that θ̄ > θ1 andθL > θ; without this assumption, the proof proceeds with some notational complexity.
First, we define functions σ 1 : [θ, θ̄∗ ] → R and σ 2 : [θ, θ̄∗ ] → R: For each θ ∈ [θ, θ̄∗ ]such that θ ∈ (θl+1 , θl ] for some l or θ ∈ [θ, θl ] for l = L,
l−1
X
1
∗ ∗
∗
σ (θ) = c(Q (θ̄ ), θ̄ ) +[c((Qm∗ (θr −))m∈M1 , (Qm∗ (θr +))m∈M2 , θr ) − c(Q∗ (θr +), θr )]2
r=1
1+ [c((Qm∗ (θ))m∈M1 , (Qm∗ (θl +))m∈M2 , θl ) − c((Qm∗ (θl +))m∈M1 , (Qm∗ (θl +))m∈M2 , θl )],
l−1
X
1
∗ ∗
∗
σ (θ) = c(Q (θ̄ ), θ̄ ) +[c(Q∗ (θr −), θr ) − c((Qm∗ (θr −))m∈M1 , (Qm∗ (θr +))m∈M2 , θr )]2
r=12+ [c((Qm∗ (θl −))m∈M1 , (Qm∗ (θ))m∈M2 , θl ) − c((Qm∗ (θl −))m∈M1 , (Qm∗ (θl +))m∈M2 , θl )],30

and for each θ ∈ (θ1 , θ̄∗ ], σ 1 (θ) = σ 2 (θ) = 12 c(Q∗ (θ̄∗ ), θ̄∗ ). Using these functions, weconstruct a function sa : ×m∈Ma Qm → R for each a = 1, 2 in the following way. (a)If q a 6≥ (Qm∗ (θ̄∗ ))m∈Ma , then let sa (q a ) = − maxq∈Q c(q, θ̄). (b) If there exists θ0 ∈[θ, θ̄∗ ] such that q a ≥ (Qm∗ (θ0 ))m∈Ma and there exists no θ < θ0 which satisfies q a ≥(Qm∗ (θ))m∈Ma > (Qm∗ (θ0 ))m∈Ma , then let sa (q a ) = σ a (θ0 ). It is easy to show thats1 ((Qm∗ (θ̄∗ ))m∈M1 ) + s2 ((Qm∗ (θ̄∗ ))m∈M2 ) − c(Q∗ (θ̄∗ ), θ̄∗ ) = 0. Moreover, the functions + s̄ is trivially upper semicontinuous on Q because Q is finite.
Next, we show that Q∗ (θ) ∈ arg maxq∈Q [s1 (q 1 ) + s2 (q 2 ) − c(q, θ)]. By constructionof s1 and s2 , q 6∈ {(Qm∗ (θ))m∈M1 | θ ∈ [θ, θ̄∗ ]} × {(Qm∗ (θ))m∈M2 | θ ∈ [θ, θ̄∗ ]} cannot be
the maximizer. Hence, it suffices to show that for each θ, θ0 , θ00 ∈ [θ, θ̄∗ ], σ 1 (θ) + σ 2 (θ) −c(Q∗ (θ), θ) ≥ σ 1 (θ0 ) + σ 2 (θ00 ) − c((Qm∗ (θ0 ))m∈M1 , (Qm∗ (θ00 ))m∈M2 , θ). Suppose first that00

θ ∈ (θl+1 , θl ) and θ0 ∈ (θl +1 , θl ) with l ≥ l0 . Then, we obtain
σ 1 (θ) − σ 1 (θ0 )=lX[c((Qm∗ (θr −))m∈M1 , (Qm∗ (θr +))m∈M2 , θr ) − c(Q∗ (θr +), θr )]r=l0 +1≥lX[c((Qm∗ (θr −))m∈M1 , (Qm∗ (θ))m∈M2 , θr ) − c((Qm∗ (θr +))m∈M1 , (Qm∗ (θ))m∈M2 , θr )]r=l0 +1≥lX[c((Qm∗ (θr −))m∈M1 , (Qm∗ (θ))m∈M2 , θ) − c((Qm∗ (θr +))m∈M1 , (Qm∗ (θ))m∈M2 , θ)]r=l0 +1= c(Q∗ (θ), θ) − c((Qm∗ (θ0 ))m∈M1 , (Qm∗ (θ))m∈M2 , θ),
where the first equality follows from the observation that Qm∗ (θl −) = Qm∗ (θ) and0Qm∗ (θl −) = Qm∗ (θ0 ), the first inequality follows from the hypothesis that c has decreasing differences in (q 1 , q 2 ) with the observation that Qm∗ (θr +) ≤ Qm∗ (θ) for each r ∈ {l0 +1, ..., l}, the second inequality follows from the assumption that c has (strictly) increasing
differences in (q, θ), the last equality follows from the observation that Qm∗ (θr −) =310Qm∗ (θr+1 +), Qm∗ (θl −) = Qm∗ (θ) and Qm∗ (θl +1 +) = Qm∗ (θ0 ).
00

Suppose next that00

θ00 ∈ (θl +1 , θl ) with l ≤ l00 . Then, the same argument as above yields
σ 2 (θ) − σ 2 (θ00 )
00

=−

l
X

[c(Q∗ (θr −), θr ) − c((Qm∗ (θr −))m∈M1 , (Qm∗ (θr +))m∈M2 , θr )]

r=l+1
00

≥−

l
X

[c((Qm∗ (θ))m∈M1 , (Qm∗ (θr −))m∈M2 , θr ) − c((Qm∗ (θ))m∈M1 , (Qm∗ (θr +))m∈M2 , θr )]

r=l+1
00

≥−

l
X

[c((Qm∗ (θ))m∈M1 , (Qm∗ (θr −))m∈M2 , θ) − c((Qm∗ (θ))m∈M1 , (Qm∗ (θr +))m∈M2 , θ)]

r=l+1= −[c((Qm∗ (θ))m∈M1 , (Qm∗ (θ00 ))m∈M2 , θ) − c(Q∗ (θ), θ)]≥ −[c((Qm∗ (θ0 ))m∈M1 , (Qm∗ (θ00 ))m∈M2 , θ) − c((Qm∗ (θ0 ))m∈M1 , (Qm∗ (θ))m∈M2 , θ)],where the last inequality follows from the hypothesis that c has decreasing differences in(q 1 , q 2 ) with the observation that Qm∗ (θ0 ) ≤ Qm∗ (θ). Therefore, we obtain[σ 1 (θ) − σ 1 (θ0 )] + [σ 2 (θ) − σ 2 (θ00 )] ≥ c(Q∗ (θ), θ) − c((Qm∗ (θ0 ))m∈M1 , (Qm∗ (θ00 ))m∈M2 , θ),
0

0

00

00

for each θ, θ0 , θ00 ∈ [θ, θ̄∗ ] such that θ ∈ (θl+1 , θl ), θ0 ∈ (θl +1 , θl ) and θ00 ∈ (θl +1 , θl ) withl00 ≥ l ≥ l0 . A similar argument applies to the other combinations of θ, θ0 and θ00 .

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score, costiso-score curve
score on quality

Figure 1: Illustration of Proposition 1

34lower envelope