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
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
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
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
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
Model Model ����� ����� …… Ren, Park, van der Schaar (UCLA) Multimedia Communications & Systems Lab October 2010 6 / 26
Model Model ����� ����� …… Network model • network service providers: S 1 and S 2 • continuum model: a large number of users Ren, Park, van der Schaar (UCLA) Multimedia Communications & Systems Lab October 2010 6 / 26
Model Model Service providers • S i : price p i and fraction of subscribers λ i ( p i , p − i ) Ren, Park, van der Schaar (UCLA) Multimedia Communications & Systems Lab October 2010 7 / 26
Model Model Service providers • S i : price p i and fraction of subscribers λ i ( p i , p − i ) • utility (revenue): R i ( p i , p − i ) = p i λ i ( p i , p − i ) Ren, Park, van der Schaar (UCLA) Multimedia Communications & Systems Lab October 2010 7 / 26
Model Model Service providers • S i : price p i and fraction of subscribers λ i ( p i , p − i ) • utility (revenue): R i ( p i , p − i ) = p i λ i ( p i , p − i ) Users • user k : u k = α k q i − p i if it subscribes to S i Ren, Park, van der Schaar (UCLA) Multimedia Communications & Systems Lab October 2010 7 / 26
Model Model Service providers • S i : price p i and fraction of subscribers λ i ( p i , p − i ) • utility (revenue): R i ( p i , p − i ) = p i λ i ( p i , p − i ) Users • user k : u k = α k q i − p i if it subscribes to S i • α k follows a distribution with PDF f ( α ) Ren, Park, van der Schaar (UCLA) Multimedia Communications & Systems Lab October 2010 7 / 26
Model Model Service providers • S i : price p i and fraction of subscribers λ i ( p i , p − i ) • utility (revenue): R i ( p i , p − i ) = p i λ i ( p i , p − i ) Users • user k : u k = α k q i − p i if it subscribes to S i • α k 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
Model Model Service providers • S i : price p i and fraction of subscribers λ i ( p i , p − i ) • utility (revenue): R i ( p i , p − i ) = p i λ i ( p i , p − i ) Users • user k : u k = α k q i − p i if it subscribes to S i • α k follows a distribution with PDF f ( α ) QoS model • q 1 is constant • q 2 = g ( λ 2 ), where g ( λ 2 ) ∈ (0 , q 1 ) is a differentiable and non-increasing function of λ 2 ∈ [0 , 1] Ren, Park, van der Schaar (UCLA) Multimedia Communications & Systems Lab October 2010 7 / 26
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
User Subscription Dynamics User Subscription • Discrete-time model { ( λ t 1 , λ t 2 ) | t = 0 , 1 , 2 · · ·} • Users’ belief model and subscription decisions • naive (or static) expectation: every user expects that the QoS in the current g k ( λ t 2 ) = g ( λ t − 1 period is equal to that in the previous period (i.e., ˜ )) 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
User Subscription Dynamics User Subscription • Discrete-time model { ( λ t 1 , λ t 2 ) | t = 0 , 1 , 2 · · ·} • Users’ belief model and subscription decisions • naive (or static) expectation: every user expects that the QoS in the current g k ( λ t 2 ) = g ( λ t − 1 period is equal to that in the previous period (i.e., ˜ )) 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
User Subscription Dynamics User Subscription • Discrete-time model { ( λ t 1 , λ t 2 ) | t = 0 , 1 , 2 · · ·} • Users’ belief model and subscription decisions • naive (or static) expectation: every user expects that the QoS in the current 2 ) = g ( λ t − 1 g k ( λ t period is equal to that in the previous period (i.e., ˜ )) 2 • a user subscribes to whichever NSP provides a higher (non-negative) utility • Dynamics of user subscriptions if p 1 p 2 q 1 > ) , then g ( λ t − 1 2 � � p 1 − p 2 h d , 1 ( λ t − 1 , λ t − 1 λ t 1 = ) = 1 − F , 1 2 q 1 − g ( λ t − 1 ) 2 � p 1 − p 2 � � p 2 � h d , 2 ( λ t − 1 , λ t − 1 λ t 2 = ) = F − F 1 2 q 1 − g ( λ t − 1 g ( λ t − 1 ) ) 2 2 Ren, Park, van der Schaar (UCLA) Multimedia Communications & Systems Lab October 2010 9 / 26
User Subscription Dynamics User Subscription • Discrete-time model { ( λ t 1 , λ t 2 ) | t = 0 , 1 , 2 · · ·} • Users’ belief model and subscription decisions • naive (or static) expectation: every user expects that the QoS in the current g k ( λ t 2 ) = g ( λ t − 1 period is equal to that in the previous period (i.e., ˜ )) 2 • a user subscribes to whichever NSP provides a higher (non-negative) utility • Dynamics of user subscriptions if p 1 p 2 q 1 ≤ ) , then g ( λ t − 1 2 � p 1 � λ t h d , 1 ( λ t − 1 , λ t − 1 1 = ) = 1 − F , 1 2 q 1 h d , 2 ( λ t − 1 , λ t − 1 λ t 2 = ) = 0 . 1 2 Ren, Park, van der Schaar (UCLA) Multimedia Communications & Systems Lab October 2010 9 / 26
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 h d , 1 ( λ ∗ 1 , λ ∗ 2 ) = λ ∗ 1 and h d , 2 ( λ ∗ 1 , λ ∗ 2 ) = λ ∗ 2 . Ren, Park, van der Schaar (UCLA) Multimedia Communications & Systems Lab October 2010 10 / 26
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 ( p 1 , p 2 ), 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 � p 1 � if p 1 p 2 λ ∗ , λ ∗ 1 = 1 − F 2 = 0 , q 1 ≤ g (0) , q 1 if p 1 p 2 λ ∗ 1 = 1 − F ( θ ∗ 1 ) , λ ∗ 2 = F ( θ ∗ 1 ) − F ( θ ∗ 2 ) , q 1 > g (0) , where θ ∗ 1 = ( p 1 − p 2 ) / ( q 1 − g ( λ ∗ 2 )) and θ ∗ 2 = p 2 / g ( λ ∗ 2 ). Ren, Park, van der Schaar (UCLA) Multimedia Communications & Systems Lab October 2010 10 / 26
User Subscription Dynamics Equilibrium Analysis Equilibrium Market Shares 1 λ 1 * 0.8 λ 2 * 0.6 λ * 0.4 0.2 0 3 3 2 2 1 1 0 0 p 2 p 1 • q 1 = 2 . 5, g ( λ 2 ) = 1 . 2 e − 0 . 5 λ 2 , and α is uniformly distributed on [0 , 1], i.e., f a ( α ) = 1 for α ∈ [0 , 1]. Ren, Park, van der Schaar (UCLA) Multimedia Communications & Systems Lab October 2010 11 / 26
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
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