Asymmetry and Revenue in Second-Price Auctions: A
Majorization Approach
Jihui Chen∗
Illinois State University
Maochao Xu†
Illinois State University
December 2, 2012
Abstract
Bidder asymmetry in second-price auctions may or may not hurt the expected
revenue, depending on the classes of value distributions. In this paper, we employ
the tool of majorization to extend and strengthen the results in Cantillon (2008).
Examples are also provided to illustrate the main findings.
JEL classification: D40; D44.
Keywords Majorization; Schur-convex; Expected revenue; Second-price auctions.
∗
Department of Economics, Illinois State University, Campus Box 4200, Normal, IL61790 U.S.A; Tel:
(309) 438-3616; Fax: (309) 438-5228; Email: [email protected].
†
Department of Mathematics, Illinois State University, Campus Box 4520, Normal, IL61790 U.S.A;
Tel: (309) 438-7674; Fax: (309) 438-5866; Email: [email protected].
1
Introduction
Bidder’s asymmetries are quite common in certain auctions where “strong” (with higher
valuations) and “weak” (with lower valuations) bidders coexist (e.g., asset sales and business procurements). Causes of the asymmetries may vary. For example, in procurement
auctions, Maskin and Riley (2000) refer to incumbent bidders as “strong” and entrant
bidders as “weak.” Also, location may lead to asymmetries where nearby firms often possess advantages while bidding for construction projects over those who are located further
away (e.g., Bajari, 2001).
However, not until recently has much attention been paid in the literature to asymmetric auctions (e.g., Maskin and Riley, 2000; Fibich et al., 2004; Cantillon, 2008). In an
important work, Cantillon (2008) states that bidder asymmetry is detrimental to the auctioneer’s expected revenue because the distribution of revenues in a symmetric benchmark
model stochastically dominates that in the asymmetric case. She also examines distribution properties of revenues under different asymmetric configurations and concludes that,
in second-price auctions (SPAs), the more asymmetric the configuration is, the lower the
expected revenue.
In this paper, we revisit the role of bidder asymmetry on revenue in the SPA using
the majorization tool to measure the asymmetries. The majorization tool is widely used
in areas including statistics, mathematics, reliability, and economics. It is a natural
tool to compare asymmetries between vectors. Many majorization concepts have been
reinvented and often rechristened in economics, such as Lorenz or dominance ordering.1
We show that under different asymmetric configurations, the conclusions in Cantillon
(2008) could be reversed in the SPA! That is, the auctioneer may actually benefit from
bidders’ asymmetries under certain classes of value distributions. This conclusion is in
line with Cheng (2011), who argues through a concrete example that bidder asymmetry
need not reduce competition and thus leads to higher revenue compared to the symmetric
case in first-price auctions (FPA).
In Section 2, we review the concepts of majorization before presenting the main results in Section 3. Also in Section 3, utilizing the majorization tool, we strengthen and
generalize the main results in Cantillon (2008) in the SPA. Specifically, we first compare
the auctioneer’s expected revenues between symmetric and asymmetric auctions and then
measure the extent to which different bidders’ asymmetries affect revenue by incorporating
power function distributions. Finally, we present a brief discussion and some suggestions
for future research in Section 4.
1
Interested readers may refer to Marshall et al. (2011) for an elaborate discussion on this topic.
1
2
Majorization
In this section, we introduce the key concepts of majorization, which are pertinent to our
main results.2
Let x(1) ≤ x(2) ≤ · · · ≤ x(n) be the increasing arrangement of the components of a
vector x = (x1 , x2 , · · · , xn ).
Definition 1 For real vectors x, y ∈ Rn , x is said to majorize y, denoted by x ≽m y, if
∑n
∑j
∑j
∑n
i=1 x(i) =
i=1 y(i) and
i=1 x(i) ≤
i=1 y(i) for j = 1, . . . , n − 1.
For example, let x = (1, 2, 5) and y = (2, 2, 4). Evidently, x has greater variability
(i.e., more asymmetry) than y. From Definition 2.1, it follows that x ≽m y. In some
∑
∑
situations where the requirement of x(i) = y(i) is too restrictive, the following weaker
concept may be used.
Definition 2 For real vectors x, y ∈ Rn , x is said to weakly supermajorize y, denoted by
w
∑
∑
x ≽ y, if ji=1 x(i) ≤ ji=1 y(i) for j = 1, . . . , n.
w
For example, if x = (1, 2, 5) and y = (2, 2, 5), then, x ≽ y.
A real-valued function Φ defined on a set A ⊆ ℜn is said to be Schur-convex on A if,
for any x, y ∈ A,
ϕ(x) ≥ ϕ(y) whenever x ≽m y
It is often difficult to verify Schur-convexity from its definition. The following result
presents a simple sufficient and necessary condition for Schur-convexity (cf. Marshall et
al. 2011).
Lemma 1 Let D ⊆ Rn be a symmetric set, and let ϕ : D 7→ R be continuously
differentiable. Then ϕ is Schur-convex on D if and only if ϕ is symmetric on D and
(
)
∂ϕ(x) ∂ϕ(x)
(xi − xj )
−
≥0
∂xi
∂xj
for all i ̸= j and x ∈ D.
3
Main results
Suppose there exist N bidders, who submit their bids X1 , . . . , XN with distribution functions F1 , . . . , FN , respectively. In the SPA, the winner, who submits the highest bid,
only pays the value of the second-highest bid. Hence, the revenue in the configuration
2
Refer to Marshall et al. (2011) for the details on the theory of majorization orders and its applications.
2
(F1 , . . . , FN ) is in fact equal to the second-highest order statistics of (X1 , . . . , XN ), denoted by X(N −1) . Following Cantillon (2008, p.4), we define the distribution of X(N −1) to
be
[
]
N
N
∏
∑
∏
S(F1 , . . . , FN ) =
(1 − Fi )
Fj +
Fi .
i=1
i=1
j̸=i
The following first-order stochastic dominance concept will be used in the sequel.
Definition 3 (Shaked and Shanthikumar, 2007) Random variable X with distribution
F is said to be smaller than random variable Y with distribution G in the first-order
stochastic dominance (denoted by X ≤st Y ), if F (x) ≥ G(x) for all x.
We first recall the following proposition from Theorem 1 of Cantillon (2008).
Proposition 1 Consider an asymmetric configuration of bidders (F1 , . . . , FN ) and its
(∏
)1/N
N
symmetric benchmark (F, . . . , F ). If F (x) =
F
(x)
, then
i=1 i
S(F1 , . . . , FN ) ≥ S(F, . . . , F ).
This result can be slightly generalized by allowing the following weaker condition on
F :
F (x) ≤
(N
∏
)1/N
Fi (x)
.
(3.1)
i=1
Under condition (3.1), the distribution of revenue in the symmetric benchmark first-order
stochastically dominates that in the asymmetric auction. Hence, the expected revenue in
the benchmark case is greater than that in the asymmetric case. This is essentially the
same conclusion as Theorem 1 of Cantillon (2008) with a slightly weaker condition.
Given Proposition 1, one may wonder whether or not the opposite inequality is plausible. That is, would the expected revenue in an asymmetric auction ever be greater than
that in the symmetric benchmark model under alternative configurations? The answer
is affirmative! The following lemma, essentially due to Ma (1999), is used to derive the
result.
Lemma 2 If X1 , . . . , Xn are independent random variables with corresponding order statistics X(1) ≤ . . . ≤ X(N ) , and Y1 , . . . , YN are independent and identical random variables
with corresponding order statistics Y(1) ≤ . . . ≤ Y(N ) , then,
X(1) ≥st Y(1) =⇒ X(N −1) ≥st Y(N −1) .
3
Now, note that
P (X(1) ≥ x) =
N
∏
P (Xi ≥ x) =
i=1
N
∏
(1 − Fi (x)),
i=1
Proposition 2 follows directly from Lemma 2.
Proposition 2 Consider an asymmetric configuration of bidders (F1 , . . . , FN ) and its
symmetric benchmark (F, . . . , F ). Then S(F1 , . . . , FN ) ≤ S(F, . . . , F ) if
(
F (x) ≥ 1 −
N
∏
)1/N
[1 − Fi (x)]
.
(3.2)
i=1
This result states that, under condition (3.2), the distribution of revenue in the asymmetric auction first-order stochastically dominates that in the symmetric auction. Hence,
the expected revenue from the asymmetric auction is greater than that from the symmetric
one.
Taken together, one may arrive at the exactly opposite conclusions regarding the
auctioneer’s expected revenue under different configurations of bidders’ asymmetries. To
further illustrate the implications of Propositions 1 and 2, we now present a numerical
example.
Example 1 Consider the following three configurations of bidders.
(a) Configuration 1: (F1 , F2 , F3 ). F1 , F2 , F3 are uniform distribution functions over
[1, 3], [2, 4], [2, 5], respectively.
(b) Configuration 2: (G1 , G1 , G1 ). G1 is defined as
G1 (x) =
( 3
∏
)1/3
Fi (x)
,
i=1
which is corresponding to Proposition 1.
(c) Configuration 3: (G2 , G2 , G2 ). G2 is defined as
(
G2 (x) = 1 −
3
∏
[1 − Fi (x)]
i=1
which is corresponding to Proposition 2.
4
)1/3
,
It is straightforward to obtain the distribution of revenue under Configuration 1


0,
x < 2,



2
(x − 2)(13x − 2x − 13)/12, 2 ≤ x < 3,
S(F1 (x), F2 (x), F3 (x)) =

(x − 2)(7 − x)/6,
3 ≤ x < 4,



1,
x ≥ 4.
For Configuration 2, it can be shown that

0,
x < 2,



1/3

2

 [(x − 2) (x − 1)/12] , 2 ≤ x < 3,
1/3
G1 (x) =
[(x − 2)2 /6] ,
3 ≤ x < 4, .


1/3

[(x − 2)/3] ,
4 ≤ x < 5,



1,
x ≥ 5.
Hence, the distribution of revenue under Configuration 2 is
S(G1 (x), G1 (x), G1 (x)) = 3(1 − G1 (x))G21 (x) + G31 (x).
For Configuration 3, we have


0,
x < 1,



1/3
1 − [(3 − x)/2] ,
1 ≤ x < 2,
.
G2 (x) =
1/3

1 − [(3 − x)(4 − x)(5 − x)/12] , 2 ≤ x < 3,



1
x ≥ 3.
Therefore, the distribution of revenue under Configuration 3 is
S (G2 (x), G2 (x), G2 (x)) = 3(1 − G2 (x))G22 (x) + G32 (x).
It follows directly from Propositions 1 and 2 that
S(G1 (x), G1 (x), G1 (x)) ≤ S(F1 (x), F2 (x), F3 (x)) ≤ S (G2 (x), G2 (x), G2 (x)) ,
which can be clearly seen in Figure 1. Therefore, the expected revenues under these three
configurations have the following relations:
ES(G1 (x), G1 (x), G1 (x)) ≥ ES(F1 (x), F2 (x), F3 (x)) ≥ ES (G2 (x), G2 (x), G2 (x)) ,
where ES(·, ·, ·) represents the expected revenue under each configuration.
It is seen that the expected revenue from the symmetric configuration (G1 , G1 , G1 ) is
higher than that from the asymmetric configuration (F1 , F2 , F3 ), which is consistent with
Proposition 1. However, at the same time, the expected revenue from the symmetric configuration (G2 , G2 , G2 ) is lower than that from the asymmetric configuration (F1 , F2 , F3 )!
That is, the more asymmetric the value distribution is, the higher the expected revenue,
which exactly illustrates the result in Proposition 2.
5
1.0
0.8
SHG2 HxL, G2 HxL, G2 HxLL
SHF1 HxL, , F2 HxL, F3 HxLL
SHG1 HxL, G1 HxL, G1 HxLL
0.6
0.4
0.2
2
4
6
8
10
Figure 1: Plot of distributions from three configurations.
3.1
Power function distributions
In this section, we generalize the above results to examine the extent to which bidder
asymmetry affects the expected revenue, taking into consideration two important classes
of power function distributions.
• Case 1: Proportional reversed hazard rate model (PRHR).
Fi (x) = F αi (x),
i = 1, . . . , N,
where F (x) is a continuous distribution function. This model is extensively used in
the statistics literature (cf. Gupta and Gupta 2007).3
• Case 2: Proportional hazard rate model (PHR).
F̄i (x) = F̄ αi (x),
i = 1, . . . , N,
where F̄ = 1 − F is a survival function. This model includes many well-known
distributions such as exponential, Weibull, and Pareto distributions (cf. Kochar
and Xu, 2007).
First consider Case 1. We now reproduce the result of Theorem 2 in Cantillon (2008,
p.6) using the concept of majorization.4
3
Cantillon (2008, p.6) also uses this model to characterize efficient collusion, joint bidding and mergers
among homogeneous bidders.
4
Although Cantillon (2008) does not explicitly mention the concept of majorization, she essentially
applies a similar condition to the majorization tool.
6
Proposition 3 Consider two configurations of bidders, α = (α1 , . . . , αN ) and β =
(β1 , . . . , βN ). Let
Fi (x) = F αi (x), Gi (x) = F βi (x).
If
(α1 , . . . , αN ) ≽m (β1 , . . . , βN ),
then
S(F1 , . . . , FN ) ≥ S(G1 , . . . , GN )
Proposition 3 implies that the more asymmetric the configuration is, the higher the
distribution of revenue. Therefore, the more asymmetric the configuration is, the lower
the expected revenue. The asymmetry softens competition among bidders, which in turn
hurts the auctioneer’s revenue (Cantillon, 2008).5 In the following, we provide a further
understanding of this result using a weaker condition.
The following lemma is needed before we proceed.
Lemma 3 ( Marshall et al., 2011) A real-valued function ϕ on Rn has the property that
w
ϕ(x) ≥ ϕ(y) whenever x ≽ y
if and only if ϕ is decreasing and Schur-convex on Rn .
Proposition 4 Consider two configurations of bidders, α = (α1 , . . . , αN ) and β =
(β1 , . . . , βN ). In the α and β configurations
Fi (x) = F αi (x),
Gi (x) = F βi (x).
If
w
(α1 , . . . , αN ) ≽ (β1 , . . . , βN ),
then
S(F1 , . . . , FN ) ≥ S(G1 , . . . , GN )
Proof: Note that
S(F1 , . . . , FN ) =
N
∑
F
∑
j̸=i
αj
(x) − (N − 1)F
∑N
j=1
αj
(x),
i=1
which is symmetric in α. Then,
[
]
∑ ∑
∑N
∂S(F1 , . . . , FN )
= log F (x)
F j̸=i αj (x) − (N − 1)F j=1 αj (x) .
∂α1
i̸=1
5
(3.3)
Cantillon (2008, p.6) presents a three-bidder example to explain the intuition behind Theorem 2.
7
Similarly,
]
[
∑ ∑
∑N
∂S(F1 , . . . , FN )
α
α
= log F (x)
F j̸=i j (x) − (N − 1)F j=1 j (x) .
∂α2
i̸=2
Therefore, it holds that
[
]
∂S(F1 , . . . , FN ) ∂S(F1 , . . . , FN )
(α2 − α1 )
−
∂α2
∂α1
∑N
[
]
= (α2 − α1 ) log F (x)F j=1 αj (x) F −α1 (x) − F −α2 (x)
≥ 0.
According to Lemma 1, S(F1 , . . . , FN ) is a Schur-convex function. Further, according to
Equation (3.3), it follows that
∂S(F1 , . . . , FN )
≤ 0.
∂α1
Hence, S(F1 , . . . , FN ) is a decreasing and Schur-convex function of α. Using Lemma 3,
the required result follows immediately.
This result shows that the distribution function of revenue is, in fact, a decreasing
and Schur-convex function. Hence, Proposition 3 follows immediately from this observation, which advances our understanding of how bidders’ asymmetries affect the revenue
distribution. To illustrate the implications of Proposition 4, we present the following
example.
Example 2 Consider the following two configurations.
• Configuration (F1 , F2 , F3 ). Fi , i = 1, 2, 3 is defined as
)αi
 (
 x−2
, 2 ≤ x ≤ 5,
Fi (x) =
,
3

0
otherwise.
where (α1 , α2 , α3 ) = (1, 2, 5).
• Configuration (G1 , G2 , G3 ). Gi , i = 1, 2, 3 is defined as
 (
)β
 x−2 i
, 2 ≤ x ≤ 5,
Gi (x) =
,
3

0
otherwise.
where (β1 , β2 , β3 ) = (2, 2, 5).
8
It is straightforward that (α1 , α2 , α3 ) m (β1 , β2 , β3 ), indicating that Proposition 3
cannot be used here. However, it can be verified that
w
(α1 , α2 , α3 ) ≽ (β1 , β2 , β3 ).
Therefore, Proposition 4 directly applies, leading to the following implication,
ES(F1 , F2 , F3 ) ≤ ES(G1 , G2 , G3 ).
That is, the more asymmetric the configuration is, the lower the expected revenue.
Next, we consider Case 2 where the configurations in the PHR models are used. Proposition 5 follows directly from Proschan and Sethuraman (1976) and Marshall et al. (2011,
p.485).
Proposition 5 Consider two configurations of bidders, α = (α1 , . . . , αN ) and β =
(β1 , . . . , βN ). In the α and β configurations
F̄i (x) = F̄ αi (x),
Ḡi (x) = F̄ βi (x).
If
w
(α1 , . . . , αN ) ≽ (β1 , . . . , βN ),
then
S(F1 , . . . , FN ) ≤ S(G1 , . . . , GN )
The implication of this result is that, under the PHR models, the more asymmetric the
configuration is, the higher the expected revenue, which is exactly opposite to Proposition
4.
The following example illustrates Proposition 5.
Example 3 Consider the following two configurations.
• Configuration (F1 , F2 , F3 ). F̄i , i = 1, 2, 3 is defined as
F̄i (x) = e−2αi x , x > 0,
where (α1 , α2 , α3 ) = (.1, .25, .4).
• Configuration (G1 , G2 , G3 ). Ḡi , i = 1, 2, 3 is defined as
Ḡi (x) = e−2βi x , x > 0,
where (β1 , β2 , β3 ) = (.15, .2, .5).
9
1.0
0.8
SHG1 HxL, G2 HxL, G3 HxLL
0.6
SHF1 HxL, F3 HxL, F3 HxLL
0.4
0.2
2
4
6
8
10
Figure 2: Plot of distributions from two configurations.
It can be verified that
w
(α1 , α2 , α3 ) ≽ (β1 , β2 , β3 ).
Hence, it follows from Proposition 4 that
S(F1 , F2 , F3 ) ≤ S(G1 , G2 , G3 ),
which is clearly illustrated in Figure 2.
Therefore, we have established that
ES(F1 , F2 , F3 ) ≥ ES(G1 , G2 , G3 ).
That is, the more asymmetric the configuration is, the higher the expected revenue.
4
Conclusion
We have shown that under different configurations, the effects of bidders’ asymmetries on
the revenue distribution (or the expected revenue) can be drastically different in the SPA.
Hence, we caution that it is important to identify the configuration of value distribution
before drawing a definite conclusion. The revenue in the SPA is, in fact, the second-highest
order statistics, which has been extensively discussed in the statistical literature (e.g.,
Balakrishnan and Rao, 1998a, 1998b). One possible extension of our analysis is to gauge
how the dependence between bidders’ asymmetric valuations affects the expected revenue.
Another area for additional work is to test these theoretical predictions through laboratory
and field experiments. There is some recent empirical evidence of pro-competitiveness of
10
asymmetric auctions (Estache and Iimi, 2010). However, the research can be extended
beyond the context of public infrastructure procurements. For example, the proxy bidding
mechanism on eBay provides a natural setting of the SPA. These interesting problems are
left for further research.
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11
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