Wireless Network Pricing Chapter 6: Oligopoly Pricing

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, 2016 1 / 69

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 (http://goo.gl/jfglai) and Amazon (http://goo.gl/jqkaeq) Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 6 November 15, 2016 2 / 69

Section 6.3: Wireless Service Provider Competition Revisited Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 6 November 15, 2016 45 / 69

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, 2016 46 / 69

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, 2016 47 / 69

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, 2016 48 / 69

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, 2016 49 / 69

Stage II: User s Demand Optimization Each user i 2I solves a user payo maximization (UPM) problem: 0 0 1 1 JX JX UPM :max@u i @ 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, 2016 50 / 69

Stage II: User s Demand Optimization Each user i 2I solves a user payo maximization (UPM) problem: 0 0 1 1 JX JX UPM :max@u i @ 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, 2016 50 / 69

Stage II: User s Demand Optimization Each user i 2I solves a user payo maximization (UPM) problem: 0 0 1 1 JX JX UPM :max@u i @ 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, 2016 50 / 69

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, 2016 51 / 69

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, 2016 51 / 69

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, 2016 51 / 69

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, 2016 52 / 69

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, 2016 53 / 69

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, 2016 54 / 69

Optimization, Game, and Algorithm Social Welfare Optimization Section 4.3.2 maximizing vector q Lagrange multipliers p (q, p ) Provider Competition Game Section 6.3.1 equilibrium user demand q equilibrium price p (q, p ) Primal-Dual Algorithm Section 4.3.3 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, 2016 55 / 69

Section 6.4: Competition with Spectrum Leasing Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 6 November 15, 2016 56 / 69

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, 2016 57 / 69

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, 2016 58 / 69

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, 2016 59 / 69

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, 2016 60 / 69

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, 2016 61 / 69

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, 2016 62 / 69

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, 2016 63 / 69

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, 2016 64 / 69

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, 2016 65 / 69

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= 0.75. Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 6 November 15, 2016 66 / 69

Section 6.5: Chapter Summary Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 6 November 15, 2016 67 / 69

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, 2016 68 / 69

References and Extended Reading J. Huang, How Do We Play Games? online video tutorial, on YouKu (http://www.youku.com/playlist_show/id_19119535.html) and itunesu (https://itunes.apple.com/hk/course/ how-do-we-play-games/id642100914) V. Gajic, J. Huang, and B. Rimoldi, Competition of Wireless Providers for Atomic Users, IEEE Transactions on Networking, vol. 22, no. 2, pp. 512-525, 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. 1706 1719, November 2012 http://ncel.ie.cuhk.edu.hk/content/wireless-network-pricing Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 6 November 15, 2016 69 / 69