Subscription Dynamics and Competition in Communications

Subscription Dynamics and Competition in

Communications Markets

Shaolei Ren, Jaeok Park, and Mihaela van der Schaar

Electrical Engineering DepartmentUniversity of California, Los Angeles

October 3, 2010

Ren, Park, van der Schaar (UCLA) Multimedia Communications & Systems Lab October 2010 1 / 26

Introduction

Outline

1 Introduction

2 Model

3 User Subscription DynamicsEquilibrium AnalysisConvergence Analysis

4 Competition in Duopoly Markets

5 Illustrative Example

6 Conclusion


Introduction

Overview of Communications Markets

UsersService

Providers

QoS

Technology

Price

Revenue

Demand

Investment

Interaction among technology, users and service providers


Introduction

Our Work

UsersService

Providers

QoS

Technology

Price

Revenue

Demand

Investment

How does the technology influence the users’ demand and the service providers’

revenues?

• We consider a duopoly communications market.

• Given prices, how does QoS affect the subscription decisions (or demand) of users?

• How are prices determined through competition between the service providers?


Model

Outline

1 Introduction

2 Model




6 Conclusion


Model

Model

……

NSP 1 NSP 2


Model

Model

……

NSP 1 NSP 2

Network model

• network service providers: S1 and S2

• continuum model: a large number of users


Model

Model

Service providers

• Si : price pi and fraction of subscribers λi (pi , p−i)


Model

Model

Service providers


• utility (revenue): Ri (pi , p−i) = piλi (pi , p−i)


Model

Model

Service providers



Users

• user k : uk = αkqi − pi if it subscribes to Si


Model

Model

Service providers



Users


• αk follows a distribution with PDF f (α)


Model

Model

Service providers



Users



assumptions on f (α)

• f (α) > 0 if α ∈ [0, β] and f (α) = 0 otherwise

• f (α) is continuous on [0, β]


Model

Model

Service providers



Users



QoS model

• q1 is constant

• q2 = g(λ2), where g(λ2) ∈ (0, q1) is a differentiable and non-increasingfunction of λ2 ∈ [0, 1]


User Subscription Dynamics

Outline

1 Introduction

2 Model




6 Conclusion



User Subscription

• Discrete-time model {(λt1, λ

t2) | t = 0, 1, 2 · · · }

• Users’ belief model and subscription decisions

• naive (or static) expectation: every user expects that the QoS in the currentperiod is equal to that in the previous period (i.e., g̃k (λ

t2) = g(λt−1

2 ))• a user subscribes to whichever NSP provides a higher (non-negative) utility



User Subscription


t2) | t = 0, 1, 2 · · · }



t2) = g(λt−1


• Dynamics of user subscriptions



User Subscription


t2) | t = 0, 1, 2 · · · }



t2) = g(λt−1



if p1q1

> p2

g(λt−12 )

, then

λt1 = hd,1(λ

t−11 , λt−1

2 ) = 1− F

(

p1 − p2

q1 − g(λt−12 )

)

,

λt2 = hd,2(λ

t−11 , λt−1

2 ) = F

(

p1 − p2

q1 − g(λt−12 )

)

− F

(

p2

g(λt−12 )

)



User Subscription


t2) | t = 0, 1, 2 · · · }

• Users’ belief model and subscription decisions• naive (or static) expectation: every user expects that the QoS in the current

period is equal to that in the previous period (i.e., g̃k (λt2) = g(λt−1



if p1q1

≤ p2

g(λt−12 )

, then

λt1 = hd,1(λ

t−11 , λt−1

2 ) = 1− F

(

p1q1

)

,

λt2 = hd,2(λ

t−11 , λt−1

2 ) = 0.


User Subscription Dynamics Equilibrium Analysis

Equilibrium Analysis

• Stabilized fraction of subscribers will stabilize in the long run

Definition

(λ∗

1 , λ∗

2) is an equilibrium point of the user subscription dynamics in the duopolymarket if it satisfies hd,1(λ

∗

1 , λ∗

2) = λ∗

1 and hd,2(λ∗

1 , λ∗

2) = λ∗

2 .



Equilibrium Analysis

• Stabilized fraction of subscribers will stabilize in the long run

Proposition (uniqueness and existence of (λ∗

1 , λ∗

2))

For any non-negative price pair (p1, p2), there exists a unique equilibrium point(λ∗

1 , λ∗

2) of the user subscription dynamics in the duopoly market. Moreover,(λ∗

1 , λ∗

2) satisfies

λ∗

1 = 1− F

(

p1

q1

)

, λ∗

2 = 0, ifp1

q1≤

p2

g(0),

λ∗

1 = 1− F (θ∗1 ) , λ∗

2 = F (θ∗1 )− F (θ∗2 ) , ifp1

q1>

p2

g(0),

where θ∗1 = (p1 − p2)/(q1 − g(λ∗

2)) and θ∗2 = p2/g(λ∗

2).



Equilibrium Market Shares

0

1

2

3

0

1

2

30

0.2

0.4

0.6

0.8

1

p1

p2

λ*

λ1*

λ2*

• q1 = 2.5, g(λ2) = 1.2e−0.5λ2 , and α is uniformly distributed on [0, 1], i.e., fa(α) = 1 for α ∈ [0, 1].


User Subscription Dynamics Convergence Analysis

Convergence of User Subscription Dynamics

• Convergence is not always guaranteed





Example: when the QoS of NSP S2 degrades fast w.r.t. the fraction of subscribers

1 suppose that only a small fraction of users subscribe to NSP S2 at period t andeach subscriber obtains a high QoS

2 a large fraction of users subscribe at period t + 1, which will result in a low QoS atperiod t + 1

3 a small fraction of subscribers at period t + 2





Theorem

For any non-negative price pair (p1, p2), the user subscription dynamics convergesto the unique equilibrium point starting from any initial point (λ0

1, λ02) ∈ Λ if

maxλ2∈[0,1]

{

−g ′(λ2)

g(λ2)·

q1q1 − g(λ2)

}

<1

K,

where K = maxα∈[0,β] f (α)α. proof



Illustration of Oscillation & Convergence

2 4 6 8 10 12 140

0.2

0.4

0.6

0.8

t

λ 2

λ2: g(λ

2)=e−2λ

2

2 4 6 8 10 12 140.3

0.32

0.34

0.36

0.38

0.4

t

λ 2

λ2: g(λ

2)=e−0.8λ

2

t = 2




2 4 6 8 10 12 140

0.2

0.4

0.6

0.8

t

λ 2

2 4 6 8 10 12 140.3

0.32

0.34

0.36

0.38

0.4

t

λ 2

λ2: g(λ

2)=e−2λ

2

λ2: g(λ

2)=e−0.8λ

2

t = 3




2 4 6 8 10 12 140

0.2

0.4

0.6

0.8

t

λ 2

2 4 6 8 10 12 140.3

0.32

0.34

0.36

0.38

0.4

t

λ 2

λ2: g(λ

2)=e−2λ

2

λ2: g(λ

2)=e−0.8λ

2

t = 4




2 4 6 8 10 12 140

0.2

0.4

0.6

0.8

t

λ 2

2 4 6 8 10 12 140.3

0.32

0.34

0.36

0.38

0.4

t

λ 2

λ2: g(λ

2)=e−2λ

2

λ2: g(λ

2)=e−0.8λ

2

t = 5




2 4 6 8 10 12 140

0.2

0.4

0.6

0.8

t

λ 2

2 4 6 8 10 12 140.3

0.32

0.34

0.36

0.38

0.4

t

λ 2

λ2: g(λ

2)=e−2λ

2

λ2: g(λ

2)=e−0.8λ

2

t = 6




2 4 6 8 10 12 140

0.2

0.4

0.6

0.8

t

λ 2

λ2: g(λ

2)=e−2λ

2

2 4 6 8 10 12 140.3

0.32

0.34

0.36

0.38

0.4

t

λ 2

λ2: g(λ

2)=e−0.8λ

2

t = 7




2 4 6 8 10 12 140

0.2

0.4

0.6

0.8

t

λ 2

2 4 6 8 10 12 140.3

0.32

0.34

0.36

0.38

0.4

t

λ 2

λ2: g(λ

2)=e−2λ

2

λ2: g(λ

2)=e−0.8λ

2

t = 15


Competition in Duopoly Markets

Outline

1 Introduction

2 Model




6 Conclusion



Cournot Competition

• We model competition between the NSPs using Cournot competition.

• each NSP chooses the fraction of subscribers independently• prices are determined such that the equilibrium market shares equate the

chosen quantities



Cournot Competition



chosen quantities

• GC = {Si ,Ri(λ1, λ2), λi ∈ [0, 1) | i = 1, 2}



Cournot Competition



chosen quantities

• GC = {Si ,Ri(λ1, λ2), λi ∈ [0, 1) | i = 1, 2}

• (λ∗∗

1 , λ∗∗

2 ) is a (pure) NE of GC (or a Cournot equilibrium) if it satisfies

Ri (λ∗∗

i , λ∗∗

−i ) ≥ Ri (λi , λ∗∗

−i ), ∀ λi ∈ [0, 1), ∀ i = 1, 2 .



Existence of NE

Lemma

Suppose that f (·) is non-increasing on [0, β]. Let λ̃i (λ−i ) be a market share thatmaximizes the revenue of NSP Si provided that NSP S−i chooses λ−i ∈ [0, 1),i.e., λ̃i (λ−i ) ∈ argmaxλi∈[0,1) Ri(λi , λ−i ). Then λ̃i (λ−i ) ∈ (0, 1/2] for all

λ−i ∈ [0, 1), for all i = 1, 2. Moreover, λ̃i (λ−i ) 6= 1/2 if λ−i > 0, for i = 1, 2.



Existence of NE

Lemma

Suppose that f (·) is non-increasing on [0, β]. Let λ̃i (λ−i ) be a market share thatmaximizes the revenue of NSP Si provided that NSP S−i chooses λ−i ∈ [0, 1),i.e., λ̃i (λ−i ) ∈ argmaxλi∈[0,1) Ri(λi , λ−i ). Then λ̃i (λ−i ) ∈ (0, 1/2] for all

λ−i ∈ [0, 1), for all i = 1, 2. Moreover, λ̃i (λ−i ) 6= 1/2 if λ−i > 0, for i = 1, 2.

• Implication

• when the strategy space is specified as [0, 1) and f (·) satisfies thenon-increasing property, strategies λi ∈ {0} ∪ (1/2, 1) is strictly dominated fori = 1, 2

• if a NE (λ∗∗

1 , λ∗∗

2 ) of G̃C exists, then it must satisfy (λ∗∗

1 , λ∗∗

2 ) ∈ (0, 1/2)2



Existence of NE

Theorem

Suppose that f (·) is non-increasing and continuously differentiable on [0, β]. Iff (·) and g(·) satisfy some conditions (Eqn. 18 and Eqn. 19 in the paper), thenthe game G̃C has at least one NE.



Existence of NE

Corollary

Suppose that the users’ valuation of QoS is uniformly distributed, i.e., f (α) = 1/βfor α ∈ [0, β]. If g(λ2) + λ2g

′(λ2) ≥ 0 for all λ2 ∈ [0, 1/2], then the game GC hasat least one NE.

• Interpretation

• if the elasticity of the QoS provided by NSP S2 with respect to the fraction ofits subscribers is no larger than 1 (i.e., −[g ′(λ2)λ2/g(λ2)] ≤ 1), the Cournotcompetition game with the strategy space [0, 1) has at least one NE

• the condition is analogous to the sufficient conditions for convergence in thatit requires that the QoS provided by NSP S2 cannot degrade too fast withrespect to the fraction of subscribers.


Illustrative Example

Outline

1 Introduction

2 Model




6 Conclusion



Numerical Results

0 5 10 15 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

t

(λ1,λ

2)

λ1

λ2

Figure: Dynamics of market shares under the best-response dynamics. Solid:g(λ2) = 1− λ2

8; dashed: g(λ2) = 1− λ2

2.



Numerical Results

0 5 10 15 200

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

t

(R1,R

2)

R1

R2

Figure: Iteration of revenues under the best-response dynamics. Solid: g(λ2) = 1− λ28;

dashed: g(λ2) = 1− λ22.


Conclusion

Outline

1 Introduction

2 Model




6 Conclusion


Conclusion

Conclusion

Study the impacts of technologies on the user subscription dynamics

• constructed the dynamics of user subscription based on myopic updates

• showed that the existence of a unique equilibrium point of the usersubscription dynamics

• provided a sufficient condition for the convergence of the user subscriptiondynamics: the QoS provided by NSP S2 should not degrade too fast as moreusers subscribe


Conclusion

Conclusion

Study the impacts of technologies on the user subscription dynamics

• constructed the dynamics of user subscription based on myopic updates

• showed that the existence of a unique equilibrium point of the usersubscription dynamics

• provided a sufficient condition for the convergence of the user subscriptiondynamics: the QoS provided by NSP S2 should not degrade too fast as moreusers subscribe

Study the impacts of technologies on competition between the NSPs

• modeled the NSPs as strategic players in a non-cooperative Cournot game

• provided a sufficient condition that ensures the existence of at least one NEof the game


Conclusion

Selected References

• J. Musacchio and D. Kim, “Network platform competition in a two-sided market: implications to the netneutrality issue,” TPRC: Conference on Communication, Information, and Internet Policy, Sep. 2009.

• D. Bertsimas and G. Perakis, “Dynamic pricing: a learning approach,” Models for CongestionCharging/Network Pricing, 2005.

• G. Bitran and R. Galdentey, “An overview of pricing models for revenue management,” Manufacturing& Service Operational Management, vol. 5, no. 3, pp. 203-229, Summer 2003.

• F. Kelly, “Charging and rate control for elastic traffic,” European Transactions on Telecommunications,vol. 8, pp. 33-37, 1997.


Conclusion

Related Publications

• S. Ren, J. Park, and M. van der Schaar, “Dynamics of Service Provider Selection in CommunicationMarkets,” accepted and to appear in Proc. IEEE Globecom 2010.

• S. Ren, J. Park, and M. van der Schaar, “Subscription Dynamics and Competition in CommunicationMarkets,” in Proc. ACM NetEcon 2010.

• S. Ren, J. Park, and M. van der Schaar, “User Subscription Dynamics and Revenue Maximization inCommunication Markets,” UCLA Tech. Report, Aug. 2010 (available athttp://arxiv.org/abs/1008.5367).


Conclusion


Proof.

1 Show that

‖hd(λ1,a, λ2,a)− hd(λ1,b, λ2,b)‖∞

= K

[

−g ′(λ2,c)

g(λ2,c)·

q1q1 − g(λ2,c)

]

|λ2,a − λ2,b|

≤ κd ‖λa − λb‖∞ .

where κd = K ·maxλ2∈[0,1] {[−g ′(λ2)/g(λ2)] · [q1/(q1 − g(λ2))]}


Conclusion


Proof.

1 Show that

‖hd(λ1,a, λ2,a)− hd(λ1,b, λ2,b)‖∞

= K

[

−g ′(λ2,c)

g(λ2,c)·

q1q1 − g(λ2,c)

]

|λ2,a − λ2,b|

≤ κd ‖λa − λb‖∞ .

where κd = K ·maxλ2∈[0,1] {[−g ′(λ2)/g(λ2)] · [q1/(q1 − g(λ2))]}

2 If maxλ2∈[0,1]

{

− g ′(λ2)g(λ2)

· q1q1−g(λ2)

}

< 1K

, then the mapping is contraction!

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Subscription Dynamics and Competition in Communications