## 3 Problem Statement and Formulation
### Problem Formulation  
In this paper, our target of optimizing the power allocation is to maximize the overall sum rate of all wireless channels under the important constraints, such as power and spectral capacity requirements. We first formulate the objective function as a Gaussian interference game (GIG) under transmit power constraint by considering the discrete frequencies [50]. For simplicity, we assume an increasing sequence of frequencies such that \(f_{o}<\cdots<f_{k}\). In particular, we adopt a discrete approximation game to model the GIG from \(j^{th}\) BS to \(i^{th}\) receiver over \(K\) frequency channel as \(GI_{\{l_{1},\ldots,l_{k}\}}\), where \(l_{k}\) is the closed interval given by \(I_{k}=\{f_{k-1},f_{k}\}\). Specifically, each BS is considered as a game player, i.e., there are \(K\) parallel frequency channels to operate as \(1,\ldots,N\) over \(K\). The player can send a power vector \(P_{i}=(P_{i}(1),\ \ldots,\ P_{i}(k))\in[0,P_{i}]^{k}\), such that \(P_{i}(k)\) is the transmit power in the closed interval \(I_{k}\). Hence, we will have \(\sum\limits_{k=1}^{K}P_{i}(k)=P_{i}\). This indicates that the set of power allocations for all receivers is a closed convex subset of the cube \(\prod_{i=1}^{N}[0,P_{i}]^{k}\), denoted by  
\[B=\prod_{i=1}^{N}B_{i} \tag{4}\]  
where \(B_{i}\) denotes the set of power allocations for player \(i\). Mathematically, \(B_{i}\) can be expressed as  
\[B_{i}=[0,P_{i}]^{k}\{(P_{i}(1),\ \ldots,\ P_{i}(k)):\sum_{k=1}^{K}P_{i}(k)=P_{i}\} \tag{5}\]  
Here, each player selects a PSD \(P_{i}=P_{i}(k):1\leq k\leq N\in B_{i}\). The spectral capacity of each player \(i\) is its sum rate \(R_{i}\), and each player has to satisfy the power and QoS requirements. Note that the QoS constraint depends on the BS's transmitting power, indicating that it shall be optimized to achieve the target of service request. Accordingly, the optimization game is formulated as:  
\[\max_{\mathbf{p}>\mathbf{0}}\sum_{i=1}^{K}w_{i}R_{i}(k)s.t.\ 0\leq P_{i}(k) \leq P_{i}^{max}SINR_{i}(k)>\gamma_{o}(k),\ k=1,\ldots,K \tag{6}\]  
where \(w_{i}\) is the given non-negative weight of the sub-channel assignment at the \(i^{th}\) BS to maximize the total sum throughput. Here, we assume \(w_{i}=1\) for each player \(i\) while \(P_{i}^{max}\) is the maximum power the BS can use it. The objective of Problem (6) is to maximize the achievable rate at the destination user so that they can obtain useful information with the desired service request, which is always defined by the constraint of the SINR higher than or equal to the given threshold value of the \(i^{th}\) user, i.e., \(SINR_{i}(k)>\gamma_{o}(k),\ k=1,2,\ldots,\ K\). In the literature, the formulation problem in (6) has been shown to be an NP-hard combinatorial problem due to the nonlinear probabilistic constraints [8]. No convex reformulation of the above problem is known, even without the QoS constraints. Therefore, the directly optimal solution is non-trivial and may not be feasible. In the following, we will show that the above problem in (6) can be unified under the Gaussian interference game (GIG) framework to simplify the formulation. Specifically, we will first analyze the payoff for each user \(i\) and provide a payoff vector \(\mathbf{R}\) for finding the GIG of Problem (6). Then, we will describe the Nash equilibrium solution to achieve a stationary solution of the given power allocation \(\mathbf{p}\). For ease of exposition, let us consider the transmit power distributions for player \(i\), such that each player selects a power \(P_{i}=P_{i}(k):1\leq k\leq N\in B_{i}\).