Discussion Paper No. 848

EXPOSURE PROBLEM
IN MULTI-UNIT AUCTIONS

Hikmet Gunay
Xin Meng

July 2012

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

EXPOSURE PROBLEM IN MULTI-UNIT AUCTIONS
Hikmet Gunay and Xin Meng∗
University of Manitoba and SWUFE-RIEM
June 24, 2012

Abstract
We characterize the optimal bidding strategies of local and global bidders for twoheterogenous licenses in a multi-unit simultaneous ascending auction. The global bidder wants to win both licenses to enjoy synergies; therefore, she bids more than her
stand-alone valuation of a license. This exposes her to the risk of losing money evenwhen she wins all licenses. We determine the optimal bidding strategies in the presenceof an exposure problem. By using simulation methods, ﬁrst, we show the frequency
of ineﬃcient allocation in the simultaneous ascending auction. Then, we show that
the Vickrey-Clarke-Groves (VCG) mechanism may generate more revenue than thesimultaneous ascending auction.

JEL Codes:D44, D82
Keywords: Multi-Unit Auctions, Vickrey Clarke Groves (VCG) mechanism, Exposure Problem, Synergies, Complementarity∗

Acknowledgements will be added later.

1

1

Introduction

In a typical American or Canadian spectrum license auction, hundreds of (heterogenous)
licenses are sold simultaneously. Each of these licences gives the spectrum usage right ofa geographical area to the winning ﬁrm. Some ‘local’ ﬁrms are interested in winning only
speciﬁc licenses in order to serve local markets while other ‘global’ ﬁrms are interested inwinning all the licenses in order to serve nationwide.1 The global ﬁrms enjoy synergies if they
win all the licenses which gives them an incentive to bid over their stand-alone valuations
for some licenses. As a result, there is a risk of incurring losses. Therefore, global bidders
lower their bids. This is known as the exposure problem.2
In a model simplifying the American and the recent Canadian spectrum license auctions,
we derive the optimal bidding strategies of local and global ﬁrms in a simultaneous ascending
auction (SAA) of two licenses. We mainly focus on the optimal bidding strategies when thereis the possibility of an exposure problem (i.e., ex-post loss) since managers of cell-phonecompanies and policy makers would be interested in such numbers. Through simulations,
we determine how frequently the exposure problem occurs. In addition, we decompose thefrequency into two cases; the case in which the exposure problem occurs when the global
bidder wins only one license, and the case in which the exposure problem occurs when the
global bidder wins all licenses.
Exposure problem indicates that the allocation may not be eﬃcient. In fact, we showthat allocation may be ineﬃcient with 7.84 per cent of the time for some parameter space.
We compare the eﬃciency and revenue properties of the simultaneous ascending auction
with those of the Vickrey-Clarke-Groves (VCG) mechanism when bidders are allowed to bid
on packages. VCG is an eﬃcient auction that gives the highest revenue among all incentive
compatible, individually rational, eﬃcient auctions. In the literature, there are examples
1
In the recent Canadian Advanced Wireless Spectrum auction, ﬁrms such as Globalive and Rogers wereinterested in all licenses whereas ﬁrms such as Bragg Communication and Manitoba Telecom Services (MTS)
were interested in East Coast and Manitoba licenses, respectively.
2
We will interchangeably use exposure problem as follows. We say that an exposure problem occurred
whenever the global bidder incurs a loss ex-post.

2which show that VCG mechanism may give extreme low revenue in complete information
settings, and low revenue is cited as one of the main reasons why VCG mechanism is not
used prevalently (e.g., Ausubel and Milgrom (2006)). We show that VCG mechanism maygive higher revenue to the seller for many parameter spaces and various distributions inincomplete information setting. For example, when local bidders win the licenses in the
SAA auction and the allocation is ineﬃcient, then VCG auction would give higher revenue
for the same private valuations. However, when the global bidder wins the license with an
ex-post loss (that is, when exposure problem occurs), then SAA auction’s revenue would behigher than the VCG mechanism. In this paper, we also show the frequency of ineﬃcient
allocation when simultaneous ascending auction is used.
The multi-unit auction literature generally assumes that global bidders have either equal
valuations (Englmaier et. al (2009), Kagel and Levin (2005), Katok and Roth (2004), Albano
et. al. (2001), Rosenthal and Wang (1996), and Krishna and Rosenthal (1996)) or very large
synergies (Albano et al. (2006)). The spectrum licenses for diﬀerent geographic areas are
not homogenous objects; hence, the equal valuation assumption does not ﬁt the Canadian or
the American spectrum license auction. Moreover, in a heterogeneous license environment,
bidders may not drop out of both auctions simultaneously. This enables us to analyze
bidding behavior in the remaining auction, and hence, the exposure problem/ex-post loss. In
addition, unlike the aforementioned multi-unit auction papers above, we allow for moderatesynergies, and our focus is on the exposure problem and the comparison of revenue andeﬃciency properties of the simultaneous ascending auction with those of the VCG auction.
In our paper, the global bidder will lower his bid because of the exposure problem; however,their optimal strategy still requires him to bid over his stand alone valuation for at least one
license. If he wins this license by receiving a potential loss, then he may need to stay in theother license auction to minimize his loss. Therefore, there are cases in which the ex-post
loss may occur even when the bidder wins all the licenses.
Goerre and Lien (2010) assume that the marginal valuation of winning a given numberof licenses is the same regardless of the composition of the particular licenses. Hence, they3

ﬁnd that the optimal drop out price is the same for both licenses. In our paper, marginalvaluations are diﬀerent; hence, the global bidder has a preference over license A and license B.
Speciﬁcally, we model situations in which winning the spectrum license for, say, Iowa City is
diﬀerent than winning the license for New York city. Hence, our paper shows that the optimal
drop out price is not the same. In addition, through simulations, we show the frequency ofineﬃcient allocation and ex-post loss for the simultaneous ascending auction. Like Goerreand Lien (2010), we ﬁnd VCG mechanism may give higher revenue than the simultaneous
ascending auction. However, we show that VCG mechanism gives higher revenue for cases
in which local bidders win licenses ineﬃciently (in the SAA auction), and SAA auction
gives higher revenue for cases in which global bidder wins licenses ineﬃciently.3 In otherwords, because of the ex-post loss possibility, global bidder does not bid too much overthe stand alone valuation; hence, local bidders win ineﬃciently. This lowers the revenue.
However, global bidder still over-bids and wins licenses with ex-post loss when local bidders
are supposed to win in an eﬃcient outcome. This increases the revenue of SAA auction
compared to VCG auction at the expense of global bidder.

2

The Model

There are 2 licenses, license 𝐴 and 𝐵 for sale.4 There are one global bidder who demands
both licenses and 𝑚𝑗 = 𝑚 − 1 local bidders who demand only license 𝑗 = 𝐴, 𝐵. We explainin detail the cases with more than one global bidder later in the paepr. Speciﬁcally, 𝑚𝑗 will
denote the number of active local bidders on the auction.5 Both local bidders and the global
bidder have a private stand alone valuation for a single license, 𝑣𝑖𝑗 , where 𝑖 and 𝑗 represent
the bidder and the license, respectively. The valuations 𝑣𝑖𝑗 are drawn from the continuous
distribution function 𝐹 (𝑣𝑖𝑗 ) with support on [0, 1] and probability density function 𝑓 (𝑣𝑖𝑗 )which is positive everywhere with the only exception that 𝑓 (0) ≥ 0 is allowed. The bidders’3

Zheng (2008) is mainly interested in showing that jump-bidding will alleviate the exposure problem. We
do not allow jump-bidding as the Canadian and the American spectrum auction has not allowed this.
4
We use two licenses like Albano et. al. (2001 and 2006), Brusco and Lopomo (2002), Chow and Yavas(2009), and Menucicci (2003).
5
Allowing diﬀerent number of local bidders per license will not change our qualitative results.

4

type, global or local, is publicly known.
We consider a setting where the licenses are auctioned oﬀ simultaneously through anascending multi-unit auction. Each license is auctioned oﬀ at a diﬀerent auction (like Krishna
and Rosenthal (1996) but unlike Kagel and Levin (2005)) but at the same time. Prices startfrom zero for both licenses and increase simultaneously and continuously at the same rate.
Bidders choose when to drop out. When only one bidder is left on a given license, the clock
stops for that license, and the sole remaining bidder wins the license at the price at which thelast bidder dropped out. If there are more than one bidder remaining on the other license,
its price will continue to increase. If n bidders drop out at the same price and nobody is left
in the auction, then each one of them will win the license with probability 𝑛1 . This is a zeromeasure event given the valuations are drawn from a continuous distribution function.
The drop-out decision is irreversible. Once a bidder drops out of bidding for a given
license, he cannot bid for this license later.6 The number of active bidders and the dropout prices are publicly known. We also assume that there is no budget constraints for thebidders.
We assume that there is a homogeneous positive synergy for the global bidder. Speciﬁcally, letting bidder 1 be the global bidder, the global bidder’s total valuation, given that itwins two licenses is, 𝑉1 = 𝑣1𝐴 + 𝑣1𝐵 + 𝛼, where the synergy term 𝛼 is assumed to be strictly
positive and public knowledge.7 His stand-alone valuation of license A or B is given by 𝑣1𝐴or 𝑣1𝐵 . Bidder 𝑖𝐴, 𝑖 = 2, 3, ...𝑚 is only interested in license A, and her private valuation
is 𝑣𝑖𝐴 . Bidder 𝑖𝐵 is only interested in license B, and her private valuation is 𝑣𝑖𝐵 .8 A local
bidder who is interested in license 𝑗 participates only in license 𝑗 auction.
We derive a symmetric perfect Bayesian equilibrium through a series of lemmas that
follow. First, we describe the equilibrium strategy of the local bidder.
6
In the real-world auctions, there is activity rule. If the bidders do not have enough highest standing
bids, then the number of licenses they may bid on is decreased (in the next rounds). Hence, when there are
two licenses, this translates into an irreversible drop-out.
7
Public knowledge assumption can be removed, and all results are still valid. We assume public knowledge
not to complicate the notation.
8
We do not assume that 𝑣𝑖𝐴 > 𝑣𝑖𝐵 since local ﬁrms are diﬀerent; hence, their eﬃciency may diﬀer.

5

Lemma 1 Each local bidder has a weakly dominant strategy to stay in the auction until the
price reaches his stand alone valuation.
This is a well-known result so we skip the proof.
Now, consider a subgame in which all the local bidder drops out of license 𝐵 auction,and hence, the global bidder wins license 𝐵 at the price 𝑝𝐵 , which in equilibrium is equal
to 𝑝𝐵 = 𝑚𝑎𝑥{𝑣2𝐵 , ..., 𝑣(𝑚)𝐵 } by lemma 1. Then, as the price for license A increases, theglobal bidder will compare the payoﬀ from dropping out from license 𝐴 auction at the clock
price 𝑝 (which is 𝑣1𝐵 − 𝑝𝐵 ) and the payoﬀ from winning license A at price 𝑝 (which is
𝑣1𝐴 + 𝑣1𝐵 + 𝛼 − 𝑝𝐵 − 𝑝). The updated optimal drop out price, 𝑝𝐴 , is found by equating these
two equations: 𝑣1𝐴 + 𝑣1𝐵 + 𝛼 − 𝑝𝐵 − 𝑝𝐴 = 𝑣1𝐵 − 𝑝𝐵 ⇒ 𝑝𝐴 = 𝑣1𝐴 + 𝛼. This is intuitive sinceby winning license A, the global bidder will also earn the synergy value. If global bidder
wins license 𝐴 ﬁrst, the updated optimal drop out price can be found symmetrically, and it
is 𝑝𝐵 = 𝑣1𝐵 + 𝛼. We state this as lemma 2.
Lemma 2 If the global bidder wins license 𝐵 (or 𝐴) ﬁrst, then it will stay in license 𝐴 (or𝐵) auction until the price reaches 𝑣1𝐴 + 𝛼 (or 𝑣1𝐵 + 𝛼)
The global bidder will not drop out before the price reaches his minimum of stand-alone
valuations. Otherwise, they will lose the chance of winning both licenses and enjoying thesynergy. In addition, if the global bidder’s average valuation, 𝑉21 = 𝑣1𝐴 +𝑣21𝐵 +𝛼 , exceeds 1,
bidding up to his average valuation will shut out the local bidders since local bidders’ standalone valuation can be at most 1. If 𝛼 is large enough, this condition will always be satisﬁed.
In such a case, the global bidder always wins both licenses in equilibrium. We summarize
these results as lemma 3.
Lemma 3 a) The global bidder stays in both license auctions at least until the price reaches
the minimum of his stand-alone valuations.
b) If his average valuation is greater than 1, the global bidder’s equilibrium strategy is tostay in until the price reaches his average valuation.
6

To calculate the optimal drop out price for the global bidder, consider ﬁrst the case inwhich 𝑣1𝐴 > 𝑣1𝐵 .9 The global bidder must compare the payoﬀs for two cases at each price p
as the clock is running: Case 1 is the payoﬀ from dropping out from license B auction at
price 𝑝 and optimally continuing on license A auction. Case 2 is the payoﬀ from winninglicense B at price p and optimally continuing on license A auction.10 At the beginning of
the auction, that is 𝑝 = 0, the second case payoﬀ is higher so the global bidder will start bystaying in the auction. We show that the diﬀerence between these two cases are monotonic
in 𝑝 ; therefore, there is a unique price that makes the global bidder indiﬀerent between thesetwo cases (assuming that the local bidders are still active).11 This is the optimal drop out
price, 𝑝∗1 . We show that this price can be calculated at the beginning of the auction. Notethat according to Lemma 3, 𝑝∗1 ≥ 𝑣1𝐵 , and the optimal updated drop out price for license A,
after winning license 𝐵 at price 𝑝, is 𝑣1𝐴 + 𝛼.
We denote the expected proﬁt of the global bidder for Case 1 by 𝐸Π11 and his expected
proﬁt for Case 2 by 𝐸Π21 , respectively.
Let 𝑝𝐴 = max{𝑣2𝐴 , ..., 𝑣𝑚𝐴 } be the price the global bidder will pay for the license A, if
he wins license A. Payoﬀs are as follows:𝐸Π11 = 𝑀 𝑎𝑥{0,𝐸Π21 =∫ 𝑣1𝐴(𝑣1𝐴 − 𝑝𝐴 )𝑔(𝑝𝐴 ∣𝑝)𝑑𝑝𝐴 }(1)𝑝∫ 𝑀 𝑖𝑛{𝑣1𝐴 +𝛼,1}∫ 1(𝑉1 − 𝑝 − 𝑝𝐴 )𝑔(𝑝𝐴 ∣𝑝)𝑑𝑝𝐴 +𝑝(𝑣1𝐵 − 𝑝)𝑔(𝑝𝐴 ∣𝑝)𝑑𝑝𝐴(2)

𝑀 𝑖𝑛{𝑣1𝐴 +𝛼,1}The explanation of equation 1 is as follows. After the global bidder drops out of theauction for license B at 𝑝, it becomes just like a local bidder, and hence, will continue to
9

The other case can be calculated symmetrically.
Assuming that it has not won license A yet, the global bidder will drop out of license B ﬁrst -or at thesame time as A- since 𝑣1𝐴 > 𝑣1𝐵 . It would not make sense to drop out from license 𝐴 ﬁrst -since the prices
of both auctions increase at the same rate-, and then continuing on the less valued license 𝐵.
11
Another way of calculating the optimal drop out price would be to maximize the expected payoﬀ.
However, this way would be much more complex; especially proving the second order conditions. Hence, we
exploit the fact that payoﬀ diﬀerence above is monotonic in 𝑝. Our results coincide with the literature when
the licenses are identical.
10

7

stay in the auction for license A until 𝑣1𝐴 . If he wins, he will pay 𝑝𝐴 since the local bidderwith highest valuation of license A will drop out last (by Lemma 1). In order to calculate
his expected proﬁt, global bidder will be using 𝐺(𝑝𝐴 ∣𝑝) (highest order statistic) which is the
distribution function of the local bidders’ highest valuation 𝑝𝐴 for license A given 𝑝. When
there are 𝑚 − 1 local bidders in license 𝐴, the distribution function 𝐺(𝑝𝐴 ∣𝑝) and its density
function 𝑔(𝑝𝐴 ∣𝑝) are:∫ 𝑝𝐴

𝑓 (𝑣)𝑑𝑣 𝑚−1
)
𝑓
(𝑣)𝑑𝑣
𝑝

(3)

𝑓 (𝑣)𝑑𝑣 𝑚−2 𝑓 (𝑝𝐴 )
)
(∫ 1
).
𝑓
(𝑣)𝑑𝑣
𝑓
(𝑣)𝑑𝑣
𝑝
𝑝

(4)

𝑝

𝐺(𝑝𝐴 ∣𝑝) = (𝐹 (𝑝𝐴 ∣𝑝))𝑚−1 = ( ∫ 1∫ 𝑝𝐴
𝑝𝑔(𝑝𝐴 ∣𝑝) = (𝑚 − 1)( ∫ 1

The ﬁrst term of 𝐸Π21 is Firm 1’s expected proﬁt of winning both licenses; assuming that
he wins license B at the price 𝑝. If the highest local bidder’s valuation 𝑝𝐴 is less than the
global bidder’s (updated) willingness to pay, 𝑣1𝐴 + 𝛼, then the global bidder wins license A
and pays 𝑝𝐴 . Since 𝑝𝐴 < 1, we use the minimum function in the upper limit of the ﬁrst
integral. The second term of 𝐸Π21 is Firm 1’s expected proﬁt of winning only license B which
can happen only if 𝑝𝐴 > 𝑣1𝐴 + 𝛼. Note that the second term is non-positive by Lemma 3(which is the exposure problem arising from winning only one license).
In Lemma 4 below, we characterize the global bidder’s equilibrium bids. It can be foundfrom 𝐸Π11 = 𝐸Π21 . Note that these payoﬀs are changing as local bidders bidding for 𝐴
are dropping out; that is, 𝑚 − 1 is changing. Therefore, the lemma below gives the global
bidder’s (updated) equilibrium drop out price as the local bidders drop out. We show, in the
proof of lemma 4, that this updated price increases as local bidders of license 𝐴 drop out.12
12

Local bidders B dropping out of auction does not aﬀect this optimal drop-out price. Seemingly surprising
result arises because of two facts. First, here the assumption is 𝑣1𝐴 > 𝑣1𝐵 so the global bidder will drop out
from license 𝐵 ﬁrst,- as explained in footnote 10. Second, when the global bidder makes the calculation, it
compares his payoﬀ of dropping out from license 𝐵 and continuing on 𝐴 at the given decision price 𝑝, andwinning license 𝐵 and continuing on license 𝐴 at the given decision price 𝑝. In both cases, the number of
local bidders 𝐵 have no eﬀect on the decision. However, number of local bidders 𝐴 will aﬀect its payoﬀ. If
𝑣1𝐵 > 𝑣1𝐴 , then ONLY the number of local bidders 𝐵 will aﬀect the global bidder’s decision. We emphasizethat local bidders B dropping out from auction aﬀect the game though. If there are fewer local bidders B, it
is more likely for the global bidder to win license B before the price reaches its optimal drop out price, andthen it will update the new optimal drop out price as 𝑣1𝐴 + 𝛼.

8Lemma 4 Suppose that the average valuation of the global bidder is less than 1 and there
are 𝑚 − 1 local bidders bidding on license 𝐴 where 𝑚 − 1 ≥ 1 and there is at least one active
local bidder 𝐵.
If 𝑣1𝐴 > 𝑣1𝐵 , the global bidder13 will drop out of license B auction at the unique optimal
drop-out price 𝑝∗1 ∈ [0, 1] that satisﬁes 𝐸Π11 = 𝐸Π21 . Moreover,∫ 𝑣1𝐴 +𝛼a) If 𝑣1𝐴 + 𝛼 < 1, and 𝑣1𝐴
𝐺(𝑝𝐴 ∣𝑝)𝑑𝑝𝐴 + (𝑣1𝐵 − 𝑣1𝐴 ) < 0, then 𝑝∗1 < 𝑣1𝐴 and the global
bidder will stay in license A auction until 𝑣1𝐴 (after dropping out from license B auction).
∫ 𝑣1𝐴 +𝛼b) If 𝑣1𝐴 + 𝛼 < 1, and 𝑣1𝐴
𝐺(𝑝𝐴 ∣𝑝)𝑑𝑝𝐴 + (𝑣1𝐵 − 𝑣1𝐴 ) > 0, then 𝑝∗1 > 𝑣1𝐴 and the global
bidder will also drop out of license A auction at 𝑝∗1 .
∫1c) If 𝑣1𝐴 + 𝛼 > 1, and 𝑣1𝐴 𝐺(𝑝𝐴 ∣𝑝)𝑑𝑝𝐴 + (𝑣1𝐵 + 𝛼 − 1) < 0, then 𝑝∗1 < 𝑣1𝐴 and the global
bidder will stay in license A auction until 𝑣1𝐴 (after dropping out from license B auction).
∫1d) If 𝑣1𝐴 + 𝛼 > 1, and 𝑣1𝐴 𝐺(𝑝𝐴 ∣𝑝)𝑑𝑝𝐴 + (𝑣1𝐵 + 𝛼 − 1) > 0, then 𝑝∗1 > 𝑣1𝐴 and the global
bidder will also drop out of license A auction at 𝑝∗1 .
Proof. See the Appendix.
We are ready to summarize our Perfect Bayesian equilibrium.
Proposition 5 (Perfect Bayesian Equilibrium)a) Local bidder of each license will stay in the auction 𝑗 until price reaches their valuation
𝑣𝑖𝑗 where 𝑗 = {𝐴, 𝐵}, 𝑖 = {2, 3, ..𝑚 − 1}.
b) A global bidder active only on license 𝑗 will bid 𝑣1𝑗 + 𝛼, if he won license −𝑗 otherthan license 𝑗. He will bid 𝑣1𝑗 when he did not win license −𝑗.
c) When 𝑣1𝐴 > 𝑣1𝐵 and the average valuation is less than one, the global bidder who isactive on both licenses and facing 𝑚 − 1 active local bidders on license 𝐴 will drop out fromlicense 𝐵 at the price that equates equations 1 and 2.
d) When 𝑣1𝐴 < 𝑣1𝐵 and the average valuation is less than one, the global bidder who isactive on both licenses and facing 𝑚 − 1 active local bidders on license 𝐵 will drop out fromlicense 𝐴 at the price that equates equations 1 and 2 (symmetrically replaced 𝑣1𝐴 with 𝑣1𝐵 ).
13

If 𝑣1𝐴 < 𝑣1𝐵 , then the proposition has to be written symmetrically9e) If the average valuation is greater than one, the global bidder will stay in both auctions
until price reaches his average valuation.14f ) Out-of-equilibrium-path beliefs: When a bidder drops out, the other bidders will see
this as an equilibrium behavior. Hence, any out of equilibrium path beliefs can be used.15
At the beginning of the game, each bidder calculates its optimal drop-out price. For local
bidders, the optimal drop out prices are their valuations. In equilibrium, it is optimal forthe global bidder to stay in the auctions for both licenses up to his optimal drop-out pricecalculated in Lemma 4. When his average valuation exceeds 1, he will stay until this averagevaluation and win both licenses. When the price reaches the minimum of these optimal
drop-out prices, that bidder drops out of license auction. If, for example, the global bidder
wins license 𝐴, the global bidder would continue to stay in the auction for license A until
the price reaches 𝑣1𝐴 + 𝛼. At this price, he ﬁnds that the payoﬀ from winning only license B
is more than the payoﬀ from winning both licenses even though it will enjoy synergy; hence,
it drops out.
The following is a corollary of Lemma 4, and is an example for the optimal drop out pricewhen 𝐹 (.) is a uniform distribution.
Corollary 6 Assume that valuations are drawn from a uniform distribution with a support[0, 1]. In addition, assume that 𝑣1𝐴 > 𝑣1𝐵 (other case is symmetrically found by exchanging
14

Price does not have to stop increasing at 1.
If local bidders drop out, global bidder will think that local bidder is using his/her equilibrium strategy
of dropping out at his/her valuation. If the global bidder drops out from license 𝐵 ﬁrst (or license 𝐴 ﬁrst),
the local bidders will think that global bidder is using his/her equilibrium strategy of dropping out fromlicense 𝐵 since he has a valuation of 𝑣1𝐵 > 𝑣1𝐴 (or 𝑣1𝐴 > 𝑣1𝐵 ) If global bidder drop out from both licenses,
then local bidders will again think that global bidder uses his equilibrium strategy and 𝑝∗1 > 𝑣1𝐴 , 𝑣1𝐵 . In
other words, sequential rationality is always satisﬁed.
15

10

When 𝑝𝐴 is here,
global bidder wins bothbut makes a loss
0 𝑣1𝐵 𝑝𝐵

𝑝∗1

When 𝑝𝐴 is here,
global bidder wins B
and makes a loss

Exposure Problem II
Exposure Problem I
𝑣1𝐴 + 𝑣1𝐵 + 𝛼 − 𝑝𝐵
𝑣1𝐴 + 𝛼1Figure 1: EXPOSURE PROBLEM

𝑣1𝐴 with 𝑣1𝐵 ), and there is one local bidder in each license.
⎧
1
1
2
2
2
+
1
−
2𝑣
−
𝛼
+
2𝑣
𝛼
+
2𝛼
−
4𝑣
𝛼)
{𝑣
+
𝛼
+
1−(𝑣
},

1𝐵
1𝐵
1𝐴
1𝐵
1𝐵
2



if 0 < 𝑣1𝐴 < 1 − 𝛼 and 2(1 − 𝑣1𝐴 )(𝑣1𝐴 − 𝑣1𝐵 ) > 𝛼2 ;



1
1
2
2

2

3 {𝑣1𝐴 + 𝑣1𝐵 + 𝛼 + 1 − ((𝑣1𝐴 + 𝑣1𝐵 + 𝛼 + 1) − 3(𝑣1𝐴 + 𝛼) − 6𝑣1𝐵 ) },
⎨
if 0 < 𝑣1𝐴 < 1 − 𝛼 and 2(1 − 𝑣1𝐴 )(𝑣1𝐴 − 𝑣1𝐵 ) ≤ 𝛼2 ;𝑝∗1 = 11
2
} 2 },{𝑣1𝐵 + 𝛼 + 1−{(𝑣1𝐵 + 𝛼 + 1)2 − 4(𝑣1𝐴 + 𝑣1𝐵 + 𝛼) + 2 + 2𝑣1𝐴

2



if 1 − 𝛼 ≤ 𝑣1𝐴 < 1 and 1 + 𝑣1𝐴 > 2(𝑣1𝐵 + 𝛼);2(𝑣1𝐴 +𝑣1𝐵 +𝛼)−1


,

3
⎩

(5)

if 1 − 𝛼 ≤ 𝑣1𝐴 < 1 and 1 + 𝑣1𝐴 ≤ 2(𝑣1𝐵 + 𝛼).

The optimal drop-out price is a function that takes a unique value deﬁned in the corollary
above. For example, case 0 < 𝑣1𝐴 < 1 − 𝛼 and 2(1 − 𝑣1𝐴 )(𝑣1𝐴 − 𝑣1𝐵 ) > 𝛼2 implies that𝑝∗1 < 𝑣1𝐴 . If we assumed 𝑣1𝐴 = 𝑣1𝐵 , then our equilibrium drop out price in the corollarywould coincide with the equilibrium drop out prices of Albano et. al (2001) and Goerre
and Yuanchuan (2010). Hence, we generalize their results, and relax the identical valuations
assumption.

2.1

Exposure Problem/Ex-post Loss

We now can discuss the exposure problem with the help of Figure 1. In the ﬁrst type of
exposure problem, the global bidder may win license B at a price above his stand alone
valuation (i.e., 𝑣1𝐵 < 𝑝𝐵 < 𝑝∗1 ) and lose the other license (i.e., 𝑝𝐴 > 𝑣1𝐴 + 𝛼). This is thetype of exposure problem Chakraborty (2004) focuses on. In the second type of exposure
problem, the global bidder wins both licenses but incurs a loss. This is the case when hewins license B at 𝑣1𝐵 < 𝑝𝐵 < 𝑝∗1 and wins license A at 𝑣1𝐴 + 𝛼 > 𝑝𝐴 > 𝑣1𝐴 + 𝛼 + 𝑣1𝐵 − 𝑝𝐵 .
Note that if he wins license A at the price 𝑣1𝐴 + 𝛼 + 𝑣1𝐵 − 𝑝𝐵 , his payoﬀ is zero. The global
bidder stays in the auction for license A in order to minimize its loss from winning only
11

Table 1: PROBABILITY OF EXPOSURE PROBLEM

Synergy
𝛼
0.2
0.4
0.6
0.8

Percentage ofPercentage of
Total
Exposure Problem 1 Exposure Problem 2
Percentage
One
One
One
Local Bidder
Local Bidder
Local Bidder
Beta Distribution with parameters 𝛼 = 1 and 𝛽 = 4
2.85
0.64
3.48
2.29
2.43
4.72
0.54
1.59
2.13
0.04
0.62
0.66
Uniform Distribution
0.40
1.07
1.51
1.25

0.2
0.4
0.6
0.8

1.17
1.13
0.69
0.36

1.57
2.20
2.20
1.61

0.2
0.4
0.6
0.8

Beta Distribution with parameters 𝛼 = 4 and 𝛽 = 1.
0.34
0.77
1.11
0.20
0.93
1.13
0.00
0.50
0.50
0.00
0.26
0.26

Percentage of
Ineﬃciency
One
Local Bidder
7.84
5.45
2.15
0.66
3.29
4.22
4.01
2.62
2.72
1.95
0.64
0.28

license 𝐵 even if the price passes 𝑣1𝐴 + 𝛼 + 𝑣1𝐵 − 𝑝𝐵 .

2.2

Simulations

In this subsection, through simulations, we determine the probability of the occurrence of
ex-post loss for the global bidder and the ineﬃcient allocation under various environments.
As we noted, we say that exposure problem occurs when the global bidder wins one or bothlicenses with a loss ex-post. We have used MATLAB to write our simulation code. This codeﬁrst draws the valuations for both the global and the local bidders from a given distributionfunction. We have one local bidder on each license and one global bidder on both licenses.
One set of valuations correspond to one auction. We calculate the optimal drop-out priceof the global bidder; local bidders’ drop-out prices are their valuations. The global bidder’sequilibrium drop-out price is updated as these local bidders drop out from the auction. If
the global bidder does not win the ﬁrst license, then no exposure problem occurs. If he wins
12the ﬁrst license, then we calculate his updated price for the remaining license (unless theglobal bidder drops out from both licenses at the same time). We next determine whether
the global bidder will win the remaining license at a positive proﬁt (no exposure problem),at a loss (exposure problem II) or lose the remaining license (exposure problem I). Dividing
the number of each of these events to the number of draws yields the probability of each
event.16
We use three diﬀerent distribution functions to draw valuations: uniform, beta distribution with 𝛼 = 1, and 𝛽 = 4, and beta distribution with 𝛼 = 4 and 𝛽 = 1. The second
distribution is ﬁrst order stochastically dominated by the uniform distribution, while thethird one ﬁrst order stochastically dominates the uniform distribution. We use one local
bidder on each license.17 We run simulations for four diﬀerent synergy levels: 0.2, 0.4, 0.6,
0.8.18 The 0.2 represents for small synergy, 0.4 and 0.6 represents middle synergy, and 0.8
represents a large synergy level. We report the results in TABLE 1.
We ﬁnd that the global bidder may face exposure problem with probability 4.72 per cent,if the valuations are drawn from beta distribution with parameters 𝛼 = 1 and 𝛽 = 4, and thesynergy level is equal to 0.4. Note that the expected stand alone valuation for each license
is 0.2 with this distribution. Hence, the global bidder’s average valuation does not exceed 1
most of the time which result in more ineﬃcient allocation and ex-post loss.
TABLE 1 also shows that the exposure problem occurs with the smallest probability
among all these diﬀerent distributions when the synergy level is 0.8. This is expected since
the global bidder, after winning the ﬁrst license, can bid very high for the remaining license(in most cases more than 1) but do not incur a loss due to high synergy level.
16

In our simulation, to simplify calculations, we only consider cases in which the global bidder values thelicense A more than license B. After 10000 draws, we select only the valuations where the global bidder’svaluation for license A is greater than license B. Hence, we are left with approximately 5000 draws. We used
UNIX system of the University of Manitoba, and our laptops for the simulations. In the UNIX machine, it
took more than four days to run each code.
17
Given the complexity of the code we use, we feel that using one local bidder in our simulations are enoughto draw reasonable conclusions though this can be extended to two local bidders in each license. Using three
local bidders or more would extremely complicate the code since one has to keep track of updated prices
every time a local bidder drops oﬀ.
18
We also write codes for a ﬁner synergy level of 0.1,0.2,...0.9 but we do not report them since no new
insight is learned.

13

Plot of Probability of Inefficiency
0.045
Probability of Inefficiency
Total Probability of Exposure Problem
Probability of Exposure Problem 1
Probability of Exposure Problem 2

0.04
0.035

probability

0.03
0.025
0.02
0.015
0.01
0.005
0
0.2

0.3

0.4

0.5
0.6
0.7
alpha (Synergy)

0.8

0.9

1

Figure 2: Valuations are drawn from Uniform Distribution.
In our simulations, Exposure problem I occurred most often when the synergy level is0.4. In this case, the global bidder overbids to enjoy the middle level synergy, so he is very
likely to make a substantial (potential) loss when he wins the ﬁrst license. His optimal dropout price for the remaining license is generally below 1; hence, the risk of losing the second
license is high. Exposure problem II generally occurs the most when the synergy level is 0.6.
After winning the ﬁrst license, the global bidder will stay in the remaining license auction
for a higher drop out price; hence, rather than exposure problem I, exposure problem II is
likely to occur.
Of the two beta distribution, the one with 𝛽 = 1 and the one with 𝛽 = 4, we observethat the exposure problem occurs much less frequently with the ﬁrst one. The reason is thatwith the former one, the valuations of the global bidder is likely to be higher; hence, evenwith a small synergy level, their optimal updated drop out price after winning one license
is more than 1 in most cases. Therefore, it is less likely to have exposure problem. On theother hand, when the global bidder’s updated optimal price is less than one, it is also more
likely that the local bidder’s valuation will be high so in such cases, one may see exposure
14

problem. As the synergy increases, this case is less likely to happen.
In TABLE 1, we show that, for the Beta distribution with 𝛽 = 1, the allocation is
ineﬃcient 7.84 per cent of the time when 𝛼 = 0.2. As 𝛼 increases, global bidder wins bothlicenses more often without facing exposure problem that much, and this is the eﬃcient
outcome. Hence, we observe ineﬃcient allocation only 0.66 per cent of the time for 𝛼 = 0.8.
Most of the ineﬃciency is due to exposure problem.
We will talk about the role of ineﬃciency on revenue in the next section. Especially,
when the ineﬃciency is due to exposure problem and when it is due to local bidders winninglicenses ineﬃciently.
Before closing this section, we quickly talk about TABLE 2 which shows the optimal dropout prices for three diﬀerent distributions. When the local bidders’ valuation is expected to
be low (when 𝛽 = 4), optimal drop out price is higher compared to the cases where the local
bidder’s valuation is expected to be high (when 𝛽 = 1).

3

Comparison with Vickrey Clarke Groves Auction

In this section, we will compare the revenue of our auction with the revenue of Vickrey Clarke
Groves (VCG) auction.19 VCG auction maximizes the expected payment of each agent
among all mechanisms for allocating multiple objects that are eﬃcient, incentive compatible,
and individually rational.20 In this auction, the seller will let the bidders bid on license A,
license B and the whole package license A and B.
We, will ﬁrst calculate the payment of each winner for all cases; that is, calculate therevenue of the seller. The payment of a winner (say player i) in this auction is the diﬀerence
between the social welfare of the others if the bidder did not participate in the auction(denote this as 𝑊 −𝑖 (𝑥−𝑖 ) where 𝑥−𝑖 denote the bid of all players other than player 𝑖), andthe welfare of the others when he participated in the auction, and bid truthfully (denote
this as 𝑊 −𝑖 (𝑥), where 𝑥 denote the bid of all players.) since truthful bidding is the weakly
19

In the literature, there are many papers on VCG mechanism but not interested in exposure problem
aspect such as Sano (2011) and Misra and PArkes (2009)20
See proposition 16.2 of Krishna (2010). In this section, we follow the notation of Krishna (2010) closely.

15

Table 2: OPTIMAL DROP OUT PRICE
GlobalBidder’s
Valuation
for License A
𝑣1𝐴

Beta
Distr.
with 𝛼 = 1and 𝛽 = 4
𝑝𝛽=4
1

Beta
Distr.
with 𝛼 = 4and 𝛽 = 1𝑝𝛽=110.2796
0.29580.32250.36
0.562

0.20270.20860.22760.289
0.4951

0.25
0.4
0.6
0.8
0.81

Synergy=0.4
0.2
0.29090.2
0.35280.2
0.45360.2
0.5551
0.4
0.73250.4026
0.4671
0.52
0.56
0.762

0.22150.25460.35240.489
0.6951

0.25
0.4
0.6
0.8
0.81

Synergy=0.6
0.2
0.37870.2
0.46670.2
0.6
0.2
0.72680.4
0.87330.51670.59120.68860.76
0.90440.28650.38740.55240.689
0.8839

0.25
0.4
0.6
0.8
0.81

Synergy=0.8
0.2
0.5
0.2
0.6
0.2
0.73330.2
0.86670.4
10.6226
0.6981
0.79870.89941

0.44510.58070.74010.8775
1

0.25
0.4
0.6
0.8
0.81

GlobalUniform
Bidder’s
Distr.
Valuation
for License B
𝑣1𝐵
𝑝𝑈
1
Synergy=0.2
0.2
0.231
0.2
0.2641
0.2
0.31270.2
0.36830.4
0.559116

Table 3: Bidders’ Valuations.
A
B
AB
𝑣1𝐴 𝑣1𝐵 𝑣1𝐴 +𝑣1𝐵 +𝛼
𝑣2𝐴
0
𝑣2𝐴
0 𝑣3𝐵
𝑣3𝐵

dominant strategy.
The table below shows the valuations of the bidders. To give an example of how payments
are calculated, let us assume that 𝑣2𝐴 + 𝑣1𝐵 > 𝑣1𝐴 + 𝑣1𝐵 + 𝛼 > 𝑣2𝐴 + 𝑣3𝐵 . VCG auction will
allocate license A to local bidder A, and license B to global bidder. Payment of local bidder
A is (𝑊 −2 (𝑥−2 )) − 𝑊 −2 (𝑥) = (𝑣1𝐴 + 𝑣1𝐵 + 𝛼) − 𝑣1𝐵 = 𝑣1𝐴 + 𝛼. If local bidder A does notparticipate in the auction, then global bidder will win the package; hence, the welfare of theothers 𝑊 −2 (𝑥−2 ) = 𝑣1𝐴 + 𝑣1𝐵 + 𝛼. When it participates in the auction, global bidder gets
only license B, and local bidder B gets nothing; hence, the welfare of others in this case is
𝑊 −2 (𝑥) = 𝑣1𝐵 .
Payment of the global bidder is: (𝑊 −1 (𝑥−1 )) − 𝑊 −1 (𝑥) = (𝑣2𝐴 + 𝑣3𝐵 ) − 𝑣2𝐴 = 𝑣3𝐵 . If the
global bidder does not participate in the auction, local bidder A and B wins each license;
hence, the welfare of others is the term inside the parenthesis. When the global bidder
participates in the auction, local bidder A wins license A but the welfare of local bidder B
is zero; hence the welfare of others in this case is just 𝑣2𝐴 .
The total revenue of the seller in this case will be 𝑣1𝐴 + 𝛼 + 𝑣3𝐵 .
We summarize the revenue of the seller for all cases in proposition 9 in appendix.
We compare the revenue of the simultaneous ascending auction with those of the VCG
auction through simulation methods. Our results are summarized in FIGURE 2. We run
the simulations for 𝛼 = 0.2, 0.4, 0.6, 0.8 and three diﬀerent distributions. All results showthat VCG mechanism gives higher revenue especially when 𝛼 = 0.2. Therefore, unlike
the complete information examples of VCG mechanism in the literature (e.g. Ausubel and
Milgrom (2006)), we show that revenue is higher with VCG mechanism when licenses are17

Table 4: Example of Revenue Comparison for Ineﬃcient Allocations.
𝑣1𝐴
0.09630.1149𝑣1𝐵
0.00080.1015

𝛼
0.2
0.2

𝑝∗1
0.07180.1677𝑣2𝐴
0.15810.31

𝑣3𝐵
0.1247
0.16

Revenue-Our Model
0.1681 = 𝑝∗1 + 𝑣1𝐴
0.47 = 𝑣2𝐴 + 𝑣3𝐵

Revenue-VCG Mechanism
0.2829 = 𝑣2𝐴 + 𝑣3𝐵
0.3628 = 2(𝑣1𝐴 + 𝑣1𝐵 + 𝛼) − 𝑣2𝐴 − 𝑣3𝐵

not identical in an incomplete information setting.
When there is ineﬃcient allocation in our auction, either global bidder wins (withexposure problem/ex-post loss), or local bidders win when global bidders would win in VCG
auction. If most of the ineﬃcient allocations are due to the ﬁrst one, then our auction’s
revenue is expected to be higher. If most of the ineﬃcient cases are due to the latter one,then VCG mechanism’s revenue is expected to be higher. For example, when we look for
Beta Distribution with 𝛽 = 4 case, exposure problem is very high with 3.48 per cent forthe synergy level of 𝛼 = 0.2; however, the overall ineﬃciency is 7.84 per cent. We canconclude that most ineﬃciency is due to local bidders winning licenses. As a result, VCG
mechanism gives a higher revenue. When 𝛼 = 0.6 or 𝛼 = 0.8, we see that most ineﬃciency
is due to exposure problem; hence, our auction gives a higher revenue compared to VCG
mechanism. When 𝛼 = 0.8, ineﬃciencies disappear, and not surprisingly the revenue levelsof both auctions converge.
To make our point, let us look at the sample valuations in TABLE 4. In the ﬁrst row, ourauction allocates goods ineﬃciently to two local bidders. With VCG, however, global bidder
wins the auction, and pays a higher price. Hence, VCG auction gives a higher revenue. In
the second row, global bidder wins both licenses with an ex-post loss. Hence, the revenue ofour model is higher than VCG mechanism’s revenue.
We prove that SAA auction increases revenue for cases in which global bidder wins
licenses with ex-post loss, and VCG mechanism increases revenue for cases in which local
bidders win the licenses but the allocation is ineﬃcient.
Proposition 7 When the global bidder wins licenses with an ex-post loss; that is, exposure
problem occurs, then revenue of the VCG auction is lower than the simultaneous ascending
auction (SAA); that is, 𝑅𝑉 𝐶𝐺 < 𝑅𝑆𝐴𝐴 .
18

Proposition 8 When the local bidder wins the licenses and the allocation is ineﬃcient, thenrevenue in VCG model is greater than SAA, that is, 𝑅𝑉 𝐶𝐺 > 𝑅𝑆𝐴𝐴 .
Unfortunately, one cannot say much when the allocation is eﬃcient with SAA auction.
There are cases in which SAA auction gives higher or lower revenue than the VCG auction
in eﬃcient allocations.
The literature writes that exposure problem lowers the revenue since the global bidder
expects that it may end up with a loss; hence, does not over bid too much. Hence, local
bidders win when the global bidder is supposed to win in an eﬃcient auction like VCG.
While this intuition is correct, there is another side of the story. The global bidder still bids
above the stand alone valuation in a SAA auction; hence, it may end up winning licenses
with a loss. This overbidding brings higher revenue compared in SAA auction compared to
VCG auction.

4

More Than One Global Bidder

In this section, we will analyze the case where there is more than one global bidder. The
diﬃculty in this case arises from the fact that each global bidder’s optimal drop out pricedepends on the other global bidders’ optimal drop out price which is a function of private
valuations. As a result, each global bidder, while calculating its own drop out price,
should at the same time calculate the distribution of the other global bidder’s dropout price. This makes deriving an analytical result impossible.
To make our case, assume that there are two global bidders, Firm 1 and Firm 2, and one
local bidder on each of license A and license B which are denoted as Firm 3 and Firm 4. Let
the valuations 𝑣𝑖𝐴 for 𝑖 = 1, 2, 3 and 𝑣𝑗𝐵 for 𝑗 = 1, 2, 4 are drawn from the same distributionfunction 𝐺(.). Everything else is the same as the previous section. In addition assume that𝑣𝑖𝐵 < 𝑣𝑖𝐴 for 𝑖 = 1, 2.21 Firm 1 will make a similar calculation as we discussed in one global
bidder case –except that the price would depend on the drop out price of the second global
21This does not have to be true but our point is even if you assume such a restrictive assumption, you
cannot calculate the optimal drop out prices.

19

Plot of Revenue Comparison
0.96
Average Revenue for Our Model
Average Revenue for efficient VCG Model

0.94
0.92

Average Revenue

0.9
0.88
0.86
0.84
0.82
0.8
0.78
0.76
0.2

0.3

0.4

0.5
0.6
0.7
alpha (Synergy)

0.8

0.9

1

Plot of Revenue Comparison
1.62
Average Revenue for Our Model
Average Revenue for efficient VCG Model

Average Revenue

1.6

1.58

1.56

1.54

1.52

1.5
0.2

0.3

0.4

0.5
0.6
0.7
alpha (Synergy)

0.8

0.9

1

Plot of Revenue Comparison
0.42
Average Revenue for Our Model
Average Revenue for efficient VCG Model

Average Revenue

0.4

0.38

0.36

0.34

0.32

0.3
0.2

0.3

0.4

0.5
0.6
0.7
alpha (Synergy)

0.8

0.9

1

20
Figure 3:
Uniform Distribution (Top), Beta Distribution with 𝛼 = 4, 𝛽 = 1, Beta Distribution with
𝛼 = 1, 𝛽 = 4 (Bottom)

bidder which we denote as 𝑝∗2 – Speciﬁcally, he should make the following calculation while
deciding to stay in the license B auction or not. This calculation is done based on a history
in which none of the other bidders have dropped out yet.
[ ∫ 𝑣1𝐴(𝑣1𝐴 − 𝑀 𝑎𝑥{𝑣2𝐴 , 𝑣3𝐴 }𝑃 (𝑝∗2 < 𝑣4𝐵 ))𝑑𝐺(𝑀 𝑎𝑥{𝑣2𝐴 , 𝑣3𝐴 }∣𝑝)𝐸Π11 = 𝑀 𝑎𝑥{0, 𝐸 ([

(6)

𝑝

+

∫ 𝑣1𝐴
𝑝

𝐸Π21 =∫ 𝑀 𝑖𝑛{𝑣1𝐴 +𝛼,1}𝑝](𝑣1𝐴 − 𝑀 𝑎𝑥{(𝑣2𝐴 + 𝛼), 𝑣3𝐴 }𝑃 (𝑝∗2 > 𝑣4𝐵 ))𝑑𝐺(𝑀 𝑎𝑥{(𝑣2𝐴 + 𝛼), 𝑣3𝐴 }∣𝑝)])∣𝑝∗2 }

(7)

∫ 1
(𝑉1 −𝑝−𝑝𝐴 )𝑑𝐺(𝑀 𝑎𝑥{𝑣2𝐴 , 𝑣3𝐴 }∣𝑝)+

𝑀 𝑖𝑛{𝑣1𝐴 +𝛼,1}(𝑣1𝐵 −𝑝)𝑑𝐺(𝑀 𝑎𝑥{𝑣2𝐴 , 𝑣3𝐴 }∣𝑝)
(8)

The explanation of equation 5 is as follows. If this global bidder drops out from license
B before the other global and local bidders, then it will continue on license A auction as alocal bidder. Hence, it can only derive a beneﬁt of 𝑣1𝐴 if it wins the license. The price it will
pay for license A depends on the result of whether the local or the other global bidder wins
license B. If the local bidder B wins license B (this happens with probability 𝑃 (𝑝∗2 < 𝑣4𝐵 )and we integrate this in the range 𝑝 to 𝑣1𝐴 ), then the price of license A will be the maximumof local bidder A’s valuation and the other global bidder’s valuation for license which is
𝑀 𝑎𝑥{𝑣2𝐴 , 𝑣3𝐴 }. Expectation is taken with respect to 𝑝∗2 .
Equation 7 analyze the case in which the other global bidder wins license B. This happens
with probability 𝑃 (𝑝∗2 > 𝑣4𝐵 ) (in the range 𝑝 to 𝑣1𝐴 ), then it can enjoy synergy and will bid
until 𝑣2𝐴 + 𝛼 for license A. Then, the price global bidder 1 will pay is 𝑀 𝑎𝑥{(𝑣2𝐴 + 𝛼), 𝑣3𝐴 }.
Equation 8 analyzes the case in which Firm 1 wins license B, and then continue optimallyon license A. This is the same as “one global bidder” case. The only diﬀerence is that theother global bidder is now a local bidder; hence, there is one more local bidder in the license
A auction compared to the “one global case”.
When we equate these integrals, we will ﬁnd 𝑝∗1 (𝑣1𝐴 , 𝑣1𝐵 , 𝛼, 𝑝∗2 (𝑣2𝐴 , 𝑣2𝐵 , 𝛼)). Global bidder, Firm 1, does not know 𝑣2𝐴 and 𝑣2𝐵 , and hence, 𝑝∗2 . Since 𝑝∗2 is not known, Firm 1 must21

use its distribution which he does not even know the functional form. At the same time,
the other global bidder, Firm 2, is making a similar calculation; however, he does not know𝑝∗1 (𝑣1𝐴 , 𝑣1𝐵 , 𝛼, 𝑝∗2 (𝑣2𝐴 , 𝑣2𝐵 , 𝛼)). In other words, two global bidders will have two non-linear
equations 𝑝∗1 (𝑣1𝐴 , 𝑣1𝐵 , 𝛼, 𝑝∗2 (𝑣2𝐴 , 𝑣2𝐵 , 𝛼)) and 𝑝∗2 (𝑣2𝐴 , 𝑣2𝐵 , 𝛼, 𝑝∗1 (𝑣1𝐴 , 𝑣1𝐵 , 𝛼)) that should be simultaneously solved, and to calculate these equations, they need to calculate the distributionof 𝑝∗𝑖 for 𝑖 = 1, 2. We are not sure whether this can be solved through simulations so it isleft as an open problem.22

5Conclusion and Discussion

We showed the optimal bidding strategies of global bidders when there are moderate synergies
and the licenses are heterogeneous. We also analyzed exposure problem and its role onrevenue extensively. Our revenue comparison of SAA and VCG auctions show that VCG
mechanism gives a higher revenue when the synergy level is low.
We were able to show exposure problem can occur even when the global bidder wins all
licenses. Literature has not studied heterogeneous license case with moderate synergies since
it is technically challenging when one uses more than one global bidder. With this paper,
we ﬁll this gap. One of our contributions is to write a complicated code to calculate theprobability of exposure problem. Our simulation results show that the exposure problem
may be minor for some distributions but may be up to 4.3 per cent for some others.
Extending the results to 𝑛 global bidders would be very complicated since the optimal
strategies of global bidders (optimal drop out prices) should be determined jointly which inturn would depend on how many local and how many global bidders are still in the auction.
Moreover, one has to know the distribution of the other global bidder’s optimal drop out
price while calculating the optimal drop out price! We leave this as an open problem, and,in this paper we follow the literature that use only one global bidder (e.g. Kagel and Levin(2005)).
22

We suspect that other papers in the literature assumes identical licenses to avoid this problem In fact,
Albano et. al (2006) writes “In fact, for intermediate values of 𝛼 ∈ (0, 1) and if 𝑣1 𝑎𝑛𝑑 𝑣2 are diﬀerent...;showing the existence of a PBE is already problematic in this case.”

22

Our other contribution is comparing the revenue and the eﬃciency properties of thesimultaneous ascending auction with those of the VCG auction. We show that when synergy
level is small (𝛼 = 0.2), the VCG mechanism generates more revenue. However, when
valuations are such that exposure problem occurs, then SAA auction gives a higher revenue.
When the local bidders win the license ineﬃciently (in SAA auction), the VCG mechanism
gives a higher revenue.

References[1] Albano, Gian L., Fabrizio Germano, and Stefano Lovo. 2001.“A Comparison ofStandard Multi-Unit Auctions with Synergies.” Economics Letters 71: 55-60.
[2] Albano, Gian L., Fabrizio Germano, and Stefano Lovo. 2006. “Ascending Auctions for Multiple Objects: the Case for the Japanese Design.” Economic Theory 28:
331-55.
[3] Ausubel, Lawrence M., and Paul Milgrom. 2006. “The Lovely But Lonely Vickrey
Auction.” Peter Cramton, Yoav Shoham, Richard Steinberg, eds. Combinatorial Auctions.
MIT Press, Cambridge, MA.
[4] Brusco, Sandro, and Giuseppe Lopomo. 2002. “Collusion via Signaling in Simultaneous Ascending Bid Auctions with Heterogeneous Objects, with and without Complementarities.” The Review of Economic Studies 69: 407-63.
[5] Brusco, Sandro, and Giuseppe Lopomo. 2009. “Simultaneous Ascending Auctions
with Complementarities and Known Budget Constraints.” Economic Theory 38: 105-24.
[6] Chakraborty, Indranil. 2004. “Multi-Unit Auctions with Synergy.” Economics Bulletin 4: 1-14.
[7] Chow, Yuenleng, and Abdullah Yavas. 2009. “Auctions With Positive Synergies:
Experimental Evidence.” Manuscript.
23

[8] Englmaier, Florian, Pablo Guillen, Loreto Llorente, Sander Onderstal, andRupert Sausgruber. 2009. “The Chopstick Auction: A Study of the Exposure Problem
in Multi-Unit Auctions.” International Journal of Industrial Organization 27: 286-91.
[9] Goerre, Jacob K., and Lien Yuanchuan. 2010. “An Equilibrium Analysis of the
Simultaneous Ascending Auction.” University of Zurich Working paper 428.
[10] Harstad, Ronald M., and Michael H. Rothkopf. 1995. “Withdrawable Bids as
Winner’s Curse Insurance.” Operations Research 43: 983-94.
[11] Kagel, John H., and Dan Levin. 2005. “Multi-Unit Demand Auctions with Synergies: Behavior in Sealed-Bid versus Ascending-Bid Uniform-Price Auctions.” Games andEconomic Behavior 53: 170-207.
[12] Katok, Elena, and Alvin E. Roth. 2004. “Auctions of Homogeneous Goods with
Increasing Returns: Experimental Comparison of Alternative Dutch Auctions.” Management Science 50: 1044-63.
[13] Krishna, Vijay, and Robert W. Rosenthal. 1996. “Simultaneous Auctions with
Synergies.” Games and Economic Behavior 17: 1-31.
[14] Porter, David P. 1999. “The Eﬀect of Bid Withdrawal in a Multi-Object Auction.”
Review of Economic Design 4: 73-97.
[15] Menicucci, Domenico. 2003. “Optimal Two-Object Auctions with Synergies.” Review
of Economic Design 8: 143-64.
[16] Mishra, Debasis, and David C. Parkes. 2009. “Multi-Item Vickrey-Dutch Auctions.” Games and Economic Behavior 66: 326-47.
[17] Rosenthal, Robert W., and Ruqu Wang. 1996. “Simultaneous Auctions with Synergies and Common Values.” Games and Economic Behavior 17: 32-55.

24

[18] Sano, Ryuji. 2011. “Non-bidding equilibrium in an ascending core-selecting auction.”
Games and Economic Behavior, Forthcoming.
[19] Wang, Ruqu. 2000. “Bidding and Renegotiation in Procurement Auctions.” European
Economic Review 44: 1577-97.
[20] Zheng, Charles Z. 2008. “Jump Bidding and Overconcentration in Decentralized
Simultaneous Ascending Auctions.” Iowa State University Department of Economics Staﬀ
General Research Papers 12698.

A

Mathematical Appendix

Proof of Lemma 4
We will prove that there is a unique optimal drop out price by solving 𝐸Π11 = 𝐸Π21 . Here,
we assume that 𝑣1𝐵 < 𝑣1𝐴 .23 We have four cases.
Case I: In this case, we will assume 𝑣1𝐴 +𝛼 < 1 and∫ 𝑣1𝐴 +𝛼𝑣1𝐴

𝐺(𝑝𝐴 ∣𝑝)𝑑𝑝𝐴 +(𝑣1𝐵 −𝑣1𝐴 ) < 0.

We will show that this implies 𝑝∗1 < 𝑣1𝐴 (which in turn implies 𝐸Π11 > 0).
First, we show that there exists a unique solution that makes equations 1 and 2 equal,
and this is the optimal drop out price 𝑝∗1 . We deﬁne a new function, 𝐽(𝑝, 𝑚) = 𝐸Π11 − 𝐸Π21 .
To prove uniqueness, we will show that this function is monotonically increasing and it isnegative when 𝑝 = 𝑣1𝐵 (by lemma 2, p cannot be less than 𝑣1𝐵 ) and is positive when 𝑝 = 𝑣1𝐴 .
Hence, there must be a unique root at the interval 𝑣1𝐵 < 𝑝 < 𝑣1𝐴 .
∫𝑣∫ 𝑣 +𝛼𝐽(𝑝, 𝑚) = 𝑝 1𝐴 (𝑣1𝐴 − 𝑝𝐴 )𝑔(𝑝𝐴 ∣𝑝)𝑑𝑝𝐴 − 𝑝 1𝐴 (𝑉1 − 𝑝 − 𝑝𝐴 )𝑔(𝑝𝐴 ∣𝑝)𝑑𝑝𝐴∫1− 𝑣1𝐴 +𝛼 (𝑣1𝐵 − 𝑝)𝑔(𝑝𝐴 ∣𝑝)𝑑𝑝𝐴 .
∫1
By using 𝑝 𝑔(𝑝𝐴 ∣𝑝)𝑑𝑝𝐴 = 1 since 𝑔(𝑝𝐴 ∣𝑝) is a probability density function on the support∫1[𝑝, 1], we have (𝑣1𝐵 − 𝑝) 𝑝 𝑔(𝑝𝐴 ∣𝑝)𝑑𝑝𝐴 = 𝑣1𝐵 − 𝑝, we can re-write it as∫ 𝑣1𝐴∫ 𝑣 +𝛼(𝑣1𝐴 − 𝑝𝐴 )𝑔(𝑝𝐴 ∣𝑝)𝑑𝑝𝐴 − 𝑝 1𝐴 (𝑣1𝐴 + 𝛼 − 𝑝𝐴 )𝑔(𝑝𝐴 ∣𝑝)𝑑𝑝𝐴 − (𝑣1𝐵 − 𝑝)𝑝∫∫
By using integration by parts twice 𝑢𝑑𝑣 = 𝑢𝑣 − 𝑣𝑑𝑢, ﬁrst we assume that 𝑢 = 𝑣1𝐴 − 𝑝𝐴
and 𝑣 = 𝐺(𝑝𝐴 ∣𝑝); then assume that 𝑢 = 𝑣1𝐴 +𝛼−𝑝𝐴 and 𝑣 = 𝐺(𝑝𝐴 ∣𝑝), thus, 𝑑𝑣 = 𝑔(𝑝𝐴 ∣𝑝)𝑑𝑝𝐴 ,23

The other case, 𝑣1𝐵 > 𝑣1𝐴 , is symmetric.

25

we have∫𝑣= (𝑣1𝐴 − 𝑝𝐴 )𝐺(𝑝𝐴 ∣𝑝) ∣𝑣𝑝1𝐴 − 𝑝 1𝐴 𝐺(𝑝𝐴 ∣𝑝)𝑑(𝑣1𝐴 − 𝑝𝐴 )∫ 𝑣 +𝛼− (𝑣1𝐴 + 𝛼 − 𝑝𝐴 )𝐺(𝑝𝐴 ∣𝑝) ∣𝑣𝑝1𝐴 +𝛼 + 𝑝 1𝐴 𝐺(𝑝𝐴 ∣𝑝)𝑑(𝑣1𝐴 + 𝛼 − 𝑝𝐴 ) − (𝑣1𝐵 − 𝑝)∫𝑣
∫ 𝑣 +𝛼∫ 𝑣1𝐴 +𝛼
= 𝑝 1𝐴 𝐺(𝑝𝐴 ∣𝑝)𝑑𝑝𝐴 − 𝑝 1𝐴 𝐺(𝑝𝐴 ∣𝑝)𝑑𝑝𝐴 − (𝑣1𝐵 − 𝑝) = − 𝑣1𝐴
𝐺(𝑝𝐴 ∣𝑝)𝑑𝑝𝐴 − (𝑣1𝐵 − 𝑝)
We take partial derivative of 𝐽(𝑝, 𝑚) with respect to 𝑝, we have,∫ 𝑣1𝐴 +𝛼∂𝐽(𝑝,𝑚)∂=[−𝐺(𝑝𝐴 ∣𝑝)] + 1 > 0
∂𝑝
∂𝑝
𝑣1𝐴
∫ 𝑣1𝐴 +𝛼∂
It is positive since the term ∂𝑝[ 𝑣1𝐴
𝐺(𝑝𝐴 ∣𝑝)] is negative. As the lower limit of theintegral increases, the value of the expression decreases (does not increase) if the term inside
is non-negative which is true since it is a cumulative distribution function. We must also
𝐴 ∣𝑝)show that ∂𝐺(𝑝≤ 0 to prove this. While one can easily see that this is correct (as p∂𝑝increases the cumulative distribution conditional on p decreases), we will give a formal proof
by using Leibniz’s rule
when necessary.
∫𝑝
𝐴 𝑓 (𝑣)𝑑𝑣)𝑚−1 ]∂[( ∫𝑝 1∂𝐺(𝑝𝐴 ∣𝑝)𝑝 𝑓 (𝑣)𝑑𝑣⇔ ∂𝑝 =∂𝑝∫∫( 𝑝𝐴 𝑓 (𝑣)𝑑𝑣)𝑚−1( 𝑝𝑝𝐴 𝑓 (𝑣)𝑑𝑣)𝑚−2= −(𝑚 − 1)𝑓 (𝑝) (∫ 1 𝑓 (𝑣)𝑑𝑣)𝑚−1 + (𝑚 − 1)𝑓 (𝑝) 𝑝(∫ 1 𝑓 (𝑣)𝑑𝑣)𝑚𝑝
𝑝∫∫𝑝
𝐴 𝑓 (𝑣)𝑑𝑣(𝑚−1)𝑓 (𝑝)( 𝑝𝑝𝐴 𝑓 (𝑣)𝑑𝑣)𝑚−2𝑝∫=[−1 + ∫ 1 𝑓 (𝑣)𝑑𝑣 ]( 𝑝1 𝑓 (𝑣)𝑑𝑣)𝑚−1𝑝∫𝑝(𝑚−1)𝑓 (𝑝)( 𝑝 𝐴 𝑓 (𝑣)𝑑𝑣)𝑚−2∫=[−1 + 𝐹 (𝑝𝐴 ∣𝑝)] < 0 (≤ 0 only if 𝑝𝐴 = 1).
( 1 𝑓 (𝑣)𝑑𝑣)𝑚−1𝑝Thus, 𝐽(𝑝, 𝑚) is monotonically increasing function of 𝑝, when 𝑣1𝐵 ≤ 𝑝 < 𝑣1𝐴 .
∫ 𝑣1𝐴 +𝛼
∫ 𝑣1𝐴
𝐺(𝑝𝐴 ∣𝑣1𝐵 )𝑑𝑝𝐴
𝐺(𝑝𝐴 ∣𝛼)𝑑𝑝𝐴 − 𝑣1𝐵
If 𝑝 = 𝑣1𝐵 , then 𝐽(𝑣1𝐵 ) = 𝑣1𝐵∫ 𝑣1𝐴 +𝛼
= − 𝑣1𝐴 𝐺(𝑝𝐴 ∣𝑣1𝐵 )𝑑𝑝𝐴 < 0. (note that we prove this for the case 𝑣1𝐴 > 𝑣1𝐵 )∫ 𝑣1𝐴 +𝛼If 𝑝 = 𝑣1𝐴 , 𝐽(𝑣1𝐴 ) = 0 − 𝑣1𝐴
𝐺(𝑝𝐴 ∣𝑝)𝑑𝑝𝐴 − (𝑣1𝐵 − 𝑣1𝐴 ) > 0, then our assumption∫ 𝑣1𝐴 +𝛼𝐺(𝑝𝐴 ∣𝑝)𝑑𝑝𝐴 + (𝑣1𝐵 − 𝑣1𝐴 ) < 0 implies that 𝐽(𝑝 = 𝑣1𝐴 ) > 0.
𝑣1𝐴
Hence, there is a unique root, 𝑝∗1 in the such that 𝑣1𝐵 < 𝑝∗1 < 𝑣1𝐴 .
Next, we show that as the number of active ﬁrms in license A auction decreases, theoptimal drop out price will increase. We will use the implicit function theorem for this:∂𝐽(𝑝∗ ,𝑚)1
𝑑𝑝∗
⇔ 𝑑𝑚1 = − ∂𝐽(𝑝∂𝑚∗1 ,𝑚)
∂𝑝∗
1

< 0.
∂𝐽(𝑝∗ ,𝑚)

We have already shown that ∂𝑝1∗ > 0.
1
∫ 𝑣1𝐴
∫ 𝑣 +𝛼Since 𝐽(𝑝, 𝑚) = 𝑝 𝐺(𝑝𝐴 ∣𝑝)𝑑𝑝𝐴 − 𝑝 1𝐴 𝐺(𝑝𝐴 ∣𝑝)𝑑𝑝𝐴 − (𝑣1𝐵 − 𝑝).

26

Since the lower limits of the two integrals are the same, we can re-write this as 𝐽(𝑝, 𝑚) =∫ 𝑣1𝐴 +𝛼
− 𝑣1𝐴 𝐺(𝑝𝐴 ∣𝑝)𝑑𝑝𝐴 − (𝑣1𝐵 − 𝑝).
By using Leibniz’s rule for diﬀerentiation under the integral sign, we take partial derivative of 𝐽(𝑝, 𝑚) with respect to 𝑚 and have,∫ 𝑣1𝐴 +𝛼 ∂𝐺(𝑝𝐴 ∣𝑝)∫ 𝑣1𝐴 +𝛼∂𝐽(𝑝,𝑚)𝐴 ∣𝑝)= − 𝑣1𝐴
𝑑𝑝𝐴 = − 𝑣1𝐴
ln(𝐹 (𝑝𝐴 ∣𝑝))𝐺(𝑝𝐴 ∣𝑝)𝑑𝑝𝐴 > 0, since ∂𝐺(𝑝
=
∂𝑚
∂𝑚
∂𝑚
ln(𝐹 (𝑝𝐴 ∣𝑝))𝐺(𝑝𝐴 ∣𝑝) < 0. Hence, we show that ∂𝐽(𝑝∣𝑚)> 0 holds.
∂𝑚
By the implicit function theorem, we show that the optimal drop out price increases as
the number of local ﬁrms, 𝑚, decreases.

∂𝐹 (𝑝∗ ,𝑚)1
𝑑𝑝∗1∂𝐽(𝑝,𝑚)∂𝑚
Since ∂𝐽(𝑝,𝑚)
>
0
and
>
0,
we
have,
=
−
∗ ,𝑚)
∂𝐹
(𝑝
∂𝑝
∂𝑚
𝑑𝑚
1
∂𝑝∗
1

Case II: In this case, we will assume that 𝑣1𝐴 +𝛼 < 1 and<0∫ 𝑣1𝐴 +𝛼𝑣1𝐴

𝐺(𝑝𝐴 ∣𝑣1𝐴 )𝑑𝑝𝐴 +(𝑣1𝐵 −𝑣1𝐴 ) ≥ 0 which we will show that this implies 𝑝∗1 ≥ 𝑣1𝐴 . And this condition in turn implies
that 𝐸Π11 = 0, since the global bidder drops from both licenses.
Now let 𝐽(𝑝, 𝑚) = 𝐸Π11 − 𝐸Π21
∫ 𝑣 +𝛼∫1𝐽(𝑝, 𝑚) = 0 − 𝑝 1𝐴 (𝑉1 − 𝑝 − 𝑝𝐴 )𝑔(𝑝𝐴 ∣𝑝)𝑑𝑝𝐴 − 𝑣1𝐴 +𝛼 (𝑣1𝐵 − 𝑝)𝑔(𝑝𝐴 ∣𝑝)𝑑𝑝𝐴∫ 𝑣 +𝛼= − 𝑝 1𝐴 𝐺(𝑝𝐴 ∣𝑝)𝑑𝑝𝐴 − (𝑣1𝐵 − 𝑝).
When 𝑝 ≥ 𝑣1𝐴 , we take partial derivative of 𝐽(𝑝, 𝑚) with respect to 𝑝, we have,∫ 𝑣 +𝛼∂𝐽(𝑝,𝑚)∂𝐴 ∣𝑝)= − ∂𝑝[ 𝑝 1𝐴 𝐺(𝑝𝐴 ∣𝑝)𝑑𝑝𝐴 ] + 1 > 0, since ∂𝐺(𝑝
< 0.
∂𝑝
∂𝑝
Thus, 𝐽(𝑝, 𝑚) is monotonically increasing function of 𝑝, when 𝑣1𝐴 ≤ 𝑝 ≤ 𝑣1𝐴 + 𝛼.
∫ 𝑣1𝐴 +𝛼
Our assumption 𝑣1𝐴
𝐺(𝑝𝐴 ∣𝑣1𝐴 )𝑑𝑝𝐴 + (𝑣1𝐵 − 𝑣1𝐴 ) ≥ 0 implies that 𝐽(𝑝 = 𝑣1𝐴 , 𝑚) ≤ 0.
If 𝑝 = 𝑣1𝐴 + 𝛼, then 𝐽(𝑣1𝐴 + 𝛼) = 0 − 0 − (𝑣1𝐵 − 𝑣1𝐴 + 𝛼) > 0. Thus, there is a uniquesolution, 𝑝∗1 , in the interval [𝑣1𝐴 , 𝑣1𝐴 + 𝛼).
Next, we show that when the number of active ﬁrms in license A auction decreases, thisoptimal drop out price will increase.
We take partial derivative of 𝐽(𝑝, 𝑚) with respect to 𝑚, we have,∫ 𝑣1𝐴 +𝛼∂𝐽(𝑝,𝑚)=−ln(𝐹 (𝑝𝐴 ∣𝑝))𝐺(𝑝𝐴 ∣𝑝)𝑑𝑝𝐴 > 0.
∂𝑚𝑝
Since ∂𝐽(𝑝,𝑚)> 0 and ∂𝐽(𝑝,𝑚)> 0, we have,∂𝑝∂𝑚∂𝐽(𝑝∗ ,𝑚)1
𝑑𝑝∗1
= − ∂𝐽(𝑝∂𝑚∗1 ,𝑚)
𝑑𝑚

<0

∂𝑝∗
1

27

Case III: In this case, we will assume that 𝑣1𝐴 + 𝛼 > 1 and∫1
𝑣1𝐴

𝐺(𝑝𝐴 ∣𝑣1𝐴 )𝑑𝑝𝐴 + (𝑣1𝐵 +𝛼 − 1) < 0 which implies 𝑣1𝐴 < 𝑝∗1 . And this condition in turn implies that 𝐸Π11 > 0.
Now let 𝐽(𝑝, 𝑚) = 𝐸Π11 − 𝐸Π21∫𝑣
∫1
𝐽(𝑝, 𝑚) = 𝑝 1𝐴 (𝑣1𝐴 − 𝑝𝐴 )𝑔(𝑝𝐴 ∣𝑝)𝑑𝑝𝐴 − 𝑝 (𝑣1𝐴 + 𝑣1𝐵 + 𝛼 − 𝑝 − 𝑝𝐴 )𝑔(𝑝𝐴 ∣𝑝)𝑑𝑝𝐴
Again, we assume that 𝑢 = 𝑣1𝐴 − 𝑝𝐴 and 𝑣 = 𝐺(𝑝𝐴 ∣𝑝) for the ﬁrst integral; then assume
that 𝑢 = 𝑣1𝐴 +𝑣1𝐵 +𝛼−𝑝−𝑝𝐴 and 𝑣 = 𝐺(𝑝𝐴 ∣𝑝) for the second integral, thus, 𝑑𝑣 = 𝑔(𝑝𝐴 ∣𝑝)𝑑𝑝𝐴 ,By using integration by parts, we have∫𝑣
= (𝑣1𝐴 − 𝑝𝐴 )𝐺(𝑝𝐴 ∣𝑝) ∣𝑣𝑝1𝐴 + 𝑝 1𝐴 𝐺(𝑝𝐴 ∣𝑝)𝑑𝑝𝐴∫1− (𝑣1𝐴 + 𝑣1𝐵 + 𝛼 − 𝑝 − 𝑝𝐴 )𝐺(𝑝𝐴 ∣𝑝) ∣1𝑝 − 𝑝 𝐺(𝑝𝐴 ∣𝑝)𝑑𝑝𝐴∫𝑣∫1= 𝑝 1𝐴 𝐺(𝑝𝐴 ∣𝑝)𝑑𝑝𝐴 − (𝑣1𝐴 + 𝑣1𝐵 + 𝛼 − 𝑝 − 1) − 𝑝 𝐺(𝑝𝐴 ∣𝑝)𝑑𝑝𝐴∫1= −(𝑣1𝐴 + 𝑣1𝐵 + 𝛼 − 𝑝 − 1) − 𝑣1𝐴 𝐺(𝑝𝐴 ∣𝑝)𝑑𝑝𝐴
We take partial derivative of 𝐽(𝑝, 𝑚) with respect to 𝑝, we have,∫1∂𝐽(𝑝,𝑚)∂𝐺(𝑝𝐴 ∣𝑝)] + 1 > 0
=
[−
∂𝑝
∂𝑝
𝑣1𝐴
∫1
∂
It is positive since the term ∂𝑝[ 𝑣1𝐴 𝐺(𝑝𝐴 ∣𝑝)] is negative. And we have shown that
∂𝐺(𝑝𝐴 ∣𝑝)
≤ 0. Thus, 𝐽(𝑝, 𝑚) is monotonically increasing function of 𝑝, when 𝑣1𝐵 ≤ 𝑝 < 𝑣1𝐴 .
∂𝑝∫1
If 𝑝 = 𝑣1𝐵 , then 𝐽(𝑣1𝐵 ) = − 𝑣1𝐴 𝐺(𝑝𝐴 ∣𝑣1𝐵 )𝑑𝑝𝐴 − (𝑣1𝐴 + 𝛼 − 1) < 0 since 𝑣1𝐴 + 𝛼 > 1.
∫1
If 𝑝 = 𝑣1𝐴 , 𝐽(𝑣1𝐴 ) = 0 − 𝑣1𝐴 𝐺(𝑝𝐴 ∣𝑣1𝐴 )𝑑𝑝𝐴 − (𝑣1𝐵 + 𝛼 − 1) > 0 since 𝑣1𝐵 < 𝑣1𝐴 and𝑣1𝐴 + 𝛼 > 1, then our assumption

∫1

𝑣1𝐴

𝐺(𝑝𝐴 ∣𝑣1𝐴 )𝑑𝑝𝐴 + (𝑣1𝐵 + 𝛼 − 1) < 0 implies that𝐽(𝑝 = 𝑣1𝐴 ) > 0.
Hence, there is a unique root in the interval 𝑣1𝐵 < 𝑝 < 𝑣1𝐴 .
We skip to show that as the number of active ﬁrms in license A auction decreases, theoptimal drop out price will increase, since we have done this in Case I.
∫1
Case IV: In this case, we will assume that 𝑣1𝐴 + 𝛼 > 1 and 𝑣1𝐴 𝐺(𝑝𝐴 ∣𝑣1𝐴 )𝑑𝑝𝐴 + (𝑣1𝐵 +𝛼 − 1) ≥ 0 which implies 𝑝∗1 ≥ 𝑣1𝐴 . And this condition in turn implies that 𝐸Π11 = 0.
Now let 𝐽(𝑝, 𝑚) = 𝐸Π11 − 𝐸Π21
∫1
𝐽(𝑝, 𝑚) = 0 − 𝑝 𝐺(𝑝𝐴 ∣𝑝)𝑑𝑝𝐴 − (𝑣1𝐴 + 𝑣1𝐵 + 𝛼 − 𝑝 − 1).
When 𝑝 > 𝑣1𝐴 , we take partial derivative of 𝐽(𝑝, 𝑚) with respect to 𝑝, we have,∫1∂𝐽(𝑝,𝑚)
∂
𝐴 ∣𝑝)
=
−
[
𝐺(𝑝𝐴 ∣𝑝)𝑑𝑝𝐴 ] + 1 > 0, since ∂𝐺(𝑝
< 0.
∂𝑝
∂𝑝 𝑝
∂𝑝
28

Thus, 𝐽(𝑝, 𝑚) is monotonically increasing function of 𝑝, when 𝑣1𝐴 ≤ 𝑝 ≤ 1.
∫1
Our assumption 𝑣1𝐴 𝐺(𝑝𝐴 ∣𝑣1𝐴 )𝑑𝑝𝐴 + (𝑣1𝐵 + 𝛼 − 1) ≥ 0 implies that 𝐽(𝑝 = 𝑣1𝐴 ) ≤ 0. If
𝑝 = 1, then 𝐽(1) = 0 − 0 − (𝑣1𝐵 + 𝑣1𝐴 + 𝛼 − 2) > 0. Since 𝑣1𝐵 + 𝑣1𝐴 + 𝛼 < 2 by our averagevaluation is less than 1 assumption in the lemma. Thus, there is a unique solution, 𝑝∗1 , in
the interval [𝑣1𝐴 , 1).
We also skip to show that when the number of active ﬁrms in license A auction decreases,
this optimal drop out price will increase which has been proven in Case II.
Proposition 9 Suppose that there is one global bidder and one local bidder bidding foreach license. In the VCG auction, the seller’s revenue will be as follows depending on thevaluations of the bidders.24
CASE I: Suppose that the valuations are such that 𝑣1𝐴 + 𝑣1𝐵 + 𝛼 < 𝑣2𝐴 + 𝑣3𝐵 .
A) (Local bidders win each license) And suppose that 𝑣1𝐴 < 𝑣2𝐴 and 𝑣1𝐵 < 𝑣3𝐵 . There
are four sub cases to consider.
i) 𝑣3𝐵 > 𝑣1𝐵 + 𝛼 and 𝑣2𝐴 > 𝑣1𝐴 + 𝛼, then the revenue is 𝑣1𝐴 + 𝑣1𝐵 .
ii) 𝑣3𝐵 < 𝑣1𝐵 + 𝛼 and 𝑣2𝐴 < 𝑣1𝐴 + 𝛼, then the revenue is 2(𝑣1𝐴 + 𝑣1𝐵 + 𝛼) − 𝑣3𝐵 − 𝑣2𝐴 .
iii) 𝑣3𝐵 > 𝑣1𝐵 + 𝛼 and 𝑣2𝐴 < 𝑣1𝐴 + 𝛼, then the revenue is 2𝑣1𝐴 + 𝑣1𝐵 + 𝛼 − 𝑣2𝐴 .
iv) 𝑣3𝐵 < 𝑣1𝐵 + 𝛼 and 𝑣2𝐴 > 𝑣1𝐴 + 𝛼, then the revenue is 𝑣1𝐴 + 2𝑣1𝐵 + 𝛼 − 𝑣3𝐵 .
B) (Local bidder wins A, and global bidder wins B) And suppose that 𝑣2𝐴 > 𝑣1𝐴 and𝑣3𝐵 < 𝑣1𝐵 . Then, the revenue is 𝑣1𝐴 + 𝛼 + 𝑣3𝐵 .
C) (Local bidder wins B, and global bidder wins A) And suppose that 𝑣2𝐴 < 𝑣1𝐴 and𝑣3𝐵 > 𝑣1𝐴 . Then, the revenue is 𝑣1𝐵 + 𝛼 + 𝑣2𝐴
CASE II: Suppose that the valuations are such that 𝑣1𝐴 + 𝑣1𝐵 + 𝛼 > 𝑣2𝐴 + 𝑣3𝐵 .
A) (Global bidder wins both licenses) And suppose that 𝑣1𝐴 < 𝑣2𝐴 and 𝑣1𝐵 < 𝑣3𝐵 . Then,the revenue is 𝑣2𝐴 + 𝑣3𝐵 .
B) (Global bidder wins both licenses) And suppose that 𝑣1𝐴 > 𝑣2𝐴 and 𝑣1𝐵 > 𝑣3𝐵 . Then,the revenue is 𝑣2𝐴 + 𝑣3𝐵 .
24

This result can easily be extended to one global and many local bidders case. Since, we use one global
bidder and one local bidder in each license in the simulations, to save the notation, we give the result for a
special case.

29C) (Global bidder wins license A, local bidder B wins license B) And suppose that 𝑣2𝐴 <𝑣1𝐴 and 𝑣3𝐵 > 𝑣1𝐵 + 𝛼. Then, the revenue is 𝑣2𝐴 + 𝑣1𝐵 + 𝛼.
D) (Global bidder wins license B, local bidder A wins license A) And suppose that 𝑣2𝐴 >𝑣1𝐴 + 𝛼 and 𝑣3𝐵 < 𝑣1𝐵 . Then, the revenue is 𝑣3𝐵 + 𝑣1𝐴 + 𝛼.
Proof of Proposition 9 Suppose that there is one global bidder and one local bidder
bidding for each license. In the VCG auction, the seller’s revenue will be as follows dependingon the valuations of the bidders.
CASE I: Suppose that the valuations are such that 𝑣1𝐴 + 𝑣1𝐵 + 𝛼 < 𝑣2𝐴 + 𝑣3𝐵 .
A) (Local bidders win each license) And suppose that 𝑣1𝐴 < 𝑣2𝐴 and 𝑣1𝐵 < 𝑣3𝐵 . There
are four sub cases to consider.
i) 𝑣3𝐵 > 𝑣1𝐵 + 𝛼 and 𝑣2𝐴 > 𝑣1𝐴 + 𝛼, then the revenue is 𝑣1𝐴 + 𝑣1𝐵 .
since local bidder A’s payment would be 𝑊 (𝑥−2 ) − 𝑊 −2 (𝑥) = 𝑣1𝐴 + 𝑣3𝐵 − (0 + 𝑣3𝐵 ) = 𝑣1𝐴
and local bidder B’s payment would be 𝑊 (𝑥−3 ) − 𝑊 −3 (𝑥) = 𝑣1𝐵 + 𝑣2𝐴 − 𝑣2𝐴 = 𝑣1𝐵ii) 𝑣3𝐵 < 𝑣1𝐵 + 𝛼 and 𝑣2𝐴 < 𝑣1𝐴 + 𝛼, then the revenue is 2(𝑣1𝐴 + 𝑣1𝐵 + 𝛼) − 𝑣3𝐵 − 𝑣2𝐴 .
since 𝑊 (𝑥−2 ) − 𝑊 −2 (𝑥) = 𝑣1𝐴 + 𝑣1𝐵 + 𝛼 − (0 + 𝑣3𝐵 ) = 𝑣1𝐴 + 𝑣1𝐵 + 𝛼 − 𝑣3𝐵 and 𝑊 (𝑥−3 ) −𝑊 −3 (𝑥) = 𝑣1𝐴 + 𝑣1𝐵 + 𝛼 − (0 + 𝑣2𝐴 ) = 𝑣1𝐴 + 𝑣1𝐵 + 𝛼 − 𝑣2𝐴iii) 𝑣3𝐵 > 𝑣1𝐵 + 𝛼 and 𝑣2𝐴 < 𝑣1𝐴 + 𝛼, then the revenue is 2𝑣1𝐴 + 𝑣1𝐵 + 𝛼 − 𝑣2𝐴 .
since 𝑊 (𝑥−2 ) − 𝑊 −2 (𝑥) = 𝑣1𝐴 + 𝑣3𝐵 − (0 + 𝑣3𝐵 ) = 𝑣1𝐴 , and 𝑊 (𝑥−3 ) − 𝑊 −3 (𝑥) =𝑣1𝐴 + 𝑣1𝐵 + 𝛼 − (0 + 𝑣2𝐴 ) = 𝑣1𝐴 + 𝑣1𝐵 + 𝛼 − 𝑣2𝐴iv) 𝑣3𝐵 < 𝑣1𝐵 + 𝛼 and 𝑣2𝐴 > 𝑣1𝐴 + 𝛼, then the revenue is 𝑣1𝐴 + 2𝑣1𝐵 + 𝛼 − 𝑣3𝐵 .
since 𝑊 (𝑥−2 ) − 𝑊 −2 (𝑥) = 𝑣1𝐴 + 𝑣1𝐵 + 𝛼 − (0 + 𝑣3𝐵 ) = 𝑣1𝐴 + 𝑣1𝐵 + 𝛼 − 𝑣3𝐵 and 𝑊 (𝑥−3 ) −𝑊 −3 (𝑥) = 𝑣1𝐵 + 𝑣2𝐴 − 𝑣2𝐴 = 𝑣1𝐵
B) (Local bidder wins A, and global bidder wins B) And suppose that 𝑣2𝐴 > 𝑣1𝐴 and𝑣3𝐵 < 𝑣1𝐵 . Then, the revenue is 𝑣1𝐴 + 𝛼 + 𝑣3𝐵 .
since 𝑊 (𝑥−2 ) − 𝑊 −2 (𝑥) = 𝑣1𝐴 + 𝑣1𝐵 + 𝛼 − (0 + 𝑣1𝐵 ) = 𝑣1𝐴 + 𝛼, and 𝑊 (𝑥−1 ) − 𝑊 −1 (𝑥) =𝑣3𝐵 + 𝑣2𝐴 − 𝑣2𝐴 = 𝑣3𝐵C) (Local bidder wins B, and global bidder wins A) And suppose that 𝑣2𝐴 < 𝑣1𝐴 and𝑣3𝐵 > 𝑣1𝐵 . Then, the revenue is 𝑣1𝐵 + 𝛼 + 𝑣2𝐴30

since 𝑊 (𝑥−3 ) − 𝑊 −3 (𝑥) = 𝑣1𝐴 + 𝑣1𝐵 + 𝛼 − (0 + 𝑣1𝐴 ) = 𝑣1𝐵 + 𝛼, and 𝑊 (𝑥−1 ) − 𝑊 −1 (𝑥) =𝑣3𝐵 + 𝑣2𝐴 − 𝑣3𝐵 = 𝑣2𝐴
CASE II: Suppose that the valuations are such that 𝑣1𝐴 + 𝑣1𝐵 + 𝛼 > 𝑣2𝐴 + 𝑣3𝐵 .
A) (Global bidder wins both licenses) And suppose that 𝑣1𝐴 < 𝑣2𝐴 and 𝑣1𝐵 < 𝑣3𝐵 . Then,the revenue is 𝑣2𝐴 + 𝑣3𝐵 .
since 𝑊 (𝑥−1 ) − 𝑊 −1 (𝑥) = 𝑣2𝐴 + 𝑣3𝐵 − (0 + 0) = 𝑣2𝐴 + 𝑣3𝐵 .
B) (Global bidder wins both licenses) And suppose that 𝑣1𝐴 > 𝑣2𝐴 and 𝑣1𝐵 > 𝑣3𝐵 . Then,the revenue is 𝑣2𝐴 + 𝑣3𝐵 .
since 𝑊 (𝑥−1 ) − 𝑊 −1 (𝑥) = 𝑣2𝐴 + 𝑣3𝐵 − (0 + 0) = 𝑣2𝐴 + 𝑣3𝐵 .
C) (Global bidder wins license A, local bidder B wins license B) And suppose that𝑣2𝐴 < 𝑣1𝐴 and 𝑣3𝐵 > 𝑣1𝐵 + 𝛼. Then, the revenue is 𝑣2𝐴 + 𝑣1𝐵 + 𝛼.
since 𝑊 (𝑥−1 ) − 𝑊 −1 (𝑥) = 𝑣2𝐴 + 𝑣3𝐵 − (𝑣3𝐵 + 0) = 𝑣2𝐴 and 𝑊 (𝑥−3 ) − 𝑊 −3 (𝑥) = 𝑣1𝐴 +𝑣1𝐵 + 𝛼 − (𝑣1𝐴 + 0) = 𝑣1𝐵 + 𝛼.
D) (Global bidder wins license B, local bidder A wins license A) And suppose that
𝑣2𝐴 > 𝑣1𝐴 + 𝛼 and 𝑣3𝐵 < 𝑣1𝐵 . Then, the revenue is 𝑣3𝐵 + 𝑣1𝐴 + 𝛼.
since 𝑊 (𝑥−1 ) − 𝑊 −1 (𝑥) = 𝑣2𝐴 + 𝑣3𝐵 − (𝑣2𝐴 + 0) = 𝑣3𝐵 and 𝑊 (𝑥−2 ) − 𝑊 −2 (𝑥) = 𝑣1𝐴 +𝑣1𝐵 + 𝛼 − (𝑣1𝐵 + 0) = 𝑣1𝐴 + 𝛼.
Proof of Proposition 7 When there is an exposure problem, then revenue in VCG
model, 𝑅𝑉 𝐶𝐺 , is lower than revenue in SAA, 𝑅𝑆𝐴𝐴 .
First, we consider when there is type 1 exposure problem, there are cases:
Case 1: when 𝑣1𝐵 < 𝑣3𝐵 < 𝑝∗1 (global bidder wins license 𝐵 with a loss), and 𝑣2𝐴 > 𝑣1𝐴 +𝛼(but loses license 𝐴), then 𝑅𝑆𝐴𝐴 = 𝑣1𝐴 + 𝛼 + 𝑣3𝐵 .
For the above parameter space, there are two possible subcases in VCG model: one iswhen 𝑣3𝐵 > 𝑣1𝐵 + 𝛼, and 𝑣2𝐴 + 𝑣3𝐵 > 𝑣1𝐴 + 𝑣1𝐵 + 𝛼 (the latter condition makes this anineﬃcient allocation), then 𝑅𝑉 𝐶𝐺 = 𝑣1𝐴 + 𝑣1𝐵 which is the Case I: A)i) from Proposition 9.
Since 𝑣3𝐵 > 𝑣1𝐵 , we have 𝑅𝑉 𝐶𝐺 < 𝑅𝑆𝐴𝐴 .
The other one is when 𝑣3𝐵 < 𝑣1𝐵 + 𝛼, and 𝑣2𝐴 + 𝑣3𝐵 > 𝑣1𝐴 + 𝑣1𝐵 + 𝛼, then 𝑅𝑉 𝐶𝐺 =𝑣1𝐴 + 𝑣1𝐵 + 𝛼 + 𝑣1𝐵 − 𝑣3𝐵 which is the Case I: A)iv) from Proposition 9. Since 𝑣3𝐵 > 𝑣1𝐵 ,31

we have 𝑅𝑉 𝐶𝐺 < 𝑅𝑆𝐴𝐴 .
Case 2: This is the symmetric case when 𝑣2𝐴 > 𝑣1𝐴 , 𝑣2𝐴 < 𝑝∗1 (global bidder wins license
𝐴 with a loss), and 𝑣3𝐵 > 𝑣1𝐵 + 𝛼 (but loses license 𝐵), then 𝑅𝑆𝐴𝐴 = 𝑣1𝐵 + 𝛼 + 𝑣2𝐴 .
For the above parameter space, there are two possible subcases in VCG model: one iswhen 𝑣3𝐵 > 𝑣1𝐵 , 𝑣2𝐴 > 𝑣1𝐴 , 𝑣2𝐴 > 𝑣1𝐴 + 𝛼, 𝑣3𝐵 > 𝑣1𝐵 + 𝛼, and 𝑣2𝐴 + 𝑣3𝐵 > 𝑣1𝐴 + 𝑣1𝐵 + 𝛼(thelatter condition makes this an ineﬃcient allocation), then 𝑅𝑉 𝐶𝐺 = 𝑣1𝐴 + 𝑣1𝐵 which is theCase I: A)i) from Proposition 9. Since 𝑣2𝐴 > 𝑣1𝐴 , we have 𝑅𝑉 𝐶𝐺 < 𝑅𝑆𝐴𝐴 .
The other one is when 𝑣3𝐵 > 𝑣1𝐵 , 𝑣2𝐴 > 𝑣1𝐴 , 𝑣2𝐴 < 𝑣1𝐴 + 𝛼, 𝑣3𝐵 > 𝑣1𝐵 + 𝛼, and𝑣2𝐴 + 𝑣3𝐵 > 𝑣1𝐴 + 𝑣1𝐵 + 𝛼, then 𝑅𝑉 𝐶𝐺 = 𝑣1𝐴 + 𝑣1𝐵 + 𝛼 + 𝑣1𝐴 − 𝑣2𝐴 which is the Case I: A)iii)
from Proposition 9. Since 𝑣2𝐴 > 𝑣1𝐴 , we have 𝑅𝑉 𝐶𝐺 < 𝑅𝑆𝐴𝐴 .
Then, we consider when there is type 2 exposure problem, there are two kinds of cases.
Case 1: When 𝑣3𝐵 > 𝑣1𝐵 , 𝑣2𝐴 < 𝑣1𝐴 + 𝛼, and 𝑣2𝐴 + 𝑣3𝐵 > 𝑣1𝐴 + 𝑣1𝐵 + 𝛼(global bidder
wins license 𝐴 and 𝐵 with a loss), then 𝑅𝑆𝐴𝐴 = 𝑣2𝐴 + 𝑣3𝐵 .
For the above parameter space, there are three possible subcases in VCG model: The ﬁrst
one is when 𝑣3𝐵 > 𝑣1𝐵 , 𝑣2𝐴 > 𝑣1𝐴 , 𝑣2𝐴 < 𝑣1𝐴 +𝛼, 𝑣3𝐵 > 𝑣1𝐵 +𝛼, and 𝑣2𝐴 +𝑣3𝐵 > 𝑣1𝐴 +𝑣1𝐵 +𝛼,then 𝑅𝑉 𝐶𝐺 = 2(𝑣1𝐴 + 𝑣1𝐵 + 𝛼) − 𝑣2𝐴 − 𝑣3𝐵 which is the Case I: A)ii) from Proposition 9.
Since 𝑣2𝐴 + 𝑣3𝐵 > 𝑣1𝐴 + 𝑣1𝐵 + 𝛼 , we have 𝑅𝑉 𝐶𝐺 < 𝑅𝑆𝐴𝐴 .
The second one is when 𝑣3𝐵 > 𝑣1𝐵 , 𝑣2𝐴 > 𝑣1𝐴 , 𝑣2𝐴 < 𝑣1𝐴 + 𝛼, 𝑣3𝐵 < 𝑣1𝐵 + 𝛼, and𝑣2𝐴 + 𝑣3𝐵 > 𝑣1𝐴 + 𝑣1𝐵 + 𝛼, then 𝑅𝑉 𝐶𝐺 = 𝑣1𝐴 + 𝑣1𝐵 + 𝛼 + 𝑣1𝐴 − 𝑣2𝐴 which is the Case I: A)iii)
from Proposition 9. Since 𝑣2𝐴 + 𝑣3𝐵 > 𝑣1𝐴 + 𝑣1𝐵 + 𝛼 and 𝑣2𝐴 > 𝑣1𝐴 , we have 𝑅𝑉 𝐶𝐺 < 𝑅𝑆𝐴𝐴 .
The third one is when 𝑣3𝐵 > 𝑣1𝐵 , 𝑣2𝐴 < 𝑣1𝐴 , and 𝑣2𝐴 + 𝑣3𝐵 > 𝑣1𝐴 + 𝑣1𝐵 + 𝛼, then 𝑅𝑉 𝐶𝐺 =𝑣1𝐵 + 𝛼 + 𝑣2𝐴 which is the Case I: C) from Proposition 9. Since 𝑣2𝐴 + 𝑣3𝐵 > 𝑣1𝐴 + 𝑣1𝐵 + 𝛼
and 𝑣2𝐴 < 𝑣1𝐴 , we have, 𝑣3𝐵 > 𝑣1𝐵 + 𝛼. So we have 𝑅𝑉 𝐶𝐺 < 𝑅𝑆𝐴𝐴 .
Case 2: This is the symmetric of case 1 above. When 𝑣2𝐴 > 𝑣1𝐴 , 𝑣3𝐵 < 𝑣1𝐵 + 𝛼,
𝑣2𝐴 + 𝑣3𝐵 > 𝑣1𝐴 + 𝑣1𝐵 + 𝛼(global bidder wins license 𝐴 and 𝐵 with a loss), then 𝑅𝑆𝐴𝐴 =𝑣2𝐴 + 𝑣3𝐵 .
For the above parameter space, there are three possible cases in VCG model: The ﬁrst
one is when 𝑣2𝐴 > 𝑣1𝐴 , 𝑣3𝐵 > 𝑣1𝐵 , 𝑣3𝐵 < 𝑣1𝐵 +𝛼, 𝑣2𝐴 < 𝑣1𝐴 +𝛼, and 𝑣2𝐴 +𝑣3𝐵 > 𝑣1𝐴 +𝑣1𝐵 +𝛼,32then 𝑅𝑉 𝐶𝐺 = 2(𝑣1𝐴 + 𝑣1𝐵 + 𝛼) − 𝑣2𝐴 − 𝑣3𝐵 which is the Case I: A)ii) from Proposition 9.
Since 𝑣2𝐴 + 𝑣3𝐵 > 𝑣1𝐴 + 𝑣1𝐵 + 𝛼, we have 𝑅𝑉 𝐶𝐺 < 𝑅𝑆𝐴𝐴 .
The second one is when 𝑣2𝐴 > 𝑣1𝐴 , 𝑣3𝐵 > 𝑣1𝐵 , 𝑣3𝐵 < 𝑣1𝐵 + 𝛼, 𝑣2𝐴 > 𝑣1𝐴 + 𝛼, and𝑣2𝐴 + 𝑣3𝐵 > 𝑣1𝐴 + 𝑣1𝐵 + 𝛼, then 𝑅𝑉 𝐶𝐺 = 𝑣1𝐴 + 𝑣1𝐵 + 𝛼 + 𝑣1𝐵 − 𝑣3𝐵 which is the Case I: A)iv)
from Proposition 9. Since 𝑣2𝐴 + 𝑣3𝐵 > 𝑣1𝐴 + 𝑣1𝐵 + 𝛼 and 𝑣3𝐵 > 𝑣1𝐵 , we have 𝑅𝑉 𝐶𝐺 < 𝑅𝑆𝐴𝐴 .
The third one is when 𝑣3𝐵 < 𝑣1𝐵 , 𝑣2𝐴 > 𝑣1𝐴 , 𝑣2𝐴 + 𝑣3𝐵 > 𝑣1𝐴 + 𝑣1𝐵 + 𝛼, then 𝑅𝑉 𝐶𝐺 =𝑣1𝐴 + 𝛼 + 𝑣3𝐵 which is the Case I: B) from Proposition 9. Since 𝑣2𝐴 + 𝑣3𝐵 > 𝑣1𝐴 + 𝑣1𝐵 + 𝛼
and 𝑣3𝐵 < 𝑣1𝐵 , we have, 𝑣2𝐴 > 𝑣1𝐴 + 𝛼. So we have 𝑅𝑉 𝐶𝐺 < 𝑅𝑆𝐴𝐴 .
Proof of Proposition 8 When the allocation is ineﬃcient (except for two types of
exposure problem) in SAA, then revenue in VCG model is greater than that in SAA, that
is, 𝑅𝑉 𝐶𝐺 > 𝑅𝑆𝐴𝐴 .
There are two cases.
Case 1: Global bidder loses both licenses in SAA model, but wins both in VCG model.
When 𝑣3𝐵 > 𝑝∗1 > 𝑣1𝐵 (global bidder loses licese 𝐵), 𝑣2𝐴 > 𝑣1𝐴 (global bidder loses license 𝐴),
and 𝑣2𝐴 + 𝑣3𝐵 < 𝑣1𝐴 + 𝑣1𝐵 + 𝛼(this condition makes an ineﬃcient allocation), if 𝑣1𝐴 > 𝑝∗1 ,then 𝑅𝑆𝐴𝐴 = 𝑝∗1 + 𝑣1𝐴 or if 𝑣1𝐴 < 𝑝∗1 , then 𝑅𝑆𝐴𝐴 = 𝑝∗1 + 𝑝∗1 .
There is one corresponding possible subcase in VCG model which is the Case I: A)iv)
from Proposition 9. when 𝑣3𝐵 > 𝑣1𝐵 , 𝑣2𝐴 > 𝑣1𝐴 , and 𝑣2𝐴 + 𝑣3𝐵 < 𝑣1𝐴 + 𝑣1𝐵 + 𝛼, then
𝑅𝑉 𝐶𝐺 = 𝑣2𝐴 + 𝑣3𝐵 . When 𝑣1𝐴 < 𝑝∗1 , since 𝑣3𝐵 > 𝑝∗1 ,and 𝑣2𝐴 > 𝑣1𝐴 , we have 𝑅𝑉 𝐶𝐺 < 𝑅𝑆𝐴𝐴 ;or when 𝑣1𝐴 > 𝑝∗1 , since 𝑣3𝐵 > 𝑝∗1 ,and 𝑣2𝐴 > 𝑣1𝐴 , we also have 𝑅𝑉 𝐶𝐺 > 𝑅𝑆𝐴𝐴 .
Case 2: Global bidder wins license A and local bidder wins license B in SAA model.25
When 𝑣3𝐵 > 𝑝∗1 (global bidder loses licese 𝐵), 𝑣2𝐴 < 𝑣1𝐴 (global bidder wins licese 𝐴), 𝑣2𝐴 > 𝑝∗1 ,and 𝑣1𝐴 + 𝑣3𝐵 < 𝑣1𝐴 + 𝑣1𝐵 + 𝛼, then 𝑅𝑆𝐴𝐴 = 𝑝∗1 + 𝛼 + 𝑣2𝐴 .
In such case, there is one possible subcase in VCG model which is the Case II: B)iv) from
Proposition 9 when 𝑣3𝐵 < 𝑣1𝐵 +𝛼, and 𝑣2𝐴 < 𝑣1𝐴 , then we can derive 𝑣2𝐴 +𝑣3𝐵 < 𝑣1𝐴 +𝑣1𝐵 +𝛼,then 𝑅𝑉 𝐶𝐺 = 𝑣3𝐵 + 𝑣2𝐴 . Since 𝑣3𝐵 > 𝑝∗1 , then we have 𝑅𝑉 𝐶𝐺 > 𝑅𝑆𝐴𝐴 .

25

Since we assume 𝑣1𝐴 > 𝑣1𝐵 , we do not have the case global bidder wins license B and local bidder license
A. If we remove the assumption, the proof can be written symmetrically.

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