An Auction Algorithm for Procuring Wireless Channel in A Heterogenous Wireless Network

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1 An Action Algorithm for Procring Wireless Channel in A Heterogenos Wireless Network N. Rama Sri Electronic Enterprises Laboratory, Department of Compter and Science and Atomation, Indian Institte of Science, Bangalore, India. nrsri@csa.iisc.ernet.in Y. Narahari Electronic Enterprises Laboratory, Department of Compter and Science and Atomation, Indian Institte of Science, Bangalore, India. hari@csa.iisc.ernet.in Abstract In this paper, we develop an action algorithm for procring wireless channel by a wireless node in a heterogeneos wireless network. We assme that the service providers of the heterogeneos wireless network are selfish and non-cooperative in the sense that they are interested in maximizing their own tilities. The wireless ser is in need of procring wireless channel to execte mltiple jobs. To solve the problem of wireless ser, we propose a reverse optimal (REVOPT) action. We characterize the expression for the expected payment by the wireless ser. Or proposed action mechanism REVOPT satisfies important game theoretic properties like Bayesian Incentive Compatibility and Individal Rationality. Keywords Heterogeneos wireless network, game theory, mechanism design, selfishness, rationality, individal rationality, Bayesian incentive compatibility, optimal action mechanisms. I. INTRODUCTION The primary goal in the wireless commnication world can be briefly smmarized as providing service for commnication anywhere, anytime, any-media and principally at high-data rates. However, this goal is in conflict with the existence of different rnning and emerging wireless systems covering almost the whole world, each one following its own architectre. The development of wireless systems evolved in an nimaginable way dring the last two decades. For example, in celllar wireless systems, the so-called First Generation (1G) is no longer in se. Crrently the dominant generations, which are nowadays attracting mch attention, are 2G, 2.5G and 3G. In Erope their representatives are GSM (Global System for Mobile Commnication), GPRS (General Packet Radio Service) and UMTS (Universal Mobile Telecommnications System) respectively and belong to the terrestrial wide area celllar systems. The circit-switched GSM provides very slow data rates ( kbps) to satisfy the brst applications, even after the appliance of High Speed Circit Switched Data (HSCSD), it does not overcome the limit of 40 kbps. Packetswitched networks, based on the access network of GSM with actal changes only in the core network (GPRS), appeared with the promise of higher bit rates (theoretically 172 kbps), bt in practice the maximm bit rate achieved is abot 45 kbps. The UMTS access network follows a different approach, in comparison to GSM and GPRS, making the achievement of higher data rates more feasible. UMTS offers data rates p to 384 kbps, even if in theory the 2 Mbps transfer rate is possible. Nevertheless, the actal performance of UMTS has still to be verified dring real operation conditions with heavy network loads. On the other hand, development in new radio access technologies and increase in ser demand for biqitos high speed access are driving the deployment of a wide array of wireless networks, ranging from wireless WAN to wireless MAN, Wireless LAN and Wireless PAN. These kind of networks provide incomparably high data rates. For example the b WLAN provides throghpt p to 5 Mbps, while the data rates in a can be p to over 25 Mbps, with the perspective to reach in the ftre the inconceivable limit of 155 Mbps. With complementary characteristics especially in terms of data rate and coverage of the varios wireless commnication technologies, the co-existence of these technologies reslts in a heterogeneos set of wireless commnications systems that can provide better commnication and service facilities to the mobile/wireless nodes. Sch heterogenos set of wireless commnication systems is called Heterogeneos Wireless Networks. There is another important reason for going towards heterogeneos wireless networks. In some type of wireless networks where there is no access point, sch as wireless ad hoc networks, protocols sffer in network performance that incldes large roting overhead, low throghpt, and large end-to-end delay. In sch networks, the isses of qality of service (QoS) are even more complicated becase of the lack of reliable methods to distribte information in the entire network. The integration of heterogeneos wireless technologies can improve the network performance, thereby meeting the demands for different qality of service (QoS). Heterogeneos wireless networks give satisfiable soltions to the problems we mentioned above. Heterogeneos wireless networks provide overlapping coverage to mobile sers. Its active components are based on different theoretical backgronds and are optimized for different

2 ranges. Heterogeneos wireless networks pose many interesting research challenges. Among which, resorce management in sch a hybrid environment is still an open problem. A. Problem Statement In this paper, we attempt to provide soltion to the following problem. We consider the sitation where a mobile/wireless ser has to perform a set of jobs, where each job is a parallel application and can be split over different wireless networks for exection. The mobile ser can access heterogeneos wireless networks provided by a set of selfish, intelligent, and noncooperative service providers. These service providers charge for allocating the wireless channel to the mobile/wireless sers. Ultimately, the problem for a mobile/wireless ser is to procre wireless channel to perform the jobs while minimizing the total amont to pay to the network service providers. So, there is a need for a mechanism that shold be optimal (i.e. in minimal sense) for the mobile ser and satisfy important game theoretic properties, say Bayesian incentive compatibility and individal rationality, so that the selfish and non-cooperative service providers participate in bidding for the time slots which are annonced by the wireless ser. B. Contribtions of the Paper As far as or knowledge is concerned, research is crrently going on developing action algorithms for wireless channel allocation in a single wireless network environment. The work presented in this paper is perhaps the earliest works in developing action based algorithms in heterogeneos wireless network environment. In this paper, we develop a noncooperative game theoretic based mechanism to solve the wireless channel procrement problem of a wireless/mobile ser having access to a heterogeneos wireless network. Or work can be organized in the following way. We first define what we call Resorce Procrement Action to explain the context of the problem. Then, we formlate Resorce Procrement Action as a mechanism design problem in a qasi-linear environment. We next design a reverse optimal (REVOPT) mechanism for Resorce Procrement Action problem. In this design, we characterize both the allocation rle and payment rle of REVOPT action mechanism. Finally, we compte an expression for the optimal (in minimal sense) total payment by the wireless/mobile ser sing or approach. C. Organization of the Paper The paper is organized in the following. Section 2 provides a review of the related literatre. In Section 3, we define Resorce Procrement Action and also formlate it as a mechanism design problem in a qasi-linear environments. We present the REVOPT action mechanism in Section 4. We conclde the paper in Section 5. II. RELATED WORK In this section, we present the related work from the literatre. As we are concerned with developing action based mechanism to the resorce procrement problem in heterogeneos wireless networks, we present the review of the literatre in two parts. In Section II(A), we present the research work in the area of heterogeneos wireless networks and resorce management. In Section II(B), we deal with the review of related literatre work on reverse actions. A. Research Work on Heterogeneos Wireless Networks Examples of integrated heterogeneos wireless networks inclde ad hoc/celllar integrated networks. W, Mkherjee, and Chan proposed mobile-assisted connection-admission (MACA) channel allocation scheme to achieve load balancing in a celllar network [1]. In MACA, some special channels are sed to connect mobile nits from different cells. When a mobile nit cannot connect to its own base station de to heavy load, it may be able to get connected to its neighboring cells base station throgh a two-hop link. A similar approach, integrated celllar and ad hoc relaying systems (icar), is proposed by W et al. in [2]. It addresses the congestion problem de to nbalanced traffic in a celllar system and provides interoperability for heterogeneos networks. The basic idea is to place a nmber of ad hoc repaying stations at strategic locations, which can be sed to relay signals between mobile hosts and base stations. In [3], Brewer, et al. present the reslts of the BAR- WAN project, which focsed on enabling trly sefl mobile networking across an extremely wide variety of real-world networks and mobile devices. The athors present the overall architectre that enables seamless roaming in a single logical overlay network composed of many heterogeneos (mostly wireless) physical networks, and provides significantly better TCP performance for these networks. It also provides complex scalable and highly available services to enable powerfl capabilities across a very wide range of mobile devices, and mechanisms for atomated discovery and configration of localized services. Topology Control in Heterogeneos Wireless Networks is addressed for the first time in [4]. This presents the possible topology control problems in these networks and provides soltions to these problems. It proposes two localized topology control algorithms for heterogeneos wireless mltihop networks with non-niform transmission ranges: Directed Relative Neighborhood Graph (DRNG) and Directed Local Minimm Spanning Tree (DLMST). In both algorithms, each node selects a set of neighbors based on the locally collected information. In [5], the athors present novel network scenarios where wired and wireless connections are melted together, a real measre of these parameters is fndamental in a planning process of new services over novel network infrastrctres. Nowadays networks are heterogeneos in terms of access network technologies (wired LAN Ethernet 10/100/1000, Wireless LAN a, b, g -, GPRS, UMTS, GSM, Bletooth,...), end-sers devices (workstation, PC desktop, Laptop/Notebook, PDA, Advanced Mobile Phone,...) and finally operating systems (Unix, Linx, Win 98/NT/2000/XP,

3 Win CE, Linx Familiar, OS Embedded,...). The athors also provide a heterogeneos network performance characterization with respect to delay and throghpt in UDP and TCP environments. Kyriazakos, et al. investigated the real-time radio resorce management in heterogeneos wireless networking environments in [6]. The athors presented a methodology and an approach for designing a hierarchical system that is agmenting the fnctionality of wireless network architectres by enforcing smooth co-operation and is capable to react when resorce shortcomings appear. Qadeer, et al. in [7] presented an approach for power management of the wireless network interfaces (WNIC) for heterogeneos wireless networks. The athors develop an integrated approach for the management of power and performance of mobile devices for these environments. Their policy decides which WNIC to employ for a given application and optimizes its sage based on the crrent power and performance needs of the system. The policy dynamically switches between WNICs dring program exection if data commnication reqirements and/or network conditions change. Sliman et al. for the first time introdced cooperative game theoretic concepts for resorce allocation in heterogeneos wireless networks in [8]. Bt, this paper does not provide the mathematical modelling of the cooperative game associated with the resorce allocation problem in which different wireless network service providers cooperate among themselves. In contrast to this approach, we consider the sitation where the service providers of the heterogeneos wireless network are selfish, intelligent, and non-cooperative in the sense that they are interested in maximizing their own tilities. We provide rigoros mathematical modelling of or model. B. Research Work on Reverse Actions Actions are concerned with the design of certain rles of interaction sing the tools of game theory and mechanism design [9], [10], for electronic transactions that will, in principle, yield some desired otcome. In the context of negotiations for procrement we reqire rles governing: (1)bidding for contracts, (2)the isses and attribtes that will be considered to determine winner(s) of the contract, (3)determination of winning sppliers, and (4)payments that will be made. English actions and Dtch actions, and sealed bid contracts are well nderstood, widely sed economic mechanisms in the context of reverse actions. Since the rles of interaction in these actions are well laid ot, they have been a natral target for atomation. A comprehensive srvey on reverse action based mechanisms appears in [11]. Other recent srveys can be fond in [12],[13]. III. RESOURCE PROCUREMENT AUCTION We consider the resorce procrement problem for a mobile ser having access to heterogeneos wireless networks provided by a set of selfish and non-cooperative service providers. We assme that the mobile ser is eqipped with mltiple network interfaces in order to access heterogeneos wireless networks. The mobile ser has a set of different jobs to be performed. The worth to the mobile ser by different jobs is also different. So, the time slots for getting access to wireless channel to perform these jobs also worth differently to the wireless/mobile ser. Withot loss of generality, we assme that the mobile ser annonces the time slots in decreasing order of the preference. It means that the vale of the mobile ser for the first time slot is more when compared to the second time slot, the vale for the second time slot is more when compared to the third time slot, so on. Once the time slots are annonced, then the service providers will sbmit bids on them. Having received the bids, the mobile ser ses a mechanism for selecting the winning bids and deciding the payment to the winning bidders. We call this action mechanism Resorce Procrement Action. The payment by the mobile ser to the winning service providers depends on the bids sbmitted by the service providers (bidders). Now, let s assme there are n service providers and N = {1, 2,..., n} represents the set of service providers. Let there are m time slots annonced by the mobile ser and M = {1, 2,..., m} represents the set of time slots to be actioned. Let θ = (θ 1, θ 2,..., θ n ) be the vector of bids received from these n service providers. Let p i (θ) be the payment to the service provider i by the mobile ser, when the vector of bids of the service providers is θ. A. Resorce Procrement Action as Mechanism Design Problem To model Resorce Procrement Action as mechanism design problem, we will make the following for assmptions. These for assmptions are treated as bench mark assmptions for the design of actions from the literatre point of view. 1) Risk Netral: The heterogeneos wireless network service providers are risk netral. 2) Independent Private Vale (IPV) Model: Each wireless network service provider knows the vale of a particlar time slot that he is going to bid and does not know the vale of the other wireless network service providers. Each service provider perceives any other service provider s valation as a draw from some probability distribtion. In the same fashion, he knows that the other service providers regard his own valation as a draw from some probability distribtion. More precisely, for service provider i, i = 1, 2,..., n, there is some probability distribtion Φ i (.) from which he draws his valation θ i for the time slot. Any service provider s valation is statistically independent from any other service provider s valation. The valation θ i can be viewed as his private vale. Let Θ i, i = 1, 2,..., n denote the set of all possible types of service provider i and assme that Θ i is a closed interval of the real line, that is Θ i = [ l, θ i ]. This implies that Φ i(.), i = 1, 2,..., n are probability distribtion fnctions of

4 the random variables Θ i, i = 1, 2,..., n. 3) Symmetry among Service Providers: The service providers are symmetric in the following sense: Θ 1 =Θ 2 =... = Θ n = Θ Φ 1 (.) = Φ 2 (.)=... = Φ n (.) = Φ(.) 4) Properties of Φ(.) and Θ: We assme that Φ(.) satisfy the following properties: Θ = [θ l, θ ] θ l > 0 φ(θ) = Φ (θ) > 0; θ l θ θ Under this setting, we can formlate the resorce procrement action as mechanism design problem. The following are the main components of the Resorce Procrement Action mechanism design problem. 1) Otcome Set X: An otcome in or mechanism design problem can be represented by a vector x = ((y ij ) i N,j M, (p i ) i N ), where y ij is the probability that service provider i is the winner for the time slot j and p i is the payment received by i th service provider. The set of all feasible alternatives is represented in the following way. X = {((y ij ) i N,j M, (p i ) i N ) y ij [0, 1], n i=1 y ij 1, m j=1 y ij 1, p i 0, i N, j M} 2) Utility Fnction of Service Providers ( i (.)): The Bernolli tility fnction of the i th service provider is given by, i (x, θ i ) = ( m j=1 y ij)(p i θ i ) 3) Social Choice Fnction (f(.)): The general strctre of the social choice fnction for this type of problems is, f(θ) = ((y ij ) i N,j M, (p i ) i N ) Note that y ij (θ) depends on allocation rle and p i (θ) depends on the payment rle. Now nder these for benchmark assmptions, a resorce procrement action can be viewed as a direct revelation mechanism Λ = ((Θ i ) i N, f(.)) in qasi linear environment, where Θ i is the type set of an service provider i and f(.) is the social choice fnction. A mechanism Λ combined with possible types of the service providers (Θ 1, Θ 2,..., Θ n ), probability density φ(.), and Bernolli tility fnctions ( 1 (.), 2 (.),..., n (.)) defines a Beyesian game of incomplete information which gets indced among the service providers after the mobile ser annonces the time slots. The indced Bayesian Game Γ b can be given in the following manner Γ b = (N, (Θ i ) i N, (Θ i ) i N, φ(.), ( i ) i N ) where i : Θ Θ R is the tility fnction of agent i and is defined in the following manner i (θ, θ) = i (f(θ ), θ i ) where Θ = i N Θ i. A strategy for the service provider i in the above game Γ b is a fnction s i : Θ i Θ i giving service provider i s contingent bidding plan of revealing his own type based on his actal type θ i. Depending on kind of mechanism the mobile ser ses, the service providers decide their bidding strategy. The strategies selected by the service providers in trn decide the payment made by the mobile ser. There comes Game theory into the sitation to analyze the existing scenario. It says that each bidder will follow the bidding strategy s i (.) sch that the strategy profile (s 1(.), s 2(.),..., s n) is a Bayesian Nash eqilibrim of the game Γ b indced by the mechanism that the mobile ser ses. Some natral qestions that arise in this context are the following. What kind of mechanism that the mobile ser imposes to procre resorces in an optimal way (i.e. minimizing the total payment)? What are the key game theoretic properties the mechanism has to satisfy? We will answer these qestions in the rest of this paper. In general, the mechanism that the mobile ser imposes shold give minimm payment (to pay to the winning service providers) and satisfy the following two important properties of a social choice fnction. Individal Rationality: The service providers participation in the action is volntary in the sense that the mobile ser shold not force them to participate in the action mechanism. As a reslt, the service provider will choose to participate in the mechanism only if he loses nothing ot of participating in the action. This is known as individal rationality constraints. So, in order to ensre the service provider i s participation in the mechanism, after he has learned his actal type as θ i, the following interim individal rationality constraints mst be satisfied U i (θ i f) = E θ i [ i (f(θ i, θ i ) θ i )] 0, θ i Θ i where U i (θ i f) is service provider i s interim expected tility nder social choice fnction f(.) when his type is θ i. Incentive Compatibility: The service providers prefer to have an action mechanism Λ for which trth telling is the eqilibrim strategy for all the service providers. The reason being for this is that some times it is very difficlt to compte the optimal strategy in closed form given the bidders are bonded rational. These constraints are called Incentive Compatibility constraints. Depending on the type of the eqilibrim concept in hand, there are two types of incentive compatibility constraints. Dominant Strategy Incentive Compatibility: The social choice fnction f(.) is said to be dominant strategy incentive compatible if the direct revelation mechanism Λ = ((Θ i ) i N, f(.)) has a dominant strategy eqilibrim s = (s 1(.), s 2(.),..., s n(.)), where s i (θ i) = θ i, θ i Θ i,, i N

5 Bayesian Incentive Compatibility: The social choice fnction f(.) is said to be Bayesian incentive compatible if the direct revelation mechanism Λ = ((Θ i ) i N, f(.)) has a Bayesian Nash eqilibrim s = (s 1(.), s 2(.),..., s n(.)), where s i (θ i) = θ i, θ i Θ i, i N In order to make the work of the service providers simpler, it is better to choose a mechanism Λ = ((Θ i ) i N, f(.)) sch that the social choice fnction f(.) is ideally dominant strategy incentive compatible. IV. REVERSE OPTIMAL MECHANISM (REVOPT) From the previos section, it is desirable for the mobile ser to have a mechanism which minimizes the expected payment along with satisfying the properties of individal rationality and Bayesian incentive compatibility. Myerson stdied sch type of action mechanisms in the context of selling (i.e. forward action settings) a single individal item [14]. Myerson called sch an action mechanism as optimal action. In this paper, we are considering the case where mltiple items (i.e. time slots) are procred (i.e. reverse action settings) by the mobile ser from the non-cooperative and selfish service providers. So, we call or proposed approach Reverse Optimal (REVOPT) action. As mentioned in Section III, we assme there are n service providers and N = {1, 2,..., n} represents the set of service providers. Let there are m time slots annonced by the mobile ser and M = {1, 2,..., m} represents the set of time slots to be actioned. Let θ = (θ 1, θ 2,..., θ n ) be the vector of bids received from these n service providers. Let p i (θ) be the payment to the service provider i by the mobile ser, when the vector of bids of the service providers is θ. The Bernolli tility fnction of service provider i is given by i (f(θ), θ i ) = ( m j=1 y ij(θ))(p i (θ) θ i ) = v i (y(θ))(p i (θ) θ i ) = t(θ i ) v i (y(θ))θ i where v i (y(θ)) = ( m j=1 y ij(θ)) is known as vale fnction of the service provider i. t i (θ) = v i (y(θ))p i (θ) denotes the payment to the service provider by the mobile ser. Now expected payment, t i (ˆθ i ), to the service provider i when he annonces his type to be ˆθ i and all the service providers j i trthflly reveal their type is given by t i (ˆθ i ) = E θ i [t i (ˆθ i, θ i )] A similar expression for the valation fnction of i th service provider is i (ˆθ i ) = E θ i [v i (y(ˆθ i, θ i ))] Now, the expected tility of i th service provider when his type θ i is given by, U i (θ i ) = t i (θ i ) i (θ i )θ i (1) A social choice fnction that the mobile ser chooses in this environment is in the form f(.) = ((y ij ) i N,j M, (t i ) i N ). We want this social choice fnction to satisfy individal rationality and Bayesian incentive compatibility, which are mentioned in the previos section. For the above mentioned social choice fnction to satisfy the interim individal rationality constraints, the following eqation mst be hold. i N, θ i Θ i U i (θ i ) = E θ i [ i (f(θ i, θ i ), θ i ) θ i ] = t i (θ i ) i (θ i )θ i 0 becase the service providers are always free to participate in the bidding of time slots. Now, we need to characterize the conditions nder which the above social choice fnction satisfies the Bayesian Incentive compatibility. The following Proposition serves the reqirement. Proposition-1: The social choice fnction, f(.) = ((y ij ) i N,j M, (t i ) i N ), chosen by the mobile ser is Bayesian incentive compatible if and only if, i N, (i) i (.) is non-increasing, (ii) U i (θ i ) = U i ( l) i l i (s)ds, θ i Θ i. Proof:(a) Necessity: Bayesian incentive compatibility implies that for each ˆθ i > θ i we have, and Ths, U i (θ i ) t i (ˆθ i ) θ i i (ˆθ i ) = U i (ˆθ i ) + (ˆθ i θ i ) i (ˆθ i ), U i (ˆθ i ) t i (θ i ) ˆθ i i (θ i ) = U i (θ i ) + (θ i ˆθ i ) i (θ i ). i (θ i ) U i(θ i ) U i (ˆθ i ) ˆθ i θ i i (ˆθ i ) This expression immediately implies that i (.) mst be nonincreasing (since we have taken ˆθ i > θ i ). In addition, letting ˆθ i θ i and sing the above expression, we have θ i and so U i (θ i) = i (θ i ) U i (θ i ) = U i ( l) i l i (s)ds, θ i Θ i. This completes the proof for the necessary conditions. (b) Sfficiency: Consider any θ i and ˆθ i and sppose withot loss of generality θ i > ˆθ i holds. If the conditions (i) and (ii) in the statement of the proposition hold, then Hence, U i (θ i ) U i (ˆθ i ) = i ˆθ i i (s)ds i ˆθ i i (ˆθ i )ds = (θ i ˆθ i ) i ( θ i ) U i (θ i ) U i (ˆθ i ) (θ i ˆθ i ) i ( θ i ) (2) = t i (ˆθ i ) θ i i (ˆθ i ) (3)

6 Similarly, we can derive the following expression for the case where ˆθ i θ i, U i (ˆθ i ) U i (θ i ) (ˆθ i θ i ) i (θ i ) (4) = t i (θ i ) ˆθ i i (θ i ) (5) Eqations (3) and (4) establish the reqired sfficiency conditions. So, f(.) is Bayesian incentive compatible. Q.E.D. A social choice fnction chosen by the mobile ser in the environment is a fnction f(.) = ((y ij ) i N,j M, (t i ) i N ) having the properties that, y ij (θ) [0, 1], i N, j M, θ Θ, j M y ij(θ) 1, i N y ij(θ) 1, and i N t i(θ) being the total payment by the mobile ser. The mobile ser s reverse optimal mechanism can be written as one of choosing fnctions (y ij ) i N,j M and (U i (.)) i N to solve i i=1 { l i (θ i )θ i + U i (θ i )} φ i (θ i )dθ i sbject to (i) (ii) (iii) (iv) i (.) is non-increasing, i N y ij (θ) [0, 1], m j=1 y ij(θ) 1, i=1 y ij(θ) 1, i N, j M, θ Θ U i (θ i ) = U i ( l) i l i (s)ds, i N, θ i Θ i U i (θ i ) 0, i N, θ i Θ i In this formlation, the objective fnction corresponds to the total expected payment by the mobile ser to all service providers. Here note that constraint (iv) is service provider s individal rationality constraints, constraint (i)&(iii) are the necessary and sfficient conditions for the social choice fnction f(θ) = ((y ij (θ)) i N,j M, (t i (θ)) i N ) to be Bayesian incentive compatible from Proposition-1. Note that if constraint (iii) is satisfied, then constraint (iv) will be satisfied if and only if (v) U i ( l) i l i (s)ds, i N So, we can replace constraint (iv) with constraint (v). Next, sbstitting for U i (θ i ) in the objective fnction from constraint (iii), we get { i i=1 l i (θ i )θ i + U i ( l) } θ i l i (s)ds φ i (θ i )dθ i Integrating by parts the above eqation, the mobile ser problem can be written as one of choosing the fnctions y ij (.) and the vales U 1 (θ1), l U 2 (θ2), l..., U n (θn) l sch that 1... n { θ1 l θn l i=1 v i(θ i )J i (θ i )} { n i=1 φ i(θ i )} dθ n... dθ 1 + i=1 U i( l) sbject to constraints (i), (ii), and (v), where J i (θ i ) = θ i 1 Φ i(θ i ) φ i (θ i ) = θ i Φ i (θ i ) φ i (θ i ). It is evident that soltion mst have U i ( l) = i l i (s)ds, i N. Hence the mobile ser s optimization problem redces to choosing fnctions y ij (.) sch that, 1... n { θ1 l θn l i=1 v i(θ i )J i (θ i )} { n i=1 φ i(θ i )} dθ n... dθ 1 + i=1 l i (s)ds sbject to (i) (ii) i (.) is non-increasing, i N y ij (θ) [0, 1], m j=1 y ij(θ) 1, i=1 y ij(θ) 1, i N, j M, θ Θ Let s ignore the constraint (i) for the moment. Then the above optimization problem indicates that y ij (.) is a soltion to this relaxed problem iff i N, we have y ij = 0 j = 1, 2,..., m : ifj i (θ i ) > 0 1 j = 1, 2,..., m < n : ifj i (θ i ) = J(j) 1 j = 1, 2,..., m n : ifj i (θ i ) = J(j) 0 : otherwise where J(j) is the j th lowest among J i (θ i ), i N. It says, if we ignore the constraint (i), then y ij is a soltion to the relaxed problem iff any the service provider for whom the vale J i (θ i ) is positive, no time slot is assigned and the rest of the service providers will be assigned to the time slots in the same order as the vales of J i (θ i ) starting from the lowest possible vale. That is, the first time slot is allocated to the service provider who has the highest negative vale for J i (θ i ), the second time slot is assigned to the service provider who has the second highest negative vale for J i (θ i ), and so on. This completes the characterization of the allocation rle of the social choice fnction chosen by the mobile ser. Now, we will characterize the payment rle in the social choice fnction. t i (θ) = t i (θ i ) = θ i i (θ i ) + U i (θ i ) = θ i i (θ i ) + U i ( l) i l i (s)ds = θ i i (θ i ) where i (θ i ) is expected vale of the i th service provider and i (θ i ) = E θ i [v i (y(θ))] = E θ i [ m j=0 y ij(θ)] This completes the analysis on the strctre of REVOPT mechanism chosen by the mobile ser. Using this action mechanism the total payment, P REV OP T, by the mobile ser for procring wireless channel is given by (nder bench mark assmptions) P REV OP T = n θ=θ l t(θ)φ(θ)dθ V. CONCLUSIONS AND FUTURE WORK In this paper, we addressed the problem of procring resorces by the wireless/mobile ser in non-cooperative heterogenos wireless networks. We have proposed a reverse optimal action (REVOPT) mechanism to solve this problem. We characterized the allocation rle and payment rle in or soltion. This proposed action mechanism satisfies important game theoretic properties like individal rationality and incentive compatibility.

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