Quantum Game Application to Spectrum Scarcity Problems
aa r X i v : . [ qu a n t - ph ] A ug Quantum Game Application to Spectrum ScarcityProblems
O.G. Zabaleta, J. P. Barrang´u and C.M. Arizmendi
Instituto de Investigaciones Cient´ıficas y Tecnol´ogicas en Electr´onica (ICYTE),Facultad de Ingenier´ıa,Universidad Nacional de Mar del Plata,Av. J.B. Justo 4302,7600 Mar del Plata, Argentina
Abstract
Recent spectrum-sharing research has produced a strategy to address spec-trum scarcity problems. This novel idea, named cognitive radio, consid-ers that secondary users can opportunistically exploit spectrum holes lefttemporarily unused by primary users. This presents a competitive scenarioamong cognitive users, making it suitable for game theory treatment. Inthis work, we show that the spectrum-sharing benefits of cognitive radio canbe increased by designing a medium access control based on quantum gametheory. In this context, we propose a model to manage spectrum fairly and ef-fectively, based on a multiple-users multiple-choice quantum minority game.By taking advantage of quantum entanglement and quantum interference, itis possible to reduce the probability of collision problems commonly associ-ated with classic algorithms. Collision avoidance is an essential property forclassic and quantum communications systems. In our model, two differentscenarios are considered, to meet the requirements of different user strate-gies. The first considers sensor networks where the rational use of energy is acornerstone; the second focuses on installations where the quality of serviceof the entire network is a priority.
Keywords:
Quantum games, minority game, spectrum allocation
1. Introduction
Modern wireless communications networks are composed of users access-ing the network through multiple devices, including cellular phones, Wi-Fi
Preprint submitted to Elsevier November 9, 2018 evices, and GPS receivers; moreover, users often operate multiple applica-tions simultaneously. The widespread use of these devices demands heavy useof network resources. The number of wireless devices and applications hasgrown exponentially in recent years, creating an almost unfathomable radiospectrum demand. Radio spectrum assignments are static and mainly as-signed to services such as TV and radio broadcasts, navigation, and so forth.As a consequence, few spectra are unused, making it a scarce and extremelyvaluable resource. Nevertheless, numerous studies have found that licensedspectrum is considerably underutilized in temporal, spatial, and frequencydomains [1]. By considering spectrum scarcity problems caused by staticspectrum allocation, cognitive radio (CR) is viewed as a novel approach forimproving the utilization of such an important resource [2]. The main ideaof CR is that users without licenses (cognitive users) can sense the spectrumin order to detect the presence or absence of licensed users (primary users);this enables them to access licensed frequency bands when primary users arenot present. Thus, in a framework of spectral opportunities, the secondaryusers must be able to make decisions and negotiate in the short term. Bythinking of the cognitive users as players competing or cooperating to accessavailable resources, the outlined scenario can be modelled by means of gametheory.
Game theory is a mathematical tool that analyses the strategic interac-tions among multiple decision makers. The generality of the theory permitsits use for modelling a wide variety of problems from different research areas[3]. The design of fair, secure, and efficient quantum information protocols isnecessary to guarantee the development of reliable quantum networks. Re-source allocation is one of the most important stages, and can be viewed asa competition in which the players are the nodes in a network that can con-trol the nodes actions. Furthermore, several authors have tackled the designof transmission protocols on classical networks by using game theory tech-niques, and have obtained interesting outcomes [4, 5]. We recently appliedquantum games to quantum wireless networks [6, 7], in order to enhancetheir efficiency.Quantum games have proven useful for solving problems encountered in thedecision sciences, in which the most relevant case is the prisoner’s dilemma.In the original classical version, the Nash equilibrium represents an inconve-nient situation for both players, while in the quantum prisoners game, a new2ash equilibrium appears that is both Pareto optimal and a better situationfor both players [8]. Moreover, for certain games, quantum strategies haveproved to be more effective than classic strategies. This is the case for theclassical penny-matching game presented by Meyer, [9] and the battle of thesexes game considered by Marinato et al., who showed that the introductionof entangled strategies leads to a unique solution, whereas in the classicalcase, the theory cannot make any unique prediction [10]. Furthermore, incites [11, 12], a quantum formulation of the dating market problem was intro-duced. In [13], the authors quantize the gamble known as Russian roulette.More recently, [14] studied the advantages of quantum strategies in evolu-tionary social dilemmas on evolving random networks, focusing on two-playergames such as the prisoner’s dilemma, snowdrift, and stag hunt. Quantumgame theory has been applied to a wide variety of phenomena where quan-tum laws rule; these include social decision theory [15], bioprocesses thatobey quantum statistical mechanics [16, 17], and quantum communications.In previous works, we analysed quantum MAC algorithms that use quan-tum game tools as a method of providing fair access to network users. In[18], a quantum algorithm was proposed to improve the current classic pro-tocols. There, under a hybrid cellular wireless network, users are capableof communicating in a centralized manner. Specifically, they communicateunder the control of a base station (BS), a device in charge of receiving andtransmitting signals to mobile devices in the network [19], or (eventually) ina distributed mode where communication among users is direct (i.e. withoutthe intervention of a third party). In the second mode, the network nodesbehave as non-cooperative game players that must decide the moment totransmit by analysing the other players actions. The channels limited ca-pacity and the multiple users wanting to transmit shape a scenario in whichchannel performance improves when fewer users are attempting to transmitin the same time slot. The minority game has been widely used in situa-tions in which players compete for limited resources, such as choosing whichevening to visit a bar that is usually overcrowded. In this context, we proposea model to fairly and effectively manage the resource allocations in cognitiveradio networks, based on a multiple-users multiple-choice quantum minoritygame [20, 21].The paper is organized as follows. Section 2 contains a description of themodel in the form of a two-user system, which acts as a preview of the moregeneral case further detailed in Section 3. In Section 4, a quantum circuit is3roposed to implement entanglement. In Section 5 two different alternativesand their implications are described. Finally, conclusions and further workare depicted in Section 6.
2. System model
The system analysed in this work has N channels and N users that mustbe assigned to one of those channels. The state of such a system in Dirac no-tation of some user j , where j = 0 , , ..., N −
1, is | c j i , with c j = 0 , , ..., N − | ψ i = | c i ⊗ | c i ⊗ ... ⊗ | c N − i = | c c ...c N − i . Thus, it must be understood that user 0 is assigned to chan-nel c , user 1 is assigned to channel c , and so on. In order to facilitateunderstanding, we present the simplest case of two users and two channels. Let 0 and 1 be the indexes of two smart devices attempting to transmitinformation through two free channels, 0 and 1. The devices are assumed tobe indistinguishable, and thus have identical transmission preferences. Thestates of the system are represented by vectors of a Hilbert space; morespecifically, the vector position corresponds to the user, and the value ineach position represents the user’s assigned channel. Collisions are avoidedwhen channels are not shared. For example, a desirable state is | c c i = | i ,which specifies that user 0 is assigned to channel c = 1; meanwhile, user 1is assigned to channel c = 0. If players play classically, the probabilities ofeach user and channel are all equal. The quantum equivalent for that case is | ψ C i = ( | i + | i + | i + | i )2 , . Then, according to the classic strategies, it is clear that they have (atbest) a 50/50 chance of avoiding collisions, and no strategy can modify thesystem in order to avoid collisions completely. Therefore, it is a classic Nashequilibrium of the system [3]. On the other hand, they can do better if theyplay quantum, because they can achieve a 100% probability of success. Inorder to take advantage of quantum computing, a one-shot quantum gameis proposed; it begins with the system in an entangled state, | ψ e i = ( | i − | i ) √ , | i means that both users are assigned to channel 0;meanwhile, state | i assigns both users to channel 1. In order to changethe initial state, the players must apply a strategy which, mathematically, isrepresented by a two-dimensional operator that we call U . Generally, playerscan choose to operate on their qubits using a classic or quantum U , in orderto increment their chances of winning. However, there is only one optimalquantum strategy (the Hadamard gate U = H ) that modifies the system toa more favourable state for the two-user example. Given the condition thatis applied by both players, the final state | ψ f i is: | ψ f i = H ⊗ · ( | i − | i ) √ | i + | i ) √ | ψ f i , it arises that the system can only collapse to ( | i ), where user 0is assigned to channel 0 and user 1 is assigned to channel 1, or to ( | i ), whereuser 0 is assigned to channel 1 and user 1 is assigned to channel 0. Thus, byquantum rules, a new Nash equilibrium arises, where the worst case is avoidedand both users transmit successfully with probability 1. Furthermore, it isa Pareto optimal solution because it is impossible to make any player betteroff without harming some other player.It is important to note that, because the studied network is composed ofindistinguishable devices, the necessary condition that all players take thesame actions is natural.
3. N-users game description
Usually, there are
N > N channels. Because none of the users has information about otherusers, there is a high probability that more than one of them will take partin a collision. When that occurs, all those involved cannot transmit, result-ing in a situation that must be avoided or, at least, minimized by meansof appropriate spectrum allocation protocols. These types of problems aredifficult to solve classically as the number of players increases, that is, theyare included in the group referred to as NP problems [22]. Accordingly, theycannot be solved in polynomial time, which generally results in inefficientsolutions. We are facing a type of decision problem consisting of agents withsimilar objectives that compete for a limited number of resources. Therefore,5he spectrum allocation problem may be modelled as a multiple-options mi-nority game.The CR concept implies that cognitive devices can make smart choices andaccess the spectrum holes left unused by primary users. Despite this promis-ing idea, it is very important to take into account that the existence of thosespectral holes is dynamic in size and limited in time, which causes difficultiesin sensing, sharing, and allocating tasks. The first constraint determines howmany users can transmit simultaneously. Meanwhile, the second constraintlimits the time that users have to select the channel they will transmit in,and the time they have to transmit. As the number of CR users increases,the decision processes become more complex, thus limiting the time the usershave to transmit. Taking the latter into account, an efficient spectrum allo-cation algorithm is absolutely necessary.Many researchers, including us, point to the use of game theory as the mostappropriate technique to model (and consequently perform) resource sharingand allocation tasks [23, 24, 25, 7]. In this same line of thought, we go astep further by proposing the use of a one-shot quantum game to minimizedecision times and the number of collisions. Figure 1: Cellular Cognitive Radio scheme
Our model considers a cellular network in which each cell has a single6ognitive BS and a group of CR users in its coverage range. The qualitativescheme of the network is shown in fig. 1. The BSs are transceivers in chargeof connecting the devices to other devices in the cell. To achieve this, theycollect the CR user reports, and prepare to allocate the radio channels. Itis assumed that the devices cooperatively sense the spectrum and recordinformation about the spectrum holes, which will eventually be provided tothe base stations. Cooperative sensing has been previously analysed by otherauthors [26].In the following, we focus on a quantum algorithm capable of managingthe spectrum allocation based on probability amplitude amplification. Morespecifically, we present two cases of interest: the first one aims to avoid allusers being assigned to the same channel, and the second one aims to enhancethe probability of quantum states that assign different channels to users.The proposed quantum medium allocation evolves by following three basicsteps:1. The cognitive quantum BS assigns a set of qubits to the cognitive usersin the cell range and prepares entangled state | ψ e i . | ψ e i = 1 √ N N − X k =0 ω k · pN | kk · · · k i , (2)where ω N = e πi/N and p is a tunable parameter that modifies theamplitude phase. Depending on p , it is possible to select BS preferencesto avoid the least favourable case, p = 1, or, on the other hand, toenhance the optimum one, p = ( N ( N − .2. Every node locally applies a one-shot strategy U to the initial state,which makes the system collapse to a new state. | ψ f i = U N · | ψ e i , (3)3. The nodes of each cell measure the final state ψ f to obtain the assignedchannel.In what follows, we present the quantum circuits for the case N=4 and de-scribe the main steps. 7 . Quantum circuit description Figure 2 shows a possible circuit to generate entangled state | ψ e i . Thesystem in base state | ... i is modified by the action of gate R applied onthe two upper qubits, R = 1 √ (cid:20) − (cid:21) generating state | ψ i = | i − | i − | i + | i . Then, thetwo upper qubits of | ψ i are the control lines of three Ctrl − F gates. Awhite circle in a control line indicates that the control qubit must be in state0; meanwhile, a black circle implies that the control must be in state 1 inorder for F k to be applied (see figure 3). Note that the range of the systemstate is N · log ( N ) and that R log ( N ) must perform the rotation on the upper log ( N ) qubits in the more general case. The extension of the circuit to N is straightforward. | i R ⊗ • •| i • •| i F F F | i | i | ψ e i| i| i| i ↑| ψ i Figure 2: Circuit that generates the initial entangled state | ψ e i . Finally, the action of gates F k on state | ψ i yields: | ψ e i = | i − | i − | i | i
5. Two alternatives - distinct purposes
One of the main functions of cognitive radio is to provide a fair spectrumscheduling method among coexisting cognitive users. In this context, the8 P F Figure 3:
Ctrl − F gate circuit, where P = ω pN · I ⊗ . Looking from top to bottom, the F operation is performed on the last six lines only if the state of the first two upper linesis | i . purpose of the proposed methods is to improve the classic methods abilityto decrease the collision probability. The objective of the first method isto increase the probability of occurrence of the no-collision state. When oneuser cannot transmit because of a collision, he must wait a lapse of time to re-manage the transmission request. System reliability and the networks qualityof service improve if collisions are avoided. The second method is focusedon wireless sensor networks, where the importance of all nodes being ableto transmit is superseded by the importance of avoiding network downtime;here, the objective is to avoid the massive collisions that occur when all usersare assigned to the same channel.As was described in Section 3, the channel assignation procedure is the samein both cases. The base station prepares the entangled state of eq. 2 withall the users in the cell, setting p = N ( N − /
2. After that, the users arepositioned to perform their strategies. Strategy U that applies each playeris represented by an N × N unitary matrix whose elements are U w = 1 √ N ( e πi/N ) r · c , where r, c = 0 , ...N − | ψ f i = U ⊗ N | ψ e i = (cid:18) √ N (cid:19) N +1 N − X k =0 ω k · pN | kk · · · k i , | ψ f i = (cid:18) √ N (cid:19) N +1 N − X k =0 N − X c =0 · · · N − X c N − =0 (cid:16) e πiN k · p e πiN k · c · · · e πiN k · c N − | c · · · c N − i (cid:17) . Thus, the state coefficients can be expressed as: α c ··· c N − = (cid:18) √ N (cid:19) N +1 N − X k =0 e πiN k ( m z }| { p + c + c + · · · + c N − . (5) The fairness of the network implies that every user has a priori the samechances to transmit. In the language of games, the BS acts as the arbiter ofthe game because it assigns the qubits to the players and creates the entan-gled state. Later on, the players strategies modify the state amplitudes andhence their chances to win. The players receive a reward, which in this caseis to succeed in transmitting.As set forth above, once spectrum holes are detected, the nodes must beassigned to one channel. Clearly, there are N ! possibilities that every playerwill be assigned to different states, with N being the number of cognitiveusers. Therefore, provided that all the cognitive users are indistinct, theprobability that all of them transmit at the same time is P c = N ! N N in theclassic world; for example, P c = 2 . × − if N = 8 , . Such a low successprobability can only be increased by means of statistical methods involvingexploration and/or a previous knowledge of the network [27], which is hardlypossible if the network is continuously changing. In this framework, we pro-pose the one-shot quantum game-based algorithm.The m sum in the phase factor of eq. 5 is analysed in order to properlyselect p . Thereby, a proper use of quantum interference makes it possible toimprove the players chances. The case where c = c = · · · 6 = c N − leads to m = p + N ( N − . Thus, in order to guarantee the constructive interference, p = N ( N − and the phase factor is e i πk ( N − . Finally, the probability of themost favourable case is P best = N · N ! N N , which is N times larger than the10lassic one, P best = N · P c . Clearly, the algorithm performance provides amore efficient use of the devices energy, extending the time of communica-tion and battery life. However, this point is even more sensitive in the typeof networks that are analysed below.
Wireless sensor networks are a type of autonomous communication net-work mainly deployed in areas where access is almost impossible. Everydevice installed at each node is a small computer in charge of monitoringphysical and environmental conditions such as temperature and air pressure.The sensed data are sent to a base station for analysis. Although sensornetworks were originally designed for military purposes, their applicationsnow include area sensing, industrial monitoring, and health care monitoring[28, 29]. One of the main challenges that communications engineers mustface is the optimization of the networks power consumption, because of thelimited lifetime of the devices batteries and the impracticality of replacingthem frequently. Therefore, in order to extend the networks lifetime, more ef-ficient communication protocols are needed. Because collisions are the maincause of unnecessary energy consumption, we propose a quantum algorithmthat prevents the most unfavourable situation. When all users are assignedto the same channel, no transmissions will be performed. There are N ofthese worst-case possibilities from a total of N N , so the probability of theworst-case is P worse = N (1 − N ) by means of classic computation, where theprobability that any user will be assigned to any channel follows a uniformdistribution.Once the eventually free channels are identified, the channel allocation pro-cedure begins. The proposed quantum algorithm considers that the cognitiveBSs have the extra ability to share a set of qubits with each node in the celland to prepare the entangled state of eq. 2, setting p so that the probabil-ity amplitude associated with states | c, c, ..., c i is reduced to zero. If playersmeasure their state directly on | ψ e i , it will collapse to one of the worst cases.Otherwise, the users perform their strategies in order to change their chances.Let us note that c = c = · · · = c N − = c leads to m = p + N · c when allusers apply U w as before. Thus, in order to guarantee the destructive inter-ference, p = 1 can be chosen. Then, the probability amplitude coefficientsare: α cc ··· c = (cid:18) √ N (cid:19) N +1 N − X k =0 e i πkc e πiN k = 0 .
11y allowing at least one node to send information at a certain time slotby using the idle spectrum holes, the sensor network avoids downtime, asignificant aspect regardless of the network structure [30]. For instance, ifthe network uses a star topology, every node communicates directly with theBS. Because the nodes can communicate only through the BS, it representsa single point of failure (SPF) that makes this topology unreliable. However,owing to its simplicity, it is frequently chosen when coverage areas are nottoo wide. In that case, the quantum BS must prepare the new allocationscheme by requesting information from the rest of the CR nodes. Althoughthe SPF problem remains unsolved because there are many failure sources,the network reliability is improved under normal BS functioning owing to theone-shot characteristic of the algorithm, which allows at least one node of thestar to always send information; this optimizes energy use. Meanwhile, in thecase of multihop systems, each node can communicate directly and is able totake distinct paths to reach the data collector, which is advantageous as thereis no SPF. On the other hand, these networks have an important disadvantagehigh power consumption. To operate, they must draw more power becauseeach node in a mesh must act as a BS. This issue is even more serious ifthe spectrum allocation task is not performed efficiently. Likewise, in ourmodel, each node of the mesh must eventually prepare the allocation schemefollowing the procedure explained above, in order to exchange informationgathered from the environment or from other nodes. The goal is to minimizethe power consumption of mesh topologies by reducing collisions, which ismade possible through the proposed quantum allocation algorithm.
6. Conclusions
In the context of spectral opportunities that constantly change with time,designing a fast and efficient method of spectrum allocation is increasinglynecessary. In order to address the problem of many users competing for acommon resource, (in this case, spectrum), many researchers have appliedstrategies based on game theory. The classic algorithms that have beenattempted up to now need exploration and learning time to allow players toselect the most favourable actions; this costs valuable time that can be used totransmit, and provides no guarantee of success. In the present work, we haveproposed a quantum game-based scheme for cognitive spectrum allocation.The model offers two alternatives. The first aims to increase the no-collisionprobability over that of the classic approaches, which is essential in networks12such as cell phone networks) where quality of service is prioritized. Theother alternative prevents all the cognitive users from converging to the samechannel. This strategy is proposed for sensor networks, where one of the mainrequirements is that they never stop working; this allows the base stationsto continuously receive data for analysing. Because of the characteristics ofthe one-shot algorithm, less time is wasted in the channel allocation process,which makes it possible to repeat the algorithm and further increase thesuccess probability. Finally, both alternatives contribute to energy savingsthrough the reduction in channel allocation times, an item that is even moresensitive in the case of sensor networks. In such a case, we considered twoactual network topologies in which the proposed allocation algorithm can besuccessfully applied. Future trends in wireless sensor networks will imposemore autonomy and less power consumption. The work we have presentedtakes advantage of cognitive radio and quantum game techniques to addressthese issues more efficiently, compared to the classic methods. Althoughcognitive radio and quantum communications are still in development, webelieve our proposal advances the adaptation of these new communicationparadigms.
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