Price War in Heterogeneous Wireless Networks

Transcription

1 Price War in Heterogeneous Wireless Networks Patrick Maillé, Bruno Tuffin Telecom Bretagne and INRIA-Centre Bretagne Atlantique Rennes, France COST IS0605 meeting, Limassol, Feb 2009 P. Maillé (Telecom-B), B. Tuffin (INRIA) Price War in Access Networks 1 / 16

2 DSL WiMAX WiFi 1 WiFi 2 P. Maillé (Telecom-B), B. Tuffin (INRIA) Price War in Access Networks Econ@tel 2 / 16

3 DSL Interactions among non-cooperative consumers: game Congested networks provide poorer quality (packet losses) P. Maillé (Telecom-B), B. Tuffin (INRIA) Price War in Access Networks 2 / 16

4 But providers play first! p 4 DSL p 3 p 1 p 2 P. Maillé (Telecom-B), B. Tuffin (INRIA) Price War in Access Networks Econ@tel 2 / 16

5 But providers play first! p 4 DSL p 3 p 1 p 2 This work: study of the two-level noncooperative game. 1 Higher level: providers set prices to maximize revenue 2 Lower level: consumers choose their provider P. Maillé (Telecom-B), B. Tuffin (INRIA) Price War in Access Networks Econ@tel 2 / 16

6 Related work Many references on network pricing, with different objectives: control congestion, ensure fairness, manage different QoS levels, maximize network revenue. But only few considering competition among providers: wireless providers playing on trans. power studies of peering agreements competition with delay-sensitive users Key & Massoulié 99, Lazar & Semret 99 Kelly et al. 98, Marbach 02 Cocchi et al. 93, Odlyzko 99 Paschalidis & Tsitsiklis 00 Felegyhazi & Hubaux 06 He & Walrand Shakkotai & Srikant 05 Acemoglu & Ozdaglar 06 Hayrapetyan et al. 06 This work: competition among providers with loss-sensitive users and minimal regulation performance of the outcome? P. Maillé (Telecom-B), B. Tuffin (INRIA) Price War in Access Networks 3 / 16

7 Outline 1 Network and pricing model Network model Pricing model 2 Game analysis Lower level: equilibrium among users Higher level: price war Social Welfare issues 3 Conclusions and perspectives P. Maillé (Telecom-B), B. Tuffin (INRIA) Price War in Access Networks Econ@tel 4 / 16

8 Competition model Infinitely small users (price-takers) N competing providers declaring price and capacity (I := {1,..., N}) Two simplified cases for the topology: P. Maillé (Telecom-B), B. Tuffin (INRIA) Price War in Access Networks Econ@tel 5 / 16

9 Competition model Infinitely small users (price-takers) N competing providers declaring price and capacity (I := {1,..., N}) Two simplified cases for the topology: 1 Common coverage area p 1 p 2 p 3 P. Maillé (Telecom-B), B. Tuffin (INRIA) Price War in Access Networks Econ@tel 5 / 16

10 Competition model Infinitely small users (price-takers) N competing providers declaring price and capacity (I := {1,..., N}) Two simplified cases for the topology: 1 Common coverage area p 1 p 2 p 3 P. Maillé (Telecom-B), B. Tuffin (INRIA) Price War in Access Networks Econ@tel 5 / 16

11 Competition model Infinitely small users (price-takers) N competing providers declaring price and capacity (I := {1,..., N}) Two simplified cases for the topology: 1 Common coverage area p 1 p 2 p 3 2 N = 2, one coverage area included in the other Prov. 1: WiMAX Prov. 2: WiFi P. Maillé (Telecom-B), B. Tuffin (INRIA) Price War in Access Networks Econ@tel 5 / 16

12 Competition model Infinitely small users (price-takers) N competing providers declaring price and capacity (I := {1,..., N}) Two simplified cases for the topology: 1 Common coverage area p 1 p 2 p 3 2 N = 2, one coverage area included in the other Prov. 1: WiMAX Prov. 2: WiFi P. Maillé (Telecom-B), B. Tuffin (INRIA) Price War in Access Networks Econ@tel 5 / 16

13 Communication model: Packet losses Time is slotted Each provider i has finite capacity C i If total demand d i at provider i exceeds C i : exceeding packets are randomly lost lost C i d i served ( P(successful transmission) = min 1, C ) i d i Expected number of transmissions = 1 P(success) = max ( 1, d ) i C i P. Maillé (Telecom-B), B. Tuffin (INRIA) Price War in Access Networks Econ@tel 6 / 16

14 Only regulation : pay for what you send The price p i at each provider i is per packet sent Marbach 02 If several transmissions are needed, the user pays several times ( p i := perceived price at i = E[price per packet] = p i max 1, d ) i C i Price p i p i p i C i Demand d i P. Maillé (Telecom-B), B. Tuffin (INRIA) Price War in Access Networks Econ@tel 7 / 16

15 Analysis of user choices: Wardrop equilibrium ( ) Users choose the provider(s) i with lowest p i = p i max 1, d i C i For a given coverage zone Z, all providers with customers from that zone end up with the same perceived price p i = p z Wardrop 52 P. Maillé (Telecom-B), B. Tuffin (INRIA) Price War in Access Networks Econ@tel 8 / 16

16 Analysis of user choices: Wardrop equilibrium ( ) Users choose the provider(s) i with lowest p i = p i max 1, d i C i For a given coverage zone Z, all providers with customers from that zone end up with the same perceived price p i = p z Wardrop 52 The total demand level in a zone z depends on that price: ( ) di,z d z = α z D( p z ), i.e. p z = }{{} v α z marg. val. function with D the total demand function, α z the population proportion in zone z, and d i,z the demand in zone z for provider i. P. Maillé (Telecom-B), B. Tuffin (INRIA) Price War in Access Networks Econ@tel 8 / 16

17 Wardrop equilibrium: illustration 1 Case of common coverage area Unit price p 4 p 3 p 2 p 1 C 1 C 2 C 3 C 4 Quantities P. Maillé (Telecom-B), B. Tuffin (INRIA) Price War in Access Networks Econ@tel 9 / 16

18 Wardrop equilibrium: illustration 1 Case of common coverage area Unit price v(q) p 4 p = p 3 p 2 p 1 d 1 d 2 d 3 Quantities P. Maillé (Telecom-B), B. Tuffin (INRIA) Price War in Access Networks Econ@tel 9 / 16

19 Wardrop equilibrium: illustration 1 Case of common coverage area 2 Case of two providers p v ( ) q 1 α Prov. 1 Prov. 2 p v ( q ) α zone A (1 α) zone B (α) Prices p 1 C 1 d 1,B Prices p 1 p 2 p 2 C 2 C 1 d 1,A q d 1,A d 2 Quantities Quantities q P. Maillé (Telecom-B), B. Tuffin (INRIA) Price War in Access Networks Econ@tel 9 / 16

20 Wardrop equilibrium: illustration 1 Case of common coverage area 2 Case of two providers Mathematical formulation For each zone z and each provider i, j, at Wardrop equilibrium ( p i = p i max 1, d ) i C i ( ) d z = α z D min p i i z If i, j z, then p i > p j d i,z = 0. P. Maillé (Telecom-B), B. Tuffin (INRIA) Price War in Access Networks Econ@tel 9 / 16

21 Wardrop equilibrium: existence and uniqueness Proposition For all price profile, there exists at least a Wardrop equilibrium. Moreover, the corresponding perceived prices of each provider are unique. NB: demand repartition among providers is not necessarily unique. P. Maillé (Telecom-B), B. Tuffin (INRIA) Price War in Access Networks Econ@tel 10 / 16

22 Higher level: price competition game Providers set their price p i anticipating users reaction Providers are Stackelberg leaders Provider i s objective: R i := p i d i. P. Maillé (Telecom-B), B. Tuffin (INRIA) Price War in Access Networks Econ@tel 11 / 16

23 Higher level: price competition game Providers set their price p i anticipating users reaction Providers are Stackelberg leaders Provider i s objective: R i := p i d i. Unit price v(q) p 4 p = p 3 p 2 p 1 d 1 d 2 d 3 Quantities P. Maillé (Telecom-B), B. Tuffin (INRIA) Price War in Access Networks Econ@tel 11 / 16

24 Price competition, main results (1/2) Case of a common coverage area Proposition Under condition (1) or (2), there exists a unique Nash equilibrium on price war among providers, given by { ( ) p i I, i = v j I C j d i = C i. Sufficient conditions: For each provider i, C i j i C j (1) D (p)p D(p) > 1, p (2) P. Maillé (Telecom-B), B. Tuffin (INRIA) Price War in Access Networks Econ@tel 12 / 16

25 Proof sketch { ( ) p 1 First step: i = v j I C j defines a Nash equilibrium; i.e. d i = C i no provider can improve its benefit by changing his price. 2 Second step: no other point can be a Nash equilibrium. We partition the provider set I into three subsets: I s := {i I : d i > C i }, I 0 := {i I : d i = C i }, I u := {i I : d i < C i }. First, I s =. If not, provider i s I s increasing (alone) its unit price p is to pi n s = p is + ε, improves its revenue. Assume then Iu. Using the fact that I s =, i u I u has interest in decreasing (alone) its price: its revenue would increase (due to increasing demand). P. Maillé (Telecom-B), B. Tuffin (INRIA) Price War in Access Networks Econ@tel 13 / 16

26 Price competition, main results (2/2) Case of 2 providers Proposition Prov. 1 Prov. 2 Under Condition (2), there exists a unique Nash equilibrium (p1, p 2 ) in the price war between providers. The Nash equilibrium is characterized as follows. ( ) C If 1 1 α C 2 α, then p 1 = v C1 1 α provider 2 by provider 1. C 1 ( ) p2 = v C2 α. Zone B is left to If 1 α > C 2 α then p 1 = p 2 = p = v(c 1 + C 2 ). Zone B is shared by the providers. Condition (2): D (p)p D(p) > 1, p P. Maillé (Telecom-B), B. Tuffin (INRIA) Price War in Access Networks Econ@tel 14 / 16

27 Social Welfare considerations A performance measure of the outcome (d 1,..., d I ) of the game = overall value of the system throughputz Social Welfare := v (x/α z ) dx, zones Z 0 }{{} users willingness-to-pay with throughput z = ( i d i,z min 1, C i d i ). Remark: under the conditions of previous Propositions, the Social Welfare maximization problem leads to the same outcome as the price war. Consequence: The Nash equilibrium corresponds to the socially optimal situation: the Price of Anarchy is 1!. P. Maillé (Telecom-B), B. Tuffin (INRIA) Price War in Access Networks Econ@tel 15 / 16

28 Conclusions and perspectives We have analyzed a pricing game among providers Characterized how demand is split (following Wardrop s principle), studied the Nash equilibrium of the pricing game (characterization, uniqueness), shown that its outcome is socially optimal, Perspectives What happens if providers partially share their capacities? What if providers play on capacities along with prices? What about the multiclass case? What about the dynamics of the model? How to drive to the equilibrium? What about more complicated topologies? P. Maillé (Telecom-B), B. Tuffin (INRIA) Price War in Access Networks Econ@tel 16 / 16