Equilibrium Analysis of a Two-Sided Market with Multiple

International Telecommunications Policy Review, Vol.19 No.3 September 2012, pp.1-22

1

Equilibrium Analysis of a Two-Sided Market with Multiple Platforms of Monopoly Provider

Dohoon Kim*

ABSTRACT

In this paper, we consider a single monopoly platform provider which operates both

platforms: an old and a new platform. These two platforms connect the user group

with the suppliers, thereby leveraging the indirect network externalities in a two-

sided market. We also incorporate a cross-platform externality which represents a

potential backward compatibility of the new platform: i.e., users joining the new

platform can also enjoy the products and services provided by suppliers using the old

platform. Users and suppliers are uniformly populated over [0, 1] interval as in the

Hotelling model, and play a subscription game to choose (exactly) one platform. The

platform determines the pricing profile for the supplier market, and users and

suppliers respond to the pricing profile. Our basic analysis for static equilibrium

indicates that it is very unlikely that an interior equilibrium is stable. Furthermore,

some specific types of boundary equilibriums, where at least one market side tips to a

single platform, are stable under certain conditions. We also present a dynamic

decision model of the platform provider, which tries to maneuver the markets toward

a target state by controlling price profiles. Our analytical results from the optimal

control theory assert that a bang-bang control with subsidization for a specific

platform eventually leads the market to the corresponding boundary equilibrium. The

cross-platform externality plays an important role for a co-existence of competing

platforms under a certain condition.

Key words: Two-sided market, Indirect network externality, Cross-platform network

externality, Monopoly platform, Technology transition, Game theory,

Unstable equilibrium, Boundary equilibrium, Dynamic optimization

※ First received, August 6, 2012; Revision received, September 18, 2012; Accepted,

September 24, 2012.

* Associate Professor, Ph.D., CPIM, School of Business, Kyung Hee University, Hoegi-dong

1, Dongdaemoon-gu, Seoul 130-701, Korea, E-mail: [email protected]

International Telecommunications Policy Review, Vol.19 No.3 September 2012

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Ⅰ. INTRODUCTION

Two-sided market models provide a new perspective to view the platform-based

industry such as credit cards, newspapers, telecommunications, Internet services,

and many more (refer to Eisenmannn et al. (2006) for more examples of two-sided

markets). In a two-sided market, two distinct parties are connected to each other

through a platform. Evans (2003) shows that a platform constitutes a set of the

institutional agreements necessary to realize a transaction between two distinct

groups. A key characteristic, here, is the presence of network externalities between

these two groups. Hagiu (2009) argues that the benefits of those transactions which

could not exist without a monopoly platform may outweigh the deadweight loss

typical of monopoly. Early studies such as Armstrong (2006), Parker & van Alstyne

(2005), and Rochet & Tirole (2002, 2006) focus on the pricing structure, where a

subsidization across the market sides serves as the main driver for maximizing

profits.

However, many studies on the two-sided markets deal with a single platform.

Even in the case of competing platforms, little attention has been paid to the

possibility that there exists a network externality crossing two platforms. This

feature may not be practical, particularly when a new technology is emerging and

replacing an old one. A backward compatibility will be an issue in such a transition,

and it can be thought of as a special type of network externality under the

framework of two-sided market. For example, when telcom operators upgrade their

network from 3G to 4G, they should make a plan for the backward compatibility in

order to take care of users and service providers joining old network (Users in 4G

network will expect to keep using the services in 3G).

The first formal modeling and analysis of this feature can be found in Mussachio

& Kim (2009), where a monopoly provider runs two platforms: old and new. They

incorporate a new type of indirect network externality, so-called ‘cross-platform

externality,’ to handle the issue of asymmetric backward compatibility. Our study,

with a different payoff structure from Mussachio & Kim (2009), presents a two-

sided market model with competing platforms which attempt to leverage both

indirect and cross-platform network externalities. The market share of each

platform in each market side is endogenously determined through interrelated

platform subscription games. We analyze not only static equilibriums together with

their stability but also dynamics in the context of optimal control theory. To our


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knowledge, there have been few studies dealing with dynamic decisions in the two-

sided market framework.

Some relevant prior studies are as follows. Kim (2009) applied a two-sided

market model to the net neutrality issues of a monopoly (or market-dominating)

platform provider. His study considered only an interior equilibrium, and did not

provide a condition under which the interior equilibrium could be stable. Musacchio

& Kim (2009) provided an off-equilibrium analysis of a two-side market operated

by a monopoly platform provider. Their research focused on interior as well as

boundary equilibriums, and found out that the latter could be more common in

practice. Kim (2010) provided a two-sided market model with competing platform

providers and analyzed a Stackelberg game where the leader tries to achieve its best

equilibrium. Kim (2011) also studied competing platform providers carrying out

myopic decisions, and tried to capture dynamic behaviors in market share changes

using a tool for system dynamics.

Our model in this paper is different from the previous ones in that we employ

not only a new dynamics but also a new type of network externalities (cross-

platform externality) that were first considered in Musacchio & Kim (2009), but not

incorporated in Kim (2010) and Kim (2011). Our study is also different from

Musacchio & Kim (2009) since we deal with a different payoff structure with

pricing decisions separated for each competing platform as in Kim (2010) and Kim

(2011). Furthermore, our approach is distinguished from all the prior works above

in that we also model and analyze the dynamic processes, in particular, without

depending on myopic decisions as in Kim (2011).

Ⅱ. STATIC MODEL AND ANALYSIS

1. Basic Model

We consider a monopoly or a dominant platform provider, which operates two

platforms A and B. Without loss of generality, we assume that the platforms A and B

are based on the current technology and new emerging technology, respectively.

Both platforms connect suppliers and users, thereby exerting the indirect network

externality in a two-sided market explained in the previous sections. Even though

both platforms share the same ownership, two platforms are virtually competing to


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achieve the market share in each side. Release of new version for operating system

will be an example of this case.

However, the new platform B is assumed to provide all the services that the

current platform A does. For example, platform B represents NGN(Next Generation

Network) which delivers QoS(Quality of Service) guaranteed services like high

resolution videos as well as typical Internet services like email and web surfing for

which the current Internet platform (so-called ‘best effort’ Internet) was designed.

Thus, platform B is ‘backward compatible’ in the sense that it may accommodate all

the services designed for platform A that was developed by past technologies.

Our supplier market is horizontally differentiated as in a typical Hotelling model

(Hotelling (1929)) for the competing platforms (refer to Figure 1). We set up the

user market in the same way as in the supplier market. That is, a user [a supplier] is

situated on an interval [l, h], and this location reveals his/her preferences toward

both platforms. Without loss of generality, the lowest extreme (l) of the interval is

supposed to represent the user [the supplier] who most prefers A to B, while the

highest extreme (h) represents the index of the player who most prefers B to A.

Following the convention of the Hotelling model, users are uniformly populated

over the line segment. We also normalize the interval (i.e., l = 0 and h = 1) since our

model focuses on the market share in each side. The same configuration is applied

to the supplier market.

Figure 1 Monopoly provider with competing platforms (A and B)


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Now, we incorporate the key features of the two-sided markets: indirect network

externalities. First, k represents the indirect network externality in the user market

for platform k (k = either A or B). That is, k represents how users in platform k

benefit from the supplier side on the same platform. Similarly, let k denote the

indirect network externality in the supplier market for platform k (k = A, B). We fix

the network effects in platform A, A and A, at 1, and focus on the relative effects

of network externalities between the platforms. In this way, we can save the

subscripts, and let and simply represent the corresponding indirect network

externalities for platform B.

We also introduce the cross-platform network externality , which measures

users’ benefits from backward compatibility of the advanced platform (i.e., B).

Users subscribing to platform B enjoy not only the services that suppliers in

platform B provide but also the ones provided by suppliers in platform A. For

example, NGN users are able to access premium services as well as traditional best

effort services. Note that this effect is asymmetric; works only for platform B

thanks to the backward compatibility, and it directly benefits users, not suppliers.1

Let xk and yk represent the current market share of platform k (k = A, B) in the

user market and the supplier market, respectively. We assume that each market is

saturated so that xB = 1 xA and yB = 1 yA.2 Therefore, any user or supplier must

join exactly one platform, which makes us simply denote x and y as the market

shares of the platform A in the user market and in the supplier market, respectively.

With notions summarized below, we define the payoffs of users and suppliers as

follows. Here, k() means the payoff of user who is situated at on [0, 1] and

chooses platform k (k = A, B). k() is similarly defined as the payoff imputed to

the supplier indexed as in platform k (k= A, B).

- : the indirect network effect working for the users joining platform B (cf.

indirect network effect for user in platform A is set at 1)

- : the indirect network effect working for the suppliers joining platform B (cf.

indirect network effect for supplier in platform A is set at 1)

- : the cross-platform network effect working for the users joining platform B

when using services from suppliers in platform A

1 But suppliers may receive indirect benefits through the indirect network effects. 2 This saturation assumption implies that even a player with negative payoff should join a

platform which minimizes his/her loss. We can also avoid this issue by assuming very large fixed

benefit common to all the players.


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- : the user index representing his/her preference to the platforms (0 means

the one extreme for platform-A-lover and 1 means the other extreme for

platform-B-lover)

- : the supplier index representing his/her preference to the platforms (0 for

extreme platform-A-lover and 1 for extreme platform-B-lover)

- Pst: the service price set by the monopoly platform provider for platform s

(either A or B) in market t (either user or supplier)

■ Payoffs in User Market (User Utilities)

UAA Py)( [Eq.1-1]

UB

UBB Py)(1P)1(y)y1()( [Eq.1-2]

■ Payoffs in Supplier Market (Supplier Profits)

SAA Px)( [Eq. 2-1]

SB

SBB Px1P)1()x1()( [Eq. 2-2]

Let’s suppose that PAU = PB

U = P (fixed) due to a policy or regulatory

requirement that price differentiation in the user market should not be allowed.3

For instance, the net neutrality legislation prohibits the network providers from

discriminating subscribers of different types of networks: NGN vs. best-effort

network etc.4 In this case, the platform provider controls only the price gap

between the user market and the supplier market in each platform: that is, A PAS

3 This assumption seems a little bit strong in practice. For example, when a new game console

like Sony’s PlayStation is released, it is likely for Sony to set a higher price for the new version.

One may relax this assumption and deal with more practical situations where platform prices in

the user market are distinguished (i.e., PAU and PB

U instead of P). However, such a modification

will increase the complexity without significantly enhancing the analytical results. This is the

reason why we focus on the pricing gaps A and B in our model. 4 There have been lots of debates around the core proposition of the net neutrality. It might be

arguable to mention the net neutrality issue as an example of a unified pricing in the user market.

However, the principle of the same pricing for users, irrespective of technology or network types,

is also widely accepted and enforced in many countries under the name of the net neutrality.


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P and B PBS P. We also consider operating costs C(x, y) incurred from

running the advanced platform B, which increases as the size of users or suppliers

joining platform B increases: that is, ∂C

∂x,

𝜕𝐶

𝜕𝑦 0 (here, x [y] represent the size of

users [supplier] in platform A).

■ Payoffs of the Platform Provider

)y,x(CP2)y1(y)y,x(CP)y1(PyP)y,x|,( BASB

SABA

[Eq. 3]

, where both Cx ≡∂C

∂x and Cy ≡

∂C

∂y are negative.

We consider a normalized market, where the entire demand is set by one. Given

P (under the regulation that no price differentiation across the platforms is allowed

in the user market), the monopoly platform decides the price gaps k PkS P (k =

A, B) so that it can maximize its profit. Table 1 summarizes all the payoffs of

players in our model.

Table 1 Payoff Functions

Markets Platform A Platform B

Users Py)(A Py)(1)(B

Suppliers )P(x)( AA )P(x1)( BB

Platform )y,x(CP2)y1(y),( BABA

In each period, our game model proceeds as follows. First, the monopoly

platform provider determines price gaps A and B. This leads us to the next stage

that we named ‘subscription game,’ where users and suppliers respond to the price

gaps and decide the platform they will join at the current period. In this stage, their

payoffs depend on the current reference players whose position represents the


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market share of the corresponding market (i.e., x or y). If the current status is out of

equilibrium then there is an incentive for some players to change their decisions in

the next period. The next section will elaborate the notions of the reference players

and the dynamics of players’ behaviors between the consecutive periods.

2. Static Equilibrium Analysis

The static analysis here focuses on the equilibrium states based on the subscription

game. Pricing decisions of the monopoly provider will be examined from a dynamic

perspective in Section 3. We start with defining some notions that will play a

fundamental role in our model. A ‘critical’ user c in the user market indicates the

user (i.e., the user location) whose payoff is indifferent across the platforms given

the current market share x and y (c =

c(x, y)). Similarly, we define the ‘critical’

supplier c (

c =

c(x, y)). Thus, critical players satisfy the following relationships:

A(c) = B(

c) and A(

c) = B(

c). [Eq. 4]

Since we are dealing with the situation where both markets are saturated, we

allow either A(c) (= B(

c)) or A(

c) (= B(

c)) to have a negative value. We can

also derive equations for c and

c in terms of x and y from [Eq. 4] as follows.

■ Equations for Identifying Critical Participants

2

y)1(1)y,x(c [Eq. 5-1]

2

x)1(1)y,x( ABc [Eq. 5-2]

If there is an interior equilibrium (xe, y

e), it will occur at the point where the

market share of each platform coincides with the corresponding critical participant

in both markets: that is, xe =

c and y

e =

c. If the current market share x and

c in

the user market [y and c in the supplier market] do not coincides then there exist

users [suppliers] whose payoffs can be raised by changing their choices (toward the

other platform). If c x [

c y], then users [suppliers] between x and

c [y and

c]


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(i.e., currently participating in platform B) will get better off by switching to

platform A. Accordingly, it is natural to define the system dynamics as follows.

■ System Dynamics

1y)1(x22

)x)y,x((x xcx

[Eq. 6-1]

BA

ycy 1y2x)1(

2)y)y,x((y

[Eq. 6-1]

subject to x, y [0, 1]

Here, j (j = x, y) represents the speed of dynamic adjustments, and we fix j = 2

in order to simplify the expressions without deteriorating the quality of the model.

We also define the following vectors and matrices in order to get a more compact

description for the system dynamics:

y

xξ

,

y

x

ξ

,

21

12M

,

1

1k

,

11

00K

,

B

AΔ .

Now, we can compactly describe the dynamics (state equations) as below:

kΔKξMξ [Eq. 7]

and x, y [0, 1].

At an ‘interior’ equilibrium (if exists), the dynamics should halt (i.e., = 0).

Thus, one can find an interior Nash equilibrium by solving the simultaneous linear

equation system, M + K k = 0, and with given (price gaps determined by the

monopoly platform at the upper stage), one gets the unique solution * = M

1(k

K). It is easy to show from the well-known facts in the dynamic system theory

that the rest (or stationary) point * (if exists) constitutes a Nash equilibrium of the

‘subscription game.’ Specifically, at the interior equilibrium, the critical players c*

and c*

are determined as follows:


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||

)1()1()1(2 BAc

M

[Eq. 8-1]

|M|

)1(2)1()1( BAc

[Eq. 8-2]

, where | M

| = 4 ( + 1)(

+

1)

.

The following Proposition provides more detailed analysis about the interior

equilibrium in the subscription game.

Proposition 1

Suppose that the following inequality holds: + 1 . Then the interior (Nash)

equilibrium in the subscription game (if exists) is unique. The equilibrium is stable

if | M

| 0, but it is unstable (a saddle point) otherwise.

(Proof) Uniqueness comes from the linearity of the model. As for the stability, it

suffices to show that all the eigenvalues of M are negative when | M

| 0. Indeed,

under the condition above, M has two distinct eigenvalues: i.e., 1 =

)1()1(2 and 2 = )1()1(2 . The largest one is max

= 1, which cannot be non-negative if | M

| 0. Therefore, if |

M

| 0 then both

eigenvalues are negative. However, if | M

| 0 then 1 is non-negative but 2 is

negative, and the interior equilibrium becomes a saddle point. ∎

First, note that the inequality condition stated in the Proposition is quite natural.

The condition requires that the cross-platform network effects should not be too

strong to overwhelm the indirect network effects. Even though the Proposition

above provides the condition for a stable interior equilibrium (i.e., | M

| 0), the

chance to attain a positive determinant of M seems quite limited since the indirect

network effects, and , are not allowed to exhibit a proper scale in order to keep |

M | positive; for example, with = 0, both and should not be larger than one for

positive | M

|, which does not fit well with the context of our model. Figure 2

shows an unstable interior equilibrium.


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Figure 2 Example of unstable interior equilibrium: x*, y

* = (73.1%, 91.2%)

Kim (2010) provides a similar observation (but in a very different context) that

Proposition 1 implies. A sort of regularity assumption, which is frequently assumed

in market analysis studies for existence of an interior solution, may not be true in a

typical two-sided market framework. Accordingly, Proposition 1 presents the

reason that we should consider ‘boundary’ equilibriums, where at least one of the

markets tips to a specific platform; for example, all users may prefer platform B to

platform A.

However, the boundary equilibrium in one side will be highly likely to affect the

other side and make it also tip to the same platform due to indirect network

externality. Thus, it will not be plausible that one side (e.g., the user market) locks

in platform A and the other side (e.g., the supplier market) locks in platform B at the

same time. Instead, we first focus on the possibilities and conditions that both

markets tip to the same platform. However, unlike Kim(2010), the cross-platform

externality makes it possible for both platforms to coexist in the supplier market


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with the user market tipped to platform B. The following Proposition presents the

conditions under which these plausible status as a boundary equilibrium (i.e., at

least one market locks in a specific platform) can or cannot occur.

Proposition 2

When a pricing gap profile = (A, B) is given, the conditions for the boundary

equilibriums (xe, y

e) = (0, 0), (1, 1) and (0, q), where 0

q

½

(B

A +

1

)

1,

are as follows:

① 0 (x

e, y

e) = (0, 0) becomes a stable Nash equilibrium if 1 and B

A + 1,

② 1 (x

e, y

e) = (1, 1) cannot be a Nash equilibrium if 0.

③ 2 (x

e, y

e) = (0, q) becomes a stable Nash equilibrium if B

A

1

min{1, B A +

1}, 2 and (

1)(B

A +

1

) 2.

(Proof) We first present the proof for 0. First, it’s easy to show that if the

condition in ① holds then A()|0 B()|0 and A()|0 B()|0 for any feasible

and . Thus, the condition serves for 0 to be a Nash equilibrium. As for the

stability of 0, we consider a small perturbation (≪

1), which put the market

shares off the boundary equilibrium. Then, the dynamics [Eq. 7] around 0 sends the

perturbation (, ) back to

0 since both x

(,) and y

(,) are negative under the

condition stated in ①. That is, the boundary Nash equilibrium 0 is locally stable

against any small under the condition.

With a positive cross-platform externality (i.e., 0), however, there is always

an incentive to deviate from 1: in particular, for users whose location is smaller

than 1 𝛿

2. That is, it’s not a best response for those users to choose platform A at

1.

Therefore, the state of market shares 1 cannot be sustained.

As for the case of 2, the overall process of the proof is also similar to the case

of 0, except that extra caution should be paid when checking out the stability

around 2. Under the condition in ③ and the specification of q, it’s easy to see that

choosing platform B is the best response for all the users and the suppliers whose

location is greater than or equal to q.

To show the stability of 2, let’s consider again sufficiently small perturbations

(, q

) and (, q

+

); we now face two possibilities, one for each perturbation.

First, consider the perturbation (, q

) and x, the off-equilibrium dynamics along


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the x-axis around 2 (y is similarly defined). That is, x = (

+

1)q

+

1

(

+

3), and it should be negative for stability along the x-axis. Indeed, the

specification of q and straightforward algebra together with the conditions in ③

reveal that (

+

1)q

+

1

0, which in turn implies there is always a

sufficiently small (and positive) so that the perturbation can move back to 2

(along the x-axis) irrespective of the sign of

+

3. A similar method can be

applied to show that for any size of the perturbation (i.e., for any 0 in q

), y

0 under the conditions in ③.

Lastly, for the perturbation (, q +

), y =

1

A +

B

2q

+

(

1), and it

should be negative for the stability of 2 (along the y-axis). Direct algebra simplifies

this requirement into (

1) 0. This inequality is satisfied for any 0 if

1

holds, which is implied by the conditions stated above (③). As for x, similar

argument leads the conclusion that one can always find a sufficiently small

perturbation which makes x negative. ∎

Implications of Proposition 2 are self-evident. First of all, we cannot expect that

both markets tip to the relatively inferior platform unless there is a negative cross-

platform externality (i.e.,

0). Instead, since we assume that the emerging

platform has a backward compatibility at least in the user market, should be

positive, which reinforces the relative advantage of platform B. However, works

as a double-edged sword. With relatively large (see the conditions in ③), some

suppliers may remain in platform A for users who use the advanced platform to

access ‘old services.’

On the other hand, both markets can tip to platform B (i.e., 0) under milder

conditions than ones for 2. What is required for tipping to platform B in both

markets is just two simple conditions on the indirect network externalities. That is,

the indirect network effect in the user market for platform B is stronger than one for

platform A (i.e.,

1) and the pricing gap for platform B is not too high to

overwhelm the indirect network effect of platform B in the supplier market (see the

second condition in ①). Comparing the conditions above, we know that 2 demands

more stringent conditions for stability: for example, weaker indirect network effect

of platform B (i.e.,

1) and (unrealistically) strong cross-platform externality (i.e.,

2). The last two requirements do not fit well with our context of the two-sided

model.


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Until now, we have focused on the static equilibriums as steady states of a

dynamic process. However, one of the equilibriums is even unstable and the others

may present only a fragile snapshot that could exist within a very short time period.

These analyses fail to provide the nature of convergent paths toward a rest point.

Thus, we need a more specific model to deal with dynamic process toward one of

the equilibrium candidates.

One may view the equilibrium dynamics of the two-sided market in various

perspectives. One example can be found in Musacchio & Kim (2009) and Kim

(2010), where a leading platform is supposed to have an ability to select its best

equilibrium (mostly a boundary one) and maneuver the markets to the state through

controlling its price. Kim (2011) presents another approach to the dynamics which

is governed by myopic platform providers. He employs a tool for system dynamics

to simulate the dynamic behaviors of interacting providers. Both studies, however,

do not incorporate specific dynamics of strategic decisions into their models. In the

next section, we take a different perspective from the previous works, and employ

an optimal control theory to build a stylized model and analyze strategic decisions

of the platform provider in a specific dynamic context.

Ⅲ. DYNAMIC MODEL AND ANALYSIS

1. Basic Analysis

In this section, we will delve into the dynamic model for a monopoly platform

which tries to maximize its profits. Presented first is an optimal control model with

A and B as control variables for the platform provider. We further assume an

acceptable range of control: that is, i [L,

U], where the lower limit L may take

negative value since subsidization across the markets is one of key features in a

two-sided market as explained in Armstrong (2006), Parker & van Alstyne (2005),

Rochet & Tirole (2002, 2006), etc. Without deteriorating the quality of analysis, we

fix L =

1 and U

=

1 for ease of analysis.

[Eq. 6] (equivalently, [Eq. 7]) governs the dynamics led by strategic decisions

that are interrelated with each other. In this study, we focus on a ‘target equilibrium,’

with which the platform provider seeks to achieve the target equilibrium while

maximizing the total accumulated profits based on [Eq. 3]. Two stable boundary


15

equilibriums in Proposition 2 constitute the candidates of the target equilibrium in

the following optimal control decisions.

Developed here is a finite time horizon model; but, the terminal time T is not

determined in advance. Our dynamic model does not need a discount factor since

the objective functional is bounded since both control range and state space are

finite. Specifically, the objective functional J 0(-) is defined as follows:

J 0

(A, B) = T

0BA dt)y,x(CP2)y1(y [Eq. 9]

, where Cx and Cy 0. State equations are subject to [Eq. 7] and x, y [0,

1] with

initial conditions x(0) = x0 and y(0) = y0 as well as target states x(T) = xT and y(T) =

yT. Though x0 and y0 are arbitrary, xT and yT are the corresponding coordinates in [0,

1][0, 1] of a target equilibrium; for example, if the target equilibrium is

0 then xT

= yT =

0. Since P is assumed to be fixed, the monopoly platform actually maximizes

the following objective functional J(-) in J 0(A, B)

=

J(A, B)

+

2PT.

J(A, B) = T

0BA dt)y,x(C)y1(y [Eq. 9’]

Then, the Hamiltonian H(-) based on J(-) can be constructed as follows:

H(A, B) =

BA2

1BA

1y2x)1(

1y)1(x2)y,x(C)y1(y

=

)1()1(y2)1(

x)1(2)y,x(C)y1()y(

2121

21B2A2

[Eq. 10]

, where 1 and 2 are co-state variables for the corresponding state equations.

Note that the Hamiltonian function is linear in the control variables. H(-) being

linear in k (k = A, B), the maximization principle leads to corner solution for i.

Indeed, differentiation of the Hamiltonian with respect to the corresponding control

variables yields


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∂H

∂∆𝐴 = y 2 and

∂H

∂∆𝐵 = 1 y + 2 . [Eq. 11]

Thus, in view of the given control region [1, 1], the natural candidate for

optimal control should be

∆𝐴∗ = 𝑠𝑖𝑔𝑛(𝑦 − 𝜔2) and ∆𝐵

∗ = 𝑠𝑖𝑔𝑛(1 − 𝑦 + 𝜔2)

, where sign() =

1

1 if

0

0

. [Eq. 12]

We now face three cases which are mutually exclusive and cover all the possible

situations: i.e., ① y 1 + 2, ② 2 y 1 + 2, and ③ y 2. According to [Eq.

12], A* = 1 and B

* = 1 in the first case ①; both A

* and B

* = 1 in ②; A

* = 1

and B* = 1 in ③. We use these results when identifying an optimal control for each

target equilibrium in the next section.

Furthermore, since each dynamic decision problem is autonomous, the optimal

Hamiltonian has a constant value over time. In the case of a finite horizon with a

fixed terminal state, the transversality condition H(-)| t=T = 0 requires H(A*, B

*)

should be 0 along the optimal path all the time. For example, in case ① above, H(1,

1) = {212(+1)}x + {1(+1)22+2}y + 1(1) 2(1+) 1

C(x,y) = 0 over [0, T]. If 0 is given as a target state of the monopoly platform then

H(1, 1)| t=T = 1(T)(1) 2(T)(1+) 1 C(0,0) = 0, which gives one boundary

value condition for the co-state equations.

The equations of motion for the co-state variables are simply ��1 = −𝜕𝐻

𝜕𝑥 and

��2 = −𝜕𝐻

𝜕𝑦 , each of which is specified as follows:

��1 = 2𝜔1 − (𝛽 + 1) ∙ 𝜔2 + 𝐶𝑥(𝑥, 𝑦) [Eq. 13-1]

��2 = −(𝛼 − 𝛿 + 1) ∙ 𝜔1 + 2𝜔2 − (∆𝐴 − ∆𝐵) + 𝐶𝑦(𝑥, 𝑦). [Eq. 13-2]

Solving the system of differential equations in [Eq. 13], we get the following

paths of co-state variables:


17

𝜔1(𝑡) = 𝐷1 ∙ 𝑒(2+Λ)𝑡 + 𝐷2 ∙ 𝑒(2−Λ)𝑡 −2𝐶𝑥+(𝛽+1)∙(𝐶𝑦−∆𝐴+∆𝐵)

4−Λ2 [Eq. 14-1]

𝜔2(𝑡) = −Θ ∙ 𝐷1 ∙ 𝑒(2+Λ)𝑡 + Θ ∙ 𝐷2 ∙ 𝑒(2−Λ)𝑡 −(𝛼−𝛿+1)∙𝐶𝑥+2(𝐶𝑦−∆𝐴+∆𝐵)

4−Λ2 [Eq. 14-2]

, where √(𝛽 + 1) ∙ (𝛼 − 𝛿 + 1), √𝛼−𝛿+1

𝛽+1, and Di’s are constants to be

determined by the transversality conditions above.

Similarly, one can solve the system dynamics (refer to [Eq. 7]) as follows:

𝑥(𝑡) = 𝐸1 ∙ 𝑒−(2+Λ)𝑡 + 𝐸2 ∙ 𝑒−(2−Λ)𝑡 +2(1−𝛼)+(𝛼−𝛿+1)∙(1−𝛽−∆𝐴+∆𝐵)

4−Λ2 [Eq. 15-1]

𝑦(𝑡) = −𝐸1

Θ∙ 𝑒−(2+Λ)𝑡 +

𝐸2

Θ∙ 𝑒−(2−Λ)𝑡 +

(1−𝛼)∙(𝛽+1)+2(1−𝛽−∆𝐴+∆𝐵)

4−Λ2 [Eq. 15-2]

, where Ei’s are constants determined by the initial states and the boundary

conditions once i’s are determined.

2. Optimal Control of Price Gaps

In this section, we consider two possible target states from Proposition 2 (except

1), each of which corresponds to a stable equilibrium in the static model. For

tractability of analysis, we specify the cost function C(x, y) as a constant function

(i.e., C(x, y) = C). Then, both Cx and Cy vanish in [Eq. 14], which simplifies the

relevant equations. For each target equilibrium state, we first find an optimal

control over a finite horizon.

■ 0 as Target Equilibrium

In this scenario, the monopoly platform wishes to find a dynamic optimal strategy

(A*(t), B

*(t)) that maneuvers the system into

0 = (0, 0), while maximizing the

objective functional. Note that all the users and suppliers join only platform B in the

given target equilibrium. Considering the shape of the Hamiltonian function, we

know that it will be a good starting point to take a bang-bang policy or its simplified

version called ‘corner solution’ as a candidate for an optimal strategy. Supposing


18

that the platform employs a corner solution, our first conjecture is that the pricing

control profile {A* =1, B

* = 1} presents an optimal control toward

0. The

following Proposition shows that the control profile above is consistent with the

necessary conditions for dynamic optimality starting from a set of initial states.

Proposition 3

Let’s suppose that ,

1,

+

1

0 and

2

4

0 (thus,

2

0). There exists

a basin of attraction in the state space, where the pricing profile {A = 1, B = 1}

becomes an optimal control for maximizing J(A, B) in [Eq. 9’] with the target

state 0.

(Proof) First note that the proposed pricing profile maximizes the Hamiltonian

defined in [Eq. 10] when case ① holds all the time: that is, y(t) 1 +

2(t) for all t

in [0, T]. We will examine this possibility by incorporating [Eq. 14-2] and [Eq. 15-2]

into the inequality above. Rearranging terms results in the following inequality,

which actually claims y(t) 1 +

2(t) for all t (with A = 1 and B = 1).

Θ ∙ (𝐷1 ∙ 𝑒(2+Λ)𝑡 − 𝐷2 ∙ 𝑒(2−Λ)𝑡) −1

Θ∙ (𝐸1 ∙ 𝑒−(2+Λ)𝑡 − 𝐸2 ∙ 𝑒−(2−Λ)𝑡)−

4+𝛿∙(𝛽+1)

Λ2−4

.

The conditions in the Proposition make the right-hand side of the inequality

negative. One can also determine Ei’s and Di’s in [Eq. 14] and [Eq. 15] (with A = 1

and B = 1) so that the left of the inequality can be positive. Indeed, at least for 𝐷1

𝐸1

Θ2 and 𝐷2

𝐸2

Θ2, the left-hand remains positive for all t. These Ei’s and Di’s are

compatible with the transversality conditions and the boundary conditions.

Therefore, there exists a basin of attraction for the target equilibrium with the

control profile proposed above. ■

■ 2 as Target Equilibrium

This scenario deals with a monopoly platform which controls the price gaps and

maneuvers the markets into 2 = (0, q). Even though the user market tips to the

advanced platform (B), there still remain some suppliers (fraction of q) dedicated to

platform A since some users will use the advanced platform only for old services.

Thus, the best conjecture for an optimal control will be to set B = 1 which


19

makes the option to stay in platform A attractive to some suppliers. The optimal

policy for users may be different from case by case and depend on the strength of

the network externality for platform B in the supplier market (i.e., the size of ).

The following Proposition provides details of possible optimal policies with 2 as

the target state.

Proposition 4

Suppose that +

+

11 and

2

4

0 hold. With 1

3, there exists a basin

of attraction, where the pricing profile {A = 1, B = 1} provides an optimal

control for maximizing J(A, B) with the target state

2 =

(0,3−𝛽

2).

(Proof) The proof follows a path similar to Proposition 3. The proposed pricing

profile maximizes the Hamiltonian defined in [Eq. 10] when case ③ holds all the

time: i.e., y 2 for all t. By incorporating [Eq. 14-2] and [Eq. 15-2] into the

inequality above and rearranging the resulting terms, we get the following

inequality, which actually claims y(t) 2(t) (with A = 1 and B = 1) for all t.

Θ ∙ (𝐷1 ∙ 𝑒(2+Λ)𝑡 − 𝐷2 ∙ 𝑒(2−Λ)𝑡) −1

Θ∙ (𝐸1 ∙ 𝑒−(2+Λ)𝑡 − 𝐸2 ∙ 𝑒−(2−Λ)𝑡)

11−𝛼∙(𝛽+1)−𝛽

Λ2−4

.

The conditions in the Proposition make the right-hand side of the inequality

positive. One can also determine Ei’s and Di’s in [Eq. 14] and [Eq. 15] (with A = 1

and B = 1) so that the left-hand side of the inequality can be negative. For

example, sufficiently small D1 and E2 restrain the left-hand from growing bigger

than the constant value of the right-hand side. Also, with 0 𝐷1

𝐸1

Θ2 and 𝐷2

𝐸2

Θ2

0, the left-hand remains negative for all t. Therefore, there exists a basin of

attraction for the target equilibrium with the control profile proposed above. ∎

Propositions 3 and 4 show only the existence of the basin of attractions for optimal

controls, each of which targets a specific equilibrium. Those Propositions also

provide the relevant conditions under which the respective pricing policy can hit the

target. For example, if the monopoly provider sets 0 as its target state then it will

maneuver the system into the target with the pricing profile {A = 1, B = 1} while

maximizing its objective. This pricing plan confirms our intuition from the previous


20

studies on the two-sided markets in that the subsidization of the suppliers in

platform B eventually results in proliferation of suppliers in platform B as well as

users in the same platform thanks to the indirect network externality. Proposition 3

also indicates the conditions for this plan; that is, the indirect network externalities

and should be sufficiently big (,

1), and the indirect network effect

working in platform A should be at least compatible with the cross platform

externality ( +

1

).

Ⅳ. DISCUSSIONS AND CONCLUSIONS

We presented a two-sided market model with two competing platforms which are

run by a monopoly provider. We identified and analyzed the Nash equilibriums in a

static setting as well as dynamic optimal pricing to attain some target equilibriums.

Our first finding was that an interior equilibrium in our static context is fragile

against a small shock. Thus, we had to specify possible boundary equilibriums

together with their conditions for a stable Nash equilibrium (in a static sense). One

interesting equilibrium type was 2 = (0, q) in Propositions 2, where the q fraction

of suppliers is designated to the old platform thanks to the backward compatibility

(). It was also shown that the tipping toward the old platform (1) is impossible

with the capability of the backward compatibility. Lastly, by employing the optimal

control theory, we formulated the dynamic version of our model, where a monopoly

platform which tries to determine an optimal path of the pricing profile leading to a

target equilibrium.

Our basic analysis reveals that a dynamic feature is essential for the two-sided

markets. Dynamic analysis in the previous section found various optimal controls,

each of which identifies an optimal path leading to the corresponding target

equilibrium. For example, a strong subsidization for suppliers choosing the

advanced platform (i.e., B = 1) will tip both markets to the platform (here,

platform B) which is assumed more profitable to the monopoly provider. Once all

the players choose platform B (i.e., 0), they get stuck there as Proposition 2 points

out; 0 is a stable equilibrium under the conditions compatible with the control

profile. On the other hand, a control profile with a subsidy to the suppliers in

platform A may provide a room for survival of some suppliers in platform A. Thus,


21

the equilibrium type of 2 in Proposition 2 can be realized as a steady state under

the conditions and control profiles in Proposition 4; it depends on the monopoly’s

preference. The backward compatibility plays a critical role here.

Our analysis presents the monopoly platform provider with some strategic

implications to maximize its overall benefits. In particular, it is interesting to

examine the role of the cross platform externality or the backward compatibility ()

in our dynamic setting. Even though we do not incorporate a multi-homing in each

market side, our model results in a possibility of coexistence of two competing

platforms (at least in the supplier market, 2) by means of . Thus, the role of the

cross-platform externality is crucial in our models. This finding can be also

interpreted as implying the undiminished significance of backward compatibility in

the two-sided markets: in particular, with (relatively) weak cross network

externalities.

In our future study, we will combine all the previous analyses and examine the

effects of important parameters such as indirect- and cross-platform externalities on

the solution paths. It may be necessary to conduct some experiments by employing

numerical simulations in order to supplement the analysis and visualize the

dynamics of the system behavior around the target equilibriums. We will also

extend our models in order to deal with oligopolistic competition of platform

providers in a two-sided market. With an extended model, we are able to compare

the effects of competition types on social welfare.

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Equilibrium Analysis of a Two-Sided Market with Multiple