Reserve Price Auctions for Heterogeneous Spectrum Sharing

Reserve Price Auctions for Heterogeneous Spectrum Sharing 1 Mehrdad Khaledi and Alhussein A. Abouzeid Department of Electrical, Computer and Systems Engineering Rensselaer Polytechnic Institute Troy, NY 12180-3590, USA Email: khalem@rpi.edu, abouzeid@ecse.rpi.edu Abstract Spectrum auction is considered a suitable approach to efficiently allocate spectrum among unlicensed users. In a typical spectrum auction, Secondary Users (SUs) bid to buy spectrum bands from a Primary Owner (PO) who acts as the auctioneer. In this paper, unlike most existing auction mechanisms, spectrum auctions are studied in a realistic setting where channels have different qualities, and SUs are allowed to express their preferences for each channel separately. That is, each SU submits a vector of bids, one for each channel. An efficient spectrum auction is proposed that maximizes the social welfare of the SUs. In addition, a reserve price auction is proposed whereby the PO imposes reserve prices on channels. The paper studies what the optimal reserve prices should be in order to maximize the PO s revenue. Optimal reserve prices are found provided that the distribution of SUs valuations is known. In the case where no prior information is available about valuation distributions, prior-free reserve prices are presented that guarantee at least half of the optimal revenue. The proposed auctions run in polynomial time and have desired economic properties that are formally proven in the analysis. Furthermore, the proposed numerical results show the effect of spectrum heterogeneity and reserve prices on the performance of spectrum auctions. Keywords Cognitive Radio Networks, Heterogenous Spectrum Sharing, Game Theory, Reserve Price Auctions.

2 I. INTRODUCTION Spectrum scarcity has become a major challenge as a result of the rapid growth in mobile wireless communications. Several studies indicate that the problem is not just the scarcity of spectrum but it is rather the inefficient use of the available wireless spectrum. Measurements reported by the FCC s Spectrum Policy Task Force show that many of the allocated bands are idle or barely used in some areas [1]. To achieve better spectrum utilization, studying efficient spectrum allocation mechanisms seems imperative. Cognitive radio network is considered as a novel communication paradigm that improves spectrum utilization by allowing dynamic spectrum sharing [2]. Dynamic spectrum sharing enables unlicensed or Secondary Users (SUs) to access idle spectrum bands that are owned by a Primary Owner (PO), enabling new methods of spectrum cooperation and competition. For this purpose, it is necessary to design mechanisms that provide incentives for both PO and SUs to participate in spectrum sharing. Auction-based mechanisms are very well-suited to the spectrum sharing problem. In an auction, the seller is not necessarily required to have prior knowledge about the value of items to the potential buyers. This is an advantage of auction mechanisms compared to the traditional pricing mechanisms. Also, with auctions efficient allocation can be easily obtained by designing a mechanism that allocates to the bidders who value the items the most. Yet another advantage of auctions is that they induce less communication overhead compared to other possible market mechanisms (e.g. bargaining games [3]), which consequently makes implementation easier and more practical. In a simple spectrum auction scenario, a PO acts as the auctioneer and sells its idle spectrum bands to SUs to make a profit, and the SUs act as bidders who want to buy spectrum bands. Unlike most existing spectrum auctions that assume identical channels (see section II for a review of prior work), we consider a more general and more realistic case where channels have different qualities. In this setting, SUs are allowed to express their preferences for each channel separately. That is, each SU submits a vector of bids, one for each channel. We define the SUs valuations as a function of channel capacities that takes into account both channel qualities and the SUs monetary values that reflect urgency for channel access. This model provides much more flexibility for SUs and is more practical compared to the existing spectrum auctions.

3 Technically, this problem can be modeled by a non-identical multiple items auction mechanism where each bidder has a different view of the available items. An auction is described by a pair of functions, namely the allocation function and the payment function. Also, it is desired for an auction mechanism to have some economic properties, such as truthfulness. We propose an auction mechanism to allocate the heterogeneous channels efficiently, with the goal of maximizing SUs valuations. The proposed auction runs in polynomial time and has proven economic properties. The challenge presented by channel heterogeneity is that we can no longer use the greedy scheme in VCG mechanism [4] for allocation. If we had m identical channels, we could take the m highest valuation SUs as winners and charge them the externality they impose on other bidders, which would be the (m + 1)th valuation. However, with m heterogeneous channels, greedily taking the highest valuation SU for each channel may result in an inefficient allocation. Also, the externalities are not simply the (m + 1)th valuation. In our proposed auction, we cast the heterogeneous channel allocation problem into a maximum weight matching in a bipartite graph and find the payments accordingly. We then consider the case where the PO can impose reserve prices on each channel. Reserve prices are minimum prices at which the PO is willing to sell channels, and are effective in increasing the PO s revenue. Auction mechanisms with no reserve prices may result in low revenues. For example, a second-price auction was used in 1990 in New Zealand for selling spectrum licenses. The winner bid $100,000 but paid only $6; in another case, the winner bid $7,000,000 but paid only $5,000 [5]. Reserve pricing is an effective way to avoid such situations. We present a reserve price auction to allocate the heterogeneous channels efficiently, maximizing the SUs valuations subject to reserve price constraints. We study the optimal reserve prices that maximize the PO s revenue. We observe that reserve prices should be set according to the SUs valuations. If reserve prices are too low compared to the SUs valuations, then the SUs may get the channels at very low prices which causes a revenue loss to the PO. On the other hand, if reserve prices are set to be too high, then a large group of SUs will be excluded and there is a risk that no SU can afford the channel(s). Therefore, some information on the SUs valuations is needed to find optimal reserve prices. With the knowledge of the distribution of SUs valuation, we find optimal reserve prices. In case no prior information exists about valuation distributions, we find prior-free reserve prices that guarantee at least half of the optimal revenue.

4 The proposed auctions run in polynomial time and have desired economic properties that we prove in the analysis. Furthermore, we provide numerical results that show the effect of spectrum heterogeneity in terms of social welfare (i.e. sum of winning SUs valuations), PO s revenue, average payments and utilities of SUs. Also, we compare different reserve prices and show how they affect the PO s revenue. The main contributions of this paper are as follows. We propose an efficient spectrum auction for cognitive radio networks with heterogeneous channel qualities which is more realistic compared to prior work. Also, SUs are given the flexibility to submit channel-specific bids taking into account both the channel quality and SUs monetary values that reflect urgency for channel access. We then present a reserve price auction for heterogeneous spectrum sharing that results in efficient spectrum allocation subject to reserve price constraints. We find optimal reserve prices that maximize the PO s revenue provided that the distribution of SUs valuations is known. For the case when the distribution is not known, we find prior-free reserve prices that guarantee at least half of the optimal revenue. We formally prove the desired economic properties of the proposed auctions. In addition, we provide numerical results that show the effect of reserve prices and channel heterogeneity. The rest of this paper is organized as follows. In Section II we review and discuss related work. Section III presents the system model used in this paper. In Section IV, we propose an efficient spectrum auction for cognitive radio networks with heterogeneous channels and prove its economic properties. In Section V, we present a reserve price auction for heterogeneous spectrum sharing. We study optimal reserve prices in Section VI. Numerical results are presented in Section VII. Finally, Section VIII concludes the paper and outlines possible avenues for future work. II. RELATED WORK Several auction mechanisms have been proposed recently for wireless spectrum sharing in different settings [6] [19]. In this section, we provide a brief overview of the most relevant studies. In [6], the authors present a spectrum auction with multiple POs. In their model, each SU selects one PO for bidding and the POs gradually raise their trading prices until the mechanism converges to an equilibrium point where no SU and PO is interested to deviate. Similarly,

5 the authors in [7] study the optimal pricing problem for two wireless service providers that work on different frequency bands. They also study the optimal service provider selection problem for SUs. The authors show that the equilibrium price and its uniqueness depend on the spectrum propagation characteristics and the SUs geographical density. In [8], Niyato et al. study the dynamics of spectrum pricing in a competitive environment with multiple POs. They use noncooperative game theory to model the competition among POs and evolutionary game theory to model the behavior of SUs. The authors in [11] consider a model in which the spectrum access opportunity is divided in frequency and time. Thus, SUs can bid for a combination of frequencies at different times. The problem then becomes a combinatorial auction and finding an efficient allocation becomes NP- Complete. The authors present approximate solutions to the general problem. In a related work [12], a core-selecting auction has been proposed in a setting that SUs can bid for a combination of channels. The auction yields at least the revenue of the VCG mechanism [4] and it is not vulnerable to shill bidding. In [10], Zhou et al. proposed a general framework, called TRUST, for truthful double spectrum auctions that provides spectrum reuse. This framework takes any reusability-driven spectrum allocation mechanism as input, and applies its own winner determination and payment rule. TAHES [9] is another truthful double auction mechanism, but works for heterogeneous spectrums. It considers spatial and frequency heterogeneity, that is, spectrums offered by different POs are available to different SUs and reside in different frequency bands. In both models, there should be an external third party who has complete information and holds the auction. In this paper, we consider heterogeneity in channel qualities, and we take that into account in the SUs valuation functions. Also, there is no need for a third party to hold the auction, since it is not a double auction. Dynamic spectrum auctions have also been studied recently. Online spectrum auctions, for instance, allow dynamic population of SUs such that SUs can enter the auction and leave at different times [13], [14]. However, an underlying assumption is that channel access has a fixed value to SUs every time they participate in the auction. The authors in [15] present a dynamic spectrum auction that allows dynamically evolving values for SUs. Recently, Chen et al. [16] proposed a truthful auction for allocating variable bandwidth spectra. They assume that SUs bid not their values but their valuation functions. Then, the PO can evaluate SUs values for any

6 SU1 SU2 SU4 PO SU3 Fig. 1. A cognitive radio network with one primary owner and four secondary users. bandwidth and make decisions accordingly. Designing optimal spectrum auctions that maximize the PO s revenue requires knowledge of distribution of SUs valuations. In [17], the authors take a prior-free approach and present a randomized auction mechanism that asymptotically achieves 1/3 of the optimal revenue in expectation. In this paper, we use reserve prices as an effective way to maximize the PO s revenue. This paper extends our previous works on heterogeneous spectrum auctions [18], [19]. With prior information on SUs valuations, we find optimal reserve prices, and for the prior-free case, we find reserve prices that guarantee at least 1/2 of the optimal revenue. III. SYSTEM MODEL In this paper, we consider a cognitive radio network consisting of a primary spectrum owner (PO) and a set of secondary users (SUs). The PO is willing to sell its idle channels to the SUs to obtain some profit, and the SUs are willing to buy channels for their services. An example of cognitive radio network is depicted in Fig. 1. The spectrum sharing process is modeled by an auction in which the PO acts as the auctioneer, and the SUs are the bidders. In our model, we consider heterogenous channels, that is, channels

7 are of different qualities. The quality of channel j is defined as the Signal-to-Noise Ratio (SNR) of the channel, and is denoted by q j. In our setting, each SU has a different view of the available channels. We allow SUs to express their preferences over each channel separately. Thus, each SU submits a vector of bids; one for each channel. Let m denote the number of available channels and n denote the number of SUs. Then, V i = (v i1, v i2,..., v im ) is the vector of bids submitted by SU i, consisting of m values for the available channels. The valuation matrix submitted to the PO will be of the following form: V 1 V 2 V =... A SU s valuation for a channel is the benefit for that specific SU of obtaining that channel. In this paper, we assume that SUs prefer channels with higher capacities. Therefore, the SUs valuations for a channel is related to the capacity of that channel, as: V n v ij θ i B log(1 + q j ), (1) where v ij represents SU i s valuation for channel j, B is the channel bandwidth and θ i is a real bounded number reflecting the urgency of channel access for SU i, the more urgent the channel access to SU i; the higher the monetary value θ i. Thus v ij takes into account both the channel quality and SU s monetary value that reflects the degree of urgency for channel access. We assume that each channel can only be used by one SU at a time. Also, each SU can only use one channel at a time. Let p i denote the payment that SU i has to make if it gets a channel. Then, the utility of SU i, denoted by u i, is defined as the difference between its valuation for the obtained channel, say channel j, and the price he has to pay, i.e. u i v ij p i. Also, u i = 0 if SU i does not get any channel. Another essential assumption in auction design is the rationality of bidders. That means the bidders want to maximize their own utilities. Therefore, an SU tries to obtain a channel with a price lower than its valuation for the channel. The auction mechanism determines the channel allocation and the payments. The channel allocation is represented by an n m matrix, denoted by X. Each element of the allocation matrix x ij {0, 1} indicates whether the channel j is allocated to SU i or not. That is, x ij = 1

8 means that the SU i has obtained the right to access channel j and x ij represent the payments by a payment vector P = (p 1, p 2,..., p n ). = 0 otherwise. We IV. THE HETEROGENEOUS SPECTRUM SHARING AUCTION In this section, we present an auction based mechanism for heterogeneous spectrum sharing with desired economic properties that we prove in subsection IV-C. The SUs compute their valuations according to (1) after the PO announces the qualities of the available channels. Then the PO holds the auction, taking into account the bids collectively. The auction mechanism takes the valuation matrix, V, as input and determines the channel allocation and the payments. The goal is to optimize the social welfare. The social welfare of an allocation X = {x ij } n m is the sum of the valuations of all the SUs for this allocation. Formally, it can be written as: S x ij v ij (2) i j The allocation that maximizes the social welfare is referred to as an efficient allocation. Formally, the efficient channel allocation problem can be written as: X = arg max S = arg max x ij v ij (3) X X s.t. x ij 1, i j x ij 1, j i x ij {0, 1}, i, j where the constraints in the above formulation are feasibility constraints for the allocation. As mentioned in the system model, we assume that each channel can only be used by one SU at a time, and each SU can only use one channel at a time. In the next subsection, we present a method to achieve an efficient allocation. i j A. Efficient Channel Allocation As mentioned in the introduction, channel heterogeneity presents a challenge in that we can no longer use the greedy scheme in VCG [4] for allocation. If we had m homogeneous channels,

9 we could greedily take the m highest bidders (SUs) as winners. However, with heterogeneous channels, greedily taking the highest valuation SU for each channel may result in an inefficient allocation. For instance, consider a network with 3 SUs indexed by SU i ; i = 1, 2, 3 and 2 channels. Let the SUs valuations for the channels be V 1 = (6, 5), V 2 = (4, 2) and V 3 = (5, 3). The greedy approach assigns the first channel to SU 1 and the second channel to SU 3, that brings social welfare of 9. However, the efficient allocation allocates the first channel to SU 3 and the second channel to SU 1 with a social welfare of 10. To cope with heterogeneity, we transform the problem of efficient channel allocation, i.e. (3), into a maximum weight matching problem in graph theory [20]. We first provide a brief review of some basic concepts from graph theory and the matching problem. A bipartite graph is a graph whose vertices can be divided into two disjoint sets V 1 and V 2, such that every edge in the graph connects a vertex in V 1 to one in V 2. A complete bipartite graph is a bipartite graph such that for any two vertices i V 1 and j V 2, ij is an edge in the graph. A weighted graph is a graph whose edges are associated with weights, usually a real number. The weight of the edge connecting vertices i and j is denoted by w ij. In a bipartite graph, a matching is a subset of edges such that they do not share an endpoint. In other words, a matching is a subset of edges such that for each vertex, there is at most one edge in the matching that is incident upon this vertex. Now, given a weighted complete bipartite graph, the problem of maximum weight matching is to find a matching with maximum weight. This is a well-studied problem in graph theory and it can be solved by the Kuhn-Munkres algorithm (also known as Hungarian algorithm) in polynomial time [21]. We do not present the details of the Kuhn-Munkres algorithm in this paper. Instead, we cast the original channel allocation problem, i.e. (3), into a maximum weight matching problem, and we show that these two problems are equivalent. We can build a complete bipartite graph G(V 1, V 2 ) by letting V 1 be the set of SUs and V 2 be the set of available channels. The edges in this graph represent bids of SUs for the channels. Since each SU submits a bid for each available channel, the graph is a complete bipartite graph. The weight of the edge ij is defined as the valuation of the SU i for the channel j, i.e. v ij. An example graph is depicted in Fig. 2 with two channels and three SUs. Proposition 1: X is an efficient channel allocation matrix if and only if M is a maximum weight matching in the constructed graph G.

10 SUs Channels 1 10 5 4 1 2 3 6 3 6 2 Fig. 2. A weighted complete bipartite graph with two channels and three SUs. Proof: First, suppose there is an efficient channel allocation matrix X. Then each nonzero element of X corresponds to an edge in the maximum weight matching M. For example, x ij = 1 means that channel j is allocated to SU i, so the edge ij will be in the matching. It should be noted that this set of edges form a matching, because each channel can only be allocated to one SU and each SU can only use one channel at a time (feasibility constraints for the allocation). Also, this is a maximum weight matching since we have an efficient allocation that maximizes summation of SUs valuations that correspond to edge weights in the graph. Conversely, suppose that we have a maximum weight matching M in graph G, then the channel allocation matrix X = {x ij } n m can be formed easily. For each edge ij in M, set its corresponding element in X to 1, i.e. x ij = 1, and set all the other elements to zero. This results in an efficient channel allocation matrix. First, according to the definition of a matching, the resulting matrix satisfies the feasibility constraints. Second, since edge weights in the graph represent SUs valuations and M is a maximum weight matching, the resulting allocation matrix is efficient. B. The Payment Rule The goal is to find a payment rule for the efficient allocation that satisfies some desired economic properties. We present the payment rule in this subsection and we discuss the economic properties in the next subsection. We use the well-known Vickrey Clarke Groves (VCG) mechanism with Clarke pivot payments

11 [4]. Based on this payment rule, SU i pays the externality it causes. In other words, SU i pays the difference between the social welfare of the others with and without its participation. In case of m homogeneous channels, the externality that winning SUs impose on others, is the (m + 1)th bid. So, the winning SUs pay the (m + 1)th bid and others pay nothing. However, with heterogeneous channels, the externalities need to be computed for each of the winning SUs. Let X = {x ij } n m and Y = {y ij } n m be efficient channel allocation matrices with and without SU i s participation, respectively. (In order to exclude SU i, we set the ith row of Y to zero.) Then, the payment for SU i is calculated by the following formula: p i = y jk v jk x jk v jk (4) j i k j i k As an example, consider the graph in Fig. 2 with two channels and three SUs. SUs valuations are V 1 = (10, 5), V 2 = (4, 6) and V 3 = (6, 3). The efficient allocation matrix X obtained by the mechanism is: 1 0 X = 0 1 0 0 That is, SU 1 gets channel 1 and SU 2 gets channel 2. To calculate p 1, we need to find the efficient allocation without SU 1 s participation, denoted by matrix Y : 0 0 Y = 0 1 1 0 Now, using (4), p 1 = 12 6 = 6. Similarly, we can find Y for SU 2 and calculate p 2 = 13 10 = 3. The heterogeneous spectrum auction is summarized in Algorithm 1. It is worth noting that in (4), the valuations of SU i are excluded in the summations and SU i does not have any control over its payment. This makes the mechanism robust against SUs strategic behaviors. In the next subsection, we discuss the economic properties of the proposed auction. C. Desired Economic Properties It is desired for an auction to have certain economic properties. First, we formally define these properties, then we show that the proposed auction satisfies them.

12 Algorithm 1 Efficient Auction for Heterogeneous Spectrum Sharing INPUT: Valuation matrix V OUTPUT: Channel allocation matrix X and payment vector P 1: Set X = 0 and P = 0 2: Build a bipartite graph G(V 1, V 2 ) with SUs as V 1, channels as V 2 and bids as edge weights 3: Run Kuhn-Munkres algorithm [21] on G to find Max weight matching M 4: for each (i, j) M do 5: x ij = 1 6: end for 7: for each SU i do 8: for each Channel j do 9: if x ij 0 then 10: Compute p i according to (4) 11: end if 12: end for 13: end for 14: return X and P Incentive Compatibility; Let V i be user i s true valuation vector and V i be the valuation vectors of all other users (excluding i). Let the utility of i be u i = j x ij v ij p i when V i and V i are declared, and be u i = j x ij v ij p i when V i and V i are declared. An auction is called incentive compatible if for every user i, every V i and every V i have u i u i. This is sometimes referred to as truthfulness, and states that the dominant strategy for users is to declare their true valuations regardless of what other users do. Individual Rationality; An auction is individually rational if for every user i, we have u i 0. That means, users do not suffer as a result of participating in the auction and the winners do not pay more than their valuations. No Positive Transfers; In an auction with no positive transfers we have p i 0, for every user i. This property prevents the auctioneer from paying users. Theorem 1: The proposed heterogeneous spectrum sharing auction is incentive compatible, we

13 individually rational and has no positive transfers. Proof: We first prove incentive compatibility. Using the payment rule, i.e. (4), utility of user i, when declaring V i and V i, is u i = x ij v ij + x jk v jk y jk v jk, j j i k j i k but when declaring V i and V i, is u i = x ij v ij + x jk v jk y jk v jk. j j i k j i k Since X maximizes social welfare among all the possible allocations, we have this inequality: x ij v ij + x jk v jk x ij v ij + x jk v jk. Now, by subtracting the term j j i k j j i k y jk v jk from both sides of the inequality, we get u i u i. Which is the incentive j i k compatibility property. Let X = {x ij } n m and Y = {y ij } n m be social welfare maximizing allocations with and without SU i s participation, respectively. To show individual rationality, consider the utility of user i: u i = j j 0 x ij v ij + x jk v jk y jk v jk j i k j i k x jk v jk y jk v jk j k k The first inequality holds since j {x ij } n m is the allocation that maximizes the social welfare, j y ij v ij 0. The second inequality holds because X = x jk v jk. k To show no positive transfers, using the payment rule (4), we have p i = y jk v jk j i k x jk v jk 0, since Y = {y ij } n m maximizes the social welfare without i s participation, j i k j i k y jk v jk. V. RESERVE PRICE AUCTION FOR HETEROGENEOUS SPECTRUM SHARING We now enable the PO to impose reserve prices on channels, which is a simple and effective way to increase a PO s revenue. Auction mechanisms with no reserve prices may result in low revenues. Several examples from past spectrum auctions can be found in [5]. In this section, we

14 present an efficient reserve price auction for heterogeneous spectrum sharing, subject to reserve price constraints. In the next section, we study optimal reserve prices that maximize the PO s revenue. The objective of the reserve price auction is to maximize the social welfare subject to reserve prices. We define the social income of an allocation X = {x ij } n m as the aggregate net profits of this allocation, where net profit being the difference of SU s valuation and the reserve price. Formally, it can be written as: S i x ij (v ij r j ) (5) j where r j denotes the reserve price on channel j. The efficient allocation problem with reserve prices can formally be written as: X = s.t. arg max X S = arg max x ij (v ij r j ) (6) X x ij 1, i j x ij 1, j i x ij (v ij r j ) 0, i, j x ij {0, 1}, i, j It should be noted that there is an extra feasibility constraint because of reserve prices; x ij (v ij r j ) 0, that ensures channels are not allocated to SUs with valuations lower than the reserve prices. Algorithm 2 shows the reserve price auction that takes the valuation matrix, V, and reserve prices r = (r 1, r 2,..., r m ) as inputs and determines channel allocation and payments. The allocation part of the auction, can be done similar to the previous section by building a bipartite graph and finding a maximum weight matching, except that we have to take into account the reserve price constraints. Here, we draw an edge between SU i and channel j only if SU i bids at least r j. Also, the weight of the edge ij is defined as the net profit of SU i getting channel j, i.e. v ij r j. It is worth noting that the graph is no longer a complete graph, because i j

15 Algorithm 2 Reserve Price Auction for Heterogeneous Spectrum Sharing INPUT: Valuation matrix V and reserve prices r = (r 1, r 2,..., r m ) OUTPUT: Channel allocation matrix X and payment vector P 1: Set X = 0 and P = 0 2: Build a bipartite graph G(V 1, V 2 ) with SUs as V 1, channels as V 2, draw an edge ij if v i j r j, use v ij r j as edge weights 3: Run Kuhn-Munkres algorithm[21] on G to find Max weight matching M 4: for each (i, j) M do 5: x ij = 1 6: end for 7: for each SU i do 8: for each Channel j do 9: if x ij 0 then 10: Compute p i according to (7) 11: end if 12: end for 13: end for 14: return X and P of reserve price constraints. But, still we can use the same argument as in previous section to show that X is a social income maximizing channel allocation if and only if M is a maximum weight matching in the constructed graph G. It should be noted that here we apply reserve prices in eager mode. Depending on when the reserve price constraints are applied, auctions can be eager or lazy. In the eager mode, the auctioneer first removes all the bidders whose valuations are less than reserve prices, then the auction is run on the remaining bidders to find the winners. However, the steps are reversed in the lazy mode, that is, the auctioneer runs the auction with all the bidders and determines the winners, then applies reserve price constraints. Fu in [22], provides a comparison between the two modes when applied to the classical VCG auction. The author shows that in a similar setting, the eager mode results in both more social welfare and more revenue than the lazy mode.

16 In order to find the payment rule, we first apply the VCG payments to net profits (i.e. v ij r j ). That is, SU i pays the difference between the social income of the others with and without its participation. We then add reservation prices to the VCG results. The payment for SU i is calculated by the following formula: y jk (v jk r k ) x jk (v jk r k ) + r j j i k j i k p i = If channel j is obtained (7) 0 If no channel is obtained where X = {x ij } n m and Y = {y ij } n m are social income maximizing allocations with and without SU i s participation, respectively. Theorem 2: The proposed reserve price auction is incentive compatible, individually rational and has no positive transfers. Proof: The proof can be obtained similar to Theorem 1, but instead of valuations v ij, net profits v ij r j need to be considered. VI. OPTIMAL RESERVE PRICES In this section, we focus on maximizing the PO s revenue and study optimal reserve prices that yield the maximum revenue. Reserve prices should be set according to SUs valuations. If reserve prices are too low compared to the SUs valuations, then the SUs may get the channels at very low prices which causes a revenue loss to the PO. On the other hand, if reserve prices are set to be too high, then a large group of SUs will be excluded and there is a risk that no SU can afford the channels. Therefore, some information on SUs valuations is required to find optimal reserve prices. We assume that SUs valuations for each channel are identical and independently distributed (i.i.d). That means, there could be m different distributions, one for each channel. We consider two different cases. In the first case that valuation distributions are known to the PO, we find the optimal reserve prices. In the second case where no prior information exists about valuation distributions, we find prior-free reserve prices that guarantee at least half of the optimal revenue.

17 A. Prior Dependant Optimal Reserve Prices In auction theory, revenue maximizing auctions (i.e. optimal auctions) are heavily influenced by the seminal work of Myerson [23] that requires prior knowledge of valuation distribution. Let F and f be the CDF and PDF of the valuation distribution respectively, then the virtual valuation function is defined as ϕ(v) = v 1 F (v). A distribution is called regular if its virtual valuation f(v) function is nondecreasing in v. A large group of distributions (e.g. exponential, uniform, some power law distributions) are regular. Myerson showed that with regular distributions, the optimal truthful auction maximizes the sum of the winning users virtual valuations (or virtual welfare). The regularity of distribution ensures monotonicity of the allocation which is needed to achieve truthfulness [4]. We also assume regularity in this section. We find reserve prices that when used in Algorithm 2, result in maximum revenue for the PO. Assume there is only one SU with valuation v drawn from distribution F and the PO wants to post a price p on its channel. If v p the SU will get the channel and pay p, otherwise the channel remains unsold and the SU pays nothing. The question is what should the price p be to maximize the PO s revenue. The expected revenue of the PO is p times the probability that v > p, which is 1 F (p). Thus, the PO s best price is arg max p [p (1 F (p))], which is called the monopoly price. Proposition 2: Using monopoly prices (for each channel) as reserve prices in Algorithm 2 results in the optimal revenue for the PO. Proposition 2 follows from that our reserve price auction with monopoly reserves is equivalent to Myerson s optimal auction. Myerson s auction allocates to bidders with highest virtual valuations, whereas in Algorithm 2 we favor highest valuation bidders. In an i.i.d environment, the bidder with highest virtual valuation is also the one with the highest valuation. Also, we want to make sure that we do not allocate to bidders with negative virtual valuations (note that virtual valuations can be negative) that requires the bidders valuations be at least ϕ 1 (0). Therefore, the auction that allocates to bidders with highest valuations and uses ϕ 1 (0)s as reserve prices is equivalent to the Myerson s optimal auction. In order to find the monopoly price p, we need to take derivative of p (1 F (p)) and set it equal to zero that yields to ϕ 1 (0) (see the definition of virtual valuations). Thus, if we use monopoly prices in our reserve price auction (Algorithm 2), the outcome is equivalent to Myerson s optimal auction that maximizes the PO s revenue. It

18 is worth noting that while using monopoly reserve prices in the classical VCG [4] is known to be optimal, Proposition 2 shows that using monopoly reserve prices in Algorithm 2, which is a modification of VCG, results in the optimal revenue. B. Prior Free Reserve Prices In a prior free setting where the PO has no information on valuation distributions, the goal is to find reserve prices that approximate the optimal revenue as close as possible. The authors in [24] proposed the idea of taking a sample from the bidders valuations to use it as a reserve price. They proposed a single-item auction (basically VCG with lazy reserve prices) that uses one bidder s valuation as a reserve price. Its revenue in an i.i.d regular environment is found to be at least 1 2 n 1 n of the optimal auction, where n 2 is the number of bidders. We adopt the sampling idea from [24] to modify our proposed reserve price auction (presented in section V) to approximate the optimal revenue in a prior free setting. In Algorithm 2, the reserve price on each channel was the same for all the SUs. Now, we allow different reserve prices for different SUs on each channel. Thus, instead of a reserve price vector r = (r 1, r 2,..., r m ), we have a reserve price matrix r = {r ij } n m. Algorithm 3 shows how we find prior free reserve prices. For each channel, we choose a SU at random (uniformly) as a sampled SU and use its valuation as the reserve price for other SUs, and we take another SU s valuation (again at random) as the reserve price for the sampled SU. Our reserve price auction (Algorithm 2) equipped with the prior free reserve prices (Algorithm 3) approximates at least half of the optimal revenue, when distributions are regular. This immediately follows from Lemma 3.5 and Lemma 3.6 in [24] that imply the sampled bidder contributes to 1 2 1 n of the optimal revenue in expectation (taken on the random choice of the sampled bidder at uniform). Also, the main result in [24] shows that non-sampled bidders contribute to 1 2 n 1 n of the optimal revenue. By combining the two, we conclude the 1 2 revenue. guarantee of the maximum The authors in [17], present a randomized auction mechanism in a prior-free setting that asymptotically achieves 1/3 of the optimal revenue. Their randomized auction does not require regular distributions. In this paper, however, we use prior-free reserve pricing as a simple and effective way to approximate the optimal revenue. With regularity assumption that holds for most of the distributions, our approach guarantees at least 1/2 of the optimal revenue.

19 Algorithm 3 Finding Prior Free Reserve Prices INPUT: Valuation matrix V OUTPUT: Reserve price matrix r = {r ij } n m 1: for each Channel j do 2: choose a sampled SU i at random (uniformly) 3: for (k = 1 : n && k i) do 4: Set r kj = v ij 5: end for 6: choose another random SU l 7: Set r ij = v lj 8: end for 9: return r VII. NUMERICAL RESULTS In this section, we provide numerical results that evaluate the effect of spectrum heterogeneity and the reserve prices discussed in the paper. We study the performance of the proposed auctions in different network scenarios with variable SUs and fixed number of channels, or with variable number of channels and fixed number of SUs. Each setting is run 1000 times in MATLAB to eliminate the effect of random initialization. At first, SUs compute their valuations according to (1) and form the valuation matrix. Then, Algorithm 1 is run for the none-reserve price auction, and Algorithm 2 is run for the reserve price auction with reserve prices described in Section VI. We assume unit bandwidth demand, i.e. B = 1, also, signal to noise ratio (SNR) of channels, that correspond to channel qualities, are randomly chosen from uniform distribution ranging from -20db to 20db. To show the effect of spectrum heterogeneity, we compare the performance of our proposed auction (Algorithm 1) with the case of identical channels where all the channel qualities are set to the mean value of SNR. Social welfare, average payment of SUs, average utility of SUs, and revenue of the PO are considered as performance metrics, where revenue of the PO is defined as the sum of SU payments i p i. When studying the effect of reserve prices, the PO s revenue is of foremost interest. We compare our reserve price auction (Algorithm 2) with the none-reserve

20 2000 1800 non identical channels identical channels 1600 Social Welfare 1400 1200 1000 800 600 2 4 6 8 10 12 14 16 18 Number of SUs Fig. 3. Social Welfare versus the number of SUs, with fixed number of channels m=4. price auction (Algorithm 1). We study both the monopoly reserve prices and prior free reserve prices (Algorithm 3). A. The effect of Spectrum Heterogeneity Fig. 3 depicts the social welfare for a fixed number of channels and variable number of SUs. As can be seen, the social welfare increases with number of SUs. With more SUs participating in the auction, we have wider range of valuations, and since the auction favors SUs with higher valuations, the winning SUs results in a higher social welfare. The average payment of SUs is depicted in Fig. 4. We observe that as the number of SUs increases and channel access becomes more competitive, payments increase. This is because with more competition, the winning SUs cause more externality, and consequently they have to pay more. This competition also benefits the PO, since its revenue increases, as shown in Fig. 5. However, this competitive environment is not favorable for SUs. Fig. 6 shows that the average utility of SUs decreases with the number of SUs. That happens because with more competition, SUs have to pay more, resulting in lower utilities. Now we consider the case of fixed number of SUs, and variable number of channels. As shown in Fig. 7, social welfare increases with the number of channels. This is clearly because with more channels available, we are adding more positive terms to the social welfare (see ( 2)). Fig. 8 depicts the average payment of SUs when the number of channels increases. As can be seen, average payments decrease with the number of channels. With more channels available,

21 450 400 non identical channels identical channels 350 Average Payments 300 250 200 150 100 50 0 2 4 6 8 10 12 14 16 18 Number of SUs Fig. 4. Average payments versus the number of SUs, with fixed number of channels m=4. 1800 1600 non identical channels identical channels 1400 Revenue of the PO 1200 1000 800 600 400 200 0 2 4 6 8 10 12 14 16 18 Number of SUs Fig. 5. Revenue of the PO versus the number of SUs, with fixed number of channels m=4. there is less competition among SUs. Therefore, the winning SUs cause less externality and pay less. It is worth noting that in the identical channels case, when the number of channels exceeds the number of SUs, the average payment drops to zero. This is because the winning SUs cause no externality when everyone gets a channel. However, in the non-identical case, winning SUs still cause externalities on each other, thus they have non-zero payments. The winner of a high quality channel causes some externality to the other winners by not allowing them to get a better quality channel. Although average payment of SUs has a decreasing trend with number of channels (as intuitively expected), the revenue of the PO increases up to some point and then drops, as

22 300 non identical channels identical channels 250 Average Utilities 200 150 100 50 2 4 6 8 10 12 14 16 18 Number of SUs Fig. 6. Average utilities versus the number of SUs, with fixed number of channels m=4. 3000 2500 non identical channels identical channels Social Welfare 2000 1500 1000 500 2 4 6 8 10 12 Number of Channels Fig. 7. Social Welfare versus the number of channels, with fixed number of SUs n=9. shown in Fig. 9. The initial increase is because the PO sells more channels and gets higher revenue, even though each channel s price is lowered. However, when the channels become too abundant, the payments considerably drop and we observe a decrease in revenue. It can be seen from Fig. 10 that the average utility of SUs increases with the number of channels. Since winning SUs pay less when the number of channels increases, we observe an increase in utilities. From all the preceding numerical results, we observe that the spectrum auction designed for heterogeneous channels (i.e. channels with different qualities) considerably outperforms that of identical channels. With non-identical channels, SUs can better express their needs and we get a wide range of valuations. Since the auction favors SUs with higher valuations, winners in the

23 400 350 non identical channels identical channels 300 Average Payments 250 200 150 100 50 0 2 4 6 8 10 12 Number of Channels Fig. 8. Average payments versus the number of channels, with fixed number of SUs n=9. 1200 1000 non identical channels identical channels Revenue of the PO 800 600 400 200 0 2 4 6 8 10 12 Number of Channels Fig. 9. Revenue of the PO versus the number of channels, with fixed number of SUs n=9. non-identical channels case have higher valuations compared to that of the identical channels case. Therefore, by a similar argument as mentioned earlier, non-identical channels auction results in higher social welfare and higher revenue for the PO, in addition to the improved utilities for SUs. B. The Effect of Reserve Prices Fig. 11 shows the PO s revenue using different reserve prices for a fixed number of channels and variable number of SUs. Also, Fig. 12 illustrates the effect of using different reserve prices on PO s revenue for a fixed number of SUs and variable number of channels. As expected and observed in both figures, monopoly reserve prices yield the maximum revenue. The prior-free

24 300 250 non identical channels identical channels Average Utilities 200 150 100 50 2 4 6 8 10 12 Number of Channels Fig. 10. Average utilities versus the number of channels, with fixed number of SUs n=9. 2500 2000 no reserve price monopoly reserve price prior free reserve price Revenue of the PO 1500 1000 500 0 2 4 6 8 10 12 14 16 18 Number of SUs Fig. 11. PO s revenue using different reserve prices, with fixed number of channels m=8. reserve prices outperform the no-reserve price scheme when competition is relatively low. This can be seen in Fig. 11 for small number of SUs and in Fig. 12 for large number of channels. An interesting observation is that, in a highly competitive environment case (i.e. large number of SUs in Fig. 11 and small number of channels in Fig. 12), the no-reserve price scheme works as good as the optimal (i.e. monopoly prices) scheme. This happens because when the environment becomes highly competitive, the externality caused by the winning SUs increases to the point that it exceeds the imposed reserve price. Thus, the reserve prices have lesser effect on payments, as the payments are based on the maximum of reserve prices and the externalities. Fig. 13 and Fig. 14 show the effect of using different reserve prices on the average payments

25 900 800 700 Revenue of the PO 600 500 400 300 no reserve price monopoly reserve price prior free reserve price 200 100 0 2 3 4 5 6 7 8 9 10 11 12 Number of Channels Fig. 12. PO s revenue using different reserve prices, with fixed number of SUs n=6. 350 300 no reserve price monopoly reserve price prior free reserve price 250 Average Payments 200 150 100 50 0 2 4 6 8 10 12 14 16 18 Number of SUs Fig. 13. Average payments using different reserve prices, with fixed number of channels m=8. with fixed and variable number of channels, respectively. We observe that the prior-free reserve price method has the highest average payments. However, it does not yield the highest revenue, according to Fig. 11 and Fig. 12. This is due to the fact that prior-free reserve prices are comparatively high, as a result, SUs may not be able to afford the channels, But when they do, they have to pay a comparatively high price. In brief, the prior-free scheme sells less channels at higher prices compared to the other two methods, because it is not aware of the valuation distributions. The numerical results show that all the reserve pricing methods are most effective when the channel access is not highly competitive.

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