Optimal auctions with multidimensional types and the desirability of exclusion

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.

1

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-multi-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 = Πmj=1[aj, bj]⊂Rm, 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 typet}

i is distributed

according to a continuous probability density fi :T →R++. Thus, for every

contained in B is given by

Pi(B) :=

Z

B

fi(t)dλ(t),

where λis the Lebesgue measure on T. Bidder ivalues the object according

to the function

Ui :T →R.

Thus, if bidderihas private informationti ∈T, then the value or utility that

theith_{bidder assigns to the object isU}i_{(t). We shall assume that the function}

Ui _{is continuously differentiable. Since the i}th _{bidder’s utility function U}i _is

real-valued and continuous, the range of utilities (or valuations), denoted by

Ui_{(T), is given by some closed bounded interval [α}

i, βi]. We shall assume

the following:

Density Assumption: The distribution function

Fiu(x) = Pi

©

t∈T :Ui(t)≤xª

= Z

{t_∈T:Ui_(t)≤x}

fi(t)dλ(t),

induced by theith _{bidder’s utility functionU}i _{has a continuous density}

fu

i (·) such thatfiu(x)>0 for all x∈(αi, βi).

The density assumption will be satisfied for bidder i if for every bidder

typet ∈T, the vector of partial derivatives (i.e., the gradient vector) is such

that,

∇Ui_{(t) =}

µ

∂Ui

∂t1

(t), . . . ,∂U

i

∂tm

(t) ¶

6

= 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

ui− 1−F

u i (ui)

fu

i (ui)

,

is increasing in the bidder’s valuation ui _{∈ [α}

i, βi], given the distribution of

valuation Fu

i , then the optimal auction delivers the object to the bidder with

Let a = (a1, a2, . . . , am). Recall that T = Πmj=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 thatfu

i (αi) = 0. Then every type

with utility near αi will be excluded. Let M be a bound forfi. Ifx > αi,

Fu

i (x) =

Z

{t_∈T;Ui_(t)≤x}

f(t)dλ(t)≤M λ¡©

t∈T;Ui_(t)≤xª¢

.

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 thatPm

j=1|tj −aj| ≤r,

Ui_(t)−Ui_{(a) =}

Z 1

0 d ds

¡

Ui_(a+s(t−a))¢

ds=

Z 1

0 ¡

Ui¢′

(a+s(t−a))·(t−a)ds ≥δ

m

X

i=1

(ti−ai).

Thus, if Ui_{(t) ≤ x then for every j, |t}

j−r| ≤ Pmi=1(ti−ai) ≤ x−U

i_(a)

δ =

x−αi

δ . Therefore

λ¡©

t;Ui(t)≤xª¢

≤λ

µ½

z ∈Rm;|zj −aj| ≤

x−αi

δ for all j

¾¶

= (x−αi)

δm

m

.

Since

fi(αi) = lim x_→αi

Fu

i (x)−Fiu(αi)

x−αi

≤ M

δm _xlim_→α

i

(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

Remark 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) = t}

1 +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− (4−₂u)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 h2,4+√10

3 i

.

Appendix

In this appendix we prove theorem 1. For convenience the auctioneer will be

bidder 0.We defineP as the set of probability distributions on {0,1, . . . , n}.

Thus P = n(qj)

n

j=0 ≥0, Pn

j=0qj = 1

o

. The interpretation is that qj is the

probability that bidder j receives the object. Thusq0 is the probability that

the auctioneer keeps the object. We consider on Tn _{the product probability}

measure. The auction proceed as follows:

1. The auctioneer announces the measurable functions1 _{q : T}n _{→ P and}

P = (Pi₎n

i=1 where Pi :Tn →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

Pi_(t).

For a given direct mechanism (q, P) we define

Qi(ti) =E₋i[qi(t)] and Pi(ti) =E₋i

£

Pi(t)¤

. (1)

The mechanisms must satisfy incentive compatibility and individual ratio-nality constraints:

Qi(ti)Ui(ti)−Pi(ti)≥Qi(t′i)U

i

(ti)−Pi

³

t′i

´

,∀t′i, ti ∈T, (IC)

Qi(ti)Ui(ti)−Pi(ti)≥0. (IR)

Let us define the auxiliary function, Vi _{:T →}

R,

Vi_(t

i) =Qi(ti)Ui(ti)−Pi(ti).

The compatibility of incentives constraints can be rewritten as

Vi_(t

i)−Vi(t′i)≥Qi(t′i)

¡

Ui_(t

i)−Ui(t′i)

¢

,∀ti, t′i ∈Ti. (IC’)

The following lemma is important:

Lemma 1 There exists a convex functionφi :Ri →Rsuch thatVi =φi◦Ui.

Moreover Qi(ti)∈∂φi(Ui(ti)).

Proof: Note first that if a and c are elements of (Ui₎−1_{(r), r ∈ R}i _then

(IC’) implies Vi_(a)−Vi_(c)≥Q

i(c)·0 = 0.Analogously, Vi(c)−Vi(a)≥0.

Thus Vi_{(a) =V}i_{(c). Therefore}

φi :Ri →R, φi(r) :=Vi(xr),wherexr ∈

¡

Ui¢−1

(r), r ∈Ri

is well defined. Since

φi(γ) =Vi(xγ) = sup x′_∈T

i

Qi(x′)γ−Pi(x′)

it follows that φi is a convex function. Moreover, ifη =Ui(ti)

φi(γ)−φi(η)≥Qi(ti) (γ −η),for all γ, η∈Ri.

Thus, we have that Qi(ti)∈∂φi(Ui(ti)). QED

For every u= (u1_{, . . . , u}n_{)∈R =}Qm

i=1Ri define

~qi(u) = E

h

qi(x)|u=

¡

Uj(xj)

¢n

j=1 i

and

~

Pi(u) = EhPi(x)¯¯u=

¡

Uj(xj)

¢n

j=1 i

Remark 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 Q~i(ui) = E[~qi(u)|ui].

We calculate first

E£

~qi(u)|ui

¤

=E£

E[qi(x)|u]|ui

¤

=E£

qi(x)|ui

¤ =

E£

E[qi(x)|xi]|ui =Ui(xi)

¤

=E£

Qi(xi)|ui =Ui(xi)

¤

∈∂φi

¡

ui¢

.

The last relation is true since Qi(xi)∈∂φi(ui) = [φ−(ui), φ+(ui)] and

φ−¡

ui¢

=E£

φ−¡

ui¢

|ui¤

≤E£

Qi(xi)|ui =Ui(xi)

¤

≤E£

φ+¡

ui¢

|ui¤

=φ+¡

ui¢

.

The incentive compatibility constraints can be rewritten as

~

Qi

¡

ui¢

ui−EhP~i(u)¯¯

¯u

ii

≥Q~i(u′)ui−E

h

~

Pi(u)¯¯

¯u

′i_{, (IC’)}

The individual rationality constraint can be rewritten

~

Qi

¡

ui¢

ui−EhP~i(u)|ui

i

≥0.

The auctioneer expected revenue is

E

" _n X

i=1

Pi(x)

# =E " E " _n X i=1

Pi(x)

¯ ¯ ¯ ¯ ¯ u ## =E " _n X i=1 ~

Pi(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,Econometrica64 (1996),

2. Hoffmann-Jørgensen, “Probability with a view towards statistics,” Vol. II, Chapman & Hall, 1994.

3. R. B. Myerson, Optimal auction design, Mathematics of operations

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