Reserve Price Auctions for Heterogeneous Spectrum Sharing
|
|
- Timothy Curtis
- 6 years ago
- Views:
Transcription
1 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 , USA 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 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 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 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 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 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 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 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 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 10 SUs Channels 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 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 = 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 = Now, using (4), p 1 = 12 6 = 6. Similarly, we can find Y for SU 2 and calculate p 2 = = 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 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 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 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 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 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 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 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 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 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 non identical channels identical channels 1600 Social Welfare 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 non identical channels identical channels 350 Average Payments Number of SUs Fig. 4. Average payments versus the number of SUs, with fixed number of channels m= non identical channels identical channels 1400 Revenue of the PO 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 non identical channels identical channels 250 Average Utilities Number of SUs Fig. 6. Average utilities versus the number of SUs, with fixed number of channels m= non identical channels identical channels Social Welfare 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 non identical channels identical channels 300 Average Payments Number of Channels Fig. 8. Average payments versus the number of channels, with fixed number of SUs n= non identical channels identical channels Revenue of the PO 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 non identical channels identical channels Average Utilities Number of Channels Fig. 10. Average utilities versus the number of channels, with fixed number of SUs n= no reserve price monopoly reserve price prior free reserve price Revenue of the PO 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 Revenue of the PO no reserve price monopoly reserve price prior free reserve price Number of Channels Fig. 12. PO s revenue using different reserve prices, with fixed number of SUs n= no reserve price monopoly reserve price prior free reserve price 250 Average Payments 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.
26 no reserve price prior free reserve price monopoly reserve price 250 Average Payments Number of Channels Fig. 14. Average payments using different reserve prices, with fixed number of SUs n=6. VIII. CONCLUSION In this paper, we studied spectrum auctions in a realistic setting where channels have different qualities, and SUs are allowed to express channel-specific bids. We proposed an efficient spectrum auction that maximizes the social welfare of the SUs. Then, we consider the case where the PO imposes reserve prices on channels, and we proposed a reserve price auction for this setting. In addition, we studied the optimal reserve prices that maximize the PO s revenue. We found optimal reserve prices providing that the distribution of SUs valuations is known. We also found prior-free reserve prices that guarantee at least half of the optimal revenue. Proposed auctions run in polynomial time and have desired proven economic properties. Furthermore, we provided numerical results to show the effect of spectrum heterogeneity and different reserve prices on the performance of spectrum auctions. ACKNOWLEDGMENT This work was funded in part by NSF and NSF REFERENCES [1] FCC Spectrum Policy Task Force, Report of the spectrum efficiency working group, Available: Nov [2] I. F. Akyildiz, W.-Y. Lee, M. C. Vuran, and S. Mohanty, NeXt generation/dynamic spectrum access/cognitive radio wireless networks: a survey, Comput. Netw., vol. 50, no. 13, pp , Sep [3] M. J. Osborne and A. Rubinstein, A course in game theory. MIT press, 1994.
27 27 [4] N. Nisan, T. Roughgarden, E. Tardos, and V. V. Vazirani, Algorithmic Game Theory. New York, NY, USA: Cambridge University Press, [5] J. Mcmillan, Selling Spectrum Rights, Journal of Economic Perspectives, vol. 8, no. 3, pp , Summer [6] L. Gao, Y. Xu, and X. Wang, MAP: Multiauctioneer progressive auction for dynamic spectrum access, Mobile Computing, IEEE Transactions on, vol. 10, no. 8, pp , [7] A. Min, X. Zhang, J. Choi, and K. Shin, Exploiting spectrum heterogeneity in dynamic spectrum market, Mobile Computing, IEEE Transactions on, vol. 11, no. 12, pp , [8] D. Niyato, E. Hossain, and Z. Han, Dynamics of multiple-seller and multiple-buyer spectrum trading in cognitive radio networks: A game-theoretic modeling approach, Mobile Computing, IEEE Transactions on, vol. 8, no. 8, pp , [9] X. Feng, Y. Chen, J. Zhang, Q. Zhang, and B. Li, TAHES: a truthful double auction mechanism for heterogeneous spectrums, Wireless Communications, IEEE Transactions on, vol. 11, no. 11, pp , [10] X. Zhou, A. Sala, and H. Zheng, Towards large-scale economic-robust spectrum auctions, 15th Annual International Conference on Mobile Computing and Networking (MobiCom 2009), [11] M. Dong, G. Sun, X. Wang, and Q. Zhang, Combinatorial auction with time-frequency flexibility in cognitive radio networks, in INFOCOM, 2012 Proceedings IEEE, 2012, pp [12] Y. Zhu, B. Li, and Z. Li, Core-selecting combinatorial auction design for secondary spectrum markets, in INFOCOM, 2013 Proceedings IEEE, [13] L. Deek, X. Zhou, K. Almeroth, and H. Zheng, To preempt or not: Tackling bid and time-based cheating in online spectrum auctions, in INFOCOM, 2011 Proceedings IEEE, 2011, pp [14] Y. Yang, J. Wu, C. Long, and B. Li, Online market clearing in dynamic spectrum auction, in Global Telecommunications Conference (GLOBECOM 2011), 2011 IEEE, 2011, pp [15] M. Khaledi and A. A. Abouzeid, ADAPTIVE: a dynamic index auction for spectrum sharing with time-evolving values, in 12th International Symposium on Modeling and Optimization in Mobile, Ad Hoc, and Wireless Networks (WiOpt), [16] T. Chen and S. Zhong, Truthful auctions for continuous spectrum with variable bandwidths, Wireless Communications, IEEE Transactions on, vol. 13, no. 2, pp , February [17] A. Gopinathan and Z. Li, A prior-free revenue maximizing auction for secondary spectrum access, in INFOCOM, 2011 Proceedings IEEE, April 2011, pp [18] M. Khaledi and A. Abouzeid, Auction-based spectrum sharing in cognitive radio networks with heterogeneous channels, in Information Theory and Applications Workshop (ITA), 2013, 2013, pp [19] M. Khaledi and A. A. Abouzeid, A reserve price auction for spectrum sharing with heterogeneous channels, in Computer Communications and Networks (ICCCN), nd International Conference on, 2013, pp [20] D. B. West, Introduction to Graph Theory (2nd Edition), 2nd ed. Prentice Hall, Sep [21] J. Munkres, Algorithms for the assignment and transportation problems, Journal of the Society for Industrial and Applied Mathematics, vol. 5, no. 1, pp. pp , [22] H. Fu, VCG auctions with reserve prices: Lazy or eager, EC, 2013 Proceedings ACM, 2013.
IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL., NO., 1. Dynamic Spectrum Sharing Auction with Time-Evolving Channel Qualities
IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL., NO., 1 Dynamic Spectrum Sharing Auction with Time-Evolving Channel Qualities Mehrdad Khaledi, Student Member, IEEE, Alhussein A. Abouzeid, Senior Member,
More informationLecture 7 - Auctions and Mechanism Design
CS 599: Algorithmic Game Theory Oct 6., 2010 Lecture 7 - Auctions and Mechanism Design Prof. Shang-hua Teng Scribes: Tomer Levinboim and Scott Alfeld An Illustrative Example We begin with a specific example
More informationChapter 17. Auction-based spectrum markets in cognitive radio networks
Chapter 17 Auction-based spectrum markets in cognitive radio networks 1 Outline Rethinking Spectrum Auctions On-demand Spectrum Auctions Economic-Robust Spectrum Auctions Double Spectrum Auctions for Multi-party
More informationSTAMP: A Strategy-Proof Approximation Auction
: A Strategy-Proof Approximation Auction Mechanism for Spatially Reusable Items in Wireless Networks Ruihao Zhu, Fan Wu, and Guihai Chen Shanghai Key Laboratory of Scalable Computing and Systems Shanghai
More informationSAFE: A Strategy-Proof Auction Mechanism for Multi-radio, Multi-channel Spectrum Allocation
SAFE: A Strategy-Proof Auction Mechanism for Multi-radio, Multi-channel Spectrum Allocation Ruihao Zhu, Fan Wu, and Guihai Chen Shanghai Key Laboratory of Scalable Computing and Systems Shanghai Jiao Tong
More informationMechanism Design in Social Networks
Proceedings of the Thirty-First AAAI Conference on Artificial Intelligence (AAAI-17) Mechanism Design in Social Networks Bin Li, a Dong Hao, a Dengji Zhao, b Tao Zhou a a Big Data Research Center, University
More informationA Reverse Auction Framework for Access Permission Transaction to Promote Hybrid Access in Femtocell Network
The 31st Annual IEEE International Conference on Computer Communications: Mini-Conference A Reverse Auction Framework for Access Permission Transaction to Promote Hybrid Access in Femtocell Network Yanjiao
More informationCS269I: Incentives in Computer Science Lecture #16: Revenue-Maximizing Auctions
CS269I: Incentives in Computer Science Lecture #16: Revenue-Maximizing Auctions Tim Roughgarden November 16, 2016 1 Revenue Maximization and Bayesian Analysis Thus far, we ve focused on the design of auctions
More informationFlexAuc: Serving Dynamic Demands in a Spectrum Trading Market with Flexible Auction
1 FlexAuc: Serving Dynamic Demands in a Spectrum Trading Market with Flexible Auction Xiaoun Feng, Student Member, IEEE, Peng Lin, Qian Zhang, Fellow, IEEE, arxiv:144.2348v1 [cs.ni] 9 Apr 214 Abstract
More informationSTRUCTURE: A Strategyproof Double Auction for Heterogeneous Secondary Spectrum Markets
STRUCTURE: A Strategyproof Double Auction for Heterogeneous Secondary Spectrum Markets Yu-E Sun 1, He Huang 2, Miaomiao Tian 3, Zehao Sun 3, Wei Yang 3, Hansong Guo 3, Liusheng Huang 3 1 School of Urban
More informationA Cooperative Approach to Collusion in Auctions
A Cooperative Approach to Collusion in Auctions YORAM BACHRACH, Microsoft Research and MORTEZA ZADIMOGHADDAM, MIT and PETER KEY, Microsoft Research The elegant Vickrey Clarke Groves (VCG) mechanism is
More informationComputationally Feasible VCG Mechanisms. by Alpha Chau (feat. MC Bryce Wiedenbeck)
Computationally Feasible VCG Mechanisms by Alpha Chau (feat. MC Bryce Wiedenbeck) Recall: Mechanism Design Definition: Set up the rules of the game s.t. the outcome that you want happens. Often, it is
More informationCompetitive Analysis of Incentive Compatible On-line Auctions
Competitive Analysis of Incentive Compatible On-line Auctions Ron Lavi and Noam Nisan Theoretical Computer Science, 310 (2004) 159-180 Presented by Xi LI Apr 2006 COMP670O HKUST Outline The On-line Auction
More information1 Mechanism Design (incentive-aware algorithms, inverse game theory)
15-451/651: Design & Analysis of Algorithms April 6, 2017 Lecture #20 last changed: April 5, 2017 1 Mechanism Design (incentive-aware algorithms, inverse game theory) How to give away a printer The Vickrey
More informationAn Auction Mechanism for Resource Allocation in Mobile Cloud Computing Systems
An Auction Mechanism for Resource Allocation in Mobile Cloud Computing Systems Yang Zhang, Dusit Niyato, and Ping Wang School of Computer Engineering, Nanyang Technological University (NTU), Singapore
More informationPrice of anarchy in auctions & the smoothness framework. Faidra Monachou Algorithmic Game Theory 2016 CoReLab, NTUA
Price of anarchy in auctions & the smoothness framework Faidra Monachou Algorithmic Game Theory 2016 CoReLab, NTUA Introduction: The price of anarchy in auctions COMPLETE INFORMATION GAMES Example: Chicken
More informationGame Theory: Spring 2017
Game Theory: Spring 2017 Ulle Endriss Institute for Logic, Language and Computation University of Amsterdam Ulle Endriss 1 Plan for Today This and the next lecture are going to be about mechanism design,
More informationActivity Rules and Equilibria in the Combinatorial Clock Auction
Activity Rules and Equilibria in the Combinatorial Clock Auction 1. Introduction For the past 20 years, auctions have become a widely used tool in allocating broadband spectrum. These auctions help efficiently
More information1 Mechanism Design (incentive-aware algorithms, inverse game theory)
TTIC 31010 / CMSC 37000 - Algorithms March 12, 2019 Lecture #17 last changed: March 10, 2019 1 Mechanism Design (incentive-aware algorithms, inverse game theory) How to give away a printer The Vickrey
More informationModified Truthful Greedy Mechanisms for Dynamic Virtual Machine Provisioning and Allocation in Clouds
RESEARCH ARTICLE International Journal of Computer Techniques Volume 4 Issue 4, July August 2017 Modified Truthful Greedy Mechanisms for Dynamic Virtual Machine Provisioning and Allocation in Clouds 1
More informationSupply-Limiting Mechanisms
Supply-Limiting Mechanisms TIM ROUGHGARDEN, Department of Computer Science, Stanford University INBAL TALGAM-COHEN, Department of Computer Science, Stanford University QIQI YAN, Department of Computer
More information1 Mechanism Design (incentive-aware algorithms, inverse game theory)
15-451/651: Design & Analysis of Algorithms April 10, 2018 Lecture #21 last changed: April 8, 2018 1 Mechanism Design (incentive-aware algorithms, inverse game theory) How to give away a printer The Vickrey
More informationOn Optimal Multidimensional Mechanism Design
On Optimal Multidimensional Mechanism Design YANG CAI, CONSTANTINOS DASKALAKIS and S. MATTHEW WEINBERG Massachusetts Institute of Technology We solve the optimal multi-dimensional mechanism design problem
More informationCS364B: Frontiers in Mechanism Design Lecture #17: Part I: Demand Reduction in Multi-Unit Auctions Revisited
CS364B: Frontiers in Mechanism Design Lecture #17: Part I: Demand Reduction in Multi-Unit Auctions Revisited Tim Roughgarden March 5, 014 1 Recall: Multi-Unit Auctions The last several lectures focused
More informationRobust Multi-unit Auction Protocol against False-name Bids
17th International Joint Conference on Artificial Intelligence (IJCAI-2001) Robust Multi-unit Auction Protocol against False-name Bids Makoto Yokoo, Yuko Sakurai, and Shigeo Matsubara NTT Communication
More informationSponsored Search Markets
COMP323 Introduction to Computational Game Theory Sponsored Search Markets Paul G. Spirakis Department of Computer Science University of Liverpool Paul G. Spirakis (U. Liverpool) Sponsored Search Markets
More informationDifferentially Private and Strategy-Proof Spectrum Auction with Approximate Revenue Maximization
Differentially Private and Strategy-Proof Spectrum Auction with Approximate Revenue Maximization Ruihao Zhu and Kang G. Shin Department of Electrical Engineering and Computer Science The University of
More informationVALUE OF SHARING DATA
VALUE OF SHARING DATA PATRICK HUMMEL* FEBRUARY 12, 2018 Abstract. This paper analyzes whether advertisers would be better off using data that would enable them to target users more accurately if the only
More informationIntro to Algorithmic Economics, Fall 2013 Lecture 1
Intro to Algorithmic Economics, Fall 2013 Lecture 1 Katrina Ligett Caltech September 30 How should we sell my old cell phone? What goals might we have? Katrina Ligett, Caltech Lecture 1 2 How should we
More informationSpringerBriefs in Electrical and Computer Engineering
SpringerBriefs in Electrical and Computer Engineering More information about this series at http://www.springer.com/series/10059 Yanjiao Chen Qian Zhang Dynamic Spectrum Auction in Wireless Communication
More informationRevenue Generation for Truthful Spectrum Auction in Dynamic Spectrum Access
Revenue Generation for Spectrum Auction in Dynamic Spectrum Access Juncheng Jia Qian Zhang Qin Zhang Mingyan Liu Computer Science and Engineering Hong Kong University of Science and Technology, CHINA {jiajc,
More informationSearching for the Possibility Impossibility Border of Truthful Mechanism Design
Searching for the Possibility Impossibility Border of Truthful Mechanism Design RON LAVI Faculty of Industrial Engineering and Management, The Technion, Israel One of the first results to merge auction
More informationCharacterization of Strategy/False-name Proof Combinatorial Auction Protocols: Price-oriented, Rationing-free Protocol
Characterization of Strategy/False-name Proof Combinatorial Auction Protocols: Price-oriented, Rationing-free Protocol Makoto Yokoo NTT Communication Science Laboratories 2-4 Hikaridai, Seika-cho Soraku-gun,
More informationCOUSTIC: Combinatorial Double Auction for Crowd sensing Task Assignment in Device-to-Device Clouds
COUSTIC: Combinatorial Double Auction for Crowd sensing Task Assignment in Device-to-Device Clouds Yutong Zhai 1, Liusheng Huang 1, Long Chen 1, Ning Xiao 1, Yangyang Geng 1 1 School of Computer Science
More informationMiscomputing Ratio: The Social Cost of Selfish Computing
Miscomputing Ratio: The Social Cost of Selfish Computing Kate Larson and Tuomas Sandholm Computer Science Department Carnegie Mellon University 5000 Forbes Ave Pittsburgh, PA 15213 {klarson,sandholm}@cs.cmu.edu
More informationIncentive-Compatible, Budget-Balanced, yet Highly Efficient Auctions for Supply Chain Formation
Incentive-Compatible, Budget-Balanced, yet Highly Efficient Auctions for Supply Chain Formation Moshe Babaioff School of Computer Science and Engineering, The Hebrew University of Jerusalem, Jerusalem
More informationOnline Ad Auctions. By Hal R. Varian. Draft: December 25, I describe how search engines sell ad space using an auction.
Online Ad Auctions By Hal R. Varian Draft: December 25, 2008 I describe how search engines sell ad space using an auction. I analyze advertiser behavior in this context using elementary price theory and
More informationTruthful Double Auction Mechanisms for Heterogeneous Spectrums and Spectrum. Group-Buying
Truthful Double Auction Mechanisms for Heterogeneous Spectrums and Spectrum Group-Buying BY SHU WANG B.S., Northeastern University, 2008 THESIS Submitted as partial fulfillment of the requirements for
More informationThree New Connections Between Complexity Theory and Algorithmic Game Theory. Tim Roughgarden (Stanford)
Three New Connections Between Complexity Theory and Algorithmic Game Theory Tim Roughgarden (Stanford) Three New Connections Between Complexity Theory and Algorithmic Game Theory (case studies in applied
More informationWireless Networking with Selfish Agents. Li (Erran) Li Center for Networking Research Bell Labs, Lucent Technologies
Wireless Networking with Selfish Agents Li (Erran) Li Center for Networking Research Bell Labs, Lucent Technologies erranlli@dnrc.bell-labs.com Today s Wireless Internet 802.11 LAN Internet 2G/3G WAN Infrastructure
More informationNon-decreasing Payment Rules in Combinatorial Auctions
Research Collection Master Thesis Non-decreasing Payment Rules in Combinatorial Auctions Author(s): Wang, Ye Publication Date: 2018 Permanent Link: https://doi.org/10.3929/ethz-b-000260945 Rights / License:
More informationGames, Auctions, Learning, and the Price of Anarchy. Éva Tardos Cornell University
Games, Auctions, Learning, and the Price of Anarchy Éva Tardos Cornell University Games and Quality of Solutions Rational selfish action can lead to outcome bad for everyone Tragedy of the Commons Model:
More informationVCG in Theory and Practice
1 2 3 4 VCG in Theory and Practice Hal R. Varian Christopher Harris Google, Inc. May 2013 Revised: December 26, 2013 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 It is now common to sell online ads using
More informationFlexible Dynamic Spectrum Allocation in Cognitive Radio Networks Based on Game-Theoretical Mechanism Design
Flexible Dynamic Spectrum Allocation in Cognitive Radio Networks Based on Game-Theoretical Mechanism Design José R. Vidal, Vicent Pla, Luis Guijarro, and Jorge Martinez-Bauset Universitat Politècnica de
More informationCOMP/MATH 553 Algorithmic Game Theory Lecture 8: Combinatorial Auctions & Spectrum Auctions. Sep 29, Yang Cai
COMP/MATH 553 Algorithmic Game Theory Lecture 8: Combinatorial Auctions & Spectrum Auctions Sep 29, 2014 Yang Cai An overview of today s class Vickrey-Clarke-Groves Mechanism Combinatorial Auctions Case
More informationCSC304: Algorithmic Game Theory and Mechanism Design Fall 2016
CSC304: Algorithmic Game Theory and Mechanism Design Fall 2016 Allan Borodin (instructor) Tyrone Strangway and Young Wu (TAs) November 2, 2016 1 / 14 Lecture 15 Announcements Office hours: Tuesdays 3:30-4:30
More informationHIERARCHICAL decision making problems arise naturally
IEEE TRANSACTIONS ON AUTOMATION SCIENCE AND ENGINEERING, VOL. 5, NO. 3, JULY 2008 377 Mechanism Design for Single Leader Stackelberg Problems and Application to Procurement Auction Design Dinesh Garg and
More informationAn Evaluation of the Proposed Procurement Auction for the Purchase of Medicare Equipment: Experimental Tests of the Auction Architecture 1
An Evaluation of the Proposed Procurement Auction for the Purchase of Medicare Equipment: Experimental Tests of the Auction Architecture 1 Caroline Kim, Brian Merlob, Kathryn Peters, Charles R. Plott,
More informationEfficiency Guarantees in Market Design
Efficiency Guarantees in Market Design NICOLE IMMORLICA, MICROSOFT JOINT WORK WITH B. LUCIER AND G. WEYL The greatest risk to man is not that he aims too high and misses, but that he aims too low and hits.
More informationNew Results for Lazy Bureaucrat Scheduling Problem. fa Sharif University of Technology. Oct. 10, 2001.
New Results for Lazy Bureaucrat Scheduling Problem Arash Farzan Mohammad Ghodsi fa farzan@ce., ghodsi@gsharif.edu Computer Engineering Department Sharif University of Technology Oct. 10, 2001 Abstract
More informationON QUADRATIC CORE PROJECTION PAYMENT RULES FOR COMBINATORIAL AUCTIONS YU SU THESIS
ON QUADRATIC CORE PROJECTION PAYMENT RULES FOR COMBINATORIAL AUCTIONS BY YU SU THESIS Submitted in partial fulfillment of the requirements for the degree of Bachelor of Science in Electrical and Computer
More informationSoftware Frameworks for Advanced Procurement Auction Markets
Software Frameworks for Advanced Procurement Auction Markets Martin Bichler and Jayant R. Kalagnanam Department of Informatics, Technische Universität München, Munich, Germany IBM T. J. Watson Research
More informationBNE and Auction Theory Homework
BNE and Auction Theory Homework 1. For two agents with values U[0,1] and U[0,2], respectively: (a) show that the first-price auction is not socially optimal in BNE. (b) give an auction with pay your bid
More informationRecap Beyond IPV Multiunit auctions Combinatorial Auctions Bidding Languages. Multi-Good Auctions. CPSC 532A Lecture 23.
Multi-Good Auctions CPSC 532A Lecture 23 November 30, 2006 Multi-Good Auctions CPSC 532A Lecture 23, Slide 1 Lecture Overview 1 Recap 2 Beyond IPV 3 Multiunit auctions 4 Combinatorial Auctions 5 Bidding
More informationFIRST FUNDAMENTAL THEOREM OF WELFARE ECONOMICS
FIRST FUNDAMENTAL THEOREM OF WELFARE ECONOMICS SICONG SHEN Abstract. Markets are a basic tool for the allocation of goods in a society. In many societies, markets are the dominant mode of economic exchange.
More informationTraditional auctions such as the English SOFTWARE FRAMEWORKS FOR ADVANCED PROCUREMENT
SOFTWARE FRAMEWORKS FOR ADVANCED PROCUREMENT A range of versatile auction formats are coming that allow more flexibility in specifying demand and supply. Traditional auctions such as the English and first-price
More informationA Two-Tier Market for Decentralized Dynamic Spectrum Access in Cognitive Radio Networks
1 A Two-Tier Market for Decentralized Dynamic Spectrum Access in Cognitive Radio Networks Dan Xu, Xin Liu, and Zhu Han Department of Computer Science, University of California, Davis, CA 95616 Department
More informationMethods for boosting revenue in combinatorial auctions
Methods for boosting revenue in combinatorial auctions Anton Likhodedov and Tuomas Sandholm Carnegie Mellon University Computer Science Department 5000 Forbes Avenue Pittsburgh, PA 15213 {likh,sandholm}@cs.cmu.edu
More informationCS364B: Frontiers in Mechanism Design Lecture #11: Undominated Implementations and the Shrinking Auction
CS364B: Frontiers in Mechanism Design Lecture #11: Undominated Implementations and the Shrinking Auction Tim Roughgarden February 12, 2014 1 Introduction to Part III Recall the three properties we ve been
More informationCompetition with Licensed Shared Spectrum
ompetition with Licensed Shared Spectrum hang Liu EES Department Northwestern University, Evanston, IL 628 Email: changliu212@u.northwestern.edu Randall A. Berry EES Department Northwestern University,
More informationCompetitive Markets. Jeffrey Ely. January 13, This work is licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License.
January 13, 2010 This work is licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. Profit Maximizing Auctions Last time we saw that a profit maximizing seller will choose
More informationFirst-Price Auctions with General Information Structures: A Short Introduction
First-Price Auctions with General Information Structures: A Short Introduction DIRK BERGEMANN Yale University and BENJAMIN BROOKS University of Chicago and STEPHEN MORRIS Princeton University We explore
More informationDiffusion Mechanism Design
1 / 24 Diffusion Mechanism Design Dengji Zhao ShanghaiTech University, Shanghai, China Decision Making Workshop @ Toulouse 2 / 24 What is Mechanism Design What is Mechanism Design? What is Mechanism Design
More informationAn Introduction to Iterative Combinatorial Auctions
An Introduction to Iterative Combinatorial Auctions Baharak Rastegari Department of Computer Science University of British Columbia Vancouver, B.C, Canada V6T 1Z4 baharak@cs.ubc.ca Abstract Combinatorial
More informationNote on webpage about sequential ascending auctions
Econ 805 Advanced Micro Theory I Dan Quint Fall 2007 Lecture 20 Nov 13 2007 Second problem set due next Tuesday SCHEDULING STUDENT PRESENTATIONS Note on webpage about sequential ascending auctions Everything
More informationApproximation and Mechanism Design
Approximation and Mechanism Design Jason D. Hartline Northwestern University May 15, 2010 Game theory has a great advantage in explicitly analyzing the consequences of trading rules that presumably are
More informationOn Optimal Tiered Structures for Network Service Bundles
On Tiered Structures for Network Service Bundles Qian Lv, George N. Rouskas Department of Computer Science, North Carolina State University, Raleigh, NC 7695-86, USA Abstract Network operators offer a
More informationAUCTION DESIGN FOR SECONDARY SPECTRUM MARKETS
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,
More informationThe Impact of Investment on Price Competition in Unlicensed Spectrum
The Impact of Investment on Price Competition in Unlicensed Spectrum Hang Zhou Mobile and Communications Group Intel Corp hang.zhou@intel.com Randall A. Berry, Michael L. Honig EECS Department Northwestern
More informationBilateral and Multilateral Exchanges for Peer-Assisted Content Distribution
1290 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 19, NO. 5, OCTOBER 2011 Bilateral and Multilateral Exchanges for Peer-Assisted Content Distribution Christina Aperjis, Ramesh Johari, Member, IEEE, and Michael
More informationCollusion-Resistant Multi-Winner Spectrum Auction for Cognitive Radio Networks
Collusion-Resistant Multi-Winner Spectrum Auction for Cognitive Radio Networks Yongle Wu, Beibei Wang, K. J. Ray Liu, and T. Charles Clancy Department of Electrical and Computer Engineering and Institute
More informationWITH limited resources and high population density,
1 Combinatorial Auction-Based Pricing for Multi-tenant Autonomous Vehicle Public Transportation System Albert Y.S. Lam arxiv:1503.01425v2 [cs.gt] 20 Sep 2015 Abstract A smart city provides its people with
More informationCombinatorial Auctions
T-79.7003 Research Course in Theoretical Computer Science Phase Transitions in Optimisation Problems October 16th, 2007 Combinatorial Auctions Olli Ahonen 1 Introduction Auctions are a central part of
More informationWireless Network Pricing Chapter 5: Monopoly and Price Discriminations
Wireless Network Pricing Chapter 5: Monopoly and Price Discriminations Jianwei Huang & Lin Gao Network Communications and Economics Lab (NCEL) Information Engineering Department The Chinese University
More informationChapter 2 Social Group Utility Maximization Framework
Chapter 2 Social Group Utility Maximization Framework 2.1 Motivation As discussed in Chap. 1, the social ties among wireless users significantly influence their interactions with each other in wireless
More informationFirst-Price Path Auctions
First-Price Path Auctions Nicole Immorlica MIT CS&AI Laboratory Cambridge, MA 2139 nickle@csail.mit.edu Evdokia Nikolova MIT CS&AI Laboratory Cambridge, MA 2139 enikolova@csail.mit.edu David Karger MIT
More informationExpressing Preferences with Price-Vector Agents in Combinatorial Auctions: A Brief Summary
Expressing Preferences with Price-Vector Agents in Combinatorial Auctions: A Brief Summary Robert W. Day PhD: Applied Mathematics, University of Maryland, College Park 1 Problem Discussion and Motivation
More informationOnline shopping and platform design with ex ante registration requirements. Online Appendix
Online shopping and platform design with ex ante registration requirements Online Appendix June 7, 206 This supplementary appendix to the article Online shopping and platform design with ex ante registration
More informationRobust Supply Function Bidding in Electricity Markets With Renewables
Robust Supply Function Bidding in Electricity Markets With Renewables Yuanzhang Xiao Department of EECS Email: xyz.xiao@gmail.com Chaithanya Bandi Kellogg School of Management Email: c-bandi@kellogg.northwestern.edu
More informationOptimizing Prices in Descending Clock Auctions
Optimizing Prices in Descending Clock Auctions Abstract A descending (multi-item) clock auction (DCA) is a mechanism for buying from multiple parties. Bidder-specific prices decrease during the auction,
More informationA Theory of Loss-Leaders: Making Money by Pricing Below Cost
A Theory of Loss-Leaders: Making Money by Pricing Below Cost Maria-Florina Balcan, Avrim Blum, T-H. Hubert Chan, and MohammadTaghi Hajiaghayi Computer Science Department, Carnegie Mellon University {ninamf,avrim,hubert,hajiagha}@cs.cmu.edu
More informationMultiagent Systems: Spring 2006
Multiagent Systems: Spring 2006 Ulle Endriss Institute for Logic, Language and Computation University of Amsterdam Ulle Endriss (ulle@illc.uva.nl) 1 Combinatorial Auctions In a combinatorial auction, the
More informationVirtual Machine Trading in a Federation of Clouds: Individual Profit and Social Welfare Maximization
IEEE/ACM TRANSACTIONS ON NETWORKING, VOL 24, NO 3, JUNE 2016 1827 Virtual Machine Trading in a Federation of Clouds: Individual Profit Social Welfare Maximization Hongxing Li, Chuan Wu, Member, IEEE, ACM,
More informationThe Price of Anarchy in an Exponential Multi-Server
The Price of Anarchy in an Exponential Multi-Server Moshe Haviv Tim Roughgarden Abstract We consider a single multi-server memoryless service station. Servers have heterogeneous service rates. Arrivals
More informationPARETO-IMPROVING CONGESTION PRICING AND REVENUE REFUNDING WITH ELASTIC DEMAND
PARETO-IMPROVING CONGESTION PRICING AND REVENUE REFUNDING WITH ELASTIC DEMAND Xiaolei Guo, Odette School of Business Cross-Border Transportation Centre University of Windsor Hai Yang, Department of Civil
More informationSpectrum Auction Design
Spectrum Auction Design Peter Cramton Professor of Economics, University of Maryland www.cramton.umd.edu/papers/spectrum 1 Two parts One-sided auctions Two-sided auctions (incentive auctions) Application:
More informationEffective Mobile Data Trading in Secondary Ad-hoc Market with Heterogeneous and Dynamic Environment
Effective Mobile Data Trading in Secondary Ad-hoc Market with Heterogeneous and Dynamic Environment Hengky Susanto,4,, Honggang Zhang 2, Shing-Yip Ho 3, and Benyuan Liu Department of Computer Science,
More informationTopics in ALGORITHMIC GAME THEORY *
Topics in ALGORITHMIC GAME THEORY * Spring 2012 Prof: Evdokia Nikolova * Based on slides by Prof. Costis Daskalakis Let s play: game theory society sign Let s play: Battle of the Sexes Theater Football
More informationTitle: A Mechanism for Fair Distribution of Resources with Application to Sponsored Search
Title: A Mechanism for Fair Distribution of Resources with Application to Sponsored Search Authors: Evgenia Christoforou, IMDEA Networks Institute, Madrid, Spain and Universidad Carlos III, Madrid, Spain
More informationCS364B: Frontiers in Mechanism Design Lecture #1: Ascending and Ex Post Incentive Compatible Mechanisms
CS364B: Frontiers in Mechanism Design Lecture #1: Ascending and Ex Post Incentive Compatible Mechanisms Tim Roughgarden January 8, 2014 1 Introduction These twenty lectures cover advanced topics in mechanism
More informationTruth Revelation in Approximately Efficient Combinatorial Auctions
Truth Revelation in Approximately Efficient Combinatorial Auctions DANIEL LEHMANN Hebrew University, Jerusalem, Israel AND LIADAN ITA O CALLAGHAN AND YOAV SHOHAM Stanford University, Stanford, California
More informationFair Profit Allocation in the Spectrum Auction Using the Shapley Value
Fair Profit Allocation in the Spectrum Auction Using the Shapley Value Miao Pan, Feng Chen, Xiaoyan Yin, and Yuguang Fang Department of Electrical and Computer Engineering, University of Florida, Gainesville,
More informationAutonomous Agents and Multi-Agent Systems* 2015/2016. Lecture Reaching Agreements
Autonomous Agents and Multi-Agent Systems* 2015/2016 Lecture Reaching Agreements Manuel LOPES * These slides are based on the book by Prof. M. Wooldridge An Introduction to Multiagent Systems and the online
More informationCopyright (C) 2001 David K. Levine This document is an open textbook; you can redistribute it and/or modify it under the terms of version 1 of the
Copyright (C) 2001 David K. Levine This document is an open textbook; you can redistribute it and/or modify it under the terms of version 1 of the open text license amendment to version 2 of the GNU General
More informationThe Need for Information
The Need for Information 1 / 49 The Fundamentals Benevolent government trying to implement Pareto efficient policies Population members have private information Personal preferences Effort choices Costs
More informationSection 1: Introduction
Multitask Principal-Agent Analyses: Incentive Contracts, Asset Ownership, and Job Design (1991) By Bengt Holmstrom and Paul Milgrom Presented by Group von Neumann Morgenstern Anita Chen, Salama Freed,
More informationThe Ascending Bid Auction Experiment:
The Ascending Bid Auction Experiment: This is an experiment in the economics of decision making. The instructions are simple, and if you follow them carefully and make good decisions, you may earn a considerable
More informationThe Sealed Bid Auction Experiment:
The Sealed Bid Auction Experiment: This is an experiment in the economics of decision making. The instructions are simple, and if you follow them carefully and make good decisions, you may earn a considerable
More informationApproximation in Algorithmic Game Theory
Approximation in Algorithmic Game Theory Robust Approximation Bounds for Equilibria and Auctions Tim Roughgarden Stanford University 1 Motivation Clearly: many modern applications in CS involve autonomous,
More informationAn Analytical Upper Bound on the Minimum Number of. Recombinations in the History of SNP Sequences in Populations
An Analytical Upper Bound on the Minimum Number of Recombinations in the History of SNP Sequences in Populations Yufeng Wu Department of Computer Science and Engineering University of Connecticut Storrs,
More information