Green Power Control in Cognitive Wireless Networks
11 Green Power Control in Cognitive WirelessNetworks
Mael Le Treust ∗ and Samson Lasaulce † and Yezekael Hayel ‡ and Gaoning He §∗ Université INRS, Centre Energie, Matériaux et Télécom, 800 rue de la Gauchetière, Montréal, Canada † Laboratoire des Signaux et Systèmes, Supélec, 3 rue Joliot Curie, 91191 Gif-sur-Yvette, France ‡ LIA/CERI, University of Avignon, Agroparc BP 1228, 84911 Avignon Cedex 9, France § HUAWEI, Central Research Institute, 2222 Xinjinqiao Rd, Pudong, 201206, Shanghai, China
Abstract —A decentralized network of cognitive and non-cognitive transmitters where each transmitter aims at maximizinghis energy-efficiency is considered. The cognitive transmitters areassumed to be able to sense the transmit power of their non-cognitive counterparts and the former have a cost for sensing.The Stackelberg equilibrium analysis of this − level hierarchicalgame is conducted, which allows us to better understand theeffects of cognition on energy-efficiency. In particular, it is proventhat the network energy-efficiency is maximized when only agiven fraction of terminals are cognitive. Then, we study a sensinggame where all the transmitters are assumed to take the decisionwhether to sense (namely to be cognitive) or not. This game isshown to be a weighted potential game and its set of equilibriais studied. Playing the sensing game in a first phase (e.g., of atime-slot) and then playing the power control game is shown tobe more efficient individually for all transmitters than playing agame where a transmitter would jointly optimize whether to senseand his power level, showing the existence of a kind of Braessparadox. The derived results are illustrated by numerical resultsand provide some insights on how to deploy cognitive radios inheterogeneous networks in terms of sensing capabilities. Index Terms —Power Control, Stackelberg Equilibrium,Energy-Efficiency.
I. I
NTRODUCTION
In fixed communication networks, the paradigm of peer-to-peer communications has known a powerful surge of interestduring the past two decades with applications such as theInternet. Remarkably, this paradigm has also been found tobe very useful for wireless networks. Wireless ad hoc andcognitive networks are two illustrative examples of this. Oneimportant typical feature of these networks is that the terminalshave to take some decisions in an autonomous or quasi-autonomous manner. Typically, they can choose their powercontrol and resource allocation policy. The correspondingframework, which is the one of this paper, is the one ofdecentralized or distributed power control (PC) or resourceallocation. More specifically, the scenario of interest is thecase of power control over quasi-static channels in cognitivenetworks [14]. In such a context, which is broader than theone of ad hoc and cognitive wireless networks, we assumethat some (possibly all) transmitters are able to sense the powerlevels of non-cognitive transmitters and adapt their power level
Part of this paper has been published in Infocom 2011. This work is alsopartially supported by L’Agence Nationale de la Recherche (ANR) within theproject ANR-09-VERS0: ECOSCELLS. accordingly. The considered model of multiuser networks isa multiple access channel (MAC) with time-selective non-frequency selective links but the methodology can be appliedto other types of interference networks. Technical issues re-lated to spectrum usage is not considered in this paper, leavingthis aspect as a relevant extension of this paper. Rather, wewant to study the effect of cognition in terms of energy usage,the potential benefits in terms of spectral efficiency havingbeen well investigated. The selected performance metric fora transmitter is derived from the energy-efficiency definitionof [16]. The authors of [16] define energy-efficiency as thenumber of bits successfully decoded by the receiver per jouleconsumed at the transmitter (in [16] the radiated power isconcerned). More specifically, the authors analyze the problemof decentralized power control in flat fading multiple accesschannels. The problem is formulated as a non-cooperative one-shot game where the players are the transmitters, the action ofa given player is her/his/its transmit power (”his” is chosenin this paper), and his payoff/reward/utility function is theenergy-efficiency of his communication with the receiver; wewill not provide here the motivations for using game theoryto study distributed power control problems but some of themcan be found e.g., in [23]. The results reported in [16] havebeen extended to the case of multi-carrier systems in [11].The framework of the present work is close in spirit to[16], [11] but differs from them in several aspects. Themost important one is that there can be a hierarchy amongthe transmitters in terms of observation capabilities, sometransmitters can be cognitive and observe the others whereasthe latter cannot observe the actions played by the former.Technically, this leads to a Stackelberg-type formulation of theproblem [34]. The closest work to the one reported here is [22]where a Stackelberg model of energy-efficient power controlproblems is introduced for the first time. The present workreports a significant extension of the framework introduced in[22]. Two games are studied in detail. The power control gamecorresponds to a generalization of the game addressed in [22] :the sensing costs are taken into account (observing/sensing theothers has a cost) and more importantly, our analysis is notlimited to one non-cognitive transmitter (i.e., a single gameleader). Then, we introduce a new game where the transmittersdecide whether to sense or not. A third game, which is is anhybrid control game and include the two mentioned games asspecial cases, is shown to be not worth being studied because a r X i v : . [ c s . I T ] A p r of the existence of a Braess paradox [8].The paper is organized as follows. In Sec. II, the assumedsignal model to describe the distributed power control problemover time-selective non-frequency selective multiple accesschannels is provided. Known results concerning the case wherethe transmitters tune their power levels from block to block ina distributed way and without observing the other transmitters(i.e., they cannot sense the powers chosen by the others) areprovided. In Sec. III, we assume that some transmitters havesensing capabilities, which creates a hierarchy in terms ofobservation capabilities between the transmitters. The effect ofthis is that choosing rational power control policies in this set-ting leads to a more efficient network outcome (a Stackelbergequilibrium), provided that the sensing cost for a cognitivetransmitter is not too high. While in Sec. III, a transmitterwas imposed to sense or not, it is assumed in Sec. IV that thischoice is left to the transmitter itself. It is shown that thereexists an optimal number of cognitive transmitters in termsof network utility and therefore having too many advancedterminals can be detrimental to the global performance. It isshown that leaving the choice to a transmitter to choose ina joint manner its power level and whether to sense is infact less energy-efficient than imposing that the transmitterschoose these two quantities separately. This shows the interestin studying the power control game (as in Sec. III) and thesensing game separately. The sensing game is a new gamewe introduce and is shown to possess attractive properties fordistributed optimization and learning algorithms. Finally, inSec. V numerical illustrations are provided and the paper isconcluded in Sec. VI.II. P ROBLEM STATEMENT
A. System model
We consider a decentralized multiple access channel witha finite number of transmitters, which is denoted by K . Thenetwork is said to be decentralized in the sense that the receiver(e.g., a base/mobile station) does not dictate to the transmitters(e.g., mobile/base stations) their power control policy. Rather,all the transmitters choose their policy by themselves and wantto selfishly maximize their energy-efficiency. In particular,they can ignore some specified centralized policies. We assumethat the users transmit their data over time-selective non-frequency selective channels, at the same time and on thesame frequency band; channels are considered to be constantover each block of data. Note that a block is defined as a se-quence of M consecutive symbols which comprises a trainingsequence that is, a certain number of consecutive symbols usedto estimate the channel (or other related quantities) associatedwith a given block. A block has therefore a duration less thanthe channel coherence time. The equivalent baseband signal atthe receiver can be written as y ( t ) = K (cid:88) k =1 h k x k ( t ) + z ( t ) (1)where k ∈ K , K = { , ..., K } , x k ( t ) represents the symboltransmitted by transmitter k at time t ∈ N , E | x k | = p i ,the noise z is assumed to be decentralized according to a zero-mean Gaussian random variable with variance σ andeach channel gain h k varies over time but is assumed to beconstant over each block ; the symbol index t will be omittedin this paper. In terms of channel state information (CSI),the receiver is assumed to know all the channel gains (globalCSI) while each transmitter only knows his own channel (localCSI). For each block, the expression of the receive signal-to-interference-plus-noise ratio (SINR) of user k is given by : γ k = g k p k σ + K (cid:88) j (cid:54) = k θ j g j p j (2)where for all j ∈ K , g j = | h j | and θ j represents a parameterdepending on the interference scenario. For example, in arandom code division multiple access (RCDMA) system withsingle-user decoding, we would have θ j = N where N isthe spreading factor [11]. This is the choice we will do.Nonetheless, note that the present work is not restricted tocode division multiple access (CDMA) systems. Indeed, bychoosing θ j = 1 (i.e., N = 1 ) the above signal model cor-responds to the information-theoretic channel model used forstudying multiple access channels [35], [9]; in this setup, goodchannel codes are assumed (see e.g., [6] for more comments onthe multiple access technique involved). Indeed, what mattersthe most in the model is that it captures the different aspectsof the problem (especially the SINR structure). At last, thecase where successive interference cancelation is used at thereceiver ( θ j ∈ { , } , depending on the decoding order) is leftas an extension of the present work. B. Performance metric
Assuming the above signal model, we assume that eachtransmitter wants to selfishly maximize the energy-efficiencyof his communication with the receiver. The used performancemetric is the one originally proposed in [16]. For a given block,transmitter k wants to maximize the following quantity : v k ( p k , p − k ) = R k f ( γ k ) p k [bit/J] (3)where R k is the transmission rate (in bit/s), f is an effi-ciency function (representing the block success rate), and thesubscript − k on vector p stands for “all the transmitters excepttransmitter k ”, i.e., p − k = ( p , . . . , p k − , p k +1 , . . . , p K ) .Note that, as a standard assumption, R k is assumed to beindependent of γ k or p k which may correspond in practice toa given choice of modulation coding scheme. As motivated in[31], [4], the efficiency function is assumed to be an increasingand sigmoidal (or S-shaped) function verifying ≤ f ( . ) ≤ with f (0) = 0 and lim x → + ∞ f ( x ) = 1 . The fact that f is sigmoidalhas at least two important consequences : the utility function v k is quasi-concave w.r.t. p k and the derivative of v k vanishesat only one point which is different from . We see that R k might be chosen to be SINR-dependent without affectingthe problem analysis provided that the product R k f be asigmoidal function. For the sake of clarity, we assume that theplayers have the same efficiency function f . In [16] and relatedworks, p k represents the power radiated by the transmitter. Interestingly, the above utility can also model situations wherethe power consumed by the whole transmitting device has tobe accounted for. Indeed, by replacing the denominator of u k by ap k + b , ( a, b ) being a pair of non-negative constants, oneobtains a first-order model of the device power consumptionwhich includes both the consumption part which does notdepend on the radiated power and the one due to the transmitpower [13]. This does not change significantly the mathemat-ical analysis of the power control problem. In order to focusour attention on the most important points of our analysis andmake the exposition as clear as possible, the original modelof [16] has been selected (i.e., ( a, b ) = (1 , ). C. Game-theoretic modeling : review of the non-cooperativegame of [16]
An appropriate model for the power control problem de-scribed above is given by a strategic form game [16]. Astrategic form game consists of an ordered triplet comprisingthe set of players, their action or strategy sets, and theirpreference orders (or their utilities when they exist, which isthe case here). The set of players is the set of transmitters K , the action set is P k = [0 , P max k ] , k ∈ K , and theutility functions are defined by (3). This describes the modelintroduced by Goodman et al in [16]. As [16] and relatedreferences such as [11], the power levels are chosen to becontinuous. This allows us to conduct a complete comparisonanalysis in terms of performance. However, this assumptionis not always suited and the case of discrete power levelsis therefore left as a complementary way of tackling theproblems under investigation. An important solution conceptfor this game is the Nash equilibrium (NE) [28], which is apower profile/vector that is robust against unilateral deviations(no player has interest in deviating if the others keep theequilibrium strategy). The unique NE of this power controlgame is : p NE k = σ g k β (cid:63) − K − N β (cid:63) , k ∈ K (4)where β (cid:63) denotes the best SINR choice for user k at theNE ; as explained in [16][11], a necessary condition for thisequilibrium to be defined is that the system load is not too high( K − N β (cid:63) < ). Note that the equilibrium SINR is common toall users. It is easy to verify that β (cid:63) is the positive solution ofthe differential equation x f (cid:48) ( x ) − f ( x ) = 0 , which is obtainedby solving max x (cid:54) =0 f ( x ) x , i.e., an equivalent problem of max p k v k (following from the assumption that R k is independent of γ k and p k ).The equilibrium solution holds if the power constraint p k ≤ P max k is satisfied, which is what we will assume throughoutthis paper (see e.g., [16] for further details about the casewhere the constraint is active). This game model, althoughleading to a decentralized solution in terms of decisions andCSI (see [16]), has one main drawback : the equilibriumsolution can be inefficient. Interestingly, introducing somehierarchy between the players in terms of observation can App. A reviews several game-theoretic notions. Note that the unconditionalexistence of a pure NE is, in part, a consequence of quasi-concavity for theutility functions. improve the game outcome, as shown in [22]. It turns outthat hierarchy is naturally present in networks where sometransmitters are equipped with a cognitive radio while theothers are not. This is one of our motivations for formulatingthe problem in decentralized cognitive networks as a two-level Stackelberg game, with arbitrary numbers of cognitiveradios, generalizing the -leader K − -follower game of [16].Compared to the latter reference [16], a second interestingfeature of the game described below is that the cost induced bysensing is accounted for in the utility function of the cognitivetransmitters. The proposed approach may be relevant in mostapplications where cognitive radio is useful. Indeed, one of themessages of this work is that if the fraction of transmitters whocan observe their environment is too high, this may degradethe global performance. To mention an existing scenario wherethis type of approaches might be applied in the future, thecase of WiFi systems can be mentioned. In France, operatorsprovides more and more advanced access points. Typically,they want to optimize channel selection (which is a specialcase of power allocation) in a more and more efficient manner.Assuming that some access points (AP) are optimized accord-ing to a Nash strategies while others implement Stackelbergstrategies allows one to provide a simplified model to accountfor the fact that advanced APs coexist with less advanced APs.Interestingly, as shown in this paper, as far as power controlis concerned, having too many advanced APs might not be asgood as the common sense would indicate.III. T HE TWO - LEVEL POWER CONTROL GAME WITHSENSING COSTS
The set of transmitters K = { , , ..., K } comprises F terminals equipped with a cognitive radio while the L = K − F other terminals have no sensing capabilities. The pair ( F, L ) is assumed to be fixed throughout the whole section; it willbe optimized in a centralized (resp. decentralized) mannerin Sec. IV-A (resp. Sec. IV-B). Without loss of generality,the set of non-cognitive (resp. cognitive) terminals will be L = { , , ..., L } (resp. F = { L + 1 , L + 2 , ..., K } ). This two-level hierarchical game is played as follows. For each block,the non-cognitive transmitters (called the leaders) choose theirpower level rationally knowing that their decisions are goingto be observed by the cognitive transmitters. The cognitivetransmitters (called the followers) react to these decisionsrationally. A choice is said to be rational in the sense thatthe transmitter maximizes his utility. To this end, we denoteby p L (cid:44) ( p , . . . , p L ) and p F (cid:44) ( p L +1 , . . . , p K ) the vectorsof actions (transmit powers) of the leaders and followers,respectively. Also denote by U (cid:63) ( p L ) the set of NE for thegroup of followers when the leaders play p L . The resultingoutcome of this interaction is a Stackelberg equilibrium (SE),which is defined as follows. Definition 3.1 (Stackelberg equilibrium): A vector of ac-tions p SE = ( p SE L , p SE F ) is called a Stackelberg equilibrium,if p SE F ∈ U (cid:63) ( p SE L ) and the actions p SE L is a Nash equilibrium for the leaders . By looking at the mathematical expression of the Stack-elberg equilibrium defined above, we can see that if the NEexists, then SE also exists. But the SE is not included in theset of NE of a non-cooperative game. There exists severalexamples like the Cournot game in which the action chosenby the leader at the SE is different compared to the actionchosen at the SE [15]. As the best-response of each playeris a scalar-valued function (see [22]), the determination ofthe Stackelberg equilibria of the game amounts to solving thefollowing bi-level optimization problem: p SE (cid:96) ∈ arg max p (cid:96) u (cid:96) (cid:0) p (cid:96) , p SE − (cid:96) , p SE L +1 ( p (cid:96) , p SE − (cid:96) ) , . . . , p SE K ( p (cid:96) , p SE − (cid:96) ) (cid:1) , ∀ (cid:96) ∈ L (5) where for all p L , p SE f ( p L ) = arg max p f u f ( p L , p SE L +1 ( p L ) , . . . , p SE f − ( p L ) , p f , ...,p SE f +1 ( p L ) , . . . , p SE K ( p L )) , ∀ f ∈ F (6) where the utility functions are given by : u k ( p ) = (cid:12)(cid:12)(cid:12)(cid:12) v k ( p ) if k ∈ L , (1 − α k ) v k ( p ) if k ∈ F . (7)The parameter α k ∈ [0 , , k ∈ F is a constant w.r.t.the power levels which accounts for the sensing cost (to beillustrative, we will choose α k = α in some places). Thisconstant has no effect on the equilibrium strategies. However,when it will come to knowing whether being a follower ornot, this constant will play a role. To elaborate further onthis constant, it can be interpreted as the fraction of time acognitive user k ∈ F spends for sensing. In order to havea good sensing capabilities , we assume that there exists acertain energy threshold ξ min (see e.g., [18]) expressed injoule : α k T min (cid:96) ∈L ( g k(cid:96) p (cid:96) ) ≥ ξ min where T is the block duration in second and p (cid:96) in Watt whereas α k and f ( . ) are unitless ; g k(cid:96) is the channel gain between anyleader (cid:96) ∈ L and the considered follower k ∈ F . If the aboveinequality holds, it means that the cognitive user k ∈ F isable to sense the presence of the primary ones. Apparently,we assume that the sensing constraint is feasible in the sensethat there exists a minimum fraction α k ≤ , k ∈ F abovewhich the minimum energy threshold for sensing is attained.A necessary condition for this is that T min (cid:96) ∈L ( g k(cid:96) p (cid:96) ) ≥ ξ min .For the sake of clarity, we suppose that the sensing cost isthe same for every player α k = α, k ∈ F . At this point,the two-level hierarchical power control game is completelydefined : the players are the L non-cognitive transmitters andthe F cognitive transmitters, their action sets are [0 , P max k ] ,and their utilities are defined by (7). We assume, w.l.o.g. two players in a non-cooperative game with one leaderand one follower. If the leader plays the NE action, then, as the followerobserves this action and plays the best-response against it, the follower willplay the NE strategy. Then, the NE strategy profile, if it exists, can be a SE.The behavior is the same when there are several leaders and followers. If theNE between the leaders corresponds to the NE of the game between leadersand followers, then the followers respond by playing the NE. Under the assumption of single-user decoding (each useful signal isdetected by considering the other signals as noise), a good sensing capabilitymeans that a follower can detect the existence of all the leaders. In particular,the leader whose link with the follower is the worst is detected.
Following the standard methodology of equilibrium analysis(see e.g., [21][23]), three important issues to be dealt with arethe existence, uniqueness, and efficiency issues for the Stack-elberg equilibrium. The next theorem provides an element ofresponse to the first two issues.
Proposition 3.1 ([22]): There always exists a Stackelbergequilibrium p SE in the two-level hierarchical game with L ≥ leaders and F ≥ followers. The power profile defined by ∀ (cid:96) ∈ L , p SE (cid:96) = σ g (cid:96) Nγ (cid:63)L ( N + β (cid:63) ) N − N ( F − β (cid:63) − [( N + β (cid:63) )( L −
1) +
F β (cid:63) ] γ (cid:63)L ∀ f ∈ F , p SE f = σ g f Nβ (cid:63) ( N + γ (cid:63)L ) N − N ( F − β (cid:63) − [( N + β (cid:63) )( L −
1) +
F β (cid:63) ] γ (cid:63)L is an SE and β (cid:63) is the positive root of xf (cid:48) ( x ) = f ( x ) , and γ (cid:63)L is the positive root of x (1 − (cid:15) L x ) f (cid:48) ( x ) = f ( x ) , with (cid:15) L = F β (cid:63) N − N ( F − β (cid:63) . (8) Moreover, the equilibrium p SE is unique if the following twoconditions hold: (1) lim x → + f (cid:48)(cid:48) ( x ) f (cid:48) ( x ) > (cid:15) L , and (2) equation x (1 − (cid:15) L x ) f (cid:48) ( x ) − f ( x ) = 0 has a single root in (0 , β (cid:63) ) . Note that the existence of such an equilibrium is ensuredfrom the properties of the Stackelberg game, especially thesigmoidness of f [22]. In [22], it also explained that the best-response of the players are scalar-valued functions, which fa-cilitates the Stackelberg equilibrium analysis. Interestingly, theprovided sufficient conditions for uniqueness can be checkedto be satisfied for two typical efficiency functions used in therelated literature namely, f ( x ) = (1 − e x ) M and f ( x ) = e − cx , c ≥ used in [16] and [4] respectively.At last but not least, we address the issue of efficiency forthe derived Stackelberg equilibrium. The key point at stakeis whether cognition helps to obtain a better decentralizednetwork in terms of global energy-efficiency. For this purpose,we first compare the utility a player would get in a systemwhere no cognitive transmitters exist with the one he wouldobtain in a system with cognitive transmitters (i.e., at theNash equilibrium corresponding to [16]). Our main results aresummarized by the following proposition. Proposition 3.2 (SE versus NE): The utility at the SE of thetwo-level hierarchical game with sensing cost of any leader isalways greater or equal to the one obtained at the NE.
If the cost for sensing is negligible, the next corollaryfollows.
Corollary 3.3 (SE versus NE with no sensing cost): Thepower profile at the SE Pareto dominates the power profile atthe NE.
The proof of Proposition 3.2 and corollary 3.3 are given in[22]. Another relevant question, initially raised in a contextwith a single leader and no sensing cost [22] is whetherit is better to follow or lead the power control game. Saidotherwise, is it beneficial for a transmitter to be equipped witha cognitive radio when sensing costs are accounted for ? Theanswer is provided below.
Proposition 3.4 (Following versus leading): At the Stackel-berg equilibrium of the two-level power control game withsensing cost, a transmitter prefers to be a follower (that is to say, to sense) if the minimum energy threshold for sensingverifies : ξ min ≤ − f ( γ (cid:63)L ) γ (cid:63)L f ( β (cid:63) ) β (cid:63) ( N + γ (cid:63)L )( N + β (cid:63) ) T min (cid:96) ∈L g f p SE (cid:96) . (9)The proof of this result is given in App. B. Interestingly, itis possible to provide an explicit lower bound on the energythreshold for a cognitive radio for being energy-efficient. For atransmitter, this bound mathematically translates the tradeoffbetween the benefit (in terms of energy-efficiency) of beinginformed about the actions played by the others and the costinduced by acquiring this knowledge. If the sensing cost isnegligible, then following becomes always better than leading,giving an incentive to equip a transmitter with a cognitiveradio. However, if all the transmitters of the network arecognitive, the network energy-efficiency is not maximized,which is what is proved in the next section.IV. T HE SENSING GAME
In the preceding section, the pair ( F, L ) and the identitiesof cognitive and non-cognitive transmitters were fixed. A quitenatural question is to ask whether the transmitters wouldeffectively sense or not in a fully decentralized network wherethe decision to sense is left to them. Providing answers to thisquestion is the purpose of this section. As a first step, we showthe existence of an optimal number of cognitive transmitters interms of social welfare (sum utility). The corresponding upperbound can be used to assess the price of anarchy of the network[19] and therefore measuring the cost of having this decisiondecentralized. As a second step, we consider a sensing gamein which each transmitter has two actions (sense/not sense).It is shown that each transmitter can learn his best decisionprovided that the number of blocks is sufficiently large. A. On the optimal number of cognitive transmitters
The global energy-efficiency of the network is measured interms of social welfare [1] or sum utility at the equilibriumwhich is defined by : w eL = K (cid:88) k =1 u k = L (cid:88) k =1 u k + K (cid:88) k = L +1 u k , (10)where e ∈ { SE , NE } . Note that a Nash equilibrium is obtainedwhen L = K , indeed in this context, there is no followersand the game is no more hierarchical. The subscript L hasbeen added to the equilibrium profiles to clearly indicate thatit is related to the number of leaders whenever L (cid:54) = K (seeequation (10). As the parameter L , which is the number ofleaders or non-cognitive transmitters, belongs to a discrete set K , the function w L : K → R + has necessarily a maximum .Is this maximum reached at the non-trivial points L = 1 or L = K ? >From Proposition 3.2, we know that it is not thecase. Indeed, if the sensing cost is small enough, then thepower profile at any SE Pareto-dominates the one of the NEobtained with L = 1 or L = K . Thus, the latter points arenot the maximizer candidates for the sum utility. However, Note that, however, we do not try to optimize the identity of the followersor leaders with respect to their channel quality. This type of issues, which isof relevant in centralized scenarios, is addressed in [22] in a special case. when the sensing cost is arbitrary, answering the questionanalytically does not seem to be trivial. This is why we solvethis maximization problem numerically in Sec. V. To still getsome insights into the problem, we study a very special caseof it. The interest in doing so is that it clearly shows that thesum utility maximizer is non-trivial and a little more about theconnection with the network load can be learned.We now consider the special case defined by the followingfour assumptions : • Assumption 1 : g k = g, R k = R for all k ∈ K . • Assumption 2 :
N >> ( K − β (cid:63) . • Assumption 3 : f ( x ) = e − cx , c ≥ . • Assumption 4 : ≤ L ≤ K − .The social welfare at the SE is given by : w SE L = (cid:88) (cid:96) ∈L R (cid:96) f ( γ (cid:63)L ) p SE (cid:96) + (cid:88) f ∈F R f f ( β (cid:63) ) p SE f = R ga L σ (cid:20) L f ( γ (cid:63)L ) γ (cid:63)L ( N + β (cid:63) ) + ( K − L ) f ( β (cid:63) ) β (cid:63) ( N + γ (cid:63)L ) (cid:21) , (11)where a L (cid:44) N − ( F − β (cid:63) − [( N + β (cid:63) )( L −
1) +
F β (cid:63) ] γ (cid:63)L N . (12)Note that in [5], c = 2 r − , where r is the spectral efficiency ofthe used channel coding scheme (in bit/s per Hz). A possiblechoice is r = RB where R is the data transmission rate and B the required bandwidth to transmit. While Assumptions and are reasonable (they respectively correspond to a small sys-tem load and the case where the efficiency function is derivedfrom the outage probability on the mutual information [5]),the first assumption is very strong and may happen in practicein specific scenarios e.g., in virtual multiple input multipleoutput (MIMO) networks with clusters of transmitters [17]with similar flow types (a voice service typically). Indeed, iffast power control is considered namely, g k represents the fastfading, this symmetry assumption will almost never be verifiedin practice. Now, if slow power control is considered, g k mayrepresent shadowing and path loss effects, and the g (cid:48) k s willbe almost equal for all users being in a given neighborhood(as explained in [17] where the notion of clusters of users isshown to be relevant). In any case, note that the symmetryassumption is very local and is only exploited to reinforce theexistence of a non-trivial optimal number of followers/leadersand provide insights on this number. Again, the goal is not toclaim a general expression for the optimal number of leaders.Rather, we just want to derive it in a special case to betterunderstand the general optimization problem. Assumption is not restrictive since the cases K = 0 and K = L are ready.Under Assumption 2, we have that a L ≈ N . From (8) wehave (cid:15) L ≈ , which implies that γ (cid:63)L ≈ β (cid:63) , f ( γ (cid:63)L ) ≈ f ( β (cid:63) ) , and f ( γ (cid:63)L ) N + β (cid:63) ≈ f ( β (cid:63) ) N + γ (cid:63)L . (13)This allows one to approximate the social welfare at theequilibrium (11) as : (cid:101) w SE L ≈ R g f ( β (cid:63) ) N ( N + β (cid:63) ) σ (cid:18) Lγ (cid:63)L + K − Lβ (cid:63) (cid:19) . (14) Note that the term g f ( β (cid:63) ) N ( N + β (cid:63) ) σ is independent of L , and theoptimal solution L (cid:63) can be approximated by : ˜ L (cid:63) = arg max L (cid:18) Lγ (cid:63)L + K − Lβ (cid:63) (cid:19) . (15)The main point in the above equation is that (cid:101) L (cid:63) is related to γ (cid:63)L which is the positive root of the equation x (1 − (cid:15) L x ) f (cid:48) ( x ) − f ( x ) = 0 . It turns out that, under Assumption 3, a very simpleexpression for γ (cid:63)L can be obtained. One can easily check that : γ (cid:63)L = c (cid:15) L c . (16)Using the above, the optimization problem given by equation(15) boils down to : (cid:101) L (cid:63) = arg max L (cid:32) L c (cid:15) L c + K − Lc (cid:33) = arg max L ( (cid:15) L L ) . Replacing the discrete variable L with a non-negative real λ ,the optimal solution of the corresponding concave functioncan be checked to be : (cid:101) λ (cid:63) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:0) Nc (cid:1) (cid:16) − (cid:113) − KN c cN +1 (cid:17) if K − N ≤ c K − κ if K − N > c (17)where κ > is arbitrary small (this constraint is added tomeet the constraint L ≤ K − ). Therefore, the optimalnumber of non-cognitive transmitters can be approximated by (cid:101) L (cid:63) = (cid:106)(cid:101) λ (cid:63) (cid:107) or (cid:101) L (cid:63) = (cid:108)(cid:101) λ (cid:63) (cid:109) depending on which number givesthe maximal social welfare. From this expression of (cid:101) L (cid:63) , someinteresting insights can be easily extracted. First of all, notethat if the load is sufficiently small compared to β (cid:63) but at thesame time greater than c , the social welfare is maximized for (cid:101) L (cid:63) = K − . Indeed, when the load is small, the interactionbetween players is not strong and the impact of a hierarchyon the overall system is small (at least for a reasonably largesystem). Thus, the social welfare is maximized at the NEpoint, which corresponds to the case where all users are leaderand play at the same time. A second type of insights can beobtained by considering the spectral efficiency related to thechannel coding namely c = 2 r − . As already mentioned,note that it is assumed that the spectral efficiency involved bythe multiple access technique is sufficiently small (Assumption2). If we go further by assuming that both the multiple accesstechnique spectral efficiency ( KN → ) and channel codingspectral efficiency ( c → ) are small, we obtain that : (cid:101) λ (cid:63) ≈ (cid:0) Nc (cid:1) KN c cN +1 ≈ K (18)which means that in the low spectral efficiency regime, theglobal energy-efficiency of the network is maximized whenhalf of the transmitters are cognitive. If c is large but theload is still small (meeting the constraint K − N ≤ c ) thesame conclusion is obtained. Conducting a deep analysisto discuss the connections between spectral efficiency andenergy-efficiency in decentralized cognitive networks in thegeneral case (arbitrary load, with cost of sensing) seems to bea relevant and non-trivial extension of the simplified analysisprovided here. B. Sensing game : description and key property
In the two-level hierarchical power control described inSec. III, the transmitter is, by construction, either a cogni-tive transmitter or a non-cognitive one and the action of aplayer consists in choosing his transmit power level. In fullydecentralized networks, it is legitimate to ask about what atransmitter would decide between sensing (being cognitive ornot) or not sensing. To analyze such a problem, we assume thattwo games occur sequentially for each block (this separationassumption will be fully justified in Sec. IV-C). First, thetransmitters decide to sense ( S ) or not to sense ( NS ) . Then,their choose their power level based on their status (followeror leader) and therefore play the two-level power control gamedescribed in Sec. III. The sensing game can therefore bedescribed by a static game whose representation is given bythe following triplet : G = ( K , ( A k ) k ∈K , ( U k ) k ∈K ) (19)where the actions sets are A k = A = { S , NS } and the utilityfunctions are those obtained at the Stackelberg equilibriumof the power control game played in the second phase. Iftransmitter k is a follower (i.e., he senses) and that there were F followers during the sensing phase of the block then hisutility is : U k ( S , s ( F,L ) − k ) = (1 − α k ) g k R k σ f ( β ∗ ) Nβ (cid:63) ( N + γ (cid:63)L +1 ) × (cid:8) N − Nβ (cid:63) − [( N + β (cid:63) ) L + ( F + 1) β (cid:63) ] γ (cid:63)L +1 (cid:9) where the notation s ( F,L ) − k means that there are F − followersand L leaders in the set of players K \ k . On the other hand,if transmitter k chooses the action NS he obtains : U k ( NS , s ( F,L ) − k ) = g k R k σ f ( γ ∗ L ) Nγ (cid:63)L +1 ( N + β (cid:63) ) × (cid:8) N − Nβ (cid:63) − [( N + β (cid:63) ) L + ( F + 1) β (cid:63) ] γ (cid:63)L +1 (cid:9) in terms of utility. The considered sensing game is a con-gestion game [30] (and therefore a potential game) understrong conditions but is always a weighted potential game.The latter property is known to be very useful for studyingexistence of pure NE and convergence of learning algorithmsor distributed iterative algorithms towards NE. For instance, in[25], [26], Monderer and Shapley proved that every weightedpotential game has the Fictitious Play Property (FPP). Thisguarantee that every learning algorithm that is Fictitious Playprocess converges in belief to equilibrium. All of this is thepurpose of the remaining of this section. For making this papersufficiently self-containing we review several useful definitionsconcerning potential games [25].
Definition 4.1 (Monderer and Shapley 1996 [25]): Thestrategic form game G is a potential game if there is apotential function V : A −→ R such that U k ( s k , s − k ) − U k ( s (cid:48) k , s − k ) = V ( s k , s − k ) − V ( s (cid:48) k , s − k ) , ∀ k ∈ K , s k , s (cid:48) k ∈ A k . The FP learning algorithm can be found in the quoted references or [36]but essentially it consists in assuming that each player observes the actions ofthe others and maximizes his average utility based on the empirical frequenciesof use of actions of the others.
Theorem 4.2: The sensing game G =( K , ( A ) k ∈K , ( U k ) k ∈K ) is an exact potential game if and onlyif one of the two following conditions is satisfied : ∀ i, j ∈ K R i g i = R j g j , (20) U L ( F + 1 , L + 1) − U L ( F, L + 2) = (1 − α )( U F ( F + 2 , L ) − U F ( F + 1 , L + 1)) , (21) where U L ( F + 1 , L + 1) is defined by σ U i ( F +1 ,L +1) R i g i whenplayer i ∈ K is one of the F +1 followers and U F ( F +1 , L +1) is defined by σ U i ( F +1 ,L +1) R i g i when player i ∈ K one of the L + 1 leaders. Condition (20) is a (strong) symmetry condition and isobtained under Assumption 1 (Sec. IV-A), which would bereasonable for a cluster of transmitters in a virtual MIMOnetwork with a common service (e.g., voice). In fact, it ismore realistic not to make these assumptions and claim for apotential property which is sufficient for key issues such asconvergence of some important learning dynamics.
Definition 4.3 (Monderer and Shapley 1996 [25]): Thestrategic form game G is a weighted potential game if thereis a vector ( µ i ) i ∈K and a potential function V : A −→ R such that : ∀ i ∈ K , ( s i , s (cid:48) i ) ∈ A i ,U i ( s i , s − i ) − U i ( s (cid:48) i , s − i ) = µ i ( V ( s i , s − i ) − V ( s (cid:48) i , s − i )) . It turns out that such a vector can be found.
Theorem 4.4: The sensing game G =( K , ( A i ) i ∈K , ( U i ) i ∈K ) is a weighted potential game with theweight vector : ∀ i ∈ K , µ i = R i g i σ . (22)The proof is given in App. F. C. Equilibrium analysis1) Existence:
First of all, note that since the sensing gameis finite (i.e., both the number of players and the sets ofactions are finite), the existence of at least one mixed NE isguaranteed [29]. Now, since the game is a weighted potentialgame, the existence of at least one pure NE is guaranteed[25]. We might restrict our attention to pure and mixed Nashequilibria. However, as it will be clearly seen in the 2-playercase study (Sec. IV-C3), this may pose a problem of fairnessand efficiency. This is the main reason why we also studythe set of correlated equilibria (App. A-A) of the sensinggame. The concept of correlated equilibrium [2] allows oneto enlarge the set of equilibrium utilities. Every utility vectorinside the convex hull of the Nash equilibrium utilities isa correlated equilibrium, which guarantees the existence ofcorrelated equilibria in general.
2) Uniqueness:
Here, we provide a brief analysis of unique-ness for the pure NE. This matters since pure NE are attractorsof important dynamics such as the replicator dynamics (whichcorresponds to the limit of important learning schemes) [7].One obvious advantage of having uniqueness of the gameoutcome is to make the game predictable, which may be usefulfrom a designer standpoint. As mentioned above, by contrast,the number of correlated equilibria is generally greater thanone and more typically infinite. The following proposition provides sufficient condition under which the sensing game(always with costs) has a unique pure NE.
Proposition 4.5: Assume the following two conditions aresatisfied : α > − ( N + γ (cid:63)K − ) ( N − ( K − β (cid:63) ) N − Nβ (cid:63) − [( N + β (cid:63) )( K − β (cid:63) ] γ (cid:63)K − (23) α > − γ (cid:63)K − ( N + β (cid:63) ) f ( β (cid:63) ) β (cid:63) ( N + γ (cid:63)K − ) f ( γ (cid:63)K − ) . (24) Then the unique Nash equilibrium of the game is ( s ∗ , s ∗ , ..., s ∗ K ) = ( NS , NS , ..., NS ) .Condition (23) insures that the non-sensing strategy NS dominates the sensing strategy S when none of the otherplayer sense. Condition (24) insures that the non-sensingstrategy NS dominates the sensing strategy S when some ofthe other player sense. Both conditions together imply thatthe sensing strategy S is always a dominated strategy foreach player. The unique Nash equilibrium of the game is ( s ∗ , s ∗ , ..., s ∗ K ) = ( NS , NS , ..., NS ) .
3) Efficiency:
In a decentralized network, since no or littlecoordination between terminals is available, an important issueis the efficiency of the network at the equilibrium state. Arethe mixed or pure NE of the sensing game efficient in terms ofutility? To be illustrative and to understand in a deep mannerthe problem under investigation, our choice, in this section,is to mainly focus on the − transmitter case but most of theprovided results can be extended to the general case K ≥ . Theorem 4.6: [Number of NE]
The matrix game has thefollowing NE : • a unique NE if and only if ( C
1) : α > β ∗ − γ ∗ − β ∗ γ ∗ ; • three NE if and only if ( C
2) : α < β ∗ − γ ∗ − β ∗ γ ∗ ; • an infinite number of NE if ( C
3) : α = β ∗ − γ ∗ − β ∗ γ ∗ . The proof of this result is provided in App. C. There isalso a strictly mixed equilibrium which can be found usingthe indifference principle. Let ( x, − x ) the mixed strategyfor player 1 and ( y, − y ) the mixed strategy for player 2. Asproven in App. D, there is a unique pair ( x ∗ , y ∗ ) satisfying theindifference principle. The corresponding distribution is givenby : x ∗ = y ∗ = (1 − α ) f ( β ∗ ) β ∗ (1 − β ∗ ) − f ( γ ∗ ) γ ∗ − γ ∗ β ∗ β ∗ X , (25) with X = (1 − α ) f ( β ∗ ) β ∗ (1 − β ∗ ) − f ( γ ∗ ) γ ∗ − γ ∗ β ∗ β ∗ + f ( β ∗ ) β ∗ (1 − β ∗ ) − (1 − α ) f ( β ∗ ) β ∗ − γ ∗ β ∗ γ ∗ and the corresponding equilibriumutilities are : U ( x ∗ , y ∗ ) = R g σ υU ( x ∗ , y ∗ ) = R g σ υ with υ = (1 − α ) f ( β ∗ ) β ∗ (1 − β ∗ ) f ( β ∗ ) β ∗ (1 − β ∗ ) X − f ( γ ∗ ) γ ∗ − γ ∗ β ∗ β ∗ (1 − α ) f ( β ∗ ) β ∗ − γ ∗ β ∗ γ ∗ X .
Fig. 1 represents the three equilibrium utility points fora typical scenario. The shaded area represents the region of feasible utilities for a given scenario (described under Fig. 1).Operating at one of the pure NE can be unfair for one ofthe transmitters and therefore inefficient for a certain fairnesscriterion [12]. Operating at the mixed NE is clearly suboptimalsince it is Pareto-dominated by some feasible pairs of utilities.A way of dealing with fairness or/and Pareto-inefficiencies isto induce correlated equilibria (CE) in the game.In practice, having a correlated equilibrium means that theplayers have no interest in ignoring (public or private) signalswhich would recommend them to play according to certainjoint distribution over the action profiles of the game. Inwireless networks, a correlated equilibrium can be induced bya common signalling from a source which is exogenous to thegame. It may be a signal generated by the receiver itself butalso an FM (frequency modulation) signal, or a GPS (globalpositioning system) signal, meaning that the additional cost foradding this signal may be zero if the terminals are already ableto decode such a signal. At last, note that such a coordinationmechanism is scalable in the sense that it can accommodate ahigh number of transmitters; in practice, physical limitationsmay arise e.g., if the signal is sampled into a finite number ofbits. If α > β ∗ − γ ∗ − β ∗ γ ∗ , as there is only one NE, the convex hullof NE boils down to a point and there does not exist any othercorrelated equilibrium other than this NE. Rather, we assumethat the sensing cost verifies condition (C1) which is the caseof interest since several NE exist (see Theorem 4.6). In thiscase the following result holds. Theorem 4.7: Any convex combination of NE is a CE. Inparticular, if there exists a utility vector ν = ( ν , ν ) and aparameter λ ∈ [0 , such that : ν = λU ( S , NS ) + (1 − λ ) U ( NS , S ) (26) ν = λU ( S , NS ) + (1 − λ ) U ( NS , S ) , (27)then ν is a correlated equilibrium.Clearly, a signal recommending the transmitters to play theaction profile ( S , NS ) (resp. ( NS , S ) ) for a fraction ofthe time equals to λ (resp. to − λ ) induces a correlatedequilibrium. This specific signalling structure leads to the setof equilibria represented by the bold segment in Fig. 1. Thefigure illustrates the potential gains which can be obtained byimplementing a simple coordination mechanism in the sensinggame with costs.We would like to end this section dedicated to the efficiencyof the equilibria of the game by mentioning the potential sub-optimality induced by playing the sensing game and powercontrol game separately (in two consecutive phases). Indeed,it would be legitimate to ask about what would happen if atransmitter were deciding jointly whether to sense or not andhis power level. In such a case the action set of a transmitterwould be : (cid:102) A k = { S k , NS k } × [0 , P max k ] . (28)An action a k = ( s k , p k ) has therefore two components. Thefirst component is discrete whereas the second component iscontinuous. This framework is referred to an hybrid control incontrol theory [10], [27]. While the control theory literatureis rich concerning hybrid control, this is not the case for hybrid control games. In particular, general existence theoremsfor Nash equilibria seem to be unavailable. This is one ofthe reasons we will only consider the special case of twotransmitters. In the − player hybrid control game it can beeasily seen that the two pure NE of the sensing game areno longer equilibria in this new game. Instead, we have thefollowing result. Proposition 4.8: The unique Nash equilibrium of the − player hybrid control game is given by : ( a ∗ , a ∗ ) = ( NS , p NE1 , NS , p NE1 ) (29) where p NE k is given by (4). This result immediately follows from the fact that actionevery action under the form ( S , p k ) is dominated by theaction ( NS , p k ) . Although the proof of this result is trivial, theinterpretation is nonetheless interesting. It shows the existenceof a Braess paradox in the hybrid control game : although theplayers have more options in the hybrid game, the equilibriumutilities are less than those obtained in the separated casewhere they first decide to sense or not and then adapt theirpower level. In additional to implementation considerations,this gives us another reason to perform the decision processin two consecutive phases.V. N UMERICAL RESULTS
In this section, numerical results are provided to validateour theoretical claims. Note that, although simple scenariosconsidered, the authors believe that most of messages andinsights conveyed by the present numerical analysis holdin more advanced simulation setups e.g., considering stan-dardized channel modulation and coding schemes (MCS),real frequency selective channel impulse responses, imperfectchannel state information, and sensing techniques accountingfor estimation noise. Indeed, as explained in [16], the choiceof a specific MCS will generally lead to a packet successrate having the assumed properties. As shown in [11], thecase of frequency selective channels is treatable once thefrequency flat case has been treated. Therefore, only the impactof channel estimation noise seems to be more uncertain andwould call for a more challenging extension of the resultsprovided here. We consider a random CDMA scenario witha spreading factor equal to N and the efficiency function ischosen to be f ( x ) = e − r − x with different parameters r [4].We consider two scenarios. The first scenario is provided inFig. 1. This scenario provides a clear understanding of thevariety of equilibria in the sensing game. The pure, mixed,and correlated equilibria are represented on the utility region.The utility region of the sensing game with two players K = 2 ,no spreading N = 1 , the sensing cost α = 20% , the sigmoidalfunction f ( x ) = e − r − x with r = 0 . and the followingparameters : R g = 2 , R g = 2 . , σ = 1 . The pureactions lead to the utilities marked by circles, the dark greenregion corresponds to the pair of utilities that is achievablewith mixed actions whereas the light green region correspondsto the utilities that are achievable only with correlated actions.The two pure equilibrium utilities, denoted by + , correspondto both upper left extremal pure utilities, the completely mixed equilibrium utility, denoted by × , is located in the interiorof the dark green region. The blue line between the twopure equilibria represents a sub-set of correlated equilibriumutilities that corresponds to the Pareto-optimal frontier. TheNash bargaining solution, denoted by ∗ , corresponds to theintersection of the hyperbolic curve with the set of correlatedequilibrium utilities. It provides a fair and optimal equilibriumsolution for the sensing game.The second scenario considers a sensing game with 17 players, N = 128 sub-carriers, the sensing cost α = 5% , the sigmoidalfunction f ( x ) = e − r − x . For simplicity, we assume an homo-geneous scenario in terms of transmission rate R k = R and r = RB bit/s per Hz for different numbers of leaders. Note thatthe value for R will not matter since only normalized/relativeperformance gains will be considered. The seemingly non-typical choice for K results from typical choices on theother parameters. Indeed, when fixing the spreading factorto N = 128 (typical e.g., in cellular systems), the spectralefficiency to r = 3 bit/s per Hz (typical in cellular systemsas well), one finds that the maximum number of admissibleusers for the Nash equilibrium to be implemented is (see thedenominator of (4)) : r = 3 ⇒ c = 7 ⇒ β ∗ = 7 ⇒ K − N < ⇒ K < where f ( x ) = e − cx , c = 2 r − , β ∗ is the uniquesolution of xf (cid:48) ( x ) − f ( x ) = 0 . Fig. 2 and 3 allows one toevaluate the improvement brought by the Stackelberg approachcompared to the Nash equilibrium approach for the utility ofone leader, one follower and the social utility. The sensingcost influences the results in two ways. First, the sensing costaffects the gain obtained by the follower compared to theleader. Indeed, in this figure, the improvement of a leader isalways larger than the improvement of one follower, whichis not true in general. Second, the improvement of the utilityof one follower compared to the Nash equilibrium utility isnegative when the number of leader is strictly more than14. In that case, the sensing cost is compensated by theimprovement due to the Stackelberg approach compared to theNash equilibrium approach. The optimal number of leaders is5 when considering both the improvement of the utility of oneleader or the improvement of the utility of one follower. Theimprovement of the sum utility of the Stackelberg equilibriumcompared to the social utility at the Nash equilibrium for thesensing game is given in figure 3. The sensing cost decreasesthe improvement of the social utility. This is especially the casewhen the number of follower is large and the number of leaderis small. The Stackelberg approach provides up to . ofimprovement compared to the Nash equilibrium approach.Fig. 4 illustrates how much the total power consumption can bereduced for the sensing game with K = 17 players, N = 128 sub-carriers, the sigmoidal function f ( x ) = e − r − x with r = 3 for different numbers of leaders : note that at the sametime, the energy-efficiency is optimized. The best reduction ofthe power consumption is achieved with the number of leadersis 5, the reduction of the power consumption is more than .We observe that the total power reduction is maximum with thenumber of cognitive users maximizes the social welfare of thesystem. Finally, the figure 5 represents the improvement of themaximal social welfare (depending on the number of cognitive users) of the sensing game compared to the social welfare atthe Nash equilibrium solution, depending on the load ( K/N ) of the system. The four curves correspond to different sensingcost α ∈ { , , , } . When the load approaches itsmaximal value /β (cid:63) + 1 /N , the improvement of the socialutility is greater than . Then, we can conclude thatour hierarchical framework with optimal number of cognitiveequipments becomes more efficient in terms of social utilitywhen a cognitive wireless network is high loaded.VI. C ONCLUSION
In this paper, we have introduced a new power controlgame where the action of a player is hybrid, one componentof the action is discrete while the other is continuous. Thefirst component is discrete since it corresponds to decidingwhether to sense the radio environment or not ; the secondcomponent is continuous because it corresponds to choosingthe transmit power level in an interval. Whereas the generalstudy of hybrid games is of independent and game-theoreticinterest and remains to be done, it turns out that in our casewe can prove the existence of a kind of Braess paradox whichallows us to restrict our attention to two separate games playedconsecutively : choosing the discrete and continuous actionsjointly is less efficient than choosing them separately overtime. The power control game is studied in detail and itis shown that there exists an optimal number of cognitivetransmitters which maximizes the network utility, meaningthat introducing too much cognition is not globally energy-efficient. This holds whether the cost of sensing is set tozero or not. From an individual point of view, the intuitionwhich consists in saying that sensing is beneficial only ifthe sensing cost is acceptable, can be proved. As distributednetworks are considered, global efficiency of the network isgenerally not guaranteed. Equilibria are indeed less energy-efficient (say in terms of sum utility) than the centralizedsolution. The (hierarchical) approach we propose can thereforebe seen as a tradeoff in terms of global performance andrequired signaling. Conducting a refined analysis in terms ofsignaling for the power control problem would be relevant.On the other hand, the sensing game can be shown to havedesirable properties like being weighted potential. This is akey property since many learning algorithms are known toconverge in such games, proving that this decision can belearned over time with partial information only. Additionally,this game is shown to have a non-trivial set of correlatedequilibria. These equilibria are very useful since they allowone to introduce some fairness among the transmitters andcan be stimulated by a public signal incurring no cost interms of extra signalling from the receiver ; in this respectthe famous Nash bargaining solution (used in the wirelessliterature for having both a fair and cooperative solution, seee.g., [20][24]) can be reached. This work therefore providesseveral new results of practical interest for cognitive wirelessnetworks but, of course, the proposed concepts would need tobe developed further to make them more appealing in termsof implementability. In particular, technical issues related tospectrum usage might be considered by introducing frequency selective channels and the corresponding power allocationproblem. Considering a more general structure of interferencenetworks, the relevance of successive interference cancelationin terms of energy-efficiency might be assessed. Of course,classical issues such as the impact of channel estimation isalso of practical interest, especially regarding to the fact thatsome learning algorithms are known to be robust against thistype of errors. A PPENDIX AR EVIEW OF SOME GAME - THEORETIC CONCEPTS
A. Correlated Equilibria
In this section, we provide the definition of correlatedequilibrium (CE). This equilibrium concept was introduced byAumann in [3] and extends the concept of Nash equilibrium.Correlated equilibria are used in section IV-C3 in order toprovide more fair equilibrium solutions.
Definition A.1: A probability distribution Q ∈ ∆( A ) is acanonical correlated equilibrium if for each player i , for eachaction a i ∈ A i that satisfies Q ( a i ) > we have : (cid:88) a − i ∈A − i Q ( a − i | a i ) u i ( a i , a − i ) ≥ (cid:88) a − i ∈A − i Q ( a − i | a i ) u i ( b i , a − i ) , ∀ b i ∈ A i . (30) The result of Aumann 1987 [3] states that for any correlatedequilibrium, it corresponds to a canonical correlated equilib-rium.
Theorem A.2 (Aumann 1987, prop. 2.3 [3]):
The utilityvector u is a correlated equilibrium utility if and only if thereexists a distribution Q ∈ ∆( A ) satisfying the linear inequalitycontraint (30) with u = E Q U . B. Potential of the Sensing Game
In this section, we provide the potential function of thesensing game presented in section IV.
Theorem A.3: The equilibria of the above potential gameare the maximizers of the Rosenthal potential function [32]. { S = ( A , . . . , A K ) | S ∈ N E } = arg max ( F,L ) Φ( F, L )= arg max ( F,L ) (cid:34) (1 − α ) F (cid:88) i =1 U ( S, i, K − i )+ L (cid:88) j =1 U ( N S, K − j, j ) The proof follows directly from the one of Rosenthal’s the-orem [32]. Let us simplify the expression of the potentialfunction, which gives: Φ( F, L ) = (cid:34) (1 − α ) F (cid:88) i =1 U ( S, i, K − i ) + L (cid:88) j =1 U ( NS, K − j, j ) (cid:35) = (1 − α ) F (cid:88) i =1 g i R i σ f ( β (cid:63) ) Nβ (cid:63) ( N + γ (cid:63) ( K − i ) ) (cid:0) N − Nβ (cid:63) − [( N + β (cid:63) )( K − i )+ ( i + 1) β (cid:63) ] γ (cid:63) ( K − i ) (cid:1) + L (cid:88) j =1 g i R i σ f ( γ (cid:63)j ) Nγ (cid:63)j ( N + β (cid:63) ) (cid:0) N − Nβ (cid:63) − [( N + β (cid:63) ) j + ( K − j + 1) β (cid:63) ] γ (cid:63)j (cid:1) . A PPENDIX BP ROOF OF PROPOSITION ξ min such that: α T min (cid:96) ∈L ( g f(cid:96) p (cid:96) ) ≥ ξ min ⇐⇒ α ≥ ξ min T min (cid:96) ∈L ( g f(cid:96) p (cid:96) ) where T is the block duration, g f(cid:96) is the channel gain betweenany leader (cid:96) ∈ L and the considered follower f ∈ F , p (cid:96) is thepower level of the leader (cid:96) ∈ L and α is the sensing cost. Theutility of the follower is maximized when the sensing cost α is minimal, that is α (cid:63) = ξ min T min (cid:96) ∈L ( g f(cid:96) p (cid:96) ) . max α ∈ [0 , U SE f ( α ) U SE l ≥ ⇐⇒ (1 − α (cid:63) ) R k f ( β (cid:63) ) p SE f R k f ( β (cid:63) ) p SE l ≥ ⇐⇒ (cid:18) − ξ min T min (cid:96) ∈L ( g f(cid:96) p (cid:96) ) (cid:19) f ( β (cid:63) ) β (cid:63) ( N + β (cid:63) ) f ( γ (cid:63)L ) γ (cid:63)L ( N + γ (cid:63)L ) ≥ ⇐⇒ f ( β (cid:63) ) β (cid:63) ( N + β (cid:63) ) f ( γ (cid:63)L ) γ (cid:63)L ( N + γ (cid:63)L ) − ≥ ξ min T min (cid:96) ∈L ( g f(cid:96) p (cid:96) ) f ( β (cid:63) ) β (cid:63) ( N + β (cid:63) ) f ( γ (cid:63)L ) γ (cid:63)L ( N + γ (cid:63)L ) ⇐⇒ (cid:18) − f ( γ (cid:63)L ) γ (cid:63)L ( N + γ (cid:63)L ) f ( β (cid:63) ) β (cid:63) ( N + β (cid:63) ) (cid:19) ≥ ξ min T min (cid:96) ∈L ( g f(cid:96) p (cid:96) ) ⇐⇒ T min (cid:96) ∈L ( g f(cid:96) p (cid:96) ) (cid:18) − f ( γ (cid:63)L ) γ (cid:63)L ( N + γ (cid:63)L ) f ( β (cid:63) ) β (cid:63) ( N + β (cid:63) ) (cid:19) ≥ ξ min This concludes the proof of Proposition 3.4.A
PPENDIX CP ROOF OF T HEOREM − player sensing game. The first important remark is that theNash equilibrium utilities are always dominated by the Stack-elberg equilibrium utilities. This implies that the followingequation holds for any parameters α ≥ . U ( NS , S ) ≤ U ( S , S ) U ( S , NS ) ≤ U ( S , S ) Thus the action ( S , S ) is not an equilibrium of the game.To compute the equilibria of this game, it remains to computethe following differences: U ( NS , NS ) − U ( S , NS ) U ( NS , NS ) − U ( NS , S ) The above differences are equal and does not depend ona particular player. We provide the proof of Theorem 4.6. Suppose first that condition (C2) is met. α < β (cid:63) − γ (cid:63) − β (cid:63) γ (cid:63) ⇐⇒ − γ (cid:63) β (cid:63) − β (cid:63) − γ (cid:63) (1 − γ (cid:63) β (cid:63) ) < − α ⇐⇒ f ( β (cid:63) ) β (cid:63) (1 − β (cid:63) ) < (1 − α ) f ( β (cid:63) ) β (cid:63) − β (cid:63) γ (cid:63) γ (cid:63) ⇐⇒ R i g i f ( β (cid:63) )(1 − β (cid:63) ) σ β (cid:63) < (1 − α ) R i g i f ( β (cid:63) )(1 − γ (cid:63) β (cid:63) ) σ β (cid:63) (1 + γ (cid:63) ) The last inequality implies that the games has two pures equi-libria ( NS , S ) , ( S , NS ) and one strictly mixed equilibrium ( x ∗ , y ∗ ) defined by the equations (25). If condition (C1) issatisfied, then the strategies ( S ) and ( S ) are dominated andthen the game has one pure equilibrium ( NS , NS ) and ifcondition (C3) is met, the game has an infinite number of NE.A PPENDIX DM IXED N ASH E QUILIBRIA
If condition (C2) is met, the sensing game has a strictlymixed equilibrium. In this section, we provide a characteri-zation of the mixed equilibrium strategies ( x ∗ , y ∗ ) using theindifference principle. R g f ( β (cid:63) )(1 − β (cid:63) ) σ β (cid:63) · y + R g f ( γ (cid:63) )(1 − γ (cid:63) β (cid:63) ) σ γ (cid:63) (1 + β (cid:63) ) · (1 − y )= (1 − α ) R g f ( β (cid:63) )(1 − γ (cid:63) β (cid:63) ) σ β (cid:63) (1 + γ (cid:63) ) · y +(1 − α ) R g f ( β (cid:63) )(1 − β (cid:63) ) σ β (cid:63) · (1 − y ) , which is equivalent to: y · [ R g f ( β (cid:63) )(1 − β (cid:63) ) σ β (cid:63) − (1 − α ) R g f ( β (cid:63) )(1 − γ (cid:63) β (cid:63) ) σ β (cid:63) (1 + γ (cid:63) )+(1 − α ) R g f ( β (cid:63) )(1 − β (cid:63) ) σ β (cid:63) − R g f ( γ (cid:63) )(1 − γ (cid:63) β (cid:63) ) σ γ (cid:63) (1 + β (cid:63) ) ]= (1 − α ) R g f ( β (cid:63) )(1 − β (cid:63) ) σ β (cid:63) − R g f ( γ (cid:63) )(1 − γ (cid:63) β (cid:63) ) σ γ (cid:63) (1 + β (cid:63) ) . Then we obtain: y = (1 − α ) f ( β (cid:63) ) β (cid:63) (1 − β (cid:63) ) − f ( γ (cid:63) ) γ (cid:63) − γ (cid:63) β (cid:63) β (cid:63) X , with X = (1 − α ) f ( β (cid:63) ) β (cid:63) (1 − β (cid:63) ) − f ( γ (cid:63) ) γ (cid:63) − γ (cid:63) β (cid:63) β (cid:63) + f ( β (cid:63) ) β (cid:63) (1 − β (cid:63) ) − (1 − α ) f ( β (cid:63) ) β (cid:63) − γ (cid:63) β (cid:63) γ (cid:63) . Replacing the above y into theindifference equation, we obtain the utility of player 1 at themixed equilibrium. U ( x , y ) = R g σ (cid:32) (1 − α ) f ( β (cid:63) ) β (cid:63) (1 − β (cid:63) ) f ( β (cid:63) ) β (cid:63) (1 − β (cid:63) ) X − f ( γ (cid:63) ) γ (cid:63) − γ (cid:63) β (cid:63) β (cid:63) (1 − α ) f ( β (cid:63) ) β (cid:63) − γ (cid:63) β (cid:63) γ (cid:63) X (cid:33) , with X = (1 − α ) f ( β (cid:63) ) β (cid:63) (1 − β (cid:63) ) − f ( γ (cid:63) ) γ (cid:63) − γ (cid:63) β (cid:63) β (cid:63) + f ( β (cid:63) ) β (cid:63) (1 − β (cid:63) ) − (1 − α ) f ( β (cid:63) ) β (cid:63) − γ ∗ β (cid:63) γ (cid:63) . The same argument applies: U ( x , y ) = R g σ (cid:32) (1 − α ) f ( β (cid:63) ) β (cid:63) (1 − β (cid:63) ) f ( β (cid:63) ) β (cid:63) (1 − β (cid:63) ) X − f ( γ (cid:63) ) γ (cid:63) − γ (cid:63) β (cid:63) β (1 − α ) f ( β (cid:63) ) β (cid:63) − γ (cid:63) β (cid:63) γ (cid:63) X (cid:33) . A PPENDIX EP ROOF OF T HEOREM
Theorem E.1: The game G is a potential game if and onlyif for every player i, j ∈ K , every pair of actions s i , t i ∈ A i and s j , t j ∈ A j and every joint action s k ∈ A K \{ i,j } , wehave that U i ( t i , s j , s k ) − U i ( s i , s j , s k ) + U i ( s i , t j , s k ) − U i ( t i , t j , s k ) + U j ( t i , t j , s k ) − U j ( t i , s j , s k ) + U j ( s i , s j , s k ) − U j ( s i , t j , s k ) = 0 Let us prove that the two conditions provided by our theoremare equivalent to the one of corollary 2.9 in [25]. We introducethe following notation defined for each player i ∈ K and eachaction T ∈ A . µ i = R i g i σ and U T ( t i , t j , s k ) = U Ti ( t i , t j , s k ) µ i . For every player i, j ∈ K , every pair of actions s i , t i ∈ A i and s j , t j ∈ A j and every joint action s k ∈ S K \{ i,j } , we havethe following equivalences: U i ( NS i , S j , S k ) − U i ( S i , S j , S k ) + U i ( S i , NS j , S k ) − U i ( NS i , NS j , S k ) = U j ( NS i , S j , S k ) − U j ( NS i , NS j , S k ) + U j ( S i , NS j , S k ) − U j ( S i , S j , S k ) ⇐⇒ µ i = µ j U L ( F + 1 , L + 1) − U L ( F, L + 2)+(1 − α )( U F ( F + 1 , L + 1) − U F ( F + 2 , L )) = 0 Thus the sensing game is a potential game if and only if oneof the two following conditions is satisfied : ∀ ( i, j ) ∈ K , R i g i = R j g j ,U L ( F + 1 , L + 1) − U L ( F, L + 2) =(1 − α )[ U F ( F + 2 , L ) − U F ( F + 1 , L + 1)] . A PPENDIX FP ROOF OF T HEOREM (cid:101) G = ( K, ( A ) i ∈K , ( ˜ U i ) i ∈K ) (31)Where the utility are defined by the following equations with µ i = R i g i σ . ˜ U i ( s i , s − i ) = U i ( s i , s − i ) µ i (32)From the above demonstration, it is easy to show that, forevery player i, j ∈ K , every pair of actions s i , t i ∈ A i and U t ili t y o f p l a y e r ( b i t/ J ) Convex Hull of the Utility RegionAchievable Utility RegionPareto − Optimal Correlated EquilibriaPure Actions ProfilesPure Nash EquilibriaMixed Nash EquilibriumNash Bargaining SolutionNash Bargaining Curve K = 2N = 1R g = 2R g = 2.5 (cid:109) = 1r = 0.9 (cid:95) = 20 %(NS ,NS )(S ,S ) (S ,NS )(NS ,S ) Fig. 1. The figure depicts, for a -transmitter scenario, the region of achiev-able utilities of the sensing game where each transmitter decides whether tosense of not to sense. The figure also shows the different equilibrium pointsof the game. One of the messages of this figure is the interest in terms offairness in stimulating a correlated equilibrium instead of a Nash equilibrium.In particular a Nash bargaining solution can be obtained. − I m p r o v e m en t o f t he SE w . r .t. t he N E : u SE / u N E − [ % ] Improvement of the utility of one leaderImprovement of the utility of one followerK = 17N = 128r = 3 (cid:95) = 5 %
Fig. 2. This figure represents the relative gain (in % ) in terms of individualenergy-efficiency obtained by equipping F = K − L transmitters with acognitive radio. For typical scenarios, we see that maximizing the number ofcognitive transmitters is not optimal. On the other hand, if there is only onecognitive radio ( F = 1 or L = 16 , as assumed in [22]) one can degrade theindividual performance for a typical value for the sensing cost ( of thetime-slot is spent for sensing). s j , t j ∈ A j and every joint action s k ∈ A K \{ i,j } , we have thefollowing equality: ˜ U i ( t i , s j , s k ) − ˜ U i ( s i , s j , s k ) + ˜ U i ( s i , t j , s k ) − ˜ U i ( t i , t j , s k ) =˜ U j ( t i , s j , s k ) − ˜ U j ( t i , t j , s k ) + ˜ U j ( s i , t j , s k ) − ˜ U j ( s i , s j , s k ) . Using Corollary 2.9 in [25], we conclude that the sensing gameis a weighted potential game. − I m p r o v e m en t o f t he SE w . r .t. t he N E : w SE / w N E − [ % ] Improvement of the social utility for (cid:95) = 0 %Improvement of the social utility for (cid:95) = 5 %Improvement of the social utility for (cid:95) = 10 %Improvement of the social utility for (cid:95) = 15 %K = 17N = 128r = 3
Fig. 3. This figure represents the improvement in terms of utility sum ornetwork energy-efficiency at the Stackelberg equilibrium compared to thecase of Nash equilibrium (i.e., no transmitter is equipped cognitive radio)with K = 17 players, N = 128 , the efficiency function is f ( x ) = e − r − x with r = 3 bit/s per Hz and for different numbers of leaders.The different curves corresponds to different values for the sensing cost : α ∈ { , , , } . When of the time has to be spent for sensing,the network energy-efficiency can be improved by whereas it is when this cost is close to zero. − − − − − − − − − R edu c t i on o f t he t o t a l po w e r c on s u m p t i on f o r t he SE w . r .t. t he N E : p SE / p N E − [ % ] Reduction of the total power consumptionK = 17N = 128r = 3
Fig. 4. From previous figures, we know that equipping the network with anadequate number of cognitive radios can significantly improve the networkenergy-efficiency. It turns out that not only efficiency is maximized by doingso but that the total consumed power is also reduced. Scenario illustrated : K = 17 players, N = 128 , the efficiency function is f ( x ) = e − r − x with r = 3 for different numbers of leaders. I m p r o v e m en t o f t he s o c i a l u t ili t y f o r t he SE w . r .t. t he N E : w SE / w N E − [ % ] Improvement of the social utility for (cid:95) = 0 %Improvement of the social utility for (cid:95) = 5 %Improvement of the social utility for (cid:95) = 10 %Improvement of the social utility for (cid:95) = 15 %Maximum loadK = 1,...,37N = 256r = 3
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Mael Le Treust
Mael Le Treust earned his DiplômedÕEtude Approfondies (M.Sc.) degree in Optimiza-tion, Game Theory and Economics (OJME) from theUniversité de Paris VI (UPMC), France in 2008 andhis Ph.D. degree from the Université de Paris Sud XIin 2011, at the Laboratoire des signaux et syst ´Rmes(joint laboratory of CNRS, Supélec, Université deParis Sud XI) in Gif-sur-Yvette, France. He was apost-doctoral researcher at the Institut d’électroniqueet d’informatique Gaspard Monge (Université Paris-Est) in Marne-la-Vallée, France and he is currentlya post-doctoral researcher at the Centre Energie, Matériaux et Télécommu-nication (Université INRS) in Montréal, Canada. He was also a Math T.A.at the Université de Paris I (Panthéon-Sorbonne) and Université de Paris VI(UPMC), France. His research interests are information theory, game theoryand wireless communications.
Samson Lasaulce
Prof. Samson Lasaulce receivedhis BSc and Agr ˝Ogation degree in Physics fromˇCcole Normale Sup ˝Orieure (Cachan) and his MScand PhD in Signal Processing from ˇCcole NationaleSup ˝Orieure des T ˝Ol ˝Ocommunications (Paris). Hehas been working with Motorola Labs for threeyears (1999, 2000, 2001) and with France T ˝Ol ˝OcomR&D for two years (2002, 2003). Since 2004, hehas joined the CNRS and Sup ˝Olec as a SeniorResearcher. Since 2004, he is also Professor at ˇCcolePolytechnique. His broad interests lie in the areasof communications, information theory, signal processing with a specialemphasis on game theory for communications. Samson Lasaulce is therecipient of the 2007 ACM VALUETOOLS, 2009 IEEE CROWNCOM,2012 ACM VALUETOOLS best student paper award and the 2011 IEEENETGCOOP best paper award. He is an author of the book ?Game Theoryand Learning for Wireless Networks: Fundamentals and Applications?. He iscurrently an Associate Editor of the IEEE Transactions on Signal Processing.
Yezekael Hayel