Multi-homing, Product Differentiation
and Competition∗
Jiangli DOU† and Bing YE‡
November 2011
∗ We
would like to thank Emmanuelle Auriol, David Bardey, Jacques Crémer,
Xiaojing Hu, Doh-Shin Jeon, Bruno Jullien, Yassine Lefouili, Jean-Marie Lozachmeur, Lucas Maestri, Alexander Rasch, Sander Renes, Patrick Rey, David Sauer,
Jean Tirole, Yaron Yehezkel and all of the participants for “2011 ECORE Summer School on Market Failure and Market Design”, “26th Annual Congress of the
European Economic Association”, “38th Annual Conference of the European Association for Research in Industrial Economics”and Seminar in Toulouse for helpful
comments and suggestions. All errors are of our own.
† Toulouse School of Economics. Contact information: [email protected].
‡ Department of Economics, Zhejiang University. Contact information: [email protected].
Abstract
This paper investigates the competition between two horizontally differentiated firms (platforms) whose products have partially overlapping functionalities. If firms make their products compatible with each other, consumers
can consume both products and derive utility from the non-overlapping functionalities. We show that the equilibrium compatibility level chosen by firms
is less than socially optimal. In addition, less differentiated products would
induce both firms and social welfare maximizer to allow purchasing both. If
we extend the model to a two-sided framework, the platform with a higher
quality product would enforce single purchase if the network externality is
large enough.
Keywords: Competition, Product Differentiation, Compatibility, Multihoming, Two-Sided Market
JEL classification: D43, D60, L13, L20, L50
1
Introduction
In reality, the environment where some consumers buy both competitive
products available on the market while some others buy only one of them
seems to be reasonable. For example, people who use Microsoft Office to
process text could install TEX on their computer, while there could be users
who use only Microsoft Office or TEX. One consumer may purchase two operating systems, Microsoft Windows and Linux, while one operating system
is enough for most of us. Some people may install Tencent QQ, Skype and
MSN on the same computer in order to chat with different friends, but some
users may use only one of those programs; game lovers may play multiple
games (WII, PS3, XBOX) during one day, but some other users only prefer
a specific one. Even if the functions of these programs overlap, buying an
extra product should increase the utility for consumers when the overlap is
not perfect. However, the larger the overlap, the smaller the extra gain that
could be obtained from installing both programs.
It happens that the consumers are restricted to buy only one product
due to the technology barriers set by firms. For example, the barriers set
between two antivirus software programs, 360 and QQ: when users install
one of them, they will be asked to uninstall the other one.1 Another example
is in the antitrust suit against Microsoft; in 2004, the European Union (EU)
competition commission cited that Microsoft withheld needed interoperability information from rival software companies which prevented them from
making software compatible with Windows; on February 2008, the EU competition commission announced its decision to fine Microsoft Corporation 899
million euros (US $1.35 billion), approximately 1/10 of the company’s net
yearly earnings, for failing to comply with the 2004 antitrust order.
Similar competition happened between two online TV broadcast software platforms: PPStream and PPLive. They are the two major software
programs which provide online TV or movies based on P2P technology in
China. Each of them has more than 200 million subscribers with an average
of 0.5 million users online simultaneously. PPStream broadcasts 17 types of
1
Even without technology barriers, the coexistence of two antivirus softwares does
not work very well because of the working principle for this kind of softwares. It was
announced by some experts that installing two antivirus softwares had a negative influence on computer functional, or even made some functional donot work. For example, the time to boot computer with 360 antivirus installed is 33 seconds, 38 seconds
with Kaspersky installed, while 51 seconds with both softwares installed. Under the
test of mass files copying, it takes 59 seconds for the computer with Kaspersky antivirus, 53 seconds with Rising antivirus, but it takes 278 seconds for the computer
with both antivirus installed to finish the copying process.The analysis can be found:
http://tech.163.com/11/0504/21/73878HBQ000915BD.html
1
programs including 250 channels with more than 10 thousand programs, and
PPLive broadcasts 11 types with 457 channels but fewer programs than PPStream. Many shows are broadcasted by both entities, but each of them has
some specific TV programs. For example, PPStream has more movies, especially recent movies and live broadcast than PPLive, but PPLive has more
sports channels and local TV channels from the mainland of China. In such
a case, we could argue that online TV broadcast programs are both horizontally differentiated and vertically differentiated. Their “quality”essentially
depends not only on the number of TV series and films they can play but
also on the content of the programs. Purchasing a second service enlarges
the number of TV series and films that a consumer can watch on his computer, so that the specific value of buying a second TV broadcast software
comes from the number of TV series and films which are specific to that
software. In 2010, PPStream set technological barriers to prevent consumers
from consuming both services from the same computer.2
The consumers are not restricted to buy only one variant in the example
of operating systems and games, and they have the option to choose between
buying one product only or buying both products (single-homing or multihoming on these products). However, the consumers can only exclusively
choose one of these products when they are restricted to consume only one
due to technological barriers. It also happens that the advertisers cannot
multi-home on a public broadcaster and a private station due to current
regulations in the EU. The “Audiovisual Media Service Directive”regulates
television advertising for all broadcasters in the EU, particularly for public
service broadcasters. For example, in the UK, the BBC must not broadcast
any advertising. The same is true in Sweden where the two public broadcasters (SVT1 and SVT2) do not air any advertisement. In Germany, the
two main public broadcasters (ARD and ZDF) are not allowed to show commercials after 8 pm, on Sundays and on public holidays.
Interesting questions arise: how do two products compete when their
contents partially overlap? Is compatibility where consumers can “multihome”and use both products in the interest of firms? Do the firms’ strategies
on compatibility enhance or harm social welfare?
To answer these questions, we consider an industry where two firms sell
2
The lawsuit between PPStream and PPLive began in April of 2010. First PPLive
copied some programs from PPStream, then PPStream set technology barrier so that
the individual who consumed PPStream would uninstall PPLive automatically. The consumers can not consume PPStream and PPLive at the same time (can not install both
on the same computer). Later on, in October of 2010 they made a joint announcement to
stop the dispute under the pressure by the Chinese Ministry of Industry and Information
Technology.
2
horizontally differentiated goods. The functionalities of their products partially overlap. The two products are located at the two end points of a
Hotelling line and they have some common programs in their products. The
benefit from consuming one product is A and B respectively, and the benefit
is A + B − C if one consumes both, where C represents the common part
or the degree of overlap between these two products. Any one of the firms
can adopt a technology barrier to block its consumers from consuming its
opponent’s product. We show that firm B, the firm which offers the lowest
quality product, is more eager than firm A to make the products incompatible. Therefore the equilibrium compatibility strategy allowing consumers
to multi-home is determined by firm B’s strategy. Furthermore: 1) When
the transportation cost is large, both firms choose to allow multi-homing
when the overlap (i.e, C) is small; only the firm whose product has quality advantage allows multi-homing when the overlap is medium; both firms
would choose to enforce single-homing when the overlap is large. 2) When
the transportation cost is small, both firms allow multi-homing. Because
the two firms do not compete directly when consumers have the option to
multi-home, and each of the firm earns a monopoly profit for its own specific
part (i.e, A − C or B − C). The larger the degree of overlap, the more similar the two products are, hence the lower the profit each firm earns under
multi-homing users. Similarly, the smaller the transportation cost, the less
difference the consumers evaluate between these two products, and the less
the consumers are willing to consume both, hence the lower the profit each
firm obtains.
In section 3.4 and 3.5, we study the optimal policy of regulator who can
enforce either single-homing or multi-homing, given that the prices are set
by the firms. We find that: 1) When the transportation cost is large, if
the regulator aims to maximize social welfare (we call him “social welfare
maximizer”), he would enforce multi-homing when the degree of overlap is
small, but if the regulator aims to maximize consumer surplus (we call him
“consumer surplus maximizer”), he enforces multi-homing for any value of
overlap. 2) When the transportation cost is small, the social welfare maximizer enforces multi-homing while the consumer surplus maximizer enforces
single-homing. It is because that if the transportation cost is small, all consumers consume both products. Even if they have more choices, they have to
pay twice, and the price effect dominates the benefit due to more consumption.
In section 4, we extend the model to a two-sided framework with users
and advertisers contacting via a platform. The number of users has a positive externality on the advertisers’ utility but not vice versa, hence the users
are more valuable to the platforms so that each of the platforms’s optimal
3
strategy is to subsidize the users and charge the advertisers. Different from
the result in the one-sided case, platform A with quality advantage product has stronger preference for multi-homing only if the externality is not
very large. Therefore the equilibrium compatibility strategy allowing multihoming is determined by platform B’s strategy if the externality is small,
but it is determined by platform A’s strategy if the externality is large. Platform A would force the users to single-home if the network externality is large
and the transportation cost is small. The larger the network externality, the
more advertisers care about the number of users on the other side of the
platform, hence the platform with product quality advantage can drive its
opponent out of the market and charge a higher price from the advertisers
in order to get a higher profit. If we use this model to analyze the competition between PPStream and PPLive, neither of them charges a positive
price to the users as the network externality is large so that each platform’s
optimal strategy is charging a positive price to the advertisers and zero price
to the users; PPLive copied some programs from PPStream as this makes
its profit larger; PPStream set the technology barrier to prevent consumers
from multi-homing as it prefers single-homing in this case.
The paper is organized as follows: we first introduce the related literature
in Section 2; then we characterize the basic model of competition and compatibility strategy from both two firms and the regulator’s point of view in
Section 3; in Section 4 we extend the basic model to a two-sided framework;
in Section 5 we analyze the special case of competition between two free TV
channels; finally we conclude in Section 6.
2
Related Literature
Competition and Multi-homing: Gabszewicz and Wauthy (2003) explore price competition between two vertically differentiated firms when consumers can multi-home by purchasing both products (they call it ”joint purchase option” in their paper), in Gabszewicz and Wauthy (2003), they do not
analyze firms strategy on multi-homing. Anderson and Coate (2005) study
how firms (TV channels) compete for advertisers when consumers (viewers)
can only single-home. Ambrus and Reisinger (2006) extend their model by
allowing consumers to multi-home, and compare the equilibrium level of advertisement to the case when consumers can only single-home. In their model,
two products are horizontally different and their basic values fully substitute
for each other. Anderson, Foros and Kind (2010) analyze the Hotelling price
competition where consumers can either single-purchase or multi-purchase.
They show that higher preference heterogeneity increases equilibrium prices
4
and profits with single-purchase, but decreases them with multi-purchase.
Reisinger (2011) analyzes the two-sided market framework where platforms
are differentiated from the users’ perspective but are homogenous for advertisers. In his framework, although there is Betrand competition for advertisers, platforms obtain positive margins in the advertising market. In addition,
platforms profits can increase in the users nuisance costs of advertising.
Competition and Compatibility: With Hotelling model, Doganoglu and
Wright (2006) study two firms’ competing strategies on compatibility and
its efficiency when consumers can (or can not) multi-home by purchasing
both horizontally differentiated products. In their paper, the fundamental
values of products fully overlap. Consumers can get the same utility from
the fundamental value no matter whether they buy both products or just
one. While consumers can enjoy the network externality of both products no
matter which product they buy if firms choose compatibility for their products. Katz and Shapiro (1985) study firms’ decisions on compatibility for
their products where consumers can only single-home in an oligopoly model.
Casadesus-Masanell and Ruiz-Aliseda (2009) extend the idea of Katz and
Shapiro (1985) into settings with two-sided platforms and indirect network
effects, and find that incompatibility gives rise to asymmetric equilibria with
a dominant platform that earns more than it earns under compatibility.
Two-Sided Markets and Multi-homing: When there is no product
differentiation on either side of the market, Armstrong and Wright (2007)
showed that competitive equilibria can be undermined when platforms can
offer exclusive contracts to one side of the market. Exclusive contracts work
by making it easier for a platform to unsettle equilibrium with multi-homing
on one side. Armstrong (2006) shows that if the first group puts more weight
on the network benefits of being in contact with the widest population of the
second group consumers than it does on the costs of dealing with more than
one platform, then the first group agents do not make an either-or decision
to join a platform. Farhi and Hagiu (2007) show that the possibility of subsidization of one side in a two-sided market can lead to fundamentally new
strategic configurations in oligopoly. Gabszewicz and Wauthy (2004) analyze
competition between two platforms where the agents are heterogenous, they
show that in equilibrium, multi-homing takes place on one side of the market only. Moreover, the only equilibrium exhibiting positive profits for both
platforms replicates the collusive outcome. Golafainy and Kovac (2007) provide a formal theory of compatibility choice between sub-sequent generations
of technology in two-sided markets. They classify the compatibility regimes
5
that can occur in two-sided markets. Poolsombat and Vernasca (2006) explore the possibility of partial multi-homing in two-sided market where on
the each side there are two types of agents. They show that in order for an
equilibrium with partial multi-homing on both sides to exist, the network
benefits of high type agents must be sufficiently higher than transportation
costs.
3
The Basic Model: Competition between
Two Firms
3.1
Set Up
Two firms sell two horizontally differentiated products located at the two
end points of the Hotelling line. These products partially overlap, and they
have some common parts. Each firm can set some technology barriers and
make its product incompatible with the product of the other firm. When the
barriers exist, consumers can only choose one of these products. While if the
products are compatible with each other, the consumers have the option to
buy both products. To ease our exploration, we assume that both the fixed
cost and the marginal production cost are zero.
We model the timing of competition between these firms in the following
way:
1. Firms decide the compatibility of their products.
2. Firms set their prices.
3. Consumers make their consumption decisions.
Formally, there is a mass 1 of individuals, uniformly distributed on the
Hotelling line. For each product, each consumer buys either one unit of the
product or none. When firms set the prices PA and PB for the two products,
the utility of the consumer located at x ∈ [0, 1] is

A − tx − PA



B − t(1 − x) − P
B
u=

A + B − C − t − PA − PB



0
if
if
if
if
he only consumes product A,
he only consumes product B,
he consumes both products,
no consumption.
(1)
We denote A and B the gross utility that can be derived from consuming
these two products respectively, C the amount of common parts for these
6
C
A
3
A
A
2
B
Figure 1: The region of parameter values.
two products and the degree of overlapping.3 When the consumer buys one
additional product when he has already bought the product from the other
firm, he can obtain more utility from the specific part of the product (A − C
or B −C ). The “transportation cost”t characterizes the horizontal difference
between these products, or the opportunity cost from not getting the most
preferred content. Before we further our analysis, we make the following two
assumptions.
Assumption 1. C ≤ B ≤ A ≤ 2B + 3C.
The region of parameters in assumption 1 is given by figure 1.
Assumption 1 expresses the facts that a) firm A has a quality advantage
and b) consuming another product will always bring a non-negative marginal
utility. The last inequality and assumption 2 ensure that all consumers are
served and there is effective competition between firms.
Assumption 2. t ≤
A+B
.
3
3
Here we can say that A and B are bundles of products, each of them contains some
different programs, hence the overlap C not only refer to the number of the common
programs but also the contents of programs contained in both bundles of products, for
example: If product A contains program a, b, c, d, e and product B contains program d,
e, f . In this case we have A = 5, B = 3 and their common program C = 2.
7
We will borrow terminology from the two-sided market literature and say
that a consumer “single-homes” when he purchases only one of the two goods
and that he “multi-homes” when he purchases both products. Note however
that before section 4, there are no network effects whatsoever in our model.
3.2
Competition Equilibrium
In subsection 3.2.1, we characterize the equilibrium in the case when consumers must single-home, and 3.2.2 the equilibrium when consumers can
multi-home.
3.2.1
Competition without Multi-homing
If one firm sets technological barriers and deters its consumers from consuming the product of the other firm, each consumer can consume at most one
product. We have the following result:4
Lemma 1. The equilibrium outcome under single-homing is:
1. If
A−B
3
<t≤
A+B
,
3
then
PAs = t+
A−B
,
3
1 A−B
s
DA
= +
,
2
6t
t A − B (A − B)2
πAs = +
+
;
2
3
18t
PBs = t−
A−B
,
3
1 A−B
s
DB
,
= −
2
6t
t A − B (A − B)2
πBs = −
+
.
2
3
18t
2. If t ≤
then
A−B
,
3
firm A becomes a monopoly and covers the whole market,
PAs = A − B − t,
s
DA
= 1,
πAs = A − B − t.
There are two different competition equilibria according to different values
of the transportation cost t. In case 1 of lemma 1, the market is fully covered
and the two firms compete for the marginal user. Firm A whose product has
quality advantage covers the whole market and drives its competitor out of
the market in case 2. Now let us analyze these two cases one by one.
It is straightforward that the market is fully covered if the price pair
charged by two firms is not very large:5
PA + PB ≤ A + B − t.
4
(2)
Here we use DIs and πIs denote the equilibria price and profit where the consumers are
enforced to single-home.
5
Assumption 2 ensures that prices in equilibrium satisfy this condition. Full analysis
can be found in Appendix A.
8
Case 1: If A−B −t < PA −PB < A−B +t, the demand for each product is
positive and the two firms compete for the marginal consumer. An individual
located at x will consume product A rather than product B if and only if
A − tx − PA ≥ B − t(1 − x) − PB .
We can get the demand for each of these two products as following:
DA =
1 A − B − PA + P B
+
,
2
2t
DB =
1 A − B − PA + PB
−
.
2
2t
Firm I maximizes its profit by setting its price:
max πI = PI ∗ DI
PI
It is easy to get
PA = t +
A−B
,
3
PB = t −
A−B
.
3
In order for A − B − t < PA − PB < A − B + t and (2) to be hold, we need
(A − B)/3 < t < (A + B)/3.6
Putting the price formula into the demand function, we have
DA =
1 A−B
+
,
2
6t
DB =
1 A−B
−
.
2
6t
Case 2: If PA − PB ≤ A − B − t, firm A becomes a monopolist and all
individuals consume product A. Therefore, the demand for each product is:
DA = 1 and DB = 0. Firm B can face zero demand even if it sets its price
at 0. Therefore, we have PA = A − B − t. In order for this equilibrium to
.
exist, we need t ≤ A−B
3
3.2.2
Competition with Multi-homing
If the consumers have an option to multi-home, the two firms do not compete for the marginal consumer who is indifferent between consuming either
product, their pricing strategy only affects their own demand. There exist
pure strategy equilibria only if t ≤ A+B−2C
. We have:7
2
6
7
The detailed analysis for segmentation of t appears in Appendix A.
Here we use DIm and πIm denote the price and profit under multi-homing users.
9
Lemma 2.
A−C
2
1. If
<t≤
A−C
,
2
B−C
=
,
2
A mass
2. If
B−C
2
A−C
2t
+
<t≤
B−C
2t
A−C
,
2
3. If t ≤
B−C
2
B−C
,
2
(A − C)2
;
4t
(B − C)2
πBm =
.
4t
m
DA
=
πAm =
− 1 of consumers buy both products.
then
PAm = A − C − t,
B−C
PBm =
,
2
A mass
then
A−C
,
2t
B−C
m
DB
=
,
2t
PAm =
PBm
A+B−2C
,
2
m
DA
= 1,
B−C
m
DB
=
,
2t
πAm = A − C − t;
(B − C)2
m
πB =
.
4t
of consumers buy both products.
all the consumers buy two products, and
PAm = A − C − t,
PBm = B − C − t,
m
DA
= 1,
m
DB = 1,
πAm = A − C − t;
πBm = B − C − t.
For different values of transportation cost t, the equilibria market segmentations are different. In case 1 of lemma 2, some consumers multi-home
on these two products and each firm faces a demand less than 1. In case 2,
all of the consumers buy product A and some of them buy product B. In
case 3, every consumer buys both products.
Case 1: An individual located at x will buy both products if and only if
the utility he obtains from using two products is greater than the utility from
consuming any single product. The following conditions must be satisfied:
A + B − C − t − PA − PB ≥ A − tx − PA ,
A + B − C − t − PA − PB ≥ B − t(1 − x) − PB .
They are equivalent to
1−
A − C − PA
B − C − PB
≤x≤
.
t
t
Therefore we get the demand functions for each firm:
m
DA
=
A − C − PA
,
t
m
DB
=
10
B − C − PB
.
t
There is no direct competition between firms if the users are not restricted
to buy only one product. Each firm’s pricing strategy affects only its own
demand but not its opponent’s. The demand formula is the monopoly demand for each firm’s own specific part A − C or B − C, there is Bertrand
competition on the common part C and both firms charge a zero price on it.
From the firms’ profit maximization problems, we have
PAm =
A−C
,
2
PBm =
B−C
.
2
Hence the demand for each products are respectively:
m
DA
=
A−C
,
2t
m
DB
=
B−C
.
2t
To be sure of the existence of this equilibrium, we need the following condition
to hold:
A−C
B−C
≤
≤ 1;
0≤1−
2t
2t
which is equivalent to (A − C)/2 ≤ t ≤ (A + B − 2C)2.
If (A − C)/2 ≤ t ≤ (A + B − 2C)2, some consumers consume both
products, that is partial multi-homing exists and each firm faces a demand
less than 1. If (A + B − 2C)/2 < t ≤ (A + B)/3, there is no pure strategy
equilibrium under multi-homing.8
Case 2: With the decrease of t, firm A’s market share increases to 1 and
firm B has a demand less than 1. We have (B − C)/2 ≤ t ≤ (A − C)/2.
Case 3: If t ≤ (B − C)/2, all individuals consume both products under
firms’ given pricing strategies. We have the following:
B − C − PB
= 0,
t
A − C − PA
= 1.
t
1−
Hence the pricing strategy for the firms are respectively:
PA = A − C − t,
8
PB = B − C − t.
The proof for non-existence of pure strategy equilibrium appears in Appendix C.
11
3.2.3
Single-homing vs. Multi-homing
The cutoffs of transportation cost t in lemma 1 are different from that in
lemma 2. The firms’ pricing strategy under single-homing depends on the
quality difference A − B. With an increase of A − B, firm A’s profit under
single-homing consumers increases but firm B’s decreases. The larger A − B
is, the larger the quality advantage of firm A and quality disadvantage of
firm B; as a result firm A earns a greater profit and firm B obtains a lower
profit. Under multi-homing, if one firm decreases its price by a small amount,
the demand and profit of the other firm do not change. What is more, under
multi-homing, each firm earns a monopoly profit of the specific part of its
own product (A − C or B − C), it is because that, if consumers have an
option to consume both products at the same time, the firms compete as
Bertrand competition on the common part C, therefore each of them charges
a monopoly price on its specific part.
If we compare the two equilibria when each firm faces a demand less
than 1, that is, case 1 in both lemmas, we note that both firms charge a
).
higher price under single-homing (PAs > PAm , PBs > PBm since t > A−C
2
The two products partially overlap each other, in order to attract users to
consume its product in addition of its opponent, each firm has to set a lower
price, that is, the price only charged on the special part of each product.
Under single-homing, firm A’s pricing strategy is increasing with the quality
difference A − B while firm B’s is decreasing with it. The larger the quality
difference is, the more advantage firm A has, hence the more the inelastic
consumers (or firm A’s original consumers) are willing to pay, therefore it
can charge a higher price to gain a higher profit. Similarly for firm B, the
larger the quality disadvantage, the less the consumers are willing to pay for
product B, hence firm B has to charge a lower price.
3.3
Compatibility Strategy
A firm will prohibit multi-homing if and only if its profit is greater under
single-homing, and the consumers will be allowed to multi-home only if both
firms have greater profits under multi-homing. For simplicity, we only consider the case where9
t≤
A + B − 2C
2
and
B−C
A−B
≤
.
2
3
This implies that t can take the values depicted in figure 2.
The value of B−C
and A−B
only change the threshold for different intervals we analyze,
2
3
the results are the same either we assume B−C
≤ A−B
or vice versa.
2
3
9
12
B−C
2
case 4
A−B
3
case 3
A−C
2
case 2
min{ A+B
, A+B−2C
}
3
2
case 1
t
Figure 2: The possible values of the transportation cost t.
Case 1: If A−C
< t ≤ A+B−2C
, from lemmas 1 and 2, it is easy to show
2
2
that firm A prefers multi-homing if and only if
C ≤A−
√
1
2[t + (A − B)],
3
while firm B prefers multi-homing if and only if
C≤B−
√
1
2[t − (A − B)].
3
(3)
The threshold C is smaller for firm B; therefore there will be multi-homing
if and only if (3) holds.
The larger the common part C, the more similar the two products are,
hence the firms have to set lower prices in order to attract the users to multihome as each user has 0 or 1 consumption. As a result, the firms earn greater
profits only if the common part C is small enough. From lemma 2, firms’
profits under multi-homing is decreasing with C. What is more, firm A has
stronger preference to allow multi-homing than firm B. The threshold of
multi-homing (3) increases with the product quality A and B.
<t≤
Case 2: If A−B
3
homing if and only if
A−C
,
2
from lemmas 1 and 2, firm A prefers multi-
A − B (A − B)2
3
−
,
C ≤A− t−
2
3
18t
while firm B’s prefers multi-homing if (3) holds. The threshold for firm B is
smaller. Therefore multi-homing exists if and only if (3) holds.
From the result of Case 1 and Case 2, firm A has a stronger preference
for multi-homing than firm B if t > A−B
. The preference for multi-homing
3
increases with product quality A and B. Because the pricing strategy under
multi-homing is the monopoly price for the specific part, the higher the
quality, the higher the price firms charge under multi-homing, and the more
consumers they attract, as a result the higher the profit firms earn under
multi-homing.
13
Case 3: If B−C
< t ≤ A−B
, if the firm sets technology barriers to deter its
2
3
rival, firm A covers the whole market but with a lower price and firm B quits,
hence firm B prefers multi-homing. While from lemma 1 and lemma 2, we
can get firm A prefers multi-homing. Because in the case of single-homing,
firm A has to set a lower price in order to drive its opponent out of the
market.
Case 4: When t ≤ B−C
, from lemma 1 and lemma 2, both firms prefer
2
multi-homing, the only difference is that firm B can attract more users under
multi-homing than case 3.
Combining all of the above cases, we get the condition for equilibrium
compatibility strategy: t ≤ (A − B)/3 or
√
A + B − 2C
1
A−B
<t≤
and C ≤ B − 2[t − (A − B)].
3
2
3
√ .10
which is equivalent to t ≤ A−B
+ B−C
3
2
We have the following result for the equilibrium compatibility strategy:
Proposition 1. The firms choose a compatibility strategy which allows some
A−B B−C
consumers to multi-home if t ≤
+ √ .
3
2
Firm A has a stronger preference for multi-homing than firm B when
t > A−B
, and both of them enforce multi-homing when t ≤ A−B
. Hence the
3
3
equilibrium compatibility strategy is determined by firm B’s strategy. What
is more, a decrease in the transportation cost t will lead both firms to value
multi-homing more.
Proposition 2. Whenever firm B accepts multi-homing, so does firm A. An
increase in product qualities A and B will never lead to more prohibition of
multi-homing. A decrease in the transportation cost t makes both firms to
value multi-homing more.
Both firms’ pricing strategy and demand are increasing with the product
quality. Firm A has a quality advantage, hence it charges a higher price
and attracts more consumers than firm B under multi-homing, as a result
firm A has a stronger preference for multi-homing than firm B. The smaller
the transportation cost t, the more severe the competition between two firms
under single-homing, each of them has to charge a lower price, hence both
firms value multi-homing more.
10
√
C ≤ B − 2[t− 13 (A−B)] ⇐⇒ t ≤
A+B−2C
.
2
A−B
3
√ . It is easy to check that
+ B−C
2
Therefore the above condition is equivalent to t ≤
14
A−B
3
+
B−C
√ .
2
A−B
3
√
+ B−C
≤
2
3.4
Social Welfare Maximization
In this subsection, we analyze the social welfare maximizer’s preference for
multi-homing. The social welfare maximizer can put pressure on firms to
enforce multi-homing but cannot set prices for the products. The firms set
the prices for their own products. We have the following:
Proposition 3. The social welfare maximizer would force the firms to allow
multi-homing if either
)
(r
7
A−C
(A − B),
t ≤ max
12
2
or
(r
t ≥ max
A−C
7
(A − B),
12
2
)
and C ≤
q
3(A + B) − 4t − 4t 14 −
7(A−B)2
48t2
6
If the transportation cost t is very large, the social welfare maximizer
prefers the users to multi-home when the common part C is not very large.
The larger C is, the more similar the two products are, therefore multihoming induces a larger social welfare loss as each consumer faces a 0 or 1
consumption. While if the transportation cost is not very large, the social
welfare maximizer prefers the users to multi-home because prohibition of
multi-homing induces a social welfare loss due to under consumption.
We use the un-weighted sum of consumer surplus and firms’ profits to
denote the total welfare. As in the above subsection, we consider different
values of transportation cost t one by one.
< t ≤ A+B−2C
, from lemmas 1 and 2, the social welfare
Case 1: If A−C
2
2
under single-homing and multi-homing consumers are respectively:
Z 1 + A−B
Z 1
2
6t
s
SW =
[A − tx]dx +
[B − t(1 − x)]dx
1
+ A−B
2
6t
0
A+B
t 5(A − B)2
− +
.
=
2
4
36t
SW
m
1− B−C
2t
Z
=
0
Z
[A − tx]dx +
A−C
2t
[A + B − C − t]dx
1− B−C
2t
1
3C 2
3(A + B)
3(A2 + B 2 )
+
[B − t(1 − x)]dx =
+ 1−
C+
.
A−C
4t
4t
8t
2t
Z
15
.
The social welfare maximizer would enforce multi-homing if and only
if the social welfare under multi-homing is greater than that under singlehoming. It is easy to show that SW m ≥ SW s if and only if
√
3(A + B) − 4t − 4t ∆
C≤
,
(4)
6
q
2
7
where ∆ = 14 − 7(A−B)
(A − B), SW m ≥ SW s
.
If
∆
≤
0,
that
is,
if
t
≤
48t2
12
always
q holds, hence the social welfare maximizer enforces multi-homing; if
7
(A − B), multi-homing is socially desirable when (4) holds. From
t ≥ 12
our assumption that each user has 0 or 1 consumption, the larger C, the less
beneficial for the consumers is multi-home, hence the social welfare maximizer
prefers the consumers to multi-home if C is not very large.
Case 2: If A−B
< t ≤ A−C
, SW s is the same as in case 1. While if
3
2
consumers multi-home, no consumer consumes only product B, all the users
and 1 consume both products.
with type between 1 − B−C
2t
SW
m
Z
1− B−C
2t
=
Z
1
[A + B − C − t]dx
[A − tx]dx +
1− B−C
2t
0
t
3(B − C)2
+A− .
8t
2
=
We can show that the social welfare maximizer would enforce multihoming if and only if
√
3B − 4t ∆
C≤
,
(5)
3
2
− 3(A−B)
+ 83 . We can show that for this value of transwhere ∆ = 5(A−B)
24t2
4t
portation cost t, ∆ < 0, hence the social welfare under multi-homing is larger
than that under single-homing. Multi-homing is socially optimal.11
Case 3: If B−C
<t≤
2
homing, then we have
A−B
,
3
firm A covers the whole market under single-
1
3(B − C)2
t
SW s = A − t ≤ SW m =
+A− .
2
8t
2
Hence the social welfare maximizer prefers consumers to multi-home. In this
case, all the consumers would like to consume product A and some of them
2
3
5 A−B
∆ = 5(A−B)
− 3(A−B)
+ 38 < 0 ⇐⇒ 24
( t − 95 )2 − 10
< 0, which is equivalent to
24t2
4t
A−B
9
6
A−B
−
<
,
that
is
<
3
which
holds
for
this
value
of
t.
t
5
5
t
11
16
would like to also consume product B, preventing multi-homing prevents
some of the consumers from consuming product B which induces a welfare
loss, as a result the social welfare maximizer prefers the consumers to multihome.
Case 4: If t ≤
homing, then
B−C
,
2
all the consumers consume both products under multi-
1
SW s = A − t,
2
SW m = A + B − C − t
The social welfare is larger under multi-homing than under single-homing.
Hence the social welfare maximizer enforces multi-homing.
3.5
Consumer Surplus Maximization
The consumer surplus is the difference between social welfare and firms’
profits. If we consider from the competition policy issue that there exists
a regulator who aims to maximize consumer surplus, we have the different
result from the firms and social welfare maximizer:
Proposition 4. The regulator who aims to maximize consumer surplus would
< t ≤ A+B−2C
or
enforce multi-homing if either A−C
2
2
A−B
A−C
<t≤
and C ≤ B − 4t −
3
2
r
2(t + A − B)2 −
16(A − B)2
.
9
We compare the consumer surplus for different cases one by one.
Case 1: If A−C
< t ≤ A+B−2C
, partial multi-homing exists, the consumer
2
2
surplus under single-homing and multi-homing are respectively:
A + B 5t (A − B)2
− +
,
2
4
36t
C2
A+B
A2 + B 2
=
+ [1 −
]C +
.
4t
4t
8t
CS s =
CS m
By comparison, the consumer surplus under multi-homing is larger than that
under single-homing. If consumers have the option to multi-home, some
consumers buy both products but they have to pay twice for prices. In this
case the consumer surplus increase due to more consumption dominates the
price effect.
17
Case 2: If A−B
< t ≤ A−C
, the consumer surplus under single-homing is
3
2
the same as in case 1. The consumer surplus under multi-homing is:
CS m =
(B − C)2
t
+C + .
8t
2
The consumer surplus under multi-homing is larger than that under singlehoming if and only if
r
16(A − B)2
.
C ≤ B − 4t − 2(t + A − B)2 −
9
Case 3: If B−C
< t ≤ A−B
, the consumer surplus under single-homing and
2
3
multi-homing are respectively:
t
CS s = B + ,
2
CS m =
(B − C)2
t
+C + .
8t
2
The consumer surplus is larger under single-homing than under multi-homing.12
Case 4: If t ≤
B−C
,
2
we have the following consumer surplus
t
CS s = B + ,
2
CS m = C + t.
The consumer surplus under single-homing is larger.
A decrease in transportation cost t makes the consumer surplus under
single-homing larger, because under multi-homing, the consumers have to
pay to both firms, which is larger than the price under single-homing. Even
if consumers have more choices under multi-homing, this is dominated by
the price effect. If there exists a regulator whose objective is to maximize
the consumer surplus, the smaller the transportation cost, the less he values
multi-homing.
3.6
Comparison
If we compare the thresholds of compatibility strategy for the social welfare maximizer and firms, we get: 1) for the parameter values when the
firms allow multi-homing on the consumers’ side, the social welfare maximizer also prefers multi-homing; 2) there exist some circumstances such
that the firms prefer the consumers to single-home while the social wel< t ≤
fare maximizer prefers them to multi-home, such as, when A−B
3
12
CS s ≥ CS m ⇐⇒ B − C ≥
(B−C)2
,
8t
which holds as we have t >
18
B−C
2 .
q
√
7
max{ 12
(A − B), A−C
} and C > B − 2[t − 13 (A − B)]. We have the
2
following:
Proposition 5. Whenever the compatibility strategy chosen by firms is multihoming, the social welfare maximizer would enforce multi-homing.
Now letqus compare the thresholds for different cases one by one. If
7
(A − B), A−C
}, the social welfare maximizer prefers multit > max{ 12
2
√
2
∆
homing if C < 3(A+B)−4t−4t
(where ∆ = 41 − 7(A−B)
); firm A whose
6
48t2
√
product has quality advantage permits multi-homing if C < A − 2[t +
1
(A − B)]; the firm with quality disadvantage B permits multi-homing
if
3
√
√
3(A+B)−4t−4t ∆
1
to show that
≥
C < B − 2[t − 3 (A − B)]. It is easy
6
√
√
√
∆
A − 2[t + 13 (A − B)] and 3(A+B)−4t−4t
≥ B − 2[t − 31 (A − B)],13 Hence
6
the social welfare maximizer has a stronger preference for multi-homing than
the two firms. That is because the social welfare maximizer cares not only
about the firms’ profit, but also the consumer surplus, which is greater under
multi-homing than that under single-homing for this value of transportation
cost t; preventing consumers from multi-homing decreases consumer surplus
by inducing less consumption.
q
7
< t ≤ max{ 12
(A − B), A−C
}, the social welfare maximizer
If A−B
3
2
prefers multi-homing no matter √
the degree of overlapping C; the two firms
prefer multi-homing if C < B − 2[t − 31 (A − B)].
, both firms and the social welfare maximizer prefer multiIf t ≤ A−B
3
homing. Because the transportation cost is small enough so that it is very
cheap to consume another product.
To summary, we use table 1 to describe the firms’ equilibrium behavior,
the social welfare maximizer’s and the consumer surplus maximizer’s preference.
From the figure in appendix D, the social welfare difference between
single-homing and multi-homing is always smaller than the profit difference
of the two firms for fixed A, B and t. That is to say, the social welfare loss
due to preventing multi-homing is always larger than the profit loss. As a
result, the equilibrium compatibility strategy chosen by firms is less than
socially optimal. Because the social welfare maximizer also cares about the
√
√
√
√
13 3(A+B)−4t−4t ∆
≥ B − 2[t − 13 (A − B)] if and only if ∆ ≤ (3−2 2)(A−B)
−1+
6
√
√
√
√ 4t
7(A−B)2
(3−2 2)(A−B)
(A−B)2
3 2
3 2 2
1
≤ [
− 1 + 2 ] , that is (58 − 36 2) t2
+ 12 ∗
2 √⇐⇒ 4 −
48t2
√ 4t
A−B
(13 2 − 18) t + 252 − 144 2 ≥ 0, which holds in any case. Similarly, we can prove
√
√
∆
that 3(A+B)−4t−4t
≥ A − 2[t + 31 (A − B)].
6
19
[0, C1 ]
[C1 , C2 ]
[C2 , C3 ]
[C3 , B]
]
(0, A−B
3
( A−B
, A−C
]
3
2
( A−C
, t̄]
2
M,M,S
M,M,S
M,M,S
M,M,S
M,M,M
M,M,S
S,M,S
S,M,S
M,M,M
M,M,M
S,M,M
S,S,M
Table 1: We summarize the equilibrium level of firms’ compatibility strategy,
social welfare maximizer’s preference of multi-homing and consumer surplus
maximizer’s preference. The items in the first horizontal line denote the value
of transportation cost t, the items in the first column denote the overlap C.
We use M to denote multi-homing and S to denote single-homing.
C1 =
√
√
16(A−B)2
2
∆1 = 2(t + A − B) −
, C2 = B − 2[t − A−B
],
B − 4t − ∆1 with
9
3
√
2
7(A−B)
3(A+B)−4t−4t ∆2
with ∆2 = 41 − 48t2 , t̄ = min{ A+B
, A+B−2C
}. When
C3 =
6
3
2
A−B
A−C
<
t
≤
and
C
≤
C
≤
B,
both
the
firms’
equilibrium
behavior
2
3
2
and consumer surplus maximizer’s are single-homing while the social welfare
maximizer would enforce multi-homing, it is because the firms’ equilibrium
behavior is determined by firm B’s strategy, multi-homing is profitable for
firm A in this case.
consumers surplus in addition to the firms’ profits. The loss in consumer surplus due to multi-homing is smaller than increase in profits, which induces a
welfare increase.
When the over-lap C is fixed, both firms prefer the users to single-home
if the differentiation t is large enough; firm A has a stronger preference for
multi-homing than firm B when t is in the middle, while if t is not very large,
firm B has stronger preference for multi-homing than firm A. It is because
that the decrease of t makes the competition more severe when single-homing,
as firm B has the quality disadvantage, it will be driven out of the market
and get nothing under single-homing. So firm B prefers multi-homing more
when t is small. While if t is large enough, the quality advantage of A is
not very large because the competition is not severe, it can not dominate
the benefit from additional users when multi-homing, so firm A has more
incentive to allow multi-homing when t is large.
When the product quality A increases, both firms prefer allowing consumers to multi-home more. Because the quality disadvantage of B becomes
more severe in the competition case, so firm B prefers multi-home more;
firm A prefers multi-home more because its price and demand is increasing with its own product quality under multi-homing, hence it gets more
profit from additional consumers in the case of multi-homing. When the
20
transportation cost t decreases, both of them value multi-homing more. The
smaller the transportation cost t, the less difference the consumers evaluate
between these two products; as a result, the less the inelastic consumers are
willing to pay; hence the competition between these two firms are more severe, therefore each of them earns a less profit when they compete under
single-homing.
Now let us compare the results with three special cases.
If A = B = C, the two firms sell the same product, the equilibrium under
single-homing in lemma 1 is the Hotelling competition equilibrium of case
1 for any positive transportation cost t; neither of them covers the whole
market to earn a positive profit; case 2 appears when t = 0 and each of
them earns a zero profit. From the character of the products, multi-homing
does not exist. Both firms and social welfare maximizer would choose singlehoming.
If A > B = C, product A includes all the contents in product B, product A is an upgrade for B. In this case, firm B definitely prefers single-homing
if t > 31 (A − B), from lemma 2, firm B earns 0 profit under multi-homing.
When t ≤ 31 (A − B), firm A prefers multi-homing as we analyzed, and firm B
is indifferent between single-homing and multi-homing, as it earns zero profit
under both cases.
If A > B and C = 0, there is no overlap between these two products.
Both firms and social welfare maximizer prefer multi-homing, which we can
see from the figure in appendix D when C = 0, both the profits and social
welfare under single-homing are smaller than that under multi-homing.
3.7
An Example
We use a numerical example to explain the relationship between the equilibrium compatibility strategy of the firms, social welfare maximizer and the
value of the transportation cost t, the value of the amount of common content
of the products, respectively. Assume A = 10, B = 8, and the transportation
cost t = 5. In this case, neither one of the firms covers the whole market
either under single-homing or multi-homing. We have
πAs − πAm =
5 2
2
(10 − C)2
+ +
−
,
2 3 45
20
5 2
2
(8 − C)2
− +
−
.
2 3 45
20
So firm A prefers multi-homing if and only if
2 √
C ≤ 10 − (5 + ) 2 ' 2.07,
3
πBs − πBm =
21
and firm B prefers multi-homing if and only if
2 √
C ≤ 8 − (5 − ) 2 ' 1.93.
3
Whenever firm B prefers multi-homing, so does firm A. The social welfare
under single-homing and multi-homing are respectively:
5 1
+ ,
4 9
3C 2 17
123
=
− C+
.
20
10
10
SW s = 9 −
SW m
Multi-homing is socially desirable if and only if
q
68
34 − 20 300
C≤
' 4.08.
6
The social welfare maximizer has a stronger preference for multi-homing than
the firms. This illuminates the result in proposition 5.
Now let us consider another case; we fix C = 6.
2
4
t 2
+ + − ,
2 3 9t t
2
1
t 2
πBs − πBm = − + − .
2 3 9t t
πAs − πAm =
From calculation, firm A permits multi-homing if t ≤ 2.16 and firm B
permits multi-homing if t ≤ 2.08. The social welfare under these two cases
are respectively:
SW s = 9 −
t
5
+ ,
4 9t
SW m =
15
+ 6.
2t
The social welfare under multi-homing is larger than that under singlehoming no matter the value of t.
4
4.1
Two-Sided Markets
Set Up
In this section, we extend the basic model to the two-sided case. For simplicity, we only consider two groups of persons who contact through one
22
platform. That is to say, we consider the case where end users and advertisers contact through TV channels.14 The users join the platform in order to
consume the contents, while sometimes they get information of new content
through the advertisement. The advertisers join the platform in order to
make more persons know about their product. The number of users has a
positive network externality on the advertiser’s utility. The more individuals
who know the advertisement, the larger probability they have more sales. If
the platform decides to exclude its opponent, it only works on the end users’
side. We denote A and B as the two platforms, they locate at the two end
points of the Hotelling line, side 1 the end users and side 2 the advertisers.
Both the end users and advertisers are uniformly distributed on [0, 1]. When
platforms charge prices PA and PB from the users’ side, the utility of an user
located at x ∈ [0, 1] is denoted by (1), which is the same as in the basic
model.
When platforms charge prices QA and QB from the advertisers, the utility
of an advertiser located at y ∈ [0, 1] is15


if it only joins platform A,
βnA − ty − QA
u = βnB − t(1 − y) − QB
if it only joins platform B, (6)


β(nA + nB − nAB ) − t − QA − QB if it joins both platforms.
We denote β as the inter-group externality, nA , nB and nAB the number of
users who join platform A, platform B and both platforms respectively, nAB
equals to 0 if no users multi-home on both platforms. The advertisers only
care about the total number of users who see the advertisement, but not the
number of times they see it. If there is no multi-homing user, the advertiser’s
utility from joining both platforms is the sum of utility from joining two
platforms, and there is no direct competition on the advertisers’ side. If
multi-homing users exist, the platforms compete also on the advertisers’ side
as the larger number of users on the other platform, the less willingness the
advertisers to join one more platform. For simplicity, there is no basic value
from joining the platform for the advertisers; they only care about the number
14
In reality, the users join the platform in order to consume the programs. They care
about the number of contents, and there exists another side of the market: the content
provider. Since the platform’s profit comes from the advertisers, it cares the number of
advertisers, while the advertisers care about the number of users. So for simplicity, we
only consider the users and advertisers in this model.
15
Here we use fixed advertising fee in stead of access fee per user so that it is easier to
denote the network externality. If we use access fee per user, the advertisers do not care
about the number of users, they will join the platform if the benefit is larger than the
access fee.
23
of users on the other side.16 The inequality A ≥ B ≥ C in assumption 1
also holds in the two-sided case. In order to ensure that the market on the
advertisers’ side is fully covered and the two platforms do compete on the
advertisers’ side, we make the following assumption
Assumption 3. β ≥ max{2t, 3(A + B − 2C)}.
We consider a similar three-stage game as in the one-sided market case. In
the first stage, the two platforms decide the compatibility of their products.
In the second stage, the platforms set their prices on the users’ side and
advertisers’ side simultaneously. In the last stage, the users and advertisers
make their decisions: advertisers decide on which platform they want to
advertise, and users decide to join the platform or not, they can join at most
one platform when the platforms are not compatible with each other.
We use this setup for the two-sided market framework by adding one more
side to the basic one-sided market model. The users’ behavior is similar as
in the basic model. We get the basic model if β = 0.
4.2
4.2.1
Competition without Multi-homing Users
The Users’ Choice
The advertisers can join both platforms, while the end users can join at
most one platform if the two platforms are not compatible with each other.
The market segmentation is similar as in 3.2.1 without advertisers: The two
platforms compete on the users’ side if the transportation cost t lies on the
middle; one of the platforms covers the whole market if the transportation
cost is small enough.
4.2.2
The Advertisers’ Choice
The users cannot multi-home on these two platforms; as a result, if one
advertiser joins one more platform, the increasing number of users who see
its advertisement is all of the users on the additional platform. There is no
direct competition between platforms on the advertisers’ side; the pricing
strategy on the advertisers’ side only affects its own demand. We use mI to
denote the number of advertisers on platform I. It is easy to show that the
16
There maybe some nuisance cost for advertisement when the user is watching TV.
But on the other hand, some advertisement on the TV channel provides information
about products, which is useful for the users. The above two effects counteract to each
other, we assume the users get zero utility from the advertisement.
24
number of advertisers joining the two platforms are respectively:
βnA − QA
βnB − QB
mA = min
,1 ,
mB = min
,1 .
t
t
4.2.3
Pricing Strategy
For different value of transportation cost t, we have different equilibrium
result denoted by lemma 5.17 If the transportation cost t is large enough, the
β
, which is larger than 1, and some advertisers
total number of advertisers is 2t
multi-home. As the two platforms compete on the users’ side, the number of
users on each platform is less than 1; for the advertisers with the middle type,
the increasing number of users seeing their advertisements by joining one
additional platform is larger than the transportation cost
√ and advertisement
fee. We have PA < P√B and QA > QB if 2t < β < 6t, while PA > PB
and QA < QB if β > 6t. Not the same as in the one-sided market, if the
externality β is very large, platform A whose product has quality advantage
charges a higher price than platform B on the users’ side and a lower price on
the advertisers’ side; if the externality β is not very large, platform A charges
a lower price on the users’ side and a higher price on the advertisers’ side
than platform B. From lemma 5, it is easy to check that PA < 0, PB < 0, this
is a normal result for the classical two-sided market models, if the advertisers
care about the number of users very much, the platform’s best strategy is to
subsidize the users and charge the advertisers.
4.3
Competition with Multi-homing Users
If the two platforms are compatible with each other, the end users have an
option to join both of them. If one advertiser joins platform B in addition
to platform A, the increasing number of users seeing its advertisement is the
users who do not join platform A, which equals 1 − nA when the market is
fully covered on the users side. Hence the larger is nA , the less willingness
the advertisers to join platform B in addition to platform A. The maximum
number of users for the advertisers network benefit is 1.
4.3.1
The Users’ Choice
2
We consider only the case 3t ≤ A + B + β2t when the market on the users’
side is fully covered so that multi-homing is possible. As in the one-sided
17
The result and detailed analysis can be found in appendix E.
25
B
A
case, the end users with type between 1 − B−C−P
and A−C−P
will join both
t
t
platforms, hence the number of users in the two platforms are respectively
A − C − PA
B − C − PB
nA = min
,1 ,
nB = min
,1 .
t
t
4.3.2
The Advertisers’ Choice
If there exist users multi-homing on both platforms, the extra utility of an
advertiser putting advertisements on one additional platform is the utility
from users who do not join this platform. If the number of users on one
platform is close to 1, for example if nA is close to 1, joining platform B
in addition to platform A is not profitable for the advertisers, as the extra
utility from the users not joining platform A is small, but they have to pay
twice the transportation cost and advertisement fee. Therefore there exist
some advertisers that join both platforms only if the number of users on each
platform is not very close to 1. We have different numbers of advertisers for
different values of parameters:18
If PA > A − C − t, PB > B − C − t, βPA − tQB ≤ t2 − βt + β(A − C),
βPB − tQA ≤ t2 − βt + β(B − C) and β(PA + PB ) − t(QA + QB ) ≥ t2 −
2βt + β(A + B − 2C), there exist some advertisers that multi-home on both
platforms, and the number of advertisers are respectively:
mA =
β(1 − nB ) − QA
,
t
mB =
β(1 − nA ) − QB
t
If PA > A−C−t, PB > B−C−t, βPA −tQB ≤ t2 −βt+β(A−C), βPB −tQA ≤
t2 − βt + β(B − C) and β(PA + PB ) − t(QA + QB ) ≤ t2 − 2βt + β(A + B − 2C),
no advertiser joins both platforms, then
mA =
β(nA − nB ) − QA + QB + t
,
2t
mB =
β(nB − nA ) − QB + QA + t
2t
If PA ≤ A − C − t, PB > B − C − t, then
mA =
β(1 − nB ) − QA + QB + t
,
2t
mB =
−β(1 − nB ) + QA − QB + t
2t
If PA ≤ A − C − t, PB ≤ B − C − t, then
mA =
18
t − QA + QB
,
2t
mB =
Detailed analysis appears in the appendix E.
26
t + QA − QB
2t
4.3.3
Pricing Strategy
The equilibria pricing strategy depends on the value of the parameters. If
the transportation cost t is large enough so that the number of users on
each platform is much smaller than 1, there exist two types of pure strategy
equilibria, one with some multi-homing advertisers and one without multihoming advertisers. If the transportation cost t is not very large, there exists
only one pure strategy equilibrium with all advertisers single-homing. The
equilibria results and detailed proof can be found in Appendix E.
From lemma 5 and lemma 6, there exist some advertisers that multi-home
on both platforms only if both the externality β and the transportation cost
t is large enough. If the transportation cost is large, the number of users on
each platform is much smaller than 1, so that joining an additional platform
gives more utility to the advertisers. The larger the externality β, the more
utility the advertisers obtain by contacting more users. From lemma 5 and
lemma 6, the platforms set negative prices on the users’ side if network
externality is large enough. In the next section, we give the constraint of
non-negative prices to analyze the case when the platforms charge a 0 price
on the users’ side.
5
5.1
Competition between Two Free Platforms
Set Up
We have the same set up as section 4 except that the users pay a 0 price. An
user located at x ∈ [0, 1] gets utility:


if he only joins platform A,
A − tx
u = B − t(1 − x)
(7)
if he only joins platform B,


A + B − C − t if he joins both platforms.
When platforms charge prices QA and QB from the advertisers, the utility
of an advertiser located at y ∈ [0, 1] is given by (6).
5.2
5.2.1
Competition without Multi-homing Users
The Users’ Choice
Similarly as in section 4, the market on the users’ side is fully covered and
shared by these two platforms if A − B < t ≤ A + B, the number of users on
27
the two platforms are respectively:
A−B+t
B−A+t
nA =
nB =
2t
2t
If t ≤ A − B, platform A covers the whole market of the end users’ side
and platform B quits.
5.2.2
The Advertisers’ Choice
The advertiser joins the platform if it obtains a positive utility on this platform. The numbers of advertisers on the two platforms are respectively:
βnB − QB
βnA − QA
,1 ,
mB = min
,1 .
mA = min
t
t
5.2.3
Pricing Strategy
If the users join the platform for free, there is no pricing effect on the users’
side. Platform A which provides a higher quality product has an advantage
on the advertisers’ side; it charges a higher price and obtains more demand
than platform B.19
Lemma 3. The equilibria outcome with single-homing users are as follows:
√
β+ β 2 +16β(A−B)
1. If
< t ≤ A + B, then
8
β
(A − B + t),
mA =
4t
β
mB =
QB = (B − A + t),
4t
√
β+ β 2 +16β(A−B)
2. If A − B < t ≤
,
8
QA =
β
(A − B + t),
4t2
β
(B − A + t),
4t2
β2
(A − B + t)2 ;
16t3
β2
πB =
(B − A + t)2 .
3
16t
πA =
then
β
(A − B + t) − t,
mA = 1,
2t
β
β
QB = (B − A + t),
mB = 2 (B − A + t),
4t
4t
QA =
β
(A − B + t) − t;
2t
β2
πB =
(B − A + t)2 .
16t3
πA =
3. If t ≤ A − B, then
QA = β − t,
mA = 1,
πA = β − t.
Some advertisers multi-home on both platforms if the transportation
cost t is large enough so that the number of users on each platform is smaller
than 1.
19
The detailed analysis can be found in Appendix F.
28
5.3
5.3.1
Competition with Multi-homing Users
The Users’ Choice
The users will join both platforms if positive utility is obtained on each
platform. We only consider the case t < A + B − 2C so that there exist
pure strategy equilibrium on the users’ side. The number of users in the two
platforms are respectively:
nA =
A−C
t
nB =
B−C
t
If t > A − C. If B − C < t ≤ A − C, all of the users join platform A and
some of them join platform B. If t ≤ B − C, each user joins both platforms.
5.3.2
The Advertisers’ Choice
If there exist multi-homing advertisers, the advertisers with type between
1 − β(1−nAt )−QB and β(1−nBt )−QA will multi-home on both platforms if the
number of users on each platform is not very close to 1. In this case, the
number of advertisers in each platform are respectively
mA =
β(1 − nB ) − QA
t
mB =
β(1 − nA ) − QB
t
In order to ensure there exist multi-homing advertisers, the following conditions should be satisfied: 0 < 1 − β(1−nAt )−QB ≤ β(1−nBt )−QA ≤ 1, which
is equivalent to tQA > tβ − β(B − C) − t2 , tQB > tβ − β(A − C) − t2 ,
t(QA + QB ) < 2tβ − β(A + B − 2C) − t2 . Combining with nA < 1, nB < 1,
we need t > A − C.
With the decrease of t, nA becomes close to 1, no advertiser joins both
platforms, the platforms compete on the marginal advertiser and the number
of advertisers in each platform are respectively:
mA =
β(nA − nB ) − QA + QB + t
2t
β(nB − nA ) − QB + QA + t
2t
In this case, we need t(QA + QB ) ≥ 2tβ − β(A + B − 2C) − t2
If B − C < t ≤ A − C, we have nA = 1, nB < 1, no advertiser joins both
platforms, the number of advertisers in these two platforms are respectively
mB =
mA =
β(1 − nB ) − QA + QB + t
2t
29
−β(1 − nB ) + QA − QB + t
2t
If t ≤ B − C, we have nA = 1, nB = 1, the two platforms are homogenous to advertisers and they compete as classical Hotelling competition, the
number of advertisers attracted by each platform are:
mB =
mA =
5.3.3
t − QA + QB
2t
mB =
t + QA − QB
2t
Pricing Strategy
If the platforms are compatible with each other, there exists pure strategy
equilibrium if and only if t ≤ A+B −2C, we have the following equilibrium:20
√
√
β− β 2 −2β(A+B−2C)
β− β 2 −3β(A+B−2C)
Lemma 4.
1. If
≤t≤
or
2
3
√
β+ β 2 −3β(A+B−2C)
≤ t ≤ A + B − 2C, there exist two equilibria, one
3
with and the other one without multi-homing advertisers,
β
∗ (t − B + C);
2t2
β
mB = 2 ∗ (t − A + C).
2t
β
(t − B + C),
2t
β
QB = (t − A + C),
2t
mA =
QA =
or
β
1
β
(A − B),
mA = + 2 (A − B);
3t
2 6t
1
β
β
mB = − 2 (A − B).
QB = t − (A − B),
3t
2 6t
√ 2
√ 2
β− β −3β(A+B−2C)
β+ β −3β(A+B−2C)
2. If
≤t≤
, there exist some multi3
3
homing advertisers in the only equilibrium,
QA = t +
β
β
(t − B + C),
mA = 2 ∗ (t − B + C);
2t
2t
β
β
QB = (t − A + C),
mB = 2 ∗ (t − A + C).
2t
2t
√
β− β 2 −2β(A+B−2C)
3. If A − C < t ≤
, there is only one equilibrium with
2
all advertisers single-homing,
QA =
β
(A − B),
3t
β
QB = t − (A − B),
3t
QA = t +
20
1
β
+ 2 (A − B);
2 6t
1
β
mB = − 2 (A − B).
2 6t
mA =
The proof of lemma 4 can be found in Appendix F.
30
4. If B − C < t ≤ A − C, all advertisers single-homing in the only equilibrium,
β
(t − B + C),
3t
β
QB = t − (t − B + C),
3t
1
β
+ 2 (t − B + C);
2 6t
1
β
mB = − 2 (t − B + C).
2 6t
QA = t +
mA =
5. If t ≤ B − C, all advertisers single-homing in the only equilibrium,
1
mA = ;
2
1
mB = .
2
QA = t,
QB = t,
There exist some advertisers multi-homing on both platforms only if the
number of users in an additional platform is large enough and the network
externality β is large enough. Multiple equilibria exist if the additional utility
from joining one more platform is not
√ very large. There is no pure strategy
equilibrium if A + B − 2C ≤ t ≤
5.4
β+
β 2 −2β(A+B−2C)
.
2
Compatibility Strategy
The platform permits users to multi-home only if it obtains a larger profit
with multi-homing users than single-homing users. We only analyze the case
t ≤ A + B − 2C so that there exists pure strategy equilibrium.
√
β−
β 2 −2β(A+B−2C)
2
√
β−
β 2 −3β(A+B−2C)
3
√
β+
β 2 −3β(A+B−2C)
≤t≤
or
≤
Case 1: If
3
t ≤ A + B − 2C, there exist two equilibria with multi-homing users. Under
the equilibrium with multi-homing advertisers, from lemma 3 and lemma 4,
it is easy to check that both platforms prefer multi-homing if and only if:
C≥
A+B−t
,
2
which is equivalent to t ≥ A + B − 2C, under which condition there is no
pure strategy equilibrium. Therefore both platforms choose compatibility
strategy to allow multi-homing in this case. Because each of them attracts
more users under multi-homing, as a result they attract more advertisers to
obtain a higher profit.
31
Under the equilibrium result without multi-homing advertisers, we have:
t
β
+ (A − B) +
2 3t
t
β
∆πB = πBm − πBs = − (A − B) +
2 3t
∆πA = πAm − πAs =
β2
(A − B)2 −
18t3
β2
(A − B)2 −
18t3
β2
(A − B + t)2 ,
16t3
β2
(B − A + t)2 .
16t3
The compatibility decision does not depend on the degree of overlap C.
Platform A has a stronger preference for multi-homing only if ∆πA ≥ ∆πB ,
which is equivalent to β ≤ 8t3 . Different from the result in the one-sided case,
platform A whose product has a quality advantage has a stronger preference
for multi-homing only if the externality β is not very large. The equilibrium
compatibility strategy is determined by platform B’s strategy only if β is
not very large. While it is determined by platform A’s strategy if β is large
enough. The platforms earn no profit from the users’ side. The larger the
externality, the more the advertisers care about the number of users, as a
result the higher the price the platforms can charge from the advertisers’
side, hence the greater the profit platform A obtains. From lemmas 3 and 4,
platform A’s charges a higher price under single-homing users than that under
multi-homing users if the externality β is large enough.
√
β−
√
β 2 −3β(A+B−2C)
3
β+
β 2 −3β(A+B−2C)
≤t≤
, there exist some
Case 2: If
3
advertisers that multi-home in the only equilibrium with multi-homing users,
the result is the same as in case 1. Both platforms choose compatibility
strategy to allow multi-homing.
√
β+
√
β 2 +16β(A−B)
8
β−
β 2 −2β(A+B−2C)
Case 3: If
<t≤
, the result is the same
2
as the one without multi-homing advertisers in case 1.
√
Case 4: If A − C < t ≤
β+
β 2 +16β(A−B)
,
8
from lemmas 3 and 4, we have:
t
β
β2
β
+ (A − B) +
(A − B)2 − (A − B + t) + t,
3
2 3t
18t
2t
2
t
β
β
β2
2
∆πB = πBm − πBs = − (A − B) +
(A
−
B)
−
(B − A + t)2 .
3
3
2 3t
18t
16t
∆πA = πAm − πAs =
Platform A has a stronger
q preference for multi-homing if and only if
2 (A−B)
2
4t
4t3 −
− 4t2 2t(A−B)
− 4(A−B)
3
3
9
β≤
.
(B − A + t)2
32
Case 5: If A−B < t ≤ A−C, the two platforms compete on the advertisers’
side, we have:
t
β
β2
β
2
= + (t − B + C) +
−
(A − B + t) + t,
(t
−
B
+
C)
−
2 3t
18t3
2t
t
β
β2
β2
2
πBm − πBs = − (t − B + C) +
(t
−
B
+
C)
−
(B − A + t)2 .
2 3t
18t3
16t3
πAm
πAs
Platform A permits multi-home if and only if
√
tβ 2 + 3t2 β + 9t3 ∆A
C≤B−
,
β2
2
2
β 2
2β t
β
β
where ∆A = ( 9t
2 + 3t ) − 9t3 [ 2 + 3 +
Platform B permits multi-home if
β2
18t
−
β 2 (A−B)2
16t3
−
β 2 (A−B)
8t2
−
β2
].
16t
√
tβ 2 − 3t2 β + 9t3 ∆B
C≤B−
,
β2
2
2
2
2
2
2
2
β (A−B)
β
β 2
2β t
β
β
β
where ∆B = ( 9t
+ β (A−B)
− 16t
].
2 − 3t ) − 9t3 [ 2 − 3 + 18t −
16t3
8t2
Platform A has a stronger
preference
for
multi-homing
than
platform
B
√
4t2 (3A+B−4C−t)−8t2
(B−C−t)(6A−2B−4C−t)
if β ≤
. If the externality β is large
3(B−A+t)2
enough, the price the advertisers are willing to pay is higher, so that platform A would like to single-home to attract more users to charge a higher
price for the advertisers.
Case 6: If B − C < t ≤ A − B, platform A covers the whole market if
multi-homing is not possible. As a result platform B prefers multi-homing.
For platform A,
πAm − πAs =
β
β2
t
+ (t − B + C) +
(t − B + C)2 − β + t.
2 3t
18t3
Platform A enforces single-home if and only if
q
9t3 − 3t2 (t − B + C) + 9t3 1 − 2(t−B+C)
−
3t
β≤
(t − B + C)2
2(t−B+C)2
9t2
.
That is, platform A would enforce multi-home only if the externality β is
not very large. If β is large enough, platform A would like to exclude its
opponent to cover the whole market so that it could charge a higher price
from the advertisers.
33
Case 7: If t ≤ B − C, platform B prefers multi-homing in any case because
it can not attract any positive profit under single-homing. Platform A would
enforce single-homing because
πAm − πAs =
3t
− β < 0,
2
as we have β ≥ 2t from assumption 3. In this case, even if everyone consumes
both products if multi-homing is possible, platform A whose product has a
quality advantage would enforce single-homing. Because if it excludes its
opponent and covers the whole market, it can charge a higher price from the
advertisers as the advertisers care much about the number of users, while
if multi-homing is possible, no advertiser would like to join both platforms,
hence competition makes the platform A faces a demand of one half and
price equals the transportation cost.
In the two-sided case, even if the users do not directly pay to the platforms, the platform may prevent them from multi-homing in some cases, we
have different result as in the one-sided case:
Proposition 6. In the two-sided case, even if the platforms do not attract
profit from the users directly, the platform whose product has a quality advantage would enforce single-homing if the network externality β is large and
the transportation cost t is small.
Different from the result in proposition 2 in the one-sided case, if the
network externality is large enough, it could happen that platform B has a
stronger preference for multi-homing so that the equilibrium compatibility
strategy is determined by platform A’s strategy.
Proposition 7. Platform A whose product has a quality advantage has a
stronger preference for multi-homing only if the externality β is not very
large.
From the equilibria profit levels of the two platforms in case 1 of lemma 4
with multi-homing advertisers, for each of these two platforms, copying programs from its opponent increases its program i (i = A, B) and the common
program C for the same amount, but decreases its opponent’s specific part
j − C. Hence copying programs from its opponent increases its own profit
while makes its opponent’s profit be nonincreasing under the equilibrium
with multi-homing advertisers. Therefore each platform has an incentive to
copy programs from its opponent in order to decrease the quality differentiation to increase its own profit. That is why PPLive copied programs from
PPStream. It was estimated that in China, about 94% of internet users use
34
the online TV program, the market is almost fully covered, and the common
program in PPLive and PPStream is very large, so that PPStream prevented
its product from being compatible with PPLive.
5.5
Social Welfare Maximization
As in the one sided case, the social welfare maximizer would enforce multihoming if and only if the social welfare is larger under multi-homing than
single-homing. By using the un-weighted sum of consumer surplus and firms’
21
profits to denote the
√ different cases one by one.
√ total welfare, we analyze
Case 1: If
√ 2
β+
β−
β 2 −2β(A+B−2C)
2
β 2 −3β(A+B−2C)
3
β−
≤t≤
or
β −3β(A+B−2C)
3
≤ t ≤ A + B − 2C, multiple equilibria exist under multihoming users. Under the equilibrium with multi-homing advertisers, the
social welfare maximizer would enforce multi-homing. Under the equilibrium
with single-homing advertisers, the social welfare maximizer would enforce
multi-homing if the overlap C is small.
√
√
β−
β 2 −3β(A+B−2C)
3
β+
β 2 −3β(A+B−2C)
≤t≤
, the social welfare
Case 2: If
3
maximizer would enforce multi-homing as under the equilibrium with multihoming advertises in case 1.
√
β+
β 2 +16β(A−B)
8
√
β−
β 2 −2β(A+B−2C)
Case 3: If
<t≤
, the result is the same
2
as the one without multi-homing advertisers in case 1.
√
β+ β 2 +16β(A−B)
Case 4: If A − C < t ≤
, the social welfare maximizer
8
enforces multi-homing if the overlap C is not very large.
Case 5: If t ≤ A−C, multi-homing is socially desirable as the social welfare
is greater with multi-homing users than that with single-homing users.
Proposition 8. The social welfare maximizer would enforce single-homing
only if both the overlap C and the transportation cost t are large enough.
6
Consumer Surplus Maximization
The regulator aims to maximize consumer surplus from the competition policy issue. In this case, multi-homing is always beneficial for the consumers.
21
Detailed analysis appears in the appendix.
35
There is no price effect on the users’ side, so that multi-homing increase
consumer surplus due to more consumption.
Proposition 9. The regulator who aims to maximize consumer surplus would
always enforce multi-homing.
If these two platforms offer symmetric products (i.e A=B), both firms
would choose compatible products, therefore multi-homing is socially desirable. Because the platforms share the market on the users’ side equally
when the products are incompatible, while compatible products leads a demand larger than one half for both platforms, therefore each of them could
charge a higher price on the advertisers’ side to obtain a greater profit.
7
Conclusion
This paper analyzes the Hotelling competition between two horizontally differentiated firms (or platforms) with partially overlapping product lines.
Each firm can set technological barriers to prevent consumers from consuming its opponent’s product. If the products are compatible with each other,
consumers can consume both products (multi-home) and obtain more utility
from the non-overlapping part. When some consumers multi-home, the firms’
pricing strategies are different. Under single-homing, the prices and profits
depend on the quality difference of the two products. Under multi-homing,
in contrast, prices and profits depend on the specific part of each product. If
the degree of overlap is sufficiently large, the additional benefit of buying a
second product might vanish. Other things equal, multi-homing will lead to
lower prices due to the overlap.
In the basic model of competition between two firms, there is no network
effect, but the firms choose incompatibility to enforce single-homing due to
the overlap and product differentiation. The firm with a higher quality product allows multi-homing more than the firm with a lower quality product,
because each firm obtains a monopoly profit of its own specific part under
multi-homing users. Therefore firm A obtains a higher profit than firm B,
so that it has a stronger preference for multi-homing. We also demonstrate
that the equilibrium compatibility level chosen by firms is less than socially
optimal. In addition, if we analyze from the regulator’s point of view who
aims to maximize consumer surplus, the smaller the transportation cost, the
less beneficial multi-homing is for consumers. As the consumers have to pay
for both products under multi-homing, even if they have more choices, this
is dominated by the pricing effect.
36
In the two-sided market model in which platforms compete for advertisers
and users, despite the fact that the users cannot be charged, the platform
that has a higher quality product would choose incompatible products to
enforce single-homing if the externality is large and the differentiation cost is
small. The larger the externality, the more advertisers care about the number
of users, hence the higher price platforms charge from the advertisers’ side;
the platform with a higher quality product would like to drive its opponent
out of the market to become a monopoly so that it can charge a higher price
from the advertisers.
In our model, the degree of overlap is exogenously given. The tradeoff
between overlap and compatibility arises if it is endogenously chosen by firms.
The greater the degree of overlap, the less benefit to firms from choosing
compatible products. This could be an interesting topic for future research:
to analyze the equilibrium degree of overlap and compatibility strategy, and
to compare it with the socially optimal one.
We have not addressed the possible endogenous choice of locations in our
model; instead we kept them as exogenous and situated the firms at the two
ends points of the Hotelling line. One topic for further research is to explore
if firms indeed have an incentive to maximize differentiate from each other.
It might be worthwhile to analyze if the degree of overlap brings effects for
firms’ differentiation incentives, or if the competition between advertisers
enhances the differentiation preference in the two-sided market case, and
how the compatibility strategy is affected by location choice as well as price
choice.
We assumed that firms set their prices simultaneously. In reality it happens that one firm comes first and then potential competitors enter the market. Different profits are earned when firms set prices sequentially. This
question could be another topic for further research: to analyze how the
equilibrium strategy might be changed under Stachelberg competition, or to
see different socially optimal result.
The online TV broadcast works due to peer-to-peer technology. Another
further topic be worthy to analyze is to consider the peer effect and competition effect on the users’ side in the two-sided market model. The platforms’
strategies could be different if two different network effects appear on the
users’ side. The pricing strategy on the advertisers’ side depends on the
network effect on the users’ side.
37
A
The Demand Function in Exclusive Networks
PA
A+B−t
A
MB
LM
A−B−t
H
MA
B−A−t
B
PB
Figure 3: The Demand Function
The demand function can be shown for four different cases in four different
areas in the price space. For the price pairs locate in the area H, it satisfies
PA + PB ≤ A + B − t and A − B − t < PA − PB < A − B + t. It satisfies
that the users locate near A can obtain more utility by consuming product
A than consuming product B. The demand for each of these two products
A +PB
are determined by the classical Hotelling demand: DA = 21 + A−B−P
and
2t
A−B−PA +PB
1
DB = 2 −
. When either one of these three conditions is violated,
2t
one of the firm becomes a monopoly. For the price pairs lie in the area LM ,
it satisfies PA + PB ≥ A + B − t, PA ≤ A and PB ≤ B. The price pair is large
enough so that some of the users will not consume either one of the product,
the market is not fully covered, each firm becomes a local monopoly in this
38
A
case. The demand is determined by the monopoly demand: DA = A−P
t
B−PB
and DB = t . When the price pairs locate in the area M A, it satisfies
PA − PB ≤ A − B − t, the price for product A is small enough so that all
of the users will consume product A, firm B will quit the market. When the
price pairs locate in the area M B, it satisfies PA − PB ≥ A − B + t, the price
for product B is small enough so that it becomes a monopoly, all of the users
will consume product B.
A.1
The consumers’ behavior under single-homing
If the firms do not permit users to consume their opponents’ product, each
of the end users can consume at most one unit of the products, they cannot
multi-home. The two firms compete on the users’ side.
We use figure 3 to depict demand in the (PA , PB ) space. If the price pair
and transportation costs are large enough relative to the basic value of the
products, not all of the users will consume one of the product, there are some
users do not consume either one of the two products, the market is not fully
covered. While in order to make demand for each product be nonnegative, it
should be that PA < A and PB < B. That is, if PA + PB > A + B − t, which
is depicted by area LM in the figure 3, each of the two products becomes a
local monopoly. The local monopoly demand for each of these two products
are respectively:
A − PA
DA =
t
B − PB
DB =
t
If PA +PB ≤ A+B−t and A−B−t < PA −PB < A−B+t, the demand for
each product is determined by the Hotelling demand, it is depicted by area
H in figure 3, in this case an individual with type x will consume product A
rather than product B if and only if:
A − tx − PA ≥ B − t(1 − x) − PB
We can get the demand for each of these two products are as follows:
DA =
1 A − B − PA + PB
+
2
2t
1 A − B − PA + PB
−
2
2t
If PA − PB ≤ A − B − t, firm A becomes a monopoly and all of the
individuals consume product A. So that DA = 1 and DB = 0. It is depicted
DB =
39
by area M A in figure 3. If PA − PB ≥ A − B + t, firm B becomes a monopoly
and all of the individuals consume product B, which is depicted by area M B
in figure 3.
A.2
The pricing strategy
When the product quality A and B are fixed, for different values of t, there
can be at most five possible types of equilibrium: (1) Each user consumes
one unit of the products, either product A or B, and the demands for both
products are positive, so that the market is fully covered and shared by
these two firms, and the marginal user who is indifferent between consuming
product A and consuming product B earns a positive utility. It is depicted
by area H in figure 3. (2) There are some users do not consume either one
of the products, and the demands for both products are positive, so that
the market is not fully covered, each of the firms becomes a local monopoly.
It is depicted by area LM in figure 3. (3) Each user consumes one unit
of the product, either product A or B, and the demands for both firms are
positive, so that the market is fully covered and shared by these two products,
and the marginal user who is indifferent between consuming product A and
consuming product B earns a zero utility. It is depicted by the intersection of
H and LM in figure 3. (4) There are only consumers for product A, nobody
consumes product B, so that product A becomes a monopoly. It is depicted
by area M A and its boundary in figure 3. (5) There are only consumers
for product B, nobody consumes product A, so that product B becomes a
monopoly and product A quits the market. It is depicted by area M B and
its boundary in figure 3. Now let us analyze these possible equilibria one by
one.
case 1: In this case, the demand is determined by the classical Hotelling
competition, the firm’s problem is to maximize its own profit.
max πI = PI ∗ DI
PI
From the maximization problem, we get
PA = t +
A−B
,
3
PB = t −
A−B
.
3
Putting the formula of price into the demand function, we have
DA =
1 A−B
+
,
2
6t
DB =
40
1 A−B
−
.
2
6t
In this case, the marginal user earns a positive utility, so we should have
A + B > 3t
⇐⇒
t<
A+B
.
3
In order to ensure that the market is shared by two firms, we need
DB =
1 A−B
−
>0
2
6t
⇐⇒
t>
A−B
.
3
case 2: In this case, the demand for each product is determined by the
A
B
local monopoly demand DA = A−P
, DB = B−P
. From the maximization
t
t
A
problem of the two firms, we get PA = 2 and PB = B2 . So the demand are
A
B
respectively DA = 2t
and DB = 2t
. Note that in the local monopoly, we have
A−PA
A−B−PA +PB +t
> t , So in order for this equilibrium to exist, we should
2t
have
A+B
t>
.
2
case 3: In this case, the marginal user earns zero utility, so we should have
A−B−PA +PB
A
+ 12 = A−P
, from the above condition, we get PA +PB = A+B−t.
2t
t
If this kind of equilibrium exists, it is optimal for the firms to keep its price.
So if firm A increases its price for a little bit, its demand will be determined
A
A
with profit PA ∗ A−P
. This is not
by the local monopoly demand A−P
t
t
A−2PA
A
profitable if
≤ 0, so we can get pA ≥ 2 . If firm A decrease its
t
price for a little bit, its demand is determined by the Hotelling competition
A +PB
demand with πA = pA ∗ [ A−B−P
+ 21 ], which is not profitable as long as
2t
A
− p2tA + A−B−P2tA +PB +t ≥ 0, based on the condition A−B−P2tA +PB +t = A−P
, we
t
2
get pA ≤ 3 A. Similar condition holds for firm B. For this kind of equilibrium
to exist, it should be
A B
+
≥1
2t 2t
⇐⇒
t≤
A+B
.
2
Compared with case 1, in order to make sure the marginal user earns zero
utility, we have t ≥ A+B
. The equilibrium pricing strategy is determined by
3
PA + PB = A + B − t,
A
2
≤ PA ≤ A,
2
3
and
B
2
≤ PB ≤ B.
2
3
case 4: In this case, firm A wants to attract all of the users, it should be
that PA − PB ≤ A − B − t. While for firm B, it has no incentive to decrease
or increase its price. So we have PA = A−B −t. In order for this equilibrium
to exist, we should have
A − B − PA + P B + t
≥1
2t
41
when PA = t +
A−B
3
and PB = t −
A−B
,
3
t≤
which is equivalent to
A−B
.
3
case 5: In this case, firm B wants to attract all of the users, it should
be that PA − PB ≥ A − B + t. While in order for this condition to be hold
when A quits the market and sets PA = 0, PB should be negative, which is
not possible. So firm A has an incentive to increase its price for a little bit
and get some nonnegative profit given that B’s profit does not decrease. So
it is not possible that firm B becomes a monopoly covering the whole market
and firm A quits.
A.3
Equilibria
From the analysis in the last subsection, we have the following equilibria
result:
1. If t >
A+B
,
2
then
A
,
2
B
PB = ,
2
PA =
2. If
A+B
3
≤t≤
A+B
,
2
A−B
3
<t<
A+B
,
3
A−B
,
3
A−B
PB = t −
,
3
PA = t +
4. If t ≤
A−B
,
3
A2
,
4t
B2
πB =
;
4t
πA =
then
2
A
≤ pA ≤ A,
2
3
3. If
A
,
2t
B
DB = ,
2t
DA =
B
2
≤ pB ≤ B,
2
3
PA + PB = A + B − t;
then
1 A−B
+
,
2
6t
1 A−B
DB = −
,
2
6t
DA =
t A − B (A − B)2
+
+
,
2
3
18t
t A − B (A − B)2
πA = −
+
;
2
3
18t
πA =
then
PA = A − B − t,
DA = 1,
42
πA = A − B − t.
B
The best-response correspondence
The best-response correspondence is the firm’s pricing strategy in order to
maximize its own profit given its opponent’s pricing strategy. The bestresponse correspondence of firm i can be written as
Bi (Pj , t) = arg max{Pi ∗ max[min(
i − j − P i + Pj + t i − Pi
,
, 1), 0]}.
2t
t
Where i, j = A, B. As the demand function is not continuously differentiable
everywhere, the best-response correspondence of player i can have different
formulas for different relationship between A, B and t. For the price pairs
locate within the area H, the best-response is determined by the classical
Hotelling best-response correspondence. While for the pairs lie outside or
the boundary of H, different cases arise. Let us consider the best-response
correspondence for firm A. First let us consider the case when A ≥ B + 3t.
Given this inequality, for any PB lies in the interval [B, +∞], no user is
willing to buy product B so firm A becomes the unique firm who survives
in the market, its best-response is the monopoly price just attract all of the
, B] (As in this case
users PA = A − t. For PB lies in the interval [2t − 2(A−B)
3
2(A−B)
2t − 3 ≤ 0, so the interval becomes [0, B].), the users locate on the other
side of A are better off buying from A than buying from B, so firm A can
cover the whole market and his best-response is setting PA = A − B − t + PB .
So the best-response for firm A in this case is:
(
A − B − t + PB if PB ∈ [0, B]
BA (PB ) =
A−t
if PB ∈ (B, +∞)
Now let us consider the case when 2t < A ≤ B + 3t. For PB smaller
than B + A2 − t (As A > 2t in this case, so B + A2 − t > B, so the interval
becomes [0, B].), the best-response is the classical Hotelling best-response.
Similar to the previous case, if PB is equal or greater than B, nobody would
like to buy from B so that A becomes the pure monopoly in the market, his
best-response is PA = A − t. So the best-response for firm A in this case is:
(
A−B+PB +t
if PB ∈ [0, B]
2
BA (PB ) =
A−t
if PB ∈ (B, +∞)
Finally, let us consider the case when A ≤ 2t. The largest quality of
product is not large enough so that not a single firm can cover the whole
market. For PB smaller than B + A2 − t, the best-response is the classical
Hotelling best-response. For PB lies in the interval [B + A2 − t], each of the
43
firms become a local monopoly, so the best-response for A is PA =
best-response for firm A in this case is:
(
A−B+PB +t
if PB ∈ [0, B + A2 − t]
BA (PB ) = A 2
if PB ∈ (B + A2 − t, +∞)
2
C
A
.
2
So the
Proof for non-existence of pure strategy
equilibrium
Proof. Under the multi-homing equilibria result in lemma 2, we have 1 −
B−C
> A−C
for t > A+B−2C
, hence no consumer buys both product, multi2t
2t
2
homing does not exist. Now let us consider the marginal users, the consumer
B
is better off consuming both products than only conwith type 1 − B−C−P
t
suming product A. From the above analysis, no one will multi-home on these
two products, as a result this consumer will not consume both, the utility he
could obtain by only consuming product B is B −(B −C −PB )−PB = C > 0,
hence he will consume product B for any price charged by firm B; similarly
A
who is better off consuming both prodfor the consumer with type A−C−P
t
ucts than only consuming product B, this consumer will consume product A
as be obtains a positive utility.
From the analysis for the marginal consumer, for any price PA and PB ,
B
A
and DB > B−C−P
. No one consumes both
the demand DA > A−C−P
t
t
products in this case, so we should have A + B − C − t − PA − PB < C,
as we have t > A+B−2C
in this case. The optimal
that is PA + PB > A+B−2C
2
2
A−C
B−C
prices the two firms would charge is 2 and 2 when face demand of
A−C−PA
B
and B−C−P
, in order to attract a higher demand, the price charged
t
t
and B−C
, which contradicts with
by the two firms should be lower than A−C
2
2
PA +PB > A+B−2C
.
Hence
there
is
no
pure
strategy
equilibrium
under multi2
homing for this value of transportation cost t. Since the profit functions of
both firms are continuous, there exists at least one mixed strategy price
equilibrium.
44
D
Firms’ strategy and social welfare
Figure 4: The relationship between the profit difference (welfare difference)
and the degree of overlapping
Figure 5: The relationship between the difference of firms’ profit and the
transportation cost (given the degree of overlapping C)
45
Figure 6: The relationship between the difference of firms’ profit and the
degree of overlapping
E
Two-Sided Markets
E.1
Proof for lemma 5
Lemma 5. If the platforms exclude multi-homing on the users’ side, the
equilibria result is:
1. If
β
4
+
βt(A−B)
2(6t2 −β 2 )
<t≤
A+B
3
+
β2
,
6t
then
β2
β 2 − 4t2
1 t(A − B)
+ 2
(A − B),
nA = + 2
;
2
4t 2β − 12t
2
6t − β 2
β2
β 2 − 4t2
1 t(A − B)
PB = t −
− 2
(A − B),
nB = − 2
;
2
4t
2β − 12t
2
6t − β 2
β 1 t(A − B)
β
1 t(A − B)
QA = ∗ [ + 2
],
mA =
∗[ + 2
];
2
2 2
6t − β
2t 2
6t − β 2
β 1 t(A − B)
β
1 t(A − B)
QB = ∗ [ − 2
],
m
=
∗
[
− 2
].
B
2 2
6t − β 2
2t 2
6t − β 2
PA = t −
46
2. If
A−B
3
+
β2
6t
<t≤
β
4
+
βt(A−B)
,
2(6t2 −β 2 )
then
t + A − B − β 4t2 β + 3tβ 2 − β 3 − β 2 (A − B) + 4t2 (A − B) − 12t3
+
,
2
2β 2 − 24t2
4t2 β + 3tβ 2 − β 3 − β 2 (A − B) + 4t2 (A − B) − 12t3
PB =
,
β 2 − 12t2
1 t(A − B)
1 t(A − B)
;
nB = − 2
;
nA = + 2
2
2
6t − β
2
6t − β 2
β tβ(A − B)
QA = +
− t,
mA = 1;
2
6t2 − β 2
β 1 t(A − B)
1 t(A − B)
β
QB = ∗ [ − 2
∗[ − 2
],
mB =
].
2
2 2
6t − β
2t 2
6t − β 2
PA =
3. If t ≤
A−B
3
+
β2
,
6t
then
PA = A − B − t,
QA = β − t,
nA = 1,
mA = 1.
Proof. For different value of the transportation cost t, the platforms have
different pricing strategy, one can show that the market on the users’ side is
fully covered if and only if PA + PB ≤ A + B − t. We analyze different cases
one by one.
Case 1: If A − B − t < PA − PB < A − B + t, the number of advertisers
on the other side of the market are respectively
mA =
β
QA
βnA − QA
= 2 (t + A − B − PA + PB ) −
t
2t
t
β
QB
βnB − QB
= 2 (t − A + B + PA − PB ) −
t
2t
t
The maximization problem for platform A and B are respectively
mB =
1 A − B − PA + P B
β
QA
max πAs = PA ∗[ +
]+QA ∗[ 2 (t+A−B−PA +PB )−
]
2
2t
2t
t
PA , QA
1 A − B − PA + PB
β
QB
max πBs = PB ∗[ −
]+QB ∗[ 2 (t−A+B+PA −PB )−
]
2
2t
2t
t
PB , QB
Taking first order derivative of πAs with respect to PA and QA , we have
A − B − 2PA + PB + t βQA
−
=0
2t
2t2
47
β A − B − PA + PB + t 2QA
∗
−
=0
t
2t
t
Rearranging the two first order conditions, we have the following:
QA =
β A − B − P A + PB + t
t
∗ (A − B − 2PA + PB + t) = ∗
β
2
2t
(8)
Similarly, from the maximization problem of platform B, we have
QB =
t
β B − A + PA − P B + t
∗ (B − A − 2PB + PA + t) = ∗
β
2
2t
Adding (8) to (9), we get
PA + PB = 2t −
β2
2t
Subtracting (9) from (8), we have
PA − P B =
β 2 − 4t2
(A − B)
β 2 − 6t2
From manipulation, we have the equilibrium results
PA = t −
β 2 − 4t2
β2
+ 2
(A − B)
4t 2β − 12t2
QA =
PB = t −
β 1 t(A − B)
∗[ + 2
]
2 2
6t − β 2
β 2 − 4t2
β2
− 2
(A − B)
4t
2β − 12t2
QB =
β 1 t(A − B)
∗[ − 2
]
2 2
6t − β 2
And the number of users on each platforms are
nA =
1 t(A − B)
+ 2
2
6t − β 2
nB =
1 t(A − B)
− 2
2
6t − β 2
mA =
β
1 t(A − B)
∗[ + 2
]
2t 2
6t − β 2
48
(9)
mB =
β
1 t(A − B)
∗[ − 2
]
2t 2
6t − β 2
In order for the conditions PA +PB ≤ A+B −t and A−B −t < PA −PB <
A − B + t and mA ≤ 1 be hold, we need
βt(A − B)
A + B β2
β
+
<
t
≤
+
4 2(6t2 − β 2 )
3
6t
βt(A−B)
βnA −QA
2t
Case 2: If t ≤ β4 + 2(6t
= 1,
2 −β 2 ) , we have nA ≥ β and mA =
t
therefore we have QA = βnA − t. Hence the maximization problem for
platform A is
1 A − B − PA + P B
1 A − B − PA + PB
max πAs = PA ∗ [ +
]+β∗[ +
] − t.
2
2t
2
2t
PA
From the first order condition with PA , we have
t + A − B + PB − 2PA − β = 0.
Combining with (9), we have
12β 2 t + 4t2 β − β 3 − 12t3
4t2 − β 2
(A − B) +
,
PB = 2
β − 12t2
β 2 − 12t2
4t2 − β 2
12β 2 t + 4t2 β − β 3 − 12t3 A − B + t − β
PA =
(A
−
B)
+
+
.
2(β 2 − 12t2 )
2(β 2 − 12t2 )
2
Case 3: If PA − PB ≤ A − B − t, all of the users join platform A if multihoming is not possible, platform A sets PA = A − B − t. Platform B quits
A
the market because no advertisers join it. We have nA = 1, mA = β−Q
= 1.
t
Hence we have QA = β − t, mA = 1. In order to have nA = 1, we need
2
t ≤ A−B
+ β6t .
3
E.2
Equilibria number of advertisers with multi-homing
users
If the market on the users side is fully covered, an advertiser with type y is
better off joining two platforms rather than either one of them if it obtains a
larger utility under multi-homing. The following conditions must be satisfied:
β − t − QA − QB ≥ βnA − ty − QA ,
β − t − QA − QB ≥ βnB − t(1 − y) − QB .
49
From calculation, the advertisers with type between 1 − β(1−nAt )−QB and
β(1−nB )−QA
will join both platforms if the number of users on each platt
form is not very close to 1. So the number of advertisers on each platform
are respectively
mA =
β(1 − nB ) − QA
,
t
β(1 − nA ) − QB
.
t
mB =
In order to ensure there exist multi-homing advertisers, we need 0 ≤ 1 −
β(1−nA )−QB
< β(1−nBt )−QA ≤ 1. Putting the formula of nA , nB into the
t
above inequality, we have βPA − tQB ≤ t2 − βt + β(A − C), βPB − tQA ≤
t2 − βt + β(B − C) and β(PA + PB ) − t(QA + QB ) ≥ t2 − 2βt + β(A + B − 2C).
Combing with nA < 1, nB < 1, we need PA > A − C − t, PB > B − C − t.
With the decrease of PA , nA becomes larger, no advertiser joins both
platforms. Because in this case, joining platform B in addition of platform
A is too costly, but the benefit is not very large as 1 − nA is close to 0. In
this case, the two platforms compete on the advertisers side, an advertiser
with type y is better off joining platform A rather than B if and only if
βnA − ty − QA ≥ βnB − t(1 − y) − QB
So the number of advertisers in both platforms are respectively:
mA =
β(nA − nB ) − QA + QB + t
,
2t
mB =
β(nB − nA ) − QB + QA + t
2t
The conditions βPA −tQB ≤ t2 −βt+β(A−C), βPB −tQA ≤ t2 −βt+β(B−C)
and β(PA + PB ) − t(QA + QB ) ≤ t2 − 2βt + β(A + B − 2C) should be satisfied.
If either one of the platforms covers the whole market on the users side,
no advertiser would like to join both platforms because it is better off joining
the platform which covers the whole market so that all users see its advertisement, and it does not need to pay twice advertisement fee and transportation
cost. If PA ≤ A − C − t, PB > B − C − t, we have nA = 1, nB < 1, no
advertiser would like to join both platforms, the two platforms compete on
the advertisers’ side. An advertiser with type y is better off joining platform
A rather than platform B if and only if
β − ty − QA ≥ βnB − t(1 − y) − QB
So the number of advertisers in these two platforms are respectively
mA =
β(1 − nB ) − QA + QB + t
,
2t
mB =
50
−β(1 − nB ) + QA − QB + t
2t
If PA ≤ A − C − t, PB ≤ B − C − t, we have nA = 1, nB = 1, an advertiser
with type y is better off joining platform A rather than platform B if and
only if
β − ty − QA ≥ β − t(1 − y) − QB
The number of advertisers in these two platforms in this case is
mA =
E.3
t − QA + QB
2t
mB =
t + QA − QB
2t
Proof for lemma 6
√
β 2 −β(A+B−2C)
2
β−
√
β+
β 2 −β(A+B−2C)
Lemma 6.
1. If
≤ t ≤
, there exist
2
two equilibria, one with and the other one without multi-homing advertisers,
B−C
A−C
,
PB =
PA =
2
2
1 B−C
1 A−C
QA = β[ −
],
QB = β[ −
]
2
4t
2
4t
A−C
B−C
nA =
,
nB =
2t
2t
β 1 B−C
β 1 A−C
mA = [ −
],
mB = [ −
]
t 2
4t
t 2
4t
or
A + B − 2C − β (3t2 − β 2 )(A − B)
+
,
PA =
4
12t2 − 2β 2
PB =
A + B − 2C − β (3t2 − β 2 )(A − B)
−
.
4
12t2 − 2β 2
2t A − B + β (3t2 − β 2 )(A − B)
2t(A − C − 2PA )
= [
−
]
QA =
β
β
2
6t2 − β 2
QB =
2t(B − C − 2PB )
2t B − A + β (3t2 − β 2 )(A − B)
= [
+
]
β
β
2
6t2 − β 2
nA =
3A − B − 2C + β (3t2 − β 2 )(A − B)
−
4t
t(12t2 − 2β 2 )
nB =
3B − A − 2C + β (3t2 − β 2 )(A − B)
+
4t
t(12t2 − 2β 2 )
mA =
1
β(A − B)
+
2 12t2 − 2β 2
51
mB =
1
β(A − B)
−
2 12t2 − 2β 2
√
(β 2 −3t2 )(A−B)
2β 2 −12t2
3A−B−2C+β
4
2. If
−
≤ t ≤
single-home in the only equilibrium,
β−
β 2 −β(A+B−2C)
,
2
all advertisers
A + B − 2C − β (3t2 − β 2 )(A − B)
+
,
PA =
4
12t2 − 2β 2
PB =
QA =
A + B − 2C − β (3t2 − β 2 )(A − B)
−
.
4
12t2 − 2β 2
2t A − B + β (3t2 − β 2 )(A − B)
2t(A − C − 2PA )
= [
−
]
β
β
2
6t2 − β 2
2t(B − C − 2PB )
2t B − A + β (3t2 − β 2 )(A − B)
QB =
= [
+
]
β
β
2
6t2 − β 2
3A − B − 2C + β (3t2 − β 2 )(A − B)
−
4t
t(12t2 − 2β 2 )
nA =
3B − A − 2C + β (3t2 − β 2 )(A − B)
nB =
+
4t
t(12t2 − 2β 2 )
β(A − B)
1
+
2 12t2 − 2β 2
mA =
2
2
2
mB =
2
1
β(A − B)
−
2 12t2 − 2β 2
2
−β t
−3t )(A−B)
3. If 6t (B−C)+3βt
< t ≤ 3A−B−2C+β
− (β 2β
, there is only one
2 −12t2
12t2 −β 2
4
equilibrium with all advertisers single-homing,
PA = A − C − t,
PB =
B − C β[6t2 − 2βt + β(B − C)]
−
2
2(12t2 − β 2 )
12t3 − 4βt2 + 2βt(B − C)
QA = 2t −
12t2 − β 2
QB =
nA = 1,
mA =
12t3 − 4βt2 + 2βt(B − C)
12t2 − β 2
nB =
6t(B − C) + 3tβ − β 2
12t2 − β 2
β
1 β(B − C) (3t2 − βt)(8t2 − β 2 )
− −
+
2t 2
12t2 − β 2
2t2 (12t2 − β 2 )
mB = −
β
3 β(B − C) (3t2 − βt)(8t2 − β 2 )
+ +
−
2t 2
12t2 − β 2
2t2 (12t2 − β 2 )
52
2
2
2
−β t
4. If t ≤ 6t (B−C)+3βt
, all advertisers single-home and the two plat12t2 −β 2
forms share the market on the advertisers’ side equally,
PA = A − C − t,
PB = B − C − t,
nA = nB = 1,
mA = mB =
QA = QB = t
1
2
Proof. We analyze different cases one by one according to different number
of advertisers.
Case 1: If there exist some multi-homing advertisers, the number of
advertisers are respectively
mA =
β(t − B + C + PB ) QA
β(1 − nB ) − QA
=
−
t
t2
t
β(1 − nA ) − QB
β(t − A + C + PA ) QB
=
−
t
t2
t
Hence the maximization problem for these two platforms are
mB =
max πAm = PA ∗
PA , QA
β(t − B + C + PB ) QA
A − C − PA
+ QA ∗ [
−
]
t
t2
t
max πBm = PB ∗
PB , QB
B − C − PB
β(t − A + C + PA ) QB
+ QB ∗ [
−
]
t
t2
t
From the first order conditions with PA and PB , we have PA = A−C
, PB =
2
B−C
. Putting the formulas of PA , PB into the first order conditions with QA
2
and QB
β(t − B + C + PB ) 2QA
−
=0
t2
t
β(t − A + C + PA ) 2QB
−
=0
t2
t
we have
1 B−C
1 A−C
QA = β[ −
],
QB = β[ −
]
2
4t
2
4t
The number of users and advertisers are respectively
nA =
A−C
,
2t
nB =
B−C
,
2t
mA =
β 1 B−C
[ −
],
t 2
4t
mB =
β 1 A−C
[ −
]
t 2
4t
In order to ensure this kind of equilibria exist, the following condition should
be satisfied: 0 ≤ 1 − mB ≤ mA ≤ 1, that is 0 ≤ 1 − βt [ 12 − A−C
] ≤ βt [ 12 −
√ 2
√4t2
β− β −β(A+B−2C)
β+ β −β(A+B−2C)
B−C
] ≤ 1, which is equivalent to
≤t≤
4t
2
2
53
if β ≥ A + B − 2C.22 If β < A + B − 2C, no advertisers multi-home on
both platforms because the network externality is not very large. Similarly
as √
in the one-sided market case, there is no pure strategy equilibrium if
β+
β 2 −β(A+B−2C)
2
2
≤ t ≤ A+B
+ β6t .
3
Case 2: With the increasing of nA , there is no advertiser joins both
platforms, the number of advertisers are respectively
mA =
β(nA − nB ) − QA + QB + t
β(A − B − PA + PB ) t + QB − QA
=
+
2t
2t2
2t
β(nB − nA ) − QB + QA + t
β(B − A − PB + PA ) t + QA − QB
=
+
2t
2t2
2t
The maximization problem for the platforms are
mB =
max πAm = PA ∗
PA , QA
β(A − B − PA + PB ) t + QB − QA
A − C − PA
+ QA ∗ [
+
]
t
2t2
2t
max πBm = PB ∗
PB , QB
β(B − A − PB + PA ) t + QA − QB
B − C − PB
+QB ∗[
+
]
t
2t2
2t
Taking first order derivatives, we have
∂πAm
A − C − 2PA βQA
=
−
=0
∂PA
t
2t2
(10)
∂πAm
β(A − B − PA + PB ) t + QB − 2QA
=
+
=0
∂QA
2t2
2t
(11)
∂πBm
B − C − 2PB βQB
=
−
=0
∂PB
t
2t2
(12)
β(B − A − PB + PA ) t + QA − 2QB
∂πBm
=
+
=0
∂QB
2t2
2t
(13)
√
22 β 1
t [2
−
B−C
4t ]
≤ 1 ⇐⇒ t ≤
β−
β 2 −4β(B−C)
4
√
or t ≥
β+
β 2 −4β(B−C)
if β > 4(B − C);
4
√
β− β 2 −4β(B−C)
holds. If
≤
4
if β ≤ 4(B − C), the condition βt [ 21 − B−C
4t ] ≤ 1 always
√
β+ β 2 −4β(B−C)
t ≤
and β > 4(B − C), the number of advertisers on platform A is 1,
4
all advertisers put advertisement on platform
√ A and some of them put
√ on platform B.
β− β 2 −β(A+B−2C)
β+ β 2 −β(A+B−2C)
β 1
β 1
A−C
B−C
1 − t [ 2 − 4t ] ≤ t [ 2 − 4t ] ⇐⇒
≤ t ≤
if
2 √
2
√
β− β 2 −4β(B−C)
β− β 2 −β(A+B−2C)
β > A + B − 2C. From comparison, we have
≥
if
4
2
√ 2
√ 2
2
β+
β
−4β(B−C)
β+
β
−β(A+B−2C)
β ≤ (A−C)
≤
.
A−B and
4
2
54
2t(B−C−2PB )
.
β
Putting
β(A − B − PA + PB ) 1 B + C − 2A − 2PB + 4PA
=0
+ +
2t2
2
β
(14)
β(B − A − PB + PA ) 1 A + C − 2B − 2PA + 4PB
=0
+ +
2t2
2
β
(15)
From (10) and (12), we have QA =
them into (11) and (13), we get
2t(A−C−2PA )
,
β
QB =
Adding (15) to (14), we have
1+
2C − A − B + 2PA + 2PB
A + B − 2C − β
= 0 ⇐⇒ PA + PB =
β
2
Subtracting (15) from (14), we have
β(A − B − PA + PB ) 3B − 3A − 6PB + 6PA
+
=0
t2
β
(β 2 − 3t2 )(A − B)
⇐⇒ PA − PB =
β 2 − 6t2
As a result the pricing strategy is
PA =
A + B − 2C − β (β 2 − 3t2 )(A − B)
+
,
4
2β 2 − 12t2
A + B − 2C − β (β 2 − 3t2 )(A − B)
PB =
−
.
4
2β 2 − 12t2
QA =
2t(A − C − 2PA )
2t A − B + β (β 2 − 3t2 )(A − B)
= [
−
]
β
β
2
β 2 − 6t2
QB =
2t(B − C − 2PB )
2t B − A + β (β 2 − 3t2 )(A − B)
= [
+
]
β
β
2
β 2 − 6t2
Putting the pricing formula into the demand, we have
nA =
3A − B − 2C + β (β 2 − 3t2 )(A − B)
−
4t
t(2β 2 − 12t2 )
3B − A − 2C + β (β 2 − 3t2 )(A − B)
+
nB =
4t
t(2β 2 − 12t2 )
mA =
1
β(A − B)
+
2 12t2 − 2β 2
mB =
55
1
β(A − B)
−
2 12t2 − 2β 2
We can check that no advertiser joins both platforms for this price, that
A +QB +t
can not get positive utility by
is the advertiser with type β(nA −nB )−Q
2t
joining platform B,
β(nA − nB ) − QA + QB + t
] − QB ≤ 0
2t
t QA + QB
nA nB
−
)− −
≤0
⇐⇒ β(1 −
2
2
2
2
β(A + B − 2C) β(PA + PB ) t QA + QB
⇐⇒ β −
+
− −
≤0
2t
2t
2
2
β(A + B − 2C)
β A + B − 2C − β 3t
⇐⇒ β −
+ ∗
−
≤0
2t
2t
2
2
β(A + B − 2C) β 2 3t
⇐⇒ β−
− − ≤ 0 ⇐⇒ 6t2 −4βt+β 2 +β(A+B−2C) ≥ 0
4t
4t 2
β(1 − nA ) − t[1 −
In this case, we have nA ≤ 1, the√transportation cost t should
√ 2 satisfy t ≥
2 −β(A+B−2C)
2
2
β−
β
β+
β −β(A+B−2C)
−3t )(A−B)
3A−B−2C+β
− (β 2β
. If
≤ t ≤
,
2 −12t2
4
2
2
there exist two types of equilibria with and without multi-homing advertisers
described by case 1 and case 2.
Case 3: If nA = 1, nB < 1, the number of advertisers are respectively
mA =
mB =
β(t − B + C + PB ) t + QB − QA
β(1 − nB ) − QA + QB + t
=
+
2t
2t2
2t
β(t − B + C + PB ) t + QA − QB
−β(1 − nB ) + QA − QB + t
=−
+
2t
2t2
2t
A
As nA = A−C−P
= 1, we have PA = A − C − t.
t
The two platforms solve their maximization problem
max πAm = QA ∗ [
QA
max πBm = PB ∗
PB , QB
β(t − B + C + PB ) t + QB − QA
+
]
2t2
2t
B − C − PB
β(t − B + C + PB ) t + QA − QB
+QB ∗[−
+
]
t
2t2
2t
Taking first order derivatives, we have
∂πAm
β(t − B + C + PB ) t + QB − 2QA
=
+
=0
∂QA
2t2
2t
∂πBm
B − C − 2PB βQB
=
−
=0
∂PB
t
2t2
56
∂πBm
β(t − B + C + PB ) t + QA − 2QB
=−
=0
+
∂QB
2t2
2t
From
m
∂πB
∂QB
m
∂πB
∂PB
= 0, we have PB =
B−C
2
−
βQB
.
4t
Putting it into
m
∂πA
∂QA
= 0 and
= 0, we get
β(B − C) β 2 QB 1 QB − 2QA
β
−
=0
−
+ +
2t
4t2
8t3
2
2t
β
β(B − C) β 2 QB 1 QA − 2QB
− +
=0
+
+ +
2t
4t2
8t3
2
2t
Adding (17) to (16), we have
1−
(16)
(17)
QA + QB
=0
2t
Subtracting (17) from (16), we have
β β(B − C) β 2 QB 3(QB − QA )
−
−
+
=0
t
2t2
4t3
2t
From the above two equations, we get
β β(B − C) β 2 QB 3QB
−
−
+
−3=0
t
2t2
4t3
t
From which, we get
QB =
12t3 − 4βt2 + 2βt(B − C)
12t2 − β 2
QA = 2t − QB = 2t −
PB =
12t3 − 4βt2 + 2βt(B − C)
12t2 − β 2
B − C βQB
B − C β[6t2 − 2βt + β(B − C)]
−
=
−
2
4t
2
2(12t2 − β 2 )
B
In order for nB = B−C−P
< 1, we need
t
of users and advertisers are respectively:
nA = 1,
mA =
nB =
6t2 (B−C)+3βt2 −β 2 t
12t2 −β 2
< t. The number
6t(B − C) + 3tβ − β 2
12t2 − β 2
β
1 β(B − C) (3t2 − βt)(8t2 − β 2 )
− −
+
2t 2
12t2 − β 2
2t2 (12t2 − β 2 )
57
β
3 β(B − C) (3t2 − βt)(8t2 − β 2 )
+ +
−
2t 2
12t2 − β 2
2t2 (12t2 − β 2 )
Case 4: If nA = nB = 1, the number of advertisers are respectively
mB = −
mA =
t − QA + QB
2t
t + QA − QB
2t
B−C−PB
=
= 1 we have PA = A − C − t,
t
mB =
A
= 1, nB
As nA = A−C−P
t
PB = B − C − t.
The maximization problem for the two platforms are
max πAm = QA ∗
QA
t − QA + QB
2t
max πBm = QB ∗
QB
t + QA − QB
2t
From the first order conditions, we get QA = QB = t. It is because that in
this case, the two platforms are identical to the advertisers, hence they act
as a classical Hotelling competition and each of them sets price equal to the
transportation cost and attracts half of the advertisers.
F
Competition between two Free Platforms
F.1
Proof for lemma 3
√
β+
β 2 +16β(A−B)
Proof. Case 1: If
< t ≤ A + B, the number of advertisers
8
on these two platforms are respectively
mA =
βnA − QA
β A − B + t QA
= ∗
−
t
t
2t
t
βnB − QB
β B − A + t QB
= ∗
−
t
t
2t
t
The maximization problem for platform A is:
mB =
β A − B + t QA
max πAs = QA ∗ [ ∗
−
]
t
2t
t
QA
From the first order condition, we get the optimal price
QA =
β
∗ (A − B + t)
4t
58
β
Similarly for platform B, QB = 4t
∗ (B − A + t). The number of advertisers
on these two platforms in this case are respectively: mA = 4tβ2 ∗ (A − B + t),
mB = 4tβ2 ∗ (B − A + t).
√
β+ β 2 +16β(A−B)
Case 2: If A − B < t ≤
, it is easy to check 2t
≤ nA ≤ 1
8
β
and mA = 1, as a result we have QA = βnA − t.
Case 3: If t ≤ A − B, platform A covers the whole market on the users
side and platform B quits. The advertisers only care about the number of
users, platform B can not attract any users so that no advertiser would like
to join platform B, it makes 0 profit in this case. The number of advertisers
A
= 1. Hence we get QA = β − t and mA = 1.
in platform A is mA = β−Q
t
F.2
Proof for lemma 4
Proof. The platforms earn profit only from the advertisers, we consider different cases one by one.
Case 1: If there exist multi-homing advertisers, the number of advertisers
− QtA ,
on these two platforms are respectively mA = β(1−nBt )−QA = β∗(t−B+C)
t2
− QtB . The maximization problem for platform A is:
mB = β∗(t−A+C)
t2
max QA ∗ [
QA
β ∗ (t − B + C) QA
−
]
t2
t
From the first order condition, we have
QA =
β
(t − B + C).
2t
QB =
β
(t − A + C).
2t
Similarly, for platform B,
The number of users in the two platforms are respectively:
nA =
A−C
,
t
nB =
B−C
.
t
β
β
∗ (t − B + C),
mB = 2 ∗ (t − A + C).
2
2t
2t
In order to ensure multi-homing exist, the following condition should be
satisfied: 0 ≤ 1 − mB ≤ mA ≤ 1, that is, 0 ≤ 1 − 2tβ2 ∗ (t − A + C) ≤ 2tβ2 ∗ (t −
√
√
β− β 2 −2β(A+B−2C)
β+ β 2 −2β(A+B−2C)
B + C) ≤ 1 from which, we need
≤t≤
2
2
if β ≥ 2(A+B−2C). If β ≤ 2(A+B−2C), no advertiser joins both platforms
mA =
59
√
β+
23
β 2 −2β(A+B−2C)
as the network externality is small. We can check
≥
2
A+
B
−
2C,
there
is
no
pure
strategy
equilibrium
if
A
+
B
−
2C
≤
t
≤
√
β+
β 2 −2β(A+B−2C)
.
2
Case 2: If the number of users on platform A is close to 1, no advertiser joins both platforms, the number of advertisers on these two platA +QB +t
= t+QB2t−QA + β(A−B)
, mB =
forms are respectively: mA = β(nA −nB )−Q
2t
2t2
β(nB −nA )−QB +QA +t
β(A−B)
t−QB +QA
=
− 2t2 . The maximization problem for the
2t
2t
two platforms are:
max QA ∗ [
QA
t + QB − QA β(A − B)
+
],
2t
2t2
max QB ∗ [
QB
t − QB + QA β(A − B)
−
].
2t
2t2
From the first order condition, we have
β
(A − B),
t
β
2QB = t + QA − (A − B).
t
2QA = t + QB +
We have the optimal price formula
QA = t +
β
(A − B),
3t
QB = t −
β
(A − B).
3t
Now let us compute the interval of transportation cost in which the above
competition price is an equilibrium. It should be satisfied that the marginal
advertiser who is indifferent between joining platform A and B can not
be better off by joining two platforms. That is, the advertiser with type
β(nA −nB )−QA +QB +t
gets non-positive utility by joining platform B. So we
2t
should have
β(1 − nA ) − t[1 −
β(nA − nB ) − QA + QB + t
] − QB ≤ 0,
2t
√
√
β 2 −8β(B−C)
β+ β 2 −8β(B−C)
or t ≥
if β ≥ 8(B − C);
4
4
√ 2
√
β− β −8β(B−C)
β+ β 2 −8β(B−C)
if β ≤ 8(B − C), mA ≤ 1 always holds. If
≤t≤
and
4
4
β
β
β ≥ 8(B − C), we have mA = 1. 1 − 2t2 ∗ (t − A + C) ≤ 2t2 ∗ (t − B + C) ⇐⇒
√
√
β− β 2 −2β(A+B−2C)
β+ β 2 −2β(A+B−2C)
≤
t
≤
if β ≥ 2(A + B − 2C). It is easy to check
2
√2
√
√
√
β− β 2 −8β(B−C)
β− β 2 −2β(A+B−2C)
β+ β 2 −8β(B−C)
β+ β 2 −2β(A+B−2C)
that
≤
and
≤
.
4
2
4
2
23 β
2t2
∗ (t − B + C) ≤ 1 ⇐⇒ t ≤
β−
60
with the above competition price. From calculation, we have β − β(nA2+nB ) −
√
√
β− β 2 −3β(A+B−2C)
β+ β 2 −3β(A+B−2C)
3t
≤ 0, which is equivalent to t ≤
or t ≥
.
2
3
3 √
√ 2
√ 2
2
β− β −2β(A+B−2C)
β+ β −3β(A+B−2C)
β− β −3β(A+B−2C)
≥
and
≤
It is easy to check that
3
2
3
√ 2
√ 2
β+ β −2β(A+B−2C)
β− β −2β(A+B−2C)
. Hence if A−C < t ≤
, the competition
2
2
price is an equilibrium pricing strategy for the two platforms. The number
of advertisers in these two platforms are respectively:
1
β
β
1
+ 2 (A − B),
mB = − 2 (A − B).
2 6t
2 6t
√ 2
√
√ 2
β− β −3β(A+B−2C)
β+ β 2 −3β(A+B−2C)
β− β −2β(A+B−2C)
≤t≤
or
≤t≤
If
2
3
3
A + B − 2C, there exist two equilibria with and without multi-homing advertisers described by case 1 and case 2.
Case 3: If B − C < t ≤ A − C, all users join platform A, while platform
B has part of the users. In this case, no one would like to put advertisement
on both platforms. The number of advertisers in these two platforms are
, mB = t−QB2t+QA − β(t−B+C)
. The
respectively: mA = t+QB2t−QA + β(t−B+C)
2t2
2t2
maximization problem for these two platforms are:
mA =
max QA ∗ [
QA
t + QB − QA β(t − B + C)
+
],
2t
2t2
max QB ∗ [
QB
t − QB + QA β(t − B + C)
−
].
2t
2t2
From the first order condition, we have
2QA = t + QB +
β
(t − B + C),
t
β
(t − B + C).
t
From the above two first order conditions, the optimal price is
2QB = t + QA −
QA = t +
β
(t − B + C),
3t
QB = t −
β
(t − B + C).
3t
The number of advertisers in these two platforms are respectively
mA =
1
β
+ 2 (t − B + C),
2 6t
mB =
1
β
− 2 (t − B + C).
2 6t
Case 4: If t ≤ B − C, all users join both platforms. For the advertisers,
it is enough to join one platform. The two platforms compete. The number
61
of advertisers in these two platforms are respectively mA =
t−QB +QA
. The maximization for these two platforms are
2t
t−QA +QB
,
2t
mB =
t − QA + QB
,
2t
t − QB + QA
max QB ∗
.
2t
QB
max QA ∗
QA
From the first order condition, we have QA = QB = t in this case. And
the number of advertisers in these two platforms in this case are respectively
mA = mB = 21 .
The advertisers join the platform in order to get maximum network effect,
if all of the users multi-home on these two networks, there is no need for the
advertisers to multi-home. So if both these two platforms get the whole
demand in the users side, they are totally symmetric on the advertisers side.
So each of them sets the price equals to the transportation cost and faces
half demand.
F.3
Social Welfare Maximization
We consider different
√ cases one by one.
√
β− β 2 −2β(A+B−2C)
β− β 2 −3β(A+B−2C)
Case 1: If
≤t≤
or
2
3
√ 2
β+ β −3β(A+B−2C)
≤ t ≤ A + B − 2C, under the equilibrium with multi3
homing advertisers, we have
s
Z
A−B+t
2t
SW =
Z
+
β(A−B+t)
4t2
0
=
β(A − B + t)
[
− ty]dy +
2t
[B − t(1 − x)]dx
A−B+t
2t
0
Z
1
[A − tx]dx +
Z
1
[
1−
β(B−A+t)
4t2
β(B − A + t)
− t(1 − y)]dy
2t
3β 2 (A − B + t)2 3β 2 (B − A + t)2 β(B − A + t) β(B − A + t)
+
+
−
32t3
32t3
4t3
4t
(A − B)(A − B + t) (A − B + t)2 A + B + t t
+
−
+
−
2t
4t
2
2
62
SW
m
Z
1− B−C
t
=
[A − tx]dx +
[A + B − C − t]dx+
1− B−C
t
0
1
Z
A−C
t
Z
Z
1−
[B − t(1 − x)]dx +
A−C
t
β(t−B+C)
2t2
β(t−A+C)
2t2
0
Z
Z
1
[β − t]dy +
[
β(t−B+C)
2t2
β(t−A+C)
1−
2t2
[
β(A − C)
− ty]dy+
t
β(B − C)
− t(1 − y)]dy
t
(A − C)2 (B − C)2
3β 2 (A − C)2 3β 2 (B − C)2
+
+C +
+
2t
2t
8t3
8t3
2
2
β(A − C) β(B − C) 3β (A − C) 3β (B − C) 3β 2
+
−
−β
−
+
+
t
t
4t2
4t2
4t
3β 2 1
2β 3β 2 3β 2 (A + B) A + B
2
=C ∗[ 3 + ]+C ∗[
+ 2 −
−
+ 1]
4t
t
t
2t
4t3
t
3β 2 (A2 + B 2 ) A2 + B 2 β(A + B) 3β 2 (A + B) 3β 2
+
−
−
+
−β
+
8t3
2t
t
4t2
4t
=
The social welfare under multi-homing users is always greater than that under
single-homing users. Because both firms obtain greater profits under multihoming users, as well as the consumer surplus. Therefore the social welfare
maximizer would enforce multi-homing.
Under the equilibrium without multi-homing users, we have:
Z A−C
Z 1
Z 1− B−C
t
t
m
[A − tx]dx +
[A + B − C − t]dx +
[B − t(1 − x)]dx
SW =
β(A−B)
1
+
2
6t2
Z
+
0
=
A−C
t
1− B−C
t
0
β(A − C)
[
− ty]dy +
t
Z
1
[
β(A−B)
1
+
2
6t2
β(B − C)
− t(1 − y)]dy
t
(A − C)2 (B − C)2
β(A − C) β(B − C) 5β 2 (A − B)2
t
+
+C +
+
+
−
3
2t
2t
2t
2t
36t
4
From comparison of these two welfare levels, the social welfare maximizer
prefers multi-home if the common program C is smaller than a threshold.
As in the Hotelling model, if C is large enough, there is no need for the users
to join both platforms.
√
√
β−
β 2 −3β(A+B−2C)
β+
β 2 −3β(A+B−2C)
Case 2: If
≤ t ≤
, there exist
3
3
some advertisers multi-homing in the only one equilibrium with multi-homing
users, the result is √
the same as in case 1. √
β+
β 2 +16β(A−B)
β−
β 2 −2β(A+B−2C)
Case 3: If
< t ≤
, the result is the
8
2
same as the one without multi-homing
advertisers
in
case
1.
√
β+ β 2 +16β(A−B)
Case 4: If A − C < t ≤
, from lemma 3 and lemma 4,
8
63
SW
m
1− B−C
t
Z
=
[A − tx]dx +
β(A−B)
1
+
2
6t2
Z
+
0
A−B+t
2t
SW =
1
+
0
[A + B − C − t]dx +
[B − t(1 − x)]dx
A−C
t
β(A − C)
[
− ty]dy +
t
Z
1
Z
[
β(A−B)
1
+
2
6t2
β(B − C)
− t(1 − y)]dy
t
1
[A − tx]dx +
[B − t(1 − x)]dx
A−B+t
2t
0
Z
1
(A − C)2 (B − C)2
β(A − C) β(B − C) 5β 2 (A − B)2
t
+
+C +
+
+
−
3
2t
2t
2t
2t
36t
4
Z
s
Z
1− B−C
t
0
=
A−C
t
Z
β(A − B + t)
[
− ty]dy +
2t
Z
1
[
1−
β(B−A+t)
4t2
β(B − A + t)
− t(1 − y)]dy
2t
(A − B + β)(A − B + t) (A − B + t)2 A + B 1 3β 2 (B − A + t)2
−
+
− +
.
2t
4t
2
2
32t3
As in the above case, the social welfare maximizer prefers multi-home if the
common program C is smaller than a threshold. If C is large enough, the
social welfare maximizer prefers single-homing to save some transportation
cost.
Case 5: If A − B < t ≤ A − C, SW s is the same as above. If the users
multi-home on these two platforms, there are some users only join platform
A, and the rest join both.
Z 1
Z 1− B−C
t
m
[A − tx]dx +
[A + B − C − t]dx
SW =
=
1− B−C
t
0
Z
+
0
β(t−B+C)
1
+
2
6t2
β(A − C)
[
− ty]dy +
t
Z
1
[
β(t−B+C)
1
+
2
6t2
β(B − C)
− t(1 − y)]dy
t
(B − C)2
t β(A − C) β(B − C)
+A− +
+
+
2t
2
2t
2t
β 2 (A − B)(t − B + C) β 2 (t − B + C)2
t
−
−
3
3
6t
36t
4
Multi-homing is socially desirable as the social welfare under multi-homing
is greater than that under single-homing.
Case 6: If B−C < t ≤ A−B, all the active users and advertisers are that
in platform A if multi-home is not possible. The welfare under multi-home
is the same as the above case.
Z 1
Z 1
s
SW =
[A − tx]dx +
[β − ty]dy = A + β − t
=
0
0
64
The social welfare under multi-homing is greater than that under singlehoming.
Case 7: If t ≤ B − C, the welfare under single-homing users is the same
as in case 6.
Z 1
Z 1
2
m
SW = A + B − C − t +
[β − ty]dy +
[β − t(1 − y)]dy
1
2
0
5
=A+B−C +β− t
4
It is easy to check that SW m > SW s . The social welfare maximizer
would enforce multi-homing.
65
References
[1] Ambrus, A. and M. Reisinger (2006) “Exclusive vs Overlapping Viewers
in Media Markets,”Working Paper
[2] Andrea, P. and S. Debapriya (2009) “Outsourcing versus technology
transfer: Hotelling meets Stachelberg,”MPRA Paper No. 15673 2009
[3] Anderson, S.P. and S. Coate (2005) “Market Provision of Broadcasting:
A Welfare Analysis,”Review of Economic Studies, 72, 947-972
[4] Anderson, S.P., O. Foros and H. Kind (2010) “Hotelling Competition
with Multi-Purchasing,”NHH Discussion paper
[5] Anderson, S.P. and J.J. Gabszewicz (2006) “The Media and Advertising:
A Tale of Two-Sided Markets,”Handbook of the Economics of Art and
Culture, Volume 1, 567-614
[6] Armstrong, M. (2006) “Competition in Two-Sided Markets,”Rand Journal of Economics, 37, 668-691
[7] Armstrong, M. and J. Wright (2007) “Two-Sided Markets, Competitive
Bottlenecks, and Exclusive Contracts,”Economic Theory, 32, 353-380
[8] Caillaud, B. and B. Jullien (2003) “Chicken and Egg: Competition
Among Intermediation Service Providers,”Rand Journal of Economics,
34, 309-328
[9] Casadesus-Masanell, R. and F. Ruiz-Aliseda (2008) “Platform Competition, Compatibility, and Social Efficiency,”HBS working paper September 2008
[10] Choi, J. P. (2006) “Broadcast Competition and Advertising with free
entry,”Information Economics and Policy, 18, 181-196
[11] Choi, J. P. (2011) “Tying in Two-Sided Markets with MultiHoming,”forthcoming in Journal of Industrial Economics
[12] Crampes, C., C. Haritchabalet and B. Jullien (2009) “Advertising, Competition, and Entry in Media Industries,”Journal of Industrial Economics, 57, 7-31
[13] Crémer, J., P. Rey and J. Tirole (2000) “Connectivity on the Commencial Internet,”Journal of Industrial Economics, Vol. 48, No. 4 (Dec.,
2000), pp. 433-472
66
[14] Doganoglu, T. and J. Wright (2006) “Multihoming and Compatibility,”International Journal of Industrial Organization 24 (2006) pp. 45–
67.
[15] Farhi, E. and A. Hagiu (2007) “Strategic Interactions in Two-sided Market Oligopolies,”working paper
[16] Farrell, J. and G. Saloner (1992) “Converters, Compatibility, and the
Control of Interfaces,”Journal of Industrial Economics, Vol. 40, No. 1
(Mar., 1992), pp. 9-35.
[17] Ferrando, J. J.Gabszewicz, D.Laussel and N.Sonnac (2008) “Intermarket Network Externalities and Competition: An Application to Media
Industries,”International Journal of Economics Theory, 4, 357-379
[18] Foros, O. and B. Hansenb (2001) “Competition and Compatibility
among Internet Service Providers,”Information Economics and Policy
13 (2001), pp. 411–425
[19] Goldfainy, K. and E. Kovac (2007) “On Compatibility in Two-Sided
Markets,”Working paper, October 2007, University of Bonn
[20] Gabszewicz, J., N. Sonnac and X. Y. Wauthy (2001) “On Price Competition with Complementary Goods,”Economic Letters, 70 (2001) 431-437
[21] Gabszewicz, J. and X. Y. Wauthy (2003) “The Option of Joint Purchase
in Vertically Differentiated Markets,”Economic Theory, 22, 817-829
[22] Gabszewicz, J. and X. Y. Wauthy (2004) “Two-Sided Markets and Price
Competition with Multi-homing,”CORE Discussion Paper No.2004/30
[23] Katz, M. and C. Shapiro (1985) “Network Externalities, Competition,
and Compatibility,”American Economic Review, 75(3), 424-440
[24] Kaiser, U. and J. Wright (2006) “Price Structure in Two-Sided Markets: Evidence from the Magazine Industry,”International Journal of
Industrial Organization, 24, 1-28
[25] Laffont, J., P. Rey and J. Tirole (1998) “Network Competition: I.
Overview and Nondiscriminatory Pricing,”The RAND Journal of Economics, Vol. 29, No. 1 (Spring, 1998), pp. 1-37.
[26] Laffont, J., P. Rey and J. Tirole (1998) “Network Competition: II.
Price Discrimination,”The RAND Journal of Economics, Vol. 29, No.
1 (Spring, 1998), pp. 38-56.
67
[27] Malueg, D. A. and M. Schwartz (1998) “Compatibility Incentives of a
Large Network facing Multiple Rivals,”Journal of Industrial Economics,
Vol. LIV No.4, December 2006, pp. 527–567
[28] Peitz, M. and T.Valletti (2008) “Content and Advertising in the Media:
Pay-TV versus Free-to-Air,”International Journal of Industrial Organization 26 (2008), 949-965
[29] Rattanasuda P. and G. Vernasca (2006) “Partial Multi-homing in TwoSided Markets,”Discussion Paper
[30] Reisinger M. (2011) “Platform Competition for Advertisers and Users
in Media Markets,”Working Paper
[31] Stuhmeie T. and T. Wenzel (2011) “Regulating Advertising in the Presence of Public Service Broadcasting,”Working Paper
[32] Valletti, T. M. and C.Cambini (1992) “Investments and Network Competition”The RAND Journal of Economics, Vol. 36, No. 2 (Summer,
2005), pp. 446-467..
[33] Valverde, C. T. and J. Robert (2004) “On Hotelling Competition with
General Purpose Products,”FACE Working Paper Series
[34] Weyl, G. and A. White (2011) “Insulated Platform Competition,”Working Paper
68