Wireless Network Pricing Chapter 6: Oligopoly Pricing

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1 Wireless Network Pricing Chapter 6: Oligopoly Pricing Jianwei Huang & Lin Gao Network Communications and Economics Lab (NCEL) Information Engineering Department The Chinese University of Hong Kong Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 6 November 15, / 69

2 The Book E-Book freely downloadable from NCEL website: http: //ncel.ie.cuhk.edu.hk/content/wireless-network-pricing Physical book available for purchase from Morgan & Claypool ( and Amazon ( Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 6 November 15, / 69

3 Section 6.3: Wireless Service Provider Competition Revisited Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 6 November 15, / 69

4 Network Model Provider 2 Provider 1 Provider 3 AsetJ = {1,...,J} of service providers I Provider j has a supply Q j of resource (e.g., channel, time, power) I Providers operate on orthogonal spectrum bands AsetI = {1,...,I} of users I User i can obtain resources from multiple providers: q i =(q ij, 8j 2J) I User i s utility function is ui PJ j=1 q ijc ij : increasing and strictly concave Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 6 November 15, / 69

5 An Example: TDMA Each provider j has a total spectrum band of W j. q ij : the fraction of time that user i transmits on provider j s band I Constraints: P i q ij apple 1, for all j 2J. c ij :thedatarateachievedbyuseri on provider j s band c ij = W j log(1 + P i h ij 2 ) 2 ij W j I Pi :useri speaktransmission power. I hij : the channel gain between user i and network j. I 2 ij : the Gaussian noise variance for the channel. u i PJ j=1 q ijc ij :useri utility of the total achieved data rate Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 6 November 15, / 69

6 Two-Stage Game Stage I: each provider j 2J announces a unit price p j I Each provider i wants to maximize his revenue I Denote p =(pj, 8j 2J) as the price vectors of all providers. Stage II: each user i 2I chooses a demand vector q i =(q ij, 8j 2J) I Each user i wants to maximize his payo (utility minus payment) I Denote q =(q i, 8i 2I) as the demand vector of all users. Analysis based on backward induction Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 6 November 15, / 69

7 Goal: Derive the SPNE Apricedemandtuple(p, q (p )) is a SPNE if no player has an incentive to deviate unilaterally at any stage of the game. I Each user i maximizes its payo by choosing the optimal demand q i (p ), given prices p. I Each provider j maximizes its revenue by choosing price p j, given other providers prices p j =(p k, 8k 6= j) andtheuserdemandsq (p ). Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 6 November 15, / 69

8 Stage II: User s Demand Optimization Each user i 2I solves a user payo maximization (UPM) problem: JX JX UPM :max@u q ij c ij A p j q ij A. q i 0 j=1 j=1 Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 6 November 15, / 69

9 Stage II: User s Demand Optimization Each user i 2I solves a user payo maximization (UPM) problem: JX JX UPM :max@u q ij c ij A p j q ij A. q i 0 j=1 j=1 Problem UPM may have more than one optimization solution q i I Since it is not strictly concave maximization problem in q i Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 6 November 15, / 69

10 Stage II: User s Demand Optimization Each user i 2I solves a user payo maximization (UPM) problem: JX JX UPM :max@u q ij c ij A p j q ij A. q i 0 j=1 j=1 Problem UPM may have more than one optimization solution q i I Since it is not strictly concave maximization problem in q i Problem UPM has a unique solution of the e ective resource x i Lemma (6.16) I For each user i 2I, there exists a unique nonnegative value x i,suchthat Pj2J c ijqij = xi for every maximizer q i of the UPM problem. I For any provider j such that q ij > 0, p j /c ij =min k2j p k /c ik. Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 6 November 15, / 69

11 Decided vs. Undecided Users Definition (Preference set) For any price vector p, useri s preference set is J i (p) = j 2J : p j p k =min. c ij k2j c ik Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 6 November 15, / 69

12 Decided vs. Undecided Users Definition (Preference set) For any price vector p, useri s preference set is J i (p) = j 2J : p j p k =min. c ij k2j c ik A decided user has a singleton preference set. An undecided user has a preference set that includes more than one provider. Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 6 November 15, / 69

13 Decided vs. Undecided Users Definition (Preference set) For any price vector p, useri s preference set is J i (p) = j 2J : p j p k =min. c ij k2j c ik A decided user has a singleton preference set. An undecided user has a preference set that includes more than one provider. One can use a bipartite graph representation (BGR) to uniquely determine the demands of undecided users. This will lead to all users optimal demand q (p) =(q i (p), 8i 2I)in Stage II. Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 6 November 15, / 69

14 Stage I: Provider s Revenue Optimization Each provider j 2J solves a provider revenue maximization (PRM) problem! PRM :max p j 0 p j min Q j, X i2i q ij(p j, p j ) Solving the PRM problem requires the consideration of other providers prices p j. Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 6 November 15, / 69

15 Benchmark: Social Welfare Optimization (Ch. 4) SWO: Social Welfare Optimization Problem maximize X u i (x i ) i2i subject to X q ij c ij = x i, 8i 2I, j2j X q ij = Q j, 8j 2J, i2i variables q ij, x i 0, 8i 2I, j 2J. Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 6 November 15, / 69

16 Stage I: Provider s Revenue Optimization Theorem Under proper technical assumptions, the unique socially optimal demand vector q and the associated Lagrangian multiplier vector p of the SWO problem constitute the unique SPNE of the provider competition game. Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 6 November 15, / 69

17 Optimization, Game, and Algorithm Social Welfare Optimization Section maximizing vector q Lagrange multipliers p (q, p ) Provider Competition Game Section equilibrium user demand q equilibrium price p (q, p ) Primal-Dual Algorithm Section lim t!1 (q(t), p(t)) = (q, p ) (q, p ) Figure: Relationship among di erent concepts Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 6 November 15, / 69

18 Section 6.4: Competition with Spectrum Leasing Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 6 November 15, / 69

19 Network Model Spectrum owner Spectrum owner Investment (leasing bandwidth) Pricing (selling bandwidth) Operator i Operator j Secondary users (transmitter-receiver pairs) Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 6 November 15, / 69

20 Three-Stage Multi-leader-follower Game Operators (leaders) Users (followers) Stage I: Leasing Game Leasing Bandwidth B1 and B2 (with unit costs C1 and C2) Stage II: Pricing Game Pricing π1 and π2 Stage III User k Chooses One Operator i and Demand wki Backward Induction Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 6 November 15, / 69

21 Stage IIII: Users Bandwidth Demands User k s payo of choosing operator i =1, 2 P max i h i u k ( i, w ki )=w ki ln n 0 w ki i w ki I High SNR approximation of OFDMA system I Optimal demand: wki ( i)=argmax wki 0 u k ( i, w ki )=g k e (1+ i ) I Optimal payo : u k ( i, wki ( i)) User k prefers the better operator: i =argmax i=1,2 u k ( i, wki ( i)) Users demands may not be satisfied due to limited resource I Di erence between preferred demand and realized demand Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 6 November 15, / 69

22 Stages II: Pricing Game Players: two operators Strategies: i 0, i =1, 2 Payo s: profit R i for operator i =1, 2: R i (B i, B j, i, j )= i Q i (B i, B j, i, j ) B i C i Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 6 November 15, / 69

23 Stage II: Pricing Equilibrium Symmetric equilibrium: 1 = 2. Threshold structure: I Unique positive equilibrium exists B 1 + B 2 apple Ge 2. B j Ge 1 ( M1) ( H ) (L) : Unique nonzero equilibrium (M1) (M3) : No equilibrium (H) : Unique zero equilibrium Ge 2 ( M 2) ( M 3) ( L) 0 2 Ge Ge 1 B i Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 6 November 15, / 69

24 Stage I: Leasing Game Players: two operators Strategies: B i 2 [0, 1), i =1, 2, and B 1 + B 2 apple Ge 2. Payo s: profit R i for operator i =1, 2: G R i (B i, B j )=B i ln B i + B j 1 C i Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 6 November 15, / 69

25 Stage I: Leasing Equilibrium Linear in wireless characteristics G = P i g i; Threshold structure: I Low costs: infinitely many equilibria I High comparable costs: unique equilibrium I High incomparable costs: unique monopoly equlibrium C j C = C + 1 j i 1 ( HI) ( HC) C = C 1 j i (L) : Infinitely many equilibria (HC) : Unique equilibrium (HI) (HI ) : Unique equilibrium ( L) ( HI ') 0 1 C i Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 6 November 15, / 69

26 Equilibrium Summary (Assuming C i apple C j ) Costs LOW HC HI C i + C j apple 1 C i + C j > 1, C j > 1+C i C j C i apple 1 equilibria Infinite Unique Unique (Bi, Bj ) ( Ge 2, (1 )Ge 2 (1+C ), j C i )G C i +C j +3, (1+C i C j )G C i +C j +3 (Ge (2+C i ), 0) 2e 2 2e 2 2 [C j, (1 C i )] ( i, j ) (1, 1) Ci +C j +1 2, C i +C j +1 2 (1 + C i, N/A) User SNR e 2 e C i +C j +3 2 e 2+C i User Payo g k e 2 g k e Ci +C j +3 2 g k e (2+C i ) Users achieve the same SNR User k s payo is linear in g k Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 6 November 15, / 69

27 Robustness of Results To obtain closed form solutions, we have assumed I All users achieve high SNR Previous observations still hold in the general case I Users operate in general SNR regime: rki (w ki )=w ki ln 1+ Pmax k h k n 0w ki Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 6 November 15, / 69

28 Impact of Duopoly Competition on Operators Benchmark: Coordinated Case I Operators cooperate in investment and pricing to maximize total profit Define Total Profit in Competition Case E ciency Ratio = Total Profit in Coordinated Case Price of Anarchy = min Ci,C j E ciency Ratio= Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 6 November 15, / 69

29 Section 6.5: Chapter Summary Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 6 November 15, / 69

30 Key Concepts Theory: Game Theory I Dominant Strategy I Pure and Mixed Strategy Nash Equilibrium I Subgame Perfect Nash Equilibrium Theory: Oligopoly I Cournot competition I Bertrand competition I Hotelling competition Application: Wireless Network Competition Revisited Application: Competition with Spectrum Leasing Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 6 November 15, / 69

31 References and Extended Reading J. Huang, How Do We Play Games? online video tutorial, on YouKu ( and itunesu ( how-do-we-play-games/id ) V. Gajic, J. Huang, and B. Rimoldi, Competition of Wireless Providers for Atomic Users, IEEE Transactions on Networking, vol. 22, no. 2, pp , April 2014 L. Duan, J. Huang, and B. Shou, Duopoly Competition in Dynamic Spectrum Leasing and Pricing, IEEE Transactions on Mobile Computing, vol. 11, no. 11, pp , November Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 6 November 15, / 69