On the Two-user Multi-carrier Joint Channel Selection and Power Control Game
aa r X i v : . [ c s . I T ] J un On the Two-user Multi-carrier Joint ChannelSelection and Power Control Game
Majed Haddad ∗ , Piotr Wiecek † , Oussama Habachi ‡ and Yezekael Hayel ∗∗ CERI/LIA, University of Avignon, Avignon, France † Faculty of Pure and Applied Mathematics, Wroclaw University of Technology,Poland ‡ XLIM, University of Limoges, Limoges, France
Abstract
In this paper, we propose a hierarchical game approach to model the energy efficiency maximizationproblem where transmitters individually choose their channel assignment and power control. We conducta thorough analysis of the existence, uniqueness and characterization of the Stackelberg equilibrium.Interestingly, we formally show that a spectrum orthogonalization naturally occurs when users decidesequentially about their transmitting carriers and powers, delivering a binary channel assignment. Bothanalytical and simulation results are provided for assessing and improving the performances in termsof energy efficiency and spectrum utilization between the simultaneous-move game (with synchronousdecision makers), the social welfare (in a centralized manner) and the proposed Stackelberg (hierarchical)game. For the first time, we provide tight closed-form bounds on the spectral efficiency of such a model,including correlation across carriers and users. We show that the spectrum orthogonalization capabilityinduced by the proposed hierarchical game model enables the wireless network to achieve the spectralefficiency improvement while still enjoying a high energy efficiency.
Index Terms
Energy efficiency; spectral efficiency; multi-carrier system; spectrum orthogonalization; game theory;Stackelberg equilibrium.
I. I
NTRODUCTION
Ecological concerns are steadily attracting more and more attention in wireless communications[1], [2]. From the operators’ perspective, energy efficiency not only has great ecological benefitsand represents social responsibility in fighting climate change, but also has significant economicbenefits. Therefore, innovative solutions that support traffic increase and maintain a limited energy consumption need to be considered at both system and device levels in order to address environ-mental and operational costs. Recently, Cisco systems have pointed out that the global mobile datatraffic will increase nearly tenfold between 2014 and 2019, giving incentive for service providers toreduce their OpEx by reducing their energy consumption [3]. This suggests to shift from pursuingoptimal capacity and spectral efficiency to efficient energy usage when designing wireless networks.Indeed, spectral efficiency has been a traditional requirement of wireless architectures, especiallywhen their access is limited to scarce spectrum. As a result, recent trends in mobile client accesstend to support both spectral and energy efficiency at the same time while addressing a widevariety of delay and throughput objectives [4].C
ONTRIBUTIONS
To address these crucial issues among others, we propose to study energy efficient wirelessnetworks in which we introduce a degree of hierarchy among users. More specifically, we considerenergy efficient multi-carrier wireless networks that can be modeled by a decentralized multipleaccess channel. The network is said to be decentralized in the sense that each user can freelychoose his power control policy and carrier assignment in order to selfishly maximize a certainindividual performance criterion, called utility (or payoff) in the context of game theoretic studies.We formally prove that the hierarchical structure of the game naturally leads to a spectrumorthogonalization pattern where the components of the network have incentive to transmit on dif-ferent carriers. This orthogonalization feature across the multiple interfering devices is particularlyappealing, not only from an interworking perspective (as a result of reduced infrastructure), butalso for increasing both network coverage and data capacity without the need to split the availablespectrum. In this sense, we prove that the advantage of the hierarchical (Stackelberg) model thatwe propose over the simultaneous-move model in [5] is rather significant.One could wonder that, as soon as the number of carriers is high, interference can be avoidedwith high probability. We show next that users still experience interference even when the numberof carriers to number of users ratio exceeds a few units, especially for synchronous decisionmakers. Moreover, to the best of our knowledge, performance bounds have never been derived inthe multiple carrier context. This allows us to provide tight closed-form bounds on the spectralefficiency of such a model. We formally prove that the spectrum orthogonalization capabilityinduced by the proposed hierarchical game model enables the wireless network to achieve thespectral efficiency improvement while still enjoying a high energy efficiency. In particular, weshow that the orthogonalization feature makes correlation over carriers suitable for energy efficientsystems as it brings more orthogonalization over the system (and thus leads to higher spectralefficiency), while correlation over users is not suited as it degrades the spectral efficiency. R ELATED L ITERATURE AND N OVELTY OF THE W ORK
To reduce the network energy consumption, [6] proposed an optimal traffic aware scheme usingan online stochastic game theoretic algorithm. In [7], authors proposed a joint transmitter andreceiver optimization for the energy efficiency in orthogonal frequency-division multiple-access(OFDMA) systems. Energy efficient power control game has been first proposed by Goodman etal. in [8] for flat fading channels and re-used by [5] for multi-carrier systems. [9] proposed anenergy efficient topology control game for wireless ad hoc networks in the presence of selfishnodes. All these works do not consider hierarchy among different actors in the system. However,as mentioned in [8] the Nash equilibrium in such games can be very energy inefficient. Note thatthe Stackelberg formulation arises naturally in many context of practical interest. For example, thehierarchy is inherent to cognitive radio networks (CRNs) where the user with the higher priority( i.e. , the leader or the primary user (PU)) transmits first, then the user with the lower priority( i.e. , the follower or the secondary user (SU)) transmits after sensing the spectral environment[10]–[12]. This is also a natural setting for heterogeneous wireless networks due to the absenceof coordination among the small cells, and between small cells and macro cells [13]–[15]. Therehave been many works on Stackelberg games [16]–[18], but they do not consider energy efficiencyfor the individual utility as defined in [8]. They rather consider transmission rate-type utilities (see e.g., [19], [20]).In a prior work [21], we proposed a hierarchical game theoretic model for two-user–two-carrierenergy efficient wireless systems. It was shown that, for the vast majority of cases, users choosetheir transmitting carriers in such a way that if the leader transmits on a given carrier, the followerhas incentive to choose the other carrier. One major motivation of this paper is to extend the originalproblem in [21] to some general models that can be widely used in practice by considering anarbitrary number of carriers.The work that is most closely related to ours is [22], where the hierarchical game was for-mulated for the energy efficiency maximization problem in the single carrier system. Notably, ithas been proved that, when only one carrier is available for the players, there exists a uniqueStackelberg equilibrium. However, multi-carrier systems have gained intense interest in wirelesscommunications, making the use of multi-carrier transmissions much more appealing for futurewireless systems, such as LTE. In fact, the multi-dimensional nature of such a problem along withthe physical properties of the transmission phenomenon make the extension to an arbitrary numberof carriers problem much more challenging than the single carrier model. We will see later in thepaper that, contrary to [22], we show that, when we come up to study multi-carrier hierarchicalgames, the degree of freedom increases and leading becomes better than following . This meansthat a player can often take advantage of playing first (as the leader), but not always. Indeed, if the players are in the same conditions, a player can improve his utility by playing after observingthe action of the other player.In the light of the above, the paper is structured as follows. The general system model ispresented in Sec. II. Sec. III reviews the simultaneous-move game and presents the hierarchicalgame problem. Then, in Sec. IV, we characterize the Stackelberg equilibrium, and we evaluate theperformance of the Stackelberg approach in Sec. V. Sec. VI provides numerical results to illustrateand validate the theoretical findings derived in the previous sections. Additional comments andconclusions are provided in Sec. VII.II. E
NERGY E FFICIENT W IRELESS N ETWORK M ODEL
We consider a wireless network, in which mobile users access to the spectrum in an asynchronousway. We assume that the overall bandwidth can be divided into an arbitrary number of narrow-bandcarriers ( K ≥ , and that the carriers are narrow enough to undergo flat fading. Let us furthersuppose that the channels are quasi-static flat fading, i.e., the channel gains are constant duringeach frame but may change from one frame to the next.Without the constraint of exclusive assignment of each carrier for users, we generally formulatethe problem of energy efficiency maximization by allowing that a carrier could be shared bymultiple users. One can think of heterogeneous networks (HetNets) or ultra-dense networks (UDNs)composed of different cellular layers and multiple access technologies. In order to improve theefficiency of spectrum use, multiple overlapping networks operate on the same frequency bands,causing (co-channel) interference, which, in turn, can cause harmful throughput degradation. Tobe specific, in the following, we will consider a decentralized multiple access channel composedof a leader – indexed by , having the priority to access the medium, and a follower – indexedby that accesses the medium after observing the action of the leader. This setting is particularlyrelevant for CRNs with the PU as the leader and the SU as the follower, with the difference thatno guarantee of service to the PU is considered while sharing the spectrum with the SU. It isalso suited for sparse mobile networks in which one may neglect the possibility of simultaneousinterference of more than two users. An extension of the proposed model to multiple users withmulti-hierarchical levels can be found in [23], where two nearly-optimal algorithms that ensurecomplete spectrum orthogonalization across users were proposed. Notice that closed-form solutionsfor the multi-user hierarchical game is in general very difficult to obtain.Accordingly, for any user n ∈ { , } and m = n , the received signal-to-noise plus interferenceratio (SINR) is expressed as γ kn = g kn p kn σ + g km p km := p kn b h kn ; for k = 1 , . . . , K (1) We will call b h kn the effective channel gain , defined as the ratio between the SINR and the trans-mission power of the other users over the k th carrier. g kn and p kn are resp. the fading channel gainand the transmitted power of user n transmitting on carrier k , whereas σ stands for the varianceof the Gaussian noise. We statistically model the channel gains g kn to be independent identicallydistributed (i.i.d.) over the fading coefficients. It follows from the above SINR expression that thestrategy chosen by a user affects the performance of other users in the network through multiple-access interference.The system model adopted throughout the paper is based on the seminal paper [8] that definesthe energy efficiency framework. In order to formulate the power control problem as a game,we first need to define a utility function suitable for data applications. Let us first define the“efficiency" function f ( · ) , which measures the packet success rate. In brief, when SINR is verylow, data transmission results in massive errors and the goodput (rate conditioned to errors) tends to ; when SINR is very high, data transmission becomes error-free and the rate grows asymptoticallyto a constant. However, achieving a high SINR level requires the user terminal to transmit at ahigh power, which in turn results in low battery life. This phenomenon is concisely captured byan increasing, continuous and S-shaped “efficiency" function f ( · ) . A more detailed discussion ofthe efficiency function can be found in [24]–[26]. The following utility function allows one tomeasure the corresponding tradeoff between the transmission benefit (total goodput over the K carriers) and cost (total power over the K carriers): u n ( p , p ) = R n · K X k =1 f ( γ kn ) K X k =1 p kn , (2)where R n is the transmission data rate of user n and p n is the power allocation vector of user n over all carriers, i.e., p n = ( p n , . . . , p Kn ) . The quantity R n can be viewed as the gross (transmission)data rate on the radio interface which only depends on the user’s application/service induced byhigh layers such as the transport and the application layers. This target rate may depend on thetype of application, but not on the physical layer or the wireless environment of the user. Theutility function u n , that has bits per Joule as units, perfectly captures the tradeoff between goodputand battery life, and is particularly suitable for applications where energy efficiency is crucial.III. T HE GAME THEORETIC FORMULATION
One proposal for designing spectrum sharing is through game theory which offers basis to modelinteractions between interacting users and develop decentralized and/or distributed algorithms forresource allocation.
A. The simultaneous-move game problem
The interaction between users can be modeled through a non-cooperative game where each usermaximizes his energy efficiency subject to interference constraints, given adversarial decisions.An important solution concept of the game under consideration is the Nash equilibrium, which isa fundamental concept in the strategic games. It is a vector of strategies (referred to hereafter andinterchangeably as actions), one for each player, p NE = { p NE , p NE } such that no player hasincentive to unilaterally change his strategy. Definition 1.
A strategy vector p NE = { p NE , p NE } is a Nash Equilibrium (NE) if and only if: ∀ p = p NE , u ( p NE , p NE ) ≥ u ( p , p NE ) and ∀ p = p NE , u ( p NE , p NE ) ≥ u ( p NE , p ) . In what follows, we define a less robust stable strategy vector for non-cooperative games inwhich the Nash equilibrium is a too strong concept. If there exists an ǫ > such that (1 + ǫ ) u n ( p n ǫNE , p ǫNE − n ) ≥ u n ( p n , p − n ǫNE ) for every action p n = p n ǫNE , we say that the vector p ǫNE = { p ǫNE , p ǫNE } is an ǫ -Nash equilibrium.The Nash equilibrium concept assumes that the players decide simultaneously. One importantframework of non-cooperative games is to assume that one player can observe the decision of theother player before deciding. This concept can be related to asymmetric information/decision innon-cooperative games and is related to the concept of Stackelberg equilibrium. B. The hierarchical game problem
Hierarchical models in wireless networks are motivated by the idea that the utility of the leaderobtained at the Stackelberg equilibrium can often be improved over his utility obtained at theNash equilibrium when the two users play simultaneously [17]. It has been proved, in [22], thatwhen only one carrier is available for the players, there exists a unique Stackelberg equilibrium inwhich both the leader and the follower improve their utilities. The goal is then to find a Stackelbergequilibrium in this bi-level game [27].In this work, we consider a Stackelberg game framework in which, a foresighted follower adaptshis power allocation vector p , based on the power vector of the leader p chosen in advance. Thepower allocation of the shortsighted leader will re-embody in the form of interference introduced tothe foresighted follower as given by Eq. (2). At the core lies the idea that, the foresighted follower The − n subscript on vector p stands for “except user n ". will extract the useful asymmetry information in order to make more efficient hierarchical decisionmaking. Definition 2. ( Stackelberg equilibrium ): A vector of actions e p = ( e p , e p ) = ( e p , . . . , e p K , e p , . . . , e p K ) is called Stackelberg equilibrium ifand only if: e p ∈ arg max p u ( p , p ( p )) , where for all p , we have p ( p ) ∈ arg max p u ( p , p ) , and e p = p ( e p ) . Remark . Note that, for sake of clarity, we will only consider the most interesting (and non-trivial)regime where the transmit powers are less than maximal power levels. However, all the resultscan be easily extended to the case of finite powers.IV. C
HARACTERIZATION OF THE S TACKELBERG EQUILIBRIUM
We first determine the best-response function of the follower depending on the action of theleader. This approach is similar to backward induction technique. This result comes directly fromProposition of [5]. For making this paper sufficiently self-contained, we review here the latterproposition. Proposition 1 (Given in [5]) . Given the power allocation vector p of the leader, the best-responseof the follower is given by p k ( p ) = γ ∗ ( σ + g k p k ) g k , for k = L ( p )0 , for all k = L ( p ) (3) with L ( p ) = arg max k b h k ( p k ) and γ ∗ is the unique (positive) solution of the first order equation x f ′ ( x ) = f ( x ) (4)Note that Eq. (4) has a unique solution if the efficiency function f ( · ) is sigmoidal [28]. Thelast proposition says that the best-response of the follower is to use only one carrier, the one suchthat the effective channel gain is the best.Let us first present a useful result that will allow us to reduce the complexity of the originalproblem (with K carriers) to a simpler one where we only focus on the two best carriers. Proposition 2.
Denote by B and S two carriers for the leader for which g k is the highest andthe second highest respectively, while by B and S the ones with two highest g k (that is, for the follower). If the Stackelberg game has an equilibrium, then it has an equilibrium where the leadertransmits on one of the carriers { B , S } , while the follower transmits on one of the carriers { B , S } . For the clarity of the exposition, all the propositions are proven in the Appendix.Given this result, we may only concentrate on strategies where each of the players uses one ofhis two best carriers. The proposition below gives the algorithm to compute the equilibrium powerallocations for both players. Before the proposition, we introduce additional notation, namely ˆ γ = g B − g S g S . Proposition 3. If B = B then equilibrium power allocation of each of the players is p kn = γ ∗ σ g kn when k = B n otherwiseIf B = B then the equilibrium power allocations of the players are computed in three steps: If ˆ γ ≤ γ ∗ then equilibrium power allocation of the leader is p k = γ ∗ σ g k when k = B otherwiseand that of the follower is p k = γ ∗ σ g k when k = S otherwiseOtherwise, go to steps 2 and 3. Find all the solutions x ≤ ˆ γ γ ∗ (1+ˆ γ ) to the equation ( x − x γ ∗ ) f ′ ( x ) = f ( x ) (5) If there are solutions different than x = 0 , choose the one for which f ( x )(1 − xγ ∗ ) x is the highest.Let β ∗ be this solution. Compare four values : V B = f ( β ∗ )(1 − γ ∗ β ∗ ) g B R β ∗ σ (1 + γ ∗ ) , W B = f (ˆ γ ) g B R ˆ γσ ,U S = f ( γ ∗ ) g S R γ ∗ σ , V B = f ′ (0) g B R σ (1 + γ ∗ ) . If V B is the greatest, then equilibrium power allocations of the leader and the follower are p k = β ∗ (1+ γ ∗ ) σ g k (1 − γ ∗ β ∗ ) when k = B otherwise Of course V can only be computed if β ∗ exists. and p k = γ ∗ (1+ β ∗ ) σ g k (1 − γ ∗ β ∗ ) when k = B otherwiseNext, if W B is the greatest, then equilibrium power allocations of the leader and the followerare p k = ˆ γσ g k when k = B otherwiseand p k = γ ∗ σ g k when k = S otherwiseIf U S is the greatest, then equilibrium power allocation of the leader is p k = γ ∗ σ g k when k = S otherwiseand that of the follower is p k = γ ∗ σ g k when k = B otherwiseFinally, if V B is (the only) greatest, then the game has no equilibrium. While the formulation of Prop. 3 is rather complicated, it can be explained in a simpler manner.It describes essentially the way the choice is made by the leader (the follower adjusts to it accordingto Prop. 1). If the best carrier of the leader is different than that of the follower, he transmits onhis best carrier with power corresponding to SINR γ ∗ . If their best carriers are the same, the leadertries to optimize his power on his best carrier B , by choosing between two powers correspondingto two values of SINR: β ∗ , which gives the highest value of the leader’s utility if the followertransmits on the same carrier as the leader, creating interference, or ˆ γ , which is the smallest valueof SINR forcing the follower to change his carrier and reduce the interference on B . If he canobtain a better utility than the best of the two on some other carrier S , he chooses to transmitthere with the power corresponding to γ ∗ . Remark . Note that the equilibria computed Prop. 3 are unique as long as channel gains fordifferent carriers are different and as long as V B = W B = U S . Also the response of the followerat equilibrium is unique as long as channel gains for different carriers are different and W B is notthe greatest value in step 3) of the algorithm described by the theorem . The matter of uniqueness Note that, in case there are multiple equilibria, because V B = U S > W B , the response of the follower to both equilibriumstrategies of the leader is unique. of the follower’s response is obviously very important, as in case there are multiple best responsesto an equilibrium strategy of the leader, the follower has no incentive to follow his equilibriumpolicy. In our case the equilibrium strategy can be imposed to the follower when he has multiplebest responses to the leader’s policy by using a simple trick: whenever W B appears to be thegreatest in step 3), the leader has to use power infinitesimally smaller than that prescribed by hisequilibrium policy. This gives him a minimally smaller utility, but at the same time makes thebest response of the follower unique. Remark . The reasoning behind Prop. 2 works also for the model where powers that players canuse are limited to the sets [0 , P max ] , so also in this case each player transmits on only one of histwo best carriers. Prop. 3 gives the form of a Stackelberg equilibrium in case each user has enoughpower in [0 , P max ] to reach the SINR γ ∗ . Otherwise, it can be shown that all the computations ofProp. 3 can be repeated under assumption that whenever the desired value of the SINR cannot bereached within the constrained regime, the users transmit at their maximum power. In that waywe also obtain an equilibrium in the model. However, considering power constraints will induceadditional cases where the equilibrium is such that some users transmit with their maximum power P max , complicating the formulation of the results, without changing their general sense.The next proposition characterizes the degenerate case when there is no equilibrium in theStackelberg game. Proposition 4.
The Stackelberg game has no equilibrium iff B = B , ˆ γ > γ ∗ and f ′ (0) > max n f (ˆ γ )(1+ γ ∗ )ˆ γ , f ( γ ∗ )(1+ γ ∗ ) γ ∗ g S g B , f ( β ∗ )(1 − γ ∗ β ∗ ) β ∗ o , (6) but for any ǫ > there are ǫ -equilibria of the form p k ( ǫ ) = α ( ǫ ) when k = B otherwisefor the leader and p k ( ǫ ) = γ ∗ ( σ + g k α ( ǫ )) g k when k = B otherwisefor the follower, where α ( ǫ ) is an arbitrarily small value, guaranteeing that the utility of the leaderis within ǫ from V B .Remark . It is important to notice that the case considered in Proposition 4 is indeed possiblefor some sigmoidal function f . One example of such f is one of the form: f ( x ) = √ − x − x ≤ √ − √ x +2 √ − ; x ≥ One can check that f is concave on interval [0 , ] and convex on [ , ∞ ) . Moreover, f and f ′ arecontinuous and lim x →∞ f ( x ) = √ , so it is definitely a sigmoidal function. It is straightforwardto compute that γ ∗ = 1 for this function. Unfortunately, Equation (5) has no solutions on (0 , ∞ ) ,which can be computed either numerically or using Taylor expansion of the function √ − x .Finally, f ′ (0) = , and so for g B ≫ g S and g B ≫ g S , the inequality (6) will be satisfied.On the other hand, any of the two following assumptions: (A1) f ′ (0 + ) = 0 , (A2) f ′ (0 + ) > and f ′′ (0 + ) f ′ (0+) > γ ∗ ,implies that (6) is never satisfied, and so the game under consideration always has an equilibrium.In particular, for the most standard form of f [5], f ( x ) = (1 − e − x ) M , M > not only there always exists an equilibrium in the Stackelberg model (because f satisfies (A1)),but also the procedure in Proposition 3 slightly simplifies, as:1) Eq. (4) can be written as M x = e x − ,2) Eq. (5) can be written as M ( x − x γ ∗ ) = e x − , moreover it has exactly one positivesolution. V. P ERFORMANCE E VALUATION
This section is dedicated to present some key properties and performances of the Stackelbergequilibrium we derived in the previous section. We first study the individual performance of eachplayer. Then, we evaluate the global performance of the system in terms of energy efficiency andspectral efficiency.
A. Individual Performance Evaluation1) Spectrum orthogonalization:
In this section, we shall first look for what values of channelgains for each of the users there is a possibility that both the leader and the follower transmit onthe same carrier. In the sequel, we will refer to the case where users transmit on the same carrieras there is no orthogonalization between users, i.e., ∃ k | p kn = 0 for n = { , } . Then, we willcompute the probability that there is no orthogonalization between the players. Proposition 5.
The set of { g n , . . . , g Kn } , n = 1 , for which there is no orthogonalization betweenusers is a proper subset of the set G of g kn s satisfying B = B and g B n n ≥ (1 + γ ∗ ) g S n n ; for n = 1 , . (7) Note that G is exactly the set of g k s for which there is no orthogonalization in the simultaneous-move game considered in [5]. Thus, introducing hierarchy in the game induces more spectrumorthogonalization than there was in the simultaneous-move scenario.In the next proposition, we will show that the probability of no orthogonalization between theplayers is always small and decreases fast as the number of carriers grows. Proposition 6.
Assume that the channel gains for different carriers of each of the users are i.i.d.Rayleigh random variables. Then, the probability that there is no orthogonalization between theplayers at the Stackelberg equilibrium is bounded above by (1 + γ ∗ ) B (1 + γ ∗ , K ) (cid:20) K − K + (1 + γ ∗ ) B (1 + γ ∗ , K ) (cid:21) ∼ O ( K − (1+ γ ∗ ) ) (8) where B denotes the Beta function, which is the exact probability of no orthogonalization in thesimultaneous-move version of the model.Remark . In the above proposition, we suppose that the channel gains of different players arenot correlated, which is typically the case when carriers are far enough [29]. Otherwise, theprobability computed there can be treated as an upper bound for respective probabilities, whenthere is a positive correlation between different carriers of each of the users, which is much morerealistic. We will see later in the paper (see Fig. 1 and 2) that, in the case of positive correlationover carriers, these probabilities will be even smaller (and so faster decreasing to ). Remark . Now, the opposite situation to that analyzed in Prop. 6 is when both users experiencethe same channel gains. The probability that there is no orthogonalization between the players inthe Stackelberg game is then bounded above by K (1 + γ ∗ ) B (1 + γ ∗ , K ) (9)which is still decreasing to as K goes to infinity, but K times bigger than the bound in Eq.(8). The intuition behind this is that, if the channels of different users are not correlated, thenwith probability ( K − /K users have different best channels and with only /K users have thesame best channels (and interference is an issue in this case). If users experience the same channelgains, they have the same best channel with probability . Also, if the number of carriers K isbig, both users will have two best carriers of similar quality as the channel gains are chosen atrandom, so the probability that they choose the same carrier becomes very small (see Fig. 3 and4).
2) Payoffs comparison:
The leader is not worse off on introducing hierarchy (which is alwaysthe case in Stackelberg games if both the leader and the follower use their equilibrium policies), but the follower loses on it in some cases. The proposition below gives more insights on what thelatter depends on. Proposition 7.
For any sigmoidal function f the following three situations are possible: B = B . Then, for both players, the payoff in the Stackelberg game is the same as that inthe simultaneous-move game. Both players use the same carrier B = B in equilibria (or ǫ -equilibria) of simultaneous-move and Stackelberg games. Then, the payoff of the follower in the Stackelberg game isalways bigger than what he receives in the simultaneous-move game. B = B and both players use different carriers in equilibria of simultaneous-move andStackelberg games: the leader in the Stackelberg game uses B , but in the simultaneous-move game he uses S in equilibrium, the follower in the Stackelberg game uses S , whilein the simultaneous-move game he uses B in equilibrium. Then, the payoff of the followerin the Stackelberg game is smaller than what he receives in simultaneous-move game.3) Comparison between leading and following: It is known from [22], that if there is only onecarrier available for the players, it is always better to be the follower than to be the leader. Thesituation changes when the number of carriers increases.
Proposition 8.
Suppose that the Stackelberg game has exact equilibria both when player 1 is theleader and when he is the follower. Then, the utility at Stackelberg equilibrium of player if heis the leader is not less than his utility if he is the follower if one of the following conditions issatisfied: B = B . B = B and min { g B g S , g B g S } ≤ γ ∗ B = B and for i = 1 , , j = i , f (cid:18) g Bii g Sii − (cid:19) g Bii g Sii − ≥ max f ( γ ∗ ) g S j j γ ∗ g B j j , f ( β ∗ )(1 − γ ∗ β ∗ ) β ∗ (1 + γ ∗ ) B = B , f ( β ∗ )(1 − γ ∗ β ∗ ) β ∗ (1 + γ ∗ ) ≥ max f ( γ ∗ ) g S γ ∗ g B , f (cid:18) g B g S − (cid:19) g B g S − and f (cid:18) g B g S − (cid:19) g B g S − ≥ max ( f ( γ ∗ ) g S γ ∗ g B , f ( β ∗ )(1 − γ ∗ β ∗ ) β ∗ (1 + γ ∗ ) ) B = B , f ( γ ∗ ) g S γ ∗ g B ≥ max f (cid:18) g B g S − (cid:19) g B g S − , f ( β ∗ )(1 − γ ∗ β ∗ ) β ∗ (1 + γ ∗ ) f ( β ∗ )(1 − γ ∗ β ∗ ) β ∗ (1 + γ ∗ ) ≥ max f ( γ ∗ ) g S γ ∗ g B , f (cid:18) g B g S − (cid:19) g B g S − and g B g S ≤ β ∗ − γ ∗ β ∗ B = B , f (cid:18) g B g S − (cid:19) g B g S − ≥ max ( f ( γ ∗ ) g S γ ∗ g B , f ( γ ∗ )(1 − γ ∗ β ∗ ) γ ∗ (1 + γ ∗ ) ) and f ( β ∗ )(1 − γ ∗ β ∗ ) β ∗ (1 + γ ∗ ) ≥ max f ( γ ∗ ) g S γ ∗ g B , f (cid:18) g B g S − (cid:19) g B g S − Although the formulation of the proposition is rather complicated, its general meaning is simple.It states that, in most of the cases, different users have different best carriers, so there is nodifference between leading and following. The two remaining cases are when both players havethe same best carrier. In the first one, each of the players has only one good carrier and thesame for both. This situation reduces to the problem considered in [22] where only one carrieris available, and so every user can obtain better energy efficient utility by decreasing its priorityfrom leading to following. The reason behind this phenomenon is basically the construction ofthe energy-efficient utility. In the simultaneous-move version of this model each user transmitswith the power corresponding to the SINR γ ∗ . Under Stackelberg regime, the leader can increasehis utility by reducing his power consumption to the level corresponding to the SINR β ∗ < γ ∗ ,which reduces the overconsumption due to interference. The optimal answer of the follower willstill be to use the power giving him the SINR of γ ∗ though. The result of the shift in the powerused by the leader without a similar change in that of the follower is that both utilities increasesimultaneously, but the increase of the utility of the follower is bigger than that of the leader. Inthe second case, both players have the same best carrier but one of them prefers to use his secondbest carrier instead (that is – the second best carrier is not much worse than the best one). In thissituation, it is the leader who is better off on introducing hierarchy, so this becomes similar tomost of the Stackelberg models . It is worth noting though that if g kn are i.i.d. Rayleigh randomvariables, one of the two first cases of Proposition 8 will occur with probability significantly biggerthan − (1 + γ ∗ ) B (1 + γ ∗ , K ) (cid:2) K − K + (1 + γ ∗ ) B (1 + γ ∗ , K ) (cid:3) , and so it will be very close to even for small values of K . We will show later in the paper (see Figure 9) that, in practice, it isthe last situation that prevails whenever the players have at least two carriers at their disposal. Notice that in basic standard economic problems, it is commonly known that a firm does always better by preempting themarket and setting its output level first (e.g., in Cournot-like competition games) [30]. B. System Performance Evaluation1) Energy efficiency:
Let us now compute the social welfare in our model, defined as the sumof utilities of both players. In the following proposition, we give upper bounds on the possibledecrease of social welfare when we introduce hierarchy in the game, as well as a bound on theratio of the maximum social welfare obtainable and that of Stackelberg equilibrium in the game.The latter can be treated as the price of anarchy [31] in our game.
Proposition 9.
The social welfare when the players apply Stackelberg equilibrium policies equalsboth maximum social welfare obtainable in the game and social welfare in Nash equilibrium ofthe simultaneous-move game whenever B = B . When B = B , the social welfare in Stackelbergequilibrium: Is at most R g B + R g B R g B + R g S times worse than that in simultaneous-move game equilibrium. Is at most R g B + R g B R g S + R g S times worse than the maximum social welfare obtainable in the game. Note that, when g kn are i.i.d. Rayleigh variables (as assumed in Proposition 6), then a) theregion where Stackelberg equilibrium is not the social optimum shrinks fast as the number ofcarriers increases; b) even in case there is no orthogonalization in Stackelberg equilibrium, theratios appearing in the above proposition are small with probability increasing with the numberof carriers.
2) Spectral efficiency:
Along with energy efficiency, spectral efficiency – defined as the through-put per unit of bandwidth – is one of the key performance evaluation criteria for wireless networkdesign. These two conflicting criteria can be linked through their tradeoff [32], [33]. Therefore,it is often imperative to make a tradeoff between energy efficiency and spectral efficiency. In thefollowing, we give a closed-form expression of the lower bound on the sum spectral efficiency ofthe proposed Stackelberg model.
Proposition 10.
The spectral efficiency in case there is a orthogonalization between the users atthe Stackelberg equilibrium is strictly bigger than log (1 + γ ∗ ) (cid:20) − (1 + γ ∗ ) B (1 + γ ∗ , K ) · (cid:18) K − K + (1 + γ ∗ ) B (1 + γ ∗ , K ) (cid:19)(cid:21) (10) which is equal to the spectral efficiency in the simultaneous-move game. The computation done in Proposition 10 holds in case there is orthogonalization between theplayers. This means that this is only a lower bound for the total spectral efficiency in our model.However, by Proposition 6, it becomes very tight as K goes to infinity. An easy consequenceof this is that the spectral efficiency in the limit model (with an infinite number of carriers) canbe computed exactly, and is equal to log (1 + γ ∗ ) . Notice that, when users experience the same P r obab ili t y o f no o r t hogona li z a t i on Nash Independent carriersCorrelated carriersTheoretical curve in Eq. (8)
Fig. 1. The probability of no orthogonalization between the players at the Nash equilibrium with correlation over carriers. P r obab ili t y o f no o r t hogona li z a t i on Stackelberg Independent carriersCorrelated carriersTheoretical upper−bound in Eq. (8)
Fig. 2. The probability of no orthogonalization between the players at the Stackelberg equilibrium with correlation over carriers.
Rayleigh channel gains, the spectral efficiency in case there is a orthogonalization between theusers at the Stackelberg equilibrium is strictly bigger than log (1 + γ ∗ ) [1 − K (1 + γ ∗ ) B (1 + γ ∗ , K )] (11)VI. S IMULATION R ESULTS
We consider the energy efficiency function, f ( x ) = (1 − e − x ) M , well-known in power allocationgames, where M = 100 is the block length in bits. For this efficiency function, γ ∗ ≃ . (or . dB). Simulations were carried out using a rate R n = 1 bps for n = { , } . We have simulated scenarios to remove the random effects from Rayleigh fading. A. The probability of no orthogonalization
Let us first consider a quasi-static correlated Rayleigh-fading channel model. Fig. 1 and 2reflect the effect of the correlation over carriers (i.e., the correlation between different carriers ofeach of the users) on the probability of no orthogonalization for the simultaneous (Nash) and thehierarchical (Stackelberg) game respectively. The correlation model follows the model in [34]. Aswe expected in Section V-A1 (see Remark 4), results show that, in the case of correlated carriers, P r obab ili t y o f no o r t hogona li z a t i on Nash θ = 0 θ = 0.5 θ = 1Theoretical curve in Eq. (8) Theoretical curve in Eq. (9) Fig. 3. The probability of no orthogonalization between the players at the Nash equilibrium with correlation over users. P r obab ili t y o f no o r t hogona li z a t i on Stackelberg θ = 0 θ = 0.5 θ = 1Theoretical upper−bound in Eq. (8) Theoretical upper−bound in Eq. (9) Fig. 4. The probability of no orthogonalization between the players at the Stackelberg equilibrium with correlation over users. the probability of no orthogonalization is smaller and so faster decreasing to even for a moderatenumber of carriers K .From now on, we will only consider the case of no correlation between different carriers ofeach of the users. In case of correlated carriers, performance results obtained in the remainderhave to be considered as a worst case performance.Fig. 3 and 4 investigate the effect of the correlation over users (i.e., the correlation betweendifferent users’ fading channel) on the probability of no orthogonalization for the Nash and theStackelberg game respectively. The correlation factor modeling the dependencies between theusers is θ . In both figures, results show that, as the correlation between different users decreases,the probability of no orthogonalization gets even smaller and so faster decreasing to , whichcorresponds to what Remark 5 claims. In order to assess the accuracy of the theoretical bounds,we also compare the simulated probability of no orthogonalization with the theoretical upper-bounds. More specifically, for i.i.d. users, we compare theoretical curve derived in Eq. (8) withsimulated curve for θ = 0 . For correlated users, we compare theoretical curve in Eq. (9) withsimulated curve for θ = 1 . We see that the simulated and theoretical curves match pretty well.Now, when we look at the Stackelberg equilibrium in Fig. 4, it is clearly illustrated that thetheoretical upper-bounds derived in Eq. (8) and Eq. (9) turn out to be greater than the simulatedprobabilities of no orthogonalization, which confirms the accuracy of the results. Remember that P r obab ili t y o f no o r t hogona li z a t i on θ = 0 Social welfareTheoretical curve in Eq. (8) StackelbergNash Fig. 5. The probability of no orthogonalization between the players as a function of the number of carriers with independentusers. P r obab ili t y o f no o r t hogona li z a t i on θ = 1 Social welfareTheoretical curve in Eq. (9) StackelbergNash Fig. 6. The probability of no orthogonalization between the players as a function of the number of carriers with correlated users. the theoretical curves derived in Eq. (8) and Eq. (9) correspond to the exact probability of noorthogonalization in the simultaneous-move game, but are only upper-bounds in the hierarchicalversion of the model, which is clearly confirmed by Fig. 3 and 4.Fig. 5 and 6 depict the probability of no orthogonalization for different schemes consideringindependent users ( i.e., for θ = 0 ) and correlated users ( i.e., for θ = 1 ) respectively. Bothcurves follow the same trend, tending to increase the orthogonalization between the users asthe number of carriers grows, which validates the obtained theoretical results. A rather significantgap between Nash and Stackelberg curves suggests that introducing hierarchy results in muchmore orthogonalization between the players. Particularly noteworthy is the fact that, at the socialoptimum, we always obtain strict orthogonalization between users. This means that, in a centralizedsystem, if maximizing the energy efficiency is the goal, introducing hierarchy moves the solutioncloser to the social optimum.To sum it up, we can argue that correlation across carriers is a suitable feature as it bringsmore orthogonalization (and thus leads to a better spectral efficiency), desirable from the socialpoint of view, while correlation across users is not suited as it increases the probability of noorthogonalization. This results is of practical interest as it suggests that designing the power A v e r qge ene r g y e ff i c i en cy [ b i t s / J ou l e ] θ = 0 Social welfareStackelbergNash Fig. 7. Average energy efficiency with independent users. A v e r qge ene r g y e ff i c i en cy [ b i t s / J ou l e ] θ = 1 Social welfareStackelbergNash Fig. 8. Average energy efficiency with correlated users. control for multi-carrier networks shall be developed tailored to the physical properties of thetransmission phenomenon.
B. Energy efficiency
We then resort to plot the average energy efficiency at the equilibrium for increasing numberof carriers K . The curves obtained in Fig. 7 for independent users ( i.e., for θ = 0 ) exhibit adifferent trend than ones in Fig. 8 for correlated users ( i.e., for θ = 1 ). Indeed, we remark that theStackelberg perform almost the same as the Nash game for θ = 0 , whereas, for θ = 1 , the gapbetween the Nash game and the Stackelberg game increases. More specifically, the Stackelbergmodel achieves an energy efficiency gain up to 25% with respect to the Nash model for K = 4 carriers. As the number of carriers K goes large, both configurations tend towards having thesame average energy efficiency. This can be justified by the fact that, when the number of carriersincreases, the probability that users transmit on different carriers is high (see Section V-A1) andthus, users are less sensitive to their degree of hierarchy in the system (see Prop. 9). Interestingly,in both the independent and correlated users’ cases, the Stackelberg game achieves almost thesame energy efficiency as at the social welfare, which tends to validate results in Prop. 9.Fig. 9 illustrates the per-user energy efficiency with independent users. Interestingly, we see from E ne r g y e ff i c i en cy [ b i t s / J ou l e ] θ = 0 Social welfare user 1Social welfare user 2Nash user 1Nash user 2Stackelberg user 1Stackelberg user 2 Fig. 9. Per-user energy efficiency with independent users. User and user in the Stackelberg game refer to the leader and thefollower respectively. Fig. 9 that, at the Stackelberg equilibrium, the energy efficiency of the follower in the Stackelberggame is smaller than in the simultaneous-move game. This suggests that, for the vast majorityof cases, Situation 3) in Prop. 7 is more likely to occur for a low number of carriers K . As K increases, Situation 1) in Prop. 7 is more likely to occur yielding the same energy efficiency forboth the leader and the follower in the Stackelberg game as in the simultaneous-move game. Thisis justified by the fact that, with probability /K , resp. ( K − /K , users have the same, resp.different, best channels. It is then easy to see that, for low K , users are more likely to have thesame best channels and interference is an issue in this case yielding to Situation 3) in Prop. 7,whereas, for sufficiently large K , users are more likely to have different best channels yieldingto Situation 1) in Prop. 7. Moreover, Fig. 9 also shows that it is profitable to be the leader whichcorresponds to what Prop. 8 points out. C. Spectral efficiency
In Fig. 10 and 11, we compare the closed-form expressions of the spectral efficiency derived inEq. (10) for i.i.d. users ( i.e., for θ = 0 ) and in Eq. (11) for correlated users ( i.e., for θ = 1 ) withthe simulated spectral efficiency. Of particular interest is the fact that the closed-form expressionsturn out to be very tight. We can also observe that the Stackelberg game performs better thanthe Nash game in terms of average spectral efficiency particularly for correlated users while stillperforming very close to the social welfare. As an example, for K = 2 carriers, the Stackelberggame yields only a negligible spectral efficiency loss . bps/Hz with respect to the social welfareand approximately . bps/Hz of spectral efficiency gain beyond the Nash game. D. Spectral efficiency – Energy efficiency Tradeoff
In order to illustrate the balance between the achievable rate and energy consumption of thesystem, we plot in Fig. 12 and 13 the spectral efficiency as a function of the energy efficiency for A v e r age s pe c t r a l e ff i c i en cy [ bp s / H z ] θ = 0 Social welfareTheoretical curve in Eq. (10) StackelbergNash Fig. 10. Average spectral efficiency with independent users. A v e r age s pe c t r a l e ff i c i en cy [ bp s / H z ] θ = 1 Social welfareTheoretical curve in Eq. (11) StackelbergNash Fig. 11. Average spectral efficiency with correlated users. independent and correlated users respectively. Surprisingly, it is clearly shown that, for both theindependent and correlated cases, the proposed Stackelberg decision approach achieves a flexibleand desirable tradeoff between energy efficiency and throughput maximization compared to thesocial welfare and the Nash model. In particular, it is shown that the Stackelberg scheme maximizesthe energy efficiency while still optimizing the spectral efficiency at the Stackelberg equilibrium.Notice that this contrasts with most related works so far in which the optimal energy efficiencyperformance often leads to low spectral efficiency performance and vice versa [35]–[38]. Thisfeature has a great impact on the network performance and provides a convincing argument thathierarchical communication is the proper context to design and optimize energy efficient wirelessnetworks. VII. C
ONCLUSION
The growing interest in energy efficient research from signal processing and communicationcommunities has spurred an increasing interest in the recent years. There have been a large numberof proposals for all communication layers, but the system infrastructure has not been clearlydefined. In this paper, we have proposed a hierarchical game to model distributed joint power andchannel allocation for multi-carrier energy efficient systems since it has the advantage of leadingtowards more realistic or even simpler distributed power control algorithms. We have established A v e r age s pe c t r a l e ff i c i en cy [ bp s / H z ] Average energy efficiency [bits/Joule] θ = 0 Social welfareNashStackelberg Fig. 12. Spectral efficiency vs. Energy efficiency with independent users. A v e r age s pe c t r a l e ff i c i en cy [ bp s / H z ] Average energy efficiency [bits/Joule] θ = 1 Social welfareNashStackelberg Fig. 13. Spectral efficiency vs. Energy efficiency with correlated users. the existence of the Stackelberg equilibrium and gave its formal expression. The proposed schemeachieves better performances as compared to those of other existing schemes, notably the Nashmodel proposed in [5]. In particular, we have proved that introducing hierarchy across usersinduces a spectrum orthogonalization which substantially improves system performances. Forthe first time, we have derived the spectral efficiency of such a model with exact expressionsfor the throughput scaling. The proposed scheme can achieve a spectral efficiency scaling of log (1 + γ ∗ ) (cid:2) − O ( K − (1+ γ ∗ ) ) (cid:3) , while a vanishing fraction of the carriers may suffer from mutualinterference as the number of the carriers goes large. Simulation results have been presentedto exhibit the effectiveness of the proposed scheme to balance the achievable rate and energyconsumption of the system. R EFERENCES [1] C. Han, T. Harrold, S. Armour, I. Krikidis, S. Videv, P. M. Grant, H. Haas, J. Thompson, I. Ku, C.-X. Wang, T. A.Le, M. Nakhai, J. Zhang, and L. Hanzo, “Green radio: radio techniques to enable energy-efficient wireless networks,”
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A. Proof of Proposition 2
Lemma 1.
For any finite sequence of n pairs ( a k , b k ) such that a k ≥ and b k > the followinginequality is true: P Kk =1 a k P Kk =1 b k ≤ max { a k b k , k = 1 , . . . , n } . The equality is only possible if each ratio a k b k is equal.Proof: We proceed by induction with respect to n . For n = 2 , let us assume that the hypothesisis not true and thus: a + a b + b > a b and a + a b + b > a b . This can be rewritten as a b + a b > a b + a b and a b + a b > a b + a b or equivalently a b > a b > a b , which is a contradiction.Next, assume that our hypothesis is true for any l < K . Then, we can proceed as follows: P Kk =1 a k P Kk =1 b k ≤ P K − k =1 a k + a K P K − k =1 b k + b K ≤ max ( P Kk =1 a k P Kk =1 b k , a K b K ) ≤ max (cid:26) max { a k b k , k = 1 , . . . , K − } , a K b K (cid:27) = max { a k b k , k = 1 , . . . , K } If there is at least one pair ( a k , b k ) , whose ratio is bigger than the other ones we can show alongthe same lines that the inequality is strong (we only need to take these a k and b k from the sums P Kk =1 a k , P Kk =1 b k in the above considerations instead of a K and b K .Now we can prove Proposition 2. Proof:
Note that by Lemma 1 u ( p , p ) = P Kk =1 R f ( γ k ) P Kk =1 p k ≤ max k R f ( γ k ) p k , so the leader in the Stackelberg game cannot use more than one carrier simultaneously, as de-creasing power to zero on every carrier different from the one realizing maximum above wouldbe beneficial. Thus he will choose only one carrier for which ˆ f k ( p k ) = f ( γ k ) p k = 1 p k f ( g k p k σ + g k p k ( p ) ) (where p k is computed according to (3)) is the greatest. Note however that since the follower willchose only one carrier, ˆ f k ( p k ) will be equal to f ( gk pk σ γ ∗ )+ γ ∗ gk pk ) p k only for one carrier, say carrier k ∗ , and for any other carrier it will be equal to f ( gk pk σ ) p k , which is maximized for p k = γ ∗ σ g k and thenequal to f ( γ ∗ ) g k γ ∗ σ . But this last value depends on the carrier only through g k , so will be maximizedfor k = B if only k ∗ = B . Thus the equilibrium strategy of the leader will put all the poweron carrier B in that case. If k ∗ = B , then the biggest value of ˆ f k ( p k ) for k = k ∗ will be for k = S , and either all the power of the leader will be put on this carrier or on k ∗ = B .As for the follower, by Proposition 1 his best response is always to put all his power on thecarrier maximizing ˆ h k ( p k ) = g k σ + g k p k , which will be equal to g k σ for all but one carrier. Now thereasoning made for the leader can be applied here as well. B. Proof of Proposition 3Proof:
First consider the case when B = B . The biggest possible value of the ratio f ( γ kn ) p kn obtainable for player n on a single carrier (when his opponent does not maximize his payoff, butalso the payoff of player n ) is f ( γ ∗ ) R n γ ∗ σ max k g kn = f ( γ ∗ ) R n γ ∗ σ g B n n . Just this is obtained by both players when they apply strategies p n defined in the theorem. Thus none of them will be interested inchanging his strategy.Now we move to the case when B = B . Suppose the leader uses only carrier B in hisequilibrium strategy. Then, by Proposition 2 the follower uses one of carriers B = B or S . Ifhe uses B then by Proposition 1 the following has to be true: ˆ h B ( p B ) = g B σ + g B p B ≥ g S σ = ˆ h S ( p S ) . Rewriting this we obtain that the follower chooses B when p B ≤ σ ( g B − g S ) g B g S (12)and S otherwise. Having this in mind, we can compute the utility of the leader at the equilibriumusing carrier B , namely R f ( g B p B σ (1+ γ ∗ )+ γ ∗ g B p B ) p B (13)when p B ≤ σ ( g B − g S ) g B g S and R f ( g B p B σ ) p B (14)otherwise. Next we need to find the values of p B maximizing (13) and (14) respectively. Beforewe obtain the first one we rewrite the SINR in that case in the following way: γ B = 1 γ ∗ −
11 + γ ∗ g B p B σ (1+ γ ∗ ) (15)and differentiate it with respect to p B , obtaining: ∂γ B ∂p B = g B σ (1 + γ ∗ )(1 + γ ∗ σ (1+ γ ∗ ) g B p B ) (16) = g B σ (1 + γ ∗ )( σ (1 + γ ∗ ) + γ ∗ g B p B ) = 1 p B σ (1 + γ ∗ ) g B p B γ ∗ ( γ B ) γ ∗ Next, we can transform (15) into γ ∗ g B p B σ (1 + γ ∗ ) = γ ∗ γ B − γ ∗ γ B . (17)and put it into (16), obtaining: ∂γ B ∂p B = 1 p B − γ ∗ γ B γ ∗ γ B ( γ B ) γ ∗ = 1 p B ( γ B − γ ∗ ( γ B ) ) . (18) Now we write the first order condition for the maximization of (13): ∂ ( R f ( γ B ) p B ) ∂p B = R − f ( γ B ) + f ′ ( γ B ) ∂γ B ∂p B p B ( p B ) . If we substitute (18) into it, we obtain the following equation: − f ( γ B ) + ( γ B − γ ∗ ( γ B ) ) f ′ ( γ B ) . (19)If we find the best solution to this equation (that is, maximizing f ( γ B ) p B ), β ∗ , we get the powerallocation of the leader in case (12), which can be computed from (17) as p ∗∗ = β ∗ σ (1 + γ ∗ ) g B (1 − γ ∗ β ∗ ) . (20)Similarly, when we write the first order condition for the maximization of (14), we obtain ∂ ( R f ( γ B ) p B ) ∂p B = − f ( γ B ) + γ B f ′ ( γ B )( p B ) , whose unique solution is γ ∗ . The corresponding value of p B is p ∗ = γ ∗ σ g B . (21)Now, we put (20) and (21) in (13) and (14) respectively, obtaining the value functions correspondingto p ∗ and p ∗∗ : V B = f ( β ∗ )(1 − γ ∗ β ∗ ) g B R β ∗ σ (1 + γ ∗ ) and U B = f ( γ ∗ ) g B R γ ∗ σ Note that the first one is always smaller than the second one (because γ ∗ maximizes the ratio f ( x ) x and γ ∗ , β ∗ > ). So, in case p ∗ satisfies the condition opposite to (12), the leader will chooseto transmit on B with this power, while the follower will choose (according to (1)) to transmiton S with power γ ∗ σ g S .Next, when p ∗ satisfies (12), the situation becomes more complex. The leader has to choosebetween one of the three possibilities: to choose the power p ∗∗ on carrier B , giving him the valueof V B , to choose power ˆ p = ˆ γσ g B on carrier B , which now gives the biggest value in case thefollower chooses to use carrier S , W B = f (ˆ γ ) g B R ˆ γσ , or to choose to use his second-best carrier S instead of B , with power p S = γ ∗ σ g S , which would give him the value U S = f ( γ ∗ ) g S R γ ∗ σ . Choosing the biggest one from V B , W B and U S will give the leader’s equilibrium payoff (andcorresponding equilibrium strategy) in the Stackelberg game, unless V B is not the biggest valueobtainable by the leader in case (12). This is only possible when the biggest value of (13) isobtained on one of the ends of the interval (0 , ˆ γ γ ∗ (1+ˆ γ ) ] . Thus, we compute these two values: V B = lim γ → R f ( γ )(1 − γ ∗ γ ) g B γσ (1 + γ ∗ ) = R f ′ (0) g B σ (1 + γ ) , V B = R f ( ˆ γ γ ∗ (1+ˆ γ ) )(1 − γ ∗ ˆ γ γ ∗ (1+ˆ γ ) ) g B γ γ ∗ (1+ˆ γ ) σ (1 + γ ∗ )= R f ( ˆ γ γ ∗ (1+ˆ γ ) ) g B ˆ γσ V B is clearly smaller than W B , so it cannot be the biggest value obtained by the leader. Thevalue V B though can be the biggest one, and so in case V B is bigger than max { V B , W B , U S } it is optimal for the leader to use the smallest power possible on carrier B (which is not anequilibrium strategy, as for any arbitrarily small power there exists a smaller power, for which thevalue function of the leader is closer to V B . The power allocations of the follower in each of thecases of (12) are computed according to Proposition 1. C. Proof of Proposition 4Proof:
The inequality (6) is a rewriting of the condition V B > max { V B , W B , U S } , appear-ing in the proof of Theorem 2, where the optimal behavior of the leader in the case when thiscondition, together with B = B and ˆ γ > γ ∗ is satisfied, was also described. The behavior of thefollower follows from Proposition 1. D. Proof of Proposition 5Proof:
By Proposition 3, no orthogonalization between the players is only possible if B = B , ˆ γ > γ ∗ (22)and max { V B , V B } > max { W B , U S } . (23)(22) can be rewritten as g B − g S g S > γ ∗ , which is then equivalent to g B > (1 + γ ∗ ) g S . Onthe other hand (23) implies that V B > U S , which can be written as f ′ (0) g B γ ∗ > f ( γ ∗ ) g S γ ∗ . Now,using the definition of γ ∗ and the fact that f ′ (0) < f ′ ( γ ∗ ) (see [28]) we can conclude that f ′ ( γ ∗ ) g B γ ∗ > f ′ ( γ ∗ ) g S , which implies g B > (1 + γ ∗ ) g S , ending the proof. E. Proof of Proposition 6Proof:
By Proposition 5, no orthogonalization is only possible if B = B and g B n n ≥ (1 + γ ∗ ) g S n n for n = 1 , (which is an exact condition for no orthogonalization in the simultaneous-move model). The probability of this can be computed as − (cid:20) K − K + P K + 1 − P K P K (cid:21) = 1 − P K (cid:18) − P K (cid:19) . (24) where P denotes the probability that for one of the players g B i i < (1 + γ ∗ ) g S i i . We can easilycompute that P = K ! Z ∞ dg Ki Z ∞ g Ki dg K − i . . . Z ∞ g i dg i Z (1+ γ ∗ ) g i g i λ K e − λ P Kk =1 g ki dg i . If we introduce new variables x = λ ( g i − g i ) , x = λ ( g i − g i ) , . . . , x K − = λ ( g K − i − g ki ) , x K = λg Ki , we can write it as K ! Z ∞ dx K Z ∞ dx K − . . . Z ∞ dx Z γ ∗ P Kk =2 x k e − P Kk =1 kx k dx and further as − K !(2+ γ ∗ ) ... ( K + γ ∗ ) . If we substitute it into the bound of no orthogonalizationprobability (24), we obtain ( K − γ ∗ ) . . . ( K + γ ∗ ) (cid:18) K − K + ( K − γ ∗ ) . . . ( K + γ ∗ ) (cid:19) = (1 + γ ∗ ) B (1 + γ ∗ , K ) (cid:20) K − K + (1 + γ ∗ ) B (1 + γ ∗ , K ) (cid:21) . It can be immediately seen that this is no less than (1 + γ ∗ ) B (1 + γ ∗ , K ) . The fact that this lastquantity is O ( K − (1+ γ ∗ ) ) is well known (see e.g., pp. 263 in [39]). F. Proof of Proposition 7Proof:
First note that the players in the Stackelberg game both use carrier B = B in( ǫ -)equilibrium when g B g S and g B g S satisfy max (cid:26) f ( β ∗ )(1 − γ ∗ β ∗ ) β ∗ (1 + γ ∗ ) , f ′ (0)1 + γ ∗ (cid:27) g B g S > f ( γ ∗ ) γ ∗ , (25) max (cid:26) f ( β ∗ )(1 − γ ∗ β ∗ ) β ∗ (1 + γ ∗ ) , f ′ (0)1 + γ ∗ (cid:27) > f ( g B g S − g B g S − , (26) which is true for g B g S and g B g S big enough (where the latter is a consequence of the fact that theRHS of (26) converges to as g B g S goes to infinity). If we intersect the set obtained with the setwhere g B g S and g B g S are bigger than − γ ∗ we get the desired set where there is no orthogonalizationin equilibria of both the Stackelberg and simultaneous-move games.Now let us compute the payoffs of the follower in this situation. The payoff in the simultaneous-move game equals f ( γ ∗ ) g B (1 − γ ∗ ) R γ ∗ σ , while that in the Stackelberg game is f ( γ ∗ ) g B (1 − γ ∗ β ∗ ) R γ ∗ σ (1+ β ∗ ) . Thelatter is bigger if − γ ∗ < − γ ∗ β ∗ β ∗ , which is equivalent to γ ∗ > β ∗ . This is always true, as anysolution to (5) has to be smaller than γ ∗ .Next, suppose that g B g S < γ ∗ and g B g S > γ ∗ , and thus player 1 uses carrier S , whileplayer 2 uses carrier B in the only equilibrium of the simultaneous-move game. Then, to obtainthe situation where it is player 1 who uses his best carrier and player 2 who uses his second-best one in the Stackelberg game, the inequality f ( gB gS − gB gS − g B g S > f ( γ ∗ ) γ ∗ has to be true. If we denote by y ( x ) the solution of the equation f ( x − x − y ( x ) = f ( γ ∗ ) γ ∗ , we may rewrite the above three inequalitiesas y ( g B g S ) < g B g S < γ ∗ , g B g S > γ ∗ . (27)Now note that since the function f is sigmoidal, f ( x − x − strictly decreases on the set x > γ ∗ .Combining this with the fact that f ((1+ γ ∗ ) − γ ∗ ) − = f ( γ ∗ ) γ ∗ , one can see that for x > γ ∗ the curve y ( x ) is strictly increasing, and thus the set of pairs ( g B g S , g B g S ) satisfying (27) is not empty.The payoffs of the follower in the simultaneous-move game and Stackelberg game (respectively)are in this situation f ( γ ∗ ) g B R γ ∗ σ and f ( γ ∗ ) g S R γ ∗ σ . Clearly the former is greater than the latter.The final case is obvious, as in case when B = B the strategies the players use are the samein the simultaneous-move and Stackelberg games. G. Proof of Proposition 8Proof:
To prove this prop., we only need to compare the utilities for player 1 when he is theleader and when he is the follower in each of the cases of Prop. 3.
H. Proof of Proposition 9Proof:
The first part of the proposition is obvious. To prove 1) of the second part first note thatthe social welfare in equilibrium of the simultaneous-move game can only be bigger than that inStackelberg equilibrium when the payoff of the follower in the Stackelberg game decreases. This isonly possible when the carrier he uses in equilibrium changes from B = B in the simultaneous-move game to S in the Stackelberg game. In such a case his utility changes from f ( γ ∗ ) g B R γ ∗ σ to f ( γ ∗ ) g S R γ ∗ σ if the leader also changes the carrier he uses from S to B or from f ( γ ∗ )(1 − γ ∗ ) g B R γ ∗ σ to f ( γ ∗ ) g S R γ ∗ σ if the leader uses carrier B in both simultaneous-move and Stackelberg equilibria. Onthe other hand the utility of the leader in the Stackelberg equilibrium is f ( γ ∗ ) g B R γ ∗ σ in the formercase and not smaller than f ( γ ∗ )(1 − γ ∗ ) g B R γ ∗ σ in the latter one (this is because this is his utility inNash equilibrium of the simultaneous game, and the utility of the leader increases in Stackelberggame). Straightforward computations yield the desired bound on the decrease of social welfare. It follows from the fact that f ( x ) x is decreasing for x > γ ∗ that there is always only one such y . To prove part 2) first note that the maximum utility that can be obtained in this game is boundedabove by f ( γ ∗ ) g B R γ ∗ σ + f ( γ ∗ ) g B R γ ∗ σ , (28)as this is the sum of maximal utilities of both players (but not obtainable at the same time if B = B ). Next note that if the leader uses carrier S in Stackelberg equilibrium, the sum ofthe utilities of both players is f ( γ ∗ ) g S R γ ∗ σ + f ( γ ∗ ) g B R γ ∗ σ , R g B + R g B R g S + R g B < R g B + R g B R g S + R g S times less than(28). On the other hand if he uses B in Stackelberg equilibrium, his utility cannot be smallerthan f ( γ ∗ ) g S R γ ∗ σ , while that of the follower not less than f ( γ ∗ ) g S R γ ∗ σ (if they were not, each of themwould change his carrier to S or S ). But the sum of these utilities is R g B + R g B R g S + R g S times less than(28). I. Proof of Proposition 10Proof:
No orthogonalization in the simultaneous-move game is possible exactly when B = B and g B n n ≥ (1 + γ ∗ ) g S n n for n = 1 , . 1 minus the exact probability of that region is computed inProposition 6, and this is also the lower bound on the same probability for the Stackelberg game.The spectral efficiency in case there is orthogonalization between the players can be computedas the expected value of log (1 + γ ) over this region. Note however that γ ≡ γ ∗ there, and sothe bound on spectral efficiency is exactly log (1 + γ ∗ ) times (the bound on) the probability oforthogonalization, which is − (1 + γ ∗ ) B (1 + γ ∗ , K ) (cid:2) K − K + (1 + γ ∗ ) B (1 + γ ∗ , K ) (cid:3)(cid:3)