Game Theory in Cooperative Communications

Transcription

1 Game Theory in Cooperative Communications Dejun Yang, Xi Fang, an Guoliang Xue Abstract Cooperative communication has great potential to improve the wireless channel capacity by exploiting the antennas on wireless evices for spatial iversity. However, applications of cooperative communication are barely seen in reality. A main obstacle blocking its wie applications is the lack of incentives for wireless noes to participate in cooperative communication. We first survey the existing game theoretic solutions for proviing cooperation incentives in cooperative communications. We then iscuss the challenges in applying game theory to cooperative communications. Keywors: I. INTRODUCTION Cooperative communication has been propose to improve the channel capacity in wireless networks. It takes avantage of the broacast nature of wireless transmission an utilizes the antennas on wireless noes to achieve spacial iversity. Depening on how the relay noe processes the overhear signal, two primary cooperative communication moes are wiely use: Amplify-an-Forwar (AF) an Decoe-an-Forwar (DF). In the AF moe, the relay noe amplifies the signal before forwaring it to the estination noe. In the DF moe, the relay noe ecoes an encoes the signal before forwaring it to the estination noe. Cooperative communication has potential applications in many ifferent networks, incluing cellular networks, a-hoc networks, an cognitive networks, as shown in Fig. 1. To achieve cooperative communication, cooperation from other noes is require. In network applications for military actions an isaster relief, cooperation among noes can be assume since the noes belong to a single authority an thus voluntarily cooperate to achieve a common goal. However, in commercial applications, where noes usually belong to ifferent inepenent entities, there is no goo reason to assume that the noes will cooperate. In fact, noes are selfish an consume their resources only when oing so can maximize their own benefits. Game theory All authors are affiliate with Arizona State University, Tempe, AZ {ejun.yang, xi.fang, xue}@asu.eu. This research was supporte in part by NSF grants an The information reporte here oes not reflect the position or the policy of the feeral government.

2 2 is naturally the most appropriate tool to moel, analyze an solve the problems in cooperative communications. Source Relay Seconary Network Relay Relay Relay Destination Relay Source Destination Source Destination Source Source Source Destination Seconary Network Relay Primary Network (a) Cellular network (b) A-hoc Network (c) Cognitive Raio Network Fig. 1. Networks where cooperation communication can be applie Game theory is the stuy that analyzes the strategic interactions among autonomous ecision makers, whose actions have mutual, probably conflicting, consequences. Originally evelope to moel problems in the fiel of economics, game theory has recently been applie to network problems, in most cases to solve the resource allocation problems in a competitive environment. The reason that game theory is an appropriate choice for stuying cooperative communications is multifol. First, noes in the network are autonomous agents, making ecisions only for their own interests. Game theory provies us sufficient theoretical tools to analyze the network users behaviors an actions. Secon, game theory primarily eals with istribute optimization, which often requires local information only. Thus it enables us to esign istribute algorithms. Finally, auction, a market game of incomplete information, allows us to esign mechanisms where relay noes can sell their resources to the source noes for cooperative communications. Such approach is esirable when neither the source noe nor the relay noe knows each other s private valuation on the resources to trae. In the following sections, we first introuce the basic concepts of cooperative communication an game theory. We then survey the existing game theoretic solutions to cooperative communication problems, with the focus on cooperation incentive provisioning. Finally, we iscuss the research challenges in applying game theory to the problems in cooperative communications. II. COOPERATIVE COMMUNICATION Depening on the availability of noes an the cooperative communication protocols, there are three ifferent communication topologies: one-to-one, one-to-many, an many-to-one, as shown

3 3 in Fig. 2. Take the simplest topology, one-to-one, as an example to illustrate the basic iea of cooperative communication. In this example, s is the source noe that transmits information, is the estination noe that receives information, an r is the relay noe that relays information to enhance the communication between the source an the estination. Let P s an P r enote the transmission power of s an r, respectively. Let W enote the banwith of the transmission channel. Assume that transmission procees in a frame-by-frame fashion, as shown in Fig. 3. Each frame is ivie into two phases. In the first phase, the source noe s transmits its ata to the estination noe. Due to the broacast nature of wireless transmission, the relay noe r can overhear the ata. In the secon phase, the relay noe r forwars the ata to the estination after processing, epening on the unerlying cooperative communication moe. r 1 r n s 1 s r s s 2 r r 2 r i s n (a) One-to-one (b) One-to-many (c) Many-to-one Fig. 2. Three cooperative communication topologies s r r r slot 1 slot 2 slot 3 slot 4 frame 1 frame 2 s r Fig. 3. Illustration of cooperative communication We next compute the achievable capacity uner cooperative communications [1]. When noe u transmits a signal to noe v with power P u, the signal-to-noise ratio (SNR) at noe v, enote by SNR uv, is SNR uv = P u N 0 u, v α, where N 0 is the abient noise, u, v is the Eucliean istance between noes u an v, an α is the path loss exponent which is between 2 an 4 in general, epening on the characteristics of the communication meium.

4 4 Amplify-an-Forwar (AF): In the amplify-an-forwar moe, the relay noe amplifies the signal transmitte by the source noe in the first time slot an then transmits the amplifie signal to the estination noe in the secon time slot. The achievable capacity from s to is C AF (s, r, P r, ) = W ( 2 log SNR s + SNR ) sr SNR r. SNR sr + SNR r + 1 Decoe-an-Forwar (DF): In the ecoe-an-forwar moe, the relay noe ecoes an estimates the signal transmitte by the source noe in the first time slot an then transmits the ata to the estination noe in the secon time slot. The achievable capacity from s to is C DF (s, r, P r, ) = W 2 min {log 2(1 + SNR sr ), log 2 (1 + SNR s + SNR r )}. III. GAME THEORY BASICS IN A NUTSHELL Game theory [2] is a iscipline aime at moeling scenarios where iniviual ecision-makers have to choose specific actions that have mutual or possibly conflict consequences. A game consists of three major components: players: The ecision makers are calle players, enote by a finite set N = {1, 2,..., n}. strategy: Each player i N has a non-empty strategy set S i. Let s i enote the selecte strategy by player i. A strategy profile s consists of all players strategies, i.e., s = (s 1, s 2,..., s n ). Obviously, we have s S = i N S i, where is the Cartesian prouct. utility/payoff: The utility of player i is a measurement function, enote by u i : S R, on the possible outcome etermine by the strategies of all players, where R is the set of real numbers. The mapping from the components in a game to the elements in cooperative communications is shown in Table I. The players of the game are assume to be rational an selfish, which means each player is only intereste in maximizing its own utility without respecting others an the system s performance. Let s i enote the strategy profile excluing s i. As a notational convention, we have s = (s i, s i ). We say that player i prefers s i to s i if u i (s i, s i ) > u i (s i, s i ). When other players strategies are fixe, player i can select a strategy, enote by b i (s i ), which maximizes its utility function. Such a strategy is calle a best response of player i. A strategy is calle a ominant strategy of player i if, regarless of what other players o, the strategy earns player i a larger utility

5 5 TABLE I COMPONENTS OF GAMES IN COOPERATIVE COMMUNICATIONS Components in the Game Elements in Cooperative Communications Players Source noes an/or relay noes Strategy Power control [3 7] Spectrum allocation [8] Relay noe(s) or source noe(s) selection [9] To cooperate or not [10] Price [3, 9, 11, 12] Utility/Payoff Data rate [6, 10] Profit, e.g., revenue minus cost [3 5, 7 9, 11] than any other strategy. In orer to stuy the interactions among players, the concept of Nash Equilibrium (NE) is introuce. A strategy profile constitutes an NE if none of the players can improve its utility by unilaterally eviating from its current strategy. To characterize an quantify the inefficiency of the system performance ue to the lack of cooperation among the players, we use the concept of price of anarchy (POA). The POA of the game is the ratio of the system performance in the worst NE to the system performance in the social optimal solution. s 1 Player 2 Player 1 Cooperate Not Cooperate Cooperate 4, 4 2, 5 s 2 Not Cooperate 5, 2 3, 3 Fig. 4. Two-player cooperative communication To illustrate the basic concepts of game theory, we use a simple two-player example. This game is essentially equivalent to the well-stuie game, Prisoners Dilemma. There are two players in this game, player 1 an player 2. Each player can choose from two strategies, Cooperate (C) an Not Cooperate (NC). If player 1 takes strategy C, it will act as a relay for player 2 for cooperative communication. Otherwise, player 1 only transmits its own ata to the estination. Therefore there are totally four ifferent strategy profiles. The utilities of ifferent profiles are shown in the table in Fig. 4. It is straightforwar to see that (NC, NC) is an NE with social

6 6 performance 6. However, the social optima is the strategy profile (C, C), which gives the social performance 8. Thus the POA of this game is 3. 4 Games can be classifie into two categories, strategic form game (or static game) an extensive form game (or ynamic game). The strategic form game is a one-shot game. In this game, the players make their ecisions simultaneously without knowing what others will o. On the contrary, the extensive form game represents the structure of interactions between players an efines possible orers of moves. The repeat game is a class of the extensive form game, in which each stage is a repetition of the same strategic game. At the beginning of each stage, players observe the past history of strategies before making ecisions. The number of stages may be finite or infinite. The utility of each player is the accumulate utility through all the stages. Therefore, players care not only the current utility but also the future utilities. The Stackelberg game is an extensive form game, which is use to moel the competition between one player, calle the leaer, an a set of players, calle the followers. In this game, the leaer takes action first an then the followers take actions. The leaer knows ex ante that the followers observe its action an take actions accoringly. The NE in the Stackelberg game is calle Stackelberg Equilibrium. As game theory stuies interactions between rational an intelligent players, it can be applie to the economic worl where people interact with each other in the market. The marriage of game theory an economic moels yiels interesting games an fruitful theoretical results in microeconomics an auction theory. Auction is a ecentralize market mechanism for allocating resources. The essence of auction is a game of incomplete information, where the players are the biers, the strategies are the bis, an both allocations an payments are functions of the bis. In an auction mechanism, each bier i has some private information t i, calle its type, an its strategy is the bi b i. A mechanism then computes an output o = o(b 1, b 2,..., b n ) an a payment vector p = (p 1, p 2,..., p n ), where p i = p i (b 1, b 2,..., b n ) is the money given to the participating agent i. For each possible output o, bier i s valuation is v i (t i, o). The utility of bier i is u i (t i, o) = v i (t i, o) + p i. Base on the number of objects auctione on the market, auctions can be categorize into single-object auction an multi-object auction. Two basic single-object auction schemes are the first-price auction an the secon-price auction. In the first-price auction, the auctioneer grants the item to the highest bier an charges the highest bi. In the secon-price auction, also

7 7 known as Vickrey auction, the auctioneer grants the item to the highest bier, but charges the secon highest bi. Multi-object auction can be homogeneous auction or heterogeneous auction, epening on whether the objects are ientical. There are three esirable economic properties while esigning an auction scheme: Truthfulness: An auction is truthful if revealing true private valuation is the ominant strategy for each bier. In other wors, no bier can improve its utility by submitting a bi ifferent from its true valuation, no matter how others submit. Iniviual Rationality: each agent participating in the auction can expect a non-negative profit. System Efficiency: An auction is system-efficient if the sum of valuations of all biers is maximize. IV. COOPERATION INCENTIVES Cooperative communication has been propose for years [1] an is gaining popularity since it has great potential to increase the capacity of wireless networks. Nevertheless, its applications are rarely seen in reality. A main obstacle blocking its wie applications is the lack of incentives for the wireless noes to serve as relay noes. There are three primary mechanisms esigne to provie such incentives: reputation-base mechanism, resource-exchange-base mechanism, an pricing-base mechanism. We will investigate these mechanisms in the following subsections. A. Reputation-base Mechanism In this mechanism, a centralize authority, e.g. base station, keeps recors of the cooperative behavior an punish non-cooperating noes. Consier a simple scenario, as shown in Fig. 4, where two noes wish to transmit ata to a common estination. Each noe is a player an its strategy is whether to cooperate with the other noe. Base on the utility table in Fig. 4, if player 1 chooses to cooperate, then player 2 will choose not to cooperate since player 2 s utility is improve from 4 to 5. If player 1 chooses not to cooperate, player 2 will also choose not to cooperate since player 2 s utility is improve from 2 to 3. Thus NC is player 2 s ominant strategy. Similarly, NC is also player 1 s ominant strategy. Therefore (NC, NC) is an NE of the static game. However, the (NC, NC) strategy profile is unesirable from the system perspective,

8 8 as it oes not efficiently utilize the system resource. Intuitively, even if a player is willing to cooperate, the other user s utility rives it to not to cooperate but rather to free-rie. In fact, a selfish player will always take avantage of a cooperating player an free-rie to maximize its own utility. In aition, such free-riing behavior has no consequence, e.g. punishment, in the static game. For these reasons, a repeate game was moele in [10]. In this game, free-riers from the previous stage will be punishe an force to reuce their transmission power, while players taking avantage of cooperative communications will be aware to transmit with higher power. Since players nee to care about their future utility, an NE in which players mutually cooperate can be achieve. B. Resource-exchange-base Mechanism In this mechanism, the source noe takes other noes as relays for cooperative communication. In return, the source noe provies its own resource to help the relay noes achieve certain objectives. For the same network in Fig. 4, a ifferent game was moele in [4]. The strategy of each player is to etermine the power for transmitting its own ata an the power for relaying the other player s ata. The utility of each player is efine as the ifference between the achieve ata rate an the energy cost. This game was moele as a Stackelberg game by taking one of the two players as the leaer an the other as the follower. It was shown that there are more benefits when cooperation is one between noe pairs who are closer to each other. In cognitive raio networks as shown in Fig. 1, the primary user (PU) can involve seconary users (SUs) as the cooperative relays. In return, the SUs obtain the opportunity to access the wireless channel for their own transmissions. In [6], the authors formulate the problem as a Stackelberg game, where the PU is the leaer an the SUs are the followers. The strategy of the PU is to ecie the portion of time it allocates to SUs an select a set of SUs as relays, base on the cooperative transmission power from the SUs. The utility of the PU is the achieve ata rate with SUs help. The strategy of the SU is the cooperative transmission power eicate to the PU, since the channel access time of each SU is proportional to the contribution it makes in the cooperative communication. The utility of each SU is efine as the ifference between its achieve ata rate an the energy cost for helping with PU s transmission. The authors prove

9 9 the existence of a Stackelberg Equilibrium an obtaine the corresponing strategies. C. Pricing-base Mechanism In this mechanism, virtual currency or tokens are assume in the network. Relay noes sell their resources, e.g., power, banwith an time, for a certain price. Source noes make payment to relay noes for using their resources. Depening on the relationship between eman an supply, the game can be formulate as a buyer s market or a seller s market. When there is one source noe an multiple relay noes, the game is formulate as a buyer s market [7]. The source noe selects a subset of relay noes for relaying base on the channel conition between itself an each relay noe an the price aske by the relay noe. Each relay noe etermines its price accoring to the conitions of the channel between itself an source noe, an the channel between itself an the estination noe, as well as the other relay noes prices. Since the game is a buyer s market, it is essentially a Stackelberg game with the source noe as the leaer an the relay noes as the followers. When there is one relay noe an multiple source noes, the market becomes a seller s market [8]. The authors assume that only the source noes are players an the relay noe has a fixe cost function, which is known to all the source noes. The strategy of each source noe is the banwith it wants to buy. A istribute algorithm was evelope to search the NE. In [3], the source noes are biers an submit bis to the relay noe. The relay noe allocates its transmission power proportional to the source noes bis. Two ifferent payments were efine, of which one is a function of the extra SNR ue to the cooperative communication an the other is a function of the power allocate to the source noe. It was prove that the NE exists an is unique. The istribute best response bi upates converge globally to the unique NE in a completely asynchronous manner. The above works only consier the selfish behavior of players, but not the cheating behavior. It has been shown both theoretically an practically that a market coul be vulnerable to market manipulation an prouce very poor outcomes if players are ishonest on their prices. Therefore truthfulness is the most critical property of the mechanism esign. In [11], the authors esigne an auction scheme for cooperative communications, which satisfies not only truthfulness, but also iniviual rationality an buget balance properties. In this auction, source noes are buyers, relay noes are sellers, an the base station is the auctioneer. Buyers bi for relaying service for

10 10 cooperative communications, while sellers offer cooperative service at the cost of their resources, e.g. energy, an receive monetary payment in return. Each buyer has ifferent valuations of the relay noes as it can achieve ifferent capacities by cooperating with ifferent relay noes. V. CHALLENGES IN APPLYING GAME THEORY While game theory has been extensively applie to moel the problems in cooperative communications, there are still many challenging research issues unsolve. We list some of them below to inspire intereste reaers on future research irections. Selection of the utility function: utility function is unoubtely a very important component in the game. It shoul precisely reflect the true valuation of the player on the outcome of the game. In the games moeling cooperative communications, ata rate an profit are wiely use as utility functions. In the esign of pricing-base mechanism, most works choose the transmission power as the cost of cooperation, which appears in the utility function as a linear term. However, in reality, the cost of transmission power may epen on the specific evice an the remaining power level, an thus is probably not linear in the transmission power. Existence an uniqueness of NE: The existence of NEs in a game is always one of the properties investigate by the researchers. The reason is that an NE is a solution concept that escribes a steay state conition of the game. If the existence of NE is not guarantee, it is possible that players oscillate their strategies to improve their utilities, generating a significant amount of communication overhea an wasting computing resources. Besies the existence, the uniqueness of NE is another esirable property, which has been largely neglecte by the existing works. If there is only one NE, players will not be confuse while selecting their NE strategies. In aition, we can preict the NE of the game an the resulting performance. Computation of NE: Once the existence of NE is prove, the next question woul be how to compute an NE. Computationally heavy algorithms for computing an NE are not esirable in networks, like cellular networks an mobile networks, where evices are powere by batteries. In such networks, computing power is a valuable resource. Therefore esigning efficient algorithms for NE computation is necessary.

11 11 Efficiency of NE: It is known that NE is usually an inefficient solution from the system s perspective. This inefficiency is capture by the concept of price of anarchy. Pricing has been aopte for steering players to converge to an equilibrium with better system performance. For example, in [9], the authors esigne a payment scheme which inuces the source noes to select the relay noes resulting in an optimal relay noe assignment. However, the esigne payment scheme is base on the conition that the optimal solution can be obtaine. This conition oes not hol in general. Therefore esigning a pricing-base mechanism to influence players to converge to an efficient NE is still a challenging task for cooperative communications. Mechanism esign: Most of the mechanism esigns in the literature only consier singlesie auction, where source noes are buyers. The only work that stuies the ouble auction for cooperative communications is presente in [11]. The authors showe that it is esirable to esign an auction satisfying truthfulness, iniviual rationality, buget balance, an system efficiency properties. Unfortunately, the impossibility theorem [13] shows that no ouble auction can simultaneously achieve all four economic properties. Thus they esigne an auction scheme, which satisfies the first three properties while ignoring the last. It is still open an very challenging to esign a ouble auction scheme, which satisfies the first three properties while approximately maximizing the system efficiency. VI. CONCLUSION In this article, we have briefly surveye the game theoretic solutions to the problems in cooperative communications, with the focus on esigning cooperation incentive mechanisms. While game theory has been extensively applie to cooperative communications, there are still many challenges that eman extra effort from researchers. REFERENCES [1] J. Laneman, D. Tse, an G. Wornell, Cooperative iversity in wireless networks: Efficient protocols an outage behavior, IEEE Trans. Inf. Theory, vol. 50, pp , [2] D. Fuenberg an J. Tirole, Game theory. MIT Press, [3] J. Huang, Z. Han, M. Chiang, an H. Poor, Auction-base resource allocation for cooperative communications, IEEE JSAC, vol. 26, pp , [4] M. Janzamin, M. Pakravan, an H. Seghi, A game-theoretic approach for power allocation in biirectional cooperative communication, in Proc. IEEE WCNC 10, pp. 1 6.

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