A Game Theory Approach: Dynamic Behaviours for. Spectrum Management in Cognitive Radio Network. Saed Alrabaee1, Mahmoud Khasawneh1, Anjali ...
GC'12 Workshop: The 4th IEEE International Workshop on Management of Emerging Networks and Services
A Game Theory Approach: Dynamic Behaviours for Spectrum Management in Cognitive Radio Network Saed Alrabaee1, Mahmoud Khasawneh1, Anjali Agarwal1, Nishith Goel2, Marzia Zaman2, 1
Department of Electrical and Computer Engineering, Concordia University, Montreal, Canada {s_alraba, aagarwal, m_khasaw}@encs.concordia.ca 2 Cistel Technology Inc., Ottawa, Canada {ngoel, Marzia}@cistel.com competition very different from an oligopoly market is that each primary user may or may not have unused spectrum available, so that a primary user who has unused spectrum is uncertain about the number of other competitive primary users. If there are few competitive primary users that have unused spectrum, the lower price will not be a good option. In the following subsections, we present the game theoretic approach, as well as the price techniques in CRNs.
Abstract—The dynamic behavior for spectrum management in cognitive radio networks is considered in this paper, which consists of spectrum trading and spectrum competition among multiple spectrum owners and spectrum leasers. The primary users adjust their behaviors in renting the spectrum to secondary users in order to achieve higher profits. The secondary users adjust the spectrum renting by observing the changes in the price and the quality of the spectrum. It is however problematic when the primary users and secondary users make the decisions dynamically. A three layer game theoretic approach is introduced in this paper to address this problem. The upper layer models the spectrum competition among primary users; a Bertrand game is formulated where the Nash equilibrium is considered as the solution. The middle layer models the spectrum trading between the primary user and secondary user; a Stackelberg game is formulated where the Nash equilibrium is considered as the solution. The lower layer models the dynamic selection strategies among secondary users in order to select the offered spectrum; an evolutionary game is formulated where the Nash equilibrium is the solution. Basically, the solution in each game is found in terms of the size of the offered spectrum to the secondary users and the spectrum price. The proposed game theory model is used to examine network dynamics under different levels of QoS where the actions of each user are made dynamically.
A. Game Theoretic Approaches in CRN Recently, game theory has been used in communication to model and analyze the interactive behaviors in a competitive area. Game theory is a useful tool that can be used for spectrum management in a cognitive radio network [1]. In cognitive radio network, some of the game theoretic models were proposed for different issues in CRNs such as power control [2], call admission control [3], spectrum trading [4], spectrum competition [4], interference avoidance [5], power allocation [6], spectrum sharing [7], and spectrum access [8] in cognitive radio networks. Many methods were proposed for dynamic spectrum access and allocation, including game theory [9]. Recently, some approaches were proposed for security issues [10]. Table I shows the different issues in CRN that have been modeled by game theory.
Keywords-component; Cognitive Radio Network; spectrum trading; spectrum competition; Stackelberg game; Bertrand game; evolutionary game; Nash equilibrium
I.
TABLE I.
Issue in CRN Power Management Spectrum Trading
INTRODUCTION
A Cognitive Radio Network (CRN) is a revolutionary technology, which aims at enhancing the efficiency of spectrum usage. With the support of radio’s functionality, each secondary node is able to sense the spectrum (channel). In a cognitive radio network, the spectrum can be traded between spectrum owners (Primary Users, PU) and spectrum leasers (Secondary Users, SU). The goal of this spectrum trading is to maximize the utility (Profit) of primary users while maximizing the utility (QoS level) of secondary users. There might be more than one primary user that offers an affordable price as well as a good quality spectrum and therefore, the competition will be advantageous to the primary users and secondary users. As a result, each user, either primary or secondary, has to control its strategy in order to reach the equilibrium point. This means that both the users (PU and SU) are provided with the best utility in terms of QoS and price. What makes the CRN spectrum trading and spectrum
978-1-4673-4941-3/12/$31.00 ©2012 IEEE
Interference Spectrum Sharing Spectrum access Security
B.
SUMMARY OF GAME MODELS FOR ISSUES IN CRN
Game Model Non-cooperative Supply and Demand functions Potential game, Stackelberg game Cooperative Non-cooperative Stackelberg
Solution Nash equilibrium Market equilibrium Nash equilibrium Nash equilibrium Nash equilibrium Nash equilibrium
Reference [1,2,6,27] [4,11-14] [5] [7,25] [8,24] [10,26]
Spectrum Managment Approaches in CRN The following works address the problem of spectrum management in a cognitive radio network. Spectrum trading is successfully formulated by economic models and competitive and cooperative pricing schemes are developed in [11]. In [12], hierarchical spectrum sharing is formulated as a unified market. Specifically, the pricing mechanism for the bandwidth allocations between the systems equates the supply to the
919
demand. In [13], the consumers’ demand functions are modeled and the Walrasian prices, which equate the demand to the supply for each good, are calculated. In [14], the Hackner utility function serves to obtain the demands for the frequency channels (which have fixed bandwidth sizes), where the SU’s demand for each channel is a function of its price having been adapted. In addition, the PUs’ competition for maximizing the profit is modeled. In [15], a game-theoretic model was presented, where SUs compete to share the bandwidth offered by the PU. However, [11-14, 24] propose a two stage model for spectrum management with one level of QoS. In [25], the authors propose one stage model with a multiple levels of QoS. In [15], the authors propose a two stage model with multiple levels of QoS.
the following equation in order to calculate the anticipated delay and we consider the idle slots and busy slots in this equation. It is called Probk, which represents the ratio of idle slots to busy slots. Probk = (nid * tid) / (nid * tid + nbusy*tbusy). where nid is the total number of idle periods, tid is the time of the idle period, nbusy is the total number of busy periods, and tbusy is the time when channel is busy. B. Second Level This is the middle level of QoS that consists of delay, as described in the previous section, as well as link robustness. This metric is very important in order to have a spectrum without any interruption by the PU, and to guarantee the stability of service. In cognitive radio network, this metric designates the presence of the PU. It is called ProbPU.
C. Motivations and Contributions In all the previous works mentioned in subsection B, different game frameworks to spectrum trading and spectrum competition are applied, and some of them formulate price models as game theory framework. There are few approaches that consider the effect of applying game theory with multiple levels of QoS. To the best of our knowledge, all the previous approaches, which are mentioned in subsection B, propose one or two levels of game theoretic approach as in Table II. We conclude the contributions as follows: (i) we propose a three layer (levels) game theoretic approach, the upper layer among PUs, the middle layer between PU and SU, and the lower layer among SUs, (ii) we introduce a novel function for a multiple levels of QoS for different secondary users, (iii) A new competition factor is defined, and (iv) the dynamic behavior in network is investigated under different levels of QoS. TABLE II.
ProbPU = Plink * Probk.
Spectrum Trading
PUs’ Competition
[11,13]
¥
¥
[12,19]
¥
[14,24]
¥
[15,17]
¥
[20,25]
¥ II.
SUs’ Competition
¥
¥
(2)
C. Third Level This is the highest level of QoS that consists of delay, link robustness, and Error Transmission Rate (ETR), which is a factor used to find links with higher transmission rates and lower bit error rates. The derivation of ETR starts with the measurements of the loss probability, denoted by Plink. ETR (link) = 1 / (Plink). III.
SUMMARY OF SPECTRUM MANAGEMENT MODELS FOR ISSUES IN CRN
Ref
(1)
(3)
SYSTEM MODEL
A. General Assumptions In this paper, we consider an overlay model in which we consider N primary users and M secondary users. The channels owned by PU are K. We define bi as a spectrum size for renting, which is offered by PUi. We assign a unique ID for each PU and for each SU.
QoS support
One level One level One level Multiple levels Multiple Levels
QOS FOR SECONDARY USERS
Most of the research that has been conducted in this area (QoS) assumes one type of service for SUs. Today, with an explosion in the diversity of real-time services, a better and more reliable communication is required. Moreover, some of these applications require firm performance guarantees from the PUs. In our model, we consider satisfying QoS for multiple levels of services. The three levels offered in our model are as follows:
Figure 1. Stages of our game theory model
B. Network Model Some existing works related to spectrum trading or leasing use one stage of dynamic game structure, as in [18], where the stage is between a primary user and a secondary user. Other existing works use two stages, where the first stage is between primary user and secondary user and the second stage is among
A. First Level The lowest level of QoS consists of delay, which is an important QoS parameter in wireless network. We introduce
920
and f(LQoS) is the function of QoS. This is a new function for QoS levels, which is defined as follows:
PUs [11, 17, and 19]. Other works define the second stage among the secondary users [16, 20], instead of PUs. In this section, we introduce our model with game theory in three stages; the first stage is between primary users, where we use the Bertrand game concept; the second stage is between primary user and secondary user, where the Stackelberg game concept is used; and the third stage is between secondary users, where the Evolutionary Game Concept is used. These three stages are illustrated in Figure 1, and we relate the game theory approaches to layers as in Figure 2.
f (LQoS) = QoSlevel * ln(b).
where QoSlevel may take a value of 1, 2 or 3, depending upon the QoS level requested, and b is the size of the spectrum traded to the secondary user. A new coefficient Į (0 < Į < 1), called the Competition Factor, is defined. A PU selects the value of this coefficient by considering the number of PUs offering that channel, the usage period of the offered channel, and the probability of its presence. The competition factor function is described in the following equations: Į = 1, 0.1 < Į < 0.99, Į = 0,
C. Spectrum Competition using Bertrand Game (Upper Layer) In Bertrand’s game [21], a player changes its behavior if it can increase its profit by changing its price, on the assumption that the other players’ prices will remain the same and their outputs will adapt to clear the market. Simply, the unit cost of production is a constant C, the same for both players (e.g. competitive PUs). Bertrand’s game has a unique Nash equilibrium, in which each firm’s price is equal to C. This concept is applied to model the competition among primary users. In this competition we assume that ϕ channels are offered by N primary users where each PUi can offer ϕi channels at a cost P. In our model, the primary users set different prices that allow secondary users to select the lowest price among the offered prices. Each primary user sets the price as given in equation 4) based on α value. Each PU’s objective is to maximize its utility (profit) by getting more customers (secondary users). Ptotal = α ∗ PPU + C Ptotal =
;Į = 1
(4a)
; 0.1 < Į < 0.99
(4b)
;Į = 0
(4c)
the normal case when there is no available channel
α = (ψ+ϕ+χ+τ) / ((1+ψ)τ+ψ+ϕ)
(7a) (7b) (7c)
(8)
where ψ is the number of PUs offering the spectrum, ϕ is the number of channels, χ is the number of SUs requesting a specific channel, and τ is the usage time of the channel. D. Spectrum Trading using Stackelberg Game (Middle Layer) In the Stackelberg game [21], it is assumed that at least one of the firms in the market is able to pre-commit itself to a particular level of supply before other firms have fixed their level of supply. Other firms observe the leader’s supply and then respond with their output decision. The firms that are able to initially pre-commit their level of output are called the market leaders and the other firms are called the followers. This concept is applied into our model specifically between the primary user and the secondary user. Hence, the primary user is the leader and the secondary user is the follower. For the leader (PU), we define QL and IL where QL is the strategy set and IL is the information set. For the follower (SU), we define QF which is the strategy set. According to Stackelberg game model, IL = QF, at the start, any strategy is chosen by the leader as the initial strategy, qLo, which belongs to the subset qL of QL. The follower will choose the reaction strategy qF*, which belongs to QF in order to maximize its own payoff UF.
The α is a new factor called the Competition Factor, which is defined in equations 7 and 8. PPU is the price function, which is defined as follows: PPU = W * ln (b) + χ * b* P + χ * f (LQoS).
when the number of primary users = 1
If no other PU offers the requested channel, Į = 1, the price remains the same. If there is no available channel, Į = 0, then there is no offered price. However, if there is a number of primary users with available channels for the request, as well as the usage period of the offered channel, the PU chooses an appropriate value for Į (0.1 < Į < 0.99) to reduce the total price, as per the following equation, this equation is derived based on study all parameters that affect the competition area:
Figure 2. Hierarchy of our game theoretic approach
Ptotal = PPU
(6)
(5)
where W is the data transmission rate, b is the size of the spectrum traded to the secondary user, χ is the number of customers (SUs that request one channel or more, P is the unit price per channel for the spectrum traded between PU and SU,
qF* = max UF (qF; qLo)
(9)
qL* = max UL (qL; qF*)
(10)
In equation (9), qLo אqL. After knowing the reaction strategy qF* of the follower, the leader will announce a strategy qL*, which belongs to the subset qL of QL, where the leader
921
where W is the data transmission rate, b is the spectrum traded, P is the price unit, and f(LQoS) is the QoS function. Replacing f(LQoS) from equation 6 gives the following, where QoSlevel is the QoS level.
maximizes its payoff UL. In equation 10, qF* אqF. In our model, the primary user PUi’s strategy set QL is defined as QL = {Pbi, i = 1, 2,……, N}, where N is the total number of PUs and Pbi is the total price, as in equations 4a) and 4b), for the allocation of spectrum b, which is used before. The secondary user SUj’s strategy set QF is defined as QF = {SjQoS, j = 1, 2, 3, …M and QoS =1, 2, or 3}, where M is the total number of SUs and SjQoS is the offered spectrum for the requested QoS level. The primary user chooses the price Pbi for spectrum b, while the secondary user selects the best size spectrum with a higher level of QoS, which optimizes its own utility function UF. In this game the PU first calculates the most probable response SrQoS (r = 1, 2, 3,……, M) from the secondary user given any of its prices Pbn (n = 1, 2… N). UL (P*; S*QoS) UL (Pbr; SrQoS)
UL = W * ln (b) + P * b – QoSlevel * ln (b)
Secondary user’s utility function consists of the spectrum size and the QoS level. For SU, the utility function is the following: UF = W * QoSlevel * ln (b) – QoSlevel2 * ln (b) * P2 * b
(14)
The derivation of SU’s utility provides the best value of spectrum size in term of QoS. Let (dUF / dQoSlevel = 0), we have:
(11)
QoSlevel = (W / 2 * b * P2)
Equation 11 shows that the reaction price is always greater than the initial price, as well as the reaction spectrum size with QoS, which is greater than the response spectrum size with QoS.
(15)
Which means, given strategy of choosing price unit P chosen by PU, the SU’s best response is to set the QoS level QoSlevel as in equation 14. Substituting the value of QoSlevel from equation 15 in equation 13 for the utility function of PU, we get:
E. Selection’s strategies using Evolutionary Game (Lower Layer) The dynamic competition of spectrum selection among secondary users is modeled as an evolutionary game [21]. This is the lower layer in our hierarchical model. This evolutionary game was originally established to describe the behavior of biological agents [22]. It was also used to model the behavior of human beings in the society [23] and entities in a market environment [24]. The strategy adaption of this game is subject to control from the primary users in terms of the size of the spectrum leased to provide spectrum for the secondary users. In addition, the primary users observe the spectrum selection of secondary users and decide the spectrum size to be leased to SUs dynamically. The secondary user can select the spectrum dynamically according to the perceived utility, which depends on the spectrum, the price, and the QoS level. We assume that a secondary user can access the spectrum from only one PU at a time. The strategy of each SU is the selection of the spectrum owner (PU) with lower price and better QoS. The utility function of each SU is a function of the spectrum, bi, and the price, Pi as discussed in the following section. When we form the utility function for SU, we take into account the following concerns; (i) a secondary user chooses the primary user that will provide the best spectrum in terms of price and level of QoS and (ii) a secondary user observes the behaviors of other users and changes the decision on spectrum selection.
UL = W * ln (b) + P * b – (W / 2 * b * P2) * ln (b)
(16)
The derivation of PU’s utility produces the best price to SU. We substitute the derivation of the SU’s utility for the PU’s utility. Now let (dUL / db = 0) we get the following equation in terms of P: 2*P3 *b2 + P2*W*b + W * (ln (b) – 1) = 0 IV.
(17)
SIMULATION RESULT
In the simulation, the cognitive nodes are randomly placed in 300x300 m2 and the transmission range is 150 meters. The length of the packet size is 64Kbit. The total number of channels is 16, the transmission rate is 100kpbs, the total number of SUs is 12 users, and the number of PUs is 2. PU1 has 10 channels and PU2 has the remaining 6 channels. The usage period is fixed for all users, and is 6 hours. The number of SUs that requests channels from PU1 is 7 and the remaining request channels from PU2. We calculate the competition factor for each primary user as per equation 8. Figure 3 shows the price unit response functions and the Nash equilibrium of the competitive pricing of two primary users with fixed spectrum. The figure shows the existence and uniqueness of the Nash equilibrium. Moreover, Figure 3 shows that the slope of the Price unit strategy of primary user PU1 is always greater than one. On the other hand, the slope of the Price unit strategy of primary user PU2 is always less than one. Hence, the unique intersection point that is called Nash equilibrium. Here, the offered price for PU1 is P1* = 9.61873 and for PU2 is P2* = 10.97288. Basically, when one primary user changes its strategy, for example the offered spectrum and
F. Utility Function of Primary User and Secondary User In our system, the primary user’s utility function consists of five parts: (i) satisfaction of its own transmission, (ii) revenue from selling spectrum to the secondary user, (iii) profit, (iv) the corresponding payment to the base station, and (v) the performance loss due to the shared spectrum with the secondary users. UL = W * ln (b) + P * b - f (LQoS)
(13)
(12)
922
price unit, the equilibrium point to achieve the highest net payoff of the other primary user changes.
price is increasing. Meanwhile, the strategy of the SU, which is the spectrum size with the QoS level to be used, is increasing, and its utility increases as well.
40 35
Price Unit 2
30 25 20 15
PU2 strategy PU1 strategy Nash Equlibrium
10 5 0
0
5
10
15
20
25
30
35
40
Price Unit 1 Figure 3. Existence of Nash equilibrium. Figure 5. PU and SU utility in QoS level 1
The best response of both primary users in terms of spectrum size is shown in Figure 4; the size of the offered spectrum in each level is changeable, while the price unit is fixed. If primary user 1 increases the size of offered spectrum, the demand will be increased by customers (SUs). However, the secondary user observes the other offered spectrum by a primary user 2 in order to select the best one in term of QoS and in term of price.
According to the two-dimensional plane indexed by two decision variables, price and spectrum size, the PU calculates the equilibrium contract (qL*; qF*) according to equations (14), (15), and (16) respectively. Then it waits until the SU announces its policy qF as qF*. In our simulation, we have calculated (qL*; qF*) = (11.5; 800) for first level of QoS, (qL*; qF*) = (13; 1440) for the second level of QoS, and (qL*; qF*) = (16; 1880) for the third level of QoS. The first value represents the best price unit for the PU and the second value represents the best spectrum size with QoS level.
1400 PU2 strategy PU1 strategy PU2 strategy PU1 strategy PU2 strategy PU1 strategy
1200
Spectrum Traded
1000
(QoS (QoS (QoS (QoS (QoS (QoS
level level level level level level
= = = = = =
1& 1& 2& 2& 3& 3&
Price unit Price unit Price unit Price unit Price unit Price unit
= = = = = =
8 8 9 9 10 10
800
600
400
200
0
0
200
400
600 800 Spectrum Traded
1000
1200
1400
Figure 4. PU1 and PU2 strategies in different QoS level and the Nash equilibrium for each level.
Figure 6. PU and SU utility in QoS level 2
Basically, if the spectrum size is increased, that means the total price will be increased, which means the profit will be increased. As shown in Figure 4, the Nash equilibrium is the intersection between each PU strategy with other PU in each level. In Figure 4, we have three Nash equilibrium points wherein each level has one point. For the sake of simplicity, we only show the size of the offered spectrum in Figure 4 and we only show the spectrum price unit strategies in Figure 3.
Note that the utility function of the PU does not achieve maximum when price unit = 11.5 in QoS level 1 or price unit = 13 in QoS level 2 or price unit = 16 in QoS level 3. However, this point called the Nash equilibrium considers the best response for both users, either PU or SU. This point (Nash equilibrium) gives satisfaction to primary user to serve maximum customers while its profit is still acceptable.
The payoff (utility) function of each player in QoS level 1, 2, and 3 is illustrated in Figures 5, 6 and 7 respectively. It shows that the utility function of PU increases, while channel
923
[8]
[9]
[10]
[11]
[12] Figure 7. PU and SU utility in QoS level 3
V.
[13]
SUMMARY
We have modeled the dynamic behavior of spectrum trading and leasing. In addition, we have presented multiple levels of QoS for different secondary users. We presented three levels of game theoretic approaches in this paper; the first game is between the PUs, the second game is between PU and SU, and the third game is between SUs. For the first game, a Bertrand game is formulated. For the second game, a Stackelberg game is formulated. For the third game, an Evolutionary game is formulated. In all games, the challenge is to obtain Nash equilibrium. The simulation results show the Nash equilibrium for each game under different system parameters and under different QoS levels. The major focus for future work will be on applying the model into routing algorithm to observe the impact of our model in the efficiency of network. In addition, we will increase the number of QoS levels. Moreover, we plan to develop our model in game theory part to monitor the behaviors of users in order to detect if there is a malicious user in the system.
[14]
[15]
[16]
[17]
[18]
[19]
REFERENCES [1]
[2]
[3]
[4]
[5]
[6]
[7]
[20]
J. W. Huang and V. Krishnamurthy, “Game theoretic issues in cognitive radio systems,” Journal of Communications, vol. 4, no. 10, pp. 790–802, November 2009. X. Zhang, and C. Li, “The security in cognitive radio networks: a survey”, International Wireless Communications and Mobile Computing Conference (IWCMC), 2009, pp. 309-313. E. Hossain, D. Niyato, I.K. Dong, “Game Theoretic Approaches for Multiple Access in Wireless Networks: A Survey”, Communications Surveys & Tutorials, IEEE, vol. 13, no. 3, pp. 372 – 395, 2011. Z. Li, L. Gao, X. Wang, X. Gao, and E. Hossain, "Pricing for uplink power control in cognitive radio networks," IEEE Transactions on Vehicular Technology, Special Section on "Achievements and the Road Ahead: The First Decade of Cognitive Radio", vol. 59, no. 4, May 2010. Y. Chen, K. Teo, S. Kishore, and J. Zhang,” A Game-Theoretic Framework for Interference Management through Cognitive Sensing”, IEEE ICC, 2008, pp. 3573-3577. E. Del Re, G. Gorni, L. Ronga, and R. Suffritti, “A Power Allocation Strategy using Game Theory in Cognitive Radio Networks”, Game Theory for Networks, 2009, pp. 117-123. Ma Liang, and Zhu Qi, “An Improved Game-theoretic Spectrum Sharing in Cognitive Radio Systems”, Communications and Mobile Computing (CMC), 2011, pp. 270-273.
[21] [22] [23]
[24]
[25]
[26]
[27]
924
Y. Xing, R. Chandramouli, and C. M. Cordeiro, “Price Dynamics in Competitive Agile Spectrum Access Markets,” IEEE Journal on Selected Areas in Communications, vol. 25, no. 3, pp. 613-621, 2007. N. Nie, and C. Comaniciu, “Adaptive channel allocation spectrum etiquette for cognitive radio networks,” in Proc. IEEE DySPAN, 2005, pp. 269-278 Saed Alrabaee, Mahmoud Khasawneh, Anjali Agarwal, Nishith Goel, Marzia Zaman, “Higher layer issues in cognitive radio network”, Proceeding ICACCI '12 Proceedings of the International Conference on Advances in Computing, Communications and Informatics Pages 325330, ISBN: 978-1-4503-1196-0, Aug,2012 D. Niyato and E. Hossain, “Market-equilibrium, competitive, and cooperative pricing for spectrum sharing in cognitive radio networks: Analysis and comparison,” IEEE Trans. Wireless Commun., vol. 7, no.11, pp. 4273–4283, Nov 2008. D. Niyato and E. Hossain, “A microeconomic model for hierarchical bandwidth sharing in dynamic spectrum access networks,” IEEE Trans. Comput., vol. 59, no. 7, pp. 865–877, Jul 2010. R. Mochaourab, and E. Jorswieck, “Exchange Economy in Two-User Multiple-Input Single-Output Interference Channels,” IEEE Journal of selected topics in signal processing, vol. 6, no. 2, April 2012. H. Mohammadian and B. Abolhassani, “A new price-based spectrum sharing algorithm in cognitive radio networks”, Software, Telecommunications and Computer Networks (SoftCOM), 2010, pp. 254-259. D. Niyato and E. Hossain, “A Game-Theoretic Approach to Competitive Spectrum Sharing in Cognitive Radio Networks”, Wireless Communication and Networking Conference, WCNC 2007. D. Niyato, P. Wang , and Z. Han, “ Dynamic Spectrum Leasing and Service Selection in Spectrum Secondary Market of Cognitive Radio Networks,” IEEE Transactions on Wireless Communications, vol.11, no.3, pp.1136-1145, Mar 2012. 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”, IEEE TMC, vol. 8, no. 8, pp. 1009-1022, Aug 2009. G.S. Kasbekar, and S. Sarkar, “Spectrum Pricing Games with Spatial Reuse in Cognitive Radio Networks,” IEEE J. Selected Areas in Comm., vol. 30, no. 1, pp.153-164, Jan 2012. L. Chen, S. Iellamo, M. Coupechoux, and P. Godlewski, “An auction framework for spectrum allocation with interference constraint in cognitive radio networks,” 2010 IEEE INFOCOM. H. Xu, J. Jin, and B. C. Li, “A secondary market for spectrum,” in Proc.2010 IEEE INFOCOM. M. J. Osborne, An Introduction to Game Theory, Oxford Univ. Press, 2003. R.A. Fisher, The Genetic Theory of Natural Selection. Clarendon Press, 1930. M.D. Sahlins, Evolution and Culture. Univ. of Michigan Press, 1970. C. Alo´s-Ferrer, A.B. Ania, and K.R. Schenk-Hoppe´, “An Evolutionary Model of Bertrand Oligopoly,” Games and Economic Behavior, vol. 33, pp. 1-19, 2000. G.I. Alptekin and A.B. Bener,” Spectrum trading in cognitive radio networks with strict transmission power control”, European Transaction on Telecommunication,2011 M. Parzy and H. Bogucka, “QoS Support in Radio Resource Sharing with Cournot Competition,” 2nd International Workshop on Cognitive Information Processing (CIP), 2010, pp.93-98. Saed Alrabaee, Anjali Agarwal, Devesh Anand, and Mahmoud Khasawneh, " Game Theory for Security in Cognitive Radio Networks," International Conference on Advances in Mobile Network, Communication and its Applications (MNCAPPS), 2012, Pages 60-63 Mahmoud Khasawneh, Anjali Agarwal, Devesh Anand, and Saed Alrabaee, " Sureness efficient energy technique for cooperative spectrum sensing in cognitive radios," International Conference on Telecommunications and Multimedia (TEMU), 2012, Pages 25-30
