An Economic Framework for Dynamic Spectrum Access and Service Pricing

1. IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 17, NO. 4, AUGUST 2009 An Economic Framework for Dynamic Spectrum Access and Service Pricing ShamikSengupta Department of Electrical and Computer Engineering, Stevens Institute of Technology, Hoboken, NJ MainakChatterjee School of Electrical Engineering and Computer Science, University of Central Florida, Orlando, FL

2. Outline • Introduction • Related Work • Spectrum Allocation Through Auctions • Service Provisioning Using Games • Estimating the Demand for Bandwidth • Channel Threshold Based Provider Selection • Numerical Result • Conclusion & Comments

3. Introduction

4. Introduction • Wireless Service Providers (WSPs) buy spectrum from the spectrum owner and use it for providing services to the end users. • It is called static spectrum allocation • However, the current practice of static spectrum allocation often leads to low spectrum utilization and results in fragmentation of the spectrum creating “white space” (unused thin bands). • Therefore, the concept of Dynamic Spectrum Allocation (DSA) is being investigated.

5. Introduction: Economic Paradigm Shift White Space

6. Introduction: Cyclic Dependency • Two problems in the trading system • Dynamic spectrum allocation (upper half of Fig. 1) • WSPs use spectrum to service end users (lower half of Fig. 1) • Cyclic Dependency (typical supply-demand scenario) • Estimation of the demand for bandwidth and the expected revenue drive the WSP’s strategies • Service pricing in turn affect the demand by the users

7. Introduction: Distribution of This Work • We answer the following questions: • How the spectrum will be allocated from the coordinated access band (CAB, a common pool of open spectrum) to the service providers • How service providers will determine the price of their services • How are the above two inter-related

8. Related Work

9. Related Work • Auction Theory • Single-unit auction [33] • Multi-unit multiple winners [1] • Second-price auction [19] • Real-time auction framework and piecewise linear demand curve [8] • Collusion issue [12] • … [1] B. Aazhang, J. Lilleberg, and G. Middleton, “Spectrum sharing in a cellular system,” in IEEE Symp. Spread Spectrum Techniques and Applications, 2004, pp. 355–359. [8] S. Gandhi, C. Buragohain, L. Cao, H. Zheng, and S. Suri, “A general framework for wireless spectrum auctions,” in Proc. IEEE DySpan, 2007, pp. 22–33. [12] Z. Ji and K. J. R. Liu, “Collusion-resistant dynamic spectrum allocation for wireless networks via pricing,” in Proc. IEEE DySpan, 2007, pp. 187–190 [19] P. Maille and B. Tuffin, “Multibid auctions for bandwidth allocation in communication networks,” in Proc. IEEE INFOCOM, 2004, ol. 1, pp. 54–65 [33]W.Vickrey, “Counterspeculation, auctions, and competitive sealed tenders,” J. Finance, vol. 16, no. 1, pp. 8–37, Mar. 1961

10. Related Work (cont’d) • Game Theory • Overview and application to networking and communication [35] • Network services have been studied with the help of stability and fairness [13] • Service admission control using game theory [17] • … [13] F. P. Kelly, A. K. Maulluo, and D. K. H. Tan, “Rate control in communication networks: Shadow prices, proportional fairness and stability,” J. Oper. Res. Soc., vol. 49, pp. 237–252, 1998 [17] H. Lin, M. Chatterjee, S. K. Das, and K. Basu, “ARC: An integrated admission and rate control framework for CDMA data networks based on non-cooperative games,” in Proc. MobiCom, 2003, pp. 326–338. [35] W. Wang and B. Li, “Market-driven bandwidth allocation in selfish overlaynetworks,” in Proc. IEEE INFOCOM, 2005, vol. 4, pp. 2578–2589.

11. Spectrum Allocation Through Auctions

12. Spectrum Allocation Through Auctions • The interaction between the spectrum broker and the WSPs. • Auction is invoked only when the total demand of spectrums exceeds the total spectrum available in the CAB. • Auction should be conducted on a periodic basis and on a small time granularity (e.g., every 1, 12, 24 hours). • Synchronous auctions will allow the spectrum brokers maximize revenue • Asynchronous auctions (WSPs can make requests at any point of time) make it possible for lower bidders win the auctions before higher bidders come and thus spectrum might be unavailable when higher bidders come.

13. Auction Issues • Spectrum auctions are multi-unit auctions (bidders bid for different amount of spectrum) • We assume that total spectrum is homogeneous and thus no band is superior or inferior than any other band • Roles in auctions • The spectrum broker – seller/auctioneer • The WSPs – buyer/bidder • Important issues • How to maximize the revenue generated from bidders. • How to maximize the spectrum usage. • How to entice bidders by increasing their probability of winning. (in simulation section) • How to prevent collusion among providers. (in simulation section)

14. Auction Rules • We assume that the WSPs need at least the spectrum amount requested (minimum requirement). • A WSP would get negative utility if he/she obtain spectrum less than the minimum requirement. • It is necessary to make the small companies interested in the auctions (encourage competition). • The problem is close to the classical knapsack problem • The sack represents the finite capacity of spectrum. • The item’s weight and value represent WSP’s requested amount and bid. • We propose the “Winning Determining Sealed Bid Knapsack Auction”.

15. Auction Rules (cont’d) • There are L WSPs competing for a total spectrum W in a particular geographic region • All the WSPs submit their demands at the same time in a sealed manner • Sealed bid auction has shown to be perform well in all-at-a-time bidding and has the tendency to prevent collusion [26] • Each WSP has no knowledge about other’s bidding quantity and price • Strategy adopted by service provider i: qi = {wi,xi}. • Amount of spectrum requested: wi • Corresponding price that the WSP is willing to pay: xi

16. Auction Rules (cont’d) • The optimization problem would be • Note that a more realistic approach would have been a multiple-choice knapsack formulation with each bidder submit a continuous demand curve. • However, optimizations with continuous demand is hard.

17. Solving the Optimization Problem • We assume that • Bids can take only integer values. • The number of bidders is typically of the order of 10. • Thus the problem can be solved through dynamic programming with reasonably low computation • P.S. Unbounded knapsack problem using dynamic programming • m[W] is the solution

18. Bidder’s Strategies • Denote the optimal allocation as M, which is the set of all the winning demand tuplesqi • The aggregate bid • Consider a particular bidder j who was all allocated spectrum. Then the aggregate bid without that bidder j is • Next consider if that bidder doesn’t exist, then the optimal allocation change from M to M*, and the aggregate bid is

19. Bidder’s Strategies (cont’d) • Therefore, minimum winning bid of bidder jmust be at least greater than • Bidder j will be • granted the request if xj > Xj • not granted the request if xj < Xj • indifferent between winning and losing if xj = Xj • This implies that if bidder j knows the bids of other bidders, he/she could govern the auctions.

20. Bidder’s Strategies (cont’d) • However, the auction is conducted in a sealed bid manner and thus bidder j would have no idea about Xj. • We want to find if there exists any Nash equilibrium strategy of the bidders. • Nash equilibrium: no player in the game finds it beneficial to change his/her strategy • Two different schemes under the knapsack auction are studied two corresponding lemmas are presented • First price scheme: winning bidders pay their bid • Second price scheme: winning bidders pay the second highest bid

21. Bidder’s Reservation Price • Definition: Bidder’s reservation price is the maximum price a bidder would be willing to pay. • WSP buys spectrum from the spectrum broker, and then sell in the form of services to end users • The revenue thus generated helps the provider to pay the cost of spectrum statically allocated and dynamically allocated. • Let the total revenue generated be R, and Rstatic goes towards the static cost, then the difference, Rdynamic, is the maximum amount that the provider can afford for the extra spectrum from CAB, i.e., • Note that Rdynamic is not the bidder’s reservation price but is a prime factor that governs this reservation price

22. Bidder’s Strategies: Second Price Scheme • Lemma 1: In the second price knapsack auction, the dominant strategy of the bidder is to bid bidder’s reservation price. • Proof • Assume that the jth bidder has the demand tupleqj = {wj, xj}. • Its reservation price for that amount wj is rj. • The request will be granted and consequently belong to optimal allocation M only if the bid is at least Xj. • And the jth bidder will pay the second price, which is Xj. (?) • Then the payoff obtained by the bidder is

23. Bidder’s Strategies: Second Price Scheme (cont’d) • Assume that the jth bidder does not bid its true evaluation, i.e., xj ≠ rj.

24. Bidder’s Strategies: Second Price Scheme (cont’d) • If bidder j wins, then the maximum expected payoff is given by Ej = rj – Xj. • And bidding any other (higher or lower) than its reservation price rj will not increase payoff. • Thus the dominant strategy of a bidder in second price bidding under knapsack model is to bid its reservation price.

25. Bidder’s Strategies: First Price Scheme • Lemma 2: In first price bidding, reservation price is the upper bidding threshold. • Proof • The expected payoff can be given by, Ej = rj – xj. • To keep Ej > 0, xjmust be less than rj. • Then the weak dominant strategy for the bidder in first price auction is to bid less than the reservation price.

26. Service Provisioning Using Games

27. Service Provisioning Using Games • In this section, we consider the model between WSPs and end users as shown in the lower half of Fig. 2, where any user can access any WSP. • Goal • investigate whether there exists any strategy that will help the users and providers to reach an equilibrium in the game • To reach the goal we • characterize the utility functions of the users and providers • and then analyze the utility functions • At last we find a pricing threshold helping both sides to reach the (Nash) equilibrium

28. Utility Function of Users • We consider L service providers that cater to a common pool of N users. • Let the unit price advertised by the service provider j, 1≦j≦L, at time t be pj(t). • Let bij(t) be the resource consumed by user i, 1≦i≦N, served by provider j. • The total resource (capacity) of provider j is Cj.

29. Utility Function of Users (cont’d) • The utility obtained by user i under provider j is • Where aij is a positive parameter that indicates the relative importance of benefit and acts as a weightage factor. • We chose the log function since it is analytically convenient, increasing, strictly concave and continuously differentiable. • The first cost component: direct cost paid to the provider for obtaining bij(t) amount of resource, that is

30. Utility Function of Users (cont’d) • The second cost component: queuing delay, assuming the queuing process to be M/M/1 at the links, and thus the delay cost component is • The third cost component: channel condition. • Assume that Qj denotes the wireless channel quality received from the base station of the jth provider and thus the third cost component is P.S. Expected waiting time in M/M/1 is 1/(μ-λ), μ: service rate, λ: arrival rate

31. Utility Function of Users (cont’d) • Combining all the components in (9), (10), (11) and (12), we get the net utility as

32. Utility Function of Service Providers • The utility of service provider j at time t is • Where Kj is the cost incurred to provider j for maintaining network resources, and is assumed to be constant for simplicity

33. Strategy Analysis • To simplify the analysis, we assume that all the users maintain a channel quality threshold. • We combine the cost components in (11) and (12), i.e., • where • We assume that bij(t) in ξ(‧) captures the behavior of channel quality.

34. Strategy Analysis: Users maximize their utility • Differentiating (15) with respect to bij(t), • Similarly, the second derivative is • If we assume the last term in (42) is positive, then Uij’’(t)<0 and Uij(t) contains a unique maximization point

35. Strategy Analysis: Users maximize their utility (cont’d) • To maximize user’s utility, the first derivative of all the users can be equated to zero: • (43) reduces to • If 1+bij(t)=mij(t), we get

36. Strategy Analysis: Providers’ price threshold • For notation simplicity, (45) can be written as • Put (46) into (41), we get • For achieving Nash equilibrium by the providers, the pricing constraint pj(t) is upper bounded by

37. Strategy Analysis: Check equilibrium • We need to investigate if this upper bound also helps the providers in maximizing their utilities. • We are interested in finding mutual best responses from both users and providers so that they don’t find better utility by deviating from the best responses unilaterally. • With users’ maximization strategy known, we find if providers’ net utility has any maximization point. (?) • Providers’ net utility can be re-written as

38. Price Threshold: Derivation (cont’d) • Differentiating (18) with respect to mIj(t), we get • Differentiating again, and we can get, Vj’’(t) < 0 (?); which implies that utility for the providers has a maximization point obeying the price bound. • Thus this pricing upper bound from the providers helps the users and providers to maximize their utilities and reach the Nash equilibrium point.

39. Estimating the Demand for Bandwidth

40. Estimating the Demand for Bandwidth • Motivation: Estimation of the resources consumed by the users and the price that is recovered from them would help a provider determine the bidding tupleqi = {wi, xi}. • We equate (19) to 0 and get the optimal value of mIj(t), which is denoted by mIj(opt)(t). • And then the optimal price can be obtained by • Further, mij(opt)(t)(or equivalently, bij(opt)(t)) can be obtained by using (46)

41. Estimating the Demand for Bandwidth (cont’d) • While knowing pj(opt)(t) and bij(opt)(t) and using a transformation function T to convert utility to a dollar value, the total revenue obtained by provider j is • (why not use mIj(opt)(t) directly?) • And thus the reservation price is governed by • Therefore the amount of demand and the corresponding reservation value are found.

42. Channel Threshold Based Provider Selection

43. Goal • Analyze whether there exists any strategy for the users in choosing wireless service providers with respect to channel condition • If there exists any channel quality threshold, i.e., any minimum acceptable channel quality below which it will not be beneficial to select a network.

44. Channel Threshold Based Provider Selection • Theorem 1: Under varying channel conditions, a rational user should be active (transmit/receive) only when the channel condition is better than the minimum channel quality threshold set by the system to achieve Nash equilibrium. • Proof: • Rewrite the net utility function of users as • where

45. Channel Threshold Based Provider Selection (cont’d) • Suppose that a user should be active with jth service provider only if its channel quality is better than a given threshold, QT. • The probability that a user is active with provider j is • where fQ(x) is the probability density function of Q. • Assume that all the other users act rationally and maintain QT, then the probability that l users out of N users in the jth network would be

46. Channel Threshold Based Provider Selection (cont’d) • The expected net utility of the ith user (active) is • And if we define • then we rewrite the expected net utility as

47. Channel Threshold Based Provider Selection (cont’d) • If the use is not active, the expected net utility is 0. • For a user, the expected net utility for being active and not being active should be equal at the threshold, i.e., • Next we show that if the threshold is set by (36), Nash equilibrium can be reached. • The achievable gain net utility considering both modes is • where

48. Channel Threshold Based Provider Selection (cont’d) • Let Q1 be the equilibrium solution to (36) • Suppose a user now decides the threshold to be Q2, while all the other users maintaining at Q1, then the difference in gain is • Study the two cases of Q1 and Q2 • Case 1: Q1 > Q2 =>

49. Channel Threshold Based Provider Selection (cont’d) • Case 1: Q1 < Q2 => • Thus, a user cannot increase his gain by unilaterally changing his/her strategy. • As a result, a channel quality threshold exists for the users and maintaining this threshold will help the users to reach Nash Equilibrium.

50. Numerical Result