Subscription Dynamics and Competition in Communications Markets - - PowerPoint PPT Presentation

▶
subscription dynamics and competition in communications
SMART_READER_LITE
LIVE PREVIEW

Subscription Dynamics and Competition in Communications Markets - - PowerPoint PPT Presentation

Feb 19, 2023 252 likes •749 views

Subscription Dynamics and Competition in Communications Markets Shaolei Ren, Jaeok Park, and Mihaela van der Schaar Electrical Engineering Department University of California, Los Angeles October 3, 2010 Ren, Park, van der Schaar (UCLA)

slide-1
SLIDE 1

Subscription Dynamics and Competition in Communications Markets

Shaolei Ren, Jaeok Park, and Mihaela van der Schaar

Electrical Engineering Department University of California, Los Angeles October 3, 2010

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

slide-2
SLIDE 2

Introduction

Outline

1 Introduction 2 Model 3 User Subscription Dynamics

Equilibrium Analysis Convergence Analysis

4 Competition in Duopoly Markets 5 Illustrative Example 6 Conclusion

Ren, Park, van der Schaar (UCLA) Multimedia Communications & Systems Lab October 2010 2 / 26

slide-3
SLIDE 3

Introduction

Overview of Communications Markets

  • Interaction among technology, users and service providers

Ren, Park, van der Schaar (UCLA) Multimedia Communications & Systems Lab October 2010 3 / 26

slide-4
SLIDE 4

Introduction

Our Work

  • 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?

Ren, Park, van der Schaar (UCLA) Multimedia Communications & Systems Lab October 2010 4 / 26

slide-5
SLIDE 5

Model

Outline

1 Introduction 2 Model 3 User Subscription Dynamics

Equilibrium Analysis Convergence Analysis

4 Competition in Duopoly Markets 5 Illustrative Example 6 Conclusion

Ren, Park, van der Schaar (UCLA) Multimedia Communications & Systems Lab October 2010 5 / 26

slide-6
SLIDE 6

Model

Model

……

  • Ren, Park, van der Schaar (UCLA)

Multimedia Communications & Systems Lab October 2010 6 / 26

slide-7
SLIDE 7

Model

Model

……

  • Network model
  • network service providers: S1 and S2
  • continuum model: a large number of users

Ren, Park, van der Schaar (UCLA) Multimedia Communications & Systems Lab October 2010 6 / 26

slide-8
SLIDE 8

Model

Model

Service providers

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

Ren, Park, van der Schaar (UCLA) Multimedia Communications & Systems Lab October 2010 7 / 26

slide-9
SLIDE 9

Model

Model

Service providers

  • Si: price pi and fraction of subscribers λi(pi, p−i)
  • utility (revenue): Ri(pi, p−i) = piλi(pi, p−i)

Ren, Park, van der Schaar (UCLA) Multimedia Communications & Systems Lab October 2010 7 / 26

slide-10
SLIDE 10

Model

Model

Service providers

  • Si: price pi and fraction of subscribers λi(pi, p−i)
  • utility (revenue): Ri(pi, p−i) = piλi(pi, p−i)

Users

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

Ren, Park, van der Schaar (UCLA) Multimedia Communications & Systems Lab October 2010 7 / 26

slide-11
SLIDE 11

Model

Model

Service providers

  • Si: price pi and fraction of subscribers λi(pi, p−i)
  • utility (revenue): Ri(pi, p−i) = piλi(pi, p−i)

Users

  • user k: uk = αkqi − pi if it subscribes to Si
  • αk follows a distribution with PDF f (α)

Ren, Park, van der Schaar (UCLA) Multimedia Communications & Systems Lab October 2010 7 / 26

slide-12
SLIDE 12

Model

Model

Service providers

  • Si: price pi and fraction of subscribers λi(pi, p−i)
  • utility (revenue): Ri(pi, p−i) = piλi(pi, p−i)

Users

  • user k: uk = αkqi − pi if it subscribes to Si
  • αk follows a distribution with PDF f (α)

assumptions on f (α)

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

Ren, Park, van der Schaar (UCLA) Multimedia Communications & Systems Lab October 2010 7 / 26

slide-13
SLIDE 13

Model

Model

Service providers

  • Si: price pi and fraction of subscribers λi(pi, p−i)
  • utility (revenue): Ri(pi, p−i) = piλi(pi, p−i)

Users

  • user k: uk = αkqi − pi if it subscribes to Si
  • αk follows a distribution with PDF f (α)

QoS model

  • q1 is constant
  • q2 = g(λ2), where g(λ2) ∈ (0, q1) is a differentiable and non-increasing

function of λ2 ∈ [0, 1]

Ren, Park, van der Schaar (UCLA) Multimedia Communications & Systems Lab October 2010 7 / 26

slide-14
SLIDE 14

User Subscription Dynamics

Outline

1 Introduction 2 Model 3 User Subscription Dynamics

Equilibrium Analysis Convergence Analysis

4 Competition in Duopoly Markets 5 Illustrative Example 6 Conclusion

Ren, Park, van der Schaar (UCLA) Multimedia Communications & Systems Lab October 2010 8 / 26

slide-15
SLIDE 15

User Subscription Dynamics

User Subscription

  • Discrete-time model {(λt

1, λt 2) | 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., ˜ gk(λt

2) = g(λt−1 2

))

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

Ren, Park, van der Schaar (UCLA) Multimedia Communications & Systems Lab October 2010 9 / 26

slide-16
SLIDE 16

User Subscription Dynamics

User Subscription

  • Discrete-time model {(λt

1, λt 2) | 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., ˜ gk(λt

2) = g(λt−1 2

))

  • a user subscribes to whichever NSP provides a higher (non-negative) utility
  • Dynamics of user subscriptions

Ren, Park, van der Schaar (UCLA) Multimedia Communications & Systems Lab October 2010 9 / 26

slide-17
SLIDE 17

User Subscription Dynamics

User Subscription

  • Discrete-time model {(λt

1, λt 2) | 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., ˜ gk(λt

2) = g(λt−1 2

))

  • a user subscribes to whichever NSP provides a higher (non-negative) utility
  • Dynamics of user subscriptions

if p1

q1 > p2 g(λt−1

2

), then

λt

1 =

hd,1(λt−1

1

, λt−1

2

) = 1 − F

  • p1 − p2

q1 − g(λt−1

2

)

  • ,

λt

2 =

hd,2(λt−1

1

, λt−1

2

) = F

  • p1 − p2

q1 − g(λt−1

2

)

  • − F
  • p2

g(λt−1

2

)

  • Ren, Park, van der Schaar (UCLA)

Multimedia Communications & Systems Lab October 2010 9 / 26

slide-18
SLIDE 18

User Subscription Dynamics

User Subscription

  • Discrete-time model {(λt

1, λt 2) | 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., ˜ gk(λt

2) = g(λt−1 2

))

  • a user subscribes to whichever NSP provides a higher (non-negative) utility
  • Dynamics of user subscriptions

if p1

q1 ≤ p2 g(λt−1

2

), then

λt

1 =

hd,1(λt−1

1

, λt−1

2

) = 1 − F p1 q1

  • ,

λt

2 =

hd,2(λt−1

1

, λt−1

2

) = 0.

Ren, Park, van der Schaar (UCLA) Multimedia Communications & Systems Lab October 2010 9 / 26

slide-19
SLIDE 19

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 duopoly

market if it satisfies hd,1(λ∗

1, λ∗ 2) = λ∗ 1 and hd,2(λ∗ 1, λ∗ 2) = λ∗ 2.

Ren, Park, van der Schaar (UCLA) Multimedia Communications & Systems Lab October 2010 10 / 26

slide-20
SLIDE 20

User Subscription Dynamics Equilibrium Analysis

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,

if p1 q1 ≤ p2 g(0), λ∗

1 = 1 − F (θ∗ 1 ) , λ∗ 2 = F (θ∗ 1 ) − F (θ∗ 2 ) ,

if p1 q1 > p2 g(0),

where θ∗

1 = (p1 − p2)/(q1 − g(λ∗ 2)) and θ∗ 2 = p2/g(λ∗ 2).

Ren, Park, van der Schaar (UCLA) Multimedia Communications & Systems Lab October 2010 10 / 26

slide-21
SLIDE 21

User Subscription Dynamics Equilibrium Analysis

Equilibrium Market Shares

1 2 3 1 2 3 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].

Ren, Park, van der Schaar (UCLA) Multimedia Communications & Systems Lab October 2010 11 / 26

slide-22
SLIDE 22

User Subscription Dynamics Convergence Analysis

Convergence of User Subscription Dynamics

  • Convergence is not always guaranteed

Ren, Park, van der Schaar (UCLA) Multimedia Communications & Systems Lab October 2010 12 / 26

slide-23
SLIDE 23

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 and

each subscriber obtains a high QoS

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

period t + 1

3 a small fraction of subscribers at period t + 2

Ren, Park, van der Schaar (UCLA) Multimedia Communications & Systems Lab October 2010 12 / 26

slide-24
SLIDE 24

User Subscription Dynamics Convergence Analysis

Convergence of User Subscription Dynamics

  • Convergence is not always guaranteed

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

1, λ0 2) ∈ Λ if

max

λ2∈[0,1]

  • −g ′(λ2)

g(λ2) · q1 q1 − g(λ2)

  • < 1

K , where K = maxα∈[0,β] f (α)α.

proof Ren, Park, van der Schaar (UCLA) Multimedia Communications & Systems Lab October 2010 12 / 26

slide-25
SLIDE 25

User Subscription Dynamics Convergence Analysis

Illustration of Oscillation & Convergence

2 4 6 8 10 12 14 0.2 0.4 0.6 0.8 t λ2 λ2: g(λ2)=e−2λ

2

2 4 6 8 10 12 14 0.3 0.32 0.34 0.36 0.38 0.4 t λ2 λ2: g(λ2)=e−0.8λ

2

t = 2

Ren, Park, van der Schaar (UCLA) Multimedia Communications & Systems Lab October 2010 13 / 26

slide-26
SLIDE 26

User Subscription Dynamics Convergence Analysis

Illustration of Oscillation & Convergence

2 4 6 8 10 12 14 0.2 0.4 0.6 0.8 t λ2 2 4 6 8 10 12 14 0.3 0.32 0.34 0.36 0.38 0.4 t λ2 λ2: g(λ2)=e−2λ

2

λ2: g(λ2)=e−0.8λ

2

t = 3

Ren, Park, van der Schaar (UCLA) Multimedia Communications & Systems Lab October 2010 13 / 26

slide-27
SLIDE 27

User Subscription Dynamics Convergence Analysis

Illustration of Oscillation & Convergence

2 4 6 8 10 12 14 0.2 0.4 0.6 0.8 t λ2 2 4 6 8 10 12 14 0.3 0.32 0.34 0.36 0.38 0.4 t λ2 λ2: g(λ2)=e−2λ

2

λ2: g(λ2)=e−0.8λ

2

t = 4

Ren, Park, van der Schaar (UCLA) Multimedia Communications & Systems Lab October 2010 13 / 26

slide-28
SLIDE 28

User Subscription Dynamics Convergence Analysis

Illustration of Oscillation & Convergence

2 4 6 8 10 12 14 0.2 0.4 0.6 0.8 t λ2 2 4 6 8 10 12 14 0.3 0.32 0.34 0.36 0.38 0.4 t λ2 λ2: g(λ2)=e−2λ

2

λ2: g(λ2)=e−0.8λ

2

t = 5

Ren, Park, van der Schaar (UCLA) Multimedia Communications & Systems Lab October 2010 13 / 26

slide-29
SLIDE 29

User Subscription Dynamics Convergence Analysis

Illustration of Oscillation & Convergence

2 4 6 8 10 12 14 0.2 0.4 0.6 0.8 t λ2 2 4 6 8 10 12 14 0.3 0.32 0.34 0.36 0.38 0.4 t λ2 λ2: g(λ2)=e−2λ

2

λ2: g(λ2)=e−0.8λ

2

t = 6

Ren, Park, van der Schaar (UCLA) Multimedia Communications & Systems Lab October 2010 13 / 26

slide-30
SLIDE 30

User Subscription Dynamics Convergence Analysis

Illustration of Oscillation & Convergence

2 4 6 8 10 12 14 0.2 0.4 0.6 0.8 t λ2 λ2: g(λ2)=e−2λ

2

2 4 6 8 10 12 14 0.3 0.32 0.34 0.36 0.38 0.4 t λ2 λ2: g(λ2)=e−0.8λ

2

t = 7

Ren, Park, van der Schaar (UCLA) Multimedia Communications & Systems Lab October 2010 13 / 26

slide-31
SLIDE 31

User Subscription Dynamics Convergence Analysis

Illustration of Oscillation & Convergence

2 4 6 8 10 12 14 0.2 0.4 0.6 0.8 t λ2 2 4 6 8 10 12 14 0.3 0.32 0.34 0.36 0.38 0.4 t λ2 λ2: g(λ2)=e−2λ

2

λ2: g(λ2)=e−0.8λ

2

t = 15

Ren, Park, van der Schaar (UCLA) Multimedia Communications & Systems Lab October 2010 13 / 26

slide-32
SLIDE 32

Competition in Duopoly Markets

Outline

1 Introduction 2 Model 3 User Subscription Dynamics

Equilibrium Analysis Convergence Analysis

4 Competition in Duopoly Markets 5 Illustrative Example 6 Conclusion

Ren, Park, van der Schaar (UCLA) Multimedia Communications & Systems Lab October 2010 14 / 26

slide-33
SLIDE 33

Competition in Duopoly Markets

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

Ren, Park, van der Schaar (UCLA) Multimedia Communications & Systems Lab October 2010 15 / 26

slide-34
SLIDE 34

Competition in Duopoly Markets

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

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

Ren, Park, van der Schaar (UCLA) Multimedia Communications & Systems Lab October 2010 15 / 26

slide-35
SLIDE 35

Competition in Duopoly Markets

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

  • 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 .

Ren, Park, van der Schaar (UCLA) Multimedia Communications & Systems Lab October 2010 15 / 26

slide-36
SLIDE 36

Competition in Duopoly Markets

Existence of NE

Lemma Suppose that f (·) is non-increasing on [0, β]. Let ˜ λi(λ−i) be a market share that maximizes the revenue of NSP Si provided that NSP S−i chooses λ−i ∈ [0, 1), i.e., ˜ λi(λ−i) ∈ arg maxλi∈[0,1) Ri(λi, λ−i). Then ˜ λi(λ−i) ∈ (0, 1/2] for all λ−i ∈ [0, 1), for all i = 1, 2. Moreover, ˜ λi(λ−i) = 1/2 if λ−i > 0, for i = 1, 2.

Ren, Park, van der Schaar (UCLA) Multimedia Communications & Systems Lab October 2010 16 / 26

slide-37
SLIDE 37

Competition in Duopoly Markets

Existence of NE

Lemma Suppose that f (·) is non-increasing on [0, β]. Let ˜ λi(λ−i) be a market share that maximizes the revenue of NSP Si provided that NSP S−i chooses λ−i ∈ [0, 1), i.e., ˜ λi(λ−i) ∈ arg maxλi∈[0,1) Ri(λi, λ−i). Then ˜ λi(λ−i) ∈ (0, 1/2] for all λ−i ∈ [0, 1), for all i = 1, 2. Moreover, ˜ λi(λ−i) = 1/2 if λ−i > 0, for i = 1, 2.

  • Implication
  • when the strategy space is specified as [0, 1) and f (·) satisfies the

non-increasing property, strategies λi ∈ {0} ∪ (1/2, 1) is strictly dominated for i = 1, 2

  • if a NE (λ∗∗

1 , λ∗∗ 2 ) of ˜

GC exists, then it must satisfy (λ∗∗

1 , λ∗∗ 2 ) ∈ (0, 1/2)2

Ren, Park, van der Schaar (UCLA) Multimedia Communications & Systems Lab October 2010 16 / 26

slide-38
SLIDE 38

Competition in Duopoly Markets

Existence of NE

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

Ren, Park, van der Schaar (UCLA) Multimedia Communications & Systems Lab October 2010 17 / 26

slide-39
SLIDE 39

Competition in Duopoly Markets

Existence of NE

Corollary Suppose that the users’ valuation of QoS is uniformly distributed, i.e., f (α) = 1/β for α ∈ [0, β]. If g(λ2) + λ2g ′(λ2) ≥ 0 for all λ2 ∈ [0, 1/2], then the game GC has at least one NE.

  • Interpretation
  • if the elasticity of the QoS provided by NSP S2 with respect to the fraction of

its subscribers is no larger than 1 (i.e., −[g ′(λ2)λ2/g(λ2)] ≤ 1), the Cournot competition game with the strategy space [0, 1) has at least one NE

  • the condition is analogous to the sufficient conditions for convergence in that

it requires that the QoS provided by NSP S2 cannot degrade too fast with respect to the fraction of subscribers.

Ren, Park, van der Schaar (UCLA) Multimedia Communications & Systems Lab October 2010 18 / 26

slide-40
SLIDE 40

Illustrative Example

Outline

1 Introduction 2 Model 3 User Subscription Dynamics

Equilibrium Analysis Convergence Analysis

4 Competition in Duopoly Markets 5 Illustrative Example 6 Conclusion

Ren, Park, van der Schaar (UCLA) Multimedia Communications & Systems Lab October 2010 19 / 26

slide-41
SLIDE 41

Illustrative Example

Numerical Results

5 10 15 20 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 t (λ1,λ2) λ1 λ2

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

8 ; dashed: g(λ2) = 1 − λ2 2 .

Ren, Park, van der Schaar (UCLA) Multimedia Communications & Systems Lab October 2010 20 / 26

slide-42
SLIDE 42

Illustrative Example

Numerical Results

5 10 15 20 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 t (R1,R2) R1 R2

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

8 ;

dashed: g(λ2) = 1 − λ2

2 .

Ren, Park, van der Schaar (UCLA) Multimedia Communications & Systems Lab October 2010 21 / 26

slide-43
SLIDE 43

Conclusion

Outline

1 Introduction 2 Model 3 User Subscription Dynamics

Equilibrium Analysis Convergence Analysis

4 Competition in Duopoly Markets 5 Illustrative Example 6 Conclusion

Ren, Park, van der Schaar (UCLA) Multimedia Communications & Systems Lab October 2010 22 / 26

slide-44
SLIDE 44

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 user

subscription dynamics

  • provided a sufficient condition for the convergence of the user subscription

dynamics: the QoS provided by NSP S2 should not degrade too fast as more users subscribe

Ren, Park, van der Schaar (UCLA) Multimedia Communications & Systems Lab October 2010 23 / 26

slide-45
SLIDE 45

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 user

subscription dynamics

  • provided a sufficient condition for the convergence of the user subscription

dynamics: the QoS provided by NSP S2 should not degrade too fast as more users 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 NE
  • f the game

Ren, Park, van der Schaar (UCLA) Multimedia Communications & Systems Lab October 2010 23 / 26

slide-46
SLIDE 46

Conclusion

Selected References

  • J. Musacchio and D. Kim, “Network platform competition in a two-sided market: implications to the net

neutrality issue,” TPRC: Conference on Communication, Information, and Internet Policy, Sep. 2009.

  • D. Bertsimas and G. Perakis, “Dynamic pricing: a learning approach,” Models for Congestion

Charging/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.

Ren, Park, van der Schaar (UCLA) Multimedia Communications & Systems Lab October 2010 24 / 26

slide-47
SLIDE 47

Conclusion

Related Publications

  • S. Ren, J. Park, and M. van der Schaar, “Dynamics of Service Provider Selection in Communication

Markets,” accepted and to appear in Proc. IEEE Globecom 2010.

  • S. Ren, J. Park, and M. van der Schaar, “Subscription Dynamics and Competition in Communication

Markets,” in Proc. ACM NetEcon 2010.

  • S. Ren, J. Park, and M. van der Schaar, “User Subscription Dynamics and Revenue Maximization in

Communication Markets,” UCLA Tech. Report, Aug. 2010 (available at http://arxiv.org/abs/1008.5367).

Ren, Park, van der Schaar (UCLA) Multimedia Communications & Systems Lab October 2010 25 / 26

slide-48
SLIDE 48

Conclusion

Convergence of User Subscription Dynamics

Proof.

1 Show that

hd(λ1,a, λ2,a) − hd(λ1,b, λ2,b)∞ = K

  • −g ′(λ2,c)

g(λ2,c) · q1 q1 − g(λ2,c)

  • |λ2,a − λ2,b|

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

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

slide-49
SLIDE 49

Conclusion

Convergence of User Subscription Dynamics

Proof.

1 Show that

hd(λ1,a, λ2,a) − hd(λ1,b, λ2,b)∞ = K

  • −g ′(λ2,c)

g(λ2,c) · q1 q1 − 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) · q1 q1−g(λ2)

  • < 1

K , then the mapping is contraction!

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