Monteiro, Paulo Klinger

Page Junior, Frank H.

Svaiter, Benar Fux

English

Research Papers in Economics

 
 
 
 
 
 
 
 
Nº 537  ISSN 0104-8910
 
Optimal auctions with multidimensional types and the 
desirability of exclusion 
  
Paulo Klinger Monteiro  
Benar Fux Svaiter 
Frank H. Page Jr 
 
 
Março de 2004 Optimal Auctions with Multidimensional
Types and the Desirability of Exclusion∗
Paulo Klinger Monteiro
EPGE-FGV
Praia de Botafogo 190 sala 1103
22250-900 Rio de Janeiro RJ
Brazil
Benar fux Svaiter
IMPA
Estrada Dona Castorina 110
22460-320 Rio de Janeiro RJ
Brazil
Frank H. Page Jr.†
Department of Finance
University of Alabama
Tuscaloosa, AL 35487
USA
March, 2004
Abstract
Within the context of a single-unit, independent private values auc-
tion model, we show that if bidder types are multidimensional, then
under the optimal auction exclusion of some bidder types will occur.
A second contribution of the paper is methodological in nature. In
particular, we identify conditions under which an auction model with
multidimensional types can be reduced to a model with one dimen-
sional types without loss of generality. Reduction results of this type
have achieved the status of folklore in the mechanism design literature.
Here, we provide a proof of the reduction result for auctions.
Keywords: Optimal Auctions, Type Exclusion, Multidimensional
Types.
∗Monteiro and Svaiter acknowledge the ﬁnancial support of CNPq-Brazil. Page ac-
knowledges the ﬁnancial support and hospitality of CERMSEM, University of Paris I and
ﬁnancial support from the Department of Finance at the University of Alabama.
†corresponding author: fpage@cba.ua.edu
11 Introduction
Many results in the mechanism design literature rely heavily upon the as-
sumption that private information is one-dimensional. What are the conse-
quences for optimality if private information is multidimensional? The main
contribution of this paper is to show that, for the case of a single-unit auction
model with independent private values, if private information (in the form of
bidder types) is multidimensional, then the exclusion of types is a necessary
condition for optimality of the auction mechanism (see Theorem 2). A simi-
lar type of result was gotten by Armstrong in his paper on nonlinear pricing
for multiproduct monopolists. In particular, Armstrong (1996) showed that
if consumer utility is increasing in all arguments (i.e., in the goods vector
and in the types vector) and convex and homogeneous of degree one in the
types vector and if the set of consumer types is strictly convex with dimen-
sion greater than one, then under the optimal nonlinear pricing mechanism
a set of types with positive measure will not be served (i.e., will be excluded)
by the monopolist. Our main result, which requires neither convexity and
homogeneity of utility with respect to types nor strict convexity of the set of
types, suggests that in a wide variety of auction environments type exclusion
may be general property of optimality whenever multidimensional private
information is present.
The second contribution of our paper is methodological in nature. In
particular, we identify conditions under which an auction model with multi-
dimensional types can be reduced to a one-dimensional model without loss of
generality. Reduction results of this type are well-known in the mechanism
design literature, and are often used without proof. Here, we provide a proof
of the reduction result for independent private values auctions (see Theorem
1).
2 The model
There is one object to be sold at an auction with n bidders. The ith bidder
bases his valuation of the object on a multidimensional parameter
ti ∈ T = Π
m
j=1 [aj,bj] ⊂ R
m, m > 1
which summarizes his private information. We shall refer to ti as the ith
bidder’s type, and we shall assume that the ith bidder’s type ti is distributed
according to a continuous probability density fi : T → R++. Thus, for every
(Borel) subset of types B ⊂ T, the probability that bidder i has type ti
2contained in B is given by
Pi (B) :=
Z
B
fi (t)dλ(t),
where λ is the Lebesgue measure on T. Bidder i values the object according
to the function
U
i : T → R.
Thus, if bidder i has private information ti ∈ T, then the value or utility that
the ith bidder assigns to the object is Ui(t). We shall assume that the function
Ui is continuously diﬀerentiable. Since the ith bidder’s utility function Ui is
real-valued and continuous, the range of utilities (or valuations), denoted by
Ui (T), is given by some closed bounded interval [αi,βi]. We shall assume
the following:
Density Assumption: The distribution function
F
u
i (x) = Pi
©
t ∈ T : U
i (t) ≤ x
ª
=
Z
{t∈T:Ui(t)≤x}
fi (t)dλ(t),
induced by the ith bidder’s utility function Ui has a continuous density
fu
i ( ) such that fu
i (x) > 0 for all x ∈ (αi,βi).
The density assumption will be satisﬁed for bidder i if for every bidder
type t ∈ T, the vector of partial derivatives (i.e., the gradient vector) is such
that,
∇U
i(t) =
µ
∂Ui
∂t1
(t),...,
∂Ui
∂tm
(t)
¶
 = 0.
For a proof, see Hoﬀmann-Jørgensen (1994) sections 8.10.3 and 8.12.
The following theorem is proved in the appendix.
Theorem 1 (Reduction Result):
The optimal auction in the multidimensional case is the same as opti-
mal auction obtained by Myerson (1981) for the one-dimensional case. In
particular, if each bidder’s marginal valuation
u
i −
1 − F u
i (ui)
fu
i (ui)
,
is increasing in the bidder’s valuation ui ∈ [αi,βi], given the distribution of
valuation F u
i , then the optimal auction delivers the object to the bidder with
the highest marginal valuation.
3Let a = (a1,a2,...,am). Recall that T = Πm
j=1 [aj,bj]. Thus, a ∈ T is
the type vector for which each component is as small as possible. Our main
result is the following:
Theorem 2 (Exclusion Result):
If the ith bidder’s utility function is such that ∂Ui
∂tj (a) > 0, j = 1,...,m,
then in the optimal auction there is exclusion of types of bidder i.
Proof. It suﬃces, by continuity, to show that fu
i (αi) = 0. Then every type
with utility near αi will be excluded. Let M be a bound for fi. If x > αi,
F
u
i (x) =
Z
{t∈T;Ui(t)≤x}
f (t)dλ(t) ≤ Mλ
¡©
t ∈ T;U
i (t) ≤ x
ª¢
.
Now by the continuity of the partial derivatives, there is a r > 0 such that
δ = min
(
∂Ui
∂tj
(t);1 ≤ j ≤ m,t ∈ T,
m X
j=1
|tj − aj| ≤ r
)
> 0.
For any t ∈ T such that
Pm
j=1 |tj − aj| ≤ r,
U
i (t) − U
i (a) =
Z 1
0
d
ds
¡
U
i (a + s(t − a))
¢
ds =
Z 1
0
¡
U
i¢′ (a + s(t − a))   (t − a)ds ≥ δ
m X
i=1
(ti − ai).
Thus, if Ui (t) ≤ x then for every j, |tj − r| ≤
Pm
i=1 (ti − ai) ≤
x−Ui(a)
δ =
x−αi
δ . Therefore
λ
¡©
t;U
i (t) ≤ x
ª¢
≤ λ
µ½
z ∈ R
m;|zj − aj| ≤
x − αi
δ
for all j
¾¶
=
(x − αi)
δm
m
.
Since
fi (αi) = lim
x→αi
F u
i (x) − F u
i (αi)
x − αi
≤
M
δm lim
x→αi
(x − αi)
m−1
and m > 1 we conclude that fi (αi) = 0 ending the proof. QED
Remark 1 Consider the one dimensional case with T = [a,b]. Then type t
is excluded if and only if tfi (t) ≤ 1 − Fi (t). For example if the distribution
is uniform on [a,b] there is no exclusion if a
b−a > 1 i.e. if 2a > b.
4Remark 2 Also, we see from our the last remark that what drives exclusion
with multidimensional types is that fi (αi) = 0 is a general property.
Remark 3 The hypothesis ∂Ui
∂tj (a) > 0 is necessary for the result. For ex-
ample if Ui (t1,t2) = t1 − t2/2 the conclusion will not be true in general.
Example 1 As an example we calculate the exclusion set in a simple case
with the uniform distribution. Consider Ui (t) = t1 + t2. We suppose that t
is uniformly distributed in [1,2]
2 . The distribution of Ui given by
Fi (u) =
(
(u−2)2
2 if 2 ≤ u ≤ 3
1 −
(4−u)2
2 if 3 ≤ u ≤ 4,
has density fU (u) =
½
u − 2 if 2 ≤ u ≤ 3
4 − u if 3 ≤ u ≤ 4.
Note that the density function is continuous and it is zero for u = 2,4. Thus,
the set of excluded types is
h
2, 4+
√
10
3
i
.
Appendix
In this appendix we prove theorem 1. For convenience the auctioneer will be
bidder 0. We deﬁne P as the set of probability distributions on {0,1,...,n}.
Thus P =
n
(qj)
n
j=0 ≥ 0,
Pn
j=0 qj = 1
o
. The interpretation is that qj is the
probability that bidder j receives the object. Thus q0 is the probability that
the auctioneer keeps the object. We consider on T n the product probability
measure. The auction proceed as follows:
1. The auctioneer announces the measurable functions1 q : T n → P and
P = (P i)
n
i=1 where P i : T n → R;
2. The bidder i conﬁdentially announces to the auctioneer types ti ∈ T;
The auctioneer forms the vector t = (ti)
n
i=1
3. The objects are delivered accordingly to q (t) ∈ P and bidder i pays
P i (t).
For a given direct mechanism (q,P) we deﬁne
Qi (ti) = E−i [qi (t)] and P
i (ti) = E−i
£
P
i (t)
¤
. (1)
1Henceforth called direct mechanisms.
5The mechanisms must satisfy incentive compatibility and individual ratio-
nality constraints:
Qi (ti)U
i (ti) − P
i (ti) ≥ Qi (t
′
i)U
i (ti) − P
i
³
t
′
i
´
,∀t
′
i,ti ∈ T, (IC)
Qi (ti)U
i (ti) − P
i (ti) ≥ 0. (IR)
Let us deﬁne the auxiliary function, V i : T → R,
V
i (ti) = Qi (ti)U
i (ti) − P
i (ti).
The compatibility of incentives constraints can be rewritten as
V
i (ti) − V
i (t
′
i) ≥ Qi (t
′
i)
¡
U
i (ti) − U
i (t
′
i)
¢
,∀ti,t
′
i ∈ Ti. (IC’)
The following lemma is important:
Lemma 1 There exists a convex function φi : Ri → R such that V i = φi◦Ui.
Moreover Qi (ti) ∈ ∂φi (Ui (ti)).
Proof: Note ﬁrst that if a and c are elements of (Ui)
−1 (r),r ∈ Ri then
(IC’) implies V i (a)−V i (c) ≥ Qi (c) 0 = 0. Analogously, V i (c)−V i (a) ≥ 0.
Thus V i (a) = V i (c). Therefore
φi : R
i → R,φi (r) := V
i (xr),where xr ∈
¡
U
i¢−1 (r),r ∈ R
i
is well deﬁned. Since
φi (γ) = V
i (xγ) = sup
x′∈Ti
Qi (x
′)γ − P
i (x
′)
it follows that φi is a convex function. Moreover, if η = Ui (ti)
φi (γ) − φi (η) ≥ Qi (ti)(γ − η),for all γ,η ∈ R
i.
Thus, we have that Qi (ti) ∈ ∂φi (Ui (ti)). QED
For every u = (u1,...,un) ∈ R =
Qm
i=1 Ri deﬁne
  qi (u) = E
h
qi (x)|u =
¡
U
j (xj)
¢n
j=1
i
and
  P
i (u) = E
h
P
i (x)
¯ ¯u =
¡
U
j (xj)
¢n
j=1
i
.
6Remark 4 The conditional expectation is only deﬁned almost everywhere.
For the step above we need to use a regular conditional distribution, so that
the conditional expectation is deﬁned everywhere (see Hoﬀmann-Jørgensen
10.3, page 120).
Note ﬁrst that
Pn
i=0   qi (u) = 1. Let us calculate   Qi (ui) = E [  qi (u)|ui].
We calculate ﬁrst
E
£
  qi (u)|u
i¤
= E
£
E [qi (x)|u]|u
i¤
= E
£
qi (x)|u
i¤
=
E
£
E [qi (x)|xi]|u
i = U
i (xi)
¤
= E
£
Qi (xi)|u
i = U
i (xi)
¤
∈ ∂φi
¡
u
i¢
.
The last relation is true since Qi (xi) ∈ ∂φi (ui) = [φ− (ui),φ+ (ui)] and
φ
− ¡
u
i¢
= E
£
φ
− ¡
u
i¢
|u
i¤
≤ E
£
Qi (xi)|u
i = U
i (xi)
¤
≤ E
£
φ
+ ¡
u
i¢
|u
i¤
= φ
+ ¡
u
i¢
.
The incentive compatibility constraints can be rewritten as
  Qi
¡
u
i¢
u
i − E
h
  P
i (u)
¯ ¯ ¯u
i
i
≥   Qi (u
′)u
i − E
h
  P
i (u)
¯ ¯ ¯u
′
i
, (IC’)
The individual rationality constraint can be rewritten
  Qi
¡
u
i¢
u
i − E
h
  Pi (u)|u
i
i
≥ 0.
The auctioneer expected revenue is
E
"
n X
i=1
P
i (x)
#
= E
"
E
"
n X
i=1
P
i (x)
¯ ¯ ¯ ¯ ¯
u
##
= E
"
n X
i=1
  P
i (u)
#
.
Thus the optimal auction problem is reduced to the one-dimensional op-
timal auction solved by Myerson. There is one twist however. Myerson’s
assumptions require that the density function have a positive minimum (i.e.,
have positive values everywhere). Already we have seen here simple exam-
ples where Myerson’s everywhere positivity assumption is not satisﬁed. One
can easily see, however, that Myerson’s proof works as long as the density is
positive and continuous in the interior of Ri and this is an assumption that
is satisﬁed by many examples including the ones considered here.
References
1. M. Armstrong, Multiproduct nonlinear pricing, Econometrica 64 (1996),
pp. 51-75;
72. Hoﬀmann-Jørgensen, “Probability with a view towards statistics,” Vol.
II, Chapman & Hall, 1994.
3. R. B. Myerson, Optimal auction design, Mathematics of operations
research, 6 (1981), pp. 58-73.
8 
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Optimal auctions with multidimensional types and the desirability of exclusion

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Optimal auctions with multidimensional types and the desirability of exclusion

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