Game Theory for PHY layer and MAC sublayer in

800

Chapter LI

Game Theory for PHY layer and MAC sublayer in Wireless Telecommunications J. Joaquín Escudero-Garzás University Carlos III de Madrid, Spain Ana García-Armada University Carlos III de Madrid, Spain

Abstract The aim of this chapter is to address the role of a novel concept in wireless telecommunications: Game theory. Game theory is a branch of applied mathematics, which has recently drawn attention as a powerful tool to solve complex problems in wireless environments. To fulfil the intended goal, this chapter introduces the most relevant concepts of game theory such as game, player, and strategy, and give an overview of the applications of game theory in wireless networks.

INTRODUCTION In telecommunications, state-of-the-art trends are towards very sophisticated services the objective of which is to provide multimedia contents to fixed or mobile users. This implies a strict quality control to guarantee the delivery of these services to the clients, ensuring high quality of service (QoS) and large data rates. To meet such requirements, telecommunications resources are intensively used. Moreover, when providing

multimedia services, their quality parameters and the available resources are interrelated, creating dependence relations between them. As one may perceive, the task of a joint management of all these variables with the aim of exploiting the network in an optimized manner requires the utilization of tools capable of tackling such complex situations, and this task frequently becomes highly intricate when layer-by-layer solutions are applied. Then, novel approaches are used to solve complex telecommunications problems. One of these novel

Copyright © 2009, IGI Global, distributing in print or electronic forms without written permission of IGI Global is prohibited.

Game Theory for PHY layer and MAC sublayer in Wireless Telecommunications

approaches is game theory, which can efficiently deal with the aforementioned problem of joint management of resources, service provisioning and network exploitation. From its beginning in the middle of 20th century game theory has been intensively applied to economics, evolutionary biology and psychology, among others. Its utilization in telecommunications started at the late 1990s for the resolution of problems related to networking (flow and congestion control, routing), but its most interesting applications came relatively recently: its potential utilization for wireless systems. The objective of this chapter is to provide an overview of game theory for telecommunications. Once the basic concepts are introduced, most relevant and recent works will be described.

GAME THEORY FOR WIRELESS COMMUNICATIONS Why may game theory be applied to wireless telecommunications? The answer to this question rests on the very nature of game theory. It formalizes the interaction among autonomous agents (players) with selfish and even opposite objectives, and defines what a solution to the stated problem may be. It is clear then that by means of game theory the design and configuration of wireless networks such as ad hoc, sensor and mesh networks may be addressed in order to cope with limitations as power constraint, decentralized operation, interference mitigation and efficient multi-hop routing. Game-theoretic concepts have mostly been applied in wireless applications to design network layer, flow control mechanisms and routing algorithms. Game theory has also demonstrated its validity for power allocation problem in wireless networks, where the network (e.g. the base station in centralized configurations) allocates the available resource (power) to the players (transmitters) with different types of constraints, e.g.

interference, energy minimization or minimum required bit-rate constraints. In a similar manner, bandwidth allocation problems may be solved for this type of networks considering constraints such as the mutual interference among the transmitters. An alternative approach to obtaining solutions in telecommunication networks, e.g. routing protocols, is the usage of pricing: pricing schemes charge each node or customer locally for the resources he has used. An overview of the most relevant works on these areas is developed in the next section. PHYSICAL (PHY) LAYER and medium access control (MAC) sublayer problems in wireless networks have not received too much attention up to recent dates in the context of game theory. For instance, in the case of PHY layer, we may consider cooperative diversity (explained in detail in a subsequent section) as a suitable technique to be studied applying game theory given the intrinsic selfish behaviour of the users of the network. With respect to the MAC sublayer, in addition to allocation applications, some game theoretic approaches have attempted to model packet radio protocols (Aloha, SlottedAloha) and channel assignment strategies. These applications demonstrate how promising game theory is for PHY and MAC LAYERS design. Game theory, as we mentioned in the introduction, is a mathematical tool, but it is not very familiar to telecommunications. Hence, we will devote the remaining of this section to introduce the most relevant concepts and elements used in the literature up to now to address wireless systems by means of game theory. Although we consider the overview provided in this section adequate for the comprehension of the rest of this chapter, the reader may find it useful to complement this introduction to game theory with (MacKenzie and DaSilva, 2006).

What is a Game? Let us start by defining the very concept of game. A game is (Osborne, 2004) “a description of 801

Game Theory for PHY layer and MAC sublayer in Wireless Telecommunications

strategic interaction that includes the constraints on the actions that the players can take and the players’ interests, but does not specify the actions that the players do take”. The ultimate aim of the game formalism is to provide a solution consisting of a set of the different players’ outcomes. Those outcomes are the result of jointly maximizing each player’s utility, by means of the assignment to each player of a certain quantity of resources (money, power), considered as variables in their benefit functions. Then, it is clear that the goal of the players is to obtain a benefit as high as possible.

To Optimize or to Play a Game? Game theory is worth being used for different telecommunication problems, but it must be handled with care. When we are trying to solve a problem in which the aim is to achieve the maximum benefit, sometimes the best tool to be used is not game theory. Therefore, we must be completely sure that we have not an optimization problem for a single decision maker, rather than a situation that involves several decision makers that are trying to get as much individual benefit as possible. To make this clear, let us see an example. Consider the situation of telecommunications monopoly market, where there is only a telecommunications services provider for the users: The company will try to maximize its benefit performing an optimization of its resources; the optimization is based on the interaction with the users. On the other hand, for an oligopoly market where other telcos have entered, pricing games among all the companies take place and they respond in this context to the decisions of the consumers.

The Basic Elements of a Game: Players, Strategies and Benefit The aim of this section is to provide to the reader an introductory overview on game theory. Then, 802

we must start by formally defining a game in its strategic form: a game G is defined by the tuple {N, {Si}, {Ui} } i ∈ N, where N = {1, ..., |N|} represents the set of players, Si = {s1,... sk} represents the set of k pure strategies available for player i, and Ui represents the benefit for player i. If S = {x1,..., x|N|}, for all xi Î Si, represents the Cartesian product of the set of strategies for all the players, then Ui : S → T, being Ta subset of R. Note that each N-tuple (x1,..., x|N|) is a possible outcome of the game. Another possibility of representing a game is the extensive form, where the game becomes a tree: the nodes of the tree stand for the different stages of the game, because of the moves of the players, and each branch represents a possible action that the player may take. Players. As we have introduced at the beginning of this section, a game is made up of three fundamental components: the players, their strategies and their benefit. The players are the decision makers interacting in the scenario where they play the game. Their final goal is to optimize the benefit they receive. In wireless communications, players model very frequently the nodes of the network or the users, but depending on the specific situation, players may be different elements of a telecommunications network, such as the links between nodes, or even telecommunications operators. Strategies. Players will have different choices or alternative actions to play the game, and the different choices for each player are known as his strategies; for instance, the strategies for a certain node in a wireless network could be the selection of a modulation and coding scheme, his transmit power level or bit-rate. Once all players have selected their strategies, this determines the outcome of the game for the player. The strategies that will yield the outcome of the game may be stated in two manners: The players of a game may act in a certain way breaking down to a set of strategies, which can be distinguished as pure and mixed strategies whether the player’s

Game Theory for PHY layer and MAC sublayer in Wireless Telecommunications

behaviour is deterministic or probabilistic respectively. Usually, the set of strategies is made finite Si = {s1,... sk}, and each sj is a pure strategy. On the other hand, a mixed strategy over the set of pure strategies is a vector p = {p1,... pk} with nonnegative values which sum up to 1, and where each coordinate pj of p represents the probability for a player to choose strategy sj Benefit: payoff and utility. Players receive a benefit when they choose a strategy, always attempting to get as much benefit as possible, and conditioned to other players’ preferences and strategies. In other words, players play the game to optimize their benefit. A players’ benefit is specified by a real value assigned to the specific actions or strategies chosen by the players. Obviously, the larger this value the higher the player’s benefit. In obtaining the benefit, there exist two possibilities: it may be defined by means of a finite set of values or payoffs, so in this case the benefit is defined by Ui : S → T, being T a finite subset of T. The second possibility is by defining a utility function, which associates a number to the chosen strategy by means of a real function, that is, the benefit is expressed as Ui : S → ℜ. As an example, a typical utility function in wireless networks is the bit-rate (bits per second) per consumed power (Watts) achieved in a particular interference situation.

Playing a Communications Game: Cooperation and Equilibrium When playing a game, players may take their decisions knowing other players’ set of strategies but without any knowledge of other players’ chosen actions. In this case, the game is non-cooperative: the players do not exchange any information with each other to achieve the maximum selfish benefit. An issue of paramount importance on a game is that the game must provide an outcome. For non-cooperative games, a stable outcome corre-

sponds to a NASH EQUILIBRIUM (Nash, 1951), a concept directly related with utility functions. A set of strategies for each player s* = {s1*,... sN*} is a Nash equilibrium of G = {N, {Si}, {Ui} } if no player can unilaterally improve his own utility while the rest maintaining their strategies fixed; mathematically, for every j player holds Uj (sj*,s-j*) ≥ Uj (sj, s-j*) for all sj ∈ Sj, where s-j* denotes de vector of strategies for all players except player j. NASH EQUILIBRIUM may also be interpreted from a second viewpoint: s* = {s1*,... sN*} is a Nash equilibrium of G = {N, {Si}, {Ui} } if sj* is the best response (strategy) to s-j*. Players may move simultaneously, or sequentially. When they play simultaneously, most of the times they have no knowledge about what the other players do, and these games are called static games. If players move sequentially, they might have some information about other players’ moves, or they might know everything; in the former case, we are talking about dynamic games of partial information, and in the latter we will refer to them as dynamic games of perfect or complete information. Nevertheless, sometimes, to achieve a solution to the game the players must interact several times, knowing the history of their previous movements and outcomes, and this interaction is modelled using repeated games. The utilization of distributed algorithms with partial information, referred sometimes in the literature as dynamic games, leads to tractable algorithms for players with processing constraints. The result is also a Nash equilibrium, since these algorithms converge to this point. We have seen the importance of NASH EQUILIBRIUM when playing games. Therefore, proving the uniqueness and existence of NASH EQUILIBRIUM is a must in practical wireless communications games. Moreover, the NASH EQUILIBRIUM, even existing and being unique, is not optimal in general for the purpose of the communications system. On the contrary, there are situations in which players may interact before

803

Game Theory for PHY layer and MAC sublayer in Wireless Telecommunications

playing the game and jointly agree their actions. In this case, we say a game is cooperative and the players can obtain optimal outcomes, namely Pareto optimal. For a cooperative game, the solution of the game s* is Pareto-optimal if there exists no other solution for which one or more players could improve their utilities without reducing the utility of any of the other players. Therefore, any other jth player’s strategy sj different from sj* will cause a decrease in, at least, the utility of one of the other players, i.e. there exists at least one j player such that Uj (sj*,s-j*) ≥ Uj (sj, s-j). In the context of game theory for wireless communications, it is widely accepted that non-cooperative game-theoretic approaches are more suitable for distributed systems, given the additional complexity intrinsic to cooperative games. On the other hand, centralized networks are commonly modelled by cooperative games, since the central node or sink asks for information to the rest of nodes of the network. We detail some examples of both cases in the next section. However, as the reader may guess, these rules are not fixed: for example, cooperative games can also be applied to distributed network problems, as we shall show by means of the use of coalitional games. Hitherto, we have considered Nash equilibria as potential solutions for a game. However, it might be appropriate for the system to model a game with the so-called Stackelberg equilibrium to define a different type of solution. When the solution comes from a Stackelberg equilibrium, the moves of the users shall follow the move of the network manager, which is acting as a “leader”.

Specific Types of Games Depending on a series of details that characterize the game, in the literature we can identify some specific game models when applying game theory to wireless communications. Among these models,

804

potential and supemodular games are particularly interesting: the existence of Nash equilibrium is not guaranteed for games in general, while Nash equilibrium always exists in these two particular types of games; furthermore, the game converges to this Nash equilibrium. In the following, we shall briefly introduce these types of games and provide some examples of application. A potential game is a game characterized in its strategic form by a function V: S → U, such that if a unilateral deviation occurs, the change in V, namely ∆V, is reflected in a change of the utility function seen by the deviating player i ∆Ui. In other words, a potential function exists and reflects exactly any unilateral change in the utility function of any player. The rationale for the use of potential games, as we have mentioned in the above paragraph, is that they converge to a NASH EQUILIBRIUM. As examples when applying potential games, we highlight the following references: Nie and Comaniciu (2006) show the application of potential games for channel allocation in wireless networks; Scutari, Barbarossa and Palomar (2006) yield a framework for power allocation using potential games; and in (Hicks, Mackenzie, Neel and Reed, 2004), interference avoidance is addressed also in the context of potential games. A game G = {N, S, {Ui}is a supermodular game if for each player i ∈ N, the strategy space Si is a nonempty and compact subset of ℜ, the utility functions Ui are continuous and has increasing differences between player i’s strategy and any other player’s strategy. Although the formulation is complicated, supermodular games posses the interesting property that there always exists a NASH EQUILIBRIUM. For this reason, this type of games has been extensively used to solve power management in wireless networks; to illustrate this we refer the reader to the works of Saraydar, Mandayam and Goodman (2002) and Altman and Altman (2003) as examples.

Game Theory for PHY layer and MAC sublayer in Wireless Telecommunications

FUNDAMENTAL RESULTS IN GAME THEORY FOR WIRELESS COMMNICATIONS In this section, we describe the relevant issues to which game theory is applied in wireless communications; for example, power allocation, the backoff attack problem or cooperative diversity. For these problems, we also summarize the most highlighted works and results that the usage of game theory has provided to date. Of course, game theory has also come up with solutions others than for the applications we mention, e. g. flow (Altman and Basar, 1998) and congestion (Hwang, Tan, Hsiao and Wu, 2005) control, quality of service (QoS) (Wang and Ramanathan, 2005) or routing (Altman, Boulogne, El-Azouzi, Jimenez and Wynter, 2006). Nevertheless, our goal is to provide a vast overview on game theory -based wireless communications for PHY and MAC LAYER. In detail, we shall analyze power management and interference avoidance in the context of PHY layer, the backoff attack problem and random access protocols for MAC sublayer, and three issues related to both PHY and MAC: spectrum management, cooperative diversity and cross-layer design.

PHYSICAL LAYER Power Control and Management Power-based design is one of the crucial wireless communications subjects, given the need of a tradeoff between the transmission power and the potential interference caused: the higher the transmit power to achieve large bit-rate, the higher the interference it may cause. Game theory has emerged, from its earliest utilization in communications, as a very adequate tool when power assignment becomes a problem if the aim is to satisfy users’ requirements fairly. Moreover,

limitation of energy of battery-powered devices in wireless networks such as sensor and ad hoc networks has arisen in the last few years as the most restricting factor of performance in these networks. To address the energy-efficient use of batteries, power must be treated as a scarce resource in the context of game theory. In the following, we will point up the most recent researches in power management, since references corresponding to the early works in this field are detailed in (Altamn et al., 2006). A major concern on power management is the level of interference a transmitter may cause over other users, especially when we face some modern wireless transmission technologies, of which MIMO (Multiple Input Multiple Output) systems and ultra-wide band (UWB) are illustrative examples. To solve the problem of interference using Game Theory, one of the two following approaches usually applies: the harmonization of selfish interests among users desiring to transmit at maximum power or directly penalizing (charging) those users employing their power unfairly. In impulse-radio (IR) UWB systems, the particular effect of self-interference must be considered and channel fading cannot be assumed flat. These systems are suitable to be analyzed from a game-theoretic perspective as done in (Bacci, Luise, Poor and Tulino, 2007), by means of a non-cooperative power control game. For interference mitigation in ad hoc MIMO systems and additional optimization of the consumption of batteries, Liang and Dandekar (2007) propose a power control game. They formulate the problem from an information-theoretic point of view: minimizing interference by using power control increases MIMO capacity and lifetime of batteries. The remarkable point in this approach is the fact that the players are the links, which choose a transmit power and want to maximize their utilities, consisting on the capacity of each link diminished by a second term proportional to the power used in this link.

805

Game Theory for PHY layer and MAC sublayer in Wireless Telecommunications

The purpose of power control for uplink of Code Division Multiple Access (CDMA) is to assure for each user a given QoS without interfering with the other users. This topic has been addressed in detail using Game Theory, and a suitable metric of satisfaction for the users is the resulting signal-to-interference-ratio (SIR) in the uplink. For example, Alpcan, Basar and Dey (2006) analyze the case of a multicell CDMA network where the utility of a player grows as the outage probability decreases and the cost function reflects the battery usage. As regards energy constraints, efficient power control has been tackled frequently through pricing, an adequate framework to deal with selfish behaviour of players (wireless terminals in our case) whose actions would affect to other players’ utilities. By using pricing, the objective is twofold: generating revenue for the whole system, and encouraging players to use system resources efficiently. The desired result through pricing has been largely obtained by means of utility functions of the form of throughput per energy (bits per Joule) or throughput per power (bits per Watt), since the utility represents the benefit obtained for one unit of consumed resources (see again (Saraydar et al., 2002)).

Interference Avoidance Interference is considered as a limitation factor of performance in the design of wireless systems, so the interest on its avoidance justifies the numerous literature in this matter. Clearly, interference analysis is closely related to power management, and they are often jointly addressed as we have seen in the precedent subsection. Let us see an example where power control games are not the only method to minimize interference in a gametheoretic framework. The Lacatus and Popescu’s (2007) proposal is a distributed adaptive algorithm for CDMA systems to obtain optimal codewords and power allocation for a required signal-to interference plus noise ratio (SINR); the game

806

is formulated for joint codeword and power adaptation and separated into two non-cooperative subgames: a codeword adaptation subgame and a power adaptation subgame. Sung, Shum and Leung (2006) introduced the concept of subgame: when the utility function of the game is separable with respect to the variables that define the users’ strategy the game may be separated into as many independent subgames as variables define the strategy; and each subgame utility function will depend exclusively on one of the variables.

MAC SUBLAYER The Backoff Attack Problem One of the security problems recently found in networks based on IEEE 802.11 standard has to do with Carrier Sense Multiple Access/Collision Avoidance (CSMA/CA), the contention mechanism of MAC sublayer Distributed Coordination Function (DCF) defined in the standard. The vulnerability, known in the literature as backoff attack, is related to the contention window (CW), which sets a countdown to transmit each time the channel is sensed as idle. Then the user will transmit with a certain probability when the countdown is over. The problem is that each user is able to set the CW locally, thus by giving a small value to the CW the user will have more opportunities to transmit: the lower the CW, the higher the throughput. This is, of course, a misbehaving procedure and those users are acting as cheaters. In order to solve the backoff attack problem, different authors have made game-theoretic approaches, and the key point is to design a game in which the cheater is discouraged on malicious behaviour. Among all works that address backoff attack, the perspective taken by Konorski (2006) is particularly interesting. He proposes a CSMA/CA repeated game (CRISP, Cooperation via Randomized Inclination to Selfish/Greedy Play), based on

Game Theory for PHY layer and MAC sublayer in Wireless Telecommunications

classical Prisoner’s Dilemma (Binmore, 2007). He proposes the utility function of user k is directly related to his previous bandwidth allocation bk and this bandwidth will be dramatically inferior if a user acts cheating about his CW instead of acting honestly, e.g., playing CRISP; consequently, users are discouraged on malicious behaviour. Equally interesting is the approach deployed by Cagalj, Ganeriwal, Aad and Hubaux (2005): initially, all the nodes are considered cheaters and they have the ability to measure the others’ throughput, and so detecting which nodes are cheating, e.g. not cooperating, and thus transmitting at higher rates than tolerated. Of course, the game is cooperative and the nodes share the necessary information to detect whether a user is cheating or not.

Random Access Protocols In wireless systems with random access, one of the most popular protocols is CSMA. In CSMA systems, user‘s access to wireless channel is probabilistic, and is characterized by the persistence probability, defined as the probability which the user will attempt to transmit with, once the channel is idle. Researchers have studied the persistence probability in the framework of Game Theory and they have mainly focused their works on calculating the persistence probabilities to optimize their transmission, either by maximizing the throughput or by minimizing the probability of collision. The key question for a user is to reach equilibrium: neither to be too aggressive, with high values of the persistence probability, nor to be too conservative. For the former case, the number of collisions would increase and for the latter, the user would miss opportunities of transmission; in both cases, the performance deteriorates. The following two examples make clear how to apply game theory to CSMA. Altman, Borkar and Kherani (2004) optimize the persistence probability for CSMA networks in this direction. The key point of the non-cooperative game is the

combination of transmission and reception probabilities through the utility function of the nodes, that is, both roles of a node, as transmitter and as receiver, are considered. Another option is by designing a MAC protocol which makes use of links as players of the game (Tang, Lee, Huang, Chiang and Calderbank, 2006), and the utility of each link will depend on his own transmission and transmissions of other links. Each link’s strategy is the persistence probability of the random access, and the utility function has the form of the expected reward (depending on the probability of successful transmission) minus the expected cost (depending on the probability of transmission failure). By its simplicity for implementation, Aloha protocols have been very popular in packet radio networks. The first and most relevant attempt on addressing Aloha protocol using game theory was provided by Wu and Wang (1995). They analyzed Slotted Aloha in a game-theoretic framework where users have different QoS requirements. The objective of the game is to achieve fairness among users when accessing the channel, trying not to favour those nodes closer to the access point. The strategies are made up by the pair packet retransmission probability (that is analogous to persistence probability of CSMA) and power transmission, resulting in a multi-objective approach, which leads to the optimization of throughput and average delay. A problem that has gained more relevance with the appearance of coexisting wireless technologies, as WiFi and WiMAX, is the presence of several access points (AP) in the same service area, based or not on the same technology. The solution to medium access in this competitive environment provides a good opportunity for game theory, given the natural selfish interest of each AP. Although involving not only MAC aspects in a sort of cross-layer protocol, a session admission and rate control (ARC) protocol with multiple access providers in a competitive market is considered in (Lin, Chatterjee, Das

807

Game Theory for PHY layer and MAC sublayer in Wireless Telecommunications

and Basu, 2005). Being competitive, the market allows performing charming to users, that is, they have the freedom of switching to a different provider. The viewpoint of providers is adopted when designing the ARC algorithm, since the objective is to maximize the provider’s revenue. For session admission, a non-cooperative game is played among each provider and his customers: the provider tries to maximize his revenue, and users want to maximize their payoffs that include a penalization in the case they decide to churn unilaterally before the end of the contract. Users are differentiated into classes according to their packet blocking rate. The rate control scheme is based on allocating power to the users: if a new admission occurs the transmission rate of users in the same class as well as in lower classes is reduced.

SPECTRUM MANAGEMENT AND SPECTRUM SHARING Given the well-known scarcity of spectrum for wireless communications, the research community has done many efforts to use it efficiently and the utilization of game theory has played a relevant role in these studies. One of the approaches deals with integration of 802.11 WLAN with 802.16 multihop mesh networks, to relay WLAN traffic to Internet; we can cite, for instance, Niyato and Hosssain (2007). These authors define the following framework to analyze the coexistence of wireless networks: a connection is admitted if and only if the global utility is improved. If the network admits the connection, all the connections play a bargaining game, being the solution a Pareto-optimal assignment of the bandwidth to the established connections. In centralized networks, a common approach is to consider mechanisms, a mathematical tool well studied in Economics, where the users communicate their preferences to a base station or a similar entity. We can see a mechanism as a three-stage

808

process. First, a leader announces the rules (named here “outcomes”) of the mechanism. At a second stage, the agents communicate to the leader, by means of messages, their private information. Finally, the leader allocates resources according to the rules and the private information provided by the agents. Price and Javidi (2007) propose a rate allocation mechanism for the downlink of singlehop networks. The ultimate aim of the algorithm is the design of a centralized scheduler considering the efficient use of uplink for the allocation of rate to the downlink. Proceeding in such a way, the users in the uplink are forced to reveal their utility truthfully, intended as private information, and they are not tempted to cheat about their needed rate, since users are charged in terms of downlink for their consumption of uplink. A different sight is to consider the following game: the players are networks deployed in the same geographical area, sharing the available channels. Wendorf and Seidenberg (2007) follow this approach, by proposing a methodology for channel assignment in spectrum-agile scenarios, where networks can dynamically switch their communication channels. These authors propose a game, which they refer to as “channel-change game”, with two networks as players. The networks can switch channel or keep it when they are using the same channel. If a network switches to a different channel, it incurs in a delay cost, but if both remain in the same channel, there is a blocking. The solution of the game leads to the use of mixed strategies, since pure strategies yield to a blocking situation.

COOPERATIVE DIVERSITY In large wireless networks, cooperation among the nodes is essential to achieve end-to-end communications when the source and the destination nodes are out of range, i.e. they cannot establish a direct link. In packet networks, end-to-end communications consist of multihop communications,

Game Theory for PHY layer and MAC sublayer in Wireless Telecommunications

and some of the intermediate nodes between the source and the destination act as relays, carrying out packet forwarding or signal forwarding. Novel techniques in wireless communications rely on cooperation among the users for the transmission of other users’ information. Cooperative diversity allows wireless nodes to collaborate with each other to complete their end-to-end transmissions in a multi-hop network: the source node sends the signal directly to the destination, and the same signal received (x1) by an intermediate node (relay) is retransmitted to the destination. The signal obtained in the receiver is a combination of both direct x2 and retransmitted received x3 signals (see Figure 1). The crucial difference then between cooperative diversity and simple relaying is the spatial diversity, due to the existence of two paths that conveys signals to the destination. Due to their implicit use of cooperation, the closeness between both cooperative diversity and relaying schemes is clear.. Consequently, in this subsection we will also refer to relay approaches based on Game Theory. As pointed out in the precedent paragraph, cooperative diversity exploits spatial diversity and the result is the improvement of the reliability of the communication in terms of, for example, bit error rate. Naturally, the first intuition is to think of cooperation from an selfish viewpoint, since a cooperative node is spending his own resources (power, bit rate) for the transmission of other’s Figure 1. Cooperative diversity transmission

information; nevertheless, the selfish and at the same time intelligent position is to cooperate as demonstrated in (Sendonaris, Erkip and Aazhang, 2003). Cooperation in cooperative diversity goes further than the simple relay action extensively used for packet forwarding, as it is explained with the following different cooperation schemes. The simplest approach is known as Amplify-andForward (AF) scheme, in which the intermediate node acts as a relay, amplifying the received signal in accordance with its own power constraints. Decode-and-Forward (DF) schemes are more sophisticated: the intermediate node performs detection and decoding for the received signal, and encodes this signal again, therefore reducing the impact of the noise. While AF and DF schemes are fixed, Selection-and-dynamic schemes allows the cooperating terminals to adapt their transmissions according to the channel state. Among cooperative diversity schemes, AF scheme is widely utilized since it is a very straightforward way of cooperation. Wang, Han and Liu (2006) formulate AF as a Stackelberg game, where the leader is the source node and the potential relays act as the followers. In the game, the source wants to buy power to the relays for retransmission, and the relays compete among themselves to provide the leader (i.e., the source) the required power at a suitable price. Also interesting is the study and application of AF and DF schemes in MIMO cellular environments, as shown in (Agustín, Muñoz and Vidal, 2004). Stimulation of AF cooperative diversity for ad hoc networks is devised as a game with pricing according to Shastry and Adve (2006). The aim is to induce forwarding, ensuring that the users and the access point (AP) benefit from cooperation: users reimburse each other for forwarding and AP charges users for transmitting data packets. Users’ preferences are to obtain as high throughput and low energy consumption as possible, so their utilities consist of the throughput per consumed energy (bits per second/joule). The game is played

809

Game Theory for PHY layer and MAC sublayer in Wireless Telecommunications

by two users with the possibility to cooperate in the form of a strategic game, with the goal to reach the Nash equilibrium, and therefore determining the power needed for cooperation. The role of the AP is to communicate to both users the reimbursement prices it has previously obtained by maximizing its own service price. As Scutari, Barbarossa and Ludovici (2003) show, the design of optimal coding strategies for multihop networks alleviates the necessary signalling required for cooperative transmission by using cooperative diversity. These authors derive the optimal coding matrix by proposing a non-cooperative game, with the objective of maximizing the information rate between transmitter and receiver in the presence of interference. In this game, the players identify the links of the network, the transmit power for each user is his strategy, and the payoff corresponds to the information rate. When using coalitional games, players are divided into “coalitions”. If cooperation is formulated in terms of a coalition game, a coalition is a group of cooperative nodes (Mathur, Sankaranarayanan and Mandayam, 2006), and the nodes of a network may take part of the coalitions. The objective of the game proposed by Mathur et al. is to maximize the receiver’s rate, based on the mutual information between the transmitters and the receivers making up the coalition. A node may decide to leave the coalition to achieve a higher rate. Nodes acting as relays may be interested in cooperation, or may not. Srinivasan, Nuggehalli, Chiasserini and Rao (2005) address how to decide about cooperation by means of a non-cooperative game where the nodes are organized into classes depending on their energy constraints and lifetime expectation. In this game, the utility function is the number of relay requests generated by a node. The nodes can use Generous Tit-For-Tat (GTFT) strategies, a variation of TFT strategies: in a dynamic game, a node playing TFT mimics the action other player has carried out in the

810

previous realization of the game; this would lead to a non-relaying action for the node playing TFT if the imitated node did not cooperate. When the nodes use GTFT they may act occasionally “generously” not mimicking the non-cooperative strategy. The implementation of GTFT entails a record of the past experience of each node. One of the advantages of the proposed game is that the solution is a Nash equilibrium that converges to a Pareto-optimal assignment of the number of connections to be relayed by a node. Felegyhazi, et al. (2005) address cooperation for packet relaying in wireless sensor networks (WSN). The goal of the game is to determine if the controllers (here referred to as authorities) pertaining to different WSNs can help each other in forwarding packets and, at the same time, increase the lifetime of their own network, since the game assumes a cost of energy for the transmission, processing and reception of a packet. Also in the context of WSN, Crosby and Pissinou (2007) propose an algorithm for cooperation in wireless sensor networks. The main difference with the game of Felegyhazi et al.’s game is that the cooperation for packet forwarding is established among classes: sensor nodes are divided into classes according to a localization criterion; therefore, the nodes pertaining to the same class perform total cooperation. Crosby and Pissinou’s algorithm is based on a type of strategy known as Grim Strategy. When using this strategy the players act as follows: as long as a player A cooperates the player B will cooperate, but if not, player A will not receive any cooperation until he shows his willingness of cooperating during a certain period of time. Crosby et al. define the Patient Grim Strategy, an adaptation of the Patient Strategy, which enforces cooperation by means of punishment: if a node rejects cooperation for n times, the non-cooperating node will abandon the network forever. The idea of enforcing cooperation, mentioned in the above paragraph, is extensible to WLAN and WWAN networks. Wei and Gitlin (2005) propose

Game Theory for PHY layer and MAC sublayer in Wireless Telecommunications

a novel approach based on scheduling schemes for 3G networks. The schemes they propose encourage cooperation for relaying, analyzing them in a game-theoretic framework. They formulate a repeated game with a punishment mechanism that avoids the pure selfish strategy: if a node does not cooperate for relaying, he will be excluded from the network and he will be admitted again after cooperating a certain period of time.

CROSS-LAYER DESIGN The classical strict layering approach to solve complex problems in engineering has a significant limitation since solving a certain problem in a given layer may produce degradation in adjacent layers, especially when it comes to wireless communications. These networks have a dynamic topology and their links exhibit a time-varying quality due to fading, shadowing and multiuser interference. If the usage of spectrum, energy or other radio resources is at a premium, we could expect a more efficient management of them if some cooperation exists between the upper (namely MAC and NET) and PHY layers. Some significant studies have emerged recently making use of game theory on cross-layer design due to its inherent capability of optimization, and addressing different types of networks by exchanging cross-layer information. Ngo and Krishnamurthy (2007) provide a remarkable example of game-theoretic cross-layer design. In multipacket reception networks the Generalized Multi-Packet Reception (G-MPR) model is widely used, providing explicit incorporation of PHY layer channel state information (CSI) into the reception of MAC packets. Hence, they define a mapping of channel states to transmission probabilities known as transmission policy. The formulation of a non-cooperative game, where the players are the transmitting nodes and the strategies are the transmission policies, provides Nash equilibrium solutions if every node adopts

transmission policies with the SINR threshold reception model.

FUTURE TRENDS We have shown that the use of game theory to solve certain problems in wireless communications, such as power control, has an important background. Even so, some specific types of games such as supermodular and potential games may help solving problems in a more precise way. Moreover, most of the used games are non-cooperative; cooperative games, on the other side, provide optimal solutions in the form of Pareto optima, but at the expense of an exchange of information that may not be easy. Thus, for some problems, it would be worth to evaluate firstly if the game can be turned into a cooperative game in order to obtain the maximum benefit. Another powerful but at the same time complex tool are mechanisms, which were also introduced in section “Spectrum Management and Spectrum Sharing”; although the application of mechanisms for very specific situations is quite recent, we may envisage their use in decision-making problems. Moreover, they may result particularly useful when a set of rules must be fulfilled; in this case, the game framework is does not apply. Cross-layer design approaches based on game theory are now emerging to deal with situations where a highly demanded resource affects the performance of other layers; in these cases, it becomes more effective to establish interlayer connections. Given that game theory is very suitable for global optimization, it appears as a very promising instrument for the future in this area. The environment of wireless communications is getting more and more complex, due to the coexistence of different types of networks and standards. Let us take the example of cellular and ad hoc networks: for the former, a regular pattern and frequency allocation are considered

811

Game Theory for PHY layer and MAC sublayer in Wireless Telecommunications

in the deployment of the base stations infrastructure, and for ad hoc networks there is no planned deployment at all. Moreover, consider diverse WLAN and WWAN standards (IEEE 802.11/802.15/802.11/802.22): for each of them, the spectrum sharing and spectrum access strategies are different. The goal of designing MAC protocols in such unfriendly situation turns out to be a hard task and game theory has shown to be a suitable choice for sharing access in more simple scenarios; so it is time to verify its effectiveness for this challenge. There exists a trend towards distributed organization in wireless systems such as ad hoc networks and Wireless Personal Area Networks (WPAN). Keeping in mind the distribution of tasks among the nodes of these networks, the most reasonable approach is to consider every node as a decision-maker that will interact with other network components. Consequently, game theory can provide the appropriate framework to address new challenges for networks based on distributed topologies, such as cooperative diversity.

CONCLUSION Game theory is an interesting field of mathematics by itself, but becomes even more attractive when applied to engineering problems such as wireless communications. We have exposed how game theory has gained popularity as a solution method for power management, interference avoidance, the backoff attack problem, medium access and cooperative diversity. We have given references to where to find the most relevant work in areas out of the scope of this chapter, such as routing and flow control. We have shown that game theory is a useful tool to tackle wireless communications in which interdependent actions may condition the performance of both the individuals and the whole network.

812

Acknowledgment The authors would like to thank Professor Bousoño-Calzón for his valuable comments and suggestions on Game Theory.

REFERENCES Agustín, A., Muñoz, O., & Vidal, J. (2004). A game theoretic approach for cooperative MIMO schemes with cellular reuse of the relay slot. IEEE International Conference on Acoustics, Speech, and Signal Processing, 4, 581-584. Altman, E., & Altman, Z. (2003). S-modular games and power control in wireless networks. IEEE Transactions on Automatic Control, 48(5), 839-842. Altman, E., & Basar, T. (1998). Multiuser ratebased flow control. IEEE Transactions on Communications, 46(7), 940-949. Altman, E., Borkar, V. S., Kherani, A. A. (2004). Optimal random access in networks with two-way traffic. 15th IEEE International Symposium on Personal, Indoor and Mobile Radio Communications, 1, 609-613. Altman, E., Boulogne, T., El-Azouzi, R., Jiménez, T., & Wynter, L. (2006). A survey on networking games in telecommunications. Elsevier Computers & Operations Research, 33(2), 286-311. Alpcan, T., Basar, T., & Dey, S. (2006). A power control game based on outage probabilities for multicell wireless data networks. IEEE Transactions on Wireless Communications, 5(4), 890-899. Bacci, G., Luise, M., Poor, H. V., & Tulino, A. M. (2007). Energy Efficient Power Control in Impulse Radio UWB Wireless Networks. IEEE Journal of Selected Topics in Signal Processing, 1(3), 508-520.

Game Theory for PHY layer and MAC sublayer in Wireless Telecommunications

Binmore, K. (2007). Playing for Real. A Text on Game Theory. New York, NY: Oxford University Press. Cagalj, M., Ganeriwal, S., Aad, I., & Hubaux, J.-P. (2005). On selfish behaviour in CSMA/CA networks. 24th Annual Joint Conference of the IEEE Computer and Communications Societies, 4, 2513-2524. Crosby, G. V., & Pissinou, N. (2007). Evolution of cooperation in multi-class wireless sensor networks. 32nd IEEE Conference on Local Computer Networks, 1, 489-495. Felegyhazi, M., Hubaux, J.-P., & Buttyan, L. (2005). Cooperative packet forwarding in multidomain sensor networks. 3rd IEEE International Conference on Pervasive Computing and Communications Workshops, 1, 345-349. Hicks, J. E., Mackenzie, A. B., Neel, J. A., & Reed, J. H. (2004). A game theory perspective on interference avoidance. IEEE Global Telecommunications Conference, 1, 257-261. Hwang, K.-S., Tan, S.-W., Hsiao, M.-C., & Wu, C.-S. (2005). Cooperative multiagent congestion control for high-speed networks. IEEE Transactions on Systems, Man and Cybernetics, Part B, 35(2), 255-268. Konorski, J. (2006). A game-theoretic study of CSMA/CA under a backoff attack. IEEE/ACM Transactions on Networking, 14(6), 1167-1178. Lacatus, C., & Popescu, D. C. (2007). Adaptive Interference Avoidance for Dynamic Wireless Systems: A Game Theoretic Approach. IEEE Journal of Selected Topics in Signal Processing, 1(1), 189-202. Liang, C., & Dandekar, K. R. (2007). Power Management in MIMO Ad Hoc Networks: A Game-Theoretic Approach. IEEE Transactions on Wireless Communications, 6(4), 1164-1170.

Lin, H., Chatterjee, M., Das, S. K., & Basu, K. (2005). ARC: an integrated admission and rate control framework for competitive wireless CDMA data networks using noncooperative games. IEEE Transactions on Mobile Computing, 4(3), 243-258. MacKenzie, A., & DaSilva, L. (2006). Game Theory for Wireless Engineers. Ft. Collins, CO: Morgan & Claypool. Mathur, S., Sankaranarayanan, L. & Mandayam, N.B. (2006). Coalitional games in receiver cooperation for spectrum sharing. 40th Annual Conference on Information Sciences and Systems, 1, 949-954). Nash, J. (1951). Non-cooperative Games. Annals of Mathematics, 5(2), 286-295. Ngo, M. H., & Krishnamurthy, V. (2007). Game theoretic cross-layer transmission policies in multipacket reception wireless networks. IEEE Transactions Signal Processing, 55(5-1), 19111926. Nie, N., & Comaniciu, C. (2006). Adaptive channel allocation spectrum etiquette for cognitive radio networks. Springer Mobile Networks and Applications, 11(6), 779-797. Niyato, D., & Hossain, E. (2007). Integration of IEEE 802.11 WLANs with IEEE 802.16-based multihop infrastructure mesh/relay networks: A game-theoretic approach to radio resource management. IEEE Network, 21(3), 6-14. Osborne, M. J. (2004). An introduction to Game Theory. New York, NY: Oxford University Press. Price, J., & Javidi, T. (2007). Leveraging downlink for efficient uplink allocation in a single-hop wireless network. IEEE Transactions on Information Theory, 53(11), 4330-4339.

813

Game Theory for PHY layer and MAC sublayer in Wireless Telecommunications

Saraydar, C. U., Mandayam, N. B., & Goodman, D. J. (2002). Efficient power control via pricing in wireless data networks. IEEE Transactions on Communications, 50(2), 291-303. Scutari, G., Barbarossa, S., & Ludovici, D. (2003). Cooperation diversity in multihop wireless networks using opportunistic driven multiple access. 4th IEEE Workshop on Signal Processing Advances in Wireless Communications, 1, 170-174. Scutari, G., Barbarossa, S., & Palomar, D. P. (2006). Potential games: a framework for vector power control problems with coupled constraints. IEEE International Conference on Acoustics, Speech and Signal Processing, 4, 241-244. Sendonaris, A., Erkip, E., & Aazhang, B. (2003). User cooperation diversity. Part I. System description. IEEE Transactions on Communications, 5(11), 1927-1938. Shastry, N., & Adve, R.S. (2006). Stimulating cooperative diversity in wireless ad hoc networks through pricing. IEEE International Conference on Communications, 8, 3747-3752). Srinivasan, V., Nuggehalli, P., Chiasserini, C.-F., & Rao, R. R. (2005). An analytical approach to the study of cooperation in wireless ad hoc networks. IEEE Transactions on Wireless Communications, 4 (2), 722-733. St. Jean, C. A., & Jabbari, B. (2004). Bayesian game-theoretic modeling of transmit power determination in a self-organizing CDMA wireless network. IEEE 60th Vehicular Technology Conference, 5, 3496-3500. Tang, A., Lee, J.-W., Huang, J., Chiang, M., & Calderbank, A. R. (2006). Reverse engineering MAC. 4th International Symposium on Modeling and Optimization in Mobile, Ad Hoc and Wireless Networks, 1, 1-11. Wang, B., Han, Z., & Liu, K. J. R.(2006). Stackelberg game for distributed resource allocation over multiuser cooperative communications

814

networks. IEEE Global Communications Conference,1, 1-5. Wang, K.-C., & Ramanathan, P. (2005). QoS assurances through class selection and proportional differentiation in wireless networks. IEEE Journal on Selected Areas in Communications, 23(3), 573-584. Wei, H.-Y., & Gitlin, R. D. (2005). Incentive scheduling for cooperative relay in WWAN/ WLAN two-hop-relay network. IEEE Wireless Communications and Networking Conference, 3, 1696-1701. Wendorf, R. G., & Seidenberg, H. B. (2007). Channel-change games for highly interfering spectrum-agile wireless networks. 2nd International Symposium on Wireless Pervasive Computing, 1, 611-616. Wu, B., & Wang, Q. (1995). A multiobjective analytic framework for slotted ALOHA wireless LANs. 4th IEEE International Conference on Universal Personal Communications, 1, 12-16.

Key terms Game: Mathematical formulation where the players choose strategies that will maximize their benefit. Game Theory: It is a branch of applied mathematics that is often used in the context of economics, which studies interactions between decision-makers. Wireless: It refers to any type of electrical or electronic operation that is accomplished without the use of a wired connection. Player: A decision-maker in the game-theoretic framework. Strategy: The different choices or alternative actions a player has to play a game. Utility Function: Function associated to each player of a game which maps the resource

Game Theory for PHY layer and MAC sublayer in Wireless Telecommunications

consumption into a number so as to represent player’s preference.

815