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How to Buy a Differentiated Product∗ - IDEI

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How to Buy a Differentiated Product∗
John Asker†and Estelle Cantillon‡
Preliminary and Incomplete: Please do not circulate
January 30, 2005
Abstract
A buyer seeks to procure a good characterized by its price and its quality from suppliers who have
private information about their cost structure (fixed cost + marginal cost of providing quality).
We solve for the optimal buying mechanism, i.e. the procedure that maximizes the buyer’s
expected utility, and discuss its properties. A scoring auction, together with a default contract
to fall back on when the scores of the submitted offers are insufficient, implements the optimal
mechanism in most cases. We also compare the performance of the optimal scheme relative to
buying procedures used in practice, namely a quasilinear scoring auction and negotiation. A key
feature of our model is that private information is multidimensional; we discuss the relationship
between our model and existing models of multidimensional screening.
Keywords: optimal auction, multi-attribute auction, differentiated product, multidimensional
screening, scoring auction.
JEL Codes: D44, D82, C78, L24, L22
в€—
We thank Mark Armstrong and Philippe Jehiel as well as seminar audiences at UCL and LSE for helpful com-
ments.
†Leonard N. Stern School of Business, NYU. Email: jasker@stern.nyu.edu
‡
FNRS, ECARES (UniversitГ© Libre de Bruxelles), Harvard Business School and CEPR. Email: ecantillon@hbs.edu
1
1
Introduction
Procurement rarely involves considerations solely based on price. Instead, concerns about the
quality of the good or service provided are often important to the final decision. In this paper, we
consider how a buyer who cares about quality should structure the purchasing process in order to
maximize his expected utility.
The two distinguishing features of our model are that suppliers’ private information about their
cost structure is multidimensional and that quality is endogenously determined as part of the procurement process. US State Highway Authorities’ procurement for highway repair jobs illustrates
these aspects of the contracting environment.1 For high density traffic areas, these agencies care
about the cost of the job and the time in which the job will be completed. A contractor may be
able to speed up the job by hiring extra labor, using some equipment more intensively, or shifting
some resources from other jobs. Hence, suppliers’ quality (here, the time they need to complete
the job) is not fixed but can be adjusted at some cost. Moreover, this adjustment cost is likely to
vary across potential contractors in a way that is not observable to their competitors. Therefore,
it represents one dimension of private information. However, there are other sources of unobserved
cost heterogeneity. These include the contractors’ backlog, and its organizational structure. Thus,
private information is likely to be better captured by a multidimensional parameter.
We build a procurement model with discrete private information. There are N potential suppliers
and each supplier has private information about two components of her cost structure: her fixed
cost and her marginal cost of providing quality. The buyer’s objective is to maximize his expected
utility subject to the suppliers’ participation and incentive compatibility constraints.
We characterize the optimal buying mechanism in this environment (Theorem 1). A useful benchmark is the optimal buying mechanism when private information is one-dimensional. Under some
regularity conditions, Che (1993) shows that the optimal buying scheme distorts the quality provided by the suppliers downwards relative to their first best levels (except for the low cost supplier).
By acting as if he does not care much about quality, the buyer reduces the differentiation between
suppliers and thus increases the level of competition among them. The optimal level of distortion
is independent of the number of suppliers, a property known as the “dichotomy between screening
and selection” (Laffont and Tirole, 1987). Finally, Che shows that a scoring auction with a scoring
rule that is linear in price implements the optimal scheme.2
1
2
See for instance Arizona Department of Transport (2002) and Herbsman et al. (1995).
A scoring auction is an auction where bidders submit bids on several dimensions (price and quality); the scoring
rule reduces these bids into one dimension: the score; and the bidder with the highest score wins. A scoring auction
is said to be quasilinear when it uses a scoring rule linear in price (or a increasing monotonic transformation thereof).
2
The optimal scheme when private information is multidimensional differs from the optimal scheme
in one-dimensional environments. First, suppliers with the same marginal cost of quality generically
supply different levels of quality in the optimal scheme. Hence, a scoring auction using a scoring rule
linear in price is generically not part of the solution, unlike in the one-dimensional case. Second, the
optimal scheme depends on the number of suppliers, i.e. the dichotomy property does not hold. The
reason is that the ordering of types, which determines the optimal quality distortion, is endogenous
in multidimensional environments. The number of suppliers, through its effect on the probabilities
of winning, affects which incentive compatibility constraints bind most and therefore the ordering
of types. Third, the optimal solution involves two sources of inefficiencies: a productive inefficiency
because quality levels differ from their first best levels and an allocative inefficiency because the
probabilities of winning differ from the first best level of probabilities of winning. There is no
allocation inefficiency in the one-dimensional model, beyond that induced by the reserve price.
Armed with the characterization of the optimal scheme, we investigate what practical mechanisms
could implement it. We find that, in most cases, a simple modification to a scoring auction will
implement the optimal scheme. The modification involves a default contract specifying the price
and quality when the supplier’s score is below a certain threshold.
Finally, we investigate how well common buying procedures perform relative to the optimal scheme.
The advantage of common buying procedures is that they are simple. Asker and Cantillon (2004)
show that a quasilinear scoring auction dominates price-only auctions with minimum quality standards, and auctions where the buyers does not announce how the offers will be evaluated, and
suppliers submit one or many price-quality offers. Hence, we use a quasilinear scoring auction and
negotiation as candidate second best solutions. Quasilinear scoring auctions are used by the US
State Highways Authorities, some independent system operators for the procurement of electricity
reserve supply and they are supported by various procurement software.3 Theorem 2 characterizes
the allocation and qualities that can be implemented by a quasilinear auction. Negotiation remains
an attractive solution for professional purchasers when more than price matters.4 The advantage
of negotiation that these buyers see is that it allows a dialogue over all the dimensions of the offers.
Following Manelli and Vincent (1995), we model negotiation as sequential take-it-or-leave it offers
and characterize the optimal negotiation process. We then compare the expected utility derived
from these two procedures to the expected utility from the optimal scheme and the Buyer-Optimal
Efficient Mechanism (BOEM).
3
See Bushnell and Oren, 1994, and Chao and Wilson, 2001 for electricity reserve auctions and the websites
of eBreviate, PurchasePro, Clarus, IBM/DigitalUnion, Oracle Sourcing, Verticalnet and Perfect for procurement
softwares.
4
See e.g. the purchaser survey in the August 2004 issue of “Purchasing” magazine.
3
Related literature. At the substantive level, this paper is related to the literature on procurement.
The key issues studied in this literature are the question of how to take other factors than prices
into account (Laffont and Tirole, 1987, Che, 1993, Branco, 1997), the impact of the potential noncontractibility of quality (Manelli and Vincent, 1995, Rezende, 2003, Morand and Thomas, 2002),
and the impact of renegotiation (Bajari and Tadelis, 2001, Bajari, McMillan and Tadelis, 2004).
Our paper belongs to the first group and we abstract from the other issues. Our contribution
to this literature is twofold. First, we extend prior analyses of optimal procurement to situation when private information is multidimensional. Our analysis shows that the assumption of
one-dimensional private information is not a trivial one: many properties of the optimal procurement process (quality distortion only a function of marginal costs, independence on the number
of suppliers, implementability through a scoring auction) are no longer valid when we move to
multidimensional settings. Second, we evaluate existing buying procedures against the benchmark
of the optimal scheme.
Our paper is also related to the multidimensional screening literature which concerns the design of
optimal contracts or auctions in the presence of multidimensional private information. Rochet and
Stole (2003) present a recent survey of the contracting applications of multi-dimensional screening.
The optimal multi-unit auction is the typical auction application:5 a seller wants to maximize his
expected revenue from the sale of several goods. Buyers’ private information corresponds to their
valuations for each good (Armstrong, 2000, Avery and Henderschott, 2000, Manelli and Vincent,
2004, Malakhov and Vohra, 2004). We show in Section 3 that the structure of our problem is
distinct from any of these two classes of problems. Unlike contracting environments, our problem
involves a resource constraint given that the contract can only be allocated to one supplier. Unlike
multi-unit auction environments, quality in our problem introduces some non-linearity. Hence,
none of the existing characterization results applies to our problem.
The key technical difficulty in multidimensional screening problems is the endogenous nature of
the ordering of agents’ types. Several approaches have been used to make the problem tractable.
For example: imposing some (strong) parametric assumptions on the primitives to simplify the
incentive compatibility constraints (e.g. Armstrong, 1996), having less instruments than dimensions
to screen over (e.g. Rochet and Stole, 2002), or discretizing the type space to reduce the number
of constraints to check for (e.g. Armstrong and Rochet, 1999, Armstrong 1999 and 2000). Our
approach belongs to the latter. The relationship between our model and other models in the
multidimensional screening literature is further discussed in Section 7.
5
An exception is Jehiel, Moldovanu and Stachetti (1999) who study optimal auctions in the presence of external-
ities.
4
2
Model
2.1
Environment
We consider a buyer who wants to buy an indivisible good for which there are N potential suppliers.
The good is characterized by its price, p, and its quality, q.
Preferences. The buyer values the good (p, q) at v(q)в€’p, where vq > 0 (we assume that vq (0) = в€ћ
to ensure an interior solution) and vqq < 0. Supplier i’s profit from selling good (p, q) is given by
p в€’ Оёi1 в€’ Оёi2 q, where Оёi1 в€€ {Оё1 , Оё1 } and Оёi2 в€€ {Оё2 , Оё2 } (Оё1 < Оё1 and Оё2 < Оё2 .).6 Given the binomial
support of Оё1 and Оё2 , there are four supplier types: (Оё1 , Оё2 ), (Оё1 , Оё2 ), (Оё1 , Оё2 ), (Оё1 , Оё2 ), which we
denote for brevity A, B, C and D. We will sometimes use (Оё1k , Оё2k ) to denote supplier type k. For
example, (Оё1A , Оё2A ) = (Оё1 , Оё2 ). Note that the buyer and the suppliers are risk neutral.
Оё2
Оё2
A
B
(Оё1 ,Оё 2 )
Оё2
D
C
(Оё1 ,Оё 2 )
Оё1
Оё1
Оё1
Figure 1: Cost structure
Social welfare. Let Wk (q) = v(q) в€’ Оё1k в€’ Оё2k q, the social welfare associated with giving the
contract to type k with quality q. Define WkF B = maxq Wk (q). Given the single crossing condition,
F B = q F B < q F B = q F B (to save on notation we will use the short-hand notation q and q to
qA
B
C
D
describe the first best levels of qualities, q < q). An expression that will play a role in the analysis
is WA (q) в€’ WC (q). Its derivative is negative,
d
dq [WA (q)
в€’ WC (q)] < 0.
Our assumptions thus far yield an incomplete ordering of types in terms of the first best levels of
welfare they generate. To simplify the analysis, we restrict attention to the case where WAF B <
WCF B . Under this specification, having a low marginal cost for delivering a higher quality product
is more important than having a low fixed cost. This case includes, as a limit, the case where firms
only differ in their marginal cost parameter, which has been studied by Laffont and Tirole (1987),
Che (1993) and Branco (1997). The natural ordering of types is thus D Г‚ C Г‚ A Г‚ B.
Information. Preferences are common knowledge among suppliers and the buyer, with the exception of suppliers’ types, (θi1 , θi2 ), i = 1, ..., N, which are privately observed by each supplier. Types
are independently and symmetrically distributed across suppliers. The probability of each type
6
The linearity of costs in quality ensures that the buyer’s optimization problem is concave - see section 3.1.
5
in the population is given by О±k > 0, k = A, B, C, D. We make the following assumption on the
distribution of types:
Assumption 1 (decreasing hazard rates):
О±D
О±A
≤
О±A +О±D
О±B
and
О±D
О±C
≤
О±C +О±D
О±B .
Assumption 1 is the discrete and multidimensional analog of the decreasing hazard rate assumption
commonly made in the auction literature and, more generally, in the screening literature. Its role
is the same: to avoid situations where types are bunched in the optimal mechanism.
2.2
Optimal buying mechanism
The buyer’s problem is to find a mechanism that maximizes his expected utility from the procurement process. For simplicity, we assume that the buyer buys with probability one (that is,
we assume non exclusion). A direct revelation mechanism in this setting is a mapping from the
announcements of all suppliers, {Оёi1 , Оёi2 }N
i=1 , into probabilities of getting the contract, the level of
quality to deliver and a money transfer.
Given that the buyer’s preference over quality levels is strictly concave, there is no loss of
generality in restricting attention to quality levels that are only a function of the supplier’s type.
Let qk denote the quality level to be delivered by a type k supplier. This, together with suppliers’
risk neutrality, implies that suppliers’ behavior only depends on their expected probabilities of
winning and their expected payment. Let xk be the expected probability that a type k supplier
wins the contract and let mk the expected payment she receives. Finally, let Uk denote type k’s
equilibrium expected utility. We have: Uk = mk в€’ xk (Оё1k + Оё2k qk ).
With these simplifications and notation, the buyer’s expected utility from the mechanism is
given by
F (xk , qk , Uk ) = N
X
k=A,B,C,D
О±k (xk Wk (qk ) в€’ Uk )
(1)
The buyer seeks to maximize this expression over contracts (xk , qk , Uk ), subject to suppliers’ incentive compatibility (IC) constraints:
Uk ≥ Uj + xj (θ1j − θ1k ) + xj qj (θ2j − θ2k )
for all k, j в€€ {A, B, C, D},
(2)
individual rationality (IR) constraints:
Uk ≥ 0
for all k в€€ {A, B, C, D},
(3)
and subject to the feasibility constraint that the probability of awarding the contract to a subset
of the types is always less than or equal to the probability of such types in the population:
X
X
N
αk xk ≤ 1 − (1 −
О±k )N for all subsets K of {A, B, C, D}
kв€€K
kв€€K
6
(4)
Finally, non exclusion imposes that
X
N
О±k xk = 1
(5)
kв€€{A,B,C,D}
Border (1991) guarantees that the feasibility constraint is both necessary and sufficient for the
expected probabilities xk to be derived from a real allocation mechanism. This ensures that the
solution to the maximization problem of (1) subject to (2), (3), (4) and (5) is implementable.
3
Preliminaries
3.1
Structure of the problem
Consider the structure of this maximization problem. For fixed xk ’s, maxqk ,Uk F (xk , qk , Uk ) subject
to (2), (3) is a well-behaved concave programming problem, akin to those studied by Armstrong
and Rochet (1999) and, especially, Armstrong (1999). The standard solution technique in this case
is to characterize the parameter space over which the first order conditions are satisfied given a set
of binding constraints.
For fixed qk ’s, the problem of maxxk ,Uk F (xk , qk , Uk ) subject to (2), (3), (4) and (5) is a linear programming problem akin to the optimal multi-unit auction studied in Armstrong (2000),
Malakhov and Vohra (2004), and Manelli and Vincent (2004). In these cases, candidates for a
solution are extreme points, and the standard technique is to characterize the parameter space over
which these extreme points are indeed solutions.
By contrast, the problem maxxk ,qk ,Uk F (xk , qk , Uk ) subject to (2), (3), (4) and (5) is neither
concave nor even quasi-concave. The reason for this is the multiplicative structure that appears
in xk Wk (qk ) and xk qk , together with the в€’Uk term. Consider however the following change of
variables: z1k = xk , z2k = xk qk . The problem becomes:
max
z1k ,z2k ,Uk
Fe(z1k , z2k , Uk ) = N
X
О±k (z1k Wk (
k=A,B,C,D
z2k
) в€’ Uk )
z1k
Uk ≥ Uj + z1j (θ1j − θ1k ) + z2j (θ2j − θ2k )
N
N
X
X
kв€€K
Uk ≥ 0
О±k z1k
s.t.
for all k, j в€€ {A, B, C, D}
for all k в€€ {A, B, C, D}
X
≤ 1 − (1 −
О±k )N for all subsets K of {A, B, C, D}
kв€€K
О±k z1k = 1
kв€€{A,B,C,D}
7
The constraints are linear in the control variables and the objective function is concave.7 The
first order conditions are thus sufficient for an optimum. In the remainder, we will continue to
work with the original formulation of the problem, but the alternative formulation guarantees
that the first order conditions of the original problem are also sufficient for an optimum. To
see this, let G(xk , qk , Uk ) gather all constraint terms of the Lagrangian of the original problem,
e 1k , z2k , Uk ) gather the constraint terms of the Lagrangian of the transformed problem.
and G(z
Suppose (xв€—k , qkв€— , Ukв€— ) solves the first order conditions of maxxk ,qk ,Uk F (xk , qk , Uk ) + G(xk , qk , Uk ).
We must show that (xв€— , xв€— q в€— , U в€— ) solves the first order condition of maxz ,z ,U Fe(z1k , z2k , Uk ) +
k
k k
k
1k
2k
k
e 1k , z2k , Uk ). The first order conditions with respect to Uk are identical. The first order condition
G(z
with respect to qk , Fqk (xв€—k , qkв€— , Ukв€— ) + Gqk (xв€—k , qkв€— , Ukв€— ) = 0, takes the form
N О±k xв€—k Wk0 (qkв€— ) в€’
X
О»l xв€—k (Оё2k в€’ Оё2l ) = 0
(where О»l are the Lagrangian multipliers of the constraints). This is equivalent to the first order
conditions of the transformed problem with respect to z2k ,
N О±k Wk0 (
X
z2k
)в€’
О»l (Оё2k в€’ Оё2l ) = 0
z1k
(6)
as long as xв€—k > 0 for all k, a consequence of the non exclusion condition (5). Finally, the first order
condition with respect to xk , Fxk (xв€—k , qkв€— , Ukв€— ) + Gxk (xв€—k , qkв€— , Ukв€— ) = 0 takes the form:
N О±k Wk (qkв€— ) в€’
X
О»l [(Оё1k в€’ Оё1j ) + qkв€— (Оё2j в€’ Оё2k )] в€’ N
X
Оі K О±k = 0
(7)
K st kв€€K
The first order condition of the transformed problem takes the form:
N О±k Wk (
X
z2k 0 z2k
z2k
) в€’ N О±k
Wk (
)в€’
О»l (Оё1j в€’ Оё1k ) в€’ N
z1k
z1k
z1k
X
Оі K О±k = 0
K st kв€€K
This is equivalent to (7) as soon as (6) holds. This yields:
Lemma 1: The first order conditions of the maximization problem (1) subject to (2), (3), (4) and
(5) are sufficient for a global optimum.
3.2
Other preliminary results
We gather here a series of preliminary characterization results for the solution.
7
вЋЎ
z2
00
О±k z2k
3 W
вЋў
The hessian is block diagonal with each block given by вЋў
вЋЈ в€’О±
8
1k
z2k
k z2
1k
0
W 00
00
в€’О±k zz2k
2 W
1k
00
О±k W
z1k
0
0
вЋ¤
вЋҐ
0 вЋҐ
вЋ¦
0
Lemma 2: Consider the feasibility constraints
X
X
N
αk xk ≤ 1 − (1 −
О±k )N for all subsets K of {A, B, C, D}
kв€€K
(8)
kв€€K
The following statements hold:
i. At the solution we cannot have more than one one-type constraint binding, more than one
two-type constraint binding and more than one three-type constraint binding.
ii. When the buyer cannot exclude suppliers, exactly one one-type constraint, one two-type
constraint and one three-type constraint bind.
iii. Finally, these binding constraints are nested, in the sense that the type in the one-type
binding constraint must belong to the two-type binding constraint, and so on.
Proof: The claim relies on the fact that the function f (t) = tN for N ≥ 2 is strictly convex.
There are two generic cases to rule out: two constraints binding with no type in common,
and two non nested constraints binding with some type in common.
Case 1: No overlap. Suppose, towards a contradiction, that the constraint for A and the
constraint for B and C bind. Then, from (8), О±A xA + О±B xB + О±C xC = 2 в€’ (1 в€’ О±A )N в€’ (1 в€’
О±B в€’О±C )N > 1в€’(1в€’О±A в€’О±B в€’О±C )N since 1+(1в€’О±A в€’О±B в€’О±C ) = (1в€’О±A )+(1в€’О±B в€’О±C )
and (1 в€’ О±A ) and (1 в€’ О±B в€’ О±C ) lie in (1 в€’ О±A в€’ О±B в€’ О±C , 1). That is, (8) is violated for
{A, B, C}. All cases with no overlap are proved in this way.
Case 2: Some overlap. Suppose, towards a contradiction that the constraint for A and B,
and that for B and C is binding. Since (8) holds for B, this means that
αA xA + αB xB + αC xC ≥ 1 − (1 − αA − αB )N − (1 − αB − αC )N + (1 − αB )N
> 1 в€’ (1 в€’ О±A в€’ О±B в€’ О±C )N by convexity
This contradicts (8) for {A, B, C}. All cases with some overlap are proved in this way.
This proves that binding constraints are nested and that they cannot be more than one
constraint of a type to bind. By Border (1991), exactly one constraint of each type must
be binding since the resulting allocation can be implemented by an allocation mechanism
and it maximizes the probability that the contract is allocated. QED.
The first best allocation is to give the contract to a type D whenever there is one, if not to a type
C if there is one. Else to a type A, or if there is no type A, a type B. Given Lemma 2, the resulting
expected probabilities are: xFDB =
xFBB =
в€’1
О±N
B
N .
1в€’(1в€’О±D )N
,
NО±D
xFC B =
(1в€’О±D )N в€’(О±A +О±B )N
,
N О±C
Lemma 3: At any solution, xA ≥ xB , xD ≥ xC and UB = 0
9
xFAB =
(О±A +О±B )N в€’О±N
B
N О±A
and
This result uses standard manipulation of the incentive compatibility constraints and the individual
rationality constraints.
4
A heuristic approach
The difficulty in multidimensional screening problems such as ours is that the “ordering” of types,
or, more precisely, the direction in which the IC constraints are binding is endogenous. Our
approach is to start with the buyer-optimal efficient mechanism, and progressively increase the
buyer’s expected utility by decreasing the informational rents of suppliers until there is no further
scope for improvement. By Lemma 1, we have reached the global optimum at that point. In many
cases, this process will lead to new IC constraints binding. These changes in the pattern of binding
IC constraints give rise to many of the interesting economic implications of the model, and are
discussed in depth.
We first illustrate our approach for the well-known case where private information is one-dimensional.
The exercise also serves to highlight the similarities and differences between the one-dimensional
problem and its multidimensional counterpart.
4.1
One dimensional private information
Suppose first that О±B = О±C = 0 so that private information is essentially one-dimensional. All
suppliers have low fixed costs. With probability О±D , they have a low marginal cost of providing
quality (Оё2 = Оё2 ); with probability О±A , they have a high cost of providing quality (Оё2 = Оё2 ). Consider
the incentive-compatible buyer-optimal efficient mechanism. To ensure efficiency, qualities must be
set equal to qA = q and qD = q. The contract is awarded to a type D supplier whenever there
is one among the suppliers, otherwise to a type A supplier. Thus, using Lemma 2, xFAB =
and xFDB =
1в€’О±N
A
N О±D .
в€’1
О±N
A
N
To ensure incentive compatibility while maximizing the buyer’s expected utility,
payments are set such that UA = 0 and UD = xFAB q(Оё2 в€’ Оё2 ) (a type A supplier has no incentive
to pretend she is of type D since xFAB < xFDB and q < q). Under this scheme, the buyer’s expected
utility is given by:
О±A xFAB WA (q) + О±D [xFDB WD (q) в€’ UD ]
or, to highlight the virtual welfare associated with each type, that is, the welfare each type generates,
corrected by the incentive cost due to its presence:
О±A xFAB [WA (q) в€’
О±D
q(Оё2 в€’ Оё2 )] + О±D xFDB WD (q)
О±A
10
(9)
(WA (q)в€’ О±О±D
q(θ2 −θ2 ) is supplier A’s virtual welfare). Now consider reducing the quality delivered by
A
A. Clearly, this would have no first order effect on WA (qA ) but this would decrease the information
rent of supplier D. In the buyer-optimal scheme, of course, qA should be decreased further until
в€—
qA
= arg max{WA (qA ) в€’
О±D
qA (Оё2 в€’ Оё2 )}
О±A
(10)
∗ (θ − θ ) and still satisfy all inThis reduces supplier D’s rent since we can now set UD = xFAB qA
2
2
centive compatibility constraints. In (10), the optimal distortion is independent of the probabilities
of winning (and thus is also independent of the number of suppliers), a property that Laffont and
Tirole (1987) have coined the “dichotomy property” between selection and incentives.
Can the buyer use the probabilities of winning as a second instrument to extract more rents from
the suppliers? Consider again (9). Given the award rule and the feasibility constraint (4), xD is
set as high as possible so there is no way to increase the buyer’s utility by changing xD . However,
в€—)в€’
supplier D’s rent is increasing in xA . If WA (qA
О±D в€—
О±A qA (Оё 2
в€’ Оё2 ), the virtual welfare generated by
type A, is negative, the buyer can increase his utility by excluding types A (and setting xA = 0).
To summarize: In the one-dimensional model, the buyer distorts the qualities of the “high cost”
suppliers downward in order to reduce the informational rents of the “low cost” suppliers. Winning
probabilities do not play any role, except for possibly excluding some of the “high cost” types from
the market.8
Che (1993) showed that this optimal buying mechanism can be implemented through a scoring
auction where the buyer announces a scoring rule that is linear in price, ve(q) в€’ p; suppliers submit
price-quality offers, and the offer generating the highest score wins. Obligations are determined, for
example, by the second score, i.e. the winner must deliver a score equal to the score of the second
best offer. The environment that Che considers has continuous types so that the equivalent of (10)
is
F (Оё)
q}
(11)
f (Оё)
where f and F denote the density and cumulative distribution of Оё respectively. Che proposes to
q в€— (Оё) = arg max{v(q) в€’ Оёq в€’
set ve(q) = v(q) − ∆(q) where the distortion term is given by
Z q ВЎ в€—в€’1 Вў
ВЈ в€—ВЎ Вў в€—
В¤
F q
(s)
ds
for
q
в€€
q
Оё
(Оё
)
,
q
∆ (q) =
2
2
в€—в€’1 (s))
0 f (q
(12)
Suppliers in scoring auctions always choose (p∗ (θ), q ∗ (θ)) to maximize the score their offer generates,
given their profit target. This means they set q ∗ (θ) = arg max{v(q) − ∆(q) − θq}. It is easy to check
8
This is true, even in a model with more than two types, because a monotone hazard rate assumption ensures
that, under the optimal choice of qualities, the ranking of the virtual welfares corresponds to the natural ranking of
types. Therefore, so does the optimal choice of probabilities of winning.
11
that this yields the same solution as problem (11) when ∆(q) is given by (12). Moreover, in the
dominant strategy equilibrium of a second score auction (where the winner must deliver a score equal
to the second highest score), suppliers submit bids generating scores equal to max{v(q)−∆(q)−θq}.
For future reference, it is useful to derive the discrete analog of Che’s scoring rule. It is clear that
a rule of the form ∆ (q) =
О±D
О±A (Оё 2
в€’ Оё2 )q will induce the optimal level of quality for Оё2 . Similarly, a
rule where ∆ (q) = 0 will induce the optimal level of quality for θ2 . This suggests a scoring rule of
the type ∆(q) =
О±D
О±A (Оё 2
− θ2 )q1{q≤qA∗ } .
The optimal scoring rule needs to satisfy three conditions: (1) Given the scoring rule, A maximizes
в€— and D maximizes her score by choosing q, (2) the scoring rule must be
her score by choosing qA
such that the equilibrium score generated by D is larger than that generated by A, (3) the payment
в€— . Condition
must be such that A gets no rent and D has an expected payoff equal to xFAB (θ2 − θ2 )qA
(2) ensures that the scoring rule allocates the contract to D in priority over A. Condition (3) is
specific to the discrete case. When types are continuous, allocations (here, the quality levels and
the probabilities of winning) pin down suppliers’ expected payoffs, a straightforward extension of
the payoff equivalence theorem (see Che, 1993 and Asker and Cantillon, 2004). This is not true
with discrete types. Two mechanisms that have the same allocation may yield different expected
utilities for the buyer. While we seek to implement the optimal buying procedure in dominant
strategies, this last point implies that we might need to adjust the second score format slightly.
Consider the following scoring rule: ve(q) = v(q) в€’
О±D
О±A (Оё 2
− θ2 )q + 1{q≥q} ε where ε > 0 is chosen
such that condition 1 is satisfied (this is always possible given their differential marginal costs).
Condition 2 is also satisfied since suppliers submit bids equal to the maximum score they can
generate in the dominant strategy equilibrium, i.e. (p, q) = arg max{e
v(q) − p} subject to p − θq ≥ 0
where Оё в€€ {Оё2 , Оё2 }. We now look at expected payments. If supplier A is asked to deliver the value
of the second score when she wins, her expected profit will be zero as required. Assuming D also
delivers the second highest score when she wins against another D type, her expected payoff is
given by
[e
v (q) в€’ Оё2 q в€’ y] =
О±Nв€’1
A
О±Nв€’1
в€—
A
(Оё2 в€’ Оё2 )qA
by condition 3
N
We need to solve for y, the score D needs to deliver when she wins against a type A. We have
y = ve(q) в€’ Оё2 q в€’
(Оё2 в€’ Оё2 ) в€—
qA
N
(N в€’ 1)
в€—
(Оё2 в€’ Оё2 )qA
N
= second highest score + correction for relative advantage
в€—
в€—
в€—
в€—
= ve(qA
) в€’ ОёqA
+ [e
v(q) в€’ Оё2 q в€’ ve(qA
) + Оё2 qA
]+
12
Note that neither the scoring rule nor the payment rule that implements the optimal buying procedure is unique. In particular, an alternative to the payment correction when a type D wins
against a type A, is a down payment for submitting a bid involving a quality higher than a certain
threshold.
4.2
Two dimensional private information
We now replicate this reasoning when private information is two-dimensional. In the Buyer-Optimal
Efficient Mechanism (BOEM), qualities must be such that qA = qB = q and qC = qD = q. The
contract is awarded giving preference to the type that generates the highest social welfare, i.e.
xk = xFk B . Let Ukj be the expected utility of a type k pretending she is of type j. Also, let
∆θl = θl − θl , l = 1, 2. To satisfy incentive compatibility, suppliers’ expected utilities must be set
such that:
UA = UAB = xFBB ∆θ1
UB = 0
UC = max{UCB , UCA } = max{xFBB q∆θ2 , −xFAB [WA (q) − WC (q)] + xFBB ∆θ1 }
UD = UDC = xFC B ∆θ1 + UC
These payoffs ensure that all incentive compatibility constraints are satisfied.9 Notice in particular,
that the incentive compatibility constraints ICAC and ICDB never bind at the efficient allocation.
In practice, this generates two cases depending on the sign of WA (q) в€’ WC (q). When WA (q) в€’
WC (q) > 0, UCB > UCA . The binding constraints are represented in Figure 2(a). When WA (q) в€’
WC (q) < 0, UCA > UCB . This case is represented in Figure 2(b).
9
The two non trivial IC constraints to check are ICAC and ICDA . ICAC : When UC = UCB , the requirement
B
FB
of UAC < UAB is equivalent to xF
C [WA (q) в€’ WC (q)] в€’ xB [WA (q) в€’ WC (q)] < 0. The first term is negative
since, by assumption, WA (q) в€’ WC (q) < 0. The second term may be positive or negative, but even when it is
B
FB
negative, xF
C [WA (q) в€’ WC (q)] < xB [WA (q) в€’ WC (q)] since
B
FB
d
[WA в€’ WC ] < 0 and xF
B < xC . When UC = UCA ,
dq
FB
в€’ WC (q)] в€’ xA [WA (q) в€’ WC (q)] < 0. This holds by exactly the same
UAC < UAB is equivalent to
B
FB
FB
FB
FB
FB
reasoning. ICDA : When UC = UCA , UDC = xF
C ∆θ 1 − xA ∆θ 1 + xA q∆θ 2 + xB ∆θ 1 > UDA = xA q∆θ 2 + xB ∆θ 1
B
B
FB
FB
FB
since xF
> xF
C
A . When UC = UCB , the claim follows from the fact that xC ∆θ 1 − xA ∆θ 2 q > xA [WA (q) − WC (q)]
FB
> xB [WA (q) в€’ WC (q)] (the inequality being implied by UCB > UCA ). (ICAD is satisfied because imitating D is
B
xF
C [WA (q)
dominated for A by imitating C).
13
A
B
AA
D
C
DD
B B
C C
(b)
(a)
WA (q ) в€’ WC (q ) > 0, that is,
WA (q ) в€’ WC (q ) < 0, that is,
∆θ1 > ∆θ 2 q
∆θ1 < ∆θ 2 q
Figure 2: Binding constraints in the efficient buyer-optimal mechanism
Consider the first case. The buyer’s expected utility,
P
k
О±k [xFk B Wk (qk ) в€’ Uk ], is given by:
αA xFAB WA (qA ) − αA xFBB ∆θ1 + αB xFBB WB (qB ) + αC xFC B WC (qC ) − αC xFBB qB ∆θ2
+αD xFDB WD (qD ) − αD xFC B ∆θ1 − αD xFBB qB ∆θ2
(where all qualities are initially equal to the first best qualities) or, to highlight the virtual welfare
generated by each supplier:
О±A xFAB WA (qA ) + О±B xFBB [WB (qB ) в€’
+О±C xFC B [WC (qC ) в€’
О±A
О±C + О±D
∆θ1 −
qB ∆θ2 ]
О±B
О±B
(13)
О±D
∆θ1 ] + αD xFDB WD (qD )
О±C
The rents of suppliers D and C depend positively on qB and the buyer can increase his expected
utility by decreasing qB , ideally until
2
= arg max{WB (qB ) в€’
qB
О±A
О±C + О±D
∆θ1 −
qB ∆θ2 }
О±B
О±B
(14)
Suppose further that, despite this, suppliers C and D still prefer to pretend they are of type B,
2 ) в€’ W (q 2 )] (this ensures that both U
that is, xFAB [WA (q) в€’ WC (q)] > xFBB [WA (qB
C B
CB > UCA and
UDC > UDA ). Now reconsider (13). There is no further scope of improvement by distorting the
qualities. Furthermore, the virtual welfare of D is clearly the largest of all so that it is optimal to
set xD = xFDB . By contrast, the relative ranking of the virtual welfare of A and C is unclear. If
WC (q) в€’
О±D
αC ∆θ 1
> WA (q), the virtual welfare generated by supplier C remains larger than that of
A so the optimal allocation is the first best allocation.
Suppose instead that the virtual welfare associated with A is larger than that associated with
C, itself larger that the virtual welfare associated with B. Then, the buyer would rather increase
the probability of giving the contract to supplier A over C. As we increase xA and decrease xC
concurrently (keeping О±A xA + О±C xC + О±D xD =
1в€’О±N
B
N
14
so that the contract is always allocated), the
buyer’s expected utility increases. Define xmax
as the expected probability with which a type A
A
as C’s expected
wins when she has priority over all the other types but D. Similarly, define xmin
C
and xmin
probability of winning when A and D have priority over C. Suppose that at xmax
A
C , UDC =
2
max
FB
xmin
C ∆θ 1 + xB qB ∆θ 2 is still larger than UDA = xA q∆θ 2 + xB ∆θ 1 . Then no other constraint
binds in the process of increasing xA and decreasing xC . The qualities and probabilities are then all
optimized given the binding constraints. By Lemma 1, we have reached the solution: xD = xFDB >
FB
FB
2
xA = xmax
> xC = xmin
A
C > xB = xB , and qk = qk , except for qB = qB .
The preceding describes cases 1.1.a and 1.1.b in Theorem 1 below. They illustrate the differences
with the one-dimensional model. First, unlike the solution to the one-dimensional model, the
probabilities of winning play a screening role beyond simple exclusion. This is intuitive. In the
one-dimensional model, we had two instruments but only one dimension to screen over. Here, we
have two dimensions to screen over, Оё1 and Оё2 . The consequence is that we now have two kinds
of inefficiencies due to screening: A production inefficiency because some qualities are distorted
downwards relative to the first best level of provision, and an allocative inefficiency because winning
probabilities differ from first best winning probabilities.
The second difference with the one-dimensional model is that which constraint binds may change
as we adjust the qualities. This is now a familiar property of all multidimensional screening models.
The third and final difference between the one-dimensional model and the two-dimensional model
is the loss of the dichotomy property between incentives and selection. Indeed, despite the fact
that the optimal choice of quality appears to be independent of the probabilities of winning in (14),
2 depends
the fact that the binding IC constraints are configured in this particular way at qB = qB
on the probabilities of winning. An implication is that, unlike in the one-dimensional model, the
optimal buying mechanism will depend on the number of suppliers.10
The full characterization of the solution is provided in the next Theorem. Figures 3 and 4 provide
a graphic representation the binding constraints in each case.
Theorem 1: Characterization of the optimal buying mechanism
2 be as defined in (14) and let q 2 = arg max{W (q ) +
Let qB
A A
A
О±C +О±D
αA ∆θ 1
в€’
О±C +О±D
αA qA ∆θ 2 }.
Part I: When WA (q) в€’ WC (q) > 0, the probabilities of winning and quality levels in the optimal
10
A tempting interpretation of the dichotomy property in the one-dimensional model is that adding an instrument
via the competition among suppliers is unnecessary because we already had one instrument and one dimension. This
interpretation is misleading. In fact, the qualitative properties of the solution would change in a model with two types
of qualities Г la Armstrong and Rochet (1999), when we add a third instrument (competition) to the two existing
ones (levels of each type of quality). The reason is the one stated in the main text. The presence of the probabilities
of winning affects the configuration of the binding constraints.
15
buying mechanisms are as given in Table 1. Details on the conditions for the subcases are given
along the proof in the Appendix.
Part II: When WA (q) в€’ WC (q) < 0, the probabilities of winning and quality levels in the optimal
buying mechanisms are as given in Table 2. Details on the conditions for the subcases are given
along the proof in the Appendix.
Table 1: Probabilities of winning and quality levels when WA (q) в€’ WC (q) > 0
Case
Probabilities of Winning
Quality Levels
2 ) − W (q 2 )] ≤ xF B [W (q) − W (q)]
Condition: xFBB [WA (qB
C B
A
C
A
1.1.a
xk = xFk B
qD = q
qC = q
qA = q
2 <q
qB = qB
1.1.b
FB
xFDB > xA = xmax
> xC = xmin
A
C > xB
qD = q
qC = q
qA = q
2 <q
qB = qB
1.1.c
FB
xFDB > xmax
≥ xA > xC ≥ xmin
A
C > xB
qD = q
qC = q
xFDB > xmax
≥ xA > xC & xB ≥ xFBB
A
qD = q
qC = q
2 , q)
qA в€€ (qA
2 , q)
qB в€€ (qB
1.2.a
xk = xFk B
qD = q
qC = q
1.2.b
xFDB > xFC B > xC > xA > xFAB > xFBB
qD = q
qC = q
2 , q)
qA в€€ (qA
2, q )
qB в€€ (qB
A
1.2.c
xFDB > xC = x = xA > xFBB
qD = q
qC = q
2 , q)
qA в€€ (qA
2, q )
qB в€€ (qB
A
1.1.d
2 ) в€’ W (q 2 )] > xF B [W (q) в€’ W (q)]
Condition: xFBB [WA (qB
C B
A
C
A
2 , q)
qA в€€ (qA
2 , q)
qA в€€ (qA
Other cases (and equivalents): 1.2.d (as 1.1.b), 1.2.e (as 1.1.c), 1.2.f (as 1.1.d)
2 , q)
qB в€€ (qB
2, q )
qB в€€ (qB
A
Additional conditions for individual cases can be found in the appendix.
Table 2: Probabilities of winning and quality levels when WA (q) в€’ WC (q) < 0
Case
Probabilities of Winning (x’s)
Quality Levels
2 ) в€’ W (q 2 )]
Condition: xFBB [WA (q) − WC (q)] ≥ xFAB [WA (qA
C A
2.1.a
xk = xFk B
qD = q
qC = q
2
qA = qA
qB = q
2.1.b
xFDB > xFC B > xC = xA > xFAB > xFBB
qD = q
qC = q
2
qA = qA
qB = q
xFDB > xFC B > xFAB > xA > xB > xFBB
qD = q
2.1.c
2 ) в€’ W (q 2 )]
Condition: xFBB [WA (q) в€’ WC (q)] < xFAB [WA (qA
C A
в€—
Cases (and equivalents): 2.2.a (as 1.2.a) , 2.2.b (as 1.2.b)в€— , 2.2.c (as
qC = q
qA в€€
2,
(qA
qB )
2 , q)
qB в€€ (qB
1.2.c)в€— , 2.2.d (as 2.1.c)
Additional conditions for the individual cases are found in the appendix
* While the conditions that characterize these cases are the same, for those cases in Part II of Theorem 1 qB
2 , q)
в€€ (qB
Tables 1 and 2 present the main features of the solution. In all cases, there is some downward
distortion of the quality provided by the high cost marginal suppliers. The quality provided by
the low marginal cost suppliers is never distorted. It is also apparent that the kind of distortion
16
that can be generated by a scoring rule linear in price (i.e. such that qD = qC and qA = qB ) is
generically not part of the solution, unlike in the one-dimensional model. Only cases 1.1.c and 1.1.d
lend themselves to such a solution as a special case.
A
D
B
A
C
D
B
A
C
D
1.1.c
1.1.d
1.2.e
1.2.f
1.1.a
1.1.b
1.2.d
B A
B
C D
1.2.a
1.2.b
C
1.2.c
Figure 3: Binding constraints at the solution when WA (q) в€’ WC (q) > 0
Probabilities of winning are also often distorted. Specifically, the probabilities of winning of the
high marginal cost suppliers are sometimes distorted upwards, whereas the probability of winning
of low marginal cost supplier C is sometimes distorted downwards. The allocation of supplier D is
never distorted.
A
B
A
B
A
B A
B
D
C
D
C
D
C D
C
2.1.a
2.1.c
2.2.a
2.2.b
2.2.d
2.1.b
2.2.c
Figure 4: Binding constraints at the solution when WA (q) в€’ WC (q) < 0.
Putting these two aspects together - productive and allocative distortions - we find no systematic
“bias against quality” in the two-dimensional model, unlike in the one-dimensional model (Laffont
and Tirole, 1987 and Che, 1993). While the economic conclusions differ, the underlying economic
motivation is the same: reduce suppliers’ rents. The qualities of the high marginal cost types are
distorted downwards to reduce the low marginal cost supplier’s benefit from imitating them. As
illustrated in Figures 3 and 4, all binding constraints between suppliers with different marginal
costs are from the low marginal cost supplier to the high marginal cost supplier so this “trick” is
effective. This is also the case in the one-dimensional model where the distortion of high-cost types’
quality lowers the informational rents of the low cost types. Similarly, the reason why supplier C’s
probability of winning is sometimes below her first best level is to reduce supplier D’s rent when
17
the incentive compatibility constraint ICDC is binding. In each case the optimal level of distortion
balances a trade-off between the costs in terms of lost social welfare and the benefits in terms of
reduced rents.
To investigate the pattern of incentive compatibility across the type space, we computed the solution
for a parameterized version of the model. Figure 5 shows the results of this simulation exercise for
1
N = 2 in an environment where αA = αC , αD = αB , v (q) = 3q 2 , ∆θ2 = 1, and θ1 = θ2 = 1. The
graph represents which case applies as a function of ∆θ1 and αA . The space is bounded on the
right by the value of ∆θ1 that equates WAF B = WCF B . It is bounded from below by αA = 0.2 (i.e.
О±B = 0.3) which corresponds to the maximum correlation allowed by assumption 1. It is bounded
from above by О±A = 0.5 (i.e. О±B = 0) which corresponds to perfect negative correlation in types.
When the number of suppliers is increased to seven (N = 7) , the partition of the type space is
depicted in Figure 6.
Comparing Figures 5 and 6 makes it clear that an increase in the number of suppliers leads to
a somewhat simpler optimal mechanism, in that many cases become essentially irrelevant. Some
divisions between cases remain unchanged, while others change considerably.
The only boundary that is completely unaffected by the change in the number of suppliers is the
boundary between Case1.2.a and 2.2.a and Case 1.2.c and 2.2.c. This boundary marks the division
between Parts I and II of Theorem 1. The condition that defines it, ∆θ1 −∆θ2 q = 0, corresponds to
the point at which the binding IC constraint for type C in the Buyer Optimal Efficient Mechanism
(BOEM) changes from binding toward A to binding toward B (ICCA ceases to bind and ICCB
binds instead). This condition is clearly independent of the number of suppliers.
Two boundaries are only somewhat affected by the increase in the number of suppliers. The
boundary between Case 1.1.a and 1.1.c is invariant to the number of suppliers, but its extension
along the boundary between Cases 1.2.a and 1.2.e & c, and Cases 2.2.a and 2.2.c does change,
although only very subtly. The invariant part, between 1.1.a and 1.1.c marks the points at which
the virtual surplus of A equals that of C after the quantities in the BOEM have been optimized.
The part that does vary reflects the same equality, but is sensitive to changes in the number of
suppliers since optimizing quantities in the BOEM leads to ICCA and ICCB both binding. This
leads to a need to set quantities so that xFAB [WA (qA ) в€’ WC (qA )] = xFBB [WA (qB ) в€’ WC (qB )] . The
sensitivity is via the x’s, which are functions of the number of suppliers. The other boundary which
is only somewhat affected is between Cases 2.1.a and 2.1.c (the invariant part), extended through
2.2.d and 2.2.a. The economics is exactly the same as the boundary already discussed.
Three boundaries are highly sensitive to the number of suppliers. These are the boundary extending
18
0.50
0.45
Case
2.1.a
Case
1.1.a
Alpha A
0.40
Case
2.1.c
0.35
Case
2.2.a
Case
1.2.a
Case
1.2.b
Case
1.1.c
0.30
Case
2.2.d
0.25
Case
2.2.b
Case
1.2.e
Case
2.2.c
0.20
0.00
0.20
0.40
Case
1.2.c
0.60
0.80
1.00
1.20
Delta 1
Figure
5: Partition of the Type Space by Case. N = 2. Other parameters are: О±A = О±C , О±D = О±B ,
1
v (q) = 3q 2 , ∆θ2 = 1, θ1 = θ2 = 1. Dashed boundaries reflect divisions between cases that represent
equivalent mechanisms.
between Cases 2.1.a and 2.2.a, the boundary extending between Cases 1.2.a and 1.1.a, and the
boundary between 1.2.c and 1.2.e. Consider the boundary between 1.2.a and 1.1.a. This marks the
points at which ICCB ceases to bind and ICCA binds when the q’s in the BOEM are optimized.
The condition for ICCB to bind,
2 ) в€’ W (q 2 )]
[WA (qB
xFAB
C B
,
≥
F
B
[WA (q) в€’ WC (q)]
xB
is a direct function of x’s and therefore of the number of suppliers. The left side of this expression
increases with N (xFBB approaches zero faster than xFAB ) while the right side is invariant to N .
Since the ratio on the right hand side is increasing in ∆θ1 ,the boundary shifts to the left in Figure
11
6.
The same intuition applies for the boundary between 1.2.c and 1.2.e which is defined by
2 ) в€’ W (q 2 )]
xFBB [WA (qB
C B
= xmax
A [WA (q) в€’ WC (q)].
Having discussed the way the boundaries work, it is now straightforward to discuss how changing
αA and ∆θ1 affect the shape of the optimal mechanism. To illustrate, in figure 5, taking αA = 0.4,
11
For the boundary extending between 2.2.a and 2.1.a, switch all subscripts on the x’s and q’s in the preceding
discussion (that is, if they read A make them B, and vice versa).
19
0.50
Case
1.1.a
Case
2.1.a
0.45
Case
2.2.a
Alpha A
0.40
0.35
Case
1.2.a
0.30
Case
1.2.c
Case
1.1.c
0.25
Case
2.2.c
Case
2.1.c
0.20
0.00
0.20
0.40
Case
1.2.e
0.60
0.80
1.00
1.20
Delta 1
Figure
6: Partition of the Type Space by Case. N = 7. Other parameters are: О±A = О±C , О±D = О±B ,
1
v (q) = 3q 2 , ∆θ2 = 1, θ1 = θ2 = 1. Dashed boundaries reflect divisions between cases that represent
equivalent mechanisms.
we can see how varying ∆θ1 affects the mechanism. We start in case 2.1.c with C’s being indifferent
between both A and B (two IC constraints binding). This is due to the fact that when the q’s
in the BOEM are optimized the virtual welfare of A is less than that from B. This induces an
incentive to decrease xA and increase xB . Doing so makes B more attractive to C than A, although
were A not bearing some of the burden of reducing C and D’s informational rents, C would prefer
to imitate A (by assumption). Thus qB and qA are distorted downward from the first best, as both
are used to reduce the informational rents of D and C. The probabilities of winning for A and B
are also distorted.
As the higher fixed cost (Оё1 ) increases, we reach a point where the virtual surplus from A is no
longer smaller than that from B after the q’s in the BOEM are optimized (recall fixed costs for A
are smaller than B).12 Thus ICCB ceases to bind and the only distortion is on the quality delivered
by A. This finds us in case 2.1.a.
This situation persists until θ1 is sufficiently large that, once the q’s in the BOEM are optimized,
the transfer to A (who has low fixed cost) becomes less attractive to C relative to that paid to B
12
Recall that in the simulation θ1 is fixed, so an increase in ∆θ1 corresponds to an increase in θ1
20
(B’s transfer compensates for a high fixed cost, possessed by C, and C is not as concerned with
the higher quality delivered by B since C has a low marginal cost of quality). So now, we are back
in a similar position to that we started with, with ICCA and ICCB binding, only here, with higher
fixed costs, there is no incentive to distort the allocation.
As Оё1 rises further we move from case 2.2.a to case 1.2.a, which are equivalent cases, and onto case
1.1.a. At this point Оё1 is so high that even if the transfer to A compensates for producing quality
2 , C would still prefer
at the first best and the transfer to B only compensates for producing at qB
to imitate B. Thus ICCA ceases to bind, and the distortion on qA disappears. Now B bears the
full brunt of reducing the informational rents of C and D.
The last transition is from case 1.1.a to case 1.1.c. Here, Оё1 has risen to the point that the virtual
surplus from A, under the BOEM, is higher than that of C (C’s virtual surplus is reduced since
D’s informational rents are being reduced via the transfer to C). As xA increases D will prefer to
switch to A over C. This results in ICDA binding, in addition to ICDC . Thus A and C share some
of the burden of extracting informational rents from D, resulting in distortions on qA , qB , xA and
xC .
5
How to buy a differentiated product
In this section, we explore the practical implications of our results. First, we investigate what
practical buying procedure could implement the optimal mechanism. Second, we explore the performance of second best but simpler mechanisms.
5.1
Practical mechanisms
Scoring auctions are often used in the environment we study. The simplest version of scoring auctions has a scoring rule that depends linearily in price and is called a quasilinear scoring auction.13
We investigate whether simple variants of these auctions could implement the optimal mechanism.
The optimal mechanism cannot, in general, be implemented by a quasilinear scoring auction. However, the characterization of the solution suggests a straightforward fix. Consider, for instance,
2.
solution 1.1.a. in Table 1. The solution is such that xk = xFk B and qk = qkF B , except for qB = qB
2 and p = Оё + Оё q 2 .
The following mechanism will implement it: There is a default contract: q = qB
1
2 B
13
A quasi-linear scoring auction is a scoring rule of the type v(q) − p, with v 0 (q) ≥ 0 together with the auction rules
where (1) bidders submit bids of the type (p, q), (2) the bidder whose bid generates the highest score wins, and (3)
the score the winner must deliver is a function of the received bids.
21
The scoring rule is ve(q) в€’ p = v(q) в€’ p, the truthful scoring rule, and the process runs as fol-
lows. Suppliers submit offers that are evaluated according to the scoring rule. If the highest score
is greater or equal to v(q) в€’ Оё1 + Оё2 q, the supplier with the highest score wins, and delivers the
quality promised. Otherwise, the default contract is randomly allocated among the suppliers. The
payment is such that the score eventually delivered is equal to the second highest score, modulo a
supplier-type adjustment.14
As before, we need to check two conditions beyond the payment to verify that this scoring auction
with default contract indeed implements the optimal procedure:
1. Given the scoring rule, suppliers maximize their score (and thus the apparent social welfare)
by choosing the assigned level of quality
2. The ranking of the resulting scores is consistent with the assigned probabilities of winning
These are clearly satisfied by the proposed scheme for case 1.1.a. Cases 1.2.a, 2.1.a and 2.2.a are
straightforward variants. A comparision of this rule to the Che rule in section 4.1 should make it
clear that the Che rule can also be expressed as a scoring rule with a default contract. In that sense,
these rules are quite similar. However, in the optimal rule two suppliers with the same marginal
cost do not provide the same quality, in contrast to the Che rule. Implementation via a scoring
rule departs even further from the Che rule for the remaining cases.
Let’s now consider case 1.1.b. The difference is that the allocation probabilities are now such that
> xC = xmin . A truthful scoring with a default contract would satisfy condition 1
xA = xmax
A
above, but not condition 2. However, as long as WA (q) в€’ WC (q) > 0, or equivalently, as long as
∆θ1 > ∆θ2 q, we can construct a scoring rule such that both conditions 1 and 2 are satisfied. This
done by the scoring rule
ve(q) = v(q)1{q≤q} + v(q)1{q>q} + 1{q≥q}
(15)
A similar rule implements cases 1.1.c, 1.2.b, 1.2.c, 1.2.d and 1.2.e.15 . For cases 1.1.d, 1.2.f, 2.1.c and
2.2.d we must confront the additional difficulty that the buyer must be made indifferent between
14
Cf. our discussion in Section 4.1. This adjustment is only due to the discrete type setting. One interpretation
of this adjustment is that it is an offer-contingent down payment. The construction of the optimal scheme ensures
that suppliers will prefer the down payment associated with their offers together with truthful bidding. Note that
incentive compatibility also insures that this down payment is weakly increasing in the score generated by the offer.
[check last statement]
15
[only for the referees] In some of these cases, the offers from suppliers A and C must be such that they generate
the same score, that is v(q) − θ1 − θ2 q = v(q) − θ1 − θ2 q. Setting ε = ∆θ1 − ∆θ2 q + θ2 (q − q) = ∆θ1 − ∆θ2 q + θ2 (q − q)
will guarantee this. This also means that Оё2 (q в€’ q) < Оµ < Оё2 (q в€’ q), a condition necessary for condition 1.
22
giving the contract to B and giving it to C, while, at the same time, C must be left with a positive
rent. Setting the minimum score equal to the score that C must deliver at equilibrium, else the
default contract is implemented, achieves the first requirement. A supplier-type specific payment
can achieve the second objective.
Finally, cases 2.1.b, 2.2.b and 2.2.c remain. While these cases look very similar to the previous
cases, the difference is that the condition WA (q) − WC (q) < 0 makes it impossible to find a scoring
rule that satisfy simultaneously conditions 1 and 2.16 This suggests an even more complicated
mechanism would be required for implementation. This difficulty in implementation, together with
the heavy informational requirements demanded, lead us to conclude that the optimal mechanism
may not yield a convenient, practical procurement mechanism. However, the optimal mechanism
still has considerable value. It provides considerable intuitive insight into the desirable qualities of
a procurement mechanism and also provides a benchmark against which to judge more practical,
commonly used mechanisms.
5.2
Comparison with commonly used procedures (incomplete)
A disadvantage of the optimal scheme is the manner in which it depends finely on the exact
parameters of the environment and does not appear to be easily implemented. This suggests
that, for practical purposes, second best solutions that are robust performers in a wide variety of
settings are likely to be more useful. Commonly used procedures are obvious candidates. They
include quasi-linear scoring auctions, price-only auctions with minimum quality standards, menu
auctions where suppliers can submit several price-quality offers, and one-to-one negotiation. Asker
and Cantillon (2004) have shown that quasi-linear scoring auctions yield a higher expected utility
to the buyer than a price-only auction with minimum standards, and that they dominate menu
auctions when a second price or ascending format is used. Hence, our contenders for a simple and
robust buying procedure are a quasi-linear scoring auction or one-to-one negotiation.
The next Theorem characterizes the set of allocations that can be implemented by a quasi-linear
scoring auction.
Theorem 2: The solution to the original problem can be implemented as a quasi-linear scoring
auction if and only if (1) qA = qB , qC = qD with qA , qB ≤ qC , qD (2) αA xA + αC xC = αA xFAB +
16
[for the referees only] Let qA be the quality level produced by A at the soution. Without loss of generality,
let v(qA ) and v(qA ) + Оµ the value of qA and q according to the scoring rule. Condition 2 requires that the scores
generated by A and C are the same, i.e. v(qA ) − θ1 − θ2 qA = v(qA ) + ε − θ1 − θ2 q, that is, ε = ∆θ1 + θ2 q − θ2 qA =
∆θ1 − ∆θ2 qA + θ2 (q − qA ) < θ2 (q − qA ). Condition 1 for C requires that v(qA ) + ε − θ1 − θ2 q ≥ v(qA ) − θ1 − θ2 qA ,
that is, ε ≥ θ2 (q − qA ). A contradiction.
23
αC xFC B , xB = xFBB and xD = xFDB , (3) ∆θ1 −∆θ2 qC ≤ 0 when xC > xmin
C and (4) ∆θ 1 −∆θ 2 qA ≥ 0
whenever the optimal solution is such that xA > xFAB .
Proof : In a scoring auction, suupliers maximize the score they generate given their profit
v(q) в€’ p} subject to p в€’ Оёi1 в€’ Оёi2 q = ПЂ. Substituting for p
target ПЂ, i.e. they solve max(p,q) {e
inside the maximizer yields
v(q) в€’ Оёi1 в€’ Оёi2 q в€’ ПЂ}
max{e
q
(16)
A property of the solution is that it is independent of ПЂ, the profit target. Moreover, a
standard incentive compatibility argument establishes that the ordering of suppliers’ winning probabilities must correspond to their ability to generate a higher score (intuitively,
v(q) − θi1 − θi2 q} as the supplier’s type). This allows us to ignore π in (16) when
define maxq {e
arguing who must win against whom. Let Sk (q) = ve(q) в€’ Оёi1k в€’ Оёi2k q.
We first prove the necessary conditions. Condition (1) follows directly from (16) given that
Оё2A = Оё2B > Оё2C = Оё2D . Condition (2) follows from the fact that D generates a strictly
higher score than either A and C for all choices of ve(q). Similarly, both A and C can
always generate a strictly higher score than B so they must win against a B type. When
xC > xmin
C , SC (qC ) ≥ SA (qA ), else A would have priority over C in the allocation. This
implies that
ve(qC ) − θ1 − θ2 qC ≥ ve(qA ) − θ1 − θ2 qA , that is,
∆θ1 − ∆θ2 qC ≤ ve(qC ) − ve(qA ) − θ2 (qC − qA )
In addition, since this is an equilibrium, A must be generating a higher score by choosing
qA than qC , i.e.
ve(qC ) − ve(qA ) − θ2 (qC − qA ) ≤ 0
Combining both inequalities yields condition (3). When xA > xFAB , SA (qA ) ≥ SC (qC ), else
C would have strict priority in the allocation. This implies
∆θ1 − ∆θ2 qA + θ2 (qC − qA ) + ve(qA ) − ve(qC ) ≥ 0
In addition, since this is an equilibrium C must be generating a higher score by choosing
qC over qA , i.e.
θ2 (qC − qA ) + v(qA ) − v(qC ) ≤ 0
Combining both inequalities yields condition (4).
24
To prove sufficiency, we construct a scoring rule that implements the intended allocation
and choice of qualities in a second score auction with type-dependent down-payment.17
Consider
ve(q) = v(q)1{q≤qA } + v(qA )1{q>qA } + 1{q≥qC }
(17)
For this scoring auction to implement the outcome, two conditions must be satisfied. First,
suppliers must be choosing the assigned qualities when they maximize their scores. Second,
the ranking of the scores must (weakly) correspond to the assigned ranking of types in the
allocation. Ties in the scores are admissible for two consecutive types, even if they are not
assigned the same probabilities of winning because the second score format makes them
indifferent about the way ties are resolved in case of ties (they are making zero profits any
way).
Given the shape of this scoring rule the two relevant choices are qA and qC . A prefers qA
to qC if and only if v(qA ) − θ1 − θ2 qA ≥ v(qA ) + ε − θ1 − θ2 qC i.e. ε ≤ θ2 (qC − qA ) (B’s
preferences yield the same condition). C prefers qC to qA if and only if v(qA ) + Оµ в€’ Оё1 в€’ Оё2 qC
≥ v(qA ) − θ1 − θ2 qA , i.e. ε ≥ θ2 (qC − qA ) (D’s preferences yield the same condition). Hence,
suppliers choose their assigned qualities if
θ2 (qC − qA ) ≤ ε ≤ θ2 (qC − qA )
(18)
which is possible by condition (1). Next, C generates a higher score if and only if SC (qC ) =
v(qA ) + ε − θ1 − θ2 qC ≥ SA (qA ) = v(qA ) − θ1 − θ2 qA i.e.
ε ≥ ∆θ1 − θ2 qA + θ2 qC = ∆θ1 − ∆θ2 qC + θ2 (qC − qA )
(19)
A generates a higher score otherwise. Condition (3) ensures (18) and (19) are always
compatible. When the solution is such that xA > xFAB , we need SA (qA ) ≥ SC (qC ) :
ε ≤ ∆θ1 − θ2 qA + θ2 qC = ∆θ1 − ∆θ2 qA + θ2 (qC − qA )
(20)
instead. It is compatible with (18) if ∆θ1 − ∆θ2 qA ≥ 0. QED.
Theorem 2 clarifies the constraints that a quasi-linear scoring auction place on the allocation and
qualities. An immediate consequence of Theorem 2 is that the Buyer-Optimal Efficient Auction
(BOEM) can be implemented by a quasi-linear scoring auction. Such a scoring auction has a
scoring rule that corresponds to the buyer’s preferences and uses a second score format (with
17
Recall from our discussion in Section 4.1. that the type-dependent down-payment is only needed because of the
discrete nature of the type space.
25
down-payments to account for the discrete nature of the type space). The theorem also articulates
the restrictions on the optimization program required to produce a rule analogous to the optimal
rule in Che (1993). However, in this richer environment such a rule can have allocative inefficiency
in addition to the productive inefficiency, interpreted in Che as a systematic bias against quality.
This bias against quality is no longer an accurate description of the optimal quasilinear rule since
the allocative inefficiency may work to the advantage of suppliers of lower quality goods.
Theorem 2 also suggests the ways in which the original problem must be adjusted in order to solve
for the optimal quasi-linear auction. Specifically, constraints (1) and (2) do not affect the nature
of the optimization problem which remains concave. In Appendix C, we use Theorem 2 to solve
for the optimal quasi-linear scoring auction for our environment. In a first step, we only integrate
constraints (1) and (2) into the problem and solve for the first order conditions. If the solution
satisfies constraints (3) and (4), it is the outcome of the optimal quasi-linear scoring auction.
Otherwise, we impose (3) and (4) and solve for the new optimum.
1
0.9
BOEM
0.8
OPT
SCORING
0.7
0.6
0.5
0.4
0.3
0.2
0.1
1.069
1.125
1.013
0.956
0.900
0.844
0.788
0.731
0.675
0.619
0.563
0.506
0.450
0.394
0.338
0.281
0.225
0.169
0.113
0.056
0.000
0
delta1
2.2.d
x AFB > x A > x B > x BFB
2.2.a
x k = x kFB
qA < qB < q
qA , qB < q
1.2.c
x A = xC
qA , qB < q
1.2.e then 1.1.c
x A > xC
qA , qB < q
в€љ
Figure 7: Mechanism performance as a function of ∆θ 1 for N = 2, θ1 = θ2 = 1, θ 2 = 2, v(q) = 3 q, αk = 0.25.
Figures 7 and 8 illustrate how the optimal quasi-linear scoring auction performs relative to the
optimal scheme and to the BOEM in an environment similar to the one studied in Figure 5. As
∆θ1 increases, fixed costs become relatively more important as a source of adverse selection and
the maximum level of welfare decreases since suppliers’ costs increase. Figure 7 suggests that the
26
expected utility from all three mechanisms are also decreasing in ∆θ1 (The kink in the BOEM
curve corresponds to WA (q) − WC (q) = 0, when the binding constraints in the efficient mechanism
switch from ICDC , ICCA and ICAB to ICDC , ICCB and ICAB : this increases the weight of ∆θ1 in
the buyer’s expected surplus. The BOEM curve is correspondingly steeper.)
The difference in expected utility between the optimal buying mechanism and the efficient buyer-
optimal mechanism represents the potential gain from being a strategic buyer. Indeed, the buyer can
always obtain the expected utility from the efficient buyer-optimal mechanism simply by revealing
his true preferences and holding a second score scoring auction. Figure 8 suggests that the optimal
linear scoring rule does pretty well in most cases. It captures a large part of the potential surplus
from being strategic, except for values of ∆θ1 between 0.48 and 0.67 where the quasilinear scoring
rule achieves less than 70% of the surplus. For values of ∆θ1 around 0.25, the optimal quasi-linear
scoring rule does basically as well as the optimal mechanism. This area corresponds to case 2.2.a
in the solution, which, as previously argued, can be implemented by a quasi-linear scoring rule for
some values of the parameters.
100
% of Optimal-BOEM revenue
90
80
70
60
50
40
2.2.d
2.2.a
1.2.c
1.2.e then
30
1.13
1.07
1.01
0.96
0.90
0.84
0.79
0.73
0.68
0.62
0.56
0.51
0.45
0.39
0.34
0.28
0.23
0.17
0.11
0.06
0.00
Delta1
Figure 8: Scoring mechanism performance (% of (Optimal - BOEM captured), as a function of ∆θ1 for
в€љ
N = 2, Оё1 = Оё2 = 1, Оё2 = 2, v(q) = 3 q, О±k = 0.25.
Note that as ∆θ1 tends to 0, the source of adverse selection reduces to one dimension, the marginal
cost. In this case, Che (1993) has shown that a quasi-linear scoring rule implements the optimal
mechanism. The reason why the expected utility from the optimal quasi-linear scoring rule does
not converge to the expected utility of the optimal mechanism in our graph is that there is some
discontinuity in the optimal quasi-linear scoring auction at ∆θ1 = 0. As long as ∆θ1 > 0, the
nature of the scoring rule imposes that A generates a strictly higher score at equilibrium than B.
27
Thus xA ≥ xFAB > xB = xFBB (see Theorem 2). This leaves some informational rent to A and
increases the rents of C and D relative to the case where xA = xB . When ∆θ1 = 0, suppliers A
and B are essentially the same (and so are suppliers C and D). The optimal scoring rule will thus
set xA = xB and leave no rent to supplier A.
The next figure plots the percentage of the surplus, that is, the difference between the expected
revenue in the optimal scheme and in the BOEM, that the quasi-linear scoring rule captures as
N increases. For this exercise, ∆θ1 is set equal to 0.6, that is, a value for which the quasi-linear
scoring rule was doing less well in Figure 7. The optimal quasi-linear scoring auction captures most
of the surplus as soon as the number of suppliers is larger than five. The reason why it does so well
is that almost all the distortion in the optimal mechanism is in the A and B types (the allocative
distortion on C also arises in some cases). As N increases the likelihood of allocating to a type
with no distortion (i.e. a D and often a C) increases dramatically. Since the optimal scheme and
the quasilinear auction are the same in these cases, it is not surprising that the revenues converge.
It is interesting that it happens so quickly.18
120%
100%
80%
60%
40%
20%
0%
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
Number of bidders
Figure 9: Performance of the optimal quasi-linear scoring rule as a percentage of the difference (optimal scheme - BOEM)
We now turn to negotiation. There are many ways to model negotiation in our environment. The
main advantage that buyers see in using negotiation over auctions when quality matters, is that
it allows them to incorporate all dimensions of the product in the purchasing process. Hence, we
18
This result is reminiscent of the results of Pesendorfer and Swinkels (2000), although their efficiency result as
N в†’ в€ћ reflects the dynamics of information acquisition in their common value setting in addition to the more
straightforward competitive forces at play here.
28
adopt a model of negotiation where, indeed, all dimensions of the product are discussed. The
following negotiation process is in the spirit of Manelli and Vincent (1995):
Negotiation consists of a sequence of take-it-or-leave-it offers. Each offer is an optimal screening
contract: it maximizes the buyer’s expected utility subject to the suppliers’ incentive compatibility constraints. The level of payment is adjusted to take into account that the contract
can be offered to another supplier in case the negotiation breaks down.
This definition requires several comments. First, to make it comparable to the other mechanisms
we need to impose that at least one buyer’s offer is always accepted. This means that the contract
offered to the last supplier (meaning all previous suppliers have rejected the offer from the buyer)
must satisfy that supplier’s individual rationality constraint. Second, the mechanism will provide
an upper bound to what negotiation can actually achieve. Indeed, it gives all bargaining power
to the buyer who can make take-it-or-leave it offers to each supplier sequentially, and it does not
restrict the number of partners the buyer negotiates with. Theorem 3 solves for the negotiation
equilibrium in our environment. [to be completed]
6
Extensions (to be written)
[Discussion of case Wa>Wc]
[Adding noise in quality realization]
7
Relationship with the multidimensional screening literature (to
be written)
8
Concluding remarks (to be written)
29
References
[1] Arizona
Department
of
Transport
(2002),
A+B
Bidding
Guide,
http://www.dot.state.az.us/roads/constgrp/A+BGuide.pdf (dated 1/18/2002).
[2] Armstrong, Mark (1996), Multiproduct Nonlinear Pricing, Econometrica, 64, 51-76.
[3] Armstrong, Mark (1999), Optimal Regulation with Unknown Demand and Cost Functions,
Journal of Economic Theory, 84, 196-215.
[4] Armstrong, Mark (2000), Optimal Multi-Object Auctions, Review of Economic Studies, 67,
455-481.
[5] Armstrong, Mark and Jean-Charles Rochet (1999), Multi-dimensional Screening: A User’s
Guide, European Economic Review, 43, 959-979.
[6] Asker, John and Estelle Cantillon (2004), Properties of Scoring Auctions, NYU Stern and HBS
mimeo.
[7] Avery, Chris and T. Henderschott (2000), Bundling and Optimal Auctions of Multiple Objects,
Review of Economic Studies, 67(3), 483-497.
[8] Bajari, Patrick, Robert McMillan and Steve Tadelis (2002), Auctions vs. Negotiation in Procurement: An Empirical Analysis, mimeo.
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Procurement Contracts, Rand Journal of Economics, 32(3), 287-307
[10] Border, Kim (1991), Implementation of Reduced Form Auctions: A Geometric Approach,
Econometrica, 59(4), 1175-1187.
[11] Branco, Fernando (1997), The Design of Multidimensional Auctions, Rand Journal of Economics, 28(1), Spring 1997, pp. 63-81
[12] Bulow, Jeremy and Paul Klemperer (1996), Auctions versus Negotiation, American Economic
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[13] Bushnell, James and Shmuel Oren (1994), Bidder Cost Revelation in Electric Power Auctions,
Journal of Regulatory Economics, 6(1), 5-26.
[14] Chao, Hung-Po and Robert Wilson (2002), Multidimensional Procurement Auctions for Power
Reserves: Incentive-compatible evaluation and settlement, Journal of Regulatory Economics,
forthcoming.
30
[15] Che, Yeon-Koo (1993), Design Competition through Multidimensional Auctions, Rand Journal
of Economics, 668-680.
[16] Dana, Jim (1993), The Organization and Scope of Agents, Journal of Economic Theory, 59,
288-310.
[17] Herbsman, Zohar, Wei Tong Chen and William C. Epstein (1995), Time is Money: Innovative Contracting Methods in Highway Construction, Journal of Construction Engineering and
Management, 121(3), 273-281.
[18] Jehiel, Philippe, Benny Moldovanu and Ennio Stachetti (1999), Multi-dimensional mechanism design for auctions with externalities, Journal of Economic Theory, 85(2), 258-94.Che,
Yeon-Koo (1993), Design Competition through Multidimensional Auctions, Rand Journal of
Economics, 668-680.
[19] Laffont, Jean-Jacques and Jean Tirole (1987), Auctioning Incentive Contracts, Journal of
Political Economy, 95(5), 921-937.
[20] Malakhov, Alexey and Rakesh Vohra (2004), Single and Multidimensional Optimal Auctions
- A Network Approach, MEDS mimeo.
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[22] Manelli, Alejandro and Daniel Vincent (2004), Multidimensional Mechanism Design: Revenue
Maximization and the Multiple-Good Monopoly, Arizona State mimeo.
[23] Morand, Pierre-Henri and Lionel Thomas (2002), Optimal Procurement Rules with Asymmetries on Quality and Efficiency, CRESE mimeo.
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[25] Rezende, Leonardo (2003), Biased Procurement, Stanford mimeo.
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31
[28] Wilson, Robert (2002), The Architecture of Power Markets, Econometrica, 70(4), 12991341.Malakhov, Alexey and Rakesh Vohra (2004), Single and Multidimensional Optimal Auctions - A Network Approach, MEDS mimeo.
32
9
Appendix A
For future reference, this appendix reproduces the optimization problem of the buyer with UB = 0
(Lemma 3) and with the subset of the IC constraints that happen to bind at the optimum.
max
{xk ,qk ,Uk }
О±A [xA W (qA ) в€’ UA ] + О±B xB W (qB ) + О±C [xC W (qC ) в€’ UC ] + О±D [xD W (qD ) в€’ UD ]
subject to:
UA ≥ xB ∆θ1
(IC 1)
UC ≥ UA − xA [WA (qA ) − WC (qA )]
(IC 2)
UC ≥ xB qB ∆θ2
(IC 3)
UD ≥ UA + xA qA ∆θ2
(IC 4)
UD ≥ UC + xC ∆θ1
P
P
N
αk xk ≤ 1 − (1 −
О±k )N for all subsets K of {A, B, C, D}
kв€€K
(IC 5)
(feasibility)
kв€€K
(We omit the non exclusion constraint). The associated Lagrangian is given by:
О±A [xA W (qA ) в€’ UA ] + О±B xB W (qB ) + О±C [xC W (qC ) в€’ UC ] + О±D [xD W (qD ) в€’ UD ]
(21)
+λ1 [UA − xB ∆θ1 ] + λ2 [UC − UA + xA (WA (qA ) − WC (qA ))]
+λ3 [UC − xB qB ∆θ2 ] + λ4 [UD − UA − xA qA ∆θ2 ] + λ5 [UD − UC − xC ∆θ1 ]
в€™
Вё
P
P
P
N
в€’ ОіK N
О±k xk в€’ 1 + (1 в€’
О±k )
kв€€K
kв€€K
(where О»i is the Lagrangian multiplier associated with IC constraint i, and Оі K is the multiplier
associated with feasibility constraint K). Figure 4 provides a graphical representation of these IC
constraints together with their associated multiplier. A dotted line means that a constraint may
bind at the optimum. A full line means it always binds.
A
О»4
D
О»1
О»2
О»5
B
О»3
C
Figure 4: Potentially binding IC constraints at
the optimum
33
The Kuhn-Tucker conditions of this program are standard. For future reference, we only reproduce
those with respect to Uk :
10
10.1
О»1 в€’ О»2 в€’ О»4 = О±A
(22)
О»2 + О»3 в€’ О»5 = О±C
(23)
О»4 + О»5 = О±D
(24)
Appendix B
Preliminaries
We first define the notation that we will be using for some of the xk variables when they take
specific values. When xA takes its maximum value conditional on D keeping priority in the contract
max is defined by the equation
allocation, we will denote it xmax
A . Formally, xA
Вў
ВЎ
+ О±D xFDB = 1 в€’ (О±C + О±B )N
N О±A xmax
A
By Border (1991), this implies the following allocation: When there is a type D, give the contract
corresponds to the value of
to D, if not, give priority to a type A if there is one. Conversely, xmin
C
C when A and D have priority over C (but maintains priority over B). Formally,
Вў
ВЎ
FB
+ О±C xmin
= 1 в€’ О±B N
N О±A xmax
A
C + О±D xD
Finally, x is defined such that xA = xC , that is
Вў
ВЎ
N (О±A + О±C )x + О±D xFDB = 1 в€’ О±B N
The proof of Theorem 1 uses the following result repeatedly:
Lemma 4: Suppose UA = xB ∆θ1 . (1) Suppose further that UCA ≥ UCB . Then, xC > xA if and
only if UDC > UDA . (2) Suppose now that UCA ≤ UCB . Then UDC > UDA when xC > xA .
Proof: The result follows directly from a comparison of UDA and UDC (when UCA ≥ UCB ) :
UDA = xA qA ∆θ2 + xB ∆θ1 UDC = xC ∆θ1 − xA ∆θ1 + xA qA ∆θ2 + xB ∆θ1
When UCB ≥ UCA , UDC = xC ∆θ1 + xB qB ∆θ2 . Since UCB > UCA is equivalent to xB [WA (qB ) −
WC (qB )] < xA [WA (qA ) в€’ WC (qA )], the condition implies UDC > UDA when xC > xA . QED.
Finally, a few words on the logic of our proof. We start from the buyer-optimal efficient mechanism
and adjust the qualities and probabilities of winning progressively in order to raise the buyer’s
34
expected utility until no further improvement is available. Along the way, new IC constraints often
become binding. The reason why we can be sure that we have reached a global maximum at the
point where no further improvement is available given the currently binding constraints lies in the
properties of the maximization problem (Lemma 1). Since the objective function is concave and
the constraints form a convex set, a local maximum is also a global maximum.
10.2
Proof of part I of Theorem 1: WA (q) − WC (q) > 0 i.e. ∆θ1 > q∆θ2
The binding constraints in the buyer-optimal first best mechanism are ICAB , ICCB and ICDC . The
buyer’s resulting expected utility is given by
О±A xA WA (qA ) + О±B xB [WB (qB ) в€’
+О±C xC [WC (qC ) в€’
О±A
О±C + О±D
∆θ1 −
qB ∆θ2 ]
О±B
О±B
(25)
О±D
∆θ1 ] + αD xD WD (qD )
О±C
Keeping the probabilities fixed at xk = xFk B , optimizing the q’s in (25) requires that only qB be
adjusted away from the first best and set equal to
2
= arg max{WB (qB ) в€’
qB
О±A
О±C + О±D
∆θ1 −
qB ∆θ2 }
О±B
О±B
(26)
This reduces the informational rents of C and D. From Lemma 4(1), we know that UDC > UDA as
long as UCB ≥ UCA . Hence, we need to consider only two scenarios:
2, U
Scenario 1: At qB
CB ≥ UCA , that is,
2
2
) в€’ WC (qB
)] ≤ xFAB [WA (q) − WC (q)]
xFBB [WA (qB
(27)
2.
In this case, all IC constraints remain satisfied as we decrease qB to qB
We now consider the optimization of the probabilities of winning. From (25) and the model assumptions, the virtual welfare associated with D is the largest. Moreover, the virtual welfare associated
with A is larger than that associated with B. Thus, we need to consider three cases depending on
the relative ranking of the virtual welfare of C with respect to the virtual welfares of B and A.
1. C Г‚ A Г‚ B : WC (q) в€’
О±D
αC ∆θ 1
2)в€’
≥ WA (q) > WB (qB
О±A
αB ∆θ 1
в€’
О±C +О±D 2
αB qB ∆θ 2
[Case 1.1.a]
The optimal probabilities of winning are xk = xFk B since the ranking of the virtual welfares
corresponds to the ranking of the first best welfares. All IC constraints are satisfied given
the arguments above and the x’s and q’s are optimized given the binding constraints; qA = q,
2 and q = q = q.
qB = qB
C
D
35
2 ) − αA ∆θ − αC +αD q 2 ∆θ
2. A Г‚ C Г‚ B : WA (q) > WC (q) в€’ О±О±D
∆θ1 ≥ WB (qB
1
2
B
О±B
О±B
C
[Cases 1.1.b
& c]
In this case, the buyer would rather give the contract to supplier A than to supplier C. When
we increase xA and decrease xC concurrently (keeping О±A xA + О±C xC + О±D xFDB constant) all
IC constraints remain satisfied, as long as xA < xC . Note that: (1) the incentives for C to
imitate A have actually decreased; (2) Lemma 4(1) continues to apply so ICDA is satisfied;
(3) the incentives for A to imitate C and D have decreased; and (4) B continues to have no
incentive to imitate B, C or D.
However, when xA > xC , ICDA will bind if
2
2
max
) в€’ WC (qB
)] ≥ xmin
xFBB [WA (qB
C ∆θ 1 − xA q∆θ 2
(a) (28) is not satisfied.
(28)
[Case 1.1.b]
This means that UDC ≥ UDA even when xA reaches its maximum. The solution is thus:
2, q = q
FB
max > xmin > x = xF B . All IC
qA = q, qB = qB
C
D = q and xD = xD > xA
B
C
B
min
max
constraints are satisfied (UCB > UCA since xmax
A [WA (q)−WC (q)] > xC ∆θ 1 −xA ∆θ 2 q
2 ) − W (q 2 )] by (28)) and the x’s and q’s are optimized given the binding
≥ xFBB [WA (qB
C B
constraints and the resulting virtual welfares.
(b) (28) is satisfied.
[Case 1.1.c]
Hence ICDA binds as we increase xA and decrease xC (this happens at xA > xC ).19 To
further increase xA while keeping UDA = UDC requires that we adjust qualities. For a
given pair of xA and xC , let qA and qB solve:
О»4
qA ∆θ2 }
О±A
О±A + О»4
О±C + О±D в€’ О»4
= arg max{WB (qB ) в€’
∆θ1 −
∆θ2 qB }
О±B
О±B
qA = arg max{WA (qA ) в€’
(29)
qB
(30)
where λ4 is such that xFBB [WA (qB ) − WC (qB )] = xC ∆θ1 − xA qA ∆θ2 . (These expressions
are derived using (21) to (24)) As xA increases, О»4 increases (so that qA decreases and
qB increases). This process increases the virtual welfare associated with C, WC (q) в€’
О±D в€’О»4
αC ∆θ 1 ,
and decreases the virtual welfare associated with A (see (29)). The process
continues until either the upper bound to xA , xmax
A , is reached or the virtual welfares
become equal:
max{WA (qA ) в€’
qA
19
О»в€—4
О±D в€’ О»в€—4
qA ∆θ2 } = WC (q) −
∆θ1
О±A
О±C
B
FB
2
2
Indeed, when xA = xC , xC ∆θ1 −xA q∆θ2 = xA [WA (q)−WC (q)] > xF
A [WA (q)−WC (q)] ≥ xB [WA (qB )−WC (qB )]
by (27).
36
whichever comes first. Thus О»4 в€€ (0, О»в€—4 ) вЉ‚ (0, О±D ) as required by (24).
≥ xA > xC ≥ xmin
> xB , qD = qC = q
This defines the solution: xD = xFDB > xmax
A
C
and qA and qB defined by (29) and (30), qA , qB < q. The x’s are optimized given the q’s
and the binding constraints. The q’s are optimized given the binding constraints. All IC
constraints remain satisfied. In particular, UCA < UCB since xFBB [WA (qB ) в€’ WC (qB )] =
xC ∆θ1 − xA qA ∆θ2 < xA [WA (qA ) − WC (qA )] when xA > xC .
2)в€’
3. A Г‚ B Г‚ C : WA (q) > WB (qB
О±A
αB ∆θ 1
в€’
О±C +О±D 2
αB qB ∆θ 2
> WC (q) в€’
О±D
αC ∆θ 1 .
[case 1.1.d]
In this case, the ideal ordering of types in the allocation is D Г‚ A Г‚ B Г‚ C. As we decrease
xC , first to the benefit of A (since she generates a higher virtual welfare), then B, we reach a
if condition (28)
point where ICDA becomes binding. It can happen before xA reaches xmax
A
and xC > xB > xFBB if condition (28) does not hold. Once this
holds, or when xA = xmax
A
happens, any change in the probabilities of winning requires some adjustment of qualities to
keep UDC = UDA . The resulting virtual welfares associated with A, B and C are:
О»4
∆θ2 qA }
О±A
О±A
О±C + О±D
О»4
max{WB (qB ) в€’
∆θ1 −
∆θ2 qB +
(∆θ2 qB − ∆θ1 )}
О±B
О±B
О±B
О±D в€’ О»4
∆θ1
WC (q) в€’
О±C
max{WA (qA ) в€’
(31)
(32)
(33)
where λ4 ∈ (0, αD ) is such that UDC = UDA i.e. xB [WA (qB ) − WC (qB )] = xC ∆θ1 − xA qA ∆θ2
for the current values of xC (xA and xB are well-defined once xC is defined given that the
allocation should go in priority to A over B). As О»4 increases, the virtual welfare associated
with A decreases, and that associated with B and C increases. Moreover, by the assumption
on the hazard rates (assumption 1) and the fact that ∆θ1 > ∆θ2 q, the virtual welfare associated with A is always larger than that associated with B. Hence the optimized probabilities
of winning are:
the virtual welfare associated with B is higher than C, the solution is
(a) If at xA = xmax
A
such that xA = xmax
and О»4 equates the virtual welfare of B and C. Thus xmin
> xC
A
C
and xB > xFBB
(b) If not, then the solution corresponds to the value of О»4 such that xA = xmax
or the
A
virtual welfare of A equals that of C, whichever comes first as xA is increased from xFAB .
This corresponds to solution 1.1.c.
All IC constraints are satisfied by construction. The qualities are optimized given the binding
37
constraints and the value of О»4 . The probabilities of winning are optimized given the resulting
virtual welfares.
2, U
FB
2
2
FB
Scenario 2: At qB
CB < UCA , that is, xB [WA (qB ) в€’ WC (qB )] > xA [WA (q) в€’ WC (q)].
In this case, ICCA becomes binding as we decrease qB . To reduce C and D’s rents further, one now
needs to decrease qA at the same time as qB in such a way that UCB = UCA , i.e., xFBB [WA (qB ) в€’
WC (qB )] = xFAB [WA (qA ) в€’ WC (qA )]. This increases the cost of rent extraction (note, the equality
UCB = UCA requires
dqA
dqB
=
B
xF
B
B
xF
A
< 1 so that qA > qB ). Formally, using (21) to (24) in Appendix A,
we let qA and qB solve:
О»в€—2
[WA (qA ) в€’ WC (qA )]}
qA
О±A
О±A + О»в€—2
(О±C + О±D в€’ О»в€—2 )
max{WB (qB ) в€’
∆θ1 −
∆θ2 qB }
qB
О±B
О±B
max{WA (qA ) +
(34)
(35)
for the value of О»в€—2 в€€ (0, О±C + О±D ) such that xFBB [WA (qB ) в€’ WC (qB )] = xFAB [WA (qA ) в€’ WC (qA )]
2 so that xF B [W (q 2 ) в€’ W (q 2 )] >
(Such value always exists. When О»в€—2 = 0, qA = q and qB = qB
A B
C B
B
2 < q = q
xFAB [WA (q) в€’ WC (q)] from the definition of scenario 2. When О»в€—2 = О±C + О±D , qA = qA
B
2 ) в€’ W (q 2 )]).
and xFBB [WA (q) в€’ WC (q)] < xFAB [WA (qA
C A
From the first best benchmark, only the rents of C and D have decreased. The IC constraint of C is
taken care of by construction, and UDC > UDA follows from Lemma 4(1). Hence, all IC constraints
remain satisfied.
We now optimize over the x’s. Notice that maxqA {WA (qA ) +
О»в€—2
О±A [WA (qA )
в€’ WC (qA )]} > WA (q).
Hence we need to consider three cases depending on the relative ranking of the virtual welfare
associated with C.
1. C Г‚ A Г‚ B : WC (q) в€’
О±D
αC ∆θ 1
≥ maxqA {WA (qA ) +
О»в€—2
О±A [WA (qA ) в€’ WC (qA )]}
[Case 1.2.a]
The optimal probabilities are thus xk = xFk B . The values of qA and qB are defined in (34) and
2 , q = q = q.
(35) and q > qA > qB > qB
C
D
2. A Г‚ C Г‚ B : maxqA {WA (qA ) +
О±A
αB ∆θ 1
в€’
О±C +О±D 2
αB qB ∆θ 2
О»в€—2
О±A [WA (qA )
в€’ WC (qA )]} > WC (q) в€’
[Cases 1.2.b,c,d & e]
О±D
αC ∆θ 1
2)в€’
≥ WB (qB
At the current value of О»2 , the buyer prefers to give the contract to A over C. As we progressively increase xA at the expense of xC , while keeping UCA = UCB , we decrease О»2 (i.e.
increase qA and decrease qB - from (34) and (35)).
38
В© 1 2 ВЄ
в€—
Define О»в€—в€—
2 в€€ [0, О»2 ) as max О»2 , О»2 , 0 where:
О»12 is defined s.t.:
О»22 is defined s.t.:
О»12
О±D
[WA (qA ) в€’ WC (qA )]} = WC (q) в€’
∆θ1 (36)
qA
О±A
О±C
xFBB [WA (qB ) в€’ WC (qB )] = xmax
(37)
A [WA (qA ) в€’ WC (qA )]
max{WA (qA ) +
(qA and qB are again defined implicitly by О»12 and О»12 through (36) and (??). We need to
consider four sub-cases.
1
в€—в€—
(a) О»в€—в€—
2 = О»2 and xA < xC at О»2
[Case 1.2.b]
At the solution, the binding constraints are ICAB , ICCB , ICCA and ICDC . The probabilities of winning are: xD = xFDB > xFC B > xC > xA > xFAB > xB = xFBB where
xA and xC are defined by xFBB [WA (qB ) в€’ WC (qB )] = xA [WA (qA ) в€’ WC (qA )] at О»в€—в€—
2 and
the feasibility constraint; qA and qB are defined by (34) and (35) where О»в€—2 should be
replaced by О»в€—в€—
2 , and qC = qD = q. The x’s are optimized given the q’s and the binding
constraints (the buyer is indifferent between A and C). The q’s are optimized given the
binding constraints and the value of О»2 . All IC constraints are satisfied (UDC > UDA by
Lemma 4 and xA < xC ).
1
в€—в€—
в€—в€—
2
(b) О»в€—в€—
2 = О»2 and xA < xC at О»2 or О»2 = О»2
[Case 1.2.c]
This means that we reach xA = xC at a value of О»2 > О»в€—в€—
2 , at which point ICDA becomes
binding. The solution is such that ICAB , ICCB , ICCA , ICDC , ICDA are binding. Based
on the expressions from (21), reworked using the equalities (22) to (24), the associated
virtual welfares are given by:
(О±C + О±D в€’ О»3 )
О±C + О»5 в€’ О»3
∆θ2 qA +
∆θ1 }
О±A
О±A
О»3
О±A + О±C + О±D в€’ О»3
max{WB (qB ) в€’
∆θ2 qB −
∆θ1 }
qB
О±B
О±B
О»5
WC (q) в€’
∆θ1
О±C
max{WA (qA ) в€’
qA
(38)
(39)
(40)
There exists values for О»3 and О»5 such that (1) x[WA (qA ) в€’ WC (qA )] = xFBB [WA (qB ) в€’
D в€’О»3 )
5 в€’О»3
∆θ2 qA + αC +λ
∆θ1 } = WC (q)− αλC5 ∆θ1
WC (qB )] and (2) maxqA {WA (qA )в€’ (О±C +О±
О±A
О±A
so that the buyer is indifferent between supplier C and supplier A.
The progressive adjustment of xA until xA = xC implies that there exists a value for О»3
that satisfies condition (1) . Once О»3 is fixed, there is a value of О»5 that ensures condition
(2). Indeed for any feasible О»3 , when О»5 = 0, the virtual welfare of C is greater. When
О»5 = О±D , so that О»2 = О±C + О±D в€’ О»3 , from the fact when ICDA becomes just binding
О»2 > О»в€—в€—
2 , so that the virtual welfare of A is larger.
39
The solution is thus: xD = xFDB > xFC B > xC = x = xA > xFAB > xB = xFBB and the q’s
solving (38) through (40) above for the values of О»3 and О»5 that satisfy conditions (1)
and (2). The x’s are optimized given the binding constraints (the buyer is indifferent
between A and C). Similarly, the q’s are optimized given the binding constraints. All
IC constraints are satisfied.
min
max
FB
2
2
(c) О»в€—в€—
2 = 0 and xC ∆θ 1 − xA ∆θ 2 q ≥ xB [WA (qB ) − WC (qB )]
[Case 1.2.d]
The solution is as in case 1.1.b.
min
max
FB
2
2
(d) О»в€—в€—
2 = 0 and xC ∆θ 1 − xA ∆θ 2 q < xB [WA (qB ) − WC (qB )]
[Case 1.2.e]
The solution is as in case 1.1.c.
3. A Г‚ B Г‚ C : WC (q) в€’
О±D
αC ∆θ 1
2)в€’
< WB (qB
О±A
αB ∆θ 1
в€’
О±C +О±D 2
αB qB ∆θ 2
[Case 1.2.f]
The solution is as in case 1.1.d.
10.3
Proof of part II of Theorem 1: WA (q) − WC (q) < 0 i.e. ∆θ1 < q∆θ2
The binding constraints in the buyer-optimal first best mechanism are ICAB , ICCA and ICDC . The
buyer’s resulting expected utility is given by
О±C + О±D
О±C + О±D
О±A + О±C + О±D
∆θ1 −
qA ∆θ2 ] + αB xB [WB (qB ) −
∆θ1 ]
О±A
О±A
О±B
О±D
∆θ1 ] + αD xD WD (qD )
(41)
+О±C xC [WC (qC ) в€’
О±C
О±A xA [WA (qA ) +
Keeping the probabilities fixed at xk = xFk B , optimizing the q’s requires that qA be set equal to
2
qA
= arg max{WA (qA ) +
О±C + О±D
О±C + О±D
∆θ1 −
qA ∆θ2 }
О±A
О±A
(42)
This reduces the informational rents of C and D. By Lemma 4(1), we know that UDC > UDA as
long as UCA ≥ UCB . Hence, we need to consider only two scenarios:
2, U
FB
2
2
FB
20
Scenario 1: At qA
CA ≥ UCB , i.e., xA [WA (qA ) − WC (qA )] ≤ xB [WA (q) − WC (q)]
2 . We now consider the
In this case, all IC constraints remain satisfied as we decrease qA to qA
optimization of the probabilities of winning. From (41), the virtual welfare associated with D is
the largest, and assumption 1 ensures that the virtual welfare associated with C is larger than that
associated with B. We need to consider three subcases:
1. C Г‚ A Г‚ B : [WC (q) в€’
О±A +О±C +О±D
∆θ1 ]
О±B
20
О±D
αC ∆θ 1 ]
2) +
> [WA (qA
[Case 2.1.a]
2
2
This implies WA (qA
) в€’ WC (qA
) < 0.
40
О±C +О±D
αA ∆θ 1
в€’
О±C +О±D 2
αA qA ∆θ 2 ]
> [WB (q) в€’
The optimal probabilities of winning are xk = xFk B since the ranking of the virtual welfares
corresponds to the ranking of the first best welfares. All IC constraints are satisfied, and the
x’s and q’s are optimized given the binding constraints.
2)+
2. A Г‚ C Г‚ B : [WA (qA
О±C +О±D
αA ∆θ 1
в€’
О±C +О±D 2
αA qA ∆θ 2 ]
> [WC (q) в€’
О±D
αC ∆θ 1 ]
[Case 2.1.b]
The buyer would like to increase xA at the expense of xC . Doing this does not affect the
2 ) − W (q 2 )] ≤ xF B [W (q) − W (q)] and
inequality UCA ≥ UCB since it requires xA [WA (qA
C A
A
C
B
2 ) − W (q 2 ) < 0. Initially, it does not affect the relative ranking of the virtual welfares
WA (qA
C A
associated with C and A either. When we reach xA = xC , UDA = UDC since UDC = xC ∆θ1 −
2 ∆θ + x ∆θ and U
2
xA ∆θ1 + xA qA
2
1
B
DA = xA qA ∆θ 2 + xB ∆θ 1 . The solution is such that xA =
xC = x (xD = xFDB and xB = xFBB ) with О»5 в€€ (0, О±D ) chosen to equate the virtual welfare
associated with A and C:
2
)+
WA (qA
(О±C + О»5 )
О±C + О±D
О»5
2
∆θ1 −
∆θ2 qA
= WC (q) в€’
∆θ1
О±A
О±A
О±C
(43)
(from (21) to (24)). Such a value for О»5 exists. When О»5 = 0, the virtual welfare associated
with C is the first best welfare. When О»5 the virtual welfare of A is bigger by assumption.
2 (from (43)), q = q and q = q = q. The qualities are optimized given the
We get qA = qA
B
C
D
value of the multipliers and the binding constraints. The x’s are optimized given the resulting
virtual welfares. All IC constraints are satisfied.
3. C Г‚ B Г‚ A : [WB (q) в€’
О±A +О±C +О±D
∆θ1 ]
О±B
2)+
> [WA (qA
О±C +О±D
αA ∆θ 1
2.1.c]
в€’
О±C +О±D 2
αA qA ∆θ 2 ]
[Case
In this case, the buyer would like to increase xB at the expense of xA . As we increase xB and
2 ) в€’ W (q 2 )] = x [W (q) в€’ W (q)], that is,
decrease xA , we reach a point where xA [WA (qA
C A
B
A
C
2 ) в€’ W (q 2 ) < 0.21 The
UCB = UCA . At that point xA > xB since [WA (q) в€’ WC (q)] < WA (qA
C A
solution is determined by the value of О»2 в€€ (0, О±C + О±D ) that equates the virtual welfares of
A and B :
max{WA (qA ) +
qA
О»2
О»2
О±A + О»2
О±C + О±D в€’ О»2
∆θ1 −
∆θ2 qA } = max{WB (qB ) −
∆θ1 −
∆θ2 qB }
qB
О±A
О±A
О±B
О±B
(44)
(from (21) to (24). Such value for О»2 exists since the virtual welfare of A is larger than
that of B at О»2 = 0, and smaller at О»2 = О±C + О±D ). By inspection of (44), this happens at
О±C +О±D в€’О»2
О±B
<
О»2
О±A
2 < q < q . Finally, we require
that is, the resulting q’s are such that qA
A
B
that UCA = UCB , that is, xA [WA (qA ) в€’ WC (qA )] = xB [WA (qB ) в€’ WC (qB )] which implies that
21
Remember also that IC compatibility (Lemma 3) requires that xA ≥ xB .
41
xA > xB . The other variables are set such that xD = xFDB , xC = xFC B , and qC = qD = q. The
q’s are optimized given the values of the multipliers and the binding constraints. The x’s are
optimized given the resulting virtual welfares. All IC constraints are satisfied (ICDA satisfied
given Lemma 4(1)).
2, U
FB
2
2
FB
Scenario 2: At qA
CB > UCA that is, xA [WA (qA ) в€’ WC (qA )] > xB [WA (q) в€’ WC (q)]
2 . To further decrease the rents to
In this case, ICCB becomes binding as we decrease qA towards qA
C and D, we now need to both decrease qA and qB , holding UCB = UCA constant. This increases
2 < q < q and q 2 < q < q. Formally, the optimal q’s are
the cost of rent extraction, leading to qA
A
B
B
defined by
О»2
О»2
∆θ1 −
∆θ2 qA }
О±A
О±A
О±A + О»2
О±C + О±D в€’ О»2
= arg max{WB (qB ) в€’
∆θ1 −
∆θ2 qB }
О±B
О±B
в€—
= arg max{WA (qA ) +
qA
(45)
в€—
qB
(46)
в€— ) в€’ W (q в€— )] = xF B [W (q в€— ) в€’ W (q в€— )].
where О»2 в€€ (0, О±C + О±D ) is chosen such that xFAB [WA (qA
C A
A B
C B
B
All other IC constraints remain satisfied.
We now consider the optimization of the probabilities of winning. From (41) and the model assumption, the virtual welfare associated with D is the largest, and assumption 1 ensures that the
virtual welfare associated with C is larger than that associated with B. We need to consider three
cases:
1. C Г‚ A Г‚ B : WC (q) в€’
О±C +О±D в€’О»2
в€—.
∆θ2 qB
О±B
О±D
αC ∆θ 1
в€—)+
> WA (qA
[Case 2.2.a]
О»2
αA ∆θ 1
в€’
О»2
в€—
αA ∆θ 2 qA
в€—)в€’
> WB (qB
О±A +О»2
αB ∆θ 1
в€’
The optimal probabilities of winning are xk = xFk B since the ranking of the virtual welfares
corresponds to the ranking of the first best welfares. The solution is as in case 1.2.a except
2 , q) since W (q) в€’ W (q) < 0.
that qB в€€ (qB
A
C
в€—)+
2. A Г‚ C Г‚ B : WA (qA
О»2
αA ∆θ 1
в€’
О»2
в€—
αA ∆θ 2 qA
> WC (q) в€’
О±D
αC ∆θ 1 .
The buyer would like to increase xA at the expense of xC . Doing this while keeping UCA = UCB
corresponds to an increase in О»2 relative to value defined in (45) and (46) (and therefore an
decrease in qA and an increase in qB ). This also decreases the virtual welfare associated with
A. We need to consider two subcases:
(a) [Case 2.2.b]
There exists a value for λ2 such that max{WA (qA )+ αλA2 ∆θ1 − αλA2 ∆θ2 qA } =
2 , q).
∆θ1 and xC > xA . Then solution is as in case 1.2.b except that qB ∈ (qB
WC (q) в€’ О±О±D
C
42
(b) [Case 2.2.c]
When xA = xC , max{WA (qA ) + αλA2 ∆θ1 − αλA2 ∆θ2 qA } > WC (q) − ααD
∆θ1 .
C
2 , q). Note that we never need to
The solution is as in case 1.2.c except that qB в€€ (qB
consider cases analogous to cases 1.2.d or 1.2.e since increasing xA increases О»2 , hence
О»2 > 0.
∗ )− αA +λ2 ∆θ − αC +αD −λ2 ∆θ q ∗ > W (q ∗ )+ λ2 ∆θ − λ2 ∆θ q ∗ .
3. C Г‚ B Г‚ A : WB (qB
1
2 B
1 О±A
2 A
A A
О±B
О±B
О±A
[Case
2.2.d]
The solution is as in case 2.1.c, noting that we arrive there by increasing xB until the virtual
welfares from A and B are equal. This occurs before xA = xB since at this point О»2 =
and qA = qB resulting in the virtual welfare from A being greater than that from B.
43
О±C +О±D
О±B
11
Appendix C: The optimal quasi-linear scoring auction (for the
referees only, not for publication)
11.1
Case 1: WA (q) в€’ WC (q) > 0 [done and coded - exposition code-oriented]
To solve for the optimal quasi-linear scoring auction, we use Theorem 2 and impose the two first
constraints, specifically, qA = qB and qC = qD , and О±A xA +О±C xC = О±A xFAB +О±C xFC B on the original
problem. Neither constraint changes the properties of the problem, which remains concave. Ex-post
we check that constraints (3) and (4) of Theorem 2 are also satisfied by the solution and adjust if
necessary.
As before, we start from the benchmark of the buyer-optimal efficient mechanism. When
WA (q) − WC (q) > 0, the binding constraints in the buyer-optimal efficient mechanism are ICDC ,
ICAB and ICCB , with the resulting expression for the buyer’s expected utility:
αA xFAB WA (qA ) − αA xFBB ∆θ1 + αB xFBB WB (qB ) + αC xFC B WC (qC ) − αC xFBB qB ∆θ2
+αD xFDB WD (qD ) − αD xFC B ∆θ1 − αD xFBB qB ∆θ2
(47)
Eye-balling this expression suggests that the optimal (constrained) qualities are qC = qD = q and
ВЅ
Вѕ
(О±C + О±D )xFBB
в€—
2
∆θ2 q ∈ (qB
, q)
(48)
qA = qB = qB = arg max v(q) в€’ Оё2 q в€’
F
B
F
B
О±A xA + О±B xB
Notice that, unlike before, the choice of quality is a function of the probabilities of winning. This
choice of quality does not affect any of the IC constraints. The relevant constraints to check are
ICDA and ICCA since only the rents of C and D are affected by the quality adjustment. Since
xFC B > xFAB , ICDA will be satisfied if ICCA is satisfied (Lemma 4(1)). UCB > UCA requires
в€— ) в€’ W (q в€— )] < xF B [W (q в€— ) в€’ W (q в€— )], which is satisfied. We next optimize the
xFBB [WA (qB
C B
A B
C B
A
winning probabilities.
1. If WC (q) в€’
О±D
αC ∆θ 1
в€— ), the first best winning probabilities are optimal. Theorem 2
≥ WA (qB
guarantees the existence of a scoring auction.
в€— ) > W (q) в€’
2. If WA (qB
C
О±D
αC ∆θ 1 ,
the buyer wants to increase xA . Changing the probabilities of
в€—
winning affects qB . Consider (48), increasing xA , while keeping the same IC constraints bind∗ (but it remains lower than q). Thus, IC
ing, increases qB
CB remains satisfied and WA remains
higher than the virtual welfare associated with C. Define qbв€— , the quality that corresponds to
in (48).
xA = xmax
A
max b∗ ∆θ , the change in the probabilities of
(a) If xFBB [WA (b
q в€— ) в€’ WC (b
q ∗ )] ≤ xmin
2
C ∆θ 1 − xA q
winning does not affect the binding constraints. The solution is thus qA = qB = qb∗ ,
44
min
qC = qD = q and xA = xmax
A , xC = xC . Theorem 2 guarantees that the result can be
implemented by a quasi-linear scoring auction.
max b∗ ∆θ , IC
q в€— ) в€’ WC (b
q в€— )] > xmin
(b) If instead, xFBB [WA (b
1
DA starts binding in the
C ∆θ 1 − xA q
process of adjusting xA . We need to reoptimize the buyer’s problem accounting for the
fact that the expression for the buyer’s maximum expected utility is no longer (47). Note
cannot be a solution because when
that ICDA , ICAB and ICCB binding and xA = xmax
A
these constraints are binding, the virtual welfare associated with C is larger. Hence, we
are looking for a solution where both ICDA and ICDC are binding. Formally, the buyer’s
expected utility is given by:
в€™
Вё
в€™
Вё
О»4
О±A + О»4
(О±C + О±D в€’ О»4 )
FB
q∆θ2 + αB xB WB (q) −
∆θ1 −
∆θ2 q
О±A xA WA (q) в€’
О±A
О±B
О±B
в€™
Вё
(О±D в€’ О»4 )
+О±C xC WC (q) в€’
∆θ1 + αD xFDB WD (q)
О±C
where we have used the fact that О»4 + О»5 = О±D . The first order condition with respect
to q is
vq (q) = Оё2 +
(О±C + О±D в€’ О»4 )xFBB + О»4 xA
∆θ2
(О±A xA + О±B xFBB )
(49)
As О»4 , the multiplier on the ICDA constraint, increases, the virtual welfare associated
with A decreases and the virtual welfare associated with C increases. The process
or the virtual welfares of A and C become equal, whichever
stops when xA = xmax
A
comes first. (The process, together with the expressions for the virtual welfares, also
ensure that there exists О»4 в€€ (0, О±D ) that satisfies this condition). The solution is thus
xA > xFAB , xC < xFC B , qC = qD = q and qA , qB solving (49). Theorem 2 guarantees the
implementation by a quasi-linear scoring auction.
b4 such that ICDA = ICDC at xmax
Algorithm: Solve for qb and О»
A
FB
max
(xmin
− xFBB )∆θ2 q
C − xB )∆θ 1 = (xA
b4 is given from the solution to (49) when xA = xmax .
and О»
A
If at that value of О»4 , the virtual welfare associated with A is still larger than C then
b4 ) and it solves
this is the solution. Otherwise, the solution is such that О»4 в€€ (0, О»
WC (q) в€’
(О±D в€’ О»4 )
О»4
∆θ1 = WA (q) −
∆θ1
О±C
О±A
where xA and q в€—в€— solve (49) and UDA = UDC for the value of О»4 . Practically,
45
(50)
i. For any candidate value of xA (xA < xmax
A ), we solve for the value of q that ensures
that UDA = UDC :
xC =
q =
(О±A xFAB + О±C xFC B в€’ О±A xA )
О±C
F
B
(xC − xB )∆θ1
(xA − xFBB )∆θ2
ii. For that value of q, we use (49) to solve for the corresponding value of О»4 . With
v(q) = aq b this yields:
О»4 (xA в€’ xFBB ) = (abq bв€’1 в€’ Оё2 )
(О±A xA + О±B xFBB )
в€’ (О±C + О±D )xFBB
∆θ2
iii. We check whether the virtual welfares are equal (cf. (50)), if not we readjust the
xA and continue.
11.2
Case 2: WA (q) в€’ WC (q) < 0
When WA (q) − WC (q) < 0, the binding constraints in the buyer-optimal efficient mechanism are
ICDC , ICCA and ICAB . The resulting expression for the buyer’s expected utility is
О±C + О±D
О±C + О±D
О±A + О±C + О±D
∆θ1 −
qA ∆θ2 ] + αB xFBB [WB (qB ) −
∆θ1 ]
О±A
О±A
О±B
О±D
+О±C xFC B [WC (qC ) в€’
∆θ1 ] + αD xFDB WD (qD )
О±C
О±A xFAB [WA (qA ) +
Eye-balling this expression suggests that the optimal constrained qualities are qC = qD = q and
в€—
= arg max{v(q) в€’ Оё2 q в€’
qA = qB = qA
(αC + αD )xFAB ∆θ2
2
q} в€€ (qA
, q)
F
B
F
B
(О±A xA + О±B xB )
(51)
в€— is a function of the probabilities of winning. Note also that q в€— < q в€— ). This
(again note that qA
A
B
adjustment reduces the rents of C and D. By Lemma 4(1), we only need to worry about ICCB .
в€— ) в€’ W (q в€— ) < 0
1. Scenario 1: WA (qA
C A
No new constraint is binding as a result of the quality adjustment: UCA > UCB requires that
∗ ) − W (q ∗ )] ≥ xF B [W (q ∗ ) − W ∗ (q ∗ )] which is satisfied under the assumption of
xFBB [WA (qA
A A
A
A
A
scenario 1.
∗ )+ αC +αD ∆θ − αC +αD q ∗ ∆θ ], the first best win∆θ1 ] ≥ [WA (qA
(a) [coded] If [WC (q)в€’ О±О±D
1
2
A
О±A
О±A
C
ning probabilities are optimal. Theorem 2 ensures that the outcome can be implemented
by a quasi-linear scoring rule.
46
(b) [never applied] If [WC (q) в€’
О±D
αC ∆θ 1 ]
в€—) +
< [WA (qA
О±C +О±D
αA ∆θ 1
в€’
О±C +О±D в€—
αA qA ∆θ 2 ],
the
в€— (cf. (51)) and thus
buyer prefers to give priority to A over C. Increasing xA decreases qA
в€— approaches q 2 in the process.
increases the virtual welfare associated with A since qA
A
It may also affect which IC constraint binds. Specifically, two new constraints could
start binding: (1) ICDA starts binding as soon as xA = xC or (2) ICCB starts binding if
в€— ) в€’ W (q в€— ) reaches zero. Let x
eA the level of xA at which this happens.
WA (qA
C A
For future reference, we write the expression for the buyer’s expected utility when ICAB ,
ICCB , ICCA , ICDA and ICDC are all potentially binding. We use equations (22) to (24)
to reduce the number of multipliers to two: О»2 and О»4 .
в€™
Вё
О»2
(О»2 + О»4 )
О±A xA WA (qA ) +
∆θ1 −
∆θ2 qA
О±A
О±A
в€™
Вё
(О±A + О»2 + О»4 )
(О±C + О±D в€’ О»2 в€’ О»4 )
+О±B xB WB (qB ) в€’
∆θ1 −
qB ∆θ2
О±B
О±B
в€™
Вё
(О±D в€’ О»4 )
+О±C xC WC (qC ) в€’
∆θ1 + αD xD WD (qD )
О±C
i. x
eA ≤ x so that ICCB first become binding. The solution is of two kinds, depending
on the parameters: either ICAB , ICCB , ICCA , ICDA , and ICDC are all binding, or
only ICAB , ICDC , ICDA and ICCB bind.
All constraints are binding ⇒ xA = x and ∆θ1 = ∆θ2 qe. λ2 and λ4 satisfy
(О»2 + О»4 )x + (О±C + О±D в€’ О»2 в€’ О»4 )xFBB
∆θ(52)
2
О±A x + О±B xFBB
О»4
= WA (e
q) в€’
∆θ1
(53)
О±A
q ) = Оё2 +
vq (e
WC (q) в€’
(О±D в€’ О»4 )
∆θ1
О±C
A value for О»4 в€€ (0, О±D ) always exists that satisfies (53). Indeed, when О»4 = О±D ,
the virtual welfare of C is higher. When О»4 = 0, the right hand-side is larger than
∗ ) + αC +αD ∆θ − αC +αD q ∗ ∆θ since q
2 , q в€— ). Thus, from the scenario ase в€€ (qA
WA (qA
1
2
A
A
О±A
О±A
sumption, it is larger than the virtual welfare of C. Similarly, a value for (О»2 + О»4 ) в€€
(0, О±C +О±D ) that satisfies (52) always exists. At zero, it yields a value for the quality
∗ > q ∗ thus ∆θ −∆θ q < 0. At α +α , ∆θ −∆θ q < 0 from the ashigher than qB
1
2
1
2
A
D
A
sumption of this subcase. However, it is not guaranteed that the value of О»2 compati-
ble with both the value of О»4 from (53) and the value of О»2 +О»4 that solves (52) is positive. When it is not, the solution is such that only ICAB , ICDC , ICDA and ICCB bind.
47
The solution is such that:
∆θ1 − ∆θ2 q > 0
(ensuring that ICCA is not binding)
xA ∆θ2 q + xFBB ∆θ1 = xC ∆θ1 + xFBB ∆θ2 q (ensuring that UDA = UDC )
О»4 xA + (О±C + О±D в€’ О»4 )xFBB
vq (q) = Оё2 +
∆θ2
О±A xA + О±B xFBB
(О±D в€’ О»4 )
О»4
WC (q) в€’
∆θ1 = WA (q) −
∆θ2 q
О±C
О±A
In both cases, Theorem 2 ensures that the solution can be implemented by scoring
auction.
Algorithm:
1. Solve for qe and qb. If qe < qb then this means that x
eA ≥ x so we are in case i.
Otherwise we are in case ii.
2. case i straightforward to solve
3. case ii: set xA = x and qe. Solve for О»4 and for О»2 + О»4 using (??) and (53). If
implied λ2 ≥ 0, this is the solution. STOP
4. ElseTake any q such that ∆θ1 − ∆θ2 q > 0
5. Solve (analytically) for the corresponding xA that ensures UDA = UDC
6. Solve (analytically) for the corresponding О»4 that makes the virtual welfares equal
7. Check whether the equation vq (q) = ... holds. If not, pick another q.
ii. x
eA > x so that ICDA first becomes binding. The solution is then one where ICAB ,
ICCA , ICDA and ICDC are binding (i.e. О»2 + О»4 = О±C + О±D ) and xA = xC = x. Let
b4 в€€ (0, О±D ) solve
qb and О»
q ) = Оё2 +
vq (b
WC (q) в€’
(О±C + О±D )x
О±A x + О±B xFBB
b4 )
b4 )
(О±D в€’ О»
(О±C + О±D в€’ О»
О±C + О±D
∆θ1 = WA (b
q) +
∆θ1 −
qb∆θ2
О±C
О±A
О±A
b4 always exist.22 The buyer is indifferent between A and C,
Such values for qb and О»
the qualities are optimized given the value of the multipliers and all IC constraints
are satisfied (ICCB is not binding).
q ) в€’ WA (b
q ) < 0 and xA > xFAB , this solution is not
However, note that since WA (b
implementable as a quasi-linear scoring auction (Theorem 2). The solution is either
в€— , or the solution to the original program to which
xA = xFAB and qA = qB = qA
22
2
Indeed, the virtual welfare of C is higher when О»4 = О±D , and, by assumption and the fact that q is closer to qA
в€—
than qA
, the virtual welfare of A is larger at О»4 = 0.
48
we have added the constraint WA (q) − WC (q) ≥ 0, whichever generates the highest
expected revenue. [IDEALLY WE SHOULD SOLVE FOR THAT SLN AS WELL]
в€— ) в€’ W (q в€— ) > 0 and W (q в€— ) в€’ W (q в€— ) < 0
2. Scenario 2: WA (qA
C A
A B
C B
ICCB becomes binding in the process of decreasing qA . From the scenario assumption, there
exists a value for О»2 в€€ (0, О±C + О±D ) in a system where ICAB , ICCB , ICCA and ICDC are
в€— , q в€— )) and
q в€€ (qA
binding such that ∆θ1 = ∆θ2 qe (e
B
vq (e
q ) = Оё2 +
О»2 xFAB + (О±C + О±D в€’ О»2 )xFBB
∆θ2
О±A xFAB + О±B xFBB
(54)
We now turn to the optimization over the probabilities of winning.
(a) [coded] If WC (q) в€’
О±D
αC ∆θ 1
≥ WA (e
q ). The first best probabilities of winning are the
optimal probabilities of winning. Theorem 2 ensures that the result can implemented
by a scoring auction.
∆θ1 < WA (e
q ), the buyer prefers A
(b) [not coded because never applied] If WC (q) в€’ О±О±D
C
to C. The only way to keep qualities optimized as we increase xA is to keep q = qe. This
requires an adjustment in О»2 which nevertheless remains within (0, О±C + О±D ).23 This
process continues until xA = xC at which point ICDA becomes binding. The solution is
such that all constraints are binding or all constraints but ICCA are binding (cfr. scenario
1(b)i). Theorem 2 ensures that the solution is implementable as a scoring auction.
в€— ) в€’ W (q в€— ) > 0 and W (q в€— ) в€’ W (q в€— ) > 0.
3. Scenario 3: WA (qA
C A
A B
C B
в€— and constraints IC
In that case, the optimal quality level is qB
AB , ICCB and ICDA are binding
when we optimize over the qualities. We now optimize over the probabilities of winning.
в€— ), the first best probabilities of winning are optimal.
∆θ1 ≥ WA (qB
(a) [coded] If WC (q)в€’ О±О±D
C
By Theorem 2, the solution can be implemented by a scoring auction. The expected
utility is given by
в€—
в€—
) + О±B xFBB (WB (qB
)в€’
О±A xFAB WA (qB
+О±C xFC B (WC (q) в€’
О±C + О±D
О±A
в€—
∆θ2 qB
в€’
∆θ1 )
О±B
О±B
О±D
∆θ1 ) + αD xFDB WD (q)
О±C
Theorem 2 ensures that it can be implemented by a scoring auction.
23
Indeed, suppose О»2 reaches zero and consider the resulting first order condition for q, vq (q) = Оё2 +
B
(О±C +О±D )xF
B
B
О±A xA +О±B xF
B
∆θ2 . A further increase in xA requires an increase in λ2 to keep q constant at q. Similarly, if λ2
has reached its upper bound, О±C + О±D , the first order condition is vq (q) = Оё2 +
increase in xA requires that we decrease О»2 .
49
(О±C +О±D )xA
B
О±A xA +О±B xF
B
∆θ2 and a further
в€— ), the buyer prefers
(b) [not coded because never applied] If WC (q) в€’ О±О±D
∆θ1 < WA (qB
C
to give priority to A over C. Increasing xA initially increases q and thus increases further
в€— < q). Two new constraints may become
the virtual welfare associated with A (since qB
binding as we increase xA . First, ICCA starts binding when q = qe. Second ICDA could
start binding.
i. Neither constraint ever binds (at xA = xmax
e): the solution is thus as in case
A , qB < q
FB
max and
1, 2(a): xA = xmax
A , qB is defined by (48) where xA has been replaced by xA
qD = qC = q. Theorem 2, ensures that it can be implemented by a scoring auction.
ii. ICDA binds first: The solution is as in case 1 2(b). At the optimum, ICDA , ICDC ,
ICAB and ICCB bind. Depending on whether the virtual welfare of A and C are
or xA < xmax
at the solution, qC = qD = q and qA = qB is defined
equal, xA = xmax
A
A
by (49).
iii. ICCA binds first. Then the solution is as in case 2, scenario 2 (b).
50
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