1. Price Differentiation for Communication Networks Shuqin (Helen) Li and Jianwei Huang, IEEE Transactions on Networking, Forthcoming

2. Revenue Management for Communication Networks 2 Authors Shuqin (Helen) Li PhD from CUHK, 2012 Now at Shanghai Bell Labs

3. Revenue Management for Communication Networks 3 Revenue Management for Comm. Networks Revenue management: how to (set the prices to) maximize the revenue? Service Provider Revenue management: to sell the right resources to the right customers at the right time and the right price. Revenue management: to sell the right resources to the right customers at the right time and the right price. Heterogeneity of users, technologies, applications…

4. Revenue Management for Communication Networks 4 Outline Introduction Network Model Complete Price Differentiation Single Price Scheme (No Differentiation) Partial Price Differentiation Conclusion

5. Revenue Management for Communication Networks 5 Network Model A Service provider S : total resource (N 1, n 1, u 1, s 1, p 1 ) (N 2, n 2, u 2, s 2, p 2 ) (N 3, n 3, u 3, s 3, p 3 ) N 2 = 3n 2 = 2 Utility: u 2 (s 2 ) =θ 2 log(1+ s 2 ) Surplus: u 2 (s 2 ) – p 2 s 2 Admitted to access One SP S: Total limited resource I groups of users Indexed by i=1,2,…I For group i: N i homogeneous users n i : admitted users u i (s i )=θ i log(1+s i ) : Utility function s i : allocated resource θ i :willingness to pay p i : Price per unit resource of group i Assume θ 1 >θ 2 >…>θ I

6. Revenue Management for Communication Networks 6 Two-stage System Model Two-stage System model (Stackelberg Game) Consider the incentives of both SP and users Stage 1: SP sets the price Stage 2: an admitted user Chooses the quantity s i Stage 2: an admitted user Chooses the quantity s i determine (p i,n i ) Complete information? Complete information? Yes No, Design a price menu SP must guarantee Complete price differentiation Single price scheme Partial price differentiation revenue complexity

7. Revenue Management for Communication Networks 7 Outline Introduction Network Model Complete Price Differentiation Single Price Scheme (No Differentiation) Partial Price Differentiation Conclusion

8. 8 Revenue Management for Communication Networks A group i user admitted in stage 1 given price p i maximize his surplus by choosing the proper resource quantity: Complete Information: Stage 2 Stage 1: SP sets the price Stage 2: an admitted user Chooses the quantity s i Stage 2: an admitted user Chooses the quantity s i (p i,n i ) Complete information? Complete information? Yes

9. Revenue Management for Communication Networks 9 Complete Price Differentiation SP’s pricing and admission control problem: Difficulties: Non-convex objective Integer variables Coupled constraint Method: decomposition Resource allocation subproblem Admission control subproblem Stage 1: SP sets the price Stage 2: an admitted user Chooses the quantity s i Stage 2: an admitted user Chooses the quantity s i (p i,n i ) Complete information? Complete information? Yes

10. 10 Revenue Management for Communication Networks Complete Price Differentiation (con’t) Solution All users are admitted There exist a group threshold K cp and λ *, such that Intuition Nonzero resourceZero resource group1group2group6group5group4group3 K cp =4 Effective market Prices are too high to deman any resource Prices can perform admission control Threshold structure Group i 123456 K cp =4 Total resource S Indicates to allocate resource to high willingness to pay users with priority θ 1 >θ 2 >…>θ I

11. Revenue Management for Communication Networks 11 Outline Introduction Network Model Complete Price Differentiation Single Price Scheme (No Differentiation) Partial Price Differentiation Conclusion

12. 12 Revenue Management for Communication Networks Single Price Scheme (No differentiation) SP charges a single price to all groups Solution All users are admitted There exist a group threshold K sp and p *, such that Remark: Similar threshold structure as the complete price differention

13. Revenue Management for Communication Networks 13 Special Properties of Single Price Scheme A smaller effective market : The effective market can be equivalent to a “Super-group” where Under single price scheme: Nonzero resourceZero resource group6group5 K sp =4 group1group2group4group3 Effective market Super-group

14. Revenue Management for Communication Networks 14 When is price differentiation most beneficial? High differentiation gain: – High willingness to pay users are minorities – Resource are comparativelly limited A three-group example But what if the number of user type is large… θ1θ1 N1N1 θ2θ2 N2N2 θ3θ3 N3N3 θ Case 19103 1802 Case 23332 1342 Case 32.2801.5101 2 Peak points indicate the chages of effective market under the single price shceme revenue complexity

15. Revenue Management for Communication Networks 15 Outline Introduction Network Model Complete Price Differentiation Single Price Scheme (No Differentiation) Partial Price Differentiation Conclusion

16. Revenue Management for Communication Networks 16 Partial Price Differentiation SP charges J( ≤ I) prices to I groups group6group5group1group2group4group3 Ex: Charge 6 groups with 2 prices p1p1 p2p2 =p 1 =p 2 =p 3 =p 4 =p 5 =p 6 General formulation: General formulation: Including CP (J=I) and SP (J=1) as special cases Difficulties: Difficulties: Combinatorial optimization problem groupsI=10I=100I=1000 S(I, J)51193306.3*10 29 8.6*10 46 5.4*10 300 Partition I groups into J clusters: S(I, J)

17. Revenue Management for Communication Networks 17 Solving Parital Price Differentiation First: reduce search space consecutive group indices – Optimal partitions involve consecutive group indices within clusters – Intuition: high willingness to pay groups have priority Transfer into three (nested) sub-problems – Optimal partition into clusters – In each cluster: single pricing problem – Among clusters: complete price differentiation problem Polynomial time (O(I J ) ) algorithm group5group1group2group4group3 0 K pp =4 group6 group2group1group3group5group4 0 group6 groupsI=10I=100I=1000 S(I, J)51193306.3*10 29 8.6*10 46 5.4*10 300 C(I-1, J-1)936994851999

18. Revenue Management for Communication Networks 18 Trade-off: Complexity Vs. Revenue Partition price differentiation : a five-group example Group i 12345 θiθi 168421 NiNi 2351080 Note: High willingness to pay users are minorities When S=100 Gain Two-price14.9% Three-price16.5% Four-price17.1% Five-price17.3%

19. Revenue Management for Communication Networks 19 Conclusion Stage 1: SP sets the price Stage 2: an admitted user Chooses the quantity s i Stage 2: an admitted user Chooses the quantity s i (p i,n i ) Complete information? Complete information? Yes No, statistical information only SP must guarantee revenue complexity Complete price differentiation Single price scheme Partial price differentiation

20. Revenue Management for Communication Networks 20 Q & A

21. Revenue Management for Communication Networks 21 CP under Incomplete Information Incomplete Information Challenge: Possible to realize price differentiation? Method: Make the users self-differentiate SP: publishes the quantity-based price menu. Users: freely choose their quantities. Stage 1: SP sets the price Stage 2: an admitted user Chooses the quantity s i Stage 2: an admitted user Chooses the quantity s i (p i,n i ) Complete information? Complete information? Yes No, statistical information only SP must guarantee I: the total number of groups N i s: the number of users in each group u i s: the utility function of each group But The SP doesn’t know which group each user belongs to. The higher the quantity, the higher the price.

22. Revenue Management for Communication Networks 22 CP under Incomplete Information (con’t) Aim: to properly set the thresholds to make the users self-differentiated to get the same result just as the complete price differentiation scheme. CP: θ 1 > θ 2, s 1 * >s 2 *, p 1 * >p 2 * θ 1 log(1+s 1 )-p 2 * s 1 θ 1 log(1+s 1 )-p 1 * s 1 High price p 1 * Low price p 2 * √ Consider two-group case: group 1 chooses higher price p 1 * at quantity s 1 * group 2 chooses lower price p 2 * at quantity s 2 *

23. Revenue Management for Communication Networks 23 CP under Incomplete Information Complete Price differentiation can be reached when the following condition is satisfied: for q=1,…K cp -1 Similar result may be extended to partial price differentiation

24. Revenue Management for Communication Networks 24 Outline Introduction Model Revenue maximization under Complete Information – Complete Price Differentiation – Single Price Scheme (No Differentiation) – Partial Price Differentiation Revenue maximization under Incomplete Information – Differentiation Schemes Conclusion

25. Revenue Management for Communication Networks 25 When is price differentiation most benifical? Differentiation Gain – Consider a simple two-group case: where Ratio of willingness to pays total number of the users the percentage of group 1 users (high willingness to pay users) the level of normalized available resource

26. 26 Revenue Management for Communication Networks Differentiation Gain in a Two-group Cases Fix α and k, for parameter t (1) K sp =2, K cp =2 (2) K sp =1, K cp =2 (3) K sp =1, K cp =1 (CD degenerates to SP) Define Differentiation gain is large when: α is small → high willingness to pay users are minorities k is small → resource is limited

27. Revenue Management for Communication Networks 27 CP under Incomplete Information Complete Price differentiation can be reached when the following condition is satisfied: for q=1,…K cp -1 where t q is the unique solution of over the domain t >1. Corollary: – t q is upper bounded by t q <t root ≈ 2.21846, for q=1,…K-1, – t root is the root of the equation Similar result can be extended to partial price differentiation