Approximation in Mechanism Design with Interdependent Values
YUNAN LI, University of Pennsylvania
The seminal work of [Myerson 1981] shows that the simple Vickrey-Clarke-Groves (VCG) mechanism
with monopoly reserves is optimal in single-item auctions where agents have independent and identically
distributed private values. [Hartline and Roughgarden 2009] and others prove approximate analogs of this
statement in the more general one-dimensional type, independent private values setting. This paper studies
the design of ex-post incentive compatible and approximately optimal mechanism in an environment with
interdependent values. We show that the VCG mechanism is ex-post incentive compatible under a matroid
feasibility constraint. We also give conditions under which the VCG mechanism with monopoly reserves
could still generate near-optimal expected revenues.
Categories and Subject Descriptors: F.0 [Theory of Computation]: General
General Terms: Economics, Theory, Algorithms
Additional Key Words and Phrases: Mechanism Design, Approximation, Interdependent Values, Revenue
Maximization
1. INTRODUCTION
In the independent private values (IPV) setting, it is well known that the VickreyClarke-Groves (VCG) mechanism is ex-post incentive compatible. In addition, when
there is a single item for sale and the agents’ values are drawn from identical distributions, the seminal work of [Myerson 1981] shows that the VCG mechanism with
monopoly reserves is revenue-maximizing. Recent works show that this idea is more
general. [Hartline and Roughgarden 2009] show that in a variety of IPV settings
monopoly reserves can be used to construct “approximately” optimal mechanisms. This
paper considers the design of approximately optimal mechanisms for an important
generalization of this setting. We relax the assumption that agents have independent
private values, and consider the case of interdependent values.
The case of interdependent values is pertinent for many practical applications, and
has received much attention in the literature in economics on auctions, starting with
[Milgrom and Weber 1982]. In an interdependent values setting, one agent’s valuation
for winning can depend on other agents’ private information, and the private information of agents may be correlated. In this paper, we consider a setting where the agent’s
private information can be summarized by a one-dimensional signal. As a motivating example, suppose the good for sale may be resold, and buyers may have different
information about future states of the world, e.g. market conditions. Then the information possessed by other buyers, if known to a particular buyer, may affect his/her
valuation of winning. For example a buyer’s value for an art piece depends not just
on his/her private consumption value, but also on his/her beliefs about the art’s resale
value, which depends on other buyers’ values. Furthermore, in many applications buyers may have different but correlated private information. A classical example is that
of a tract of oil for sale, where each potential buyer surveys the tract and estimates the
A full version of this paper is available at: https://economics.sas.upenn.edu/sites/economics.sas.upenn.edu/files/ec41li.pdf
See the Acknowledgements section before REFERENCES.
Author’s addresses: Yunan Li, Economics Department, University of Pennsylvania, Philadelphia, PA, USA,
yunanli@sas.upenn.edu.
Copyright is held by the author/owner(s).
EC’13, June 16–20, 2013, Philadelphia, PA, USA.
ACM 978-1-4503-1962-1/13/06.
extractable oil — buyers’ estimates are therefore statistically correlated. Additionally
a buyer’s private information will impact others’ valuations — for example if buyer 1
finds that buyer 2 has a lower estimate on the amount of extractable oil, he/she may
revise his/her own valuation downwards. Several seminal applied papers on auctions
use an interdependent values setting (see, e.g. [Hendricks and Porter 1988] and [Hendricks et al. 2003]).
Instead of considering that there is a single item for sale, we consider the environment where there is a system of feasible sets specifying which subset of buyers can
win simultaneously. For example, in the k-unit good auction with unit-demand buyers, the seller can sell to at most k buyers. The feasible sets are precisely those subsets
which contain no more than k buyers. Another example is a combinatorial auction with
single-minded buyers, where a feasible set corresponds to a subset of buyers seeking
mutually disjoint bundles. In such environments, identifying the optimal mechanism
remains an open question. This paper investigates a simpler question: can we find
a simple mechanism that is ex-post incentive compatible and performs “reasonably”
well?
Specifically, we study the performance of the VCG mechanism with monopoly reserves, which is the optimal mechanism in the IPV, single unit environment. As is
standard in the literature studying ex-post incentive compatibility for interdependent
value problems, we assume that agents’ valuations satisfy a single-crossing condition.
Under this condition, the VCG mechanism is ex-post incentive compatible in singleitem auctions (see, e.g. [Ausubel 2000]). Unfortunately, this need not be true once we
leave the single-item auction setting. One contribution of this paper is to show that
when the system of feasible sets is a matroid (described below), the VCG mechanism
is still ex-post incentive compatible. We exhibit a novel example where the VCG mechanism is not ex-post incentive compatible when the system of feasible sets is not a
matroid. The matroid feasibility constraint covers many interesting cases. Examples
include single-item auctions, the allocation of homogeneous goods ([Ausubel 2004]) and
digital good auctions ([Goldberg et al. 2001]), among others.
If in addition the agents’ valuations and the distribution of their private information satisfy a generalized monotone hazard rate condition, we prove that the seller’s
expected revenue by employing the VCG mechanism with monopoly reserves is at least
1/e of the full surplus, where e is the base of the natural logarithm. The proof uses the
fact that distributions meeting the monotone hazard rate condition have tails no heavier than that of an exponential distribution (which has a constant hazard rate). The
identified bound is tight for arbitrary number of buyers. It is attained in a single-item
auction when there is only one “serious” buyer with an exponentially distributed valuation, and all the other buyers’ valuations of the item are negligible.
This paper is broadly related to the large literature on revenue-maximizing mechanism design (see, e.g. [Krishna 2009]). The works most closely related to the current
paper are those on approximation guarantees for simple mechanisms. The literature
can be classified into two branches.
One branch studies the general one-dimensional type environments where agents
have independent private values. [Chawla et al. 2007], [Hartline and Roughgarden
2009] and [Dhangwatnotai et al. 2010] examine the extent to which simple mechanisms can achieve good approximations of the optimal one. This paper is closely related
to [Hartline and Roughgarden 2009]. In their paper, they also study the general onedimensional type environments, but focus on the case where agents have independent
private values. They consider two classes of environments, one in which the system of
feasible sets is downward-closed and the valuation distributions satisfy the increasing
hazard rate condition, and one in which the system of feasible sets is a matroid and the
valuation distributions are regular. This paper only considers matroid environments
since the ex-post incentive compatibility of the VCG mechanism fails in the more general downward-closed environments even if we assume the agents’ valuations satisfy
the single-crossing condition. They prove that the seller’s expected revenue by employing the VCG mechanism with buyer specific reserves is at least 1/2 of the optimal
revenue in both settings. See [Hartline 2012] for a thorough survey on this.
The other branch focuses on the single-item auctions where agents have correlated
private values. [Ronen 2001] proposes a mechanism that can achieve at least half of the
optimal revenue. [Ronen and Saberi 2002] prove that no deterministic polynomial time
ascending auction can achieve an approximation ratio better than 3/4. The paper most
related to the current paper is [Neeman 2003], which compares the expected revenue
generated by the English auction (with or without reserves) with the full surplus. He
considers the environments where there are n agents and each agent’s expected value
is as least α of his/her maximum possible value, and quantifies the fraction of surplus
extracted as a function of both n and α.
Contemporaneously with the present paper, [Roughgarden and Talgam-Cohen forthcoming] study single sample mechanisms in interdependent value settings, with positive results but slightly different assumptions. The reader is encouraged to see their
paper for an alternative approach to approximately optimal mechanism design in these
settings.
The focus of the current paper is on the design of approximate revenue-maximizing
mechanisms, but a number of papers have analyzed mechanisms that achieves social
efficiency in the interdependent value cases. [Ausubel 2000] gives an elegant extension
of the Vickrey auction that achieves efficiency when there are n buyers and k identical goods for sale. He does not restrict attention to unit-demand buyers, but assumes
buyers’ marginal value from an additional unit of good is decreasing. The VCG mechanisms considered in this paper are a generalization of his to accommodate the more
general feasibility constraint faced by the seller. [Dasgupta and Maskin 2000] constructs a detail-free mechanism achieves efficiency even if the goods for sale are heterogenous. In the homogeneous-goods downward sloping demand environment, [Perry
and Reny 2002] proposes another simpler modification of Vickrey’s auction which consists of a collection of second-price single-unit auctions carried out over two rounds and
can achieve an socially efficient outcome. [Bikhchandani et al. 2011] study ascending
auctions that implement the Vickrey outcome when the seller is constrained to sell
bases of a matroid.
The rest of the paper is organized as follows. We present the model in Section 2.
In Section 3, we show that the VCG mechanism is ex-post incentive compatible, and
approximately optimal when supplemented with monopoly reserves. We conclude in
Section 4 with a discussion of future work.
2. THE MODEL
There are a finite set N of buyers, where n = |N |, and a collection I ⊆ 2N of feasible
sets of buyers, which are the subsets of buyers that can simultaneously win. For example, in a k−unit good auction with unit-demand buyers, a feasible set is any subset
of N containing at most k buyers. We assume that (N , I) is a matroid, i.e.:
Definition 2.1 ([Oxley 2006]). We say (N , I) is a matroid if
(1) ∅ ∈ I;
(2) For all S 0 ⊆ S ⊆ N , if S ∈ I then S 0 ∈ I;
(3) For all S, S 0 ∈ I, if |S| > |S 0 |, then there exists i ∈ S\S 0 such that S 0 ∪ {i} ∈ I.
A subset S of N is said to be independent if S ∈ I, and dependent if it is not independent. A set B ∈ I is called a base of the matroid if it becomes dependent upon adding
any element of N . Examples of matroid environments include digital good auctions
where I = 2N and k-unit good auctions with unit-demand buyers where I is the collection of all subsets of N which contains at most k buyers. The seller’s reservation
value is 0.
Each buyer i ∈ N receives a private signal (buyer i’s type) ti ∈ Ti ≡ [ti , t̄i ] ⊆ R+ . Let
t ≡ (t1 , · · · , tn ) and T ≡ Πni=1 Ti . Let v : T → Rn+ , where vi (t) ∈ R+ denotes buyer i’s
valuation of winning. We assume the payoff to buyer i when he/she does not win (and
does not pay) is 0. We assume that vi (·) is non-negative, differentiable and increasing
in all its elements, and strictly increasing in ti . Furthermore, we assume that v(·)
satisfies the following single-crossing condition:
Assumption 2.2. For all i, j (i 6= j), t−i and t0i > ti , we have
vi (ti , t−i ) ≥ vj (ti , t−i ) =⇒ vi (t0i , t−i ) > vj (t0i , t−i ),
and
vi (t0i , t−i ) ≤ vj (t0i , t−i ) =⇒ vi (ti , t−i ) < vj (ti , t−i ).
This single-crossing condition is typically assumed in most work that study ex-post
incentive compatibility for interdependent value problems (e.g. [Crmer and McLean
1985] and [Ausubel 2000]). However, in the standard interpretation, vi is a reduced
form utility function that defines agent i’s expected valuation of winning under some
circumstances given the agents’ signals. Agent i’s valuation depends on others’ signals only to the extent that the signals provide information about the state of nature. [McLean and Postlewaite 2006] show that the conditions on the primitive utility
functions that would ensure that the reduced form utility functions satisfy the singlecrossing condition are stringent.
Let F : T → [0, 1] denote the cumulative distribution of t, with a joint density f such
that f (t) > 0 for all t ∈ T . Let Fi (·|t−i ) denote the corresponding conditional cumulative
distribution of ti given t−i , with a conditional density fi (·|t−i ). We assume that v(·) and
F (·) satisfy the following generalized monotone hazard rate condition:
Assumption 2.3.
1 − Fi (ti |t−i ) ∂vi (t)
is decreasing in ti for all i and t−i .
fi (ti |t−i )
∂ti
It is easy to see that the IPV settings where each buyer’s value is drawn from an increasing hazard rate distribution satisfy Assumptions 2.2 and 2.3. First, when buyers
have private values, Assumption 2.2 is trivially satisfied since a buyer’s value only depends on his/her own type. Second, usually in the private values setting, we normalize
vi (t) = ti . Then ∂vi (t)/∂ti = 1 and Assumption 2.3 is satisfied if ti follows a distribution
with an increasing hazard rate.
There are also other interesting cases that satisfy Assumptions 2.2 and 2.3. Note
that one easy way to make sure Assumption 2.3 to be satisfied is to make sure that for
all i and t−i :
∂vi (·, t−i )
is decreasing; and
∂ti
(2) Fi (·|t−i ) has an increasing hazard rate.
(1)
We first give examples of v(·) that satisfy Assumption 2.2 and have strictly positive
decreasing partial derivatives, ∂vi (t)/∂ti , and second we give examples of distributions
F whose conditionals have increasing hazard rates.
Example 2.4.
P For all i ∈ N , buyer i’s valuation of winning takes the following form:
vi (t) = ti + β j6=i tj with 0 < β < 1.
Note that for any (v, F ), we can define (ṽ, F̃ ) such that ṽi (g(t)) = vi (t) and F̃ (g(t)) =
F (t), where gi : Ti → Si ⊂ R+ is strictly increasing and g(t) = (g1 (t1 ), · · · , gn (tn )).
Then (v, F ) and (ṽ, F̃ ) represent the same environment. Clearly, vi is non-negative,
increasing and strictly increasing in ti if and only if ṽi is non-negative, increasing and
strictly increasing in si = gi (ti ). Suppose further that gi is differentiable, then
∂ṽi (s) 0
∂vi (t)
g (tj ) =
.
∂sj j
∂tj
This implies that ṽ(·) satisfies the single-crossing condition if and only if v(·) does.
Finally, suppose both v and g are twice differentiable, then we have
−1 2
−1 2
∂vi (t)
∂vi (t)
∂ ṽi (s)
∂ vi (t) gi0 0 (t)
gi0 (ti )2
=
− 0 .
(1)
2
∂ti
∂si
∂ti
∂t2i
gi (t)
h
i0
Suppose for all i and ti , supt−i ln ∂v∂ti (t)
< +∞, then we can define gi such that
i
0
∂vi (t)
0
[ln gi (ti )] ≥ sup ln
.
∂ti
t−i
Then it follows from (1) that ∂ 2 ṽi (s)/∂s2i ≤ 0. Hence without much loss of generality
we can assume that ∂vi (t)/∂ti is decreasing.
We now turn to the condition that for all i and t−i the conditional distribution
Fi (·|t−i ) has an increasing hazard rate. There is a literature in mathematics about
conditionally specified distributions, i.e. whether there exists a unique joint distribution compatible with given families of conditional distributions. [Arnold et al. 2001]
provides a thorough survey on this topic. In the special case where the conditional distributions belong to exponential families, which includes many of the most common
distributions including Poisson, binomial, exponential, gamma, multivariate normal,
Weibull and many others, it is known that the joint distribution exists and must be of
some specific form. In the literature about auctions with interdependent values, it is
common to assume that t are affiliated and therefore positively correlated. For the bivariate case, [Arnold 1990] gives sufficient conditions on conditionals for the negative
or positive dependence of the two variables, and his results can be extended to multivariate cases. Here are some examples of distributions F whose conditionals Fi (·|t−i )
have increasing hazard rates.
Example 2.5. Suppose t follows a multivariate normal distribution, then the conditional distribution of ti given t−i is normal, which has an increasing hazard rate.
Example 2.6. Suppose t is two-dimensional and has a joint density positive on the
positive orthant and zero elsewhere. [Arnold and Strauss 1988] prove that the conditional distributions f1 (t1 |t2 ) and f1 (t2 |t1 ) are both exponential if and only if the joint
distribution is proportional to exp(−λt1 − µt2 − νt1 t2 ) for some λ, µ > 0 and ν ≥ 0. It
is well know that the exponential distribution has a constant hazard rate. Note that
under this distribution t1 and t2 are either independent or negatively correlated.
Without loss of generality we focus on direct mechanisms, in which each buyer reports his/her own signal directly. A mechanism M is a pair (q, p), where the allocation rule q maps every vector of reported signals t̃ to a probability mass function,
q(t̃) ∈ ∆|I|−1 , and the payment rule p maps every t̃ to a payment vector (in Rn+ ). Let
P
xi (t̃) ≡ {S∈I:i∈S} q(t̃)(S) denote the probability with which buyer i is one of the winners.
If the seller had complete information about the buyers’ valuations of winning and
full bargaining power, the maximum total surplus he could extract from the buyers is:
"
#
X
E max
vi (t) .
(2)
S∈I
i∈S
We are interested in obtaining a lower bound on the ratio between the expected revenue the seller can obtain by employing the VCG mechanism with monopoly reserves
(described in detail later) and the full surplus given by (2). Obviously, the full surplus
provides an upper bound for the expected revenue the seller can obtain by employing
the optimal mechanism. It is attainable in some environments. For example, when the
buyers’ information are correlated, [Crmer and McLean 1988] and [McAfee and Reny
1992] provide conditions under which the optimal mechanism succeeds in extracting
the full surplus. Ideally, we want to compare the VCG mechanism with monopoly reserves with the optimal mechanism. However, for the general environment considered
here, a characterization of the optimal mechanism is unavailable. To the extent that
the optimal mechanism falls short of extracting the full surplus, our measurement
provides a lower bound on the performance of the mechanism relative to the optimal
one.
3. VCG MECHANISMS
This section studies the incentive compatibility and approximate optimality of the
VCG mechanisms. In Section 3.1 we define the mechanisms and show that they are
ex-post incentive compatible if v(·) satisfies the single-crossing condition and the system of feasible sets is a matroid. We show in Section 3.2 that if in addition the generalized monotone hazard rate condition is satisfied, the VCG mechanism with monopoly
reserves is approximately optimal.
3.1. Ex-Post Incentive Compatibility
Definition 3.1. We say a mechanism M = (q, p) is ex-post incentive compatible if
and only if for all i, t−i and ti
vi (t)xi (t) − pi (t) ≥ vi (t)xi (t̃i , t−i ) − pi (t̃i , t−i ), ∀t̃i ∈ Ti .
(3)
We say a mechanism is Bayesian incentive compatible if E[vi (t)xi (t) − pi (t)|ti ] ≥
E[vi (t)xi (t̃i , t−i ) − pi (t̃i , t−i )|ti ] for all i, ti and t̃i . Note that if a mechanism is ex-post
incentive compatible then it is Bayesian incentive compatible. The Bayesian incentive
compatible mechanism has been very well characterized in the IPV settings by [Myerson 1981], and the same method can be easily extended to characterize ex-post incentive compatible mechanism in the IPV settings (see, e.g. [Archer and Tardos 2001]).
The following lemma extended their results to the case where agents have interdependent values and correlated types:
L EMMA 3.2. A mechanism M = (q, p) is ex-post incentive compatible if and only
if xi (·, t−i ) is increasing for all i and t−i , and the payment function satisfies, for each t
and each buyer i,
Z ti
∂vi (s, ti )
xi (s, t−i )ds − ui (ti , t−i ),
(4)
pi (t) = vi (t)xi (t) −
∂s
ti
where ui (t) ≡ vi (t)xi (t) − pi (t).
P ROOF. This proof follows [Myerson 1981]. Suppose M = (q, p) is ex-post incentive
compatible, then, for all i, t−i and ti > t0i , we have
ui (ti , t−i ) ≥ ui (t0i , t−i ) + [vi (ti , t−i ) − vi (t0i , t−i )]xi (t0i , t−i ),
and
ui (t0i , t−i ) ≥ ui (ti , t−i ) + [vi (t0i , t−i ) − vi (ti , t−i )]xi (ti , t−i ).
Combine the above two inequalities and we get
[vi (t) − vi (t0i , t−i )]xi (t) ≥ ui (t) − ui (t0i , t−i ) ≥ [vi (t) − vi (t0i , t−i )]xi (t0i , t−i ).
(5)
vi (t) − vi (t0i , t−i )
Since vi (·, t−i ) is strictly increasing by assumption, we have
> 0. Then
(5) implies that xi (ti , t−i ) ≥ xi (t0i , t−i ). That is, xi (·, t−i ) is increasing. Dividing (5) by
ti −t0i and taking t0i → ti proves that for any t at which xi (·, t−i ) is continuous, ∂ui (t)/∂ti
exists and
∂ui (t)
∂vi (t)
=
xi (t).
∂ti
∂ti
Finally, ui (·, t−i ) is absolutely continuous, since vi is differentiable and
|ui (t) − ui (t0i , t−i )| ≤ |vi (t) − vi (t0i , t−i )| max{xi (t0i , t−i ), xi (t)} ≤ |vi (t) − vi (t0i , t−i )|.
Thus,
Z
ti
ui (t) = ui (ti , t−i ) +
ti
∂vi (s, ti )
xi (s, t−i )ds.
∂s
Hence (4) holds by the definition of ui . It is straightforward to verify that when xi (·, t−i )
is increasing for all i and t−i and the payment function satisfies (4), the mechanism
M = (q, p) is ex-post incentive compatible.
A direct corollary of Lemma 3.2 is that for any ex-post incentive compatible mechanism, the expected payment of buyer i is
E[pi (t)] = −E[ui (ti , t−i )] + E [xi (t)ϕi (t)] ,
where
ϕi (t) ≡ vi (t) −
1 − Fi (ti |t−i ) ∂vi (t)
.
fi (ti |t−i )
∂ti
(6)
Here ϕi is buyer i’s “virtual value” in the interdependent value case. Hence the seller
gets the same expected revenues from any two ex-post incentive compatible mechanisms which have the properties that (1) given t, a buyer is one of the winners with
the same probabilities, and (2) a buyer with the lowest signal gets the same expected
utilities.
Consider now the allocation rule of the VCG mechanism with no reserves. Given a
vector of reportedP
signals t̃, the mechanism chooses a feasible set S ∈ I that maximizes
the total surplus i∈S vi (t̃). Let x∗i denote buyer i’s probability of winning implied by
the mechanism, then x∗i (t) ∈ {0, 1} for all t since the mechanism is deterministic. In
general, x∗i (·, t−i ) need not be increasing, and therefore the mechanism need not be
ex-post incentive compatible:
Example 3.3. Buyer 1 is “ big” and his/her valuation is v1 (t) = 5+t1 . Buyers 2, · · · , 11
are “small” and their valuations are vi (t) = ti + 0.5t1 . Then v(·) satisfies the singlecrossing condition. The feasible sets are precisely those that do not contain both the
big buyer and a small buyer. Let ti = 0 for i ≥ 2. Suppose t1 = 1, then v1 (t) = 6 and
P
VCG mechanism will allocate to the big buyer. Suppose t1 = 2,
i≥2 vi (t) = 5, and the
P
then v1 (t) = 7 and i≥2 vi (t) = 10, and the VCG mechanism will allocate to the set
of small buyers. The probability with which the big buyer wins falls as his/her signal
increases.
x∗i (·, t−i ) is increasing when the system of feasible sets is a matroid:
L EMMA 3.4. Suppose (N , I) is a matroid, and v(·) satisfies Assumption 2.2. Let x∗i
denote the probability with which buyer i is one of the winners under the VCG mechanism with no reserves. Then for all i and t−i , x∗i (·, t−i ) is increasing. Furthermore,
let
t̂i (t−i ) ≡ inf{t0i |x∗i (t0i , t−i ) = 1}.
(7)
Then x∗i (t) = 1 if ti > t̂i (t−i ) and x∗i (t) = 0 if ti < t̂i (t−i ).
The proof of Lemma 3.4 relies on the following well-known property of matroids:
L EMMA 3.5 ([O XLEY 2006]). Let (N , I) be a matroid, and B denote the collection of
its bases. For every A, B ∈ B and a ∈ A\B, there exists b ∈ B such that A\{a} ∪ {b} ∈ B
and B\{b} ∪ {a} ∈ B.
P ROOF OF L EMMA 3.4. Fix i and t−i . Suppose that there exist t0i > ti such that
= 1 and x∗i (t0i , t−i ) = 0. Let W ≡ {j|x∗j (ti , t−i ) = 1} and W 0 ≡ {j|x∗j (t0i , t−i ) = 1}
denote the set of winners under (ti , t−i ) and (t0i , t−i ), respectively. Then i ∈ W , but i ∈
/
W 0 . Clearly, W and W 0 are bases of the matroid. By Lemma 3.5, there exits j ∈ W 0 \W
0
such that S1 ≡ W ∪ {j}\{i} ∈ I and
P S2 ≡ W ∪ {i}\{j}
P ∈ I. Since W is surplus
maximizing given (ti , t−i ), we have i∈W vi (ti , t−i ) ≥ i∈S1 vi (ti , t−i ), which implies
that
x∗i (ti , t−i )
vi (ti , t−i ) ≥ vj (ti , t−i ).
0
(t0i , t−i ),
(8)
Similarly,
W is Psurplus maximizing given
P
0
0
v
(t
,
0
i
i t−i ) ≥
i∈W
i∈S2 vi (ti , t−i ), which implies that
and therefore we have
vj (t0i , t−i ) ≥ vi (t0i , t−i ).
(9)
However, by the single-crossing condition, (8) implies that
vi (t0i , t−i ) > vj (t0i , t−i ),
which contradicts with (9). Hence x∗i (·, t−i ) is increasing. It then follows directly that
x∗i (t) = 1 if ti > t̂i (t−i ) and x∗i (t) = 0 if ti < t̂i (t−i ).
Given Lemma 3.4, we can define the payment rule of the VCG mechanism with no
reserves as following: a buyer pays if and only if he/she is one of the winners, and in
case of winning he/she pays vi (t̂i (t−i ), t−i ). It is easy to verify that this is consistent
with (4) and therefore by Lemma 3.2 the VCG mechanism with no reserves is ex-post
incentive compatible. This is also intuitive. On the one hand, a buyer has no incentive
to underreport since his/her winning probability is increasing in his/her own type, but
his/her payment upon winning is independent of his/her own type. On the other hand,
a buyer has no incentive to overreport since otherwise he/she might win, but pay an
amount that is higher than his/her value. Note also that the VCG mechanism with no
reserves is ex-post efficient and ex-post individually rational.
In the rest of the paper, we focus on the following class of VCG mechanisms with
reserves (VCGr):
Definition 3.6. The VCG mechanism with reserves (VCGr), r̃i (·), is a direct mechanism in which buyers simultaneously report their signals, and given the vector of
reported signals t̃, the set of winners and their payments are decided in the following
three steps:
P
(1) Choose a set W ∗ ∈ arg maxS∈I i∈S vi (t̃), and let x∗i denote the probability with
∗
which buyer i is in W .
(2) Delete all the buyers i in W ∗ with t̃i such that t̃i < r̃i (t̃−i ), where r̃i : T−i → Ti .
Then the set of winners is W = {i ∈ W ∗ |t̃i ≥ r̃i (t̃−i )}.
(3) Charge each winner the larger of vi (r̃i (t̃−i ), t̃−i ) and vi (t̂i (t̃−i ), t̃−i ), where t̂i (·) is
given by (7) using x∗i in step 1.
The VCG mechanism with no reserves is a special VCGr mechanism with r̃i (t−i ) = ti
for all i and all t−i . The VCGr mechanisms are ex-post incentive compatible under the
matroid feasibility constraint. To see this let x denote the allocation rule of a VCGr,
then xi (t) = 1 if and only if x∗i (t) = 1 and ti ≥ r̃i (t−i ). Suppose xi (t) = 1 and consider t0i > ti . First, x∗i (t0i , t−i ) = 1 by Lemma 3.4. Second, t0i > ti ≥ r̃i (t−i ). Hence
xi (t0i , t−i ) = 1. That is, xi (·, t−i ) is increasing. Finally, buyer i’s payment function in the
VCGr satisfies:
0
if ti < max{t̂i (t−i ), r̃i (t−i )},
pi (t) =
vi (max{t̂i (t−i ), r̃i (t−i )}, t−i ) if ti > max{t̂i (t−i ), r̃i (t−i )}.
One can easily verify that the payment scheme satisfies (4). Hence by Lemma 3.2 the
VCGr mechanisms are ex-post incentive compatible.
3.2. Approximate Optimality
The monopoly reserves, ri (·), are defined such that ϕi (ri (t−i ), t−i ) = 0. They are well
defined since under Assumption 2.3 buyer i’s virtual value, ϕi (t), is strictly increasing
in ti . The VCG mechanism with monopoly reserves is the VCGr mechanism in which
the reserves are given by the monopoly reserves: r̃i (·) = ri (·). It generates the highest
expected revenue among all the VCGr mechanisms. To see this, consider the seller’s
expected revenue from a VCGr mechanism:
X
X
X E[pi (t)] = −
E[ui (ti , t−i )] +
E x∗i (t)1{ti ≥r̃i (t−i )} (t)ϕi (t) ,
i
i
i
where 1{ti ≥r̃i (t−i )} (·) is an indicator function which equals 1 if and only if ti ≥ r̃i (t−i ).
The expected revenue is maximized when a buyer in W ∗ wins if and only if his/her
virtual value is positive, i.e. when r̃i (·) = ri (·). The main result of the paper is the
following theorem:
T HEOREM 3.7. For the class of v(·) and F (·) satisfying Assumption 2.3, the seller’s
expected revenue by employing the VCG mechanism with monopoly reserves is at least
1/e of the full surplus.
The proof of Theorem 3.7 relies on the following lemma:
L EMMA 3.8 (L EMMA 3.9 IN [D HANGWATNOTAI ET AL . 2010]). Suppose a nonnegative random variable v follows a cumulative distribution F with a density function
f (v)
∗
f , and 1−F
(v) is increasing. Then for all v̂ ≥ 0 and s = max{v̂, v }
Z
1
s[1 − F (s)] ≥
vdF (v),
e v≥v̂
where v ∗ is defined by v ∗ − [1 − F (v ∗ )]/f (v ∗ ) = 0.
P ROOF OF T HEOREM 3.7. Let x∗i denote buyer i’s winning probability under the
VCG mechanism with no reserves, then the full surplus is given by
X
X
X
E[
x∗i (t)vi (t)] =
E[x∗i (t)vi (t)] =
E[E[x∗i (t)vi (t)|t−i ]].
(10)
i
i
i
Similarly, the seller’s expected revenue by employing the VCG mechanism with
monopoly reserves is given by
X
X
X
E[
pi (t)] =
E[pi (t)] =
E[E[pi (t)|t−i ]].
(11)
i
i
i
We now prove that for all i and t−i ,
1
E[x∗i (t)vi (t)|t−i ].
(12)
e
Once we prove (12), the theorem follows immediately from taking expectations over
t−i and summing over all buyers.
Fix i and t−i . Let Gi (vi (t)|t−i ) ≡ Fi (ti |t−i ), i.e. Gi (·|t−i ) is the cumulative distribution
function of vi conditional on t−i . Let gi (·|t−i ) denote the corresponding density function.
Then
−1
fi (ti |t−i )
∂vi (t)
gi (vi |t−i )
=
,
1 − Gi (vi |t−i )
1 − Fi (ti |t−i )
∂ti
E[pi (t)|t−i ] ≥
which is increasing by Assumption 2.3 and by the fact that vi (·, t−i ) is increasing. Let
v̂i ≡ vi (t̂i (t−i ), t−i ) and vi∗ ≡ vi (ri (t−i ), t−i ), then
vi∗ −
gi (vi∗ |t−i )
= 0.
1 − Gi (vi∗ |t−i )
Let s = max{v̂i , vi∗ }. Then the LHS of (12) can be rewritten as
E[pi (t)|t−i ] = s[1 − Gi (s|t−i )],
The RHS of (12) multiplied by e is
E[x∗i (t)vi (t)|t−i ]
Z
t̄i
vi (s, t−i )f (s|t−i )ds,
=
Zt̂i
vi dGi (vi |t−i ).
=
(13)
vi ≥v̂i
By Lemma 3.8,
s[1 − Gi (s|t−i )] ≥
1
e
Z
vi dGi (vi |t−i ).
vi ≥v̂i
(12) follows immediately.
Considering the single-item auction where there is a single buyer with an exponentially distributed valuation shows that the bound in Theorem 3.7 is tight. Without
further restrictions on F , this bound is also tight for an arbitrary number of buyers.
Consider the single-item auction with n buyers where one buyer’s valuation is exponentially distributed and the other n − 1 buyers have valuation 0. This is essentially
an auction with a single buyer. This result is similar to [Neeman 2003]. He finds that
in the general private value environment, the worst-case ratio of the expected revenue
from an English auction to full surplus stays bounded away from 1 as the number
of buyers tends to infinity even when the seller sets reserve prices optimally. This is
because in the worst case there is only one “serious” buyer, it is not surprising that increasing in the number of buyers has a negligible effect on the revenue. This example
also shows that reserve prices are important for revenue guarantee, since in this case
the expected revenue from the VCG mechanism with no reserves is 0.
4. CONCLUSION
In the general one-dimensional type setting where agents have interdependent values and correlated types, this paper shows that (1) when the system of feasible sets
is a matroid and the agents’ valuations satisfy the single-crossing condition, the VCG
mechanisms are ex-post incentive compatible; (2) if in addition the valuation distribution satisfies the generalized monotone hazard rate conditions, the VCG mechanism
with monopoly reserves is approximately optimal, i.e. the expected revenue from the
VCG mechanism with monopoly reserves is at least 1/e of the full surplus.
Looking toward future work, one limitation of the present paper is that the main
result heavily depends on the increasing hazard rate assumption. In the IPV settings
[Hartline and Roughgarden 2009] proves approximation result under matroid feasibility constraints and regular valuation distributions. Their analysis is facilitated by the
fact that in the IPV settings the optimal revenue is given by the expected total virtual
surplus. However, this is no longer true when we allow buyers’ types to be correlated
(see, e.g. [Crmer and McLean 1988]). Therefore in this paper we compare the expected
revenue of the VCG mechanism with monopoly reserves with the full surplus. It is not
clear in this case how to connect the full surplus with virtual surplus under the weaker
regularity assumption.
Another disadvantage of the VCG mechanism with monopoly reserves studied in
this paper is that it is not anonymous. [Hartline and Roughgarden 2009] show that
there is an anonymous random reserve price such that the expected revenue of VCG
with that reserve is at least 1/4 of the expected revenue of the optimal auction. Their
result is based on [Bulow and Klemperer 1996] type result. [Bulow and Klemperer
1996] shows that in a single-item auction where buyers have interdependent values
but independent types, the VCG mechanism with no reserves is more profitable in
expectation than the optimal mechanism with one less buyer.
[Hartline and Roughgarden 2009] proves an approximate analog of this statement
in the general one-dimensional type environments where agents have independent
private values. Specifically, they consider the following duplication of the original environment: Each buyer i is replaced by a pair i, n + i of buyers. For ease of notation
let i0 = n + i. Let N d ≡ {1, 2, · · · , 2n} and I d denote the collection of the feasible sets
of the duplicated environment. Then for each S d ⊂ N d , S d ∈ I d if and only if (1) for
each i ∈ N , at most one buyer from the pair {i, i0 } is selected; (2) S d , when naturally
interpreted as a set of buyers from the original environment, is a feasible set in that
environment. It is easy to verify that (N d , I d ) is a matroid if (N , I) is a matroid. They
show that ratio of the expected revenue from the VCG mechanism with no reserves in
the duplicated environment to the optimal revenue in the original environment is at
least 1/3 when the system of feasible sets is downward-closed and the the valuation
distributions have increasing hazard rates, and 1/2 when the system of feasible sets is
a matroid and the the valuation distributions are regular.
A direct extension of their results to the case where agents have interdependent
values and correlated types is difficult for several reasons. First, how to define a buyer’s
valuation of winning properly when the total number of buyers changes? Let t and t0
denote the types of the original buyers and their duplicates, respectively. One way to
deal with this question is to assume v̄i (t) = E[vi (t, t0 )|t] as buyer i’s valuation when
there are only n buyers (see [Bulow and Klemperer 1996]). The other is to assume that
a buyer’s value only depends on his/her own type and the state of the world, and the
other buyers’ types affect his/her expected valuation only to the extent that they are
correlated with the state of the world (see, e.g. [McLean and Postlewaite 2002] and
[McLean and Postlewaite 2004]). Second, consider a buyer i and his/her duplicate i0 ,
we need to make assumptions so that not only their valuations depend on the other
buyers’ types in a similar way, but also their types affect the other buyers’ valuations in
a similar way. Finally, the correlation between i and i0 ’s types and how their valuations
depend on each other’s type also matters for the approximation result.
ACKNOWLEDGMENTS
I am grateful to Steven Matthews, Mallesh Pai, Andrew Postlewaite and Tim Roughgarden for extremely
helpful discussions. All remaining errors are my sole responsibility.
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