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 *The Wealth of Nations*, Adam Smith (a Scottish moral philosopher, pioneer of political economy, and father of modern economics) expounded how rational self-interest and competition can lead to economic prosperity and well-being through macroeconomic adjustments. For example, all states today have some form of macroeconomic control over the market that removes the free and unrestricted direction of resources from consumers and prices such as tariffs and corporate subsidies.

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 . Write user 's average payoff as (Without loss of generality, we consider user in the following context, since the problem is symmetric for user )

Now, is transformed into a function of , write it as . To maximize the average achievable rate, user needs to solve the following optimization problem, as mentioned in (4)

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, , which is our object in this section.

But unfortunately, may not be a convex function [18], so the single-user problem may not be a convex optimization problem. It can be further verified that is convex under some special conditions, depending on all the parameters , and . Here, we will not discuss all the convex cases, but only focus on the high SNR regime (meaning that the noise can be omitted compared to the signal strength). In this case, we have

This function is strict convex. To be more precise, it is decreasing on and increasing on , and the solution is given by

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 () channel states. Note that the result of this subsection can also be applied to the case of two channel states.

Assumption.

Each user's channel gain has positive states, which are with probability , respectively (Without loss of generality,), and we have *.*

Based on Assumption 6, we define , as the transmit power when a user's channel gain is . As previously mentioned, our object in this part is to maximize the sum ergodic capacity of the system, that is, . Under the symmetric assumption, this sum-ergodic-capacity maximization problem is equivalent to the following single-user maximization problem

where is now defined as . This optimization problem is difficult, since the objective function is again nonconvex in . However, we can consider a relaxation of the optimization by introducing a lower bound [21]

where and are chosen specified as

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

Let us consider the lower bound (denoted as ) by using the relaxation (26) to the objective function in (25)

which is still nonconvex, and so it is not concave in . However, with a logarithmic change of the following variables and constants: and , we can turn the geometric programming [18] associated with the objective function (28) into the following problem:

where is defined as

Now, it is easy to verify that the lower bound is concave in the transformed set , since the log-sum-exp function is convex. The constraints of the optimization problem are such that Slater's condition is satisfied [18]. So, the Karush-Kuhn-Tucker (KKT) condition of the optimization is sufficient and necessary for the optimality. Given the following Lagrangian dual function (denoted by ):

the KKT conditions are

where , and is a dual variable associated with the power constraint in (29).

Define , the equivalent KKT conditions can be simply written as a quadratic equation

where the parameters are expressed as

Note that and are functions of , we can write them as and . Since , the solution to the KKT conditions can only be one of the roots to the quadratic equation, that is,

where is chosen such that . Thus, for some fixed value of , we can directly apply (35) to maximize the lower bound (28). Then, it is natural to improve the bound periodically. Based on the discussion above, we propose the following algorithm, namely Lower Bound Tightening (LBT) algorithm

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.