A Power Allocation Strategy using Game Theory in Cognitive RadioNetworks

Enrico Del Re and Gherardo GorniDepartment of Electronics and

Telecommunications (DET)University of Florence

Florence, Italy{enrico.delre, gherardo.gorni}@unifi.it

Luca Ronga and Rosalba SuffrittiNational Inter-University Consortium for

Telecommunications (CNIT)University of Florence

Florence, Italy{luca.ronga, rosalba.suffritti}@cnit.it

Abstract— The Cognitive Radio approach can be consideredas a promising and suitable solution to solve in an efficient andflexible way the increasing and continuous demand of servicesand radio resources. This paper investigates how the adoptionof a cognitive radio strategy can help in the coexistence problemof two wireless networks operating on the same spectrum of fre-quencies. A DVB-SH based satellite network will be consideredas primary system, while an infrastructured wireless terrestrialnetwork will constitute the cognitive radio based secondarysystem. In this work it will be presented a power resourceallocation technique based on Game Theory, considering mainlyPotential Games. We will show the proposed approach issuitable for distributed implementation, furthermore it providesperformances comparable to an heuristic allocation methodrepresenting the optimum allocation. The comparison betweenthese two resource allocation methods will be provided as resultof this work.

I. INTRODUCTION

Because of the increasing and continuous demand ofservices and radio resources, the traditional communicationsystems which imply an a priori association of the frequencyband, the service assigned to it and the used technology,need to become much more flexible, efficient and easy-to-usedynamic systems able to cope with the requirements and con-straints of the environment and the users. A Cognitive Radio(CR) approach can be considered as a promising and suitablesolution to solve this problem. The term cognitive radio wasintroduced in [1] with reference to a communication systemable to observe and learn from the surrounding environmentas well as to implement and adapt its own transmissionmodalities also to user requirements. The concept of CR isoriginated from the contrast between an increasing demandof broadband services and the scarcity of radio resources.Recent studies of the FCC Spectrum Policy Task Force[2] demonstrated that a large amount of licensed bands areunder-utilized, i.e., a lot of spectral resources are reserved forspecific services, but, actually, they remain unused for mostof the time or in several locations. From these studies, thepossibility of a CR is envisaged, i.e., a system able to sensethe electromagnetic environment (spectrum sensing), detectthe spectral resources actually occupied in a given temporalinterval and in a given location, and use the free bands(spectrum holes) for its own communications [3]. The searchfor available resources is not limited to spectrum portionsdedicated to unlicensed communications, but is also extended

to licensed bands. This paper shows the potential benefits ofthe adoption of a cognitive radio strategy to the coexistenceproblem. The developed cognitive radio strategy it has beenformulated according the mathematical discipline of GameTheory, with particular reference to Potential Games [4].

Game Theory has already been considered in radio re-source management of wireless networks as well as in thedevelopment of access control and routing techniques. In[5], an extensive study about the adoption of Game Theorybased methods in wireless networks has been done. PotentialGames applications in CDMA power control problems havebeen investigated in [6] and [7], while in [8] this mathe-matical framework has been used to model an interferenceavoidance scheme. However it is in Cognitive Radio net-works analysis that Potential Games are being used widely,as firstly noticed in [9]. The PhD thesis [10] investigatesvarious applications in Cognitive Radio contexts; it proposesas well a distributed frequency selection algorithm developedthrough Potential Games. In [11], a spectrum sharing game isformulated in order to perform a distributed adaptive channelallocation in a CR network, while in [12] the potential gameframework encompasses both power control and channelselection.

In this paper a Game Theory based strategy is employedto perform power allocation in an OFDM Cognitive Radionetwork. Such power allocation is aimed to the up-link com-munication toward the CR local base station, in a scenariosimilar to the one envisaged by IEEE 802.22 Draft Standardfor WRANs [13]. The main contribution of this work is toachieve the solution of the Potential Game in closed form forthe case of a two-carriers OFDM system, and then, to extendsuch solution to the considered OFDM system. It will bealso showed in Section III-A, how the considered potentialfunction is readily available for distributed implementation.

The paper is organised as follows: in Section II thecoexistence scenario will be described while Section III willexplain the Game Theory based framework. In the Section IVthe achieved results and the comparisons with the heuristicallocation method will be presented. The concluding remarkswill be given in Section V.

978-1-4244-4177-8/09/$25.00 ©2009 IEEE 117

Secondary SystemBase Station

Primary SystemSatellite

Primary SystemNetwork

Secondary SystemWireless Network

Fig. 1. The considered scenario

II. THE COEXISTENCE SCENARIO

The considered scenario is constituted by a primary li-censed system and a secondary system whose terminalsimplement cognitive resource allocations, i.e. the CognitiveRadio based network. In particular it has been considereda mobile satellite system compatible with DVB-SH stan-dard [14] as primary system in the envisaged scenario. Assecondary system, it has been considered an infrastructuredwireless terrestrial network, i.e. all the secondary terminalscommunicate with a local base station. The two systemswork in the L band frequency spectrum (0,39-1,55 GHz),which in this context is considered as the main radio re-source. In Fig.1 it is showed the proposed scenario. Forboth systems it has been supposed, as representation inthe frequency domain of the transmitted OFDM symbols,a complex vector of length K, which is the actual numberof subcarriers used to transmit the signal. Considering theDVB-SH standard specifics, K has been set equal to 853.

The two communication systems require the definition oftwo distinct channel models. For the satellite based primarysystem at L band it has been considered the Lutz et al.propagation model [15]. This model is based on a twostate, GOOD-BAD, Markov chain for the fading process.Both slow fading and fast fading are kept into account.In particular, slow fading events due to large obstacles aremodeled as a finite state machine; fast fading events instead,caused by irregular obstacles (e.g. vegetative shadowing) andmultipath phenomena, they have been superimposed as arandom variation with a specific probability density functionfor each state of Markov chain. The propagation modeldescribed shows a flat frequency response. A path loss termhas been also considered in the primary system propagationmodel. The path loss is modeled by:

L = 10 log10

(4πdλ0

)αdB (1)

where α is the attenuation depending on the distance, λ0

is the wave-length at central frequency of the consideredfrequency band, and d is the distance between the transmitterand the receiver.

The propagation channel model considered for the ter-restrial secondary system keeps into account mainly twophenomena: the path loss and the multipath fading. Thepath loss term is given by the expression used into thechannel model for the primary system, i.e. the equation(1). The multipath fading due to the operating environmenthas been considered through a tapped-delay model withRayleigh distributed coefficients. The power delay profile isexponential as follows:

σ2n = e−βn (2)

where σ2n is the variance of the n-th coefficient and β is

computed for a normalized mean power response. In thetime domain the propagation model adopted exibits a finiteimpulse response of l samples resulting in a frequencyselective channel response.

III. THE POWER ALLOCATION GAME

The Game Theory based allocation approach in the sec-ondary network implies the definition of a proper framework.Indeed Game Theory, if formulated in the correct way, canlead to a stable equilibrium point known as Nash Equilibriumof the game. Such framework needs the definition of theplayers involved in the game that in this case are M cognitiveradio terminals. Furthermore a set of available strategies hasto be defined. The strategies are the actions each playercan choose from in order to adapt both to the operatingenvironment and to the opponent players choices. It hasbeen considered as possible strategies for the players, theamounts of power that secondary terminals allocate on theK OFDM subcarriers to communicate effectively with thesecondary system base station. The strategies actually playedcan be represented through a power allocation matrix, PII,containing all the amounts of power the cognitive terminalsallocate. The power allocation matrix has some mandatoryconstraints to respect. In fact each cognitive radio terminalhas to keep into account two constraints per subcarrier plusa general one due to the maximum power available on boardfor transmission, P II

max. The constraint on the maximum

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power for the i-th user can be stated in the following way:

K∑k=1

p IIi (k) 6 P II

max (3)

where the term p IIi (k) stands for the amount of power the

i-th cognitive user is allocating on the k-th subcarrier. As faras it is concerned about the constraints on each subcarrier,two constraints are needed: one establishing the minimumamount of power to allocate in order to respect the secondarysystem target BER (i.e. the lower bound), the second oneis needed in order to guarantee and protect the primarysystem functioning and it establishes the maximum amountof power on a certain subcarrier (i.e. the upper bound).The lower bound relative to the k-th subcarrier for the i-th terminal is given by equation (4) at the top of the nextpage, where SINR II

min is the signal-to-noise-interferenceratio corresponding to the maximum BER tolerable by acognitive terminal in order to use a QPSK modulation, hII

i (k)is the channel coefficient between the i-th terminal and thebase station of the secondary network relative to the k-thsubcarrier,

∑Mj=1,j 6=i h

IIj (k)p II

j (k) is the interference on thek-th subcarrier due to the other terminals of the secondarynetwork, c I→II(k)P I(k) is the disturb due to the transmissionof the satellite system on the k-th subcarrier, and N(k) is thenoise power on the k-th subcarrier. The former constraintsdefine a multidimensional real subspace within the availablestrategies stay. Given such particular constraints the realsubspace considered is compact and convex.

The upper bound relative to the k-th subcarrier for the i-th terminal is given by equation (5), where SINR I

min isthe signal-to-noise-interference ratio corresponding to themaximum BER tolerable by receivers of the primary system,h I(k) is the coefficient describing the channel betweenthe satellite and the primary terminal nearest to the i-thsecondary user, P I(k) is the power the primary systemsatellite is transmitting on the k-th subcarrier, c II→I

i (k) is thechannel coefficient considering the impairment effects of thepower allocation p II

i (k) on the primary receiver nearest to thei-th terminal. In the case inequalities (4) and (5) don’t havea common interval of real values, it is clear no power willbe allocated on that subcarrier. This possibility identifies asituation where a primary system receiver is particularly nearto a cognitive terminal or it is particularly sensible on somesubcarriers to impairments effects due to cognitive terminals.

Finally opportune utility functions have to be defined.They represent the future benefit a player will achieveadopting a certain strategy, i.e. power allocation. In this case,the utility functions map the power allocation strategies forthe i-th player into a real number considering also the powerallocated by other terminals, i.e. ui : P II → R. Makingthe assumption of rationality for all implies each player willchoose the strategy producing the highest utility. Thus, itis a coherent hypothesis since the players are electronicdevices suitably programmed. The choice of utility functionswith specific properties has a crucial importance in thedevelopment of the game and especially in the existence

of Nash Equilibrium Point. The utility function for the i-thcognitive player has been defined in the following way:

ui(P II) =K∑k=1

B log2

[1 + SINRi

(P II(k)

) ]−

M∑m=1

c II→Im (k)p II

m(k) (6)

where SINRi(PII(k)

)represents the signal-to-noise-

interference ratio the i-th terminal is achieving on the k-th subcarrier at the base station of the secondary network.SINRi

(PII(k)

)is so defined:

SINRi(P II(k)

)=

=h IIi (k)p II

i (k)∑Mj=1,j 6=i

h IIj (k)p II

j (k) + c I→II(k)P I(k) +N(k)(7)

In the previous equations PII is the power allocation matrixconcerning all the secondary terminals. PII has dimension-ality KxM , where K is the number of subcarriers and M isthe number of secondary terminals. Moreover in equation (6)the term B is the channelization bandwidth value consideredfor every subcarrier. The sum of the terms c I→II(k)P I(k) andN(k) in (7) has been assumed the same for all the secondaryusers; the reason is due to the mathematical conditionsrequired to develop analitically the game. The chosen utilityfunctions have as domain the real subspace given by thepower constraints introduced before. It has to be pointed outthe utility functions chosen have an attractive property: theyare twice differentiable. Concerning the meaning of (6), thefirst part is the sum of capacity values achieved by i-th playerwith the power allocation vector p II

i : higher this sum willbe, higher the bit rate achieved by a terminal in the uplinkwill be. The sum

∑Mm=1 c

II→Im (k)p II

m(k), in the second partof (6), considers instead the disturbs caused on the primaryusers and it counterbalances the secondary terminal tendencyto use as much power as possible.

The defined game will be since now referred syntheticallyas Γ = (M,P, {ui}). In order to determine the solutionpoint of the game Γ, i.e. the Nash Equilibrium, the utilityfunctions have been modified to highlight the consideredgame is a Potential Game. According [4], a game, thatrespects the definition of Potential Game, has a potentialfunction encompassing the gains all the players perceivechanging their own strategy. The only technical requirementto assure the convergence of the game is that at eachstage at most one player can change its strategy. The lastcondition is mandatory in order to achieve a best responsestrategy increasing at each stage the value of the potentialfunction. In the considered scenario the requirement canbe fullfilled, imposing to terminals to allocate power in around robin way for example. For potential games it hasbeen also demonstrated that the Nash Equilibrium point iscorresponding to the strategies maximizing such potentialfunction. Potential games have been analyzed for applica-tions in wireless networks by MacKenzie and DaSilva in

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p IIi (k) >

SINR IImin

(∑Mj=1,j 6=i

h IIj (k)p II

j (k) + c I→II(k)P I(k) +N(k))

h IIi (k)

(4)

p IIi (k) 6

h I(k)P I(k)SINR I

mincII→Ii (k)

−

∑Mj=1,j 6=i

c II→Ij (k)p II

j (k)

c II→Ii (k)

− N(k)c II→Ii (k)

(5)

[16]. Afterwards some interesting results in [16] will beused to prove that Γ is a potential game and to identify theassociated potential function.

Through opportune manipulations over the utility func-tions, it is possible to show the following condition issatisfied for all i and j:

∂2ui∂p II

i ∂pIIj

=∂2uj

∂p IIi ∂p

IIj

(8)

where pi and pj are the power allocation strategies on thek subcarriers made by respectively the i-th and j-th users.A property required to satisfy the former condition is thetwice differentiability of utility functions, but the chosenset owns it as already stated. In [4] authors have presentedequation (8) as mandatory condition for a game to be anExact Potential Game. The considered game for cognitivesecondary terminals will be then managed as a potentialgame. It has to be underlined equation (8) is verified becauseof the former assumption about the sum of c I→II(k)P I(k)and N(k).

The potential function V : P II → R for the game Γ hasto be, for all players and strategies, such that:

V (p IIi ,p

II−i)− V (p′ IIi ,p

II−i) = ui(p II

i ,pII−i)− ui(p′

IIi ,p

II−i)(9)

where p IIi and p′ IIi are two different power allocation vectors

for the i-th user, while p II−i is the power allocation matrix

for all the opponents of the i-th user in the game. Exploitinga result reported in [16] about the definition of potentialfunction for exact potential games, it has been determinedthe potential function V (P II) for the proposed game. Thisfunction is given by:

V (P II) =K∑k=1

B log2

[N(k) + c I→II(k)P I(k) +

+M∑m=1

h IIm(k)p II

m(k)]−

M∑m=1

c II→Im (k)p II

m(k) (10)

A. The solution of the game

The definition of the potential function implies that thesolution of Γ can be found maximizing V (P II). Such powerallocation matrix, representing a profile of strategies use byplayers, will lead to the Nash Equilibrium. It has to bepointed out that we aren’t dealing anymore with a gamein the classic mathematical sense, in fact the definition ofa potential function allows to treat Γ as an optimization

problem with several constraints given by equations (3), (4)and (5). Furthermore this optimization problem is a convexproblem since the potential function V (P II) is convex andthe real subspace defined by constraints is convex as well.Recalling the assumption that only one player can change itspower strategy at a certain time, we have that the value ofpotential function will increase step by step till the maximumvalue will be reached. This means the maximization of (10) iscarried out by each cognitive terminal optimizing a potentialfunction where the terms depending on the other playersare considered constant. Then the potential function eachcognitive terminal will maximize will be:

V (p IIi ) =

K∑k=1

B log2

[αi(k) + h II

i (k)p IIi (k)

]−

−c II→Ii (k)p II

i (k)− βi(k) (11)

where αi(k) and βi(k) are costant terms, for the k-thsubcarrier, of the function relative to the i-th user. The termsαi(k) and βi(k) are actually the following quantities:

αi(k) = N(k) + c I→II(k)P I(k) +

+M∑

m=1,m6=i

h IIm(k)p II

m(k) (12)

βi(k) =M∑

m=1,m6=i

c II→Im (k)p II

m(k) (13)

In order to show a fully distributed resource allocationtechnique has been achieved, equations (12) and (13) need anexplanation. In (12) there are three terms at second memberof the equation, the first two terms, N(k) and c I→II(k)P I(k),give a constant sum as formerly assumed for mathematicalnecessities. The third term

∑Mm=1,m6=i h

IIm(k)p II

m(k) is actu-ally the interference the signal from the i-th user is perceivingat the base station. In order to achieve a fully distributedtechnique, the value of this sum can be communicatedeach time by the base station to the terminal performingthe allocation during the downlink. Differently the term∑Mm=1,m6=i c

II→Im (k)p II

m(k) in (13) encompasses the channelstate coefficients, c II→I

m , between secondary terminals andtheir relative nearest primary receiver. It is reasonable to as-sume that each secondary terminal achieve the channel stateinformation regarding its nearest primary devices throughspecific sensing and monitoring activities. Then it can beassumed the estimation done is communicated to secondarysystem base station as information in the packet overhead.

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The solution vector for the i-th terminal, p IIi , can be found

or with iterative methods, like gradient based methods, oreven in closed form through Lagrange multipliers mehod,[17], when the value of K is low. In fact the complexity ofthe problem is asymptotic with 2 · 3K . Since the number ofcarriers K constitutes an issue for the optimization, we havefound the closed form solution when K = 2 and we haveextended to higher number of subcarriers, K � 2, througha Divide and Conquer based algorithm.

IV. SIMULATION RESULTS

The whole system has been simulated in MatLabr envi-ronment. Three different working points, in terms of Eb/No,have been considered for the primary system receivers, i.e.10 dB, 15 dB, 20 dB. The working points represent threealternatives for the primary system operations. The primaryreceivers achieved rate depends only on the Eb/No, since thesecondary system allocation strategy has always to preservethe primary system users. As far as secondary system isconcerned, different BER target values have been consideredin the range (5 · 10−5 ÷ 10−2) for the secondary systemterminals.

10

20

30

40

30

210

60

240

90

270

120

300

150

330

180 0P1P2

CR1CR2

CR3 CR4

Primary System TerminalsSecondary System TerminalsSecondary System Base Station

Fig. 2. The terminals placement.

The scenario considered for the simulations is the onerepresented in Fig.2: two primary system terminals andfour cognitive radio terminals located around the secondarysystem base station at the centre of scenario. The powerallocation has been performed over the 853 subcarriers men-tioned in Section II. Fig.3 shows the actual BER achievedby cognitive terminals. All the curves are widely below theimposed target BER. The target BER required in advance hasbeen satisfied with a relevant margin. In Fig.4 the aggregaterate of the cognitive radio network has been plotted as

0 0.5 1 1.5 2 2.510-7

10-6

10-5

10-4

10-3

10-2

Rate [Mbps]

BE

R

Receiver no.1Receiver no.2Receiver no.3Receiver no.4

Fig. 3. The Bit Error Rate actually achieved at the base station by secondaryterminals as function of the average rate.

10-5 10-4 10-3 10-21.5

2

2.5

3

3.5

4

4.5

5

Secondary System Target BER

Rat

e [M

bps]

QPSKMultiLevel QAM

Fig. 4. The aggregate rate of the secondary system as function of theimposed target BER.

function of the target BER a-priori imposed. In the graphtwo aggregate rates have been compared: a first one achievedusing the QPSK as originally assumed, and a second oneachieved employing the multilevel QAM modulation with2, 4 or 6 bits transmitted per symbol according the signal-to-noise ratio obtained at the base station. Multilevel QAMdemonstrates to be more efficient than the simple QPSK, andthen it will be considered later on for the comparisons withan heuristic allocation algorithm. In Fig. 4 is also evident asecond aspect, the graph in fact shows how the aggregaterate rises up relaxing the requirement for the secondarysystem target BER. This trend is valid also for each cognitiveradio. The two graphs in Fig.5 show the signal-to-noise-and-interference ratio of receivers no.1 and no.4 over theconsidered frequency band. Cognitive terminals no.2 and

121

390 392 394 396 398-100

-80

-60

-40

-20

0

20

40Receiver no.1

Frequencies [MHz]

SIN

R [d

B]

390 392 394 396 398-120

-100

-80

-60

-40

-20

0

20

40Receiver no.4

Frequencies [MHz]

SIN

R [d

B]

Fig. 5. Signal-to-noise-interference ratio versus frequency for the secondaryterminals no.1 and no.4

10-5 10-4 10-3 10-22

3

4

5

6

7

8

9

10

Secondary Target BER

Rat

e [M

bps]

Game Theory ApproachHeuristic Approach

Fig. 6. The comparison between game theoretic approach and heuristicapproach in terms of aggregate rate.

no.3 have not been considered in this representation dueto the scarce resources they were achieving. This fact hasto be remarked since both receivers no.1 and no.4 haveprimary terminal no.1 as nearest primary receiver. It canbe explained assuming that in this representation the pri-mary no.1 leaves unoccupied wide portions of the spectrumdue to a channel particularly impaired. Finally Figs.6-8display the comparisons in terms of terminal rate, Figs.7-8, and aggregate rate, Fig.6, between the game theoreticapproach presented in this paper and an heuristic powerallocation method. The heuristic approach considered, it isan extension to a multi-user context of the power allocationstrategy proposed in [18]. Differently from the analyzedgame theory approach, the heuristic method has a centralizedimplementation and it needs a huge quantity of information

10-5 10-4 10-3 10-20.2

0.4

0.6

0.8

1

1.2

1.4

1.6Secondary Receiver no.1

Secondary Target BER

Rat

e [M

bps]

Game Theory ApproachHeuristic Approach

10-5 10-4 10-3 10-20.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0.22Secondary Receiver no.2

Secondary Target BER

Rat

e [M

bps]

Game Theory ApproachHeuristic Approach

Fig. 7. The comparison between game theoretic approach and heuristicapproach in terms of rate for terminals no.1 and no.2 .

10-5 10-4 10-3 10-20.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8Secondary Receiver no.3

Secondary Target BER

Rat

e [M

bps]

Game Theory ApproachHeuristic Approach

10-5 10-4 10-3 10-21

2

3

4

5

6

7Secondary Receiver no.4

Secondary Target BER

Rat

e [M

bps]

Game Theory ApproachHeuristic Approach

Fig. 8. The comparison between game theoretic approach and heuristicapproach in terms of rate for terminals no.3 and no.4 .

to be forwarded on the network. Fig.6 clearly shows whatis the difference of performance between the two methods:the heuristic allocation outperforms the Game Theory basedallocation by a consistent amount, e.g. it doubles game theoryapproach aggregate rate. This fact has not to be interpreted asa failure for the Game Theory allocation because the heuristicmethod is actually representing the optimal allocation, i.e. theone achieving the highest rate. The detail of the comparisonfor each terminal is given in Figs.7-8, where it is possibleto notice how the difference of rate between the methodsincreases severely when the rate achieved is above 1 Mbps,i.e. receivers no.3 and no.4; receiver no.1 manifests a similarbehaviour, while receiver no.2 shows an unusual occurrenceachieving higher rate through game theory approach thanthrough heuristic method. This result is going against the

122

general tendency, and it lets assume that game theory strat-egy is more fair than heuristic method toward those usersexperiencing lower rates.

V. CONCLUDING REMARKS

In this paper it has been presented a power allocationstrategy for Cognitive Radio networks. The framework ofthe proposed technique is based on Game Theory. It hasbeen showed how the game concerning power allocation isa Potential Game. The potential function has been found andit has been discussed how it can be implemented in a dis-tributed way. Results achieved with the proposed techniquehave been presented for a specific scenario. Performancesof the game theoretic approach have been given in terms ofrate, actual BER and signal-to-noise ratio. Finally the methodhas been compared with the results achieved by an heuristicapproach to the power allocation, representing the optimum.In the comparison the heuristic method outperformed gametheory based one; however this result has not to be interpretedas a failure for the Game Theory allocation since, respectto heuristic method, it can be implemented in a distributedway. A final issue emerged, and it lets assume the proposedmethod is intrinsically more oriented to fairness than theheuristic one.

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