Optimal procurement when
suppliers are divided into groups
and the buyer is constrained to buy
from all the groups
P. Samuel NJIKI
Université de Montréal and CIREQ
May 2008
Abstract
We consider an agency willing to achieve a project. For example the project can be the building of equipments or a research
project. The project is divided into smaller projects and contracts for the completion of these projects are granted through a
procurement auction. The participants are agents coming from
di¤erent groups. We assume that the agency is committed to
allocate the contracts so that agents from all groups win. We
study the problem of optimally choosing a procurement mechanism to complete the project. Sellers have private information on
their cost and their payo¤s are linear. We derive a mechanism
that minimizes the expected price paid by the buyer when he is
committed to "all the groups". Under this optimal mechanism
the buyer allocates the suprojects to a set of agents coming from
all possible groups and having the smallest aggregate virtual cost
(the sum of the ironed out virtual costs of agents in the set)
among all sets of agents who may win the project.
Keywords: Optimal mechanism, Procurement, (Aggregate)
Virtual costs, Iron out, Compatible suppliers, Groups.
JEL classi…cation: C72, D44, D82.
1
Introduction
A buyer willing to procure some units of heterogeneous items will often seek to minimize the total price of the desired quantity. Beside this
objective, a buyer facing potential suppliers from di¤erent groups may
sometimes want to procure the items in such a way that all the groups
are represented by the actual suppliers. For example, consider a research
project commanded by a government agency. The agency may divide
ps.njiki.njiki@umontreal.ca
1
the project into subprojects and award the contracts related to these
subprojects through a procurement auction. Assume that participants
are researchers a¢ liated to institutions in di¤erent provinces of the country. A government willing to encourage the research in all the provinces
may design a procurement mechanism so that winners come from all the
provinces. Another example is that of an international institution seeking to hire a given number of individuals among job candidates coming
from di¤erent countries. If the institution is created and/or …nanced by
these countries. Then it is natural for the institution to hire from all
these countries.
In this paper we derive a procurement mechanism that minimizes
the expected price of the units when the buyer is constrained to buy the
units from actual suppliers coming from all the groups he faces. We also
derive a mechanism that minimizes the expected price of the units when
the buyer wants to procure each item independently of the others. Then,
assuming that the buyer can implement these mechanisms, we compare
the expected prices of the units under the two mechanisms.
The literature on optimal auctions has ‡ourished since the early
1980’s. Economists have designed various revenue maximizing (or cost
minimizing, in the case of procurement auctions) mechanisms in the contexts of single unit auctions (ex. Myerson, 1981), multi-unit auctions (ex.
Branco, 1996) and multi-object auctions (ex. Armstrong, 2000). To the
best of our knowledge, the available literature does not consider design
problems where potential buyers (suppliers in the case of procurement
auctions) are not always compatible; that is while every potential supplier may win a priori, only some subsets of the potential buyers (or
suppliers) may win simultaneously. This aspect of the problem is considered in the present paper using the approach laid in (Myerson, 1981).
The notion of virtual value appears to be crucial in the literature on
optimal auction. The equivalent notion in procurement auction is that
of virtual costs. We …nd that virtual costs are also a key instrument
in the optimal mechanism under the constraint of the groups. Virtual
costs are summed over every potential winning subset of suppliers and
the aggregate virtual costs are used to order these subsets; contracts are
then awarded to suppliers of one of the maximal subsets.
The model we adopt is the same as in (Armantier and Njiki, 2008).
Instead of searching for optimal auctions (Armantier and Njiki, 2008)
are interested in particular auctions. On the one hand they consider two
speci…c procurement auctions where contracts are awarded so that all
the groups are represented by the winners of the auction, and on the
other hand they consider traditional procurement auctions where items
are procured independently through second price auctions and …rst price
2
auctions. The expected prices of the units under Nash equilibria of the
games induced by these auctions are then compared. They identi…ed a
mechanism under constraint of the groups that yields a lower expected
price than two independent second price auctions, under assumptions
that imply costs correlation or asymmetry. Due to the assumption that
costs are independent in our model, we …nd that the optimal independent
mechanism yields a lower expected price than the optimal mechanism
under the constraint of the groups. Nevertheless we provide conditions
under which the two prices are equal.
We use a simple model with a single buyer willing to buy two heterogeneous items (a unit of each), and four potential suppliers coming
from two di¤erent groups. There each supplier can only supply one item
and for each item there are two potential suppliers from di¤erent groups.
Suppliers are supposed to have private information about their supply
costs and these costs are independent. Our results however generalize
when there is more than two groups or items. The abstract approach
used here to solve the problem can be applied to more complex models.
The rest of the paper is organized as follows. In section 2 we present
the model and introduce some useful de…nitions. In section 3 we clarify
what mechanisms are considered feasible and focus on direct mechanisms. The optimal mechanisms are derived in section 4. Examples and
applications are considered in section 5. Then we conclude in section 6.
Notation 1 (Note) Scalars and scalar functions are denoted by lowercase letters. Vectors and vector functions are denoted by boldface lowercase letters. u i denotes the vector u without the component of order
i: u = (ui ; u i ). Finally conditional expectations are denoted by the
uppercase of the same letter e.g.: X(ci ) = Ec i [x(ci ; c i )].
2
The model
We consider two suppliers (1 and 2) from a group l and two suppliers
(3 and 4) from a group h. Suppliers 1 and 3 produce the same item
and suppliers 2 and 4 produce another item. A buyer would like to
buy a unit of each of these two items through a procurement auction.
If supplier i 2 N = f1; 2; 3; 4g wins the procurement it will cost ci to
provide one unit. Costs are independently distributed and supplier i’s
cost is distributed according to the probability density function fi ; and
the cumulative density function Fi with support i = [ci ; ci ]. Denote =
¯
i the Cartesian product of these supports. The costs’distributions
i2N
and the groups of the suppliers are common knowledge to the buyer and
the suppliers. Every supplier observes privately his own cost but not the
3
other suppliers’ costs. There is no cooperation between the suppliers.
Note that in this model suppliers win the procurement in pairs: the
buyer can only buy from one of the following pairs f1; 2g ; f2; 3g ; f3; 4g
or f4; 1g :1
There are many di¤erent mechanisms for procuring the two units.
For example, the buyer can buy the units through two simultaneous
…rst price auctions: suppliers are asked to bid the price they are willing
to accept, and the buyer buys the …rst unit from the supplier with the
lowest bid among suppliers 1 and 3 at a price equal to the lowest bid,
and he buys the second unit from the supplier with the lowest bid among
suppliers 2 and 4 at a price equal to this lowest bid: He can also buy
each unit through a second price auction (for each unit the potential
suppliers submit their bids, and the item is bought from the supplier
with the lowest bid at a price equal to the second lowest bid). The
buyer can even practice di¤erent kind of auctions on each item. In the
mechanisms considered so far the two items are bought independently.
The buyer’s decision concerning the supplier to whom he purchases the
…rst item and the …rst item’s unit price is independent on his decision
concerning the supplier to whom he purchases the second item and the
second item’s unit price. In that sense, the procurement of one unit does
not dependent on the procurement of the other unit.
There exist mechanisms under which the procurements of the two
units are not independent. For example, consider the following rule: "the
buyer must buy the units from suppliers coming from all the groups".
We shall refer to this rule as the rule (R). Under this rule the buyer
can buy only from the pairs f1; 4g and f2; 3g:2 Thus if he buys the
…rst item from supplier 1 he must buy the second unit from supplier 4:
Consider for example the auction format where supplier i submits a bid
bi and the units are bought from the pair f1; 4g if b1 + b4 < b2 + b3 ; and
from the pair f2; 3g if b1 + b4 > b2 + b3 . In any case the winners are
paid a price equal to their bid but the other suppliers receive nothing.
This auction format is studied (among others) in (Armantier and Njiki,
2008) in order to model procurement at the European space agency. We
introduce some basic de…nitions in the rest of this section.
Let P = ff1; 2g ; f2; 3g ; f3; 4g ; f4; 1gg : We already mentioned that
the pair of suppliers who win the procurement is necessarily one of the
pairs in P.
In a typical procurement mechanism game, the buyer …rst announces
1
The pair f1; 3g is excluded because 1 and 3 sell the same good and the buyer
needs only a single unit of it. f2; 4g is excluded for the same reason.
2
The pair f1; 2g is excluded because 1 and 2 come from the same country though
they sell di¤erent goods. f3; 4g is excluded for the same reason.
4
the procurement rules to the suppliers. Suppliers observe privately their
own costs and place their bids. Finally the buyer collects the bids and
buys the items according to the rules set before.
Let i be the set of all possible bids for supplier i 2 N ; and let
4
Q
=
is determined by the type of information that
i : The set
i=1
the buyer requires to each supplier during the procurement process. So
we may assume that the buyer knows the set
of the information he
requires. Therefore suppliers cannot bid out of that set.
A strategy for supplier i is a function i : i ! i transforming i’s
cost into a bid.
An allocation rule is a function q = (qG )G2P : !R4+ such that:
X
for any bid vector b 2 ;
qG (b) = 1:
G2P
In other words q(b) is a probability distribution on P and qG (b) is the
probability that pair G wins.
The individual allocation rule associated with the allocation rule q
is a function x = (xi )i2N : !R4+ such that:
for any supplier i 2 N and any bid vector b 2
; xi (b) =
X
qG (b):
G2P=i2G
xi (b) is supplier i’s winning probability with a bid vector b.3 Note
that this de…nition implies that x1 (b) + x3 (b) = 1 = x2 (b) + x4 (b):
A payment rule is a function t = (ti )i2N : !R4 such that ti (b) is
the amount of money paid to supplier i when the bid vector is b. Note
that this payment can be negative, meaning that supplier i will have to
make a transfer to the buyer rather than receive from him.
A procurement mechanism is de…ned by a set of bids, an allocation
rule and a payment rule. We use the notation ( ; q; t) to refer to a
procurement mechanism with a set of bids , an allocation rule q and a
payment rule t: It is important to note that we do not mention the individual allocation rule in the de…nition of a mechanism since it is uniquely
determined by the allocation rule. On the contrary a given individual
allocation rule might be associated with more than one allocation rule.4
What is usually referred to as the allocation rule in auction literature
3
For example x1 (b) = qf1;2g (b) + qf4;1g (b):
Indeed let x = (xi )i2N : !R4+ be such that x1 (b)+x3 (b) = 1 = x2 (b)+x4 (b):
Consider a function : ![0; 1] such that max(0; x3 + x4 1)
min(x3 ; x4 );
exists because x3 ; x4 2 [0; 1] ) 0 max(0; x3 + x4 1) min(x3 ; x4 ) 1:
Take for example qf3;4g = ; qf4;1g = x4
; qf3;2g = x3
; qf1;2g = 1 x3 x4 + ;
4
5
actually corresponds to our individual allocation rule. The reason for
this di¤erence is that our de…nition is designed to grasp the possibility
of an additional constraint on the compatibility of the suppliers. The
rule (R) is an example of such constraints. In this case suppliers from
the same group are not compatible in the sense that they cannot win
simultaneously.
A procurement mechanism ( ; q; t) is independent if there exists two
4
4
functions : 1
: 2
3 ! R and
4 ! R such that, for any
b2 ;
(i) (x1 (b); x3 (b); t1 (b); t3 (b)) = (b1 ; b3 );
(ii) (x2 (b); x4 (b); t2 (b); t4 (b)) = (b2 ; b4 ):
To understand this de…nition, remember that suppliers 1 and 3 are
selling the same item as well as 2 and 4; so the conditions (allocation
and payment) under which one item is purchased depend solely on the
message sent by the suppliers of that item and not the message sent by
the suppliers of the other item.
An allocation rule q follows the rule (R) if:
for any b 2
; qf1;4g (b) + qf2;3g (b) = 1:
In that case if x is the associated individual allocation rule, for all
b2 ;
x1 (b) = x4 (b) = qf1;4g (b) and x2 (b) = x3 (b) = qf2;3g (b):
A procurement mechanism ( ; q; t) follows the rule (R) if q follows
the rule (R).
Our concern in the next section is to de…ne the set of all possible
mechanisms.
3
Direct mechanisms
The set of bid vectors
can be a complex object, depending on the
information the buyer requires from the suppliers. This makes di¢ cult
the problem of optimally choosing a mechanism. Direct mechanisms are
a particular class of mechanisms where each supplier is asked to directly
X
It is easy to see that
qG = 1 and for any G 2 P; qG 0: Thus q is an allocation
rule. Moreover the individual allocation rule associated with q is precisely x; but the
allocation rule q depends on the selection .
6
report his cost. Formally a direct mechanism is a mechanism where the
set of bid vectors is : When ( ; q; t) is a direct mechanism we shall
simply denote it (q; t):
3.1
The revelation principle
A procurement mechanism induces a game of incomplete information
between the suppliers, and the notion of direct mechanism has been de…ned in the broader context of games with incomplete information. In
such games players observe privately an information considered as their
types, they send a message and resources are allocated on the basis of the
messages sent and prede…ned rules. The search for an optimal mechanism can be simpli…ed if one can restrict attention to direct mechanisms.
The well known revelation principle allows us to make such a restriction
without loss of generality. This principle states that: given a game of
incomplete information and a Bayesian Nash equilibrium (BNE), there
exist a direct mechanism (with the same outcomes as the …rst game)
for which it is a BNE to report honestly the types. For the interested
reader we provide (in appendix) a version of the proof of the revelation
principle in the framework we have set earlier.
Proposition 2 Revelation principle
Given a mechanism ( ; q; t) and an equilibrium for that mechanism
, there exists a direct mechanism (q; t) in which it is an equilibrium for
each supplier to report honestly his cost and the outcomes are the same
as in the equilibrium of the …rst mechanism.
3.2
Incentive compatible and individually rational
direct mechanisms
Suppliers need not report their true costs in a direct mechanism since
this information is private; if the buyer cares about truth, he must choose
a procurement mechanism that gives them incentives to do so. This condition imposes further restrictions on mechanisms that may be chosen:
a procurement mechanism must be incentive compatible and individually
rational. Before we de…ne these two concepts we need to introduce some
more notations.
Consider a direct mechanism (q; t) with an individual allocation rule
x:
Let
Z
Xi (mi ) =
xi (mi ; c i )f i (c i )dc
i
7
i
(1)
be the (interim) winning probability of supplier i if he reports the value
mi given that the other suppliers report their true costs.
And let
Z
Ti (mi ) =
ti (mi ; c i )f i (c i )dc
(2)
i
i
be the (interim) expected payment received by supplier i if he reports
the value mi given that the other suppliers report their costs honestly.
The (interim) expected pro…t of supplier i when he reports mi (rather
than ci ) and the other suppliers report their true costs is:
i (mi ; ci )
= Ec i [ti (mi ; c i )
xi (mi ; c i )ci ] = Ti (mi )
Xi (mi )ci : (3)
In particular:
i (ci ; ci )
Ti (ci )
(4)
Xi (ci )ci :
is the expected pro…t when supplier i reports his true cost
ci . If honesty (reporting the true cost) is an equilibrium then i (ci ; ci ) is
supplier i’s pro…t at equilibrium: Xi (ci ) and Ti (ci ) are respectively the
winning probability and the expected payment of supplier i at equilibrium.
i (ci ; ci )
A mechanism is individually rational (IR) if:
for any i 2 N and ci 2
i;
i (ci ; ci )
0:
(5)
This means that even the suppliers with the worst costs will make
non negative pro…ts if they participate in the procurement honestly when
all the other players do so.
A mechanism is incentive compatible (IC) if:
for all i 2 N and ci 2
i;
i (ci ; ci )
= max i (mi ; ci ) = fTi (mi ) Xi (mi )ci g:
mi 2
i
(6)
This means that reporting honestly his cost give a supplier the highest expected pro…t when the other suppliers report their true costs. In
other words the mechanism (q; t) is incentive compatible if honesty is
an interim Bayesian Nash Equilibrium (BNE) of the game induced by
(q; t): The next proposition characterizes an IC mechanism by the winning probabilities and the expected payment functions.
8
Proposition 3 A mechanism (q; t) is IC if and only if:
for all i 2 N ;
the function Xi is decreasing
(7)
and,
for all ci 2
i ; Ti (ci ) = Ti (ci )
Xi (ci )ci + Xi (ci )ci +
Z
ci
Xi (ti )dti : (8)
ci
Thus, when honesty is a BNE, suppliers with the lowest costs have the
highest interim winning probabilities. And these winning probabilities
determine the expected payments up to a constant. equation (8) is well
known in the literature on auction design as the revenue equivalence
theorem. A proof is available in appendix.
In the next section we suppose the buyer has the choice of the procurement mechanism. We look for the mechanism he will choose depending on whether he is looking for an independent mechanism or a
mechanism that follows the rule (R). We also assume that the buyer is
motivated by the minimization of the expected price of the two units.
4
Designing the optimal mechanisms
In this section we …nd mechanisms that minimize the expected amount
paid by the buyer (i.e. the expected price of the two units) among
IC and IR direct mechanisms. We do this by restricting our search to
mechanisms satisfying the rule (R) and then to independent mechanisms.
Consider an incentive compatible direct mechanism (q; t) with an
individual allocation rule x: Below is an expression of the expected price
P (q; t) of the units under this mechanism.
X
X
X
X
P (q; t) = E
ti (c) =
E[ti (c)] =
EEc i [ti (ci ; c i )] =
ETi (ci )
i2N
i2N
i2N
i2N
using (8),
E[Ti (ci )] = Ti (ci )
+
Z
ci
c
¯i
Xi (ci )ci +
Z
Z
ci
Xi (ci )ci fi (ci )dci
c
¯i
ci
Xi (ti )dti fi (ci )dci
ci
Fubini’s theorem implies,
9
Z
ci
c
¯i
Z
ci
Xi (ti )dti fi (ci )dci =
ci
=
Z
Z
ci
c
Z¯ ici
ti
fi (ci )dci Xi (ti )dti
c
¯i
Fi (ti )Xi (ti )dti :
c
¯i
Therefore,
E[Ti (ci )] = Ti (ci )
Xi (ci )ci +
Z
Z
ci
[Xi (ci ) ci fi (ci ) + Fi (ci )Xi (ci )]dci
c
¯i
ci
Fi (ci )
)Xi (ci ) fi (ci )dci
fi (ci )
ci
Z¯
F (ci )
)xi (c) f (c)dc (using (1)).
= i (ci ; ci ) + (ci + i
fi (ci )
=
i (ci ; ci ) +
(ci +
It follows that,
P (q; t) =
X
i (ci ; ci )
+
Z (X
)
i2N
i2N
Fi (m)
fi (m)
(9)
Hi (ci )xi (c) f (c)dc
where Hi (m) = m +
for all i 2 N and m 2 i ; the function Hi
is usually called the virtual cost of supplier i:
We can also express the expected price in terms of the allocation rule
rather than the individual allocation rule:
8
Z <X
9
=
Hi (ci )
qG (c) f (c)dc
P (q; t) =
i (ci ; ci ) +
:
;
i2N
i2N
G2P=i2G
8
9
Z
<
=
X
X X
=
Hi (ci )qG (c) f (c)dc
i (ci ; ci ) +
:
;
i2N
i2N G2P=i2G
)
Z (X X
X
=
Hi (ci )qG (c) f (c)dc
i (ci ; ci ) +
X
i2N
=
X
X
G2P i2G
i (ci ; ci )
i2N
+
Z (X
qG (c)
G2P
X
i2G
)
Hi (ci ) f (c)dc:
We de…ne the aggregate virtual cost of the pair G as SG (c)
P
Hi (ci ):
i2G
And the price can be written as:
P (q; t) =
X
i2N
i (ci ; ci )
+
Z (X
G2P
10
)
qG (c)SG (c) f (c)dc:
(10)
For any i 2 N , consider the function Ki : [0; 1] ! R; Ki (zi ) =
1
(zi )
Hi (t)fi (t)dt: Note that Ki0 (Fi (ci )) = Hi (ci ):
0
^ i : [0; 1] ! R be the convex hull5 of Ki : K
^ i is di¤erentiable
Let K
^
^
^ 0 (Fi (ci )):
almost surely. Let Hi : i ! R be such that Hi (ci ) = K
i
Then it is known that:
^ i (0) = Ki (0) and K
^ i (1) = Ki (1);
(a) K
^
(b) Ki (zi ) Ki (zi ) for any zi 2 [0; 1] ;
^ i (zi ) < Ki (zi ) then K
^ i0 is constant in some neighborhood of
(c) if K
^
zi ; hence Hi is constant in some neighborhood of Fi 1 (zi ).
^ i (ci ) is called the ironed out virtual cost of supplier i: S^G (c)
PH
^ i (ci ) will be called the aggregate ironed out virtual cost.
H
R Fi
i2G
Let
P^ (q; t) =
X
i (ci ; ci )
+
i2N
Z (X
)
^ i (ci )xi (c) f (c)dc:
H
i2N
(11)
P^ is obtained from the expression of P by replacing virtual costs by
the ironed out virtual costs. Thus we can also write:
)
Z (X
X
qG (c)S^G (c) f (c)dc:
(12)
P^ (q; t) =
i (ci ; ci ) +
G2P
i2N
The following lemma will also be useful.
Lemma 4 for any IC mechanism (q; t): P^ (q; t)
Proof.
P^ (q; t)
P (q; t) =
Z (X
)
^ i (ci )xi (c) f (c)dc
H
i2N
XZ n
^ i (ci )
=
H
i2N
=
XZ
ci
i2N
ci
i2N
ci
i2N
c
¯i
n
^ i (ci )
H
P (q; t):
Z (X
i2N
o
Hi (ci ) xi (c)f (c)dc;
¯
X Z ci n
^ i Fi )0 (ci )
=
(K
o
^ i ),
Ki0 (Fi (ci )) Xi (ci )f (ci )dci (by de…nition of Hi and H
o
(Ki Fi )0 (ci ) Xi (ci )dci .
i.e. the greatest convex function g : [0; 1] ! R such that g(zi )
zi 2 [0; 1] :
11
Hi (ci )xi (c) f (c)dc;
o
Hi (ci ) Xi (ci )f (ci )dci (using (1)),
¯
X Z ci n
^ 0 (Fi (ci ))
=
K
i
5
)
Ki (zi ); for any
Integrating by part,
Xn
^ i Fi )(ci )
P^ (q; t) P (q; t) =
(K
i2N
Z
ci
ci
P^ (q; t)
X ¯n
^ i (1)
P (q; t) =
K
i2N
Z
ci
c
¯i
P^ (q; t)
n
^i
(K
P (q; t) =
n
^ i Fi )(ci )
(K
XZ
i2N
o
^ i Fi )(ci ) + (Ki Fi )(ci )
(K
¯
¯
o
Fi )(ci ) (Ki Fi )(ci ) dXi (ci );
o
^ i (0) + Ki (0)
Ki (1) K
ci
c
¯i
(Ki Fi )(ci )
n
^ i (Fi (ci ))
K
o
(Ki Fi )(ci ) dXi (ci );
o
Ki (Fi (ci )) dXi (ci ) (using (a)).
Since the mechanism is IC, proposition 3 implies that dXi (ci )
0.
^
Furthermore property (b) implies Ki (Fi (ci )) Ki (Fi (ci )) 0. It follows
that P^ (q; t) P (q; t) 0:
We now derive the optimal mechanism under the rule (R).
4.1
Optimal mechanism under the rule (R)
Consider the allocation rule qR de…ned as follow:
8
R
>
< qf1;4g (m) = 1 if S^f1;4g (m) S^f2;3g (m);
R
for any m 2 ; qf2;3g
(m) = 1 if S^f1;4g (m) > S^f2;3g (m);
>
: q (m) = 0 if G 2 ff1; 2g ; f3; 4gg:
G
R
R
Note that qf1;4g
(m) + qf2;3g
(m) = 1 for any m 2 . Under this allocation rule the buyer buys from the pair with the smallest aggregate
ironed out virtual cost between the pairs f1; 4g and f2; 3g. Thus qR
follows the rule (R).
Let xR be the individual allocation rule associated with qR ; xR is
such that, for any m 2 ,
R
R
R
R
R
xR
1 (m) = x4 (m) = qf1;4g (m) and x2 (m) = x3 (m) = qf2;3g (m): (14)
Finally consider the following payment rule:
Z ci
R
R
for any m 2 ; ti (m) = xi (m)mi +
xR
i (t; m i )dt:
(15)
mi
R
Note that we have in particular tR
i (ci ; m i ) = xi (ci ; m i )ci for all
m i 2 i . Then taking the expectation TiR (ci ) = XiR (ci )ci and …nally
12
(13)
the expected pro…t of supplier i with costs ci when he bids honestly is
R
R
XiR (ci )ci = 0: The consequence is that the mechai (ci ; ci ) = Ti (ci )
nism (qR ; tR ) always yields a worse expected pro…t than any individually
rational mechanism (q ; t) for the supplier i with costs ci when he bids
honestly: i (ci ; ci )
0 = R
i (ci ; ci ): And we may prove the following
result.
Proposition 5 The direct mechanism (qR ; tR ) minimizes the expected
price of the two units among all direct mechanisms that are IC, IR and
follow the rule (R). Moreover the minimum expected price is given by
R
= E[min(S^f1;4g (c); S^f2;3g (c))]:
Proof. If (q; t) is IC and follows the rule (R) then for any c 2 ; qf1;4g (c)+
qf2;3g (c) = 1 and qG (c) = 0 if G 2 ff1; 2g ; f3; 4gg ; Then we have
Z (X
G2P
)
Z n
o
qG (c)S^G (c) f (c)dc =
qf1;4g (c)S^f1;4g (c) + qf2;3g (c)S^f2;3g (c) f (c)dc
Z
=
Z
Z
qf1;4g (c) min(S^f1;4g (c); S^f2;3g (c))
f (c)dc
+qf2;3g (c) min(S^f1;4g (c); S^f2;3g (c))
qf1;4g (c) + qf2;3g (c) min(S^f1;4g (c); S^f2;3g (c))f (c)dc
R
R
qf1;4g
(c) + qf2;3g
(c) min(S^f1;4g (c); S^f2;3g (c))f (c)dc
Z n
o
R
R
^
^
=
qf1;4g (c)Sf1;4g (c) + qf2;3g (c)Sf2;3g (c) f (c)dc
)
Z (X
R
=
qG (c)S^G (c) f (c)dc:
=
G2P
If (q; t) is also IR we just showed that
P
i2N
i (ci ; ci )
0=
P
R
i (ci ; ci ):
i2N
We conclude that P^ (q; t)
P^ (qR ; tR ) for any IC and IR mechanism
(q; t) following the rule (R). Putting this with lemma 4 we obtain that,
for any IC and IR mechanism (q; t) following the rule (R) P (q; t)
P^ (qR ; tR ): It will be su¢ cient to show that P^ (qR ; tR ) = P (qR ; tR ) and
that (qR ; tR ) is an IC and IR mechanism following the rule (R).
R
By de…nition tR
xR
i (m)
i (m)mi and (integrating over m i ) Ti (mi )
XiR (mi )mi ; i.e. R
0: So (qR ; tR ) is IR.
i (mi )
R R
We show that (q ; t ) is IC using The two conditions
R ci ofRproposition
R
R
R
3. First taking the de…nition of t : ti (c) = xi (c)ci + ci xi (t; c i )dt;
13
integrating this equality over c i leads to TiR (ci ) = XiR (ci )ci +
R ci and
R
R
Xi (t; c i )dt and since R
XiR (ci )ci = 0; TiR (ci ) =
i (ci ; ci ) = Ti (ci )
ci
R
c
TiR (ci ) XiR (ci )ci + XiR (ci )ci + cii XiR (t; c i )dt:
We will show that XiR is decreasing when i = 1; the proof is similar
for any i 2 N .6
^ 1 is increasing we have H
^ 1 (s) H
^ 1 (t)
Let s; t 2 1 : s < t; since H
^ 1 (s)+ H
^ 4 (c4 ) H
^ 1 (t)+ H
^ 4 (c4 ): Therefore the following implication
and H
is true:
^ 1 (t) + H
^ 4 (c4 )
H
^ 2 (c2 ) + H
^ 3 (c3 ) ) H
^ 1 (s) + H
^ 4 (c4 )
H
^ 2 (c2 ) + H
^ 3 (c3 ):
H
In other words:
S^f1;4g (t; c 1 )
S^f2;3g (t; c 1 ) ) S^f1;4g (s; c 1 )
S^f2;3g (s; c 1 );
or
R
R
qf1;4g
(t; c 1 ) = 1 ) qf1;4g
(s; c 1 ) = 1:
R
R
So qf1;4g
(s; c 1 )
qf1;4g
(t; c 1 ); and …nally (using 14): xR
1 (s; c 1 )
R
x1 (t; c 1 ): Since c 1 is arbitrary we may take the integral over c 1 and
obtain X1R (s) X1R (t): Thus X1R is decreasing.
Since (qR ; tR ) is IC then, by equation 13:
o
X Z ci n
R R
R R
^
^
Ki (Fi (ci )) Ki (Fi (ci )) dXiR (ci );
P (q ; t ) P (q ; t ) =
i2N
c
¯i
^ i (Fi (ci )) Ki (Fi (ci )) 6= 0 then K
^ i (Fi (ci )) Ki (Fi (ci )) < 0 and
If K
^ i (ci ) is constant in some neighborhood of ci (property (c)) and so,
H
XiR (ci ) is also constant7 (i.e.
dXiR (ci ) = 0) in o
some neighborhood
R ci n
^ i (Fi (ci ) Ki (Fi (ci ) dXiR (ci ) = 0 and
of ci : We then conclude c K
i
¯
P^ (qR ; tR ) P (qR ; tR ) = 0:
6
The large inequalities in the implications below must be replaced by strict inequalities if i = 2 or i = 3: This is necessary because of the de…nition of qR where
ties are solved in favor of the pair f1; 4g :
7
Without loss of generality suppose i = 1:
^
^
^ 2 (c2 ) + H
^ 3 (c3 ). By
Recall that xR
H
1 (c1 ; c 1 ) = 1R if and only if H1 (c1 ) + H4 (c4 )
R
R
de…nition we have X1 (c1 ) = 3 x1 (c1 ; c 1 )f 1 (c 1 )dc 1 . Since the integral is taken
^ i )i6=1 and on H
^ 1 (c1 ). Thus it is
over c 1 we conclude X1R (c1 ) depends only on (H
^
constant if H1 (c1 ) is constant.
14
Finally the expected price is:
Z n
o
X
R
R
R
R R
^
^
^
qf1;4g (c)Sf1;4g (c) + qf2;3g (c)Sf2;3g (c) f (c)dc
= P (q ; t ) =
i (ci ; ci ) +
i2N
Z n
o
R
R
^
^
=
qf1;4g (c)Sf1;4g (c) + qf2;3g (c)Sf2;3g (c) f (c)dc (because i (ci ; ci ) = 0)
Z n
o
R
R
(c) min(S^f1;4g (c); S^f2;3g (c)) f (c)dc
(c) min(S^f1;4g (c); S^f2;3g (c)) + qf2;3g
=
qf1;4g
h
i
R
^
^
= E min(Sf1;4g (c); Sf2;3g (c)) .
4.2
Optimal independent mechanism
For any m 2 let
xI1 (m) =
^ 1 (m1 ) H
^ 3 (m3 )
1 if H
^ 1 (m1 ) > H
^ 3 (m3 ) ;
0 if H
(16)
xI2 (m) =
^ 2 (m2 ) H
^ 4 (m4 )
1 if H
^ 2 (m2 ) > H
^ 4 (m4 ) ;
0 if H
(17)
xI3 (m) = 1
xI1 (m) and xI4 (m) = 1
and
tIi (m)
=
xIi (m)mi
+
Z
xI2 (m);
(18)
ci
xIi (t; m i )dt:
(19)
mi
We showed in section 2 that there exist at least one allocation rule
I
)G2P such that the vector function xI = (xIi )i2N is the inq = (qG
dividual allocation rule associated with qI :8 The mechanism (q I ; tI ) is
independent since xI3 ; xI1 ; tI3 ; tI1 depend only on m1 or m3 ; and xI2 ; xI4 ; tI2 ; tI4
depend only on m2 or m4 : We see that in the sense of (Myerson 1981)
(xI3 ; xI1 ; tI3 ; tI1 ) is a procurement mechanism for the …rst item. Likewise
(xI2 ; xI4 ; tI2 ; tI4 ) is a procurement mechanism for the second item. By de…nition any independent mechanism is actually the sum of two independent mechanisms for each item. Moreover it is known that (xI3 ; xI1 ; tI3 ; tI1 )
I
(resp. (xI2 ; xI4 ; tI2 ; tI4 )) is the optimal procurement mechanism of the …rst
(resp. second) item. Because the expected price P (q; t) is an additively
separable function of (x3 ; x1 ; t3 ; t1 ) and (x2 ; x4 ; t2 ; t4 ), (q I ; tI ) is optimal
among independent mechanisms. Our next result is that this mechanism
actually minimizes the expected price of the items among all IC and IR
mechanisms, including those satisfying the rule (R).
8
We …x qI once for all.
15
Proposition 6 The direct mechanism (qI ; tI ) minimizes the expected
price of the two units among all IC and IR mechanisms. Moreover
^ 1 (c1 ); H
^ 3 (c3 )) +
the minimum expected price is given by I = E[min(H
^ 2 (c2 ); H
^ 4 (c4 ))]:
min(H
The proof is similar to the proof of proposition 5 and is found in the
appendix. A consequence of this is that the expected price of the units
when the buyer implements optimally the rule (R) is higher than the
expected price when he implements independent mechanisms optimally.
R
i.e I
. In the next proposition we characterize conditions under
I
which = R or I < R :
^ 1 (c1 ) H
^ 3 (c3 )][H
^ 2 (c2 ) H
^ 4 (c4 )]
Proposition 7 I = R if and only if [H
0 for
almost every c 2 :
^ 1 (c1 )
An equivalent result would be: I < R if and only if [H
^ 3 (c3 )][H
^ 2 (c2 ) H
^ 4 (c4 )] > 0 with positive measure:
H
^ 1 (c1 ) H
^ 3 (c3 )] and [H
^ 2 (c2 ) H
^ 4 (c4 )]
The last condition means that [H
have the same sign. Assume suppliers are …rms and interpret the supply
cost as their technology levels. In the optimal mechanisms derived the
buyer is interested in …rms’virtual cost (that is his actual vision of the
technologic di¤erence between the two) rather than their costs. And the
^ 1 (c1 ) H
^ 3 (c3 )][H
^ 2 (c2 ) H
^ 4 (c4 )] > 0 means that one country
condition [H
dominates the other in the buyer’s actual view. That this happens with
positive probability is a su¢ cient and necessary condition for I < R .
A proof is provided in appendix.
We illustrate these results in the particular case of power distributions.
5
Example: power distributions
We suppose Fi (ci ) =
ci c i
¯
ci c i
¯
ai
and ai
1 for all i 2 N and ci 2
probability density functions are given by fi (ci ) =
ai
ci c i
¯
ci c i
¯
ci c i
¯
ai 1
the virtual cost of supplier i is increasing in ci : Hi (ci ) = ci +
^ i.
(1 + a1i )ci a1i ci . Since Hi is increasing we have Hi = H
¯
The condition for I = R writes
^ 1 (c1 )
[H
^ 3 (c3 )][H
^ 2 (c2 )
H
^ 1 (c1 )
The domain of H
^ 1 (c1 )
[H
¯
^ 4 (c4 )]
H
0 almost surely.
^ 3 (c3 ) is
H
^ 3 (c3 ); H
^ 1 (c1 )
H
16
^ 3 (c3 )] = [x; x]:
H
¯
¯
i.
The
, and
Fi (ci )
fi (ci )
=
^ 2 (c2 )
Likewise the domain of H
^ 2 (c2 )
[H
¯
^ 4 (c4 ) is
H
^ 4 (c4 ); H
^ 2 (c2 )
H
^ 4 (c4 )] = [y; y]:
H
¯
¯
^ i ’s are di¤erentiable and increasing, the set
Because the H
n
o
^ 1 (c1 ) H
^ 3 (c3 )][H
^ 2 (c2 ) H
^ 4 (c4 )] 0
c 2 : [H
is of full measure if and only if the set (x; y) 2 [x; x] [y; y] : xy 0
¯
¯
is also of full measure.
The later is true if and only if [x 0 and y 0] or [x 0 and y 0] .
¯
¯
[x
¯
^ 1 (c1 ) H
^ 3 (c3 ) and H
^ 2 (c2 ) H
^ 4 (c4 )]
0] , [H
¯
¯
1
1
1
1
c]
c and (1 + )c2
c
(1 + )c3
a3
a3 ¯ 3
a2
a2 ¯ 2 ¯ 4
1
1
c3 + (c3 c3 ) and c2 + (c2 c2 ) c4 ]:
¯
¯
¯
a3
a2
0 and y
, [c1
¯
, [c1
¯
Note that ci >ci and condition c1 c3 + a13 (c3 c3 ) means that supplier
¯
¯
¯
1’s minimal unit cost c1 is su¢ ciently higher than his direct opponent’s
¯
maximal unit cost c3 . And the second condition means that supplier
4’s minimal unit cost c4 is su¢ ciently higher than his direct opponent’s
¯
maximal unit cost c2 . In other word group h = f3; 4g has a better
technology for producing item 1 (via supplier 3) while the group l =
f1; 2g has a better technology for producing item 2 (via supplier 2).
Similarly the other condition [x 0 and y 0] implies that group l =
¯
f1; 2g has a better technology for producing
item 1 (via supplier 1)
while group h = f3; 4g has a better technology for producing item 2 (via
supplier 4). None of the group has a better technology for producing
both items.
In the rest of the section we assume ci = 1 and ci = 0 for i 2 N . Let
¯
"i = (1 + a1i ). We provide simple expressions of the optimal mechanisms
I
and examine the evolution of R
when the parameters ai increase.
We know that I < R . Since Fi (ci ) = cai i , an increase in ai (a decrease
in "i ) results in an increase in the probability of having low costs. Thus
the parameter ai somehow describes the technology level of supplier i or
at least what is believed about his technology level. The greater ai the
better i’s technology.
5.1
Optimal mechanisms
Using the expressions of the ironed out virtual costs the allocation rule
qR is such that, for any m 2 :
17
R
R
"2 m 2 + "3 m 3
xR
1 (m) = x4 (m) = qf1;4g (m) = 1 if "1 m1 + "4 m4
R
R
R
x2 (m) = x3 (m) = qf2;3g (m) = 1 if "1 m1 + "4 m4 > "2 m2 + "3 m3 :
Given an arbitrary seller i 2 N we denote the other sellers i0 , j, and
j 0 so that i and j (resp. i0 and j 0 ) sell the same good, and i and i0 (resp.
j and j 0 ) belong to the same group.9
Thus we may show that supplier i 2 N receives payment:
tR
i (m)
=
min(ci ;
"j mj +"j 0 mj 0 "i0 mi0
)
"i
0
if xR
i (m) = 1
R
if xi (m) = 0:
So supplier i reports a unit cost of mi and the buyer uses a subjective
correction factor "i in order to compare the bids of the suppliers. We
shall call "actualized bid" the numbers "i mi . So the buyer compares
actualized bids and allocates the contracts to the pair with the lowest
(aggregate) actualized bid. Then if supplier i wins, he is paid an amount
equal to his actualized bid plus the absolute value of the gap between
aggregate actualized bids of the two pairs10 (all of this "disactualized"),
provided that it does not exceed the maximal cost ci .
Similarly we have
xIi (m) =
1 if "i mi < "j mj
0 if "i mi > "j mj
with the ties (i.e. situations where "i mi = "j mj ) solved in favor
of suppliers 1 and 4. But this is irrelevant in the computation of the
payment rule that can be written as:
tIi (m)
=
min(ci ;
0
"j mj
)
"i
if xIi (m) = 1
if xIi (m) = 0:
We see that in this case i wins if his actualized bid is lower than his
opponent’s. Then he is paid an amount equal to his opponent’s actualized bid "disactualized" using supplier i’s correction factor, provided
that it does not exceed the maximal cost ci .
9
We use the same notation throughout this section.
for example, if i = 1 then i 0 = 2; j = 3 and j 0 = 4:
10
Indeed, "j mj + "j 0 mj 0 "i0 mi0 = "i mi + j("j mj + "j 0 mj 0 ) ("i0 mi0 + "i mi )j: If
i = 1 for example, "3 m3 + "2 m2 "4 m4 = "1 m1 + j("3 m3 + "2 m2 ) ("1 m1 + "4 m4 )j:
18
5.2
Expected prices and changes in technology
We know from the proof of proposition 7 in appendix that
R
I
1 ^
^
^
^
^
^
^
^
= [jH
1 (c1 ) H3 (c3 )j+jH2 (c2 ) H4 (c4 )j j(H1 (c1 ) H3 (c3 )) (H2 (c2 ) H4 (c4 ))j]:
2
Substituting we obtain:
R
I
1
= E fj"1 c1 "3 c3 j + j"2 c2 "4 c4 j j("1 c1 "3 c3 ) ("2 c2
2
1
= E fmin[j"1 c1 "3 c3 j; j"2 c2 "4 c4 j]:1[("1 c1 "3 c3 )("2 c2
2
"4 c4 )jg
"4 c 4 )
0]g ;
where 1(A) = 1 if assertion A is true, and 1(A) = 0 otherwise.
We impose some more structure in the costs’distributions. Assume
that "1 = "2 = "l > "h = "3 = "4 , that is, supplier from the same group
have the same technology level but the group h has in general a higher
technology than group l. We may think of the groups as countries.
Consider an improvement of the technologies in both countries. This
corresponds to an increase in ai , a decrease in "i and also an increase
in "i ai = (1 + ai ): We make the additional assumption that this change
in the parameters leaves constant the ratio ""hl . This implies that the
technology level increases faster in the high-tech country than in the
low-tech country. The following shows that such changes in technologies
I
result in an increase of R
. i.e within this context an increase in
the technology gaps between the two countries results in an increase in
the di¤erence between the expected price when the rule (R) is optimally
implemented and the expected price when items are optimally procured
independently.
Indeed the gap between the expected prices under the two rules depends on the parameters "l and "h as follows:
R
I
"h
"h
"h
"h
1
c3 j; jc2
c4 j]:1[(c1
c3 )(c2
c4 ) 0]
= E "l min[jc1
2
"l
"l
"l
"l
Z
"l min[jc1 ""hl c3 j; jc2 ""hl c4 j]:
1
=
dc
1[(c1 ""hl c3 )(c2 ""hl c4 ) 0]al ca11 1 al ca22 1 ah ca33 1 ah ca44 1
2
Z
"l al min[jc1 ""hl c3 j; jc2 ""hl c4 j]:
1
=
dc;
1[(c1 ""hl c3 )(c2 ""hl c4 ) 0]ca11 1 al ca22 1 ah ca33 1 ah ca44 1
2
and the conclusion follows immediately.
19
6
Conclusion
In this paper we have derived a procurement mechanism minimizing
the expected price when the buyer is constrained to buy the units from
suppliers coming from the many groups he faces. Under the optimal
mechanism, virtual costs are summed over every potential winning subset of suppliers and the aggregate virtual costs are used to order these
subsets; then contracts are awarded to suppliers of one of the maximal
subsets. We have also derived an optimal mechanism when each item is
procured independently of the others. Due to the assumption of costs
independence the optimal independent mechanism consists of independent Myerson auctions on each item and yields a lower expected price
than the optimal mechanism under the constraint of the groups. In the
simple model with only two groups, we have found that if the probability that a group dominates the other in terms of virtual costs is positive
then the two mechanisms are not equivalent. They are equivalent in the
opposite case.
Future directions of research may include relaxing the assumption of
independent costs and allow in the model situations where some suppliers
may supply more than one item.
7
References
Armantier,O., Njiki, P.S. Fair return principle : e¢ ciency and cost effectiveness. Working paper, 2008.
Armstrong, M. Optimal multi-object auctions. The Review of Economic Studies, 67(3):455-481, 2000.
Branco, F. Multiple unit auctions of an indivisible good. Economic
Theory, 8:77-101, 1996.
Myerson, R. Optimal auction design. Mathematics of Operations Research, 6(1):58-73, 1981.
A
Appendix
Proof of the revelation principle. Consider a mechanism ( ; q; t)
and an equilibrium ; de…ne
qG (c) = qG ( (c)) and ti (c) = ti ( (c));
And let x be the individual allocation mechanism associated with q:
Suppose that under the direct mechanism (q; t) the other suppliers
(than i) report their true costs, supplier i’s expected pro…t when he bids
20
mi 2
i
rather than the true cost ci is:
Ec i [ti (mi ; c i )
xi (mi ; c i )ci ];
And we have,
ti (mi ; c i )
xi (mi ; c i )ci = ti ( (mi ; c i )) xi ( (mi ; c i ))ci
= ti ( i (mi ); i (c i )) xi ( i (mi );
i (c i ))ci :
Observe that Ec i [ti ( i (mi ); i (c i )) xi ( i (mi ); i (c i ))ci ] is supplier i’s expected pro…t when he bids i (mi ); and the other suppliers
strategy is i under the mechanism ( ; q; t): Since is an equilibrium,
Ec i [ti ( i (mi );
Ec i [ti ( i (ci );
= Ec i [ti (ci ; c i )
i (c i ))
xi ( i (mi ); i (c i ))ci ]
xi ( i (ci ); i (c i ))ci ]
i (c i ))
xi (ci ; c i )ci ]:
Thus,
Ec i [ti (mi ; c i )
xi (mi ; c i )ci ]
Ec i [ti (ci ; c i )
xi (ci ; c i )ci ];
implying that supplier i’s best response is to report his true cost.
Proof of proposition 3.
If (q; t) is an IC mechanism then the
functions i are convex. Indeed let 2 [0; 1]; and u; v 2 i :
i(
u + (1
)v) = max fTi (mi )
mi 2
i
mi 2
i
Xi (mi )( u + (1
)v)g
= max[ fTi (mi )
Xi (mi )ug + (1
)fTi (mi )
maxfTi (mi )
Xi (mi )ug + (1
) max fTi (mi )
=
mi 2
i
i (u)
+ (1
) i (v)
mi 2
Xi (mi )vg]
Xi (mi )vg
i
i being convex is di¤erentiable almost everywhere in the interior
of its domain and the derivative is increasing. The envelope theorem
applied to condition (6) implies that:
0
i (ci )
=
Xi (ci ):
Thus Xi is increasing and therefore Xi is decreasing.
(20) also implies that:
Z ci
Xi (ti )dti :
i (ci ; ci ) = i (ci ; ci ) +
ci
21
(20)
(21)
Replacing i (ci ; ci ) by Ti (ci ) Xi (ci )ci leads to (8).
Conversely, from (8) we can have the expressions of Ti (ci ) and Ti (mi ):
Substituting these expressions in what follows, we have:
Z mi
[Ti (ci ) Xi (ci )ci ] [Ti (mi ) Xi (mi )ci ] =
Xi (ti )dti Xi (mi )(mi ci );
ci
or equivalently,
i (ci ; ci )
[Ti (mi )
Xi (mi )ci ] =
Z
mi
Xi (ti )dti
Xi (mi )(mi
ci ):
ci
This last expression is positive if the function Xi is decreasing. Therefore condition (6) holds and the mechanism is IC.
proof of proposition 6. If (q; t) is an IC mechanism then for any
c 2 , Using x1 (c) + x3 (c) = 1 = x2 (c) + x4 (c),
h
i h
i
X
^ i (ci )xi (c) = H
^ 1 (c1 )x1 (c) + H
^ 3 (c3 )x3 (c) + H
^ 2 (c2 )x2 (c) + H
^ 4 (c4 )x4 (c)
H
i2N
h
i h
i
^ 1 (c1 ); H
^ 3 (c3 )) + min(H
^ 2 (c2 ); H
^ 4 (c4 ))
min(H
h
i h
i
I
I
I
I
^
^
^
^
= H1 (c1 )x1 (c) + H3 (c3 )x3 (c) + H2 (c2 )x2 (c) + H4 (c4 )x4 (c)
X
^ i (ci )xI (c):
H
=
i
i2N
Therefore P^ (q; t)
P^ (qI ; tI ) and because of lemma 4 P (q; t)
I
I
P^ (q ; t ): We know from equation (13) that if (qI ; tI ) is IC then:
o
X Z ci n
I I
I I
^
^
P (q ; t ) P (q ; t ) =
Ki (Fi (ci )) Ki (Fi (ci )) dXiI (ci );
i2N
c
¯i
^ i (Fi (ci )) Ki (Fi (ci )) 6= 0 then K
^ i (Fi (ci )) Ki (Fi (ci )) < 0
If K
^ i (ci ) is constant in some neighborhood of ci (property (c)) and
and H
I
so Xi (ci ) is also constant11n(i.e. dXiI (ci ) = 0) in some
neighborhood
o
R ci
I
^ i (Fi (ci )) Ki (Fi (ci )) dX (ci ) = 0 and
of ci : We may conclude c K
i
¯i
11
I
Without loss of generality suppose i = 1. Recall that
1 ; c 1 ) = 1 if and only
R x1 (c
I
I
^ 1 (c1 )
^ 3 (c3 ). By de…nition we have X (c1 ) =
if H
H
x
(c
1 1 ; c 1 )f 1 (c 1 )dc 1 .
1
1
I
^ i )i6=1 and
Since the integral is taken over c 1 we conclude X1 (c1 ) depends only on (H
^ 1 (c1 ). Thus it is constant if H
^ 1 (c1 ) is constant.
on H
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P^ (qI ; tI ) = P (qI ; tI ): Thus P (q; t)
P (qI ; tI ) for any IC mechanism
(q; t):
We need to show that (qI ; tI ) is IR and actually IC. First, note that
xIi (m)mi ; taking the
by de…nition of the payment rule tI : tIi (m)
I
I
0; and
Xi (mi )mi i.e. Ii (mi )
integral over m i ; we have Ti (mi )
I I
(q ; t ) is IR.
We show that (qI ; tI ) is IC using proposition 3; consider two suppliers
i and j selling the same item. We will show that XiI is decreasing in the
case i 2 f1; 2g : The proof is similar in the case where i 2 f3; 4g except
that the large inequalities in the implication below must be replaced by
strict inequalities.
^ i (t)
^ i (s) and the
Let t; s 2 i and suppose t
s: Therefore H
H
following assertion is true:
^ i (s)
H
^ j (mj ) ) H
^ i (t)
H
^ j (mj );
H
i.e.
xIi (s; m i ) = 1 ) xIi (t; m i ) = 1;
or equivalently
xIi (s; m i )
xIi (t; m i ):
XiI (t); and XiI is deSince m i is arbitrary we also have XiI (s)
creasing.
Using the de…nition of the payment rule and taking the integral over
m i ; we deduce:
Z ci
I
I
Ti (mi ) = Xi (mi )mi +
XiI (t; c i )dt
mi
and
I
i (ci )
XiI (ci )ci = 0;
Rc
and …nally: TiI (mi ) = Ii (ci )+XiI (mi )mi + mii XiI (t; c i )dt. Using Proposition 3 we conclude (qI ; tI ) is IC.
From the …rst lines of our proof we have I = P (qI ; tI ) = P^ (qI ; tI ) =
^ 1 (c1 ); H
^ 3 (c3 )) + min(H
^ 2 (c2 ); H
^ 4 (c4 ))]:
E[min(H
proof of proposition 7.
R
I
= TiI (ci )
^ 1 (c1 ) + H
^ 4 (c4 ); H
^ 2 (c2 ) + H
^ 3 (c3 ))]
= E[min(H
^ 1 (c1 ); H
^ 3 (c3 )) + min(H
^ 2 (c2 ); H
^ 4 (c4 ))]
E[min(H
^ 1 (c1 ) + H
^ 4 (c4 ); H
^ 2 (c2 ) + H
^ 3 (c3 ))
= E[min(H
^ 1 (c1 ); H
^ 3 (c3 )) min(H
^ 2 (c2 ); H
^ 4 (c4 ))];
min(H
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Using the equality min(a; b) =
R
I
1 ^
= [jH
1 (c1 )
2
^ 1 (c1 )
j(H
a+b ja bj
2
we obtain:
^ 3 (c3 )j + jH
^ 2 (c2 )
H
^ 3 (c3 ))
H
^ 2 (c2 )
(H
^ 4 (c4 )j
H
^ 4 (c4 ))j]:
H
The triangular inequality implies that:
^ 1 (c1 ) H
^ 3 (c3 )j+jH
^ 2 (c2 ) H
^ 4 (c4 )j j(H
^ 1 (c1 ) H
^ 3 (c3 )) (H
^ 2 (c2 ) H
^ 4 (c4 ))j
jH
I
^ 1 (c1 ) H
^ 3 (c3 )j + jH
^ 2 (c2 )
Then we have R
= 0 if and only if jH
^ 4 (c4 )j = j(H
^ 1 (c1 ) H
^ 3 (c3 )) (H
^ 2 (c2 ) H
^ 4 (c4 ))j almost surely. Using
H
the square of both sides of the equality we obtain the equivalent assertion
^ 1 (c1 ) H
^ 3 (c3 ))jj(H
^ 2 (c2 ) H
^ 4 (c4 ))j = 2(H
^ 1 (c1 ) H
^ 3 (c3 ))(H
^ 2 (c2 )
2j(H
^ 4 (c4 )) almost surely. And this simply means (H
^ 1 (c1 ) H
^ 3 (c3 ))(H
^ 2 (c2 )
H
^
H4 (c4 )) 0 almost surely.
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0