1. Dynamic Spectrum Management: Optimization, game and equilibrium Tom Luo (Yinyu Ye) December 18, WINE 2008

2. Outline Introduction of Dynamic Spectrum Management (DSM) Social Utility Optimization Noncooperative Nash Game Competitive Spectrum Economy Pure exchange market Budget Allocation Channel Power Production The objective is to apply algorithmic game/equilibrium theory to solving real and challenging problems

3. Dynamic Spectrum Management Communication system DSL, cognitive radio, cellular networks, cable TV networks, Multiple users (each has a utility function) access multiple channels/tones 2/3 allocated spectrum is not being used at any given times An efficient spectrum management scheme is needed

4. Spectrum Allocation Problem Model Each user i has a physical power demand Each channel/tone j has a power supply maximize system efficiency and utilization... user 1 user 2 power allocation user 3 channel D1D1 D2D2 D3D3 power supply s3s3 s2s2 s1s1 snsn s4s4

5. Shannon Utility Function x ij : the power allocation to user i on channel j x-bari: power allocations to all users other than i б ij : the crosstalk ratio to user i on channel j a i kj : the interference ratio from user k on channel j They may time varying and stochastic

6. Spectrum Management Models From the optimization perspective, the dynamic spectrum management problem can be formulated as 1. Social utility maximization May not optimize individual utilities simultaneously Generally hard to achieve 2. Noncooperative Nash game May not achieve social economic efficiency 3. Competitive economy market Price mechanism proposed to achieve social economic efficiency and individual optimality

7. 1. Social Utility Maximization

8. Social Utility Maximization - a two user and two channel example u1=u1= u2=u2= 1 Demand 1 1 user 1 user 2 1 Channel 1 2

9. Difficulty of the problem Even in the two user case, the problem is NP- hard. No constant approximation algorithm even for one channel and multiple users. Problems under the Frequency Division Multiple Access (FMDA) policy can be solved efficiently Luo and Zhang 2007

10. 2. Noncooperative Nash Game Model Each user maximize its own utility under a physical power demand constraint Ciofi and Yu 2002, etc.... user 1 user 2 power allocation user 3 channel D1D1 D2D2 D3D3

11. Individual rationality The basic game assumes that there is no limit on power supply for each channel. IWF: iterative water filling algorithm converges in certain cases

12. Spectrum Nash Game - the same toy example u1=u1= u2=u2= 1 Demand 1 1 user 1 user 2 1 Channel 1 2

13. Results on the problem No bound on ``price of anarchy’’ Can be solved as finding solution of a linear complementarity problem, so that it’s PPAD hard in general There is a FPTAS under symmetric interference condition There is a polynomial time algorithm under symmetric and strong weak interference condition Key to the proofs: the LCP matrix is symmetric Luo and Pang 2006, Xie, Armbruster, and Y 2008

14. 3. Competitive Spectrum Market  The problem was first formulated by Leon Walras in 1874, and later studied by Arrow, Debreu, and Fisher, also see Brainard and Scarf.  Agents are divided into two categories: seller and buyer.  Buyers have a budget to buy goods and maximize their individual utility functions; sellers sell their goods just for money.  An equilibrium is an assignment of prices to all goods, and an allocation of goods to every buyer such that it is maximal for the buyer under the given prices and the market clears.

15. Market Equilibrium Condition I

16. Market Equilibrium Condition II Physical Constraint: The total purchase volume for good j should not exceed its available supply:

17. Market Equilibrium Condition III Walras Law:

18. Competitive Communication Spectrum Economy What’s the ``budget’’ in DSM?

19. 3.1 Competitive Equilibrium in Spectrum Economy for Fixed Budget and Power Supply

20. Spectrum Management Channel Price Adjustment ( p j ) Channel Power Allocation ( s j ) Budget Allocation ( w i ) Objectives Fixed and given Improve channel power utilization

21. Competitive Spectrum Economy Model Each user buys channel powers under her budget constraint and maximize her own utility Price control goal Avoid congestion Improve resource utilization budget... user 1 user 2 power allocation user 3 channel w1w1 w2w2 w3w3 Price p1p1 p2p2 p3p3 p4p4 pnpn

22. Problem Formulation m users, each has a budget w i n channels, each with power capacity s j Design variable x ij Power allocation for i th user in jth channel p j Price for j th channel (Nash Equilibrium: p j =1 fixed) User utility (Shannon utility function )

23. Competitive Equilibrium Model Theorem A competitive equilibrium always exists for the spectrum management problem Y 2007 based on the Lemma of Abstract Economy developed by Debreu 1952

24. Equilibrium Properties Every channel has a price: All power supply are allocated: All budget are spent

25. Weak-Interference Market Weak-interference environment: the Shannon utility function of user i is In the weak-interference environment, An equilibrium can be computed in polynomial time. The competitive price equilibrium is unique. Moreover, if the crosstalk ratio is strictly less than 1, then the power allocation is also unique. (Y 2007)

26. Two methods of solving competitive equilibrium Centralized Solving the equilibrium conditions Decentralized Iterative price-adjusting

27. user 1 user 2 s 2 =2 s 1 =2 budget w 1 =1 w 2 =1 power supply Competitive Equilibrium Model user 1 user 2 Nash Equilibrium Model power constraint 5/3 7/3 Competitive Equilibrium Model - the same toy example equilibrium price p 1 =3/5 p 2 =2/5 5/3 1/3 2 u 1 =0.3522u 2 =0.2139 Social utility=0.5661 power allocation 5/3 u 1 =0.2341 u 2 =0.2316 Social utility=0.4657 1 4/3 power allocation

28. Computational Results Compare competitive equilibrium and Nash equilibrium Evaluate the performance in Individual utility and Social utility In most cases, CE results in a channel allocation Have a higher social utility value Make more users achieve higher individual utilities

29. 3.2 Budget Allocation in Competitive Spectrum Economy

30. Spectrum Management Channel Price Adjustment ( p j ) Channel Power Allocation ( s j ) Budget Allocation ( w i ) Objectives Fixed and given Make each user meet minimum power demand or utility value threshold Improve channel power utilization Lin, Tasi, and Y 2008

31. Budget Allocation in Competitive Spectrum Economy Budget allocation aims to satisfy a minimum physical power demand d i for each user i or satisfy a minimum utility value u i for each user i ; e.g., all users achieve an identical utility value Theorem: Such a budget equilibrium always exists.

32. Two methods of solving competitive equilibrium Centralized Solving entire optimal conditions which may be nonconvex Decentralized Iterative budget-adjusting

33. Budgeting for demand - computational results Number of (budget-adjusting) iterations required to achieve individual power demands

34. Budgeting for demand - computational results Number of iterations and CPU time (seconds) required to satisfy individual power demands in large scale problems, error tolerance=0.05

35. General cases: background noise randomly selected from (0,m], crosstalk ratio randomly selected from [0,1] In all cases, the social utility of CE is better than that of NE. Budgeting for demand - CE and NE comparison results

36. Budgeting for demand - More CE and NE comparison results In special type of problems, the competitive equilibrium performs much better than the Nash equilibrium does. For instance, the channels being divided into two categories: high-quality and low-quality. (In simulations, one half of channels with background noise randomly selected from the interval (0; 0,1] and the other half of channels with background noise randomly selected from the interval [1;m].)

37. Two-tier channels CE with power demands v.s. NE Budgeting for demand - More CE and NE comparison results

38. Budget allocation to balance utilities - Computational results Number of iterations and CPU time (seconds) required to balance individual utilities in large scale problems, difference tolerance=0.05

39. Two-tier channels CE with balanced utilities v.s. NE Budgeting to balance utilities - CE and NE comparison results

40. Comparison result summaries Compare with NE, in most cases, CE with minimum power demands results in power allocation Have a higher social utility Compare with NE, in most cases, CE with balanced utilities demands results in a power allocation Have a higher social utility Make more users have higher individual utilities Have a smaller gap between maximal individual utility and minimal individual utility In special type of problems, for instance, two tiers of channels, CE performs much better than NE does.

41. 3.3 Channel Power Production in Competitive Spectrum Economy

42. Spectrum Management Channel Price Adjustment ( p j ) Channel Power Allocation ( s j ) Budget Allocation ( w i ) Objectives To achieve higher social utility Fixed and given Improve channel power utilization

43. Produce power supply to increase social utility: the same toy example u1=u1= u2=u2=

45. Future Work How to systematically adjust channel power supply capacity to increase social utility? The convergence of the iterative variable- adjusting method for general setting Real-time spectrum management vs optimal policy at top levels