AUCTION DESIGN FOR SECONDARY SPECTRUM MARKETS

Transcription

1 AUCTION DESIGN FOR SECONDARY SPECTRUM MARKETS by Yuefei Zhu A thesis submitted in conformity with the requirements for the degree of Master of Applied Science, Department of Electrical and Computer Engineering, at the University of Toronto. Copyright c 2012 by Yuefei Zhu. All Rights Reserved.

2 Auction Design for Secondary Spectrum Markets Master of Applied Science Thesis Edward S. Rogers Sr. Dept. of Electrical and Computer Engineering University of Toronto by Yuefei Zhu 2012 Abstract Opportunistic wireless channel access by non-licensed users has emerged as a promising solution for addressing the bandwidth scarcity challenge. In this thesis, we first design both a deterministic heuristic auction and a randomized auction with a provable performance bound with the guarantee of truthfulness, for networked secondary users. We then turn our attention to mobility support for the secondary users. We introduce two-dimensional bids that reflect a secondary user s willingness to pay for exclusive and nonexclusive channel usage, for the single-channel and multiple-channel scenarios, under which we prove their performances under desired equilibria, respectively. We also devise core-selecting auctions in a combinatorial setting, where secondary users can submit flexible preferences on channels. These auctions can resolve VCG s vulnerability to collusion and shill bidding, and improves seller revenue. ii

3 To my parents

4 Acknowledgments I must first express my gratitude towards my advisor, Professor Baochun Li. Without his leadership, support, patience and guidance, this two year s study journey cannot be smoothly completed. Throughout my paper-writing period, he provided sound advice and good teaching. I would have been lost without his assistance. I would also like to thank Professor Zongpeng Li, who provided lots of suggestions on exploring interesting and insightful outcomes during my research. I am indebted to all the members in iqua Research Group at the University of Toronto, who not only offered insightful suggestions to my research but also created a stimulating and fun environment in which to learn and grow. I am especially grateful to Wei Wang, who was particularly helpful, patiently assisting me with the analysis of the problem in my first research project. I would also like to thank Prof. Wei Yu and Prof. Shahrokh Valaee for their valuable suggestions on revising the thesis to its final completion. Their questions have largely helped me think through and revise the motivations, the correctness of the problem settings, and the details of the technical materials. Last, and most importantly, I wish to express my love and gratitude to my family my parents, Bingkui Zhu and Juane Wang, my brother Yuejie Zhu, for their understanding and endless love. When I felt stressed, it was their continued support that encouraged me to get through those difficult times. iii

5 Contents Abstract ii Acknowledgments iii List of Tables vii List of Figures ix 1 Introduction Spectrum Auctions for Secondary Networks Spectrum Auctions for Mobile Users Spectrum Auctions with Heterogenous Channels Thesis Organization Background and Related Work Auction Design Related Work Truthful Spectrum Auction Design for Secondary Networks Preliminaries iv

6 CONTENTS CONTENTS Truthful Auction Design System Model A Heuristic Truthful Auction Channel Allocation Payment Calculation A Truthful Auction for Approximately Maximizing Social Welfare Decomposing the Fractional Solution Studying the Integrality Gap A Randomized Approximation Auction Simulation Results Auction Efficiency under Various Settings Comparison with the Naive Greedy Auction Illustration for the Randomized Auction Performance Discussions on Intra-SN Interference Summary Designing Spectrum Auctions with Mobility Support Preliminaries GR 2D : Enabling Exclusivity for the Single Channel Case The design of GR 2D Analysis of GR 2D VCG 2D : Enabling Exclusivity for the Multiple Channel Case GR 2D Is Not Suitable for Auctioning Multiple Channels Network Partitioning for Interference Control Design of VCG 2D for Multiple Channels v

7 CONTENTS CONTENTS Analysis of VCG 2D for Multiple Channels Simulation Studies Summary Core-Selecting Auctions for Secondary Spectrum Markets Network Model and Preliminaries Core Selection and Its Necessity The Core of An Auction Revenue Lower Bound of A Core-Selecting Auction Payment Rules Payment Rule of the VCG Mechanism Revenue-Minimization Rule VCG-Nearest Rule Simulation Results Simulation Environment Allocation Results Revenues of Core-Selecting Auctions Summary Conclusion 91 Bibliography 95 vi

8 List of Tables 3.1 List of notations Agents with non-zero fractional flows vii

9 List of Figures 1.1 A secondary spectrum market with 3 SNs and 2 channels A exclusivity-enabled secondary spectrum market with 4 SUs and 2 channels Procedure of channel assignment. Dots and squares represent source and destination nodes respectively The circumstance formed by interference regions of one link The circumstance formed by interference regions of two links Auction efficiency with different numbers of bidders enrolled Auction efficiency under different interference situations Auction efficiency with different sizes of networks Comparison of three different auction settings Comparing our heuristic auction with the NAIVE-b auction, under different evaluation metrics. Fig. 3.8a - Fig. 3.8c assume uniformly random topologies and Fig. 3.8d - Fig. 3.8f assume clustered topologies A histogram of fractional solutions of the LPR A simple illustration for Gopinathan s auction Uniformly coloured hexagons, with 7 colours viii

10 LIST OF FIGURES LIST OF FIGURES 4.3 Minimum distance between two points in co-coloured hexagons Performance evaluation of the 2D auctions A geometric illustration of the core Four SUs bidding for 2 channels Three SUs bidding for 2 channels, where bidder c is using shills. An edge between bidders 2 and 4 is introduced Performance of the allocation result Impact of interference on revenues Comparisons of revenues Influence of bid distribution: Revenue vs Number of bids Influence of bid distribution: Revenue vs Number of channels ix

11 Chapter 1 Introduction Recent years have witnessed substantial growth in wireless technology and applications, which rely crucially on the availability of bandwidth spectrum. Traditional spectrum allocation is static, and is prone to inefficient spectrum utilization in both temporal and spatial domains: large spectrum chunks remain idling while new users are unable to access them. Such an observation has prompted research interest in designing a secondary spectrum market, where new users can access a licensed channel when not in use by its owner, with appropriate remuneration transferred to the latter. In a secondary spectrum market, a spectrum owner or primary user (PU) leases its idle spectrum chunks (channels) to secondary users (SUs) through auctions [1, 2]. SUs submit bids for channels, and pay the PU a price to access a channel if their bids are successful. A natural goal of spectrum auction design is truthfulness, under which an SU s best strategy is to bid its true valuation of a channel, with no incentive to lie. A truthful auction simplifies decision making at SUs, and lays a foundation for good decision making at the PU. Another important goal in spectrum auction design is social welfare maximization, i.e., maximizing the aggregated happiness of everyone in the system. 1

12 CHAPTER 1. INTRODUCTION 2 Such an auction tends to allocate channels to SUs who value them the most. In economic theory, auctions are a well studied protocol for allocating scarce resources amongst competing, selfish agents. However, when it comes to network resource allocation problems like wireless spectrum distribution, directly applying existing auction mechanisms is often infeasible some characteristics from a network setting requires a specific, brand-new designed auction. One of the most important underlying reasons in wireless spectrum auction design is, agents may operate the spectrum in a geographically restricted area, and the usage of the same spectrum chunk by multiple closely located nodes can cause interference. That is, we are under the premise of spectrum sharing, where the same spectrum chunk can be reused amongst different users that are far enough from each other. Therefore special care must be given to ensure that the allocation is interference-free. Especially in a secondary spectrum market, an auction is often hold for a short period (e.g., within one week), whose result must be calculated in real time after the arrivals of all the bids (e.g., within tens of minutes or even shorter). This can incur many interesting and complex problems when we are facing concrete network requirements. In this thesis, our goals are to design different auctions for various network and economic environments, which are briefly summarized as in the following. Existing works on spectrum auctions often assume the simplest model of a SU: a single node, or a single link, similar to a single hop transmission in cellular networks [2 4]. However, SU may very well comprise multiple nodes forming a multi-hop network, which we refer to as a secondary network (SN). These include scenarios such as users with multihop access to base stations, or users with their own mobile ad hoc networks. SNs

13 1.1. SPECTRUM AUCTIONS FOR SECONDARY NETWORKS 3 require coordinated end-to-end channel assignment, and in general benefit from multichannel diversity along its path. The SN model subsumes the SU model as the simplest special case. Auctioning channels to SNs for successful transmission needs to make judicious joint routing and channel assignment decisions. We next turn our attention to mobility support. Instead of employing the traditional assumption that all the SUs are static, we propose that in practice, some of the SUs prefer not to compromise their ability to communicate on the move after all, being mobile has been one of the original driving forces behind wireless communication [5]. A highly mobile SU incurs potential interference with all other SUs within its mobility range. We design auctions with an additional bid dimension to provide exclusive usage of a channel, thereby supporting mobility. Finally, when the scale of the auction is limited which is often the case in reality we resort to combinatorial auctions to provide flexibilities for bidding. That is, SUs are allowed to bid for combinations of channels in a single round of auction, given their desired technology requirements or the heterogeneity of channels. This is quite different from most of the existing proposed auctions in the literature [1, 3, 6], where channels are assumed to be identical. Under such a combinatorial setting, we resort to the recent proposed core-selecting auctions to provide high revenues and economic robustness. In what follows, we give a brief overview of each problems, and present how this thesis is organized. 1.1 Spectrum Auctions for Secondary Networks In a secondary spectrum market, a unique feature of spectrum auctions is to enable spectrum reuse, without incurring interference. A channel can be reallocated to multiple users

14 1.1. SPECTRUM AUCTIONS FOR SECONDARY NETWORKS 4 given that they are far apart. Optimal channel assignment for social welfare maximization is equivalent to the graph colouring problem, and is NP-hard [7], even assuming truthful bids are given for free. Existing works on spectrum auctions often focus on resolving such a challenge (e.g., [3, 4]) while assuming the simplest model of a SU: a single node, or a single link, similar to a single hop transmission in cellular networks [2 4]. However, this assumption may not hold in practice for the users who have multihop transmission demands (SNs). Even we can hold an auction multiple times for those users, a successful end-to-end path for a specific user may not be formed, and the auction itself would be quite inefficient. In addition, spectrum can be highly under-utilized if the auctioneer is unaware of the SNs requirements. SN1 SN2 3 1 SN3 2 Figure 1.1: A secondary spectrum market with 3 SNs and 2 channels. Fig. 1.1 depicts three co-located SNs, SN1, SN2 and SN3, which have interference with one another, because their network regions overlap. The primary network (PN) has two channels, Ch1 and Ch2, which have been allocated to SN1 and SN2, respectively. Now SN3 wishes to route along a two-hop path Under existing singlechannel auctions for SUs, SN3 cannot obtain a channel, because each channel interferes with either SN1 or SN2. However, a solution exists by relaxing the one channel per user assumption, and assigning Ch1 to link 1 2 and Ch2 to the link 2 3. In general,

15 1.2. SPECTRUM AUCTIONS FOR MOBILE USERS 5 taking multichannel, multihop transmissions by SNs into consideration can apparently improve channel utilization and social welfare. Note here that the model in which an SN bids for multiple channels is inapplicable, because due to the unawareness of other SNs information, an SN cannot know the number of channels to bid for, to form a feasible path. Designing truthful auctions for SNs is an interesting problem, but by no means an easy one. We note that it is hard for an SN to decide by itself an optimal or good path to bid for. Such decision making requires global information on other SNs as well, and is naturally best made by the auctioneer, i.e., the primary network (PN) owner. Consequently, a bid from an SN includes just a price it wishes to pay, with two nodes it wishes to connect using a path. Furthermore, SNs now interfere with each other in a more complex manner. Not only that they transmit along multihop paths, but each path can be assigned with distinct channels at different links. The PN, after receiving bids, needs to make judicious joint routing and channel assignment decisions. 1.2 Spectrum Auctions for Mobile Users As one of the most efficient allocation mechanisms, spectrum auctions have attracted strong research attention in recent years. A salient feature of spectrum auctions, as compared to classic auctions from the field of economics, is the need to handle wireless interference among SUs properly. Most existing spectrum auctions in the literature [3,8 10] are designed to appropriately model the externality resulting from such interference, for better channel reuse and hence a larger social welfare. Unfortunately, to the best of our knowledge, all existing spectrum auction designs are based on the implicit assumption that SUs are static. Although the auction may be held periodically [3], and the location

16 1.2. SPECTRUM AUCTIONS FOR MOBILE USERS 6 of an SU may vary from one round to another, generally all SUs are required to be static at least within one round of the auction. In practice, a secondary network may indeed be heterogeneous. While some SUs are content with static communication, others prefer not to compromise their ability to communicate on the move after all, being mobile has been one of the original driving forces behind wireless communication [5]. A highly mobile SU incurs potential interference with all other SUs within its mobility range. It would demand for exclusive usage of a channel (mobile) Primary User 2 Figure 1.2: A exclusivity-enabled secondary spectrum market with 4 SUs and 2 channels. In the economy of a secondary spectrum market, whether exclusivity is a desirable choice depends on the behaviour of all SUs, particularly, on their submitted bids. Intuitively, if an SU s bid is high enough to exclude all the other users from accessing a specific channel, it may own that channel with the guarantee that no one else will reuse it in the entire region. For example, in Fig. 1.2, there are 4 SUs (SU 1, 2, 3, 4) and two available channels (ch1, ch2) in the region. SUs whose transmission regions overlap interfere with one another. Imagine SU 3 has the demand to move within the entire region without communication conflict, and SU 3 submits a dominantly high bid leading itself to the exclusive use of channel 1. The other users, given their relatively uncompetitive bids, can still share (reuse) the other channel available. One possible scenario is that SU

17 1.3. SPECTRUM AUCTIONS WITH HETEROGENOUS CHANNELS 7 1 and 2 reuse channel 2, while SU 4 is not allocated with a channel due to interference. To enable such exclusive channel access to support mobility, we adopt a two-dimensional bidding language, assuming SUs have two-dimensional valuations: one for exclusively accessing a channel, and another for reusing a channel. Let s denote a 2D bid as (b, b). If an SU prefers either exclusive channel access or no access at all, it may submit a bid of zero for reuse ( b = 0). An SU without mobility demand may submit two identical values in its bid (b = b). More generally, an SU may submit two non-zero values in its bid (b b 0), a higher value for exclusive, mobile channel access and a lower value for non-exclusive, static channel access. A traditional spectrum auction design essentially has the second bid b (for channel reuse) only, or can be viewed as a special case of our 2D auction where SUs are required to bid b = b. The additional dimension of information in b helps our 2D spectrum auction decide whether channels are to be allocated exclusively or reused. For each channel, we have two possible allocations: S, where the channel is entitled to a single SU in the region, and M, where the channel is shared by multiple SUs. Designing a spectrum auction that accommodates such 2D bids and judiciously decides between the two possible channel allocation outcomes for the goal of optimal social welfare becomes a new, intriguing subject of study. 1.3 Spectrum Auctions with Heterogenous Channels Existing literature on secondary spectrum auctions often treats wireless channels as identical goods [1, 6, 11]. Given the heterogeneity of channels and different technology requirements from real-world settings, secondary users are likely to desire combinations of channels in practice. For instance, the channels may experience different levels of fading

18 1.3. SPECTRUM AUCTIONS WITH HETEROGENOUS CHANNELS 8 at different locations, and two users may value the same channel quite differently. As another example, LTE and WiMAX require paired and unpaired channels respectively [12]. An SU aiming to provide an LTE-based service will be willing to bid for two paired channels, while a WiMax-based SU will not. If the PU holds multiple auctions to sell these heterogenous channels and paired channels, that usually undermines the efficiency in spectrum assignment [12]. Combinatorial auctions enable expressive bids for requesting bundles of channels, and are especially useful when the PU has no a priori information on how SUs plan to utilize the channels. A classic auction that guarantees truthfulness is the celebrated VCG mechanism [13 15]. It is the only auction mechanism that is both truthful and efficient [16,17]. Despite a myriad of interests in theoretical research, VCG mechanisms witness less enthusiasm in actual implementations [18 20]. Part of the hurdle was attributed to the requirement for solving the allocation problem to optimality, which is often NP-hard, as in interferencefree channel allocation. This motivated the design of truthful spectrum auctions with a compromise in efficiency (social welfare) [1, 3, 21, 22]. However, our studies reveal that, given a representative secondary spectrum market, winner determination and channel allocation can be formulated into a linear integer program of modest size (on the order of 1000 variables and constraints), which can be solved to optimality in seconds over today s average computing platform. Sacrifices in efficiency are therefore less justified. The VCG mechanism suffers from two other problems that are economic instead of computational. The first is that it turns to generate a low revenue for the auctioneer, under-exploiting the payment potential of bidders. The second is that a VCG mechanism is susceptible to a form of strategic bidding known as shill bidding, or false-name bidding, in which a single bidder desires a set of items impersonate multiple bidders, each bidding

19 1.3. SPECTRUM AUCTIONS WITH HETEROGENOUS CHANNELS 9 for a subset of those items [23]. For example, consider three SUs (SU1, SU2, SU3) bidding for two different channels ch1 and ch2, and they desire {ch1}, {ch2} and {ch1, ch2}, respectively. Each SU is willing to pay $10 for acquiring what it desires, and $0 otherwise. The VCG mechanism allocates ch1 to SU1, and ch2 to SU2, with zero charges (because neither of them would cause any loss to social welfare by not participating in the auction, see Sec. 5.3 for details). The zero income is by no means satisfactory, given that each SU has expressed a willingness to pay up to $10, manifesting the low revenue problem. Furthermore, assume that SU1 and SU2 are indeed controlled by a single SU0 who has a valuation of $10 for ch1, $10 for ch2, and $20 for {ch1, ch2}. Knowing the rule of the auction, SU0 can reduce its payment for winning {ch1, ch2} from $10 (because SU3 would win the two channels if SU0 was ruled out, creating social welfare $10, the payment for SU0 is $20 $10) to $0 via impersonation, manifesting the shill bidding problem. The vulnerabilities of the VCG mechanism are so severe that it rarely made to a direct application in practice. Since VCG is the only truthful and efficient mechanism [16, 17], any other efficient auction aimed at addressing the two economic problems inherent in VCG will have to relax the requirement of absolute truthfulness. A promising direction of research is core-selecting auctions [24 26]. An auction outcome is in-core if no group of participants (including the auctioneer) is motivated to secede to settle for their own solution. Taking the group as the entire set of participants, this implies efficiency (social welfare is maximized). Given guaranteed efficiency, the auctioneer can further judiciously select from the core an auction (actually a payment rule, see Sec. 5.3) that maximizes the likelihood of truthful bidding. Representing the state-of-the-art of a pragmatic combinatorial auction, core-selecting auctions recently enjoyed real-world applications, including

20 1.4. THESIS ORGANIZATION 10 spectrum auctions at the primary spectrum market level [12]. The primary and secondary spectrum markets differ fundamentally due to concerns on wireless interference absent in the former and present in the latter. 1.4 Thesis Organization The remainder of this thesis is organized as the following. We present related work in the literature in Chapter 2. Chapter 3 presents the design of truthful auctions for secondary networks where end-to-end multihop transmission is supported. In Chapter 4, we introduce the design of two-dimensional auctions to enable mobility support for secondary users. Chapter 5 shows how to ensure the economic robustness in combinatorial settings in secondary spectrum auctions with respect to optimal social welfare, and enhancing seller s revenue at the same time. Chapter 6, we conclude our work and propose future work.

21 Chapter 2 Background and Related Work In this chapter, we first briefly introduce some background knowledge on auction design and then present some further reading related to this work. 2.1 Auction Design In an auction, agents compete over a set of items through a bidding system. Generally, it can be described as the following [27]: 1. A finite set O of allowed outcomes. 2. Each agent has a privately-known real value, called its valuation, which quantifies the bidder s benefit from the outcome. 3. Bidders are required to submit/declare their valuations in terms of bids. The bidders may lie about their valuations. Thus the bid of an agent may not be equal to its valuation. 11

22 2.1. AUCTION DESIGN An auction chooses an outcome o based on some criteria over the vector of declared bids. 5. In addition to determining an outcome, the auction also charges each bidder a certain amount of currency. 6. The utility of each bidder is the difference between its true valuation and its payment, based on the outcome. In this thesis, we will focus on a natural and important goal of auction design, social welfare maximization. Social welfare is defined as the sum of all the winning agents valuations, which can be viewed as the aggregated happiness (utility) of everyone in the system, including the auction holder. Adopting conventional assumptions in the literature, we assume that each agent is selfish and rational. A selfish agent is one that acts strategically to maximize its utility. An agent is said to be rational when it always prefers the outcome that brings itself a higher utility. Hence, an agent may lie about its valuation if doing so yields a higher utility. An auction is said to be truthful when bidders optimal behaviour is to report their true valuations, regardless of others bids, i.e., declaring their valuation truthfully can maximize their utilities in such an auction setting. The only general auction that aims at optimizing social welfare and guarantees truthfulness is due to Vickrey, Clarke, and Groves (VCG) [13 15]. Informally, the celebrated VCG auction finds the optimal outcome o that maximizes the social welfare, and charges each winning agent i an amount equal to the total damage that it causes to the other bidders, i.e., the difference between the social welfare of the others with and without i s participation [28]. However, in a spectrum auction, it requires optimal channel allocation, whose computation is NP-complete [29], and hence real-time spectrum auctions often rely on other heuristic algorithms.

23 2.2. RELATED WORK Related Work Auctions serve as an efficient mechanism for distributing scarce resources to competing participants in a market. To simplify the strategical behaviour of agents and hence encourage participation, truthfulness is desired. A celebrated work is the VCG mechanism due to Vickrey [13], Clarke [14], and Groves [15]. However, the VCG mechanism is only suitable when optimal solutions are computationally feasible, and is not directly applicable for secondary spectrum auctions, because interference-free channel allocation is NP-Hard. The large series of work on spectrum auction design dates back to almost a decade ago. Huang et al. [30] propose two auction-based mechanisms for sharing spectrum, highlighting the unique challenge from wireless interference constraints. Another early solution is due to Buddhikot and Ryan [31], in which spectrum access is coordinated and controlled by a spectrum broker. VERITAS [1] is the first spectrum auction based on a monotonic channel allocation rule, and is thus truthful. Zhou and Zheng propose TRUST [6], a truthful double auction with multiple sellers (licensed users). For spectrum auctions that take interference among secondary users into consideration, Wu et al. [2] develop a semi-definite programming based mechanism, which is truthful and resistant to bidder collusion. Gopinathan [3] et al. propose auctions that incorporate fairness considerations into channel allocation. Their goal is to maximize social welfare, while ensuring a notion of fairness among bidders when the auction is repeatedly held. A truthful and scalable spectrum auction enabling both sharing and exclusive access is proposed by Kash et al. [4]. This auction handles heterogeneous agent types with different transmission powers and spectrum needs. We note that, all the works mentioned above focus on single-hop users bidding for a single channel only. Our

24 2.2. RELATED WORK 14 work essentially generalizes the problem to multi-hop users, who may enjoy multi-channel paths. To make the auction expressive enough to support mobility for wireless users, we need to introduce another dimension of externality except interference. We found that it has been considered more generally in economics. Jehiel et al. [32] design multi-dimensional auctions where winners not only care about just winning, but also who else wins. While general and expressive enough, this design does not take computational challenges into account. A number of work have introduced externalities into their deigns for online advertisement [33 36]. However, such designs cannot be directly applied to spectrum auctions, as the externalities in spectrum auctions are far more complicated when wireless interference is considered. Kash et al. [4] propose a spectrum auction enabling the sharing of a channel within an SU s interference range. By introducing a binary variable into the valuation of each SU to indicate the willingness to share, this auction primitively models exclusive/non-exclusive access of a channel for static SUs of heterogeneous types, with different transmission power and spectrum needs. However, their design is less expressive than ours, and more importantly, is still limited to static SUs. Deek et al. [37] design Topaz, an online spectrum auction that adopts three-dimensional bids including channel valuation and claimed arrival and departure time. Their goal is to discourage bidders from misreporting both their valuations and channel access time window. Our approach, as a two-dimensional auction, is quite different from Topaz, since the extra dimension(s) are from exclusive and mobility channel access instead of from the temporal domain. The common drawback in most of the existing spectrum auctions is, channels are

25 2.2. RELATED WORK 15 assumed to be identical and bidders are not allowed to bid for combinations of different channels, which may not be feasible for modern spectrum auction design [12]. To overcome this, combinatorial auctions emerged as a feasible and efficient solution. In this scope, Hoefer et al. propose a randomized combinatorial auction that is truthful in expectation, with a guaranteed performance bound on social welfare [22]. A recent solution due to Dong et al. employs a combinatorial auction as well [?], allowing bidders to have more flexible bids to require not only the channels, but also the time periods to use them.

26 Chapter 3 Truthful Spectrum Auction Design for Secondary Networks In this chapter, we first design a simple heuristic auction for spectrum allocation to SNs, which guarantees both truthfulness and interference-free channel allocation, providing winning SNs with end-to-end multihop paths, with a channel assigned to each hop. The heuristic auction enables multi-channel assignment along a path, thereby reducing the possibility that a path is blocked due to interference. To achieve truthfulness, we employ a greedy, monotonic allocation rule and design an accompanying payment scheme, by referring to Myerson s characterization of truthful auctions [38]. The heuristic auction provides no hard guarantee on social welfare. We next design a randomized auction, which is truthful in expectation, and is provably approximate optimal in social welfare. We note that absolute optimal social welfare is impossible, since the joint routing-channel assignment problem is already NP-hard with truthful bids given for free. We relax an integer program (IP) formulation to the social welfare maximization problem into a linear program (LP), and prove an upper-bound on the integrality gap. 16

27 3.1. PRELIMINARIES 17 We then employ the decomposition technique due to Lavi and Swamy [39] to decompose an LP solution into a set of feasible IP solutions (allocations). A pair of primal-dual LPs are formulated, for computing a probability distribution over the allocation set. Based on the set and the probabilities, an approximation algorithm is finally designed, for computing a feasible solution to the original IP with a provable performance guarantee. Such a solution assigns channels to paths constructed for winning SNs. We then apply the classic VCG [13 15] payment scheme, to conclude the design of a truthful auction that approximately maximizes social welfare for spectrum allocation to SNs. We also consider the intra-sn interference, which can affect the allocation outcome and social welfare. The remainder of this chapter is organized as follows. We present preliminaries in Sec A heuristic truthful auction is designed in Sec In Sec. 3.3, we propose and analyze a randomized auction with performance bound. Simulation studies are presented in Sec We discuss the influence of more complex interference in Sec 3.5. Sec. 3.6 summarizes the chapter. 3.1 Preliminaries In this section, we first introduce some background in truthful auction design in Sec , then describe our system model in Sec Truthful Auction Design Auction theory is a branch of economics that studies how people act in an auction and analyzes the properties of auction markets. We first introduce some basic and most

28 3.1. PRELIMINARIES 18 related concepts, definitions and theorems from auction design. An auction allocates items or goods (channels in our case) to competitive agents with bids and private valuations. We adopt w i as nonnegative valuations of each agent i, which is often private information known only to the agent itself. Besides determining an allocation, an auction also computes payments/charges for winning bidders. We denote by p(i) and b i the payment and bid of agent i, respectively. Then the utility of i is a function of all the bids: w i p(i) if agent i with bid b i gets an item u i (b i, b i ) = 0 otherwise where b i is the vector of all the bids except b i. We first adopt some conventional assumptions in economics here. Truthfulness is a desirable property of an auction, where reporting true valuation in the bid is optimal for each agent i, regardless of other agents bids. If agents have incentives to lie, other agents are forced to strategically respond to these lies, making the auction and its analysis complex. A key advantage of a truthful auction is that it simplifies agent strategies. Formally, an auction is truthful if for any agent i with any b i w i, any b i, we have u i (w i, b i ) u i (b i, b i ) (3.1) An auction is randomized if its allocation decision making involves flipping a (biased) coin. The payment and utility of an agent are then random variables. A randomized auction is truthful in expectation if (3.1) holds in expectation. Besides, we also prefer an auction to be individually rational, in which agents pay no more than their gain (valuations).

29 3.1. PRELIMINARIES 19 As discussed, the classic VCG mechanism for truthful auction design requires the optimal allocation to be efficiently computable, and is not practical for spectrum auctions, since optimal channel allocation is NP-hard. If we aim to design a tailored, heuristic truthful auction, then we may rely on the characterization of truthful auctions by Myerson [38]. Theorem 3.1 Let P i (b i ) be the probability of agent i with bid b i winning an auction. An auction is truthful if and only if the followings hold for a fixed b i : P i (b i ) is monotonically non-decreasing in b i ; Agent i bidding b i is charged b i P i (b i ) b i 0 P i(b)db. Given Theorem 3.1, we see that once the allocation rule P( ) = {P i (b i )} i N is fixed (N is the set of bidders), the payment rule is also fixed. For the case where the auction is deterministic, there are two equivalent ways to interpret Theorem 3.1: (i) there exists a minimum bid b i, such that i will win only if agent i bids at least b i, i.e., the monotonicity of P i (b i ) implies that, there is some critical bid b i, such that P i (b i ) is 1 for all b i > b i and 0 for all b i < b i ; (ii) the payment charged to agent i for a fixed b i should be independent of b i (formally, p i (b i ) = b i b i b i db = b i ) System Model We assume there is a set of SNs, N. Each SN has deployed a set of nodes in a geographical region, and has a demand for multihop transmission from a source to a destination. A PN has a set of channels, C, available for auctioning in the region. We refer to SNs as agents and the PN as the auctioneer. Each node within an SN is equipped with a radio

30 3.1. PRELIMINARIES 20 Table 3.1: List of notations w i valuation of agent i b i bid of agent i b i bid of all agents except b i φ(i) virtual bid of agent i p(i) payment of agent i u i a node in SN i luv i link from u i to v i fuv i flow rate on link luv i O(w) objective function of S(w) objective function of the IP (3.5) LPR I s (i) the set of SNs that interfere with SN i along its path x(c, luv) i binary var: whether channel c is allocated to link luv i G i (E i, V i ) connectivity graph of SN i with link set E i, node set H(E H, V H ) V i conflict graph of links of all the SNs with edge set E H and vertex set V H that is capable of switching between different channels. SNs do not collaborate with each other, and nodes from different SNs are not required to forward traffic for each other. We assume nodes from each SN i form a connected graph G i (E i, V i ), which also contains node locations. We use node and link for the connectivity graphs and vertex and edge for the conflict graph introduced later. To better formulate the joint routing-channel assignment problem, we incorporate the concept of network flows. Let u i be a node in SN i and s i, d i be the source and the destination in SN i. We use l i uv to denote the link from node u i to node v i belonging to SN i, and f i uv to denote the amount of flow on link l i uv. Later we connect d i back to s i with a virtual feedback link lds i, for a compact formulation of the joint optimization IP. We define a conflict graph H(E H, V H ), whose vertices correspond to links from all the connectivity graphs. We use (l i uv, l j pq) to denote an edge in E H, indicating that link l i uv and link l j pq interfere if allocated a common channel. Before the auction starts, each SN i submits to the auctioneer a compound bid, defined as B i = (G i (E i, V i ), s i, d i, b i ). Then the conflict graph can be centrally obtained by the auctioneer. We denote by w i the

31 3.1. PRELIMINARIES 21 private valuation of SN i for a feasible path between s i and d i, and p(i) its payment. b i, w i and p(i) all represent monetary amounts. Note that we assume agents only have incentives to lie about their valuations. We denote by R T (u i ) and R I (u i ) the transmission range and interference range of node u i, respectively. We assume that R I(u i ) R T (u i ) = and R T (u i ) R max for any node u i where 1. Since no inter-sn collaboration is assumed, links from different SNs do not participate in joint MAC scheduling, and cannot be assigned the same channel if they interfere. We assume the MAC protocol, carrier sense multiple access (CSMA) with ready-to-send/clear-to-send/acknowledgment (RTS/CTS/ACK) is used to protect unicast transmissions. Thus, as a result of carrier sensing, two links l i uv and l j pq interfere if a node in {u, v} is within the interference range of a node in {p, q}, and cannot be assigned the same channel if i j. Formally, let a binary variable x(c, l i uv) {0, 1} denote whether channel c C is assigned to link l i uv for user i. If for channel c C, x(c, l i uv) = x(c, l j pq), then (l i uv, l j pq) / E H. Hence, for the joint routing-channel assignment problem we have the Channel Interference Constraints: x(c, l i uv) + x(c, l j pq) 1, (l i uv, l j pq) E H, c C (3.2) We also need Flow Conservation Constraints, i.e., at any node in V i, the total incoming and outgoing flows are equal (recall the virtual feedback link): fuv i = fvu, i v V i (3.3) u V i u V i

32 3.1. PRELIMINARIES 22 Assuming each channel has the same unit capacity 1, we have the Capacity Constraints: u V i \{d i }f i uv c C x(c, l i uv) 1 (3.4) which also ensures that a link can be assigned a single channel only. An agent needs an end-to-end path between its source and destination. This corresponds to a network flow of rate 1. Note that the link flow on the feedback link fds i equals the end-to-end flow for SN i. We formulate the joint routing-channel assignment problem for SNs into an IP: maximize O(w) = i N w i f i ds (3.5) subject to x(c, luv) i + x(c, lpq) j 1, fuv i = fvu, i u V i u V i uv u V i \{d }f i x(c, luv) i 1, i c C fuv, i x(c, luv) i {0, 1}. (l i uv, l j pq) E H, c C v V i v V i where O(w) denotes the objective function of the IP. Note that w i s are not known to the auctioneer, but if we can design a truthful auction to elicit the true valuations of the bidders, we can replace w i with b i by simply assuming all the bidders are bidding truthfully, since it is the optimal strategy for all the bidders. However, solving this IP to optimal is an NP-hard problem. Therefore, we first introduce a heuristic auction in Sec. 3.2, which is based on the technique of monotonic allocation and critical bids, and

33 3.2. A HEURISTIC TRUTHFUL AUCTION 23 is simple and truthful. However, it does not provide any bound on the social welfare generated. A more sophisticated, randomized auction with a proven bound is studied next, where the LP relaxation of IP (3.5) is solved as a first step. 3.2 A Heuristic Truthful Auction In this section, we design an auction with a greedy style allocation and a payment scheme to ensure truthfulness. The auction consists of two phases: Algorithm 3.1 determines the channel assignment and winning bidders, and Algorithm 3.2 computes the payments for winning agents. d 1 1 d d a c 1 1 b a c 1 (a) Assign channels to SN a (b) Assign channels to SN b (c) Assign channels to SN d Figure 3.1: Procedure of channel assignment. Dots and squares represent source and destination nodes respectively. 1 2 b a c b Channel Allocation As discussed in Sec. 3.1, the key to designing a truthful auction is to have a non-decreasing allocation rule. Prices can then be calculated by the critical bids to make the auction truthful. A simple method is to sort all agent bids in a non-decreasing order, and greedily assign channels to agents in this order, subject to interference constraints [10]. However, ranking agents only according to their bids is inefficient. An agent with high bid may

34 3.2. A HEURISTIC TRUTHFUL AUCTION 24 heavily interfere with others (a large number of other agents are closely located), so assigning channels to it with higher priority is potentially detrimental to social welfare. Our solution is also a greedy allocation adopted in Algorithm 3.1. Assume channels are indexed by 1, 2,... C. For a simple heuristic auction, we first compute the shortest path for each agent as its end-to-end path. Let I s (i) be the set of SNs that interfere with i along the path, including i itself. We define the virtual bid of SN i as φ(i) = b i I s (i) (3.6) The rationale behind scaling the bid by I s (i) is to take i s interference with other agents into consideration, for heuristically maximizing social welfare. Then we greedily assign minimum indexed available channels along the paths to each link, according to a nonincreasing order of virtual bids φ(i). Fig. 3.1 shows an example to illustrate the channel assignment procedure. There are four SNs, a, b, c and d, where φ(a) > φ(b) > φ(c) > φ(d). Two channels are available for allocation. In the figure, two intersecting links also interfere with each other. If two links from two different SNs intersect, they cannot be allocated with the same channel. The algorithm first assigns Channel 1 to SN a. As a result, it cannot assign Channel 1 to the first link of SN b, which receives Channel 2 instead, as shown in Fig. 3.1b, leaving SN c without a channel it is impossible to assign either channel to c s first link. However, SN d wins, and receives a channel assignment along its path without introducing interference to a or b. We now prove that the greedy auction is monotone. Lemma 3.1 Algorithm 3.1 is monotone. That is, the probability of bidder i with bid b i winning the auction is non-decreasing in b i, and critical bids for winning agents exist.

35 3.2. A HEURISTIC TRUTHFUL AUCTION 25 Proof Bidding higher can only increase an agent s virtual bid, and therefore increase its rank in Algorithm 3.1. Hence, the probability of assigning a channel to the agent is non-decreasing. Besides, Algorithm 3.1 is deterministic, so a critical bid b i exists for a winning bidder i, such that agent i always wins if it bids b i b i. Algorithm 3.1 A greedy truthful auction channel allocation. 1. Input: Set of channels C, all the compound bids B i = (G i (E i, V i ), s i, d i, b i ), conflict graph H(E H, V H ) 2. for all i N do 3. I s (i) {i}; 4. Compute the shortest path P i from s i to d i ; 5. for all i N do 6. for all luv i along path P i do 7. x(c, luv) i 0 c C; 8. if (luv, i lpq) j E H then 9. I s (i) I s (i) {j}; 10. φ(i) b i I s(i) 11. Win(i) TRUE; 12. for i N in non-increasing order of φ(i) do for all luv i along path P i do Let Tuv i C; 15. for all c T i uv do 16. if x(c, lpq) j = 1 with (luv, i lpq) j E H, p, q then 17. Tuv i Tuv\{c}; i 18. if Tuv i = then 19. Win(i) FALSE; 20. if Win(i) = TRUE then for all luv i along path P i do Choose the minimum indexed channel c m in Tuv; i 23. x(c m, luv) i 1; Payment Calculation Algorithm 3.2 computes payments for the winning agents. The payment scheme design is where we ensure the truthfulness of an auction. Algorithm 3.2 aims to find a critical

36 3.2. A HEURISTIC TRUTHFUL AUCTION 26 bidder with critical bid b i for a winning agent, such that i is guaranteed to win as long as i s virtual bid φ(i) φ (i). Here φ (i) = b i I s(i) is the critical virtual bid for i. If b i is independent from b i, then charging agent i b i will ensure that the auction is truthful, which we will argue formally later. Algorithm 3.2 A greedy truthful auction payment calculation. 1. Input: Set of channels C, all the compound bids B i = (G i (E i, V i ), s i, d i, b i ), conflict graph H(E H, V H ), all the routing paths P i and channel assignment from Algorithm for i N in non-increasing order of φ(i) do 3. p(i) 0; 4. if Win(i) = 1 then 5. Set b i 0; 6. Run Algorithm 3.1 on (b i, b i ); 7. if Win(i) = FALSE then 8. Let φ (i) + ; for all luv i along path P i do Let Tuv i C; 11. for all c T i uv do 12. if x(c, lpq) j = 1 with (luv, i lpq) j E H then 13. Tuv i Tuv\{c}; i 14. if Tuv i = then 15. A {j (luv, i lpq) j E H, p, q; Win(j) = TRUE}; 16. φ (i) min(φ (i), min j A φ(j)); 17. p(i) φ (i) I s (i) ; We now explain how Algorithm 3.2 works. It first clears a winning agent i s bid, and hence its virtual bid, to 0. Then we run Algorithm 3.1 based on (0, b i ). In Algorithm 3.1, an agent loses only if a link along its shortest path is unable to receive any channel. If we are unable to accommodate agent i, there must exist at least one link along its shortest path whose neighbouring links (neighbouring vertices in the conflict graph) have used all the channels. From all the agents that block links of agent i, we find out an agent j with the minimum virtual bid, set it as i s critical bidder, and compute i s payment. We

37 3.2. A HEURISTIC TRUTHFUL AUCTION 27 claim that φ(i) φ(j), because otherwise agent i would not be a winning agent among agents in I s (i) {i}. Agent i s payment can be computed as follows: p(i) = φ (i) I s (i) = φ(j) I s (i) (3.7) For the example in Fig. 3.1, we first set SN a s bid to 0, and run Algorithm 3.1 based on the new bid vector. After assigning channels to agent c, we find that there are no available channels for the second link of agent a. Hence, agent c becomes the critical bidder of agent a, which leads to a s payment p(a) = φ(c) I s (a). The rule applies to the other two winning agents b and d as well, where p(b) = φ(c) I s (b) and p(d) = 0. We next show that the auction is individually rational and truthful. Lemma 3.2 The auction shown in Algorithms 3.1 and 3.2 is individually rational. Proof Assume agent i wins by bidding b i, and let j be the critical bidder of i. Then we have φ(i) φ(j), so p(i) = φ(j) I s (i) φ(i) I s (i) = b i. Theorem 3.2 The auction in Algorithms 3.1 and 3.2 is truthful. Proof Fix i, b i. Let w i and w i be agent i s bid when being truthful and not, respectively. We need to show that for agent i with valuation w i, the utility of bidding w i is no less than the utility of bidding w i. We analyze the auction case by case. Let 1 if agent i with bid b i receives a channel; r(b i ) = 0 if agent i with bid b i doesn t receive a channel. First, assume that w i < w i. According to Lemma 3.1, it is impossible for agent i to have r(w i ) = 0 and r(w i) = 1. If r(w i ) = 0 and r(w i) = 0, there is no incentive to lie.