HIAS-E-45

Which good to sell first in a sequential auction?
Hikmet Gunay
Hitotsubashi Institute for Advanced Study, Hitotsubashi University, 2-1, Naka, Kunitachi, Tokyo 186-8601, Japan
Department of Economics, University of Manitoba, Winnipeg, MB, R3T 5V5, Canada

Xin Meng
CIBO, Dongbei University of Finance and Economics, Dalian, 116025, China

May 2017

Hitotsubashi Institute for Advanced Study, Hitotsubashi University
2-1, Naka, Kunitachi, Tokyo 186-8601, Japan
tel:+81 42 580 8604
http://hias.ad.hit-u.ac.jp/
HIAS discussion papers can be downloaded without charge from:http://hdl.handle.net/10086/27202
https://ideas.repec.org/s/hit/hiasdp.html

All rights reserved.

Which good to sell first in a sequential auction?
Hikmet Gunay ⇤Hitotsubashi Institute for Advanced Study, Hitotsubashi University, Tokyo 186-8601, Japan
Department of Economics, University of Manitoba, Winnipeg, MB, R3T 5V5, Canada

Xin Meng
CIBO, Dongbei University of Finance and Economics, Dalian, 116025, China

May 31, 2017

Abstract
In a sequential auction, we analyze whether selling a stochastically more valuable
good in the first or second auction generates more revenue and welfare. One of thebuyers is a global bidder who enjoys synergy if she wins both goods. The others are
local bidders interested in one specific good. After deriving the equilibrium, we showthat there are cases in which selling the less valuable good in the first auction generates
higher revenue and/or welfare. We also show the impact of inefficient allocations onrevenue.

JEL Codes: D44, D82
Keywords: Sequential Auctions, Inefficiencies, Multi-dimensional values, Simulations⇤

Corresponding author, Tel.:(204) 474-8915, E-mail address: Hikmet.Gunay@umanitoba.ca.

11Introduction

There are auctions in which one good is more valuable than the other one, such as spectrum
license auctions (e.g., Meng and Gunay, 2017). In some of these auctions, there are local
bidders bidding only for one specific good, and global bidders bidding for all goods (e.g.
Krishna and Rosenthal, 1996). Global bidders enjoy synergies if they win all goods.
In this paper, we analyze whether selling a stochastically more valuable good in thefirst or second auction generates higher revenue and/or welfare in a second-price sequential
auction. By using simulation methods, we find cases where selling the more valuable goodfirst might result in less revenue and/or welfare. We also show how the inefficient allocationsa↵ect the revenue in these auctions.
In the literature, it has been assumed that the goods are either stochastically equivalent(Krishna and Rosenthal, 1996; Branco, 1997), or the second good becomes more valuable to
the winner of the first auction (Jeitschko and Wolfstetter, 2002; De Silva 2005; Leufkens et.
al., 2010), and thus, the order of selling goods has no impact on revenue and welfare. One
exception is Benoit and Krishna (2001). They find that selling the more valuable good first
generates more revenue in a model with budget constrained (global) bidders in a complete
information game.

2

The Model

Two goods, A and B, are sold in a second-price sequential auction. The goods has zero
value to the seller. There is one risk-neutral global bidder, G, who bids for both goods,
and enjoy a synergy of ✓ > 0 if wins both goods.1 There are also Ni > 0 risk neutral local
bidders bidding for good i = A, B. Ni + 1 independent draws from the distribution function
Fi determines the private valuation, vki , for each bidder, k = G, 1, 2.., Ni , and i = A, B. The
distribution function Fi , i = A, B, has a twice di↵erentiable density function fi > 0 on the1
Assuming one global bidder when a bidder has multi-dimensional valuations is not uncommon in theliterature (see Meng and Gunay, 2017 and the references therein) since the equilibrium strategy for multiple
global bidders have not been calculated unless they have single types.

2interval (0, 1] with fi (0)0.

Definition: We say that A is stochastically more valuable than B if (i) FA (x)  FB (x)and (ii) [FA (x)]NA  [FB (x)]NB for all x.

The definition guarantees that, [FA (x)]NA +1  [FB (x)]NB +1 , the expected value of thefirst highest order statistics for A is greater than for B.
We use symmetric subgame perfect Bayesian equilibrium like Leufkens et. al., (2010).
The equilibrium strategy for local bidders is bidding their valuations truthfully in bothauctions (in weakly undominated strategies). The global bidder’s equilibrium strategy inthe second auction is bidding her marginal valuation truthfully; hence, she bids vGj + ✓ if
won good i in the first auction, and bid vGj otherwise, where i, j = A, B and i 6= j.
To derive the global bidder’s equilibrium strategy in the first auction for good i, we haveto maximize her payo↵ given the sequential rationality. Let pi = max{vki }, k = 1, 2.., Ni
denote the maximum valuation of local bidders for good i = A, B. Then, the distributionfunction for pi is Gi (.) = [Fi (.)]Ni for i = A, B. The expected payo↵ for the global bidder
when she bids p is

M axp

Z p

(vGi

pi )dGi (pi ) + P r(p > pi )0

Z min{vGj +✓,1}(vGj + ✓Z vGj+P r(p < pi )(vGjpj )dGj (pj )0pj )dGj (pj )(1)

0

The first integral is the expected profit from winning i in the first auction, second is the
expected profit from winning j after winning i, and the third is the expected profit fromwinning j after losing i.
Equation 2 is the first order condition, and it gives the equilibrium bidding price which
we denote as pij when good i is auctioned first, and j second. The global bidder compares
two payo↵s when he decides on pij ; the first one is the expected payo↵ from winning the firstauction, and the second one is the expected payo↵ from losing the first auction. If the first
expected payo↵ is higher, she should increase pij until both payo↵s are equal.

3

(vGi
|

pij ) +Z min{vj +✓,1}0(vGj + ✓{zpj )dGj (pj )}=Expected profit from winning the first auction

Z vj
|

(vGj

0

pj )dGj (pj )
{z
}

(2)

and losing the first auction

By using integration by parts and equation 2, we derive the global bidder’s equilibrium
bid.2
Proposition 1 The global bidder’s equilibrium bid, pij in the first auction for good i isa) If vGj + ✓ < 1, then pij (vGi , vGj , Nj ) = vGi +Z vGj +✓Gj (p, Nj )dp

vGjb) If vGj + ✓1, then pij (vGi , vGj , Nj ) = vGi + (vGj + ✓

1) +

Z 1

Gj (p, Nj )dp

vGj

We have some observations based on proposition 1. First, we have pij 6= pji in ourmodel. This is di↵erent than the aforementioned papers in the literature (except the complete information model of Benoit and Krishna, 2001) since we use multi-dimensional types
and di↵erent number of bidders on each auction. Second, the global bidder’s equilibrium
incentives are conditioned on winning or losing the first auction, and its continuation payo↵;hence, pij is a function of the number of second-auction local bidders but not the number offirst-auction local bidders. Third, the global bidder bids over her stand-alone valuation which
exposes her to the ex-post loss as well known in the literature (e.g. Krishna and Rosenthal,
1996). Fourth, the global bidder does not bid above the possible highest marginal valuation
for license A. That is, pij < vGi + ✓ holds since Gi (x) < 1. Fifth, when vGA + vGB + ✓2,

the global bidder bids above 1 in both auctions, and wins both goods.3
Next, we show that the global bidder is expected to bid higher in the first auction if themore valuable good is auctioned first.
Proposition 2 Assume that good A is stochastically more valuable good, and vGA > vGB .
Then pAB

pBA .

2

See the discussion paper of Gunay and Meng (2017) for the proof.
Since this implies vGj + ✓2 vGi
1, we have to use part b of proposition 1. But then pij
vGi + (vGj + ✓ 1) 1, which guarantees that global bidder wins the first auction.
3

4

While the global bidder’s bid is higher in the AB auction where more valuable A is soldfirst compared to the BA auction where B is sold first, we show that this does not guarantee
a higher expected revenue in the AB auction in the next section.

3

Simulations

Since we have multi-unit valuations, calculating ex-ante revenue, welfare, and probabilityof inefficient allocations are extremely difficult; hence, we use simulations like Krishna andRosenthal (1996) to compare the AB and BA auctions.4 We draw 50000 valuations for eachbidder in MATLAB, and calculate the equilibrium bidding prices, pAB and pBA with thehelp of proposition 1. Table 1 shows all possible outcomes, and the corresponding revenue
and welfare in AB and BA auctions which helps us in calculating them ex-post. The tableshows that there are four di↵erent types of inefficiency. Two of them are the global bidder
winning one or both goods with an ex-post loss (rows 2 and 4 in Table 1); one of them isthe global bidder winning (one good) inefficiently with a profit (row 6), and the last one isthe local bidders winning both goods inefficiently (row 8). There cannot be an inefficient
outcome in which the global bidder wins the first good and loses the second one with profit.
In the next proposition, we prove this.
Proposition 3 There is no inefficient outcome in an ij auction such that global bidder wins
i with profit but loses j.

1.
2.
3.
4.
5.
6.
7.
8.

License i won by
Global Bidder
Global Bidder
Global Bidder
Global Bidder
Local Bidder i
Local Bidder i
Local Bidder i
Local Bidder i

License j won by
Global Bidder
Global Bidder
Local Bidder j
Local Bidder j
Global Bidder
Global Bidder
Local Bidder j
Local Bidder j

Global bidder makes
ProfitLoss
ProfitLoss
ProfitProfitZero Profit
Zero Profit

Allocation is
Efficient
Inefficient
Efficient
Inefficient
Efficient
Inefficient
Efficient
Inefficient

Revenue ispi + pj
pi + pj
pi + v j + ✓pi + v j + ✓p⇤ij + pj
p⇤ij + pj
p⇤ij + vj
p⇤ij + vj

Welfare isvGi + vGj + ✓vGi + vGj + ✓vGi + pjvGi + pj
pi + vGj
pi + vGj
pi + pj
pi + pj

Table 1: All possible outcomes in an ij auction when Ni = Nj = 1
4
Krishna and Rosenthal (1996) use simulations except for uniform distribution but they have single type
for the global bidder.

5Synergy
Synergy
Synergy
with
with
with
✓ = 0.2✓ = 0.5✓ = 0.8
NA = NB = 1 NA = NB = 1 NA = NB = 1
FA = Beta Distribution with ↵ = 3 andpExp.Loss
AB
Inef f.(GW )
pAB
f.
pInef
AB
pExp.LossBA
Inef f.(GW )
pBA
f.
pInef
BA

4.5060 %2.8760 %8.0320 %2.0560 %1.5600 %4.0020 %0.1447 %-0.1325 %pExp.Loss
AB

2.9080 %1.8920 %5.5460 %4.5060 %2.8960 %8.1080 %0.2639 %0.1081 %

4R%
4SW %

6.3200 %1.7920 %9.3060 %2.3340 %1.2320 %4.3080 %0.7650 %-0.3432 %3.6480 %0.3680 %4.5400 %1.5620 %0.5860 %2.6760 %0.6961 %-0.1847 %Synergy
Synergy
Synergy
with
with
with
✓ = 0.2✓ = 0.5✓ = 0.8
NA = NB = 2 NA = NB = 2
NA = NB = 2= 1; FB =Uniform Distribution
4.6300 %2.8260 %8.4380 %1.8080 %1.1220 %3.5280 %-0.1020 %-0.1369 %7.7980 %1.8820 %12.3580 %2.4240 %1.0380 %4.8760 %-0.2140 %-0.4394 %FA =Uniform distribution; FB = Beta Distribution with ↵ = 1 andInef f.(GW )

pAB
f.
pInef
AB
Exp.LosspBA
Inef f.(GW )
pBA
f.
pInef
BA
4R%
4SW %

3.6200 %1.7380 %6.4820 %6.1260 %1.7700 %9.2380 %0.2127 %0.2520 %2.3720 %0.7560 %3.7680 %3.5200 %0.3400 %4.3920 %-0.1359 %0.1179 %2.8700 %1.7320 %5.6820 %3.5220 %2.6720 %7.0700 %0.2451 %0.0608 %5.2860 %2.1180 %6.9220 %6.6720 %2.1760 %11.6740 %1.0171 %0.1343 %5.6860 %0.3780 %7.8860 %1.9720 %0.6120 %3.7120 %0.0981 %-0.3155 %=3
4.5160 %1.1640 %7.7240 %5.7500 %0.4740 %8.4120 %0.6351 %0.1197 %pExp.Loss=100*(Sum of outcomes in row 2 and 4 in Table 1)/50000. Probability of the global bidder having ex-post loss.
ij
Inef.(GW )pij=100*(Outcomes in row 6 in Table 1)/50000. Probability of the global bidder winning inefficiently with profit.
f.
pInef=100*(Sumof outcomes in row 2,4,6, and 8 in Table 1)/50000. Probability of all inefficient allocations.
ij
4R% =100 ⇤ (RAB RBA )/RBA . Percentage change in revenue between AB and BA auctions.
4SW % =100 ⇤ (SWAB SWBA )/SWBA . Percentage change in social welfare (SW) between AB and BA auctions.

Table 2: Simulation Results
Simulations are done for three di↵erent synergy levels of 0.2, 0.5 and 0.8 for NA = NB = 1
and then NA = NB = 2.
Simulation results are summarized in Table 2. The main observation is that there arecases where selling more valuable good A first might generate less revenue and/or welfare
compared to selling it second.
Another observation is that whenever 4R% < 0 we have pExp.Loss> pExp.Loss
and/orBA
AB

Inef.(GW )pBA

Inef.(GW )< pAB. In the next two propositions, we link these types of inefficient

allocations to the revenue of AB and BA auctions. Let Rij denote the (ex-post) revenue inthe ij auction with i, j = A, B and i 6= j.
Proposition 4 Consider a set of valuations vGA , vGB , pi , and pj . If the global bidder wins
6one or both good(s) with an ex-post loss in an ij auction, then RijRji .

Proposition 4 states that, as pExp.Loss
increases, the revenue di↵erence Rijij

Rji (weakly)

increases.
Proposition 5 Consider a set of valuations vGA , vGB , pi , and pj . If the global bidder wins
one good with profit and the allocation is inefficient in an ij auction, then Rij  Rji .
Inef.(GW )Proposition 5 states that as the di↵erence pij
Rij

Rji (weakly) increases.

4

Conclusion

decreases, the revenue di↵erence

The main message of the simulations is that selling more valuable good first does not necessarily generate higher revenue and/or welfare compared to selling it second. While themodels are not directly comparable, this is di↵erent than Benoit and Krishna (2001) whoshow that selling more valuable good generate higher revenue.
It is clear that choosing which good to sell first has an impact on revenue, welfare, andprobability of inefficient allocations. The revenue di↵erence was more than 1 per cent insome of our examples.

5

Appendix-Proofs

Proof of proposition 1
We take the derivative of equation 1 by using the fact that P r(p > pi ) = Gi (p). After
equating the derivative to zero, and cancelling g(p) from the equation, we get equation 2p ⌘ pij = vGi +Z min{vj +✓,1}(vj + ✓pj )dGj (pj )

0

Z vj

(vGj

pj )dGj (pj )0

We can re-write this by using integration by parts, by letting dv = dGj (pj ) in both
integrals. When vGj + ✓ < 1,7

pij = vGi +Z vGj +✓Z vGj

Gj (pj )dpj0

Gj (pj )dpj

=

vGi +

0

Z vGj +✓Gj (p)dp

vGj

When vGj + ✓ > 1, the proof is similar, and omitted.

Finally, we show that the SOC is satisfied when vGj + ✓ < 1.
R min{vj +✓,1}R vj
F OC = [vGi p + 0(vj + ✓ pj )dGj (pj )(vGj pj )dGj (pj )]gi (p)0
R vj +✓
R vj= [vGi p + 0
Gj d(pj )Gj d(pj )]gi (p)0
R vj +✓R vj
0
SOC = [vGi p + 0
Gj d(pj )Gj d(pj )]gi (p) gi (p) < 0, since F OC = vGi p +0
R vj +✓Rv
Gj d(pj ) 0 j Gj d(pj ) = 0. Since there is a unique equilibrium pij , and SOC is negative0
at pij , SOC is satisfied.

When vGj + ✓ > 1, the proof is similar, and omitted.

Proof of proposition 2: First, assume that vGA + ✓ < 1. Then, by using proposition
1,
pAB
vGA

vGB +

Z vGB +✓

pBA = vGAvGB +Z vGA +✓GA (p)dpvGA

Z vGB +✓vGB

GA (p)dp >vGAGA (p)dp = vGA

vGA

vGA

GB (p)dpvGB

Z vGA +✓Z vGA +✓
vGB +✓vGB

Z vGA +✓GA (p)dp

vGB +✓

1dp = 0 ) pAB > pBA

The first inequality is written by using GA  GB since A is stochastically more valuable
and vGA < vGB by assumption. The last inequality is written since GA (.)  1
Second, assume that vGB + ✓ < 1 < vGA + ✓. Then, by using proposition 1,

pAB
Z vGB +✓
vGB

pBA = vGA +GB (p)dp +Z vGB +✓

vGB
Z 1

vGB +✓GB (p)dp

1dp

Z 1

vGA

[vGB + vGA + ✓
Z 1

1+

Z 1

vGA

GA (p)dp] =Z 1

GA (p)dp >GA (p)dp
GA (p)dp
vGB
vGA
Z vGA
=
GA (p)dp > 0 ) pAB > pBA
vGB

The first inequality is written by using GA  GB and GA  1. The third case, vGB +✓ > 1
is similar.
8

Proof of proposition 3:
If the global bidder wins i with profit but loses j in an ij auction,
pij > vGi > pi and pj > vGj + ✓(3)should hold. Revenue in the ij auction is Rij = pi + vGj + ✓. There is no inefficient outcome
since the winner of license i with vGi > pi and the winner of license j with valuation pj suchthat pj > vGj + ✓, and vGi + pj > vGi + vGj + ✓.

Proof of Proposition 4:

First we prove this when the global bidder wins one goodand makes an ex-post loss. For this to happen, in the ij auction,

pij > pi > vGi and vGj + ✓ < pj

(4)should hold. Revenue in the ij auction is Rij = pi + vGj + ✓.
If instead, ji auction is conducted with these valuations, global bidder will lose j sincepji < vGj + ✓ < pj since bidding price can never be above the marginal valuation vGj + ✓and vGj + ✓ < pj by equation 4. The global bidder will also lose license i since pi > vGi byequation 4. Hence,
Rji = pji + vGi < Rij = pi + vGj + ✓Second, we prove this when the global bidder wins both goods and makes an ex-post loss.

pij > pi > vGi and vGj + ✓ > pj

(5)should hold. Revenue in the ij auction is Rij = pi +pj . i) Assume that pji < pj . By equation5, pi > vGi , hence, local bidders win both goods. Rji = pji + vGi < pi + pj = Rij .
ii) Now assume that pji > pj . The global bidder wins j auction and hence, bids vGi + ✓in the i auction and wins it since vGi + ✓ > pij > pi . The last inequality is from equation 5.
Hence, Rij = Rji .

9

Proof of Proposition 5: In ij auction, local bidder wins i, global bidder wins j, andthis is an inefficient allocation; hence,pi > pij and vGj > pj and pi + vGj < vGi + ✓ + vGj(6)We have Rij = pij + pj . If this is sold in ji auction, global bidder wins j auction sincepji > vGj > pj and i auction since vGi + ✓ > pi by equation 6. Hence, Rji = pj + pi >pj + pij = Rij by equation 6.

6

Appendix-B
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.

License i won by
Global Bidder
Global Bidder
Global Bidder
Global Bidder
Global Bidder
Global Bidder
Local Bidder i
Local Bidder i
Local Bidder i
Local Bidder i
Local Bidder i
Local Bidder i
Local Bidder i
Local Bidder i
Local Bidder i
Local Bidder i
Local Bidder i
Local Bidder i

License j won by
Global Bidder
Global Bidder
Local Bidder j
Local Bidder j
Local Bidder j
Local Bidder j
Global Bidder
Global Bidder
Global Bidder
Global Bidder
Local Bidder j
Local Bidder j
Local Bidder j
Local Bidder j
Local Bidder j
Local Bidder j
Local Bidder j
Local Bidder j

Global bidder makes
ProfitLoss
ProfitLoss
ProfitLoss
ProfitProfitProfitProfitZero Profit
Zero Profit
Zero Profit
Zero Profit
Zero Profit
Zero Profit
Zero Profit
Zero Profit

Allocation is
Efficient
Inefficient
Efficient
Inefficient
Efficient
Inefficient
Efficient
Inefficient
Efficient
Inefficient
Efficient
Inefficient
Efficient
Inefficient
Efficient
Inefficient
Efficient
Inefficient

Revenue is
v1i + v1j
v1i + v1j
v1i + vGj + ✓v1i + vGj + ✓v1i + v2j
v1i + v2j
pij + v1j
pij + v1j
v2i + v1j
v2i + v1j
pij + vGj
pij + vGj
pij + v2j
pij + v2j
v2i + vGj
v2i + vGj
v2i + v2j
v2i + v2j

Table 3: All possible outcomes when NA = NB = 2

Acknowledgements
Authors contributed equally to this paper. Authors acknowledges the support of NSFC
(National Natural Science Foundation of China under Grant no. 71302180 ), SSHRC and the
support of DUFE-CIBO (Subject Research Project Fund under Grant no. LNTX2015001).

10

References
Benoit J.P., Krishna V., 2001. Multiple-object auctions with budget constrained bidders.
Review of Economic Studies 68(1), 155-179.
Branco F., 1997. Sequential auctions with synergies: An example. Economics Letters
54(2), 159-163.
Chakraborty, I., 2004. Multi-unit auctions with synergy. Economics Bulletin 4, 1-14.
De Silva, D.G., 2005. Synergies in recurring procurement auctions: an empirical investigation. Economic Inquiry 43(1),55-66.
Jeitschko, T.D., Wolfstetter, E., 2002. Scale Economies and the Dynamics of Recurring
Auctions. Economic Inquiry 40(3), 403-414.
Krishna, V., Rosenthal, R.W., 1996. Simultaneous auctions with synergies. Games andEconomic Behavior 17, 1-31.
Leufkens, K., R Peeters,R., Vermeulen,D., 2010. Sequential auctions with synergies: The
paradox of positive synergies. Economics Letters 109(3),139-141.
Meng, X., Gunay, H., 2017. Exposure problem in multi-unit auctions. International
Journal of Industrial Organization 52, 165-187.

11