Do Agency Contracts Facilitate Upstream Collusion?∗
Yoshifumi Hino†1,3 , Susumu Sato‡2 , and Yusuke Zennyo§1
1

Graduate School of Business Administration, Kobe University
2

3

Graduate School of Economics, The University of Tokyo

Business Administration Program, Vietnam–Japan University

This version: July 1, 2019

Abstract
We examine whether agency contracts facilitate collusion among upstream manufacturers, ascompared to traditional wholesale contracts. We consider an infinitely repeated game witha monopoly platform and multiple manufacturers. We show that the critical discount factor,
above which the upstream collusion can be sustainable by Nash-reversion trigger strategies, ishigher under the agency contract than under the wholesale contract. This result indicates thatthe agency contract does not facilitate upstream collusion in the monopoly platform market. By
contrast, in an extended model with competing platforms, we show that the agency contract facilitates upstream collusion. These results provide much implications for several recent antitrust
cases on collusion and coordination in online platforms, such as an e-book cartel orchestratedby Apple and a high commission rate imposed by Apple and Google in the mobile application
market.

Keywords: agency contract, cartel stability, upstream collusion, wholesale contract
JEL Classification: L13, L42, L81, D21, D43, D86, K21∗

Special appreciation is extended to Jay Pil Choi, Daisuke Hirata, Akifumi Ishihara, Toshihiro Matsumura, Noriaki
Matsushima, Hodaka Morita, and Naoki Watanabe. The authors also thank seminar participants at Hitotsubashi
University and Kyoto University, and participants at the 2018 annual conference of the JAAE in Kyoto and the 2018
Annual Meeting of the APIOC in Melbourne for their helpful comments.
†
E-mail: y.hino@vju.ac.vn‡
E-mail: susumusato@live.jp§
Corresponding author. E-mail: xyzennyo@b.kobe-u.ac.jp1

1

Introduction

In March 2016, the U.S. Supreme Court denied a hearing an appeal lodged by Apple Inc., leaving
the conclusion that Apple violated Section 1 of the Sherman Act by orchestrating a horizontal
conspiracy among five of the six largest U.S. book publishers (Hachette, HarperCollins, Macmillan,
Penguin, and Simon & Schuster) to raise e-book prices.1 The U.S. Department of Justice (DOJ)
initially filed its complaint that agency agreements played an instrumental role.2 Along with theagency contract, Apple organized a price cartel among the publishers by interacting and sharing
information with them. That was a per se violation of the Sherman Act. However, it is noteworthy
that the district court conceded that the agency agreement is not inherently illegal, and is anentirely lawful contract.
In the case of the e-book cartel, because Apple assembled “a horizontal price-fixing conspiracy”
consisting of a group of competitors, it was a per se violation of the Sherman Act. Moreover, thedistrict court concluded that even if this case were analyzed under the rule of reason, it would
still constitute an unreasonable restraint of trade in violation of §1. However, it has not been
evaluated whether the agency contract itself facilitates collusion among upstream publishers. This
article provides an economic explanation for when and how the agency contract facilitates collusion
among publishers.
To this end, we first develop a stylized model of an infinitely repeated game comprising amonopoly platform and multiple manufacturers that sell diﬀerentiated products via the platform.
The manufacturers can be interpreted as publishers in the example of the e-book market given
above. Other examples include e-commerce platform markets (many manufacturers’ products are
sold on online platforms such as Amazon, eBay, Rakuten in Japan, and Taobao in China) andmobile application markets (application developers distribute their “apps” through Apple App
Store and Google Play Store).
We investigate two contract forms: a wholesale contract and an agency contract, which are
the two simplest and most popular contracts adopted by platforms.3 In the stage-game, for each1

See U.S. v. Apple, Inc. for more details.
This agency agreement was a contract between a platform and publishers, by which the publishers can set final
retail prices. Sales revenue obtained through the platform is split between publishers and the platform subject to a
fixed revenue-sharing rule. Recently, such agency contracts have been used widely on other online platforms as wellas e-book platforms.
3
In mobile application markets, agency contracts have been most widely used. By contrast, both the wholesale22contract, the monopoly platform and manufacturers move sequentially explained below. Under
the wholesale contract, each manufacturer chooses its wholesale price before the platform sets theretail prices. Under the agency contract, the platform chooses a revenue-sharing rate (a so-calledroyalty rate or commission rate) before each manufacturer sets the retail price directly. For eachcontract, we obtain the value of the critical discount factor, a minimum discount factor at which
joint profit maximizing collusion among manufacturers is sustainable by Nash-reversion trigger
strategies. Then, we compare the two critical discount factors to assess whether the agency contract
facilitates upstream collusion, or not. We can say that the agency contract facilitates upstream
collusion if it reduces the critical discount factor.
Our analysis of the monopoly platform case shows that the critical discount factor is higher under the agency contract than under the wholesale contract, implying the agency contract does notfacilitate upstream collusion. The presence of upstream collusion exacerbates the double marginalization problem, which in turn decreases the profit of the platform. Under the agency contract,
due to its leadership in the stage-game, the monopoly platform can block the formation of collusion
among upstream manufacturers that move later.
In addition, we also examine the extended model with two competing platforms in an eﬀort to
explore how the existence of platform competition alters the eﬀect of agency contracts on upstream
collusion. Fixing a revenue-sharing rate both the platforms charge, due to revenue-sharing agreements, collusive manufacturers are willing to maximize the total profit across the whole channel,
which enables the platforms to obtain the largest revenue-share for the fixed revenue-sharing rate.
This, which is in a sharp contrast with the monopoly model, makes deviation less attractive forcompeting platforms. Therefore, the agency contract facilitates upstream collusion in the markets
with competing platforms.
The result obtained implies that introducing agency contracts not only facilitates upstream collusion, but also provokes coordination between competing platforms. In fact, in mobile application
markets with competition between Apple App Store and Google Play Store, although developers set
the prices of their apps, both Apple and Google impose a 30% commission rate on each purchase.4
and agency contracts are widely used, in e-commerce markets. For example, according to Amazon’s full-year 2017financial results, net sales from online stores (i.e., the wholesale model) were $108.4 billion, whereas those fromthird-party seller services (i.e., the agency model) were $31.9 billion.
4
In 2016, Apple introduced a new revenue-sharing rule, such that the 30% commission rate will drop to 15% foreach subscriber active for at least one year. Google also followed this revenue-sharing rule from the beginning of 20183

That high commission rate led to a key antitrust case that reached the U.S. Supreme Court inNovember 2018.
The results derived and presented herein represent several important contributions to competition policy. On the one hand, the agency contract itself is not anticompetitive: when the market isserved by a monopoly platform, the agency contract does not facilitate upstream collusion. On theother hand, with platform competition, the introduction of agency contracts has an anticompetitive
concern to facilitate upstream collusion, and to mitigate platform competition. Therefore, strong
revenue-sharing rules should be monitored carefully along with the formation of upstream collusion,
which is the second and the most important message of the article.

1.1

Case of e-book cartel

According to U.S. v. Apple, Inc., we summarize the case of the e-book cartel in relation to the
purpose of this article.5
Amazon, a famous e-book platform, started selling its Kindle e-book reader in 2007. First,
Amazon signed wholesale contracts with e-book publishers, whereby each publisher sets a wholesale
price of its e-book before Amazon charges retail prices for those e-books. In practice, Amazon
charged $9.99 for certain new releases and bestselling e-books. When Apple entered the e-bookmarket with its new iPad device in 2010, it convinced publishers to adopt agency agreements bywhich publishers can control the prices of their e-books, with Apple receiving a 30% commission,
to break Amazon’s monopolistic grip on the market. In addition, Apple included a most-favorednation (MFN) clause in their contract with major publishers, which allowed Apple to sell e-books
at its competitors’ lowest price (i.e., Amazon’s $9.99). The existence of the MFN clause forced
publishers to negotiate the signing of an agency contract with Amazon.6 In consequence, Amazon
also moved to the agency contract.
The industrial movement toward agency contracts engendered an increase in e-book retail pricesof about 18%, on average. Particularly, the price of New York Times bestsellers rose by about 40%.
What caused such an increase in the retail prices of e-books? On September 5, 2013, the districtin a move to compete better with Apple’s oﬀering for iOS developers.
5
Other examples of digital cartels on online platforms include the case of poster cartels and the Lithuanian Eturas
case (Rusu, 2016).
6
The analyses presented in this article do not incorporate the MFN clause into the model, instead emphasizing
how the contract structure aﬀects the sustainability of the upstream price cartel.

4

court issued a final injunctive order prohibiting Apple from enforcing the MFN clause with e-bookpublishers and requiring Apple to modify its agency agreements. However, it is noteworthy thatthe district court conceded that both the agency agreement and MFN clause are not inherently
illegal. Therefore, in this case, the increase in e-book retail prices might have resulted from anothercause. The district court found evidence indicating that the publishers joined in a horizontal pricefixing conspiracy, undertaken by Apple, to increase the retail price of e-books by eliminating pricecompetition. Apple’s central role in the conspiracy was proven as a per se violation of the Sherman
Act.

1.2

Literature

After an increase in e-book prices associated with Apple’s entry into the e-book market in 2010,
studies of whether agency agreements raise the retail prices were launched. A seminal work isthat of Johnson (2017), who shows that the shift from the wholesale model to the agency model
lowers retail prices, which benefits retailers and consumers, but which harms suppliers. Gaudin andWhite (2014) investigate the bilateral monopoly model with an upstream e-book publisher and adownstream retailer. They show that the wholesale contract engenders a higher retail price if andonly if the elasticity of demand strictly decreases as quantity increases. Additionally, they consider
an extended case in which the retailer has monopoly market power on an e-book reader device that
consumers must purchase to use e-books. In this case, under the wholesale contract, the retailer
sets the e-book price at the wholesale price it pays to the publisher and extracts residual surplus
through the device price (i.e., two-part tariﬀ pricing). Consequently, no double marginalization
problem exists. By contrast, under the agency contract, the double marginalization problem arises
because the retailer charges a positive commission fee if the elasticity of demand strictly decreases
as quantity increases. As a result, the agency contract can engender a higher retail price.
Foros et al. (2017) consider platform competition with a bilateral duopoly market consisting of
two upstream suppliers and two downstream platforms. In their model, under both wholesale andagency contracts, sales revenues are shared between the upstream and the downstream firms. Theyshow that the agency contract engenders higher retail prices than the wholesale contract if and only
if the degree of substitution between the platforms is high compared to the degree of substitution
between the suppliers. Lu (2017) also analyzes the bilateral duopoly model to demonstrate the pro5

competitive eﬀect of agency contracts. He shows that the retail prices are lower and the demands
are greater under the agency contract than under the wholesale contract, i.e., consumers benefit
from the agency contract. The diﬀerence from the model of Foros et al. (2017) is the setting of thewholesale contract. He assumes that the upstream firms charge their linear wholesale prices before
the downstream firms choose the retail prices, unlike the revenue-sharing contract in Foros et al.
(2017). The setting of the stage-game in this article is similar to that of Lu (2017). However, theexisting literature on this issue has not considered repeated interactions. This article is the first to
examine whether agency contracts facilitate or obstruct upstream collusion.
The present article is also related to the literature on tacit collusion in vertically related markets(e.g., Bernheim and Whinston, 1985; Cooper, 1986; Neilson and Winter, 1993).7 Recent studies
investigate how the sustainability of upstream collusion is aﬀected by retail-price maintenance(RPM) agreements (Jullien and Rey, 2007), by vertical mergers (Nocke and White, 2007), andby channel structures (Reisinger and Thomes, 2017). Huang (2017) studies how the sustainabilityof downstream collusion is aﬀected by stationary two-part tariﬀ contracts oﬀered by an upstream
supplier before downstream retailers engage in collusive actions infinite-repeatedly. They focus on
the traditional wholesale agreements between upstream and downstream firms. That is, unlike thisarticle, none of them considers upstream collusion under the agency contract.
Nevertheless, no study described in the relevant literature addresses the question of whether
agency contracts facilitate upstream collusion as compared to traditional wholesale contracts. This
study contributes to the literature by showing when and how the agency contract facilitates upstream collusion. The results obtained from this study can present important policy implications.

2

Monopoly Platform

In this section, we examine a model with a monopoly platform and multiple manufacturers. Section
2.1 explains analysis of the stage-game. Section 2.2 presents examination of an infinitely repeatedgame to derive the critical discount factors under the wholesale and agency contracts; then they
are compared to show whether the agency contract facilitates upstream collusion, or not. Detailed
proofs are relegated to the Appendix.
7
The framework used by Bernheim and Whinston (1985) and Cooper (1986) is a finite horizon; it is not what wemean by collusion in our manuscript.

6

2.1

The stage-gameWe describe the setting of the stage-game; then we derive the static equilibrium and the collusive
equilibrium, respectively, under the wholesale and agency contracts. Lastly, those stage-gameoutcomes are compared.

Setting of the stage-game

We consider a vertically related market with a monopoly platform

P and n manufacturers (n ≥ 2). Each manufacturer i (i = 1, · · · , n) produces good i and sells it
through platform P . Let pi and qi be the price and the quantities of good i sold on the platform.
The utility function of a representative consumer is given asu(q1 , · · · , qn ) =

∑
i

aqi −

∑b
2

i

qi2 −

∑

bθqi qj −pi qi ,

(1)

i

i̸=j

which yields the inverse demand function as pi = a − bqi − bθ∑∑j̸=i qj . Solving those inverse demand

functions with respect to quantity yields the demand function of good i as
qi = α − βpi + γ

∑

pj ,

(2)

j̸=i
{1+(n−2)θ}aθ
where α ≡ b{1+(n−1)θ}, β ≡ b(1−θ){1+(n−1)θ}, and γ ≡ b(1−θ){1+(n−1)θ}. We let σ ≡ β − (n − 1)γ > 0,

which means that the own-price eﬀect dominates the sum of cross-price eﬀects.
This article presents examination of two contract forms: wholesale and agency. The stage-gameof each contract is presented as follows. Under the wholesale contract, each manufacturer i sets its
wholesale price wi in the first round before platform P charges a pair of retail prices (p1 , · · · , pn )in the second round. We assume that manufacturers incur a constant marginal cost of productionc > 0. The profits of manufacturer i and of platform P are given respectively as πi = (wi − c)qi and∑Π = i (pi − wi )qi . In contrast, under the agency contract, the platform oﬀers a revenue-sharing
rule s ∈ [0, 1] for all manufacturers in the first round before each manufacturer i sets its retail pricepi in the second round. The profits of manufacturer i and of platform P are given respectively as∑
πi = {(1 − s)pi − c}qi and Π = i spi qi .
In the symmetric equilibrium (i.e., pi = p and qi = q for all i), consumer surplus is given as
CS = nq 2 /(2σ). Additionally, we interpret social welfare as a sum of consumer surplus and profits

7

Table 1: Stage-game equilibrium and collusive equilibrium under the wholesale contractk = {N, C}

Stage-game Nash equilibrium (N )Collusive equilibrium (C)

∗
wW
k

α+βc
β+σ

α
c
2σ + 2

p∗W k

α+βc
α
2σ + 2(β+σ)β(α−σc)2(β+σ)
β(α−σc)22(β+σ)22 (α−σc)2n · β4σ(β+σ)
22(α−σc)2n · β8σ(β+σ)
22n · β(3β+4σ)(α−σc)8σ(β+σ)2

3α+σc
4σ

∗
qW
k
∗
πW
k

Π∗W k
∗
CSW
k
∗
SWW
k

of all firms, i.e., SW = CS + Π +α−σc
4(α−σc)28σ
2n · (α−σc)16σ2n · (α−σc)32σ
2n · 7(α−σc)32σ∑i πi .

Analysis of the stage-game under the wholesale contract One can consider the situationin which the game above is played only once under the wholesale contract. The stage-game is solved
using backward induction. In the second round, given a pair of wholesale prices w = (w1 , · · · , wn ),platform P chooses retail prices (p1 , · · · , pn ) to maximize its profit. Solving ∂Π/∂pi = 0 for all i
implies that pi (wi ) = α/(2σ) + wi /2.
Then, the maximization problem faced by manufacturer i in the first round is written asmax πi (w) = (wi − c) α − βpi (wi ) + γwi∑pj (wj ) .

(3)

j̸=i

∗
Solving ∂πi /∂wi = 0 for all i implies that wW
N = (α + βc)/(β + σ). We here use subscript W

to indicate the wholesale contract, subscript N to denote the stage-game Nash equilibrium, andsuperscript ‘∗’ to represent the equilibrium result of the monopoly platform model. The equilibrium
retail price, profit of manufacturers, profit of the platform, consumer surplus, and social welfare
are presented in Table 1.
Next, we derive the collusive stage-game equilibrium. The best response of platform P at thesecond round is unchanged, i.e., pi (wi ) = α/(2σ) + wi /2. In the first round, manufacturers collude∑∑in their wholesale prices to maximize their joint profit, which is given as π C = i πi = i (wi −c)qi .

8The maximization problem of cartel party (i.e. manufacturers) is the following.

max

w1 ,··· ,wn∑∑
pj (wj )
π C (w) =(wi − c) α − βpi (wi ) + γ
i

(4)

j̸=i

∗
Solving this problem yields wW
C = α/(2σ) + c/2. We use subscript C to denote the collusive

equilibrium. Other outcomes are presented in Table 1.
By comparing the stage-game Nash equilibrium with the collusive result obtained under thewholesale contract, we derive several results indicating how upstream collusion aﬀects the vertical
relation.
Lemma 1. Consider the case with a monopoly platform.

Under the wholesale contract, up-∗∗∗∗
stream collusion increases the wholesale and retail prices (i.e., wW
N < wW C and pW N < pW C ),∗∗which decreases the quantities demanded, consumer surplus, and social welfare (i.e., qW
N > qW C ,
∗
∗
∗
∗
CSW
N > CSW C , and SWW N > SWW C ). By colluding, the manufacturers receive greater profit∗∗∗∗(i.e., πW
N < πW C ), although the profit of the platform declines (i.e., ΠW N > ΠW C ).

Under the wholesale contract, collusion among manufacturers increases the wholesale price,which in turn induces the platform to choose higher retail prices. The wholesale price increased by
the collusion raises the manufacturers’ margin per unit sold, which increases their profits. Simultaneously, the resulting higher retail price shrinks the quantities demanded; it thereby decreases the
profit of the platform, consumer surplus, and social welfare.
Analysis of the stage-game under the agency contract As in the case of the wholesale
contract, we consider a situation in which the stage-game is played only once under the agency
contract. In the second round, each manufacturer i chooses its retail price pi independently, given
the revenue-sharing rule s chosen by the platform. Solving ∂πi /∂pi = 0 for all i implies pAN (s) =()
c
α + β 1−s/(β + σ), where we use subscript A to denote the agency contract. The corresponding
quantities, profit of manufacturers, and profit of platform P are given respectively as()
cβ α − σ 1−s

,
β+σ
(
c )2
α − σ 1−sπAN (s) = β(1 − s),β+σqAN (s) =

9

(5)
(6)

(
)
c
c
β
α
−
σ
α
+
β
∑
1−s 
1−s
ΠAN (s) =
·
s ·
.
β+σ
β+σ


(7)

i

In the first round, the platform sets the revenue-sharing rule s∗AN which satisfies the first-ordercondition, which is equivalent to (∂ΠAN (s∗AN )
nβ

=
·
2
∂s
(β + σ) 

α + β 1−sc∗)(AN

α − σ 1−sc∗

)

AN

{
α(β − σ) − 2βσ 1−sc∗AN(1 − s∗AN )2
nβ
1
⇐⇒
·
· ξ(s∗AN ) = 0,2
(β + σ) (1 − s∗AN )3s∗AN c

+




= 0,
} 


(8)

{
}
where ξ(s) ≡ α2 (1 − s)3 + (β − σ)αc + βσc2 (1 − s) − 2βσc2 . s∗AN is unique over s ∈ (0, 1 − σc/α)
and satisfies the second-order condition.8 The equilibrium outcomes of the stage game are written∗∗= qAN (s∗AN ), πAN= πAN (s∗AN ), and Π∗AN = ΠAN (s∗AN ),with s∗AN , that is, p∗AN = pAN (s∗AN ), qAN

which are presented in Table 2.9
Next, we derive the collusive stage-game equilibrium. Consider collusion among manufacturers
in the second round, where the manufacturers seek to maximize their joint profit for any given∑∑
revenue-sharing rule s set by the platform. Let π C =i πi =i {(1 − s)pi − c}qi . Solving()
c∂π C /∂pi = 0 for all i implies that pAC (s) = α + σ 1−s/(2σ). The corresponding quantities,
profit of manufacturers, and profit of platform P respectively denote given as shown below.

qAC (s) =c
α − σ 1−s2πAC (s) = (1 − s)

8

(

,

(9)

c
α − σ 1−s

)2

4σ

(10)

(s)
= sign ξ(s) for all s ∈ (0, 1). We have
It holds that sign ∂ΠAN∂s

ξ(0) =(α + βc)(α − σc) > 0,
σc )
c2 σ
ξ 1−
=−
(σ + β) (α − σc) < 0,
α
α
{
}
′
2
ξ (s) = − 3α (1 − s)2 − (β − σ)αc + βσc2 < 0.
(

The first two inequalities ensure the existence of the solution in (0, 1 − σc/α). The last inequality ensures uniquenessof the solution which satisfies the second-order condition.
9
The fact that sAN ∈ (0, 1 − σc/α), which we have already shown in footnote 8, ensures all outcomes to be positive(i.e., pAN > 0, qAN > 0, πAN > 0, and ΠAN > 0).

10

Table 2: Stage-game equilibrium and collusive equilibrium under the agency contractk = {N, C}

Stage-game Nash equilibrium (N )Collusive equilibrium (C)

s∗Ak

∗
CSAk

s∗AN in equation (8)
(
)
1
c
α
+
β
∗
β+σ
1−sAN
)
(
β
c
α
−
σ
∗
β+σ
1−sAN
)2
(
β(1−s∗AN )cα−σ1−s∗AN(β+σ)2()()βs∗ANn · (β+σ)2 α + β 1−sc∗α − σ 1−sc∗
AN)
AN
(
2
nβ 2
c
α − σ 1−s∗2σ(β+σ)2s∗AC in equation (12)
(
)
1
c
α
+
σ
∗
2σ
1−sAC
)
(
1
c
α
−
σ
∗
2
1−sAC
)2
(
1−s∗ACcα−σ
4σ
1−s∗AC)(s∗AC
σ 2 c2n · 4σ α2 − (1−s
∗ )2
(
)2AC
n
c
8σ α − σ 1−s∗

∗
SWAk

∗
∗
CSAN
+ Π∗AN + n · πAN∗ + Π∗ + n · π ∗
CSAC
AC
AC

p∗Ak
∗
qAk
∗
πAk

Π∗Ak

AN

AC


ΠAC (s) =

∑
s ·


c
α + σ 1−s

2σ

i

·

c
α − σ 1−s


2



(11)

In the first round, the platform sets the revenue-sharing rule s∗AC which satisfies the following
first-order condition.10)][(
σ 2 c22σ 2 c2
2
∗
α −
− sAC
=0
(1 − s∗AC )2(1 − s∗AC )3}{(1 + s∗AC )σ 2 c2n=0⇐⇒
· α2 −
4σ(1 − s∗AC )3∂ΠAC (s∗AC )
n
=
·
∂s
4σ

(12)

∗
The collusive outcomes of the stage game are written with s∗AC , i.e., p∗AC = pAC (s∗AC ), qAC
=
11
∗
∗ =π
∗
∗
qAC (s∗AC ), πACAC (sAC ), and ΠAC = ΠAC (sAC ), which are presented in Table 2.

By comparing the stage-game Nash equilibrium with the collusive one under the agency contract,
we derive the following lemma.

10s∗AC is unique over (0, 1 − σc/α) and satisfies the second-order condition. We obtain
n
∂ΠC
A
(0) =
(α + σc)(α − σc) > 0,
∂s
4σ
)
C (
σc
2nα2
∂ΠA
1−
=
(α − σc) < 0,
∂s
α
σc
(
2 C
∂ ΠA
σc )
.
(s) < 0, ∀s ∈ 0, 1 −2∂sαTherefore, s∗AC is uniquely decided over (0, 1 − σc/α) and satisfies the second-order condition.
11
The fact that s∗AC ∈ (0, 1 − σc/α), which we have already shown in footnote 10, ensures that all outcomes are∗∗
positive (i.e., p∗AC > 0, qAC> 0, πAC> 0, and Π∗AC > 0).

11

Lemma 2. Consider the case with a monopoly platform. Under the agency contract, upstream
collusion decreases the revenue-sharing rule (i.e., s∗AN > s∗AC ), but it increases the retail price(i.e., p∗AN < p∗AC ), which engenders lower demand, consumer surplus, and social welfare (i.e.,∗∗ , CS ∗
∗
∗
∗
qAN
> qAC
AN > CSAC , and SWAN > SWAC ). By colluding, the manufacturers receive∗∗ ), whereas the profit of the platform is lower in the presence of thegreater profit (i.e., πAN< πACmanufacturers’ collusion (i.e., Π∗AN > Π∗AC ).
The monopoly platform’s optimal revenue-sharing rule is lower in the presence of the manufacturers’ collusion. This is because upstream collusion exacerbates the double marginalization
problem, making it unattractive for the platform to impose a high revenue-sharing rule. Lemma
2 also shows that, even though upstream collusion induces the platform to set the lower revenuesharing rule, it results in the higher retail price eventually.
Upstream collusion enables manufacturers to win the favorable revenue-share and to charge thehigher retail price. Therefore, they can gain the greater profit in the collusive equilibrium. By
contrast, the platform loses its profit and therefore has no incentive to foster the collusion.
Finally, as in the wholesale contract, upstream collusion shrinks consumer demand. It therefore
degrades consumer surplus and social welfare.

Comparison of collusive outcomes under the two contracts Comparing the collusive outcome under the wholesale contract with the one under the agency contract, we have the following
lemma.
Lemma 3. Consider the case with a monopoly platform. Presume also that the upstream manufacturers collude to maximize their joint profit. Compared to the wholesale contract, the retailprice is lower under the agency contract (i.e., p∗W C > p∗AC ), which engenders greater demand (i.e.,∗∗qW
C < qAC ). Then, consumer surplus and social welfare are higher under the agency contract∗∗∗∗(i.e., CSW
C < CSAC and SWW C < SWAC ). The manufacturers receive greater profit under∗∗
the wholesale contract (i.e., πW
C > πAC ), although the platforms prefer the agency contract (i.e.,Π∗W C < Π∗AC ).
When the manufacturers collude to maximize their joint profit, the retail price is lower under
the agency contract than under the wholesale contract, which generates the greater demand and12then improves both consumer surplus and social welfare. This is because revenue-sharing under theagency contract mitigates the double-marginalization problem. In this respect, the agency contract
is not necessarily anticompetitive, but rather improves social welfare.
Furthermore, the manufacturers prefer the wholesale contract to the agency contract, whereas
the platform has the opposite preference. In other words, every firm prefers a contract that guarantees itself the first move. This result is also presented by Johnson (2017), although upstream
collusion is not considered in his model. Consequently, the presence of upstream collusion does notalter the preference of each firm related to the contract type.

2.2

Infinitely repeated game

This subsection presents examination of an infinitely repeated game in which the monopoly platform
and manufacturers play the above stage-game over period (t = 1, 2, · · · , ∞). The game is of common
knowledge and perfect monitoring. Let δ ∈ (0, 1) be a common discount factor. The payoﬀ of eachplayer is given as the sum of the discounted stage-game payoﬀ. They maximize their expected
payoﬀ.
We assume that manufacturers sustain their joint profit maximizing collusion through infinite
Nash-reversion, where any deviation by an upstream manufacturer is followed by the infinite repeated play of the subgame-perfect equilibrium of the stage game. We determine the value of thecritical discount factor, which is the lowest discount factor with which manufacturers’ joint profit
maximizing collusion can be sustained.12 We stipulate that the agency contract facilitates upstream
collusion if it reduces the critical discount factor.
Critical discount factor under the wholesale contract∗ to denote the critical
We use δW

discount factor under the wholesale contract. Manufacturers collude to maximize their joint profit
before the platform moves. This is the standard case of the literature on collusion in the verticallyrelated market.
We have already derived the stage-game Nash equilibrium and the collusive equilibrium in
Section 2.1. Consequently, here, we compute the deviation payoﬀ. The best response of platform
12

Generally, many studies in the literature on Repeated Games are interested in the derivation of the set of theequilibrium payoﬀs for a fixed discount factor. In contrast, we follow the recent Industrial Organization literature on
collusion in vertical related markets (e.g., Nocke and White, 2007; Reisinger and Thomes, 2017) that are interested
in the derivation of the value of the critical discount factor.

13

P at the second round is pi (wi ) = α/(2σ) + wi /2. We consider the deviation by manufacturer 1 in
the first round, with no loss of generality. Consequently, the other manufacturers set the collusive∗
wholesale price (i.e., wj = wW
C for j = 2, · · · , n). Then, the profit of manufacturer 1 can be

written as
π1D (w1 ) = (w1 −c) α − βp1 + γ

∑
j̸=1



(
{
)}
(α
∗
wW
w1 )
α
C

+
+ (β − σ)+pj = (w1 −c) α − β
.
2σ
2
2σ
2
(13)

∗
Solving ∂π1D (w1 )/∂w1 = 0 yields wW
D = {α(β + σ) + σ(3β − σ)c} /(4βσ). We use subscript D∗
2
to denote the deviation outcome. The deviation profit can be computed as πW
D = (β + σ) (α −σc)2 /(32βσ 2 ).
Then collusion among manufacturers is sustainable if and only if the following inequality holds.
∗
∗
δπW
πW
∗
N
C
≥ πW
+
D
1−δ
1−δ

(14)

Solving condition (14) with equality determines the critical discount factor under the wholesale∗ .
contract, δW

Proposition 1. Consider the case with a monopoly platform. Under the wholesale contract, collusion among manufacturers is sustainable if and only if the discount factor is suﬃciently high tosatisfy∗δ ≥ δW≡β 2 + σ 2 + 2βσ.
β 2 + σ 2 + 6βσ(15)∗ increases with n and γ and that it
One can readily show that the critical discount factor δW∗ /∂n > 0, ∂δ ∗ /∂γ > 0, and ∂δ ∗ /∂β < 0. The tougher competition (e.g., adecreases in β, i.e., ∂δWWW∗ ,larger number of manufacturers and/or a higher degree of substitution) engenders the higher δWwhich is relevant to a well-known result that the collusion becomes more diﬃcult to sustain as thecompetition gets tougher.13
Critical discount factor under the agency contract Next, we derive the critical discount∗ . Unlike the wholesale contract, under the agency
factor under the agency contract, denoted as δA
13

The opposite result is obtained in the common retailer case of Reisinger and Thomes (2017). That is, they
demonstrated that the critical discount factor decreases in γ when each manufacturer oﬀers a two-part tariﬀ contract
to the common retailer.

14

contract, one player moves before the formation of collusion. In other words, the monopoly platform can design its action strategically to influence collusion among manufacturers. The platformmight have an incentive to hinder or foster the collusion by manipulating its action. Thus, weneed to consider the equilibrium path by which the platform deters manufacturers’ joint profit
maximization.
Because this situation largely diﬀers from the standard of the literature, let us formally describe
all players’ strategies. We consider two states: State C (collusion) and State P (punishment). The
initial period begins with State C. Players’ strategies for each state are described as follows.
State C: Monopoly platform chooses a revenue-sharing rule s. All manufacturers take a symmetricaction that maximizes their joint profit for the given s, that is, pi = pAC (s) for all i. If
a diﬀerent revenue-sharing rate s′ is chosen by the platform, manufacturers play the Nashequilibrium strategy, i.e., pi = pAN (s′ ) for all i. The state turns to State P if any manufacturer
sets pi ̸= pAC (s) even as the platform chooses s.
State P: The subgame perfect equilibrium of the stage-game is played. The state of the next
period is State P.
We assume that, if manufacturers observe a diﬀerent revenue-sharing rate in a period, they stop
colluding tentatively in that period and play the Nash equilibrium strategy. In other words, themonopoly platform is allowed to temporarily deter the formation of upstream collusion by charging
a diﬀerent revenue-sharing rate. Note that, even if the platform chooses a diﬀerent revenue-sharing
rate, the state remains to be State C.
Because Lemma 2 shows Π∗AN > Π∗AC , the platform has an incentive to set a diﬀerent revenuesharing rule s′ (= sN ) in an eﬀort to hinder the collusion in every period.
Proposition 2. Consider the case with a monopoly platform. Under the agency contract, theplatform deters the formation of upstream collusion in every period. Therefore, as compared to the∗ < δ ∗ = 1).
wholesale contract, the agency contract does not facilitate upstream collusion (i.e., δW
A

Basically, the presence of upstream collusion exacerbates the double marginalization problem,
which in turn decreases the profit of the platform. By definition, under the agency contract, theplatform can temporarily block the formation of upstream collusion simply by charging a diﬀerent
15revenue-sharing rule. On the equilibrium path, in every period, the monopoly platform deters cartel∗ = 1 holds.
formation because of the potential danger of upstream collusion. Thus, δA

As one would expect, this result largely depends on the definition of strategy. However, the∗ < δ ∗ , which implies that the agency contract does not facilitate upstream collusion
main result δWA

in the monopoly platform market, can be robust even under more relaxed definitions of strategy,
which is demonstrated in Section 4.1.

3

Platform Competition

In the previous section, we show that the agency contract does not facilitate upstream collusionin the monopoly platform market. The purpose of this section is to address how the presence of
platform competition aﬀects the sustainability of upstream collusion.

3.1

The stage-game

Herein, after we describe only those components of the stage-game that diﬀer from those of themonopoly platform case, we analyze static equilibrium and collusion among manufacturers under
the two contracts. Subsequently, we compare those stage-game outcomes.

Setting of the stage-game

We consider two upstream manufacturers (u = 1, 2) and two down-

stream platforms (d = 1, 2). Both manufacturers produce diﬀerentiated goods with a constantmarginal cost, which is normalized to zero. Let pud and qdu be the price and the quantities ofmanufacturer u’s good demanded at platform d.
Following Dobson and Waterson (1996), we assume a representative consumer with the following
utility function:14

U=

∑
u,d

∑1
−u
−u
u
u −u
u −u
qdu −
(q u )2 −λ(qdu q−d+qd−u q−d)−µ(qdu qd−u +q−dq−d )−λµ(qdu q−d+q−dqd ) for u, d = 1, 2,
2 d
u,d

(16)
where µ ∈ (0, 1) and λ ∈ (0, 1) respectively represent the degrees of substitution between the
upstream manufacturers and between the downstream platforms.
14
This utility function and the resulting demand function have been used widely in the literature (e.g., Johansen
and Vergé, 2017; Foros et al., 2017; Lu, 2017).

16

Table 3: Stage-game equilibrium and collusive equilibrium under the wholesale contract with platform competitionk = {N, C}

Stage-game Nash equilibrium (N )Collusive equilibrium (C)

∗∗
wW
k

1−µ
2−µ
(1−λ)(2−µ)+1−µ(2−λ)(2−µ)121(1+λ)(1+µ)(2−λ)(2−µ)
2(1−µ)(1+λ)(1+µ)(2−λ)(2−µ)22(1−λ)(1+λ)(1+µ)(2−λ)2 (2−µ)212(1+λ)(1+µ)(2−λ)2(1+λ)(1+µ)(2−λ)2 (2−µ)22(7−4λ−4µ+2λµ)(1+λ)(1+µ)(2−λ)2 (2−µ)212(1+λ)(1+µ)(2−λ)2

p∗∗
Wk
∗∗
qW
k
∗∗
πW
k

Π∗∗
Wk
∗∗
CSW
k
∗∗
SWW
k

3−2λ
2(2−λ)12(1+λ)(1+µ)(2−λ)
1−λ2(1+λ)(1+µ)(2−λ)2

7−4λ2(1+λ)(1+µ)(2−λ)2Utility-maximization subject to the budget constraint yields the inverse demand functions:u ) − µ(q −u + λq −u ) for u, d = 1, 2. Solving for quantities, we can derive thepud = 1 − (qdu + λq−dd−ddemand function as follows.

qdu =

−u
u
(1 − λ)(1 − µ) − pud + µp−ud + λ(p−d − µp−d )(1 − λ2 )(1 − µ2 )for u, d = 1, 2

(17)

Analysis of the stage-game under the wholesale contract Under the wholesale contract,
manufacturer u sets wholesale prices (w1u , w2u ) in the first round; then platform d sets retail prices(p1d , p2d ) in the second round.
First, we derive the stage-game Nash equilibrium. In the second round, solving the platforms’
profit maximization problems, we obtain the second-stage price pud (w). Using this, the corresponding second-round quantities are given as

qdu (w) =λ(wdu − µwd−u ) − (2 − λ2 )(wdu − µwd−u )
1
+
.
2
2
2
(4 − λ )(1 − λ )(1 − µ )(2 − λ)(1 + λ)(1 + µ)

(18)

In the first round, the stage-game equilibrium can be derived by solving the following problem∑
for each manufacturer: max(wdu )d=1,2 d′ (wdu′ − c)qdu′ (w), as analyzed by Lu (2017). The outcomesof the stage-game Nash equilibrium are presented in Table 3. We use superscript ‘∗∗’ to denote theequilibrium in the model with the two competing platforms.
Second, we consider joint profit maximization by manufacturers, which can be derived by solv-

17

ing the following problem: max(wdu )u,d=1,2

∑ ∑
u′

u′
u′
d′ (wd′ − c)qd′ (w).

The resulting outcomes of thecollusive equilibrium are also presented in Table 3.
By comparing the stage-game Nash equilibrium with the collusive one under the wholesale
contract, we derive the following lemma.
Lemma 4. Consider a market with two competing platforms. Under the wholesale contract, up∗∗ < w ∗∗ and p∗∗ < p∗∗ ), which
stream collusion increases wholesale and retail prices (i.e., wW
N
WC
WN
WC
∗∗ > q ∗∗ , CS ∗∗ >decrease the quantities demanded, consumer surplus, and social welfare (i.e., qW
N
WC
WN
∗∗
∗∗ , and SW ∗∗
∗∗
∗∗
CSW
C
W N > SWW C ). The manufacturers receive greater profit (i.e., πW N < πW C ),∗∗whereas the profit of platforms declines (i.e., Π∗∗
W N > ΠW C ).

Lemma 4 presents the same result as that of Lemma 1. Therefore, under the wholesale contract,the eﬀects of upstream collusion are irrelevant to the existence of platform competition. As in
the monopoly platform case, the upstream collusion is profitable only to the manufacturers; it isdeleterious to the platforms, consumer surplus, and social welfare.

Analysis of the stage-game under the agency contract

Under the agency contract, platformd sets a revenue-sharing rule sd in the first round; then manufacturer u sets retail prices (pu1 , pu2 ) in
the second round.
We first derive the stage-game Nash equilibrium. In the second round, given (s1 , s2 ), eachmanufacturer u solves the following problem:max (1 − s1 )pu1 q1u + (1 − s2 )pu2 q2u .

upu
1 , p2(19)

Solving the maximization problem above, we derive that p1d = p2d holds for d = 1, 2, implying thatcompeting manufacturers charge the same retail price for each platform. Then, we use pNd (sd , s−d )and qdN (sd , s−d ) to represent the resulting price and the corresponding quantities for platform dgiven (sd , s−d ), respectively.
In the first round, each platform chooses revenue-sharing rule sd to maximize ΠNd (sd , s−d ) =N∗∗
2sd pNd (sd , s−d )qd (sd , s−d ). By solving this problem, we obtain a symmetric Nash equilibrium sAN .

This stage-game Nash equilibrium has been analyzed by Foros et al. (2017) and Lu (2017), as
presented in Table 4.
18

Table 4: Stage-game equilibrium and collusive equilibrium under the agency contract with platformcompetitionk = {N, C}

Stage-game Nash equilibrium (N )Collusive equilibrium (C)s∗∗Ak(2−µ)(1−λ2 )1 − λ22−µ−λµ
1−µ
2−µ

p∗∗
Ak
∗∗
qAk
∗∗
πAk

Π∗∗
Ak
∗∗
CSAk
∗∗
SWAk

1
2

1
(1+λ)(1+µ)(2−µ)
2λ(1−µ)(2λ−µ−λµ)(1+λ)(1+µ)(2−µ)2 (2−µ−λµ)2(1−λ)(1−µ)(1+µ)(2−µ)(2−µ−λµ)12(1+λ)(1+µ)2(1+λ)(1+µ)(2−µ)26−4µ(1+λ)(1+µ)(2−µ)212(1+λ)(1+µ)λ2
2(1+λ)(1+µ)
1−λ2(1+µ)32(1+λ)(1+µ)

Second, we consider joint profit maximization by manufacturers. In this case, given (s1 , s2 ),each manufacturer u solves the following problem:
∑

max

(pu
d )u,d=1,2 u=1,2{(1 − s1 )pu1 q1u + (1 − s2 )pu2 q2u } .

(20)Solving the maximization problem above, we derive the retail prices, which are denoted as pC
d (sd , s−d )for d = 1, 2. Let qdC (sd , s−d ) be the resulting quantities.
In the first round, platform d solves the maximization problem: maxsd∑

CC
u=1,2 sd pd (sd , s−d )qd (sd , s−d ).

The first-order condition is given as
∑

(
C
pC
d qd + sd

u=1,2

{

∂pC
d
∂sd

(

)
C
∂pC
∂q C
−d ∂qd+qdC + d pC
pC
d∂pd
∂sd ∂p−d d

})
= 0.

(21)

2
∗∗
Invoking symmetry, sd = s−d = s, we obtain s∗∗
AC = 1 − λ > sAN . The other resulting outcomesare also presented in Table 4.
By comparing the stage-game Nash equilibrium with the collusive one under the agency contract,
we derive the following lemma.
Lemma 5. Consider the case with two competing platforms. Under the agency contract, therevenue-sharing rule and the retail price are higher in the presence of the manufacturers’ collusion∗∗∗∗∗∗(i.e., s∗∗
AN > sAC and pAN < pAC ), which decrease the quantities demanded, consumer surplus, and19

∗∗ > q ∗∗ , CS ∗∗ > CS ∗∗ , and SW ∗∗ > SW ∗∗ ). Both the manufacturerssocial welfare (i.e., qAN
AC
AN
AC
AN
AC
∗∗ < π ∗∗ and Π∗∗ > Π∗∗ ).
and the platforms receive greater profits (i.e., πAN
AC
AN
AC

Lemma 5 presents the same result as that of Lemma 2. Therefore, also under the agency
contract, the eﬀects of upstream collusion are irrelevant to the existence of platform competition.
The upstream collusion is profitable only to the manufacturers; it is deleterious to the platforms,
consumer surplus, and social welfare.

Comparison of collusive outcomes under the two contracts Here, we compare the collusive
outcome under the wholesale contract with the outcome obtained under the agency contract.
Lemma 6. Suppose that two platforms compete in the market. Presume that the upstream manufacturers collude to maximize their joint profit. Compared to the wholesale contract, the retail∗∗price is lower under the agency contract (i.e., p∗∗
W C > pAC ), which engenders greater demand, con∗∗ < q ∗∗ , CS ∗∗ < CS ∗∗ , and SW ∗∗ < SW ∗∗ ). The
sumer surplus, and social welfare (i.e., qW
AC
WC
AC
WC
AC
C
∗∗ > π ∗∗ ), although the
manufacturers receive greater profit under the wholesale contract (i.e., πWCAC∗∗platforms prefer the agency contract (i.e., Π∗∗
W C < ΠAC ).

Lemma 6 shows qualitatively the same result as that obtained in Lemma 3. Therefore, whenmanufacturers collude to maximize their joint profit, the existence of platform competition does
not aﬀect the comparison between the two contracts.

3.2

Infinitely repeated gameIn this subsection, we consider an infinitely repeated game in which two platforms and two manufacturers play the above stage-game over period (t = 1, 2, · · · , ∞) in order to derive the values of
the critical discount factor for the respective contracts

Critical discount factor under the wholesale contract∗∗ to denote the critical
We use δW

discount factor under the wholesale contract. As in the monopoly case, we consider the standard
Nash-reversion trigger strategies in the literature on collusion in the vertically related market.
We have already derived the stage-game Nash equilibrium and the collusive equilibrium in Section 3.1. Therefore, we compute the deviation payoﬀ here. The platforms’ best response strategies

20

given a pair of wholesale prices are unchanged if a manufacturer deviates. Then, the first-ordercondition for the best deviation from the joint profit maximization can be written as
u
u
∂π u
u
u ∂qd
u ∂q−d
=
q
+
w
+
w
=0
d
d
−d
∂wdu
∂wdu
∂wdu

s.t. wd−u = wC

for d = 1, 2.

(22)∗∗ = (2 − µ)/4 and the corresponding
Solving this equation, we obtain the deviation strategy wWD∗∗ = (2 − µ)2 /{8(2 − λ)(1 + λ)(1 − µ2 )}.
profit of the deviating manufacturer πWD∗∗ is computed as follows.
Consequently, the critical discount factor δW

∗∗
δW
=

∗∗ − π ∗∗πW(2 − µ)2
D
WC
=
∗∗
∗∗
πW D − πW N
8 − 8µ + µ2

(23)

Proposition 3. In the model with two competing platforms, collusion under the wholesale contract∗∗ .
among manufacturers is sustainable if and only if δ ≥ δW

Critical discount factor under the agency contract Next, we derive the critical discount∗∗ . In order to confine our attention to symmetricfactor under the agency contract, denoted as δA

equilibrium, we consider the following Nash-reversion trigger strategies.
State C: Platforms choose the same revenue-sharing rule, (s1 , s2 ) = (s, s). All manufacturers
take a symmetric action that maximizes their joint profit for the given s. If a platform dsets a diﬀerent revenue-sharing rate s′ , manufacturers play the Nash equilibrium strategy for(sd , s−d ) = (s′ , s). The state turns to State P if any manufacturer takes a diﬀerent action
even under (s1 , s2 ) = (s, s).
State P: The subgame perfect equilibrium of the stage-game is played. The state of the next
period is State P.
The definition of strategy implies that, as in the monopoly platform case, platforms can temporarily deter the formation of upstream collusion by charging a diﬀerent revenue-sharing rule. In
what follows, we will show that the agency contract facilitate upstream collusion even with suchthe definition of strategy that seems to make the cartel formation diﬃcult.
We have already derived the stage-game Nash equilibrium in Section 3.1. In addition, toderive the critical discount factor, we need to consider the manufacturers’ collusive strategy given21

(s1 , s2 ) = (s, s), and the optimal deviation from that collusive situation.
First, let us take a closer look at the manufacturers’ collusive pricing given (s1 , s2 ) = (s, s).
The first-order condition for joint profit maximization given (s1 , s2 ) = (s, s) is written by
(
)
−u
−u
u
u
∂q
∂q
∂q
∂q
∂(π 1 + π 2 )−dd= (1 − s) qdu + pudA du + p−u+ pu−dA −d
+ p−u
= 0.
d
d
∂pudA
∂pd
∂pud
∂pud
∂pud

(24)

Under the symmetric revenue-sharing rate, all retail prices are the same (i.e., p11 = p21 = p12 = p22 ).
Thus, we use pC (s, s) to denote the resulting retail price. The resulting price and the corresponding
quantities, profit of manufacturers, and profit of platforms are given as follows.

pC (s, s) =12

12(1 + λ)(1 + µ)
1−s
π C (s, s) =2(1 + λ)(1 + µ)s
ΠC (s, s) =2(1 + λ)(1 + µ)
q C (s, s) =

(25)
(26)
(27)
(28)

Next, we look at the optimal deviation from the above collusive pricing. Given (s1 , s2 ) = (s, s),
the optimal deviation from joint profit maximization is obtained by solving the following problem.
−uCmax (1 − s)(pu1 q1u + pu2 q2u ) s.t. p−u
1 = p2 = p (s, s)

u
pu
1 ,p2

(29)

Solving the above maximization problem, we can show that the best deviation is charging pD (s, s) =(2 − µ)/4. The resulting profit for the deviating manufacturer is given by π D (s, s) = (1 − s)(2 −µ)2 /{8(1 + λ)(1 − µ2 )}.
Now, let us consider the manufacturers’ incentive for deviation given (s1 , s2 ) = (s, s). Their
joint profit maximization is sustainable if and only if the following inequality holds:
δπ ∗∗
π C (s, s)≥ π D (s, s) + AN
1−δ1−δ
(1 − s)(2 − µ)2 µ2 (2 − µ − λµ)⇐⇒ δ ≥≡ δA (s)(1 − s)(2 − µ)4 (2 − µ − λµ) − 16λ(1 − µ)2 (2λ − µ − λµ)The critical discount factor given s, δA (s), has the following property.
22

(30)
(31)

′ (s) > 0 and (ii) δ (s∗∗ ) = δ ∗∗ .
Lemma 7. The following statements hold: (i) δA
A NW

Lemma 7 (i) implies that the upstream collusion becomes increasingly diﬃcult to sustain as theplatforms set a higher revenue-sharing rule and (ii) shows that, when the platforms set s1 = s2 =s∗∗N , the critical discount factor becomes equal to the critical discount factor under the wholesale
contract.
Finally, we consider the platforms’ incentives for charging a diﬀerent revenue-sharing rate in∗∗order to temporarily block the cartel formation. Although Lemma 5 shows Π∗∗
AN > ΠAC , for a fixeds, the following lemma holds.
C
Lemma 8. ΠN
d (s, s) < Π (s, s) for any s ∈ (0, 1).

This lemma states that the platforms are better oﬀ with upstream collusion for any fixed s,which would discourage the competing platforms from taking unexpected actions. As a result,the presence of platform competition can radically alter the eﬀect of agency contracts on collusion
among manufacturers: The agency contract facilitates upstream collusion, which is summarized in
the following proposition.
Proposition 4. When two platforms compete in a market, agency contracts facilitate upstream∗∗ < δ ∗∗ is satisfied.
collusion. Formally, δAW

The detailed proof is delegated to the Appendix. Some intuition can be provided for why
the presence of platform competition drastically alters the attitude of platforms towards upstream
collusion.
When upstream manufacturers collude, they seek to concentrate their sales into a platform
with lower s. Thus, competing platforms have a stronger incentive to set a lower s than their rival.
That is, upstream collusion makes platform competition fiercer, resulting in lower platform profits,
as shown in Lemma 5. In contrast, fixing (s1 , s2 ) = (s, s), the presence of upstream collusion
heightens profits of competing platforms, as shown in Lemma 8, which is a sharp contrast to themonopoly platform case. The reason is the following. Given (s1 , s2 ) = (s, s), due to revenue-sharing
agreements, the objective that collusive manufacturers maximize is equivalent to the total profit
across the whole channel. Thus, upstream collusion enables the platforms to obtain the largest
revenue-share for the fixed revenue-sharing rate, which makes it less attractive for them to deviate
from (s1 , s2 ) = (s, s).
23Interestingly, repeated interactions provoke not only collusion among the cartel party (i.e.,manufacturers), but also coordination among the non-cartel party (i.e., platforms). In this respect,
the result obtained might be related to the fact that Apple actively organized an e-book pricecartel among five of the six largest U.S. book publishers. Additionally, the result could also be
helpful to understand the high commission rate of 30% imposed by Apple and Google in the mobile
application market. In fact, this attention-getting antitrust case reached the U.S. Supreme Court.

4

Discussion

In this section, we discuss several issues missing from the main analysis.

4.1

On the deviation by monopoly platform

In Proposition 2, we show that the agency contract does not facilitate upstream collusion in themonopoly platform market. However, players’ strategies considered are strict about cartel formation
among upstream manufacturers: Any deviation by monopoly platform temporarily breaks down
upstream collusion. One might expect that the result of Proposition 2 crucially relies on this strong
assumption. The purpose of this subsection is to confirm the robustness of the result even undermore relaxed definition about players’ strategies.
Before describing each player’s strategy, let us define a set S(x) as follows.
Definition 1. S(x) is a set of revenue-sharing rates at which manufacturers sustain their joint
profit maximization given that monopoly platform will continue to charge s = x from the next period
onwards.
In this subsection, using S(x) defined above, we weaken the extent to which a deviation bymonopoly platform blocks the formation of upstream collusion. In particular, we assume the following manufacturers’ response to deviations by the platform. Suppose that monopoly platform
deviates from charging s and chooses s′ in a period. If s′ lies within S(x), then upstream manufacturers sustain their joint profit maximizing collusion in that period. Otherwise, if s′ ∈/ S(x),
manufacturers play the Nash equilibrium strategy for the given s′ at that period. Note that ouranalysis in Section 2.2 is corresponding to the situation in which S(x) is defined as S(x) = {x}.
In sum, we consider the following strategy for each player.
24

State C: Monopoly platform chooses a revenue-sharing rule s. All manufacturers take a symmetric action that maximizes their joint profit as long as the revenue-sharing rule set by theplatform lies within S(s). If the platform sets a diﬀerent revenue-sharing rate s′ ∈/ S(s), thenmanufacturers play the Nash equilibrium strategy for the given s′ . The state turns to StateP if any manufacturers do not choose the collusive action even as the platform chooses arevenue-sharing rate that lies within S(s).
State P: The subgame perfect equilibrium of the stage-game is played. The state of the next
period is State P.
Moreover, we focus on the platform-preferred revenue-sharing rate s∗ that maximizes the profitof the monopoly platform under the strategies stated above. Formally, the platform-preferred
revenue-sharing rate can be described as follows.
s∗ = arg max ΠA (s, S(s)),
swhereΠA (s, S(s)) =ΠAC (s)if s ∈ S(s)ΠAN (s)if s ∈/ S(s)(32)(33)

We investigate whether the upstream collusion can be sustainable or not under the platformpreferred revenue-sharing rate, which depends on the value of discount factor δ. Therefore, we
explore the minimum value of discount factors at which the platform-preferred revenue-sharing
rate is determined so as to make the upstream collusion sustainable.
Proposition 5. The platform-preferred revenue-sharing rule s∗ is given by

∗

s =




s∗

AN

∗ ,
if δ < δA



s∗AC

∗ ,
if δ ≥ δA(34)∗ = δ∗ .
where δAW

The proof is delegated to Appendix B. This proposition shows that the critical discount factor
under the agency contract is the same as that under the wholesale contract, which confirms the25robustness of Proposition 2 that the agency contract does not facilitate upstream collusion.

4.2

Observability of actions

For the main analysis, we have assumed perfect monitoring, i.e., manufacturers can observe all
actions of the others. In reality, however, there might be situations in which wholesale prices set
by other manufacturers are not publicly observable, although the retail prices are easier to observe.
It is well-known in the literature of collusion under imperfect monitoring that the more diﬃcult it
is to observe each player’s action, the more diﬃcult it is to sustain the collusion.15 Consequently,
under the wholesale contract, one would have the higher critical discount factor if the wholesale
prices were not publicly observable. That fact enhances the robustness of our results such that theagency contract might be more likely to facilitate collusion among the manufacturers.

5

Conclusion

This article addresses the important issue of the sustainability of a digital cartel. To address therelation between cartel sustainability and the contract form, we develop a stylized model of theinfinitely repeated game. Then we obtain and compare the critical discount factors, a minimum
discount factor at which the price cartel among upstream manufacturers can be sustainable, undera wholesale contract and an agency contract.
The central message from our study is that the agency contract does not facilitate upstream collusion in the case of monopoly platform, although it facilitates upstream collusion with the platformcompetition. Our results are expected to contribute to the literature by providing important policy
implications. The agency contract is not necessarily per se illegal, but competition authorities must
be more concerned about it when several platforms compete in the market.
From the other viewpoint, however, given upstream collusion to be sustained, revenue-sharing
agreements under agency contracts mitigate the double marginalization problem, which generates
higher consumer surplus and social welfare compared to the wholesale contract. In this regard,
the agency contract can not necessarily be characterized as anticompetitive even in markets withplatform competition.
15

See Jullien and Rey (2007) for this argument applied to retail-price maintenance.

26

We conclude by describing the limitations of our model and by discussing potential avenues for
future research. First, we do not examine the eﬀect of Most-Favored-Nation (MFN) clauses. In thereal case of the e-book cartel, however, the clause played a central role in the proceedings along withthe agency agreements.16 Secondly, we assume that, in the stage-game of the wholesale contract,
manufacturers move before the platform(s). In the e-book market before Apple’s entry, Amazon
had set retail prices of most e-books at $9.99 and had committed to such a pricing policy. It might
be valuable to analyze the wholesale contract with realistic timing of moves, where platforms move
before manufacturers.17 Lastly, in the model of platform competition, we compare the sustainabilityof collusion between two symmetric situations where both platforms impose the same contract(wholesale or agency) on their manufacturers. As typically observed in Amazon’s behavior in thee-book market, however, platforms can choose their contract forms strategically. In this regard, itwould be interesting to consider the platforms’ endogenous choices about which contract to select,
and to confirm whether and when the agency contract is adopted in equilibrium. These extensions
require more complex analysis, which is beyond the scope of this article.

Appendix A: Proofs
Proof of Lemma 2.
First, we show s∗AN > s∗AC . We focus on the first-order conditions with respect to s, i.e.,equations (8) and (12). Consider that the revenue-sharing rule is set at the one in the collusive
case (i.e., s = s∗AC ). The derivative of platform’s profit under the punishment phase is computedby (∂ΠAN (s∗AC )
nβ

·
=

∂s
(β + σ)2 

α + β 1−sc∗AC)(

α − σ 1−sc∗
AC

)

{
+
α(n − 1)γ − 2βσ 1−sc∗AC(1 − s∗AC )2[]∗2(1 + sAC )βσcnβα(β − σ)c=
· α2 +−∗
22(β + σ)(1 − sAC )(1 − s∗AC )3s∗AC c

16




} 


(A.1)

(A.2)

For example, Boik and Corts (2016), Johansen and Vergé (2017), Johnson (2017), and Maruyama and Zennyo(2018) are noteworthy for providing analyses of so-called a Most-Favored-Customer clause or a price parity clause.
17
Timing of moves of this type has been analyzed in Marketing Science and Operations Research, and so-called
retailer-stackelberg game (e.g., Choi, 1991).

27

[
]
(1 + s∗AC )σ 2 c2
(1 + s∗AC )βσc2nβα(β − σ)c=·+−(β + σ)2(1 − s∗AC )3(1 − s∗AC )2(1 − s∗AC )3(β − σ)cnβ=·· {(1 − s∗AC )α − (1 + s∗AC )σc} ,2(β + σ) (1 − s∗AC )3

(A.3)
(A.4)

where the third equality follows from equation (12).
If ∂ΠAN (s∗AC )/∂s > 0, then s∗AN > s∗AC . Therefore, we suﬃce to show ∂ΠAN (s∗AC )/∂s > 0,
which is equivalent to s∗AC < (α − σc)/(α + σc). By looking at the derivative of platform’s profit
under collusion with respect to s at s = (α − σc)/(α + σc), we have()α−σc∂ΠAC s = α+σc∂s=−n α(α − σc)2
α − σc·< 0 ⇐⇒ s∗AC <,4σ
4σc
α + σc(A.5)

which was what we wanted.
Second, we show p∗AN < p∗AC . It holds that pAN (s) < pAC (s) for all s ∈ (0, 1 − σc/α). Because
both pAN (s) and pAC (s) are increasing function in s, there exists s′ > s∗AC such that pAN (s′ ) =pAC (s∗AC ) holds. Substituting s = s′ into ∂ΠAN (s)/∂s and then simplifying it with equation (12),
we have



∂ΠAN
∂s

(s′ )

=−

β 2 − σ24(1 − s∗AC )3 β 2 σ

2 2
2 2
∗
 (1 − sAC ) α (β − σ) − (β + σ)σ c +2(1 − s∗AC )α(s∗AC β + σ)σc

 < 0,(A.6)

which implies that s∗AN < s′ . Because pAN (s) is increasing function, it follows that pAC (s∗AC ) =pAN (s′ ) > pAN (s∗AN ). The higher retail price under the collusive equilibrium directly leads the lower∗∗ , CS ∗∗∗demand, consumer surplus, and social welfare (i.e., qAN> qAC
AN > CSAC , and SWAN >
∗ ).
SWAC
∗
∗ . It holds that π
Third, we show πAN< πAC
AN (s) < πAC (s) for all s ∈ (0, 1 − σc/α). The∗ , it follows that π∗∗manufacturers’ profit is decreasing in s. Because s∗AN > sAC
AN (sAN ) < πAC (sAC ).

Lastly, we show Π∗AC ≤ Π∗AN . As above, we consider s = s′ > s∗AC such that pAN (s′ ) =pAC (s∗AC ). The same retail price leads the same total output, which also implies the same grossrevenue in the channel. Because s′ is larger than s∗AC , the platform gains the larger share of thegross revenue, that is, ΠAC (s∗AC ) < ΠAN (s′ ). Finally, from the definition of s∗AN , the platform canobtain the greater profit by charging s∗AN than s′ , that is, ΠAN (s′ ) ≤ ΠAN (s∗AN ). Therefore, it

28

holds that Π∗AN > Π∗AC .
Proof of Lemma 3.
∗∗
First, we can show that p∗W C > p∗AC and qW
C < qAC by simple calculations, which in turn∗∗∗∗
imply that CSW
C < CSAC and SWW C < SWAC .
′′∗∗Next, we show that πW
C > πAC holds. There exists a unique s ∈ (0, 1 − σc/α) such that
∗
′′
′′
′′
πW
C = πAC (s ). Substituting s = s into ∂ΠAC (s)/∂s, we have ∂ΠAC (s )/∂s > 0. BecauseΠAC (s) is concave, it holds that s′′ < s∗AC . Moreover, because πAC (s) is decreasing in s, it holds∗′′∗∗
that πAC (s′′ ) > πAC (s∗AC ). In sum, we can derive that πW
C = πAC (s ) > πAC (sAC ) = πAC holds.
∗ )/p∗
Lastly, we show that Π∗W C < Π∗AC holds. Set s′ such that s′ = (p∗W C − wWC
W C . IfpAC (s′ ) < p∗W C holds, then we havep∗W C q(p∗W C ) < pAC (s′ )q(pAC (s′ ))(A.7)

by the concavity of pq(p) in p and the inequality arg maxp pq(p) < pAC (s′ ) < p∗W C , where q(p) ≡qi (p, . . . , p) for i = 1, . . . n. This in turn implies thatΠ∗AC = max ΠAC (s) ≥ ΠAC (s) =s∗p∗W C − wWC
pAC (s′ )q(pAC (s′ )) > Π∗W C .
p∗W C

(A.8)

What remains to be shown is that pAC (s′ ) < p∗W C . By the fact that p∗W C = (3α + σc)/4σ,cα∗wW
C = 2σ + 2 , and pAC (s) = {α + σc/(1 − s)}/2σ, we havep∗W C − pAC (s′ ) =α(α − σc)
> 0.
4σ(α + σc)(A.9)

Thus, we have pAC (s′ ) < p∗W C , which completes the proof.
Proof of Proposition 2.
Here, we show that the monopoly platform always has an incentive to deter the formation of
upstream collusion for all s ∈ [0, 1].
First, let us consider the case in which the monopoly platform sets s ̸= sN and upstream
manufacturers sustain their collusion. The monopoly platform gains ΠAC (s) for every period. If theplatform deviates from charging s, the best deviation is to charge s = sN because manufacturers
who observe any revenue-sharing rate diﬀerent from s will start to play the stage-game Nash29

equilibrium strategy (i.e., pi = pAN (s) for all i). This optimal deviation enables the platform toobtain ΠAN (sN ) = Π∗AN over subsequent future periods. From the result of Lemma 2, the following
string of inequalities holds: ΠAC (s) < Π∗AC < Π∗AN . Therefore, the deviation to s = sN always
improve the profit of the monopoly platform. In other words, the platform has an incentive to deter
the upstream collusion by setting s = sN .
Next, we look at the remaining case of s = sN . As above, by deviating to s = sN ± ε where εrepresents an arbitrary small number, the monopoly platform can deter the cartel formation andthen derive the profit of ΠAN (sN ± ε) ≃ Π∗AN , which is strictly greater than ΠAC (sN ). Thus, theplatform has an incentive to deter the upstream collusion even for s = sN .
In total, the monopoly platform always has an incentive to deter the upstream collusion for alls ∈ [0, 1]
Proof of Lemma 7
The derivative of δA (s) can be computed as
′
δA
(s) =

16λµ2 (1 − µ)2 (2 − µ)2 (2λ − µ − λµ)(2 − µ − λµ){(1 − s)(2 − µ)4 (2 − µ − λµ) − 16λ(1 − µ)2 (2λ − µ − λµ)}2,(A.10)

which is greater than zero when all the stage-game Nash equilibrium outcomes in Table 4 arepositive (i.e., 2λ − µ − λµ > 0 and 2 − µ − λµ > 0).
Proof of Lemma 8s
2s(1 − µ)−2(1 + λ)(1 + µ) (1 + λ)(2 − µ)2 (1 + µ)sµ2>0=(1 + λ)(2 − µ)2 (1 + µ)

ΠC (s, s) − ΠNd (s, s) =(A.11)

Proof of Proposition 4∗∗ < δ ∗∗ . To this end, because Lemma 7 (i) shows that δ (s) is an increasing
Here, we will show δAAW∗∗function and Lemma 7 (ii) shows δA (s∗∗N ) = δW , it suﬃces to show that neither platform has an∗∗incentive to deviate from (s1 , s2 ) = (s∗∗N − ε, sN − ε).
∗∗C ∗∗ ∗∗Let us first consider (s1 , s2 ) = (s∗∗N , sN ), in which competing platforms gain Π (sN , sN ) in

every period. If a platform deviates from charging s∗∗
N in a period, then manufacturers play the30Nash equilibrium strategy in that period. Thus, the best deviation is charging s∗∗
N ± ε. The
∗∗
∗∗
∗∗
′
deviating platform will gain ΠNd (sN ± ε, sN ) = ΠAN − ε in that period, which is strictly lessthan the payoﬀ that the platform would gain if it had not deviated, as shown in Lemma 8 (i.e.,∗∗ ∗∗C ∗∗ ∗∗
ΠNd (sN , sN ) < Π (sN , sN )). Because even the best deviation does not pay, neither platform has∗∗
an incentive to deviate from (s1 , s2 ) = (s∗∗N , sN ) in every period.
∗∗
Due to continuity of all variables, this argument holds for a neighborhood of (s1 , s2 ) = (s∗∗N , sN ).

Consequently, it turns out that neither platform has an incentive to deviate from (s1 , s2 ) = (s∗∗N −ε, s∗∗
N − ε) in every period. This completes the proof.

Appendix B: Relaxation on deviations by monopoly platform
First, we derive the deviation payoﬀ given a revenue-sharing rule s. Without loss of generality,
we consider the deviation by manufacturer 1 in the second round, i.e., the other manufacturers
charge the collusion price for a given s (i.e., pj = pAC (s) for j = 2, · · · , n). Then, the profit ofmanufacturer 1 can be expressed as presented below.
π1D (p1 ) = {(1 − s)p1 − c} {α − βp1 + (n − 1)γpAC (s)}

(B.1)

}
{
c
/(2β). The deviation profit
α + (n − 1)γpC + β 1−s()2cof manufacturer 1 is computed as πAD (s) = (1 − s)(β + σ)2 α − σ 1−s/(16βσ 2 ).
Solving ∂π1D (p1 )/∂p1 = 0 yields pAD (s) =Next, let us characterize S(s∗ ). The condition for which neither manufacturer has deviation
incentive given the platform will continue to choose s∗ from the next period onwards is given by
πAD (s) − πAC (s) ≤δ{πAC (s∗ ) − πAN (s∗AN )} .
1−δ

(B.2)

We use s̃(δ, s∗ ) to denote the value of revenue-sharing rule s such that condition (B.2) holds withequality. Given a pair of parameters (α, β, σ, c), one can interpret s̃ as a function of discount factor δand the platform-preferred revenue-sharing rule s∗ .
In the same vein, let δA (s, s∗ ) be the value of δ such that condition (B.2) holds with equality
for any given s and the platform-preferred revenue-sharing rule s∗ . That is, δA (s, s∗ ) is an inverse
function of s̃ with respect to δ at a given s∗ . Then, one can infer that condition (B.2) holds if and31

only if δ ≥ δA (s, s∗ ). We have the following lemma on the property of δA (·, ·).
Lemma 9. The following statements hold;(i) δA (s, s∗ ) is increasing in s∗ and decreasing in s.
(ii) δ̃A (s) := δA (s, s) is increasing in s.
∗ .
(iii) δ̃A (s∗AN ) = δW

Proof. (i) First, δA (s, s∗ ) is increasing in s∗ simply because πAC (s∗ ) is decreasing. Second, δA (s, s∗ )is decreasing in s because∂πAD (s) ∂πAC (s)
∂
−
=
∂s
∂s
∂s

{

(
(1 − s) α − σc1−s)2 } {(β + σ)2
1
−
2
16βσ
4σ

}
< 0.

(B.3)

(ii) Formally, δ̃A (s) is computed as follows.

δ̃A (s) = δA (s, s) =πAD (s) − πAC (s)∗πAD (s) − πAN(B.4)

The derivative of δ̃A (s) with respect to s is given by()∂πAD (s)∗ } − {π{πAD (s) − πAN
AD (s) − πAC (s)}
∂s
{
}
2
∗
πAD (s) − πAN()()(β−σ)2 ∗cc1
α − σ 1−s
4σ α + σ 1−s4βσ πAN,={}2∗
πAD (s) − πAN∂δA (s)=∂s(s)∂πAD (s)− ∂πAC
∂s
∂s

(B.5)

(B.6)

which is greater than 0. Therefore, δ̃A (s) is an increasing function.
∗ .
(iii) Substituting s = s∗AN into δ̃A (s) directly yields δ̃A (s∗AN ) = δW

Part (i) of Lemma 9 first shows that the collusion among manufacturers is harder to sustain
as the platform charges a higher equilibrium revenue-sharing rule. Additionally, it implies that
upstream collusion can be sustained more easily if the platform charges a revenue-sharing rulehigher than the equilibrium level in a period. The intuition is that an increase in the future
revenue-sharing rule s∗ decreases the future value of sustaining collusion, whereas an increase in
the current revenue-sharing rule decreases the current net value of deviating from the collusion.

32

This result also implies that collusion among manufacturers is sustainable if and only if s ≥s̃(δ, s∗ ). In other words, s̃(δ, s∗ ) is the smallest revenue-sharing rule at which the upstream collusion
can be sustained.
Similarly, as shown in Lemma 9 (ii), there exists threshold s̃A (δ) below which the collusion
among manufacturers can be sustainable. That is, if s∗ ≤ s̃A (δ), then the manufacturers cansustain their joint profit maximizing collusion. This result generates tension between fosteringand hindering collusion in the platform’s decision on s∗ . Setting a high revenue-sharing rule forsubsequent future periods hinders the formation of upstream collusion overall, but in each period,
the platform should set a low revenue-sharing rule to hinder collusion.
In sum, given the equilibrium revenue-sharing rule s∗ , a maximal set S ∗ (s∗ ) is determined asS ∗ (s∗ ) = {s ∈ [0, 1] | s > s̃(δ, s∗ )}..

(B.7)

Furthermore, given the maximal S ∗ , the platform determines the equilibrium revenue-sharing
rule s∗ to satisfys∗ = arg max ΠA (s, S ∗ ),
s

where
∗

ΠA (s, S ) =




Π

AC (s)

if s ∈ S ∗ΠAN (s)if s ∈
/ S∗

(B.8)

(B.9)

Finally, a pair of the revenue-sharing rule and the maximal set (s∗ , S ∗ (s∗ )) characterizes theequilibrium under consideration. Solving the above system, we can derive the platform-preferred∗ , as shown in Proposition 5.
revenue-sharing rate s∗ and the critical discount factor δA

References
Bernheim, B. D. and Whinston, M. D. (1985), ‘Common marketing agency as a device for facilitating collusion’, The RAND Journal of Economics 16(2), 269–281.
Boik, A. and Corts, K. S. (2016), ‘The eﬀects of platform most-favored-nation clauses on competition and entry’, The Journal of Law and Economics 59(1), 105–134.

33

Choi, S. C. (1991), ‘Price competition in a channel structure with a common retailer’, Marketing
Science 10(4), 271–296.
Cooper, T. E. (1986), ‘Most-favored-customer pricing and tacit collusion’, The RAND Journal of
Economics 17(3), 377–388.
Dobson, P. W. and Waterson, M. (1996), ‘Product range and interfirm competition’, Journal of
Economics & Management Strategy 5(3), 317–341.
Foros, Ø., Kind, H. J. and Shaﬀer, G. (2017), ‘Apple’s agency model and the role of most-favorednation clauses’, The RAND Journal of Economics 48(3), 673–703.
Gaudin, G. and White, A. (2014), ‘On the antitrust economics of the electronic books industry’,
Available at SSRN: https://ssrn.com/abstract=2352495.
Huang, Z. (2017), ‘The power of upstream contracting over downstream collusion’, Job Market
Paper.
Johansen, B. O. and Vergé, T. (2017), ‘Platform price parity clauses with direct sales’, Working
Paper Series No. 1(17).
Johnson, J. P. (2017), ‘The agency model and MFN clauses’, The Review of Economic Studies
84(3), 1151–1185.
Jullien, B. and Rey, P. (2007), ‘Resale price maintenance and collusion’, The RAND Journal of
Economics 38(4), 983–1001.
Lu, L. (2017), ‘A comparison of the wholesale model and the agency model in diﬀerentiated markets’, Review of Industrial Organization 51(2), 151–172.
Maruyama, M. and Zennyo, Y. (2018), ‘Platform most-favored-customer clauses and investment
incentives’, Available at SSRN: https://ssrn.com/abstract=3099305.
Neilson, W. S. and Winter, H. (1993), ‘Bilateral most-favored-customer pricing and collusion’, The
RAND Journal of Economics 24(1), 147–155.
Nocke, V. and White, L. (2007), ‘Do vertical mergers facilitate upstream collusion?’, American
Economic Review 97(4), 1321–1339.
34

Reisinger, M. and Thomes, T. P. (2017), ‘Manufacturer collusion: Strategic implications of thechannel structure’, Journal of Economics & Management Strategy 26(4), 923–954.
Rusu, C. S. (2016), ‘Eturas: Of concerted practices, tacit approval, and the presumption of innocence’, Journal of European Competition Law & Practice 7(6), 396–398.

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