Coalition Game Theoretic Power Allocation Strategy for Multi-Target Detection in Distributed Radar Networks

Abstract

This paper studies a coalition game theoretic power allocation algorithm for multi-target detection in radar networks based on low probability of intercept (LPI). The main goal of the algorithm is to reduce the total radiated power of the radar networks while satisfying the predetermined target detection performance of each radar. Firstly, a utility function that comprehensively considers both target detection performance and the radiated power of the radar networks is designed with LPI performance as the guiding principle. Secondly, it causes a coalition to form between cooperating radars, and radars within the same coalition share information. On this basis, a mathematical model for power allocation in radar networks based on coalition game theory is established. The model takes the given target detection performance as a constraint and maximizing system energy efficiency and optimal power allocation as the optimization objective. Furthermore, this paper proposes a game algorithm for joint coalition formation and power allocation in a multi-target detection scenario. Finally, the existence and uniqueness of the Nash equilibrium (NE) solution are proven through strict mathematical deduction. Simulation results validate the effectiveness and feasibility of the proposed algorithm.

1. Introduction

In recent years, distributed radar networks have attracted extensive academic attention [1,2,3]. Compared with the traditional single-base radar, distributed radar networks have many potential advantages, such as superior waveform diversity gain [4], spatial diversity gain [5,6], and better target detection and tracking performance [7,8,9]. Due to their advantages in signal and spatial diversity, they are widely used in target detection [10], target localization [11,12], target tracking [13], waveform design [14], parameter estimation [15,16], sensor selection [17,18], and information extraction [19]. Currently, they are in the transition stage from theory to practice.

With the rapid development of passive detection technology, low probability of intercept (LPI) technology has become an essential element in the design of radar systems. To achieve better LPI performance, it is critical to strictly regulate the radiated power of radar systems [20,21,22].

In recent years, domestic and foreign scholars have applied game theory methods to the resource management of networked radar systems [23,24,25]. Reference [26] studies distributed power allocation algorithms based on the framework of game theory for multi-input multi-output (MIMO) radar networks. They divide the radar networks into clusters aiming to achieve a specific signal-to-disturbance ratio (SDR) and consider power adaptive allocation based on SDR estimation to enable communication-free operation among radars in the cluster. It has been proven that non-cooperative game algorithms can converge to equilibrium even in the presence of SDR estimation errors. Reference [27] studies a game-theory-based power allocation scheme considering the radar networks consisting of multiple clusters, with the main objective of minimizing transmission power while satisfying predetermined detection criteria. Due to the absence of communication between distributed clusters, convex optimization algorithms and non-cooperative game techniques based on SDR estimation are employed to address the power allocation problem. Consequently, each cluster independently determines its optimal power allocation result in a distributed manner. Furthermore, the existence and uniqueness of the Nash equilibrium (NE) solution are proven. Reference [28] proposes a Bayesian game-based technique for maximizing the signal-to-interference-plus-noise ratio (SINR) in distributed radar networks. In the radar networks, each radar’s primary objective is to maximize its SINR under the constraint of its maximum radiated power. The existence and uniqueness of the Bayesian NE in this game model are also demonstrated.

As research on game theory in radar network resource management has matured, scholars have started to focus on applying game theory to multi-objective scenarios. They have begun to explore different game approaches to adapt to specific scenarios, aiming to further improve LPI performance and maximize resource utilization in networked radar systems. Reference [29] addresses the problem of inconsistent global resource allocation due to network topology switches and information delays in decentralized radar networks. They formulate the decentralized radar node power allocation problem as a cooperative game model with the SINR as the characteristic function. They improve the computation of cooperative game Shapley values using the weighted graph concept to reduce computational complexity and propose a fast-solving algorithm based on cooperative game theory. Reference [30] tackles the issue of low resource utilization in networked radar systems. They propose a power allocation method based on the Stackelberg game and derive the Fisher information matrix determinant for target state estimation as a measure of tracking accuracy. They establish a leader subgame to minimize power consumption and a follower subgame to optimize tracking accuracy. The problem is modeled as a two-level Stackelberg game, and the optimization model is solved using the barrier function method. By improving resource utilization while ensuring tracking performance, the system’s efficiency is enhanced. Reference [31] aims to overcome the limitations of robustness, high computational cost, and limited flexibility in centralized optimization methods for task assignment in multi-objective tracking scenarios of radar networks. They redefine the task assignment problem of radar networks in multi-target tracking scenarios as a multi-intelligent decision and learning problem and propose a multi-intelligent stochastic Fourier feature reinforcement learning algorithm to find the optimal solution of the game in a high-dimensional state space. Moreover, the convergence and effectiveness of the algorithm are demonstrated.

From the above research, it can be found that game theory has been widely studied and applied in the power allocation problem of radar networks, laying a solid foundation for subsequent research. However, there are still many aspects that need to be further investigated: (a) the current study only investigates the task allocation distribution problem of networked radar systems in multi-target tracking scenarios based on the coalition game and does not consider the problem of resource allocation; (b) an algorithm that integrates coalition formation and power allocation has not yet been proposed; and (c) the effects of several system factors on the coalition division and power allocation results have not yet been analyzed. Therefore, we will investigate the above issues in this paper which, as far as we know, have not been discussed prior to this work.

Unlike the non-cooperative game, the cooperative game emphasizes group rationality, efficiency, justice and fairness, and the pursuit of maximizing collective gains. Among them, the cooperative game of transferable utility is the coalition game. Therefore, in order to maximize the energy efficiency of the system, we can model the interactions between radars in the radar networks as a coalition game model. Maximizing the energy efficiency of the system can be accompanied by obtaining the optimal coalition structure and minimizing the total radiated power of the system. This paper contributes by proposing a game algorithm for joint coalition formation and power allocation and rigorously proving the existence and uniqueness of the NE solution.

The main contributions of this paper are summarized as follows:

• A framework for a coalition game of LPI-based power allocation in radar networks is developed with the aim of improving LPI performance by reducing the radar’s radiated power while satisfying a predefined target detection performance requirement.

• We propose a game algorithm for joint coalition formation and power allocation to solve this coalition game model and rigorously prove the existence and uniqueness of the NE.

• Simulation results show that compared with other existing algorithms, the proposed algorithm can quickly form a stable coalition structure and achieve the maximum system energy efficiency under the current coalition structure and then effectively reduce the total radiated power of the radar networks under the condition of meeting the preset target detection performance requirements.

• We demonstrate the relationship between the coalition formation results and the power allocation results with two factors: the target radar cross section (RCS) and the relative position relationship between the target and the radar.

The remainder of this paper is presented below. In Section 2, we give some information about the system model of radar networks, which is needed in the rest of the paper. Section 3 gives a detailed description of the model and a coalition game analysis. A game algorithm for joint coalition formation and power allocation is proposed, and the existence and uniqueness of the NE solution are rigorously proved. The simulation results and analysis of the corresponding scenarios are given in Section 4, and finally, the concluding remarks are presented in Section 5.

2. System Model

In this paper, as shown in Figure 1, we examine radar networks comprising M radars. We assume that there are Q targets to be detected in the given scenario, where all radars operate within the same frequency band.

Each radar’s antenna is transceiver co-located, which means that the same antenna is used for both transmitting and receiving signals [32,33]. Additionally, the radar antennas can receive echoes reflected from each target independently, as well as signals transmitted directly from other radars without any target reflection. In the case of detecting the presence of a target, the received signal at radar i can be expressed mathematically as follows:

(1)$x_i={\alpha }_i\sqrt{p_i}{s}_i+{\displaystyle \sum _{j=1.j\ne i}^M{\beta }_{j,i}}\sqrt{p_j}{s}_j+w_i$

where $s_i={\phi }_i{\nu }_i$ denotes the transmitted signal of radar I, ${\nu }_i=[1,{\mathrm{e}}^{\mathrm{j}2\pi f_D^i},\dots ,{\mathrm{e}}^{\mathrm{j}2\pi (L-1)f_D^i}]$ denotes the Doppler guidance vector of radar i with respect to the detected target, L denotes the number of pulses received by the radar during the dwell time, $f_D^i$ denotes the Doppler shift of the detected target with respect to radar i, ${\phi }_i$ denotes the transmit waveform of radar i, $p_i$ denotes the gain between radar i and radar j, and $w_i$ denotes an additive noise. It is assumed that ${\alpha }_i\sim N(0,h_{i,i}^{\mathrm{t}})$, ${\beta }_{j,i}\sim N(0,c_{j,i}(h_{j,i}^{\mathrm{t}}+h_{j,i}^{\mathrm{d}}))$, and $w_i\sim N(0,{\sigma }^2)$. $c_{j,i}$ denotes the correlation coefficient between radar j and radar i. Where $h_{i,i}^{\mathrm{t}}$, $h_{j,i}^{\mathrm{t}}$, and $h_{j,i}^{\mathrm{d}}$ can be expressed as follows [1]:

(2)$\left\{\begin{array}{c}{h}_{i,i}^{\mathrm{t}}=\frac{G_{\mathrm{t}}{G}_{\mathrm{r}}{\sigma }_{i,i}^{i,\mathrm{RCS}}{\left(\lambda \right)}^2}{{\left(4\pi \right)}^3{R}_{i,i}^4},\\ h_{j,i}^{\mathrm{t}}=\frac{G_{\mathrm{t}}{G}_{\mathrm{r}}{\sigma }_{j,i}^{j,\mathrm{RCS}}{\left(\lambda \right)}^2}{{\left(4\pi \right)}^3{R}_{j,i}^2{R}_{j,j}^2},\\ h_{j,i}^{\mathrm{d}}=\frac{G_{\mathrm{t}}{}^{\prime }{G}_{\mathrm{r}}{}^{\prime }{\left(\lambda \right)}^2}{{\left(4\pi \right)}^2{d}_{j,i}^2},\begin{array}{cc}& \end{array}\end{array}\right.$

where $h_{i,i}^{\mathrm{t}}$ denotes the gain of the propagation path of radar i–target–radar i, $h_{j,i}^{\mathrm{t}}$ denotes the gain of the propagation path of radar j–target–radar i, $h_{j,i}^{\mathrm{d}}$ denotes the gain of the propagation path of radar j–radar i, $G_{\mathrm{t}}$ is the main lobe transmission antenna gain of a radar, $G_{\mathrm{t}}{}^{\prime }$ is the side lobe transmission antenna gain of a radar, $G_{\mathrm{r}}$ is the main lobe receiving antenna gain of a radar, and $G_{\mathrm{r}}{}^{\prime }$ is the side lobe receiving antenna gain of a radar. ${\sigma }_{i,i}^{i,\mathrm{RCS}}$ is the radar cross section (RCS) of radar i returning to itself past its detected target, ${\sigma }_{j,i}^{j,\mathrm{RCS}}$ is the RCS of radar j passing through the target to radar i, $\lambda$ is the signal wavelength, $R_{i,i}$ is the distance between radar i and its detected target, $R_{j,i}$ is the distance between radar i and the target detected by radar j, $R_{j,j}$ is the distance between radar j and its detected target, and $d_{j,i}$ is the distance between radar j and radar i.

The miss-detection probability $P_{\mathrm{MD},i}$ and false alarm probability $P_{\mathrm{FA},i}$ of radar i are [34].

(3)$\left\{\begin{array}{c}{P}_{\mathrm{MD}, i}\left({\delta }_i,{\gamma }_i\right)=1-{\left(1+\frac{{\delta }_i}{1-{\delta }_i}\cdot \frac{1}{1+L{\gamma }_i}\right)}^{1-L},\\ P_{\mathrm{FA}, i}\left({\delta }_i\right)={\left(1-{\delta }_i\right)}^{L-1},\hfill \end{array}\right.$

where ${\delta }_i$ denotes the target detection threshold of radar i, and L denotes the number of pulses received by radar i during the dwell time. ${\gamma }_i$ denotes the SINR of radar i, which can be expressed as follows:

(4)${\gamma }_i=\frac{h_{i,i}^{\mathrm{t}}{p}_i}{{\displaystyle {\sum }_{j=1,j\ne i}^{N_j}{c}_{j,i}\left[h_{j,i}^{\mathrm{d}}{p}_j+h_{j,i}^{\mathrm{t}}{p}_j\right]+{\sigma }^2}},$

where $p_i$ is the radiated power of the ith radar, and $c_{j,i}$ denotes the correlation coefficient between the jth radar and the ith radar. The ambient noise is zero-mean additive white Gaussian noise with variance ${\sigma }^2$, and $N_j$ is the set of other radars outside the coalition in which radar i is located. It can be seen from Equation (4) that the numerator of ${\gamma }_i$ describes the return signal of the radiation on the target, while the denominator consists of interference and noise. Therefore, Equation (4) can be equated as follows:

(5)${\gamma }_i=\frac{h_{i,i}^{\mathrm{t}}{p}_i}{I_{-i}},$

where $I_{-i}={\displaystyle {\sum }_{j=1,j\ne i}^{N_j}{c}_{j,i}\left[h_{j,i}^{\mathrm{d}}{p}_j+h_{j,i}^{\mathrm{t}}{p}_j\right]+{\sigma }^2}$ is the total interference and noise received by radar i. We can obtain the corresponding ${\delta }_i$ by equating $P_{\mathrm{MD}, i}\left({\delta }_i,{\gamma }_i\right)$ and $P_{\mathrm{FA}, i}\left({\delta }_i\right)$ to the parameters determined by the desired target detection performance. Then, using the obtained ${\delta }_i$, we can calculate the value of ${\gamma }_i$ for each radar. In order to obtain the optimal coalition structure and optimal power allocation scheme of the radar networks, this paper uses the coalition game model to solve the problem. In the coalition game, the radar networks reduce the radiated power by maximizing its system utility function under the condition that the predefined SINR requirement is satisfied.

3. Game Theoretic Formulation

In this section, we study LPI-based coalition formation and power allocation for radar networks. The objective is to reduce the total radiated power of the radar networks while guaranteeing a certain target detection requirement in the radar networks. And the coalition game is a powerful tool to solve this problem due to its collective rationality property. Firstly, we define an LPI performance-oriented utility function to evaluate the effect of power allocation and develop it as a coalition game model. Then, we propose a game algorithm for joint coalition formation and power allocation in a multi-target detection scenario. And the existence and uniqueness of the NE is proved through a rigorous mathematical derivation.

3.1. Coalition Game Power Allocation Model

In the multi-target detection scenario, we have to consider radar-target selection, the energy efficiency of the radar networks, and the total system radiated power. Therefore, the power allocation problem of radar networks in a multi-target scenario can be modeled as a coalition game model. However, there is mutual interference between radars within different coalitions, resulting in reduced LPI performance of the radar networks. To solve this problem, a balance can be found between the radiated power and SINR requirements. The game model can be expressed as follows:

(6)$\mathcal{l}=\left(N,S,u\right)$

where N is the set of participants consisting of radars in the radar networks, S is the set of strategies of the participants, and u is the set of the utility function of the radar networks.

The interactions of radars in the radar networks can be modeled as a coalition game. Radars in a game with conflicting interests will act in a selfish and rational manner to maximize their own utility functions. This will inevitably lead to mutual interference between different radars, which in turn will degrade the LPI performance of the radar networks. To solve this problem, it is necessary to find a balanced relationship between radiated power and target detection performance requirements, and the power allocation problem is formulated as a coalition game from a game-theoretic perspective.

When using the coalition game, it is important to choose an ideal utility function. The utility function is the basis of game theory, and it is necessary to derive the game algorithm from it. In order to balance the radiated power and target detection performance of the radar networks, both should be reflected in the utility function. The main goal of the radar networks is to obtain the optimal coalition structure and the optimal radiated power allocation scheme under the condition that the predefined target detection performance requirements are met. Therefore, a new utility function is proposed in this paper as follows:

(7)$v_i\left(p_i,p_{-i}\right)=a_i\ln \left({\gamma }_i-{\gamma }_i^{\mathrm{th}}\right)+\frac{b_i{h}_{i,i}^{\mathrm{t}}}{n+1}\left[\left(n+1\right)p_{\max }-p_i-{\displaystyle \sum _{i=1}^n{p}_i^d}\right]$

where $p_{-i}$ denotes the power vector of all radars except radar i, ${\gamma }_i^{\mathrm{th}}$ denotes the predefined SINR threshold of radar i, and $p_{\max }$ is the maximum radiated power of the radars in the radar networks. ${\displaystyle {\sum }_{i=1}^n{p}_i^d}$ denotes the sum of the radiated power of radar i in the last n iterations. $a_i$ and $b_i$ denote the pricing variables associated with target detection performance and radar radiated power, respectively. From Equation (7), it can be seen that this utility function consists of two components. The first part $a_i\ln \left({\gamma }_i-{\gamma }_i^{\mathrm{th}}\right)$ is the benefit term, and the second part $\frac{b_i{h}_{i,i}^{\mathrm{t}}}{n+1}\left[\left(n+1\right)p_{\max }-p_i-{\displaystyle \sum _{i=1}^n{p}_i^d}\right]$ is the penalty term. It is worth mentioning that $h_{i,i}^{\mathrm{t}}$ is the parameter used to ensure fairness among all radars in the radar networks.

The utility function of the radar networks can be expressed as follows:

(8)$u\left(p_i,p_{-i}\right)={\displaystyle \sum _i^M{\nu }_i\left(p_i,p_{-i}\right)}$

where $M$ denotes the total number of radars in the radar networks.

In order to enable radar transfer between coalitions in a multi-target detection scenario, it is necessary to find out what drives radar transfer. In general, the greater utility gained by the radar in the coalition game is one of the reasons for its transfer. In addition, the total system utility is also an important reason. Therefore, two transfer conditions for the coalition game $\mathcal{l}$ are set for the above properties as follows:

(9)$\left\{\begin{array}{c}{\nu }_i{}^{\prime }\ge {\nu }_i\\ u^{\prime }\ge u\end{array}\right.$

The above equation indicates that the utility ${\nu }_i{}^{\prime }$ obtained after the transfer of radar i is not less than the utility ${\nu }_i$ before the transfer, and the system utility $u^{\prime }$ under the new coalition structure formed after the transfer of radar i is not less than the system utility $u$ before the transfer.

The main objective of this paper is to maximize the energy efficiency of the system by choosing a suitable coalition structure and power allocation scheme. Therefore, considering the predefined SINR requirement and the constraint of maximum radar radiated power, the power allocation model based on the coalition game can be expressed as follows:

(10)$\begin{array}{l}\max u\left(p_i,p_{-i}\right)\\ \mathrm{s}.\mathrm{t}.:\left\{\begin{array}{c}{\gamma }_i^{\mathrm{th}}\le {\gamma }_i\le {\gamma }_{\max },\begin{array}{cc}& \end{array}\hspace{1em}\hspace{1em}\hspace{1em}\forall i\in \mathbb{N}\\ 0\le p_i\le p_{\max },\begin{array}{cc}& \end{array}\hspace{1em}\hspace{1em}\hspace{1em} \forall i\in \mathbb{N}\end{array}\right.\end{array}$

The first constraint indicates that the SINR obtained as a result of the power distribution of each radar should be greater than or equal to a predetermined SINR threshold. The second constraint indicates that the radiated power of each radar is constrained by the maximum radiated power. It is worth pointing out that the pricing variables $a_i$ and $b_i$ are chosen to be very important, when $a_i/b_i\ge 1$, more focused on target detection performance, and when $a_i/b_i<1$, more focused on radiated power control, where ${\gamma }_{\max }$ denotes the upper bound of the SINR. When ${\gamma }_i\le {\gamma }_{\max }$, $b_i$ remains constant; otherwise, $b_i=b_i{\left({\gamma }_i/{\gamma }_i^{\mathrm{th}}\right)}^2$ is adaptively adjusted to reduce the radiant power of the radar by increasing the penalty term for the game participants.

3.2. Coalition Game Power Allocation Algorithm

Here, we propose a game algorithm for joint coalition formation and power allocation under a multi-target detection scenario. To obtain the NE solution, the first-order derivative of $u$ with respect to $p_i$ is found:

(11)$\frac{\partial u\left(p_i,p_{-i}\right)}{\partial p_i}=\frac{a_i}{{\gamma }_i-{\gamma }_i^{\mathrm{th}}}\frac{h_{i,i}^{\mathrm{t}}}{I_{-i}}-\frac{b_i{h}_{i,i}^{\mathrm{t}}}{n+1}$

Then, let the first-order derivative of $u$ with respect to $p_i$ be 0; we have

(12)$\frac{a_i}{{\gamma }_i-{\gamma }_i^{\mathrm{th}}}\frac{h_{i,i}^{\mathrm{t}}}{I_{-i}}=\frac{b_i{h}_{i,i}^{\mathrm{t}}}{n+1}$

After a simple algebraic operation, we have

(13)${\gamma }_i={\gamma }_i^{\mathrm{th}}+\frac{a_i(n+1)}{b_i{I}_{-i}}$

Bringing ${\gamma }_i=h_{i,i}^{\mathrm{t}}{p}_i/I_{-i}$ into Equation (13), we are able to obtain the optimal solution for the radiated power of radar i for the current coalition structure as follows:

(14)$p_i=\frac{I_{-i}}{h_{i,i}^{\mathrm{t}}}{\gamma }_i^{\mathrm{th}}+\frac{a_i(n+1)}{b_i{h}_{i,i}^{\mathrm{t}}}$

Clearly, according to Equations (5) and (14), Equation (14) can be used to obtain $p_i$ by iteration, as follows:

(15)$p_i^{(\mathrm{ite}+1)}=\frac{p_i^{(\mathrm{ite})}}{{\gamma }_i^{\left(\mathrm{ite}\right)}}{\gamma }_i^{\mathrm{th}}+\frac{a_i^{(\mathrm{ite})}(n+1)}{b_i^{(\mathrm{ite})}{h}_{i,i}^{\mathrm{t}}}\triangleq ℝ(p_i^{(\mathrm{ite})})$

where ite denotes the number of iterations. On the basis of Equation (15), the expression for the radiated power of radar i can be expressed as

(16)$p_i^{(\mathrm{ite}+1)}=\left\{\begin{array}{cc}ℝ(p_i^{(\mathrm{ite})}),\hfill & 0<p_i^{(\mathrm{ite}+1)}<p_{\max },\\ p_{\max },\hspace{1em}\hfill & p_i^{(\mathrm{ite}+1)}>p_{\max }.\end{array}\right.$

The algorithmic flow of the proposed coalition game power allocation algorithm is summarized in Algorithm 1, as follows.

Algorithm 1: Joint coalition formation and power allocation

1: Initialization: All radars in the radar networks select a random target for detection and record the current coalition structure as ${\pi }_N$. Initialize the parameters SINR threshold ${\gamma }_i^{\mathrm{th}}$ and radiated power $p_i\left(0\right)$, as well as $a_i$ and $b_i$. Make the iteration coefficients $\mathrm{ite}=1$, and $\epsilon >0$ ($\epsilon$ is a very small positive constant) to obtain the corresponding channel gain coefficients.2: for $i=1,\cdots ,M$, do   for $\mathrm{ite}=1,\cdots ,{\mathrm{ite}}_{\max }$, do     Calculate $p_i^{(\mathrm{ite})}$ by solving Equation (16);     if ${\gamma }_i^{(\mathrm{ite})}>{\gamma }_i^{\mathrm{th}}$       Adjust the level of $b_i^{(\mathrm{ite}+1)}=b_i^{(\mathrm{ite})}{\left({\gamma }_i^{(\mathrm{ite})}/{\gamma }_i^{\mathrm{th}}\right)}^2$ adaptively;     else       $b_i^{(\mathrm{ite}+1)}$ remains unchanged;     end if   end for   update $p_i^{\ast }\leftarrow p_i^{(\mathrm{ite})}$  Calculate ${\nu }_i$ by solving Equation (7);end for Calculate $u$ by solving Equation (8); 3: The energy efficiency $v_i$ of radar i and the total system energy efficiency $u$ when the coalition structure is ${\pi }_N$ are obtained. 4: while ${\pi }_N$ is unstable, do   for $i=1,\cdots ,M$, do     Record Radar i’s current coalition $C_k$;     for $k^{\prime }=1:Q$, do       if $k^{\prime }\ne k$, then       Repeat step 2 to calculate energy efficiency $v_i{}^{\prime }$ and total system energy efficiency $u^{\prime }$ after radar i specificity to $C_k{}^{\prime }$;         if the conditions of (9) are satisfied, then         $v_i=v_i{}^{\prime }$;         $u=u^{\prime }$;         ${\pi }_N={\pi }_N{}^{\prime }$;         else $v_i$, $u$ and ${\pi }_N$ remain unchanged;         end if       end if     end for  end for end while5: After achieving stable coalition structure ${\pi }_N$, update $p_i^{\ast }\leftarrow p_i^{(\mathrm{ite})}$.

3.3. Existence and Uniqueness of NE

In order to analyze the validity of the proposed power allocation model based on the coalition game, this section introduces the fundamental theorem to prove the existence and uniqueness of the NE solution of this game model.

Theorem 1 (Existence).

The proposed coalition game model $\mathcal{l}$ has at least one NE solution, so the following conditions should be met:

• Radiated power $p_i$ is a non-empty, closed, bounded convex set on the Euclidean space;

• The utility function $v_i$ is a continuous concave function with respect to $p_i$.

Proof of Theorem 1.

Since the value range of $p_i$ is $0\sim p_{\max }$, it is not difficult to see that the power distribution model of the coalition game proposed in this paper satisfies condition 1. And the utility function $v_i$ is a linear function of $p_i$, so it is continuous. Finding the second derivative of $v_i$ with respect to $p_i$ yields.

(17)$\frac{{\partial }^2{v}_i\left(p_i,p_i\right)}{\partial {\left(p_i\right)}^2}=-\frac{a_i{\left(h_{i,i}^{\mathrm{t}}\right)}^2}{{\left[I_{-i}\left({\gamma }_i-{\gamma }_i^{\mathrm{th}}\right)\right]}^2}<0$

Therefore, the utility function $v_i$ is a concave function of $p_i$. So, it satisfies condition 2. This proves the existence of the NE solution in the proposed power distribution model of the coalition game. □

Theorem 2 (Uniqueness).

The NE solution of the game model $\mathcal{l}$ proposed in this paper is unique. The best response strategy function for radar i is $f\left(p_i\right)={\gamma }_i^{\mathrm{th}}\frac{p_i}{{\gamma }_i}+\left(n+1\right)\frac{a_i}{b_i{h}_{i,i}^{\mathrm{t}}}$ , which should satisfy the following conditions:

• Positivity: For $\forall i\in \mathbb{N}$ , $f\left(p_i\right)>0$ ;

• Monotonicity: If $p_i^1>p_i^2$ , $f\left(p_i^1\right)>f\left(p_i^2\right)$ ;

• Scalability: For $\forall \chi >1$ , $\chi f\left(p_i\right)>f\left(\chi p_i\right)$ .

Proof of Theorem 2.

Since $p_i$ is in the range of $0\sim p_{\max }$, it is clear that when $\forall i\in \mathbb{N}$, $f\left(p_i\right)>0$. So $f\left(p_i\right)$ satisfies condition 1. When $p_i^1>p_i^2$, $f\left(p_i^1\right)-f\left(p_i^2\right)=\frac{p_i^1-p_i^2}{{\gamma }_i}{\gamma }_i^{\mathrm{th}}$. Thus, we can obtain $f\left(p_i^1\right)>f\left(p_i^2\right)$. So $f\left(p_i\right)$ satisfies condition 2. When $\forall i\in \mathbb{N}$,

(18)$\chi f\left(p_i\right)-f\left(\chi p_i\right)=\chi \left(\frac{p_i}{{\gamma }_i}{\gamma }_i^{\mathrm{th}}+\left(n+1\right)\frac{a_i}{b_i{h}_{i,i}^{\mathrm{t}}}\right)-\left(\frac{\chi p_i}{{\gamma }_i}{\gamma }_i^{\mathrm{th}}+\left(n+1\right)\frac{a_i}{b_i{h}_{i,i}^{\mathrm{t}}}\right)=\left(n+1\right)\frac{a_i\left(\chi -1\right)}{b_i{h}_{i,i}^{\mathrm{t}}}$

Since $\forall \chi >1$, we can obtain $\chi f\left(p_i\right)-f\left(\chi p_i\right)>0$, or $\chi f\left(p_i\right)>f\left(\chi p_i\right)$. So $f\left(p_i\right)$ satisfies condition 3.

To sum up, $f\left(p_i\right)$ satisfies the above three conditions. This proves the uniqueness of the NE solution in the proposed power distribution model of the coalition game. □

4. Simulation Results

In this paper, the following simulations are conducted to verify the feasibility and effectiveness of the algorithm proposed in this paper. We think about the radar networks consisting of $M=9$ radars and assume that there are $Q=2$ targets to be detected in the scenario. The system parameters are set as follows: environmental noise ${\sigma }^2={10}^{-18} \mathrm{W}$, radar preset SINR thresholds ${\gamma }_i^{\mathrm{th}}=10 \mathrm{dB}$, ${\gamma }_{\max }=11 \mathrm{dB}$, maximum radiant power of the radar $p_{\max }=3000 \mathrm{W}$, $\epsilon ={10}^{-15}$, radar main flap transmit antenna gain and receive antenna gain $G_{\mathrm{t}}=G_{\mathrm{r}}=30 \mathrm{dB}$, radar side flap transmit antenna gain and receive antenna gain $G_{\mathrm{t}}{}^{\prime }=G_{\mathrm{r}}{}^{\prime }=-30 \mathrm{dB}$, pricing variables $a_i=3$ and $b_i={10}^{20}$ associated with target detection performance and the radiated power of the radar. The correlation coefficient between radar i and radar j is $c_{i,j}=0.01 \left(i\ne j\right)$. The miss-detection probability and false alarm probability of radar i are $P_{\mathrm{MD},i}=2.7\times {10}^{-3}$, $P_{\mathrm{FA},i}={10}^{-6}$. In order to analyze the effect of the relative position relationship between the target and each radar in the radar networks and the RCS on the power allocation results and the coalition formation results, the following three scenarios are considered in this section.

4.1. Scenario 1

The initialization of the radar-target selection is shown in Figure 2. The distribution of the RCS for the targets is assumed to be as shown in Figure 3. Other simulation parameters are shown above.

Figure 4 depicts the convergence of the utility function of the radar networks. From the figure, it can be seen that with random initialization of the radar-target selection, the proposed algorithm in this paper can make the utility function of the radar networks converge to the optimal value quickly. This means that the optimal coalition structure can be formed rapidly, and the entire process can be completed in at most M iterations. Figure 5 displays the outcome of the coalition formation when the energy efficiency of the system attains its maximum value.

In order to analyze the convergence performance of the proposed algorithm, Figure 6 plots the convergence performance of the proposed algorithm. From Figure 6a, it can be seen that the algorithm converges to the NE solution in only 6–10 iterative processes. Moreover, since the NE solution of the game model proposed in this paper is unique, the proposed algorithm will converge to this NE solution independent of the initial radiated power value. It is obvious from Figure 6b that the obtained SINR converges quickly after very few iterations of computation, satisfying the predefined SINR requirement. In addition, several curves in the figure almost overlap, which indicates that the algorithm is able to satisfy the SINR requirements for each radar. This also confirms that the coalition game model can guarantee fairness among all radars in the radar networks.

As shown in Figure 7, to further demonstrate the superiority of the proposed power allocation algorithm, the total radiated power of the system and the SINR of each radar obtained from this algorithm are compared with two other power allocation algorithms: the non-cooperative power allocation (NCPA) algorithm in [34] and the Koskie and Gajic’s (K-G) algorithm in [35]. It is worth pointing out that the algorithm proposed in this paper obtains the maximum system energy efficiency under the current coalition structure and effectively reduces the total radiated power of the system while meeting the preset SINR requirement of each radar in the radar networks. Although the K-G algorithm requires the least amount of radiated power, the SINR obtained by the K-G algorithm is not ideal to meet the requirements of target detection performance. This is because the K-G algorithm is biased toward minimizing the radiated power of the radar without considering the desired target detection performance. In particular, as the radiated power pricing variable increases, both the target detection performance and the radiated power consumption decrease. As a result, the predefined SINR requirements for each radar are difficult to meet. On the other hand, the NCPA algorithm requires a large radiated power due to the non-cooperative behavior of the radars during the gaming process. Specifically, in the non-cooperative game scenario, each radar is more concerned about its own utility function and maximizes its own utility function as much as possible. In turn, the utility function of other radars in the radar networks is reduced, and the radiated power of each radar is increased in order to meet the preset SINR requirement, which leads to higher total radiated power of the radar networks. In turn, this leads to a reduction in the LPI performance of the radar networks. This further demonstrates the advantage of using the proposed algorithm to improve LPI performance in radar networks.

From Figure 7b, we can see that the SINR values of both the algorithm proposed in this paper and the NCPA algorithm can satisfy the predefined target detection performance requirements, but the SINR values of the K-G algorithm are not satisfactory, and the SINR of most of the radars is below the SINR threshold. We can see that the algorithm proposed in this paper is satisfying the predefined SINR requirements while the SINR of each radar is converging toward the SINR threshold as much as possible, which indicates that our proposed algorithm can overcome the near–far effect. Overall, as mentioned earlier, the K-G algorithm requires the least radiated power, but it cannot meet the requirements of the target detection performance required by the radars. On the other hand, the NCPA algorithm can meet the requirements of the predefined target detection SINR performance, but the required radiated power is higher, and the resulting SINR is slightly smaller than that of the algorithm proposed in this paper. Overall, these results show that the proposed algorithm not only satisfies all the radars’ predefined SINR requirements but also obtains the optimal coalition structure and improves the LPI performance of the radar networks.

4.2. Scenario 2

The initialization of the radar-target selection is shown in Figure 8. The distribution of the RCS for the targets is assumed to be as shown in Figure 9. Other simulation parameters are the same as in scenario 1.

Figure 10 depicts the convergence of the utility function of the radar networks. From the figure, it can be seen that the algorithm proposed in this paper can still make the utility function of the radar networks converge to the optimal value rapidly with changing the radar target location distribution and randomly initializing the radar-target selection, and the whole process can be completed in only four iterations. It further illustrates the low complexity and robustness of the proposed algorithm in this paper. Figure 11 shows the result of the coalition formation when the energy efficiency of the system reaches its maximum value. In addition to this, it can be seen from Figure 11 that Radar 2, Radar 5, and Radar 8 have the same distance to Target 1 and Target 2, while the coalition attribution for these three radars is not exactly the same. Comparing Figure 9, it can be found that although the radars have the same distance to both targets, there are differences in the RCS for different targets. The comparative analysis reveals that when the distances between radars and different targets are the same, the radars will prefer the target with the larger RCS.

Figure 12 plots the convergence of the algorithm proposed in this paper under scenario 2. From Figure 12a, it can be seen that the algorithm converges to the NE solution in only 6–10 iterative processes. As can be seen in Figure 12b, the SINR obtained in this scenario can still converge quickly and satisfy the predefined SINR requirement after very few iterations of computation. Moreover, several curves in the figure almost overlap, which indicates that the algorithm is able to satisfy the SINR requirement for each radar. This also further confirms that the coalition game model can guarantee fairness among all radars in the radar networks and exclude experimental chance.

Figure 13 shows the results of comparing the total radiated power of the system and SINR of each radar obtained by this algorithm with two other power allocation algorithms in the case of scenario 2. It is worth pointing out that the proposed algorithm in this paper still obtains the maximum system energy efficiency under the current coalition structure after the change in radar and target position distribution, effectively reducing the total radiated power of the system while satisfying the predefined SINR requirement for each radar in the radar networks. The SINR obtained by the K-G algorithm is still unsatisfactory and cannot satisfy the predefined target detection SINR performance requirement. On the other hand, the NCPA algorithm still requires a higher radiated power than the proposed algorithm, and the resulting SINR is still lower than the proposed algorithm. This proves the superiority and effectiveness of the algorithm proposed in this paper. Therefore, the algorithm proposed in this paper can not only guarantee the intended target detection SINR performance requirements of all radars but also obtain the optimal coalition structure and effectively improve the LPI performance of the radar networks.

4.3. Scenario 3

The initialization of the radar-target selection is shown in Figure 14 to further analyze the impact of the target RCS on the coalition structure. The distribution of the RCS for the targets is assumed to be as shown in Figure 15. Other simulation parameters are the same as in scenario 1.

Figure 16 depicts the convergence of the utility function of the radar networks. It can be seen from the figure that, compared to scenario 2, the utility function of the radar networks can still be made to converge to the optimal value rapidly with the change in the target RCS, and the whole process can be completed in only four iterations. This further confirms the low complexity and robustness of the algorithm proposed in this paper. Figure 17 shows the results of the coalition formation when the energy efficiency of the system reaches its maximum value. In addition to this, it can be seen from Figure 15 that Radar 2, Radar 5, and Radar 8 are selected for the larger RCS target 2. This further proves the conclusion in scenario 2. That is, when the distances between the radar and different targets are the same, the radar will tend to favor the target with a larger RCS.

Figure 18 shows the convergence of the algorithm proposed in this paper under scenario 3. Combining the analysis of the results in scenario 1 and scenario 2, it can be seen that the proposed algorithm can converge quickly in terms of radiated power and the SINR and meet the requirements of target detection performance of all radars. The stability and effectiveness of the algorithm are also demonstrated by the simulation experiments in several scenarios. In addition to this, the analysis combined with Figure 15, Figure 17 and Figure 18a shows that within the same coalition, with the same distance from the target and the same RCS, the closer the radar is to the other coalitions, the higher the radiated power required, and at a distance farther away, the lower the radiated power required. This is because the radars of other coalitions will cause some interference with the radars in the current coalition, and the interference is higher when the distance is close, so the radars closer to the other coalitions need higher radiated power.

Figure 19 shows the results of the total radiated power of the system and the SINR of each radar obtained by this algorithm in comparison with the other two power allocation algorithms in the case of scenario 3. It can be found that the results obtained by the algorithm proposed in this paper are still optimal.

The superiority and effectiveness of the algorithm proposed in this paper are proved by analyzing the results obtained by the algorithm proposed in this paper with those obtained by the other two algorithms under different scenarios. Therefore, the algorithm proposed in this paper can not only guarantee the predefined target detection SINR performance requirements of all radars but can also obtain the optimal coalition structure and effectively improve the LPI performance of radar networks.

5. Concluding Remarks

The purpose of the algorithm proposed in this paper is to enhance the LPI performance of the radar networks by reducing the radiated power of the radars while ensuring that the predefined SINR threshold and power constraints are satisfied. In addition, a mathematical model of power allocation for the radar networks based on the coalition game theory is established, with the optimization objectives of maximizing the system energy efficiency and achieving optimal power allocation, subject to the constraint of target detection performance. Based on this, a game algorithm for joint power allocation and coalition formation is proposed to solve the optimization problem, and the optimal coalition structure and power allocation scheme are obtained. Furthermore, we rigorously prove the existence and uniqueness of the NE solution. Simulation results demonstrate that the proposed algorithm can maximize system energy efficiency and effectively reduce the total radiated power of the system, thus improving its LPI performance while satisfying the predefined SINR performance requirement in a multi-target detection scenario. The next research work will focus on the multi-domain resource allocation strategy of radar networks based on the coalition game in multi-target tracking scenarios. In the case of multi-target tracking, the radar-target selection situation needs to be updated in real time; therefore, how to reduce the algorithm complexity to meet the requirement of timeliness is also a very important part of future research work.

Author Contributions

X.D., C.S. and Z.W. conceived and designed the experiments; X.D., C.S. and Z.W. performed the experiments; J.Z. analyzed the data; X.D. wrote the paper; J.Z. contributed to data analysis revision; J.Z. contributed to English language corrections. All authors of the article provided substantive comments. All authors have read and agreed to the published version of the manuscript.

Data Availability Statement

Not applicable.

Conflicts of Interest

The authors declare no conflict of interest.

Footnotes

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

Figures

Figure 1. System model.

Figure 2. Radar-target selection initialization.

Figure 3. The distribution of RCS for the targets.

Figure 4. The convergence of the utility function.

Figure 5. Coalition formation result.

Figure 6. The convergence of the proposed algorithm: (a) transmit power allocation, (b) the SINR of each radar.

Figure 7. Comparisons with other power allocation algorithms: (a) total transmit power, (b) the achieved SINR of each radar.

Figure 8. Radar-target selection initialization.

Figure 9. The distribution of RCS for the targets.

Figure 10. The convergence of the utility function.

Figure 11. Coalition formation result.

Figure 12. The convergence of the proposed algorithm: (a) transmit power allocation, (b) the SINR of each radar.

Figure 13. Comparisons with other power allocation algorithms: (a) total transmit power, (b) the achieved SINR of each radar.

Figure 14. Radar-target selection initialization.

Figure 15. The distribution of RCS for the targets.

Figure 16. The convergence of the utility function.

Figure 17. Coalition formation result.

Figure 18. The convergence of the proposed algorithm: (a) transmit power allocation, (b) the SINR of each radar.

Figure 19. Comparisons with other power allocation algorithms: (a) total transmit power, (b) the achieved SINR of each radar.

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