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

Outline 2 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

Introduction 3

4 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.

Introduction: Economic Paradigm Shift 5 White Space

Introduction: Cyclic Dependency 6 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

Introduction: Distribution of This Work 7 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

Related Work 8

9 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

Related Work (cont’d) 10 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 overlay networks,” in Proc. IEEE INFOCOM, 2005, vol. 4, pp. 2578–2589.

Spectrum Allocation Through Auctions 11

Spectrum Allocation Through Auctions 12 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.

Auction Issues 13 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)

Auction Rules 14 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”.

Auction Rules (cont’d) 15 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: q i = {w i, x i }. Amount of spectrum requested: w i Corresponding price that the WSP is willing to pay: x i

Auction Rules (cont’d) 16 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.

Solving the Optimization Problem 17 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

Bidder’s Strategies 18 Denote the optimal allocation as M, which is the set of all the winning demand tuples q i 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

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

Bidder’s Strategies (cont’d) 20 However, the auction is conducted in a sealed bid manner and thus bidder j would have no idea about X j. 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

Bidder’s Reservation Price 21 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 R static goes towards the static cost, then the difference, R dynamic, is the maximum amount that the provider can afford for the extra spectrum from CAB, i.e., Note that R dynamic is not the bidder’s reservation price but is a prime factor that governs this reservation price

Bidder’s Strategies: Second Price Scheme 22 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 tuple q j = {w j, x j }. Its reservation price for that amount w j is r j. The request will be granted and consequently belong to optimal allocation M only if the bid is at least X j. And the jth bidder will pay the second price, which is X j. (?) Then the payoff obtained by the bidder is

Bidder’s Strategies: Second Price Scheme (cont’d) 23 Assume that the jth bidder does not bid its true evaluation, i.e., x j ≠ r j. OptionsCasesWin?Expected payoff x j < r j r j > x j > X j O(r j – X j ) r j > X j > x j X0 X j > r j > x j X0 x j > r j x j > r j > X j O(r j – X j ) x j > X j > r j O(r j – X j ) < 0 (*) X j > x j > r j X0

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

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

Service Provisioning Using Games 26

Service Provisioning Using Games 27 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

Utility Function of Users 28 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 p j (t). Let b ij (t) be the resource consumed by user i, 1 ≦ i ≦ N, served by provider j. The total resource (capacity) of provider j is C j.

Utility Function of Users (cont’d) 29 The utility obtained by user i under provider j is Where a ij 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 b ij (t) amount of resource, that is

Utility Function of Users (cont’d) 30 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 Q j 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

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

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

Strategy Analysis 33 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 b ij (t) in ξ ( ‧ ) captures the behavior of channel quality.

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

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

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

Strategy Analysis: Check equilibrium 37 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

Price Threshold: Derivation (cont’d) 38 Differentiating (18) with respect to m Ij (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.

Estimating the Demand for Bandwidth 39

Estimating the Demand for Bandwidth 40 Motivation: Estimation of the resources consumed by the users and the price that is recovered from them would help a provider determine the bidding tuple q i = {w i, x i }. We equate (19) to 0 and get the optimal value of m Ij (t), which is denoted by m Ij(opt) (t). And then the optimal price can be obtained by Further, m ij(opt) (t) (or equivalently, b ij(opt) (t)) can be obtained by using (46)

Estimating the Demand for Bandwidth (cont’d) 41 While knowing p j(opt) (t) and b ij(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 m Ij(opt) (t) directly?) And thus the reservation price is governed by Therefore the amount of demand and the corresponding reservation value are found.

Channel Threshold Based Provider Selection 42

Goal 43 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.

Channel Threshold Based Provider Selection 44 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

Channel Threshold Based Provider Selection (cont’d) 45 Suppose that a user should be active with jth service provider only if its channel quality is better than a given threshold, Q T. The probability that a user is active with provider j is where f Q (x) is the probability density function of Q. Assume that all the other users act rationally and maintain Q T, then the probability that l users out of N users in the jth network would be

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

Channel Threshold Based Provider Selection (cont’d) 47 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

Channel Threshold Based Provider Selection (cont’d) 48 Let Q1 be the equilibrium solution to (36) Suppose a user now decides the threshold to be Q 2, while all the other users maintaining at Q 1, then the difference in gain is Study the two cases of Q 1 and Q 2 Case 1: Q 1 > Q 2 =>

Channel Threshold Based Provider Selection (cont’d) 49 Case 1: Q 1 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.

Numerical Result 50

Goal 51 In this section, we want to Simulate our auction model Show how our model outperforms the classical highest bid auction models Model the interaction between WSPs and users

Numerical Result: Spectrum Auctioning 52 Parameter setting Total amount of spectrum in CAB: 100 units. Minimum and maximum amount of spectrum requested by each bidder: 11 and 50 units. (!) Minimum bid per unit of spectrum: 25 units. Number of bidders: 10. Models compared Knapsack auction v.s. Classical highest bid auction (!) Knapsack synchronous v.s. Knapsack asynchronous (!)

Numerical Result: Spectrum Auctioning (cont’d) 53 Fig. 3(a) and (b) compares revenue and spectrum usage for knapsack synchronous and classical highest bid strategies for each auction round

Numerical Result: Spectrum Auctioning (cont’d) 54 In Fig. 4(a) and (b) we compare revenue and spectrum usage for both the synchronous and asynchronous strategies

Numerical Result: Spectrum Auctioning (cont’d) 55 Figs. 5(a), (b) and 6(a), (b) show the average revenue and spectrum usage with varying number of bidders for both comparisons

Numerical Result: Spectrum Auctioning (cont’d) 56 Fig. 7(a) and (b) show that the proposed Knapsack auction model encourages the low potential bidders

Numerical Result: Spectrum Auctioning (cont’d) 57 Fig. 8(a), (b) and Fig. 9 show that the proposed auction model discourages the presence of collusion.

Numerical Result: Pricing 58 How the pricing strategies proposed for WSP and end users interaction work as incentives for both We consider two cases: Fixed number of users Consider the equal weightage factor a ij = 1.5 and C j varies from 1 to 100 Increasing number of users Consider the ratio of a Ij and C j is fixed and C j varies form 1 to 100

Numerical Result: Pricing (fixed) 59 Fig. 10 shows how the provider must decrease the price per unit of resource if the total amount of resources increases with the same user base

Numerical Result: Pricing (fixed) (cont’d) 60 With the number of users fixed, we observe that the total profit of the provider increases till a certain resource and then decreases Allow us to estimate the maximum point

Numerical Result: Pricing (fixed) (cont’d) 61 More resources is an incentive of users

Numerical Result: Pricing (increasing) 62 The resource is initially scarce (raising high price) and slightly decreasing after a certain point

Numerical Result: Pricing (increasing) (cont’d) 63 With users increasing proportionally with resources, the total profit is always increasing which presents an incentive for the providers

Numerical Result: Pricing (increasing) (cont’d) 64 The net utility increases with increasing resources; thus providing incentive for the users (but it saturates earlier than the case of fixed number of users)

Conclusion & Comments 65

Conclusion 66 We provide a framework based on auction and game theories that capture the interaction among spectrum broker, service providers, and end-users. Knapsack Auction (synchronous, second price sealed manner) Utility function of WSPs and users and Nash Equilibrium Method for providers to estimate spectrum needed and reservation price Channel quality threshold Simulation result illustrating the collusion issue

Comments 67 Auction model makes bidders truly report their reservation prices which seems hard to get at first. Knapsack auction works well with the integer demand and bidding price (simply leaves to bidders) How to characterize Nash Equilibrium in mathematical equation Remind the issue of collusion Incomplete information for setting up the simulation, e.g., How to model the behavior of bidders through auction rounds? How to model the behavior of bidders if they colludes? Etc.