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Chun-Han Ko, and Hung-Yu Wei, Member, IEEE ...... pp. 796–801. [3] X. Xiao, L. Yang, W. Wang, and S. Zhang, “A broadcasting retransmission approach based ...
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[8] C. Snow, L. Lampe, and R. Schober, “Impact of WiMAX interference on MB-OFDM UWB systems: Analysis and mitigation,” IEEE Trans. Commun., vol. 57, no. 9, pp. 2818–2827, Sep. 2009. [9] W. Coo, A. Nollonothon, and C. Choi, “Exact bit error rate analysis of direct sequence ultra-wide band multiple access systems in lognormal multipath fading channels,” IET Commun., vol. 2, no. 3, pp. 410–421, Mar. 2008. [10] A. Erdlyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions. New York: McGraw-Hill, 1953.

Incentive Compatible Configuration for Wireless Multicast: A Game Theoretic Approach Fu-Yun Tsuo, Student Member, IEEE, Jen-Ping Huang, Chun-Han Ko, and Hung-Yu Wei, Member, IEEE

Abstract—Multicast and broadcast service (MBS) is a service offered by the base station (BS) to multiple receivers requesting the same information. Such a BS must be kept updated with the receivers’ feedback information (e.g., packet loss rates) to configure the MBS. We propose MBS operation schemes that are dominant-strategy incentive compatible in game theory, i.e., the schemes induce dominant-strategy equilibria, where all selfish receivers reveal their true information. Moreover, the induced equilibria are Pareto efficient and max-min fair. To conclude, the proposed schemes can elicit true feedback information from the receivers, avoiding any manipulation and, thereby, ensuring an efficient and fair system operation. Index Terms—Broadcast, feedback signaling, game theory, mechanism design, multicast.

I. I NTRODUCTION Designing an efficient wireless multicast and broadcast mechanism is the core for information delivery in next-generation wireless systems. An efficient multicast and broadcast service (MBS) often uses forward error correction, whereby redundant packets are added to the transmission to allow the receivers to correct errors without requesting retransmission [1]. In this paper, we consider a wireless broadcast system where the base station (BS) adopts the technique of network coding. As the channel conditions are variable, the receivers incur different levels of packet loss and thus request different numbers of redundant packet (redundancy level). However, the BS cannot satisfy all the receivers’ requests at the same time since only a single redundancy level is allowed in MBS transmission. Under such conditions, designing appropriate MBS operation schemes to determine the number of redundant packets is critical. Typically, the channel state information plays a crucial role in designing MBS operation schemes [2]. The BS requires all receivers to report their information (e.g., packet loss rate, PLR) since local Manuscript received August 2, 2010; revised February 5, 2011 and May 19, 2011; accepted June 8, 2011. Date of publication June 20, 2011; date of current version September 19, 2011. This work was supported in part by the National Science Council, by National Taiwan University, and by Intel Corporation under Grant NSC99-2911-I-002-001, Grant 99R70600, Grant 10R70500, and Grant AE00-00-04. The review of this paper was coordinated by Dr. H. Jiang. F.-Y. Tsuo, J.-P. Huang, and C.-H. Ko are with the Department of Electrical Engineering, National Taiwan University, Taipei 106, Taiwan. H.-Y. Wei is with the Graduate Institute of Communication Engineering and the Department of Electrical Engineering, National Taiwan University, Taipei 106, Taiwan (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TVT.2011.2160104

information is only available to the receiver themselves. With correct information about receivers, the BS can easily make beneficial decisions for the whole system when determining the number of redundant packets. Most of the existing works are based on the assumption that the BS either can collect immediate feedback or already acquires such information [1], [3]. However, the receivers’ divergent selfinterests are not considered in these works. A receiver may untruthfully report to manipulate the BS’s MBS configuration into the receiver’s preferable number of redundant packet. In response to the receivers’ self-interested tendency, we formulate the untruthful feedback problem into game theoretical models since game theory is suitable to model selfish user behaviors. The main contributions of this work are to consider the untruthful feedback problem of MBS system and to propose MBS operation schemes that not only solve the problem but achieve the desired properties as well. At first, we show a naive MBS operation scheme that leads to an undesired consequence of receivers untruthfully reporting. Then, we propose MBS operation schemes that achieve desirable operational properties, including incentive compatibility (i.e., prompt receivers to truthfully report), Pareto efficiency (i.e., no other schemes can improve all receivers’ utility simultaneously), and max-min fairness (i.e., prioritize the demand of bad-state receivers). II. R ELATED W ORKS Liu et al. applied the Nash Bargaining Solution to select the operating point in wireless video multicast [4]. Krishnaswamy and van der Schaar modeled the video transmission for 802.11a networks, where the BS can choose different modulation and coding schemes as the strategy to maximize the effective physical layer throughput [5]. Jorswieck and Mochaourab applied the d’Aspremont and Gérard-Varet (AGV) mechanism to prevent players from manipulating the Nash equilibrium [6]. Rama Suri et al. used the Vickrey–Clarke–Groves (VCG) mechanism to prompt selfish nodes to forward packets and to voluntarily reveal their private information [7]. Chen et al. derived a unique proportionally fair Nash equilibrium in the rate allocation game for wireless multimedia MBS systems and proposed a distributed cheat-proof convergence algorithm [8]. Lin et al. designed incentivebased strategies to stimulate cooperation among users for peer-to-peer live-streaming social networks [9]. Different from these works, we investigate truthful mechanisms for selection in wireless multicast operation. In [4] and [5], the authors did not focus on the truthful mechanism design. In addition, we avoid using the VCG pricing adopted in [6] and [7] due to its impractical assumption that the number and the utility function of receivers must be known to the BS. Compared to [8] and [9], our schemes directly achieve a truth-telling equilibrium, excluding the need of an algorithm or a strategy design. In addition, we refer to [10] to provide a guideline to select the optimal scheme from the series of schemes we propose. III. S YSTEM M ODEL We consider an MBS system where a BS provides MBS to n receivers requesting identical M information packets (Fig. 1). The receiver derives its (packet) loss rate, which is denoted by θi , based on the past experience of channel measurement. The loss rate remains constant and private in this game. The BS adopts random linear network coding to encode these M information packets into λ encoded packets (in general, λ ≥ M ), which can increase the efficiency of broadcast. Accordingly, every received encoded packet contributes one information packet until M information packets are received [11].

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Fig. 1. In a wireless MBS system, the receivers report their PLRs to the BS, but the reports might not conform with the truth. With the proposed MBS operation schemes, the BS can collect the true information to determine the number of encoded packets λ.

We introduce two different kinds of MBS system, i.e., loss-tolerant MBS and loss-sensitive MBS. The receivers can tolerate some packet loss in the former one, whereas the receivers are sensitive to packet loss in the latter one. In both kinds of MBS system, the receivers benefit from receiving information packets and bear the receiving cost c per encoded packet. The cost is a linear function since the packets are equally important when encoded by random linear network coding.1 Assume c < 1 − max{θ1 , θ2 , . . . , θn } to ensure positive utility of receivers. The utility functions of loss-tolerant and loss-sensitive receivers are defined here. 1) Loss-tolerant data receiver: uLT i (λ, θi ) = min {(1 − θi )λ, M } − cλ where (1 − θi )λ is receiver i’s expected number of received encoded packets if the probability of packet loss follows the Bernoulli distribution [1]. The utility function increases along with the growing number of received encoded packets, which stops once M information packets are received. The losstolerant data receivers refer to dividable Internet applications, such as online TV series or photos. 2) Loss-sensitive data receiver: uLS i (λ, θi ) = I(λ, θi , M )M − cλ where the indicator function I(λ, θi , M ) = 1 if (1 − θi )λ ≥ M or I(λ, θi , M ) = 0 otherwise. The first term of utility is either M or zero, depending on whether the number of received encoded packets exceeds M or not. The loss-sensitive data receivers refer to the most critical data-downloading application (e.g., software update), which needs M received encoded packets to decode the whole data file.

IV. M ULTICAST AND B ROADCAST S ERVICE O PERATION S CHEMES In an MBS system with feedback, the BS has to collect the loss rates from the receivers to determine the number of encoded packets. The receivers, at the other end, report the information to the BS. Denote the reported loss rates by p = (p1 , p2 , . . . , pn ). Here, we use λ(p), instead of λ, as it is a function of p, i.e., the configuration is based on the collected information. However, the collected information may not be true since the receivers may falsely report in the hopes of increasing their utility by manipulating the MBS operation. To avoid such manipulation, we propose a series of MBS operation schemes, i.e., the kth selection scheme (KSS) and max-min fairness scheme (MFSS), as follows: 1) KSS: λ(p) = M/1 − s(k, p), where s(k, p) represents the k th lowest element in p. The physical meaning and the guideline to choose k are provided in the Appendix. The KSS scheme is incentive compatible in loss-tolerant MBS systems. 2) MMFS: λ(p) = M/1 − s(n, p). When choosing k = n in the KSS, we obtain a special scenario called the MMFS scheme. The MMFS is the only scheme that is incentive compatible in losssensitive MBS systems. Moreover, the MMFS scheme induces the max-min fair equilibrium.2 The proposed schemes are incentive compatible, whereas other schemes may not be the case. An intuitive scheme λ(p) = M/1 − p¯, where p¯ is the arithmetic mean of p, is incapable of motivating the receivers to truthfully report (the proof method is described in Section VI), which we call the mean scheme (MS). The comparison between the MS and the proposed schemes is shown in the simulation section. The notations are summarized in Table I. V. G AME M ODELS AND S OLUTION C ONCEPTS

1 To generalize the usage of our model, we address the issue of cost with the following explanations: First, the more packets the BS broadcasts, the longer the receiver waits. In other words, the cost is directly proportional to the number of encoded packets. Second, even when the receiver stops receiving, maintaining the basic operation still consumes energy, which incurs an incremental energy cost. Third, the receivers are also charged for requesting MBS from the BS, according to the number of encoded packets. With any of the preceding explanations, we can justify that the cost is a linear function of the number of encoded packets.

Since there exists competition for common interest among receivers, we apply game theory, which is a mathematical tool for modeling self2 Note that the number of broadcast packet λ may not be an integer. However, the BS can still exactly broadcast λ without rounding. Define the function x as the greatest integer of x. If the BS broadcasts λ encoded packets with probability 1 and an extra encoded packet with probability λ − λ, then the expected number of broadcast packets is λ.

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∗ LT of other players, uLT i (λ(pi , p−i ), θi ) ≥ ui (λ(pi , p−i ), θi ), ∀pi = p∗i , and ∀p−i ∈ P−i . Definition 2 (DSE): A strategy profile p∗ = (p∗1 , p∗2 , . . . , p∗n ) is a DSE if p∗i is a dominant strategy for all i ∈ N. For each player, the dominant strategy always leads to higher utility than other available strategies, regardless of the strategy that other receivers choose. If every player plays the dominant strategy, the set of dominant strategy constitutes a DSE. Moreover, if truthfully reporting is a dominant strategy for all receivers, the game reaches to a truthtelling DSE. In this case, the scheme prompting truth-telling DSE is dominant-strategy incentive-compatible (DSIC). Definition 3 (Dominant-Strategy Incentive Compatibility): A scheme is DSIC if the game induced by the scheme has a DSE p∗ = (p∗1 , p∗2 , . . . , p∗n ), where p∗i = θi ∀i ∈ N.

TABLE I NOTATIONS IN THE LOSS-TOLERANT AND LOSS-SENSITIVE DATA MBS SYSTEMS

VI. G AME -T HEORETIC A NALYSIS A. Dominant-Strategy Incentive Compatibility

ish decision makers’ interactions, to show that the proposed schemes enable receivers to truthfully report. In the following section, we construct MBS game models induced by the proposed schemes and introduce the solution concepts to such games. A. MBS Games For the loss-tolerant data MBS system under the KSS, we construct a loss-tolerant data MBS game GLT = N, P, θ, ULT  consisting of four components. 1) player set N = {1, 2, . . . , n} consisting of n loss-tolerant data receivers;  2) strategy space P = i∈N Pi , where Pi = {pi : 0 ≤ pi ≤ 1} is receiver i’s strategy set and pi is receiver i’s reported PLR; 3) set of true PLRs θ = {θi }i∈N , where θi ∈ [0, 1] is receiver i’s true PLR, which is its private information; 4) utility function set ULT = {uLT i (λ(p), θi )}i∈N , where receiver i’s utility uLT i (λ(p), θi ) under the KSS is



uLT i

(λ(p), θi ) = min

M (1 − θi ) ,M 1 − s(k, p)





cM . 1 − s(k, p)

In this section, we prove that both the MBS games GLT and GLS have a unique DSE, where the receivers truthfully report their PLRs. Therefore, the KSS is DSIC in the loss-tolerant data MBS system, and the MMFS is DSIC in the loss-sensitive data MBS system. Theorem 1: The KSS is DSIC in loss-tolerant data MBS systems. Proof: First, we show that the strategy pi = θi uniquely maximizes the utility, provided that any other receivers’ strategy profile is p−i . Therefore, the strategy pi = θi is a unique dominant strategy for every receiver. The proof is developed into three possible cases. Case 1 s(k, (θi , p−i )) = θi .If pi = θi , receiver i’s utility is uLT i (λ(θi , p−i ), θi ) = M −

If pi < θi , we have s(k, p) < s(k, (θi , p−i )), and therefore uLT i (λ(p), θi ) =

M (1 − θi ) − cM < uLT i (λ(θi , p−i ), θi ) . 1 − s(k, p)

If pi > θi , we have s(k, p) > s(k, (θi , p−i )), and therefore uLT i (λ(p), θi ) = M −

cM < uLT i (λ(θi , p−i ), θi ) . 1 − s(k, p)

Case 2 s(k, (θi , p−i )) < θi .If pi = θi , receiver i’s utility is uLT i (λ(θi , p−i ), θi ) =

Similarly, for the loss-sensitive data MBS system under the MMFS, we construct a loss-sensitive data MBS game GLS = N, P, θ, ULS . The game setting is the same as the loss-tolerant data MBS game GLT , except that the utility function set is ULS = {uLS i (λ(p), θi )}i∈N , where loss-sensitive data receiver i’s utility uLS i (λ(p), θi ) under the MMFS is



uLS i (λ(p), θi ) =

cM M − 1−s(n,p) , if θi ≤ s(n, p) cM − 1−s(n,p) , otherwise.

cM . 1 − s (k, (θi , p−i ))

(1)

B. DSE and Incentive Compatibility We adopt a robust solution concept known as the dominant-strategy equilibrium (DSE). Let p−i = (p1 , . . . , pi−1 , pi+1 , . . . , pn ) denote the strategy of every player, except for player i. (To simply state, we use the notations for the loss-tolerant data MBS games in all the succeeding definitions.) Definition 1 (Dominant Strategy): Player i’s strategy p∗i is a dominant strategy if it maximizes player i’s utility for all possible strategies

M (1 − θi ) − cM . 1 − s (k, (θi , p−i ))

If pi = θi , we have s(k, p) ≤ s(k, (θi , p−i )), and therefore uLT i (λ(p), θi ) =

M (1 − θi ) − cM ≤ uLT i (λ(θi , p−i ), θi ) . 1 − s(k, p)

Case 3 s(k, (θi , p−i )) > θi .If pi = θi , receiver i’s utility is uLT i (λ(θi , p−i ), θi ) = M −

cM . 1 − s (k, (θi , p−i ))

If pi = θi , we have s(k, p) ≥ s(k, (θi , p−i )), and therefore uLT i (λ(p), θi ) = M −

cM ≤ uLT i (λ(θi , p−i ), θi ) . 1 − s(k, p)

Due to the uncertainty of others’ loss rates, the receiver is not sure which case it belongs to. As a result, the receiver would choose pi = θi , which maximizes its utility for all possible cases (i.e., the dominant strategy). Accordingly, the truth-telling strategy profile p∗ = θ is a unique DSE. 

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Although the KSS is DSIC in loss-tolerant data MBS systems, it is not DSIC in loss-sensitive data MBS systems. Instead, the special-case MMFS is DSIC in loss-sensitive data MBS systems. Theorem 2: The MMFS is DSIC in loss-sensitive data MBS systems. The proof follows the same procedure as that in Theorem 1.

B. Pareto Efficiency and Max-Min Fairness In addition to the DSIC property, the KSS also prompts a Paretoefficient DSE in loss-tolerant data MBS systems, whereas the MMFS even further prompts a max-min fair DSE in both loss-tolerant and loss-sensitive data MBS systems. Since both Pareto efficiency and max-min fairness are the most commonly discussed efficiency and fairness criteria in the context of communication networks, an outcome (or equilibrium) that is both Pareto efficient and max-min fair would be the most favorite in terms of system design. A Pareto-efficient outcome efficiently utilizes the system resources in the sense that there is no alternative outcome that makes some receivers better off without worsening some other receivers. On the other hand, a max-min fair outcome assigns more resources to the receivers having higher PLR and fewer resources to the receivers having lower PLR. In the succeeding sections, we refer to any number of encoded packets as an outcome in the MBS systems for short. Note that an equilibrium is a kind of outcome where players will not deviate from the current strategy. Definition 4 (Pareto Efficiency): An outcome λ∗ is Pareto efficient LT ∗ if, for any other λ such that uLT i (λ, θi ) > ui (λ , θi ), there must exist LT LT at least one receiver j such that uj (λ, θi ) < uj (λ∗ , θi ). A Pareto-efficient outcome can efficiently utilize the system resources in the sense that there is no alternative outcome that makes some receivers better off without making some other receivers worse off. Note that, in the definition, every possible outcome has been considered, including the outcomes induced by the proposed schemes. Theorem 3: In the loss-tolerant data MBS system, the DSE induced by the KSS is Pareto efficient. Proof: We use λ∗ to denote the outcome at the DSE θ for short. Note that λ∗ = M/(1 − s(k, θ)) or, equivalently, s(k, θ) = 1 − M/λ∗ . Our proof shows that any λ = λ∗ worsens some of the receivers’ utilities. When the BS broadcasts λ∗ encoded packets (i.e., when the out∗ come is λ∗ ), the loss-tolerant data receiver i’s utility is uLT i (λ , θi ) = ∗ ∗ (λ , θ ) = (1 − θ M − cλ∗ if θi ≤ 1 − M/λ∗ or uLT i i − c)λ i otherwise. When the BS broadcasts λ > λ∗ , receiver i, whose θi ≤ 1 − ∗ M/λ∗ , obtains lower utility uLT i (λ, θi ) = M − cλ < M − cλ = ∗ uLT (λ , θ ). On the other hand, when the BS broadcasts λ < λ∗ , i i ∗ LT receiver i, whose θi > 1 − M/λ , obtains lower utility ui (λ, θi ) = ∗ (1 − θi − c)λ < (1 − θi − c)λ∗ = uLT i (λ , θi ). Therefore, the DSE θ induced by the KSS is Pareto efficient.  Definition 5 (Max-Min Fairness): An outcome λ∗ is max-min fair LT ∗ if, for any other λ such that uLT i (λ, θi ) > ui (λ , θi ), there must LT ∗ ∗ exist at least one receiver j such that ui (λ , θi ) ≥ uLT j (λ , θi ) and LT ∗ uLT (λ, θ ) < u (λ , θ ). i i j j A max-min fair outcome maximizes the utility of the receiver whose utility is originally minimal. In other words, it assigns more resource to the receivers having higher packet loss rate, and fewer resources to the receivers having lower packet loss rate. Moreover, a max-min fair equilibrium must be Pareto efficient. Theorem 4: In both the loss-tolerant and loss-sensitive data MBS systems, the DSE induced by the MMFS is max-min fair. Proof: Following the proof of Theorem 3, if k = n (MMFS), ∗ LS ∗ the receivers obtain the same utility uLT i (λ , θi ) = ui (λ , θi ) =

Fig. 2. Graphical example of Pareto efficiency and max-min fairness.

M − cλ∗ for both loss-tolerant and loss-sensitive MBS, where λ∗ = M/(1 − s(n, θ)). Moreover, Theorem 3 has claimed that any deviation of λ∗ worsens some receivers’ utility. In this case, since receivers have the same utility, Pareto efficiency directly implies maxmin fairness.  Next, to show an graphic example for Pareto efficiency and maxmin fairness, we consider a loss-tolerant data MBS system composed of two receivers in Fig. 2. The line segments are composed of utility pairs (u1 (λ), u2 (λ)) induced by different outcome (i.e., different λ). Note that we consider every possible λ rather than p since the Pareto efficient outcome is superior to any other possible outcomes, including the outcomes induced by the proposed schemes with any possible input p. Accordingly, the utility pairs are composed of three line segments: Ok1 , k1 k2 , and k2 O. At the point k1 , it is the outcome λ1 induced by the KSS with k = 1, and at the point k2 is λ2 induced by the KSS with k = 2 (MMFS). The point O consists of two extreme outcomes: λ = 0 and λ = M/c. The former case indicates no broadcast packet while the latter excess packets, both resulting in zero utility of receivers. The line segment Ok1 is composed of the outcomes λ < λ1 . Similarly, the line k1 k2 is composed of the outcomes λ1 < λ < λ2 , and the line k2 O is λ > λ2 . Obviously, the outcomes on Ok1 and k2 O are not Pareto efficient since there exists an outcome on k1 k2 that simultaneously increase the utility pair. Yet, the outcomes on k1 k2 are Pareto efficient, including the end points k1 and k2 –the outcomes induced by the proposed schemes. Moreover, the outcome at k2 is max-min fair since it locates at the 45◦ line where the minimal utility has been maximized. VII. E XTENSION —L AYER -E NCODED M ULTIMEDIA M ULTICAST AND B ROADCAST S ERVICE G AME We introduce a layer-encoded multimedia MBS system as an extension of the loss-tolerant and loss-sensitive MBS system. The layer-encoded multimedia receivers request layer-encoded multimedia stream with two types of information packet—the base-layer and the enhanced-layer packets. The base layer, which is the most important layer, provides the most basic and the lowest multimedia quality, and the enhanced layers provide higher multimedia quality only when the base layer is completely decoded. Assume that the multimedia stream is encoded into single base layer and E enhanced layers. The base layer contains M 0 information

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packets and the tth enhanced layer contains M t information packets for t = 1, 2, . . . E. Similarly, we also have the number of broadcast packets λt for each layer t = 0, 1, . . . , E, the true packet loss rate θit 3 , the reported PLR pti , and the cost of receiving packets ct for each layer t. Accordingly, the utility of layer-encoded multimedia receiver is defined as 1) Layer-encoded multimedia receiver: (λ, θi ) = uMul i

E  







I λ0 , θi0 , M 0 vit λt , θit − ct λt

t=0

where





vit λt , θit =



M 0, min {(1 − θit ) λt , M t } ,

t=0 t = 1, 2, . . . , E.

The valuation function vit (λt , θit ) represents the benefit acquired from encoded packets of the t th enhanced (or base) layer. Note that the valuation function of the enhanced layer relies on its counterpart in the base layer. To decode the enhanced layers, the base layer must be completely decoded so that the indicator function I(λ0 , θi0 , M 0 ) is equal to 1 and the first term of utility is nonzero. Obviously, the receiver is loss sensitive to base-layer packets and loss tolerant of enhanced-layer packets. We can apply the previous results to prove that the scheme composed of the MMFS for the base layer and the KSS for the enhanced layers is DSIC in the layer-encoded multimedia MBS system.

VIII. N UMERICAL R ESULTS In this section, we numerically verify the properties of the proposed schemes. First, we apply the MMFS to a loss-tolerant data MBS system (see Fig. 3). Receiver 3, who has the highest loss rate, can maximize its utility only if truthfully reporting [Fig. 3(c)]. Receivers 1 and 2, on the other hand, have lower utility when reporting higher loss rates. By contrast, when reporting lower loss rates, their utility is the same as truthfully reporting. [See Fig. 3(a) and (b)]. As a result, all receivers truthfully report as Theorem 1 states. In addition, we justify the value of incentive compatibility of the proposed schemes in Fig. 3, where the MMFS is compared to the MS described in Section IV. Under the MS, the utility of receiver is denoted by dashed curves. Receiver 1 would choose to report p1 = 0 rather than the true one [see Fig. 3(a)]. Receivers 2 and 3 prefer to report p2 = 0.22 and p3 = 0.7, respectively, rather than their true information [see Fig. 3(b) and (c)]. The BS, therefore, cannot collect true information. Finally, Fig. 4 shows a multimedia MBS system with 25 receivers. The parameters are N = 25, E = 2, (M 0 , M 1 , M 2 ) = (15, 6, 5), (c0 , c1 , c2 ) = (0.55, 0.3, 0.2), and uniformly random θ. The BS adopts the MMFS for the base layer and the KSS with different k’s for the enhanced layers. The result shows that all receivers obtain the same utility for the base layer, as shown in Theorem 4, whereas four receivers obtain relatively lower utility for the first enhanced layer and 11 receivers for the second enhanced layer, as we will explain in the Appendix.

3 The packet loss rate of each layer is different from each other if the unequal error protection (UEP) technique is used, which provides stronger protection to the base-layer packets than the enhanced-layer packets.

Fig. 3. Loss-tolerant MBS system: N = 3, M = 15, c = 0.35, and (θ1 , θ2 , θ3 ) = (0.12, 0.23, 0.35). (a) Receiver 1 prefers to falsely report p1 = 0 under the MS but to truthfully report under the MMFS. (b) Receiver 2 prefers to falsely report p2 = 0 under the MS but to truthfully report under the MMFS. (c) Receiver 3 prefers to falsely report p3 = 0 under the MS but to truthfully report under the MMFS.

IX. C ONCLUSION In this paper, we have formulated game-theoretic models for heterogeneous MBS: loss-tolerant data MBS, loss-sensitive data MBS, and layer-encoded multimedia MBS. We have proposed MBS operation schemes, i.e., the KSS and the MMFS, to decide the number of network-coding packets to broadcast based on the information that the receivers report. The proposed schemes are dominant-strategy incentive compatible as they induce truthful feedback of PLRs from all receivers. Moreover, the proposed schemes further ensure that the BS will broadcast the proper number of network-coding packets to guarantee a Pareto-efficient and max-min fair system operation.

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[6] [7] [8] [9] [10] [11]

Fig. 4. Receivers with different loss rates obtain different values for each layer in a layer-encoded multimedia MBS system.

A PPENDIX A P HYSICAL M EANING OF THE KSS The KSS enables the receiver with k th lowest loss rate (suppose it is receiver j) to receive all information packets. Note that s(k, p) represents receiver j’s loss rate. If λ = M , on average, receiver j loses s(k, p)M encoded packets. To compensate the lost packets, we need λ = M + s(k, p)M . However, on average, receiver j loses s(k, p)2 M encoded packets again. Accordingly, the number of broadcast packet should be λ = M + s(k, p)M + s(k, p)2 M + s(k, p)3 M + · · · = M/(1 − s(k, p)). A PPENDIX B S ELECTION OF PARAMETER k Assume that the probability density function of loss rate f (θ) is known, where 0 ≤ θ ≤ 1. Denote by Θ = (θ1 , . . . , θn ) a set of random variables following the distribution f (θ). Then, there exists a proper parameter k∗ such that the expected overall utility is maximized, i.e., k∗ = arg max



k

EΘ [ui (λ(Θ), θi ) |θi ]

i∈N

where λ(Θ) = M/1 − s(k, Θ). The optimal parameter k∗ can be found by examining k from 1 to N , which formulates an algorithm with complexity polynomial in N . R EFERENCES [1] D. Nguyen, T. Tran, T. Nguyen, and B. Bose, “Wireless broadcast using network coding,” IEEE Trans. Veh. Technol., vol. 58, no. 2, pp. 914–925, Feb. 2009. [2] D. Komosny, P. Moravek, R. Burget, and K. Ganeshan, “Feedback transmission in large-scale IPTV sessions,” in Proc. ICCIT, 2009, pp. 796–801. [3] X. Xiao, L. Yang, W. Wang, and S. Zhang, “A broadcasting retransmission approach based on random linear network coding,” in Proc. ICYCS, 2008, pp. 457–461. [4] Z. Liu, Z. Wu, P. Liu, H. Liu, and Y. Wang, “Layer bargaining: Multicast layered video over wireless networks,” IEEE J. Sel. Areas Commun., vol. 28, no. 3, pp. 445–455, Apr. 2010. [5] D. Krishnaswamy and M. Van der Schaar, “Adaptive modulated scalable video transmission over wireless networks with a game theoretic

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approach,” in Proc. IEEE Workshop Multimedia Signal Process., 2004, pp. 107–110. E. Jorswieck and R. Mochaourab, “Power control game in protected and shared bands: Manipulability of Nash equilibrium,” in Proc. Int. Conf. GameNets, 2009, pp. 428–437. N. Rama Suri, Y. Narahari, and D. Manjunath, “An efficient pricing based protocol for broadcasting in wireless ad hoc networks,” in Proc. Int. Conf. Comsware, 2006, pp. 1–7. Y. Chen, B. Wang, and K. J. R. Liu, “Multiuser rate allocation games for multimedia communications,” IEEE Trans. Multimedia, vol. 11, no. 6, pp. 1170–1181, Oct. 2009. W. S. Lin, H. V. Zhao, and K. J. R. Liu, “Incentive cooperation strategies for peer-to-peer live multimedia streaming social networks,” IEEE Trans. Multimedia, vol. 11, no. 3, pp. 396–412, Apr. 2009. W. Ge, J. Zhang, and S. Shen, “A cross-layer design approach to multicast in wireless networks,” IEEE Trans. Wireless Commun., vol. 6, no. 3, pp. 1063–1071, Mar. 2007. T. Ho, M. Medard, R. Koetter, D. R. Karger, M. Effros, J. Shi, and B. Leong, “A random linear network coding approach to multicast,” IEEE Trans. Inf. Theory, vol. 52, no. 10, pp. 4413–4430, Oct. 2006.

On Generalized MIMO Y Channels: Precoding Design, Mapping, and Diversity Gain Ning Wang, Zhiguo Ding, Member, IEEE, Xuchu Dai, and Athanasios V. Vasilakos, Senior Member, IEEE

Abstract—In this paper, we first study the design of network coding for the generalized multiple-input–multiple-output (MIMO) Y channels, where K users wish to exchange information with each other within two time slots. Precoding at each user and the relay is carefully constructed to ensure that the signals from the same user pair are grouped together and that cross-pair interference can be canceled. In addition, a simple mapping function is proposed to ensure low-complexity detection at the relay. Exact expressions of symbol error rate (SER) are then developed to establish the explicit relationship between the diversity gain and the number of node antennas. Monte Carlo simulation is also provided to demonstrate the performance of the proposed scheme. Index Terms—Cooperative diversity, modulation mapping, network coding.

I. I NTRODUCTION Recently, there has been growing interest in the application of network coding to wireless communications due to its superior capability to increase system throughput [1], [2]. The key idea of network coding is to ask relays to broadcast a mixture of many source messages simultaneously. Provided that each destination has a priori information about the mixture, it is possible that one relay transmission can help

Manuscript received September 16, 2010; revised March 18, 2011 and June 17, 2011; accepted July 1, 2011. Date of publication July 14, 2011; date of current version September 19, 2011. The work of N. Wang and X. Dai was supported in part by the National Natural Science Foundation of China under Grant NSFC-61071094. The work of Z. Ding was supported by the UK Engineering and Physical Sciences Research Council under Grant EP/I037423/1. The review of this paper was coordinated by Prof. H. Liu. N. Wang and X. Dai are with the Department of Electronic Engineering and Information Science, University of Science and Technology of China, Hefei 230027, China. Z. Ding is with the School of Electrical, Electronic, and Computer Engineering, Newcastle University, NE1 7RU Newcastle upon Tyne, U.K. A. V. Vasilakos is with the Department of Computer and Telecommunications Engineering, University of Western Macedonia, Kozani 50100, Greece. Digital Object Identifier 10.1109/TVT.2011.2162011

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