Motorola Labs, Parc Les Algorithmes, 91193 Gif sur Yvette, France

Alcatel-Lucent Chair on Flexible Radio, 3 Rue Joliot-Curie, 91192 Gif sur Yvette, France

INRIA, 2004 Route des Lucioles, 06902 Sophia Antipolis, France

A Bayesian game-theoretic model is developed to design and analyze the resource allocation problem in

1. Introduction

The fading multiple access channel (MAC) is a basic wireless channel model that allows several transmitters connected to the same receiver to transmit over it and share its capacity. The capacity region of the fading MAC and the optimal resource allocation algorithms have been characterized and well studied in many pioneering works with different information assumptions [1–4]. However, in order to achieve the full capacity region, it usually requires a central computing resource (a scheduler with comprehensive knowledge of the network information) to globally allocate the system resources. This process is centralized, since it involves feedback and overhead communication whose load scales linearly with the number of transmitters in the network. In addition, with the fast evolution of wireless techniques, this centralized network infrastructure begins to expose its weakness in many aspects, for example, slow reconfiguration against varying environment, increased computational complexity, and so forth. This is especially crucial for femto-cell networks where it is quite difficult to centralize the information due to a limited capacity backhaul. Moreover, the high density of base stations would increase the cost of centralizing the information.

In recent years, increased research interest has been given to self-organizing wireless networks in which mobile devices allocate resources in a decentralized manner [5]. Tools from game theory [6] have been widely applied to study the resource allocation and power control problems in fading MAC [7], as well as many other types of channels, such as orthogonal frequency division multiplexing (OFDM) [8], multiple input and multiple output (MIMO) channels [9, 10], and interference channels [11]. Typically, the game-theoretic models used in these previous works assume that the knowledge, for example, channel state information (CSI), about other devices is available to all devices. However, this assumption is hardly met in practice. In practical wireless scenarios, mobile devices can have local information but can barely access to global information on the network status.

A static noncooperative game has been introduced in the context of the two-user fading MAC, known as "waterfilling game" [7]. By assuming that users compete with transmission rates as utility and transmit powers as moves, the authors show that there exists a unique Nash equilibrium [12] which corresponds to the maximum sum-rate point of the capacity region. This claim is somewhat surprising, since the Nash equilibrium is in general inefficient compared to the Pareto optimality. However, their results rely on the fact that both transmitters have complete knowledge of the CSI, and in particular, perfect CSI of all transmitters in the network. As we previously pointed out, this assumption is rarely realistic in practice.

Thus, this power allocation game needs to be reconstructed with some realistic assumptions made about the knowledge level of mobile devices. Under this consideration, it is of great interest to investigate scenarios in which devices have "incomplete information" about their components, for example, a device is aware of its own channel gain, but unaware of the channel gains of other devices. In game theory, a strategic game with incomplete information is called a "Bayesian game." Over the last ten years, Bayesian game-theoretic tools have been used to design distributed resource allocation strategies only in a few contexts, for example, CDMA networks [13, 14], multicarrier interference networks [15]. The primary motivation of this paper is therefore to investigate how Bayesian games can be applied to study the resource allocation problems in the fading MAC. In some sense, this study can help to design a self-organizing femto-cell network where different frequency bands or subcarriers are used for the femto-cell coverage, for example, different femto-cells operate on different frequency bands to avoid interference.

In this paper, we introduce a Bayesian game-theoretic model to design and analyze the resource allocation problem in a fading MAC, where users are assumed to selfishly maximize their ergodic capacity with incomplete information about the fading channel gains. In such a game-theoretic study, the central question is whether a Bayesian equilibrium exists, and if so, whether the network operates efficiently at the equilibrium point. We prove that there exists exactly one Bayesian equilibrium in our game. Furthermore, we study the network sum-rate maximization problem by assuming that all users coordinate to an optimization-based symmetric strategy. This centralized strategy is important when the fading processes for all users are relatively stationary and the global system structure is fixed for a long period of time. This result also serves as an upper bound for the unique Bayesian equilibrium.

The paper is organized in the following form: In Section 2, we introduce the system model and state important assumptions. In Section 3, the

2. System Model and Assumptions

2.1. System Model

We consider the uplink of a single-cell network where

where

where

In this study, we consider the wireless transmission in fast fading environments, that is, the coherence time of the channel is small relative to the delay constraint of the application. When the receiver can perfectly track the channel but the transmitters have no such information, the codewords cannot be chosen as a function of the state of the channel but the decoding can make use of such information. When the fading process is assumed to be stationary and ergodic within the considered interval of signal transmission, the channel capacity in a fast fading channel corresponds to the notion of ergodic capacity, that is,

where

2.2. Assumption of Finite Channel States

Before introducing our game model, we need to clarify a prior assumption for this section.

Assumption.

On the one hand, our assumption is closely related to the way how feedback information is signalled to the transmitters. In order to get the channel information

On the other hand, this is a necessary assumption for analytical tractability, since in principle the functional strategic form of a player can be quite complex with both actions and states being continuous (or infinite). To avoid this problem, in [15] the authors successfully modelled a multicarrier Gaussian interference channel as a Bayesian game with discrete (or finite) actions and continuous states. Inspired from [15], we also model the fading MAC as a Bayesian game under the assumption of continuous actions and discrete states.

3. Game Formulation

We model the

In such a communication system, the natural object of each user is to maximize its ergodic capacity subject to an average power constraint, that is,

where

For a given set of power strategies

where the dual variable

The

(i)Player set:

(ii)Type set:

(iii)Action set:

(iv)Probability set:

(v)Payoff function set:

In games of incomplete information, a player's type represents any kind of private information that is relevant to its decision making. In our context, the fading channel gain

4. Bayesian Equilibrium

4.1. Definition of Bayesian Equilibrium

What we can expect from the outcome of a Bayesian game if every selfish and rational (rational player means a player chooses the best response given its information) participant starts to play the game? Generally speaking, the process of such players' behaviors usually results in a Bayesian equilibrium, which represents a common solution concept for Bayesian games. In many cases, it represents a "stable" result of learning and evolution of all participants. Therefore, it is important to characterize such an equilibrium point, since it concerns the performance prediction of a distributed system.

Now, let

Definition 2 (Bayesian equilibrium).

The strategy profile

where we define

From this definition, it is clear that at the Bayesian equilibrium no player can benefit from changing its strategy while the other players keep theirs unchanged. Note that in a strategic-form game with complete information each player chooses a concrete action, whereas in a Bayesian game each player

4.2. Characterization of the Bayesian Equilibrium Set

It is well known that, in general, an equilibrium point does not necessarily exist [6]. Therefore, our primary interest in this paper is to investigate the

Theorem 3.

There exists a unique Bayesian equilibrium in the

Proof.

It is easy to prove the existence part, since the strategy space

In order to prove the uniqueness part, we should rely on a sufficient condition given in [19]: a non-cooperative game has a unique equilibrium, if the nonnegative weighted sum of the payoff functions is

Definition 4 (diagonally strictly concave).

A weighted nonnegative sum function

where

We start with the following lemma.

Lemma 5.

The weighted nonnegative sum of the average payoffs

Proof.

Write the weighted nonnegative sum of the average payoffs as

where

the transmit power of player

where

Now, we can write the pseudogradient

where the function

To check the diagonally strictly concave condition (10), we let

where

Since

Since our sum-payoff function

5. Optimal Symmetric Strategies

The Bayesian game-theoretic approach provides us a better understanding of the wireless resource competition existing in the fading MAC when every mobile device acts as a selfish and rational decision maker (this means a device always chooses the best response given its information). The advantage of this model is that it mathematically captures the behavior of selfish wireless entities in strategic situations, which can automatically lead to the convergence of system performance. The introduced Bayesian game-theoretic framework fits very well the concept of self-organizing networks, where the intelligence and decision making is distributed. Such a scheme has apparent benefits in terms of operational complexity and feedback load.

However, from the global system performance perspective, it is usually inefficient to give complete "freedom" to mobile devices and let them take decisions without any policy control over the network. It is very interesting to note that a similar situation happens in the market economy, where consumers can be modeled as players to complete for the market resources. In the famous literature

In particular, wireless service providers would like to design an appropriate policy to efficiently manage the system resource so that the global network performance can be optimized or enhanced to a certain theoretical limit, for example, Shannon capacity or capacity region [20]. Apparently, a centralized scheduler with comprehensive knowledge of the network status can globally optimize the resource utility. However, this approach usually involves sophisticated optimization techniques and a feedback load that grows with the number of wireless devices in the network. Thus, the optimization-based centralized decision has to be frequently updated as long as the wireless environment varies, or the system structure changes, for example, a user joins or exits the network.

In this section, we consider that the channel statistics (fading processes) for all wireless devices are jointly stationary for a relatively long period of signal transmission, and the global system structure remains unchanged. In addition, we neglect the problem of computational complexity at the scheduler and the impact of feedback load to the useful data transmission rate. In this case, the network service provider would strictly prefer to use a centralized approach, that is, a scheduler assigns some globally optimal strategies to the wireless devices, guiding them how to react under all kinds of different situations. Based on the Bayesian game settings, we provide a special discussion on the optimal symmetric strategy design. Note that this result can be treated as a theoretical upperbound for the performance measurement of Bayesian equilibrium.

We now introduce a necessary assumption.

Assumption.

Mobile devices are designed to use the same power strategies, that is, they send the same power if their observations on the channel states are symmetric. In addition, we assume that the mobile devices have the same average power constraint, that is,

5.1. Two Channel States

For simplicity of our presentation, We first consider the scenario of two users with two channel states. In fact, the analysis of multiuser MAC can be extended in a similar way. According to Assumption 6, we define

and we have

Now,

Under Assumption 6, it can be shown that (due to the symmetric property) this single-user maximization problem is equivalent to the multiuser sum average rate maximization problem, that is,

But unfortunately,

This function is strict convex. To be more precise, it is decreasing on

Note that in this setting the choice of the optimal symmetric strategy is to concentrate the full available power on a single channel state. The selection of the channel state on which to transmit depends not only on the channel conditions but also on the probability of the channel states. This result implies that, in the high SNR regime, the optimal symmetric power strategy is to transmit information in an "opportunistic" way. For a better understanding of the "opportunistic" transmission, the interested reads are referred to [2].

5.2. Multiple Channel States

In this subsection, we discuss the extension to arbitrary

Assumption.

Each user's channel gain

Based on Assumption 6, we define

where

where

we say that the lower bound (26) is tight with equality at a chosen value

Let us consider the lower bound (denoted as

which is still nonconvex, and so it is not concave in

where

Now, it is easy to verify that the lower bound

the KKT conditions are

where

Define

where the parameters

Note that

where

The algorithm convergence can be easily proved, since the objective is monotonically increasing at each iteration. However, the global optimum is not always guaranteed, due to the nonconvex property.

6. Numerical Results

In this section, numerical results are presented to validate our theoretical claims. For Figures

The uniqueness of Bayesian equilibrium.

**The uniqueness of Bayesian equilibrium.** (a)

Average network sum-rate.

**Average network sum-rate.** (a)

First, we show the existence and uniqueness of Bayesian equilibrium in the scenario of two-user fading MAC. In Figure

Second, we investigate the efficiency of Bayesian equilibrium from the viewpoint of global average network performance. The

Finally, we show the convergence behavior of the lower bound tightening (LBT) algorithm. In Figure

The convergence of the lower bound tightening (LBT) algorithm.

**The convergence of the lower bound tightening (LBT) algorithm.**

7. Conclusion

We presented a Bayesian game-theoretic framework for distributed resource allocation in fading MAC, where users are assumed to have only information about their own channel gains. By introducing the assumption of finite channel states, we successfully found a analytical way to characterize the Bayesian equilibrium set. First, we proved the existence and uniqueness. Second, the inefficiency was shown from numerical results. Furthermore, we analyzed the optimal symmetric power strategy based on the practical concerns of resource allocation design. Future extension is considered to improve the efficiency of Bayesian equilibrium through pricing or cooperative game-theoretic approaches.

**Algorithm 1:**Lower Bound Tightening (LBT).

**Initialize**

**repeat**

**repeat**

**for** **do**

update

**end for**

**until**

**for** **do**

**end for**

**until** converge