A Mathematical Modeling Approach of Target Damage Strategy on the Intersection Confrontation

Abstract

The efficiency of target damage is to measure the effect of projectile attacks. Because there is antagonism at the intersection of projectiles and targets, it is very difficult to establish scientific damage assessment strategy model and a numerical calculation method. To scientifically evaluate the target damage effect when the projectile attacks the aircraft target, this paper introduces a game confrontation mechanism and proposes a numerical calculation method of aircraft target damage game strategy under the intersection confrontation. According to the theory of two-person non-cooperative games, the projectile is regarded as one of the players in the confrontation damage game, and the aircraft target is regarded as the other player. The damage gain model and gain function of the intersection confrontation of both sides are established. The effective expectation function and the minimum and maximum theorem are used to obtain the optimal Nash equilibrium solution of the game matrix strategy, and the Nash equilibrium point of the strategy space set is derived. Through calculation and comparative analysis, the results show that the optimal gain degree composed of the damage strategy selected by both the projectile and aircraft target under the Nash equilibrium solution brings the greatest gain. The proposed hybrid game strategy of damage assessment can be an effective reference for air combat decision making in the state of intersection confrontation.

1. Introduction

In a space confrontation between missiles and aircraft targets, the target damage effectiveness that is caused by a missile intercepting an aircraft target evaluates the power effectiveness of the missile, which is not only a research focus of weapon test evaluation but also a difficult problem in the target damage test between the missile and aircraft target. In the existing research on aircraft target damage assessment methods, some studies have introduced the calculation method of aircraft target damage caused by warhead fragments, such as the damage effect calculation method of the blast-fragmentation warhead against the ground target based on the shot-line model [1], the damage assessment method of airplane targets by defining the damage rules for aircraft target components [2], the damage assessment method based on adaptive neuro-fuzzy inference [3] and the method of battle damage assessment under the uncertain environment based on the subpixel morphological anti-aliasing (SMAA) technique [4]. Otherwise, Tian et al. [5] researched the fuze explosion point effect on ammunition damage assessment; Fu et al. [6] used the fragment centroid tracking method to establish the calculation model of fragment distribution density and killing area and researched a comprehensive performance analysis system for the numerical simulation of a blast-fragmentation warhead power field; Xu et al. [7] established the equivalent damage of an energetic fragment warhead to a target by examining the coverage area ratio function of an equivalent target struck by a warhead fragment in conjunction with the missile’s function, structure and damage mechanism and analyzed the equivalent damage of an energetic fragment warhead to a missile target. In the calculation of target damage in the existing literature, only the state of the fragment hitting the target itself is considered; it is passive damage. In fact, in the process of missile explosion damaging an aircraft target, there is not only the passive damage of the aircraft target but also the guidance system of the missile itself and the detection system of the aircraft target, which are important factors in damage evaluation. When a missile attacks an aircraft target, it uses its detection guidance function to form a warhead fragment power field to damage the aircraft target under the condition of satisfying the missile’s explosion condition, which makes the aircraft target lose the combat capability. On the other hand, the aircraft target has a recognition system; it can change its flight path based on the attitude of the incoming missile. Therefore, the damage efficiency formed by the intersection of the missile and the aircraft target can be regarded as their antagonistic damage in space, so it is possible to form the game damage of the missile and the aircraft target by using game theory.

In the confrontation damage game, the projectile (missile) and the aircraft target (damage target) are regarded as two opposing players; a two-person non-cooperative game function is used to establish the damage game’s strategy model with the target’s damage factor and damage degree. Considering the strategic combination of the projectile and the aircraft target, the gain function of both parties involved in the game can be obtained. At the same time, according to the different vulnerability weights of the aircraft target, the aircraft target is divided into multiple damage cabin compartments, and the area ratio of each damage cabin compartment covered by warhead fragments is used to obtain the gain matrix’s parameters. According to the damage mixed strategy of the two-person non-cooperative game of the projectile and the aircraft target, the mathematical expectation of the players in the game model is calculated to obtain the Nash equilibrium solution of the mixed strategy.

At present, there are few studies on the game damage of projectiles and aircraft targets in terms of the antagonism game strategy research. There are indeed many relevant reports and descriptions in some studies—for example, the land defense weapon and the target assignment [8] using game theory to optimize decision making [9], the UAV offensive/defensive game [10,11], multiple UCAVs in the air-to-ground attack game [12], the IADS non-cooperative game [13], the dynamic game with incomplete information [14], the underwater game based on cooperative confrontation [15], a method of analysis of the game equilibrium of the frog-man carrier anti-destruction performance [16] and so on. However, the game relationship between the projectile and the aircraft target is not a conventional ideal state game confrontation relationship, so it cannot be directly calculated by the existing mature game strategy model. Because the control and discrimination thinking of the projectile is not very sufficient, and many judgment and identification parameters are uncertain and incomplete in numerical terms, how to effectively calculate the attack and damage effect of the aircraft target from the damage parameters and warhead fragments by the projectile itself is indeed a very difficult problem.

There are very few studies on the game strategy between projectiles and aircraft targets, mainly because smart missiles have less of a representation of antagonistic factors formed by themselves. Meanwhile, their ability to discriminate surrounding targets is not as obvious as that of UAVs, and many antagonistic information is vague. Based on the current demand for the damage effectiveness of air defense projectiles to intercept incoming aircraft targets, this paper attempts to form a game relation between projectiles and aircraft targets, explores a new calculation method for the effective damage of projectiles to aircraft targets and sets up a numerical calculation method of aircraft target damage game strategy on the intersection confrontation between projectiles and aircraft targets. The main contributions of this work are as follows:

(1) According to the principle of two-person non-cooperative games, the projectile is regarded as one of the players in the confrontation damage game, and the aircraft target (damage target) is regarded as another player in the confrontation damage game. A damage game strategy model based on the intersection confrontation of the projectile and the aircraft target has been set up. In the model, the warhead fragment formed by the projectile explosion hitting the aircraft target is taken as the gain of the projectile participant or the loss of the aircraft target participant. By combining the game relationship between the projectile and the aircraft target, the gain matrix function based on their strategies is formed.

(2) The damage strategy problem of projectiles and aircraft target games is boiled down to solve the Nash equilibrium solution of the payoff function in the confrontation game. By taking the warhead fragment attacking the aircraft target surface effectively as the gain element, the aircraft target damage gain matrix has been established.

(3) According to the damage mixed strategy of the two-person non-cooperative game of the projectile and the aircraft target, the mathematical expectation of the players in the game model has been calculated to obtain the Nash equilibrium solution of the mixed strategy.

The remainder of this paper is organized as follows. Section 2 states the damage game basic model on the projectile and aircraft target based on the principle of two-person non-cooperative games. Section 3 states the payoff matrix of the confrontation game between the projectile and the aircraft target based on the strategy combination of both parties. Section 4 states the calculation method of the Nash equilibrium solution. Section 5 gives the calculation and analysis. Finally, Section 6 concludes this paper.

2. Target Damage Game Strategy Modeling Based on the Intersection Confrontation of Projectiles and Aircraft Targets

Regarding the target damage assessment on the intersection of the projectile and the aircraft target, there are two main features: one is that, when the projectile finds the aircraft target, the control and guidance system of the projectile are required to complete the task of explosion; another is that the warhead fragments formed by the projectile explosion must have an intersection with the aircraft target. These two characteristics are the prerequisites for the aircraft target being destroyed. Figure 1 is the intersection state of the projectile and the aircraft target.

According to the Figure 1, both the projectile and the aircraft target have the characteristics of active attack; $\theta$ is the intersection angle between the projectile and the aircraft target. $K_1$ presents the projectile damage side of the damage game players; $K_2$ presents the aircraft target confrontation side.

If the projectile wants to cause the greatest damage to the aircraft, it needs to consider its position, the intersection angle, the flying attitude and other factors relative to the aircraft target. The minimum loss of the aircraft target is related to its ability to sense the movement of the incoming projectile and make a quick response. Therefore, the projectile and the aircraft target have the characteristics of confrontation between the two sides. Due to the uncertainty of the intersection of the projectile and the aircraft target, the explosive position of the projectile is not fixed, and the number of warhead fragments attacking the aircraft target is not very certain, which makes it more difficult to directly calculate the damage result. In order to express the damage effect more comprehensively, we propose a calculation method of game damage based on the confrontation features of the projectile and the aircraft target under the state of intersection.

This paper regards the projectile as one of the players in the confrontation damage game and the aircraft target as the other player in the confrontation damage game, and it takes the warhead fragments that cannot hit the aircraft target as the loss of the participant of the projectile. The warhead fragments that penetrate the aircraft target are taken as the gain. On the contrary, the gain of the aircraft target is defined such that the warhead fragments do not hit it. When the warhead fragments hit the aircraft target, the aircraft target will incur losses. Under the definition of gains and losses for projectiles and aircraft targets, a two-person non-cooperative game function [17,18,19] is used to establish a damage game strategy model with the target’s damage factor and damage degree. Formula (1) is their basic game model.

$$G=\left\{K_1,K_2;M_1,M_2;u_1,u_2\right\}$$

(1)

where $M_1$ is a set of strategies to minimize projectile losses so as to maximize benefits under the condition that the projectiles damage the aircraft target; $M_2$ is a set of strategies to minimize the loss caused by the damage under the condition that the projectile confront the aircraft target; $u_1$ is the corresponding gain set for all the damage strategies of the projectile side; $u_2$ is the corresponding gain set for all the confrontation strategies of the aircraft target side. The main factors of the projectile and the aircraft target intersection confrontation damage are composed of the damage factor and the damage degree strategy set.

Suppose $\left\{H_1,H_2,\cdots ,H_i,\cdots ,H_n\right\}, i=1,2\cdots ,n$ is the set of the projectile, $x_i$ indicating that the projectile damages the aircraft target. When $x_i=1$, it indicates that the aircraft target is damaged by the projectile. $x_i=0$ indicates that the aircraft target is not damaged by the projectile. The set of aircraft target segments is $\{A_1,A_2,\cdots ,A_j,\cdots ,A_m\}, j=1,2\cdots ,m$. $y_j$, indicating that the aircraft target eludes projectile damage if $y_j=1$, which indicates that the aircraft target succeeds in confronting. $y_j=0$ indicates that the aircraft target fails to confront. From the basic game model, the main purpose of the damage game strategy on the projectile and the aircraft target is to set up a damage strategy set and a non-cooperative game strategy set. The problem of the damage game strategy focuses on forming the gain matrix and finding the gain matrix elements, and then, the fundamental problem of the confrontation game is to solve the Nash equilibrium solution of the confrontation game payoff function matrix according to the confrontation gain brought by the different strategies of the projectile and the aircraft target.

Considering the strategic combination of the projectile and the aircraft target, we obtain the gain function of both parties involved in the game. At the same time, according to the different vulnerability weights of the aircraft target, the aircraft target is divided into multiple damage cabin compartments, and the area ratio of each damage cabin compartment covered by warhead fragments is used to obtain the gain matrix’s parameters. Based on the damage mixed strategy of the two-person non-cooperative game of the projectile and the aircraft target, the mathematical expectation of the players in the game model obtaining the Nash equilibrium solution of the mixed strategy can be calculated.

3. The Establishment of the Damage Game Gain Function Model

For the damage game formed in the state where the projection and the aircraft target intersect, it is not exactly the same as the conventional game. The damage characteristics are also different in $\{A_1,A_2,\cdots ,A_j,\cdots ,A_m\}$. In order to simplify the equilibrium solution, the same damage weight calculation method for each cabin segment on the aircraft target is used. According to the two-person non-cooperative game, the change of any strategic action by which the players involved in the game seek the maximization of the gain will change the choice of the other party’s strategic action and will lead to the change of the overall relative gain of the game, which will eventually affect the gain of both players [20,21,22,23,24]. Treat each intersection state as a generalized strategy according to different warhead fragment parameters and different intersection attitude parameters. A damage game strategy set can be formed. The key point of the confrontation game between the two sides (the projectile and the aircraft target) is to calculate the difference of the damage gain generated by the two sides after the game and to construct the damage gain function of the confrontation game between the two sides based on the vulnerable segments of the aircraft target.

The gain function of the two-person non-cooperative game $\left\{M_{n-pay},M_{m-pay}\right\}$ can be expressed by Formula (2).

$$\{\begin{array}{l}{M}_{n-pay}=N_n\left(G_H{e}^m-{\sigma }_e{\omega }_b\right)-N_m\left[\left(1-{\sigma }_e\right){\omega }_b-G_H{e}^m\right]\\ M_{m-pay}=N_m\left({\omega }_b-G_H{e}^m\right)-N_n\left(G_H{e}^m-{\omega }_p\right)\end{array}$$

(2)

where $N_n$ is the projectile countermeasure decision variable, $N_m$ is the aircraft target countermeasure decision variable, $G_H$ is the damage probability under the intersection of the projectile and the aircraft target, ${\omega }_p$ is the combat warhead fragment gain, ${\sigma }_e$ is the reliability of the projectile’s task execution, $e^m$ is the damage gain of the aircraft target vulnerable segment [25,26] and ${\omega }_p={\sigma }_e\cdot {\omega }_b$, ${\omega }_b$ is the damage gain of the projectiles, which is obtained by combing with the guidance accuracy of the weapon system and the explosion probability of the projectile fuze; it is expressed by using Formula (3).

$${\omega }_b= \frac{P_{zd} \cdot P_{yz}}{l_d} \cdot J$$

(3)

where $J$ is the final complex correlation aggregation empirical constant, $l_d$ is the final complex correlation degree, $P_{zd}$ is the guidance accuracy and $P_{yz}$ is the explosion probability of the projectile fuze.

The damage gain of the aircraft target vulnerable segment $e^m$ is the core variable of the damage game strategy, and it is related to the damage weight of the aircraft target, the penetration area ratio of the warhead fragments and the damage vulnerability coefficient of the aircraft target. $e^m$ can be expressed by Formula (4).

$$e^m= a{K}_{EMF}\frac{S_{CM}{S}_{BM}}{S_{MAX}^2}$$

(4)

where $S_{CM}$ is the damage area of the cabin segment of the aircraft target, $S_{MAX}$ is the maximum cross-sectional area of the aircraft target, $S_{BM}$ is the damage area of the internal parts of the cabin segment of the aircraft target, $K_{EMF}$ is the joint physical quantity of the maneuver efficiency and endurance efficiency on the aircraft target and $a$ is the damage coefficient of the aircraft target [27].

In the state of the intersection confrontation and correlation of the projectile and the target, the final damage probability is comprehensively affected by the number of fragments, the damage velocity and the damage incidence angle. Supposing $G_H$ is the damage probability under the intersection confrontation of the projectile and the aircraft target, it can be obtained by Formula (5).

$$\begin{array}{l}{G}_H=G_{LT} \times G_K\\ ={\displaystyle \underset{-\infty }{\overset{+\infty }{\int }}{\displaystyle \underset{-\infty }{\overset{+\infty }{\int }}{\displaystyle \underset{-\infty }{\overset{+\infty }{\int }}\left\{f_{{\phi }_{mt}}\left(x,y,z\right)f_{v_r}\left(x,y,z\right)f_{{\tilde{m}}_p}\left(x,y,z\right)\times \frac{e\left(x,y,z\right)\overline{\Delta N}{D}^2{v}_B\sin \phi \sin \alpha }{\rho \left(\phi ,\xi \right)N_W} \right\}}}}dxdydz\end{array}$$

(5)

where $G_{LT}$ is the intersection confrontation state model with the target coordinate damage probability as the intersection confrontation condition, $G_{LT}=f({\phi }_{mt},v_r,{\tilde{m}}_p)$. Among them, ${\phi }_{mt}$ is the intersection confrontation angle, $v_r$ is the intersection confrontation velocity and ${\tilde{m}}_p$ is the miss parameter [28]. $G_K$ is the damage model of the warhead fragments against the target, which is related to the flow density $e\left(x,y,z\right)$ of the warhead fragments under the intersection confrontation, the average number of warhead fragments of the damaged target $\overline{\Delta N}$ [29], the explosion point distance D, the target entry velocity $v_B$, the damage incidence angle $\alpha$, the total number of warhead fragments $N_W$ and the intersection countermeasure fragment flow density $\rho \left(\phi ,\xi \right)$, which is determined by the inclination $\phi$ and the azimuth $\xi$.

Take the aircraft target as a cylinder and divide it into m equal unit cabin compartments, as shown in Figure 2.

The damage gain for the whole aircraft target is superimposed by the damage gain of m equal unit cabin compartments. Then, the damage gain for the whole aircraft target can be obtained by Formula (6).

$$E_{total}=\frac{a{K}_{EMF}}{S_{MAX}^2}{\displaystyle \sum _{j=1}^m{S}_j^{DM}}$$

(6)

where $S_j^{DM}$ is the damage area of the j-th unit cabin compartment.

Based on Formulas (2), (3) and (6), the damage gain function of the two sides involved in the game can be obtained by Formula (7).

$$\{\begin{array}{l}{M}_{n-pay}=\frac{\left(N_m{\sigma }_e-N_n{\sigma }_e-N_m\right){\sigma }_e{P}_{zd}{P}_{yz}J}{l_d}+\frac{a\left(N_n+N_m\right)G_H{K}_{EMF}{S}_{DM}}{S_{FMAX}^2}\\ M_{m-tpay}=\frac{\left(N_m+N_n{\sigma }_e\right)P_{zd}{P}_{yz}J}{l_d}-\frac{a\left(N_n+N_n\right)G_H{K}_{EMF}{S}_{DM}}{S_{MAX}^2}\end{array}$$

(7)

The countermeasure decision variable of the projectile participant $N_n$ is a binary decision. Under the condition of the intersection confrontation of the projectile and the aircraft target, when the decision value variable of the projectile participant is 1, it indicates that the projectile must have caused damage to the aircraft target. On the contrary, when the decision variable value of projectile participant is 0, there is no damage and no confrontation between the projectile and the aircraft target.

The countermeasure decision variable of the aircraft target participant N_m is also a binary decision. When the decision value variable of the aircraft target side is 1, it indicates that the aircraft target and the projectile side have intersected and that the projectile and the aircraft target have formed confrontation damage. When the decision value variable of the aircraft target side is 0, it indicates that the target does not intersect with the projectile side or that the aircraft target intersects with the projectile but does not form confrontation damage with the projectile.

In the damage game strategy of the intersection of the projectile and the aircraft target, the damage gain is not only reflected by the total damage area of the aircraft target, but it is also related to the countermeasure decision variable of the projectile and the countermeasure decision variable of the aircraft target. The game gain matrix is established by using the adopted strategies that both sides approved, which is a strategic representation of the simultaneous actions of the players in a game. It can be expressed by Formula (8).

$$\begin{array}{r}\begin{array}{cc}\begin{array}{cc}& {\gamma }_1\end{array}& \end{array} \begin{array}{cccccc}\begin{array}{cc}& \end{array}\begin{array}{c}\begin{array}{cc}& {\gamma }_2\end{array}\end{array}& \begin{array}{c}\begin{array}{cc}& \cdots \end{array}\end{array}& & \begin{array}{c}{\gamma }_h\end{array}\begin{array}{c}\begin{array}{cc}& \end{array}\end{array}& \cdots & \begin{array}{c}{\gamma }_n\end{array}\end{array}\begin{array}{c}\begin{array}{cc}& \end{array}\end{array}\\ h_{\lambda \gamma }=\begin{array}{c}{\lambda }_1\\ {\lambda }_2\\ ⋮\\ {\lambda }_k\\ ⋮\\ {\lambda }_m\end{array}\left[\begin{array}{cccccc}\left(x_{11},y_{11}\right)& \left(x_{12},y_{12}\right)& \cdots & \left(x_{1h},y_{1h}\right)& \cdots & \left(x_{1n},y_{1n}\right)\\ \left(x_{21},y_{21}\right)& \left(x_{22},y_{22}\right)& \cdots & \left(x_{2h},y_{2h}\right)& \cdots & \left(x_{2n},y_{2n}\right)\\ \begin{array}{cc}& ⋮\end{array}& \begin{array}{cc}& ⋮\end{array}& & \begin{array}{cc}& ⋮\end{array}& & \begin{array}{cc}& ⋮\end{array}\\ \left(x_{k1},y_{k1}\right)& \left(x_{k2},y_{k2}\right)& \cdots & \left(x_{kh},y_{kh}\right)& \cdots & \left(x_{kn},y_{kn}\right)\\ \begin{array}{cc}& ⋮\end{array}& \begin{array}{cc}& ⋮\end{array}& & \begin{array}{cc}& ⋮\end{array}& & \begin{array}{cc}& ⋮\end{array}\\ \left(x_{m1},y_{m1}\right)& \left(x_{m2},y_{m2}\right)& \cdots & \left(x_{mh},y_{mh}\right)& \cdots & \left(x_{mn},y_{mn}\right)\end{array}\right]\end{array}$$

(8)

where $\left\{{\lambda }_1,{\lambda }_2,\cdots ,{\lambda }_m\right\}$ is the confrontation damage strategy set of the aircraft target, $s_1=\left\{{\lambda }_1,{\lambda }_2,\cdots ,{\lambda }_m\right\}$; $\left\{{\gamma }_1,{\gamma }_2,\cdots ,{\gamma }_n\right\}$ is the damage strategy set of the projectile damages, $s_2=\left\{{\gamma }_1,{\gamma }_2,\cdots ,{\gamma }_n\right\}$. $\left\{x_{1n},x_{2n},\cdots ,x_{mn}\right\}$ indicates the damage gain of the vulnerable cabin compartments when the projectile adopts the strategies of $({\lambda }_1,{\lambda }_2,\cdots ,{\lambda }_m)$ under the ${\lambda }_k$-th strategy. $\left\{y_{m1},y_{m2},\cdots ,y_{mn}\right\}$ indicates the damage gain of the projectile when the aircraft target adopts the strategies of $({\gamma }_1,{\gamma }_2,\cdots ,{\gamma }_{\mathrm{n}})$ under the ${\gamma }_h$-th strategy. $(x_{kh},y_{kh})$ refers to the game gain of the confrontation between the projectile and the aircraft target when the target adopts the pure strategy ${\lambda }_k$ and the projectile adopts the pure strategy γ_h Each line vector of the gain matrix is a corresponding strategy of the aircraft target. Each column vector is a corresponding strategy of the warhead fragments that is formed by the projectile explosion, which contains the number and velocity of warhead fragments that can penetrate the aircraft target, and it is the total set of warhead fragments that cause the aircraft target damage [30,31].

4. Calculation of the Nash Equilibrium Solution of the Damage Game Strategy

For the gain matrix of the damage game strategy under the intersection status of the projectile and the aircraft target, their game relationship is not a conventional ideal state game confrontation relationship, so it cannot be directly calculated by the existing mature game strategy model because the control and discrimination thinking of the projectile is not very sufficient, and many judgment and identification parameters are uncertain and incomplete in numerical terms. So, the strategies of both players have uncertainty and random selectivity, and the gain matrix has no solution in a purely strategic sense.

Considering the strategic combination of the projectile and the aircraft target, the gain function of both parties involved in the game can be obtained, and $h_{\lambda \gamma }$ is used to denote the gain matrix. At the same time, according to the different vulnerability weights of the aircraft target, the aircraft target is divided into multiple damage cabin compartments, and the area ratio of each damage cabin compartment covered by warhead fragments is used to obtain the gain matrix’s parameters. It is very obvious that the fundamental problem of the confrontation game is to solve the Nash equilibrium solution of the confrontation game payoff function matrix according to the confrontation benefits brought by the different strategies of the projectile and the aircraft target.

Based on Formulas (7) and (8), the Nash equilibrium solution is obtained in the sense of hybrid strategy based on von Neumann’s minimum–maximum theory [32], which is the most favorable gain solution of the game strategy. This is also the difference between the hybrid game strategy and the existing strategy model. In addition, the characteristics of the hybrid game are used to find the incomplete information of the players involved, and the hybrid game strategy is determined by using the expected utility function, which is also the optimal damage of the projectile to the aircraft target.

The actual game shows that the players (the projectile and the aircraft target) randomly choose different strategies with a certain probability distribution under the condition of complete information. At this time, the game strategy turns into a hybrid game strategy. Then, the two-person non-cooperative game damage hybrid game strategy can be obtained. Consider ${\mathfrak{A}}_{+}^c$ as the nonnegative orthant of a c-dimensional Euclidean space. Let ${\epsilon }_l$ be the probability that the projectile participant selects the strategy of ${\lambda }_k$. ${\eta }_g$ is the probability that the aircraft target participant selects the strategy of ${\gamma }_h$ for l ∈ ${\Delta }_1$ and g ∈ ${\Delta }_2$, where ${\Delta }_1=\left\{1,2,\cdots ,c\right\}$ and ${\Delta }_2=\left\{1,2,\cdots ,r\right\}$. If ${\displaystyle \sum _{l=1}^c{\epsilon }_l=1}$ and ${\displaystyle \sum _{g=1}^r{\eta }_g=1}$, for $(\epsilon ,\eta )\in {\mathfrak{A}}_{+}^c\times {\mathfrak{A}}_{+}^r$, where $\epsilon =\left({\epsilon }_1,{\epsilon }_2,\cdots ,{\epsilon }_l,\cdots ,{\epsilon }_c\right)$ and $\eta =\left({\eta }_1,{\eta }_2,\cdots ,{\eta }_g,\cdots ,{\eta }_r\right)$, then $\epsilon$ and $\eta$ are the c damage strategies of the projectile and the t damage strategies of the aircraft target, respectively. We use Formulas (9) and (10) to denote them.

$$S_1=\left\{\epsilon =\left({\epsilon }_1,{\epsilon }_2,\cdots ,{\epsilon }_l,\cdots ,{\epsilon }_c\right)\in {\mathfrak{A}}_{+}^c,{\epsilon }_l\ge 0,l=1,2,\cdots ,c|{\displaystyle \sum _{l=1}^c{\epsilon }_l=1}\right\}$$

(9)

$$S_2=\left\{\eta =\left({\eta }_1,{\eta }_2,\cdots ,{\eta }_g,\cdots ,{\eta }_r\right)\in {\mathfrak{A}}_{+}^r,{\eta }_g\ge 0,g=1,2,\cdots ,r|{\displaystyle \sum _{g=1}^r{\eta }_g=1}\right\}$$

(10)

Formulas (9) and (10) are the hybrid game strategies about the player of the projectile’s warhead fragments and the player of the aircraft target, respectively [33].

For the matrix $h_{\lambda \gamma }$, it is a strategic representation of the simultaneous actions of the players in the game. Then, the hybrid game strategy matrix is defined as $R(S_1,S_2;h_{\lambda \gamma })$. From this, the hybrid game strategy matrix $R(S_1,S_2;h_{\lambda \gamma })$ is supposed to call as a game matrix $h_{\lambda \gamma }$.

Due to the randomness and unknowns of each participant strategy choice, the gain matrix function of the hybrid game strategy is uncertain. To scientifically calculate the effective optimal solution of the hybrid game strategy, the expected utility function is used to determine the gain matrix hybrid strategy. The mathematical expectation of the participants under the hybrid strategy can be obtained by Formula (11).

$$E\left(\epsilon ,\eta \right)={\displaystyle \sum _{l=1}^c{\displaystyle \sum _{g=1}^r{r}_{lg}{\epsilon }_l{\eta }_g}}={\epsilon }^T{h}_{\lambda \gamma }\eta$$

(11)

where $r_{lg}$ is the sufficient condition of the solution of the gain matrix hybrid strategy in the sense of pure strategy.

For the hybrid strategy of the Nash equilibrium solution of the two-person non-cooperative game, the hybrid strategy of maximizing the expected utility function is adopted to obtain the optimal combination of the two players.

According to von Neumann’s minimum–maximum theory, define $S_1^{*}=\underset{\epsilon \in S_1}{\max } \underset{\eta \in S_2}{\min }E\left(\epsilon ,\eta \right)$ and $S_2^{*}=\underset{\eta \in S_2}{\min } \underset{\epsilon \in S_1}{\max }E\left(\epsilon ,\eta \right)$. Then $S_1^{*}$ and $S_2^{*}$ are called the projectile’s gain floor and aircraft target’s loss ceiling. For the two players, $S_1^{*}$ and $S_2^{*}$ are the Nash equilibrium solution of the hybrid strategy on the projectile and the aircraft target, which satisfies Formula (12) [34,35].

$$\{\begin{array}{l}\underset{\eta \in S_2}{\min }\left\{E(\epsilon ,\eta )\right\}\underset{\_}{\prec }E(\epsilon ,\eta ), \epsilon \in S_1\\ \underset{\epsilon \in S_1}{\max }\left\{E(\epsilon ,\eta )\right\}\underset{\_}{\succ }E(\epsilon ,\eta ), \eta \in S_2\end{array}$$

(12)

Hence, for any $\epsilon \in S_1$ and $\eta \in S_2$, it can be described by Formula (13).

$$\underset{\epsilon \in S_1}{\max } \underset{\eta \in S_2}{\min }\left\{E\left(\epsilon ,\eta \right)\right\}\underset{\_}{\prec }\underset{\eta \in S_2}{\min } \underset{\epsilon \in S_1}{\max }\left\{E\left(\epsilon ,\eta \right)\right\}$$

(13)

If there exists some $({\epsilon }^{*},{\eta }^{*})\in S_1\times S_2$, satisfying Formula (14), then $({\epsilon }^{*},{\eta }^{*})$ is the Nash equilibrium solution of the gain matrix, and $\Lambda \left({\epsilon }^{*},{\eta }^{*}\right)$ is the value of the gain matrix in the sense of hybrid strategy [36,37,38].

$$\underset{\epsilon \in S_1}{\max }\underset{\eta \in S_2}{\min }\left\{{\epsilon }^T{h}_{\lambda \gamma }\eta \right\}=\underset{\eta \in S_2}{\min }\underset{\epsilon \in S_1}{\max }\left\{{\epsilon }^T{h}_{\lambda \gamma }\eta \right\}={\epsilon }^{*}{}^T{h}_{\lambda \gamma }{\eta }^{*}=\Lambda \left({\epsilon }^{*},{\eta }^{*}\right)$$

(14)

where ${\epsilon }^{*}$ and ${\eta }^{*}$ are the optimal hybrid strategies of the participants on the projectile and the aircraft target, respectively.

If $v$ is the value of the hybrid game strategy $R(S_1,S_2;h_{\lambda \gamma })$ and ${\eta }^{*}$ is the optimal strategy of the aircraft target, when the projectile chooses the pure strategy ${\lambda }_k$, the expected gain of the projectile in the game is not satisfied $v$, which indicates that the strategy ${\lambda }_k$ is meaningless. Then, any optimal strategies ${\epsilon }^{*}$ of the projectile in the game must abandon the pure strategy. That is, if it is known that one optimal strategy ${\epsilon }^{*}$ of the projectile satisfies ${\epsilon }^{*}>0$, it must be the case that $E({\lambda }_k,{\epsilon }^{*})=v$. If it is determined that the sum of the optimal strategies of the projectile and the aircraft target is greater than zero, there is an inequality group, which can be expressed by Formulas (15) and (16) [39,40,41].

$$\{\begin{array}{l}{\displaystyle \sum _{l=1}^c{r}_{lg}{\epsilon }_l\underset{\_}{\succ }v,l\in {\Delta }_1,g\in {\Delta }_2}\hfill \\ {\epsilon }_l\ge 0\hfill \\ {\displaystyle \sum _{l=1}^c{\epsilon }_l=1}\hfill \end{array}$$

(15)

$$\{\begin{array}{l}{\displaystyle \sum _{g=1}^r{r}_{lg}{\eta }_g\underset{\_}{\prec } v,l\in {\Delta }_1,g\in {\Delta }_2}\hfill \\ {\eta }_g\ge 0\hfill \\ {\displaystyle \sum _{g=1}^r{\eta }_g=1}\hfill \end{array}$$

(16)

According to the inequality group of the optimal strategy between the projectile and the aircraft target, it is transformed into the problem of solving two equations.

The matrix strategy is the basic expression of the two-person non-cooperative damage game strategy for the intersection of the projectile and the aircraft target. For the solution of the optimal solution of the matrix strategy, the matrix strategy is transformed into an equivalent inequality group. According to the property that the optimal solution remains unchanged after adding a constant to the elements of the matrix strategy, the inequality group problem can be formed into the solution of two linear programming problems.

The traditional methods for solving programming methods include the Lagrange multiplier method, the feasible direction method, the constraint function method and so on. However, these methods often can only solve simple optimization problems, and it is difficult to solve the hybrid strategy problem based on the gain matrix. Therefore, this paper uses the particle swarm optimization algorithm to solve the Nash equilibrium of the game gain matrix of the hybrid strategy. Due to the limited space, this paper does not give the specific solution steps. The following will describe the process of using the particle swarm optimization algorithm to solve the Nash equilibrium of the game gain matrix.

The optimal particle of each individual is the best position that the corresponding individual has reached, and its update method is described as: Assuming that the individual optimal particle of the previous generation is $p_{\epsilon }(t)$ [42,43], the current individual optimal particle is $P_{\epsilon }(t+1)$, and the newly generated particle is $X_{j^{\prime }}(t+1)$. If $P_{f(X_{j^{\prime }}(t+1))>f(P_{\epsilon }(t+1))}>0.5$, then $P_{\epsilon }(t+1)=X_{j^{\prime }}(t+1)$, where $f(x)$ represents the fitness function. If $P_{f(X_{j^{\prime }}(t+1))>f(P_{\epsilon }(t+1))}\le 0.5$, then $P_{\epsilon }(t+1)=P_{\epsilon }(t)$.

The specific processing algorithm steps are as follows:

(1) In the entire search space, the randomly generated position and velocity are used to initialize the particle swarm. Regarding the current particle as the individual optimal, the corresponding fitness of each particle is obtained, and then the global optimal is obtained according to the interval number ranking method.

(2) The particles are updated to obtain a new generation of particles, and the corresponding fitness of each particle is calculated.

(3) Each particle and its corresponding individual optimal particle are sorted to obtain a new individual optimal particle, and all individual optimal particles are sorted to obtain a new global optimal particle.

(4) If the iteration termination condition is not satisfied, the particles are updated again, and the fitness of each particle is determined again. If the maximum number of iterations is reached, the global optimal particle is output after the end of the cycle.

The position corresponding to the output global optimal particle is the Nash equilibrium point to be found.

In hybrid game strategy, we mainly consider two aspects of the measure attribute: first, the number of effectively allocated warhead fragments in each cabin of the aircraft target; second, the vulnerability characteristics of each cabin of the aircraft target. According to the two measure attributes, the game payoff function matrix under hybrid game strategy can be obtained.

In the intersection confrontation game damage, the projectile and the aircraft target are regarded as two players who independently participate in the game damage to form the continuous confrontation game behavior of both sides such that their own maximum gain can be obtained, that is, both sides will adopt different strategies to deal with each other in order to maximize their own damage gain. Due to the rational driving of the game principle, both sides finally achieve a balance of the game gain in the confrontation game, and their respective gains are maximized at this time. Based on this, a two-person non-cooperative damage game strategy model is established. The mathematical modeling process of the target damage strategy based on a two-person non-cooperative game is shown in Figure 3.

5. Simulation and Analysis of Confrontation Game Damage Strategy

According to the established hybrid game strategy set, the gain degree can be expressed by Formula (17).

$$M(\tau )=\frac{M_{n-pay}}{M_{m-apy}}$$

(17)

Because the projectile can form damage to the aircraft target, the gain strategy mainly depends on the warhead fragments formed by the projectile explosion to the aircraft target. Based on the hybrid strategy set and the damage gain function model of the confrontation damage game model, the mixed strategy of three-game single-damage game is based on the number of warhead fragments, the warhead fragment velocity and the intersection angle, respectively. The vulnerable damage level of the aircraft target is divided and its damage degree is quantified by Bayesian estimation. The aircraft target is finally divided into less, moderate and more corresponding level III damage, level II damage and level I damage.

The hybrid strategies of the damage game based on the warhead fragment number, warhead fragment velocity and intersection angle are defined as follows: (1) The hybrid strategy of the damage game based on the number of warhead fragments is embodied in the strategy of the different number of warhead fragments under the condition of the damage gain of the confrontation game corresponding to different damage degrees; (2) The hybrid strategy of the damage game based on the warhead fragment velocity is embodied in the strategy of different velocities of warhead fragments under the condition of the damage gain of the confrontation game corresponding to different damage degrees; (3) The hybrid strategy of the damage game is based on the intersection angle of the warhead fragments, and the aircraft is embodied in the strategy of the different intersection angles under the condition of the damage gain of the confrontation game corresponding to different damage degrees.

In order to obtain the game damage effect of the warhead fragments formed by the projectile explosion on the aircraft target, the quantitative warhead fragments data are used to calculate and analyze the game damage effect. In the calculation, it is assumed that the evaluation of the damage effectiveness of each cabin compartment of the aircraft target is the same. According to the gain degree, the aircraft damage strategy can be attributed to the gain degree reflected by the number of warhead fragments hit.

Supposing the numbers of warhead fragments hitting the aircraft target are 1500, 2000 and 2500, respectively, the aircraft target has nine strategies. The damage gain of the confrontation game is shown in

Table 1. The gain degree under different numbers of warhead fragments hitting the aircraft target.

Strategic Factors		Gain Degree
		Less	Moderate	More
Number of warhead fragments hitting the aircraft target	1500	0.1	0.2	0.4
	2000	0.3	0.5	0.7
	2500	0.5	0.7	0.9

.

Based on the gain degree of the number of warhead fragments hitting the aircraft target, player 1 is aircraft target side whose strategy set is {X1, X2, X3, X4, X5, X6, X7, X8, X9} and player 2 is projectile side whose strategy set is {Y1, Y2, Y3, Y4, Y5, Y6, Y7, Y8, Y9}. Among them, when the number of warhead fragments hitting the aircraft target is 1500, the strategy X1 of the aircraft target indicates a small gain degree, the strategy X2 of the aircraft target indicates a moderate gain degree and the strategy X3 of the aircraft target indicates more of a gain degree. When the number of warhead fragments hitting the aircraft target is 2000, the strategy X4 of the target indicates a small gain degree, the strategy X5 of the aircraft target indicates a moderate gain degree and the strategy X6 of the aircraft target indicates more of a gain degree. When the number of warhead fragments hitting the aircraft target is 2500, the strategy X7 of the aircraft target indicates a small gain degree, the strategy X8 of the aircraft target indicates a moderate gain degree and the strategy X9 of the aircraft target indicates more of a gain degree.

Under the numbers of warhead fragments hitting the aircraft target, the strategy set of the aircraft target is {X1, X2, X3, X4, X5, X6, X7, X8, X9}, and the strategy set of the projectile is {Y1, Y2, Y3, Y4, Y5, Y6, Y7, Y8, Y9}. Based on Formulas (6) and (7), the calculation result of the game gain matrix is:

$$\left[\begin{array}{rrrrrrrrr}\hfill \left(77.0,-37.0\right)& \hfill \left(92.5,-42.1\right)& \hfill \left(101.0,-58.3\right)& \hfill \left(82.1,-52.5\right)& \hfill \left(66.1,-26.1\right)& \hfill \left(90.1,-52.9\right)& \hfill \left(98.3,-61.0\right)& \hfill \left(94.7,-50.1\right)& \hfill \left(62.3,-22.3\right)\\ \hfill \left(82.1,-52.5\right)& \hfill \left(97.5,-57.6\right)& \hfill \left(106.1,-73.8\right)& \hfill \left(87.2,-68.0\right)& \hfill \left(71.2,-41.6\right)& \hfill \left(95.2,-68.4\right)& \hfill \left(103.4,-76.5\right)& \hfill \left(99.8,-65.6\right)& \hfill \left(67.4,-37.8\right)\\ \hfill \left(98.3,-61.0\right)& \hfill \left(113.8,-66.1\right)& \hfill \left(122.3,-82.3\right)& \hfill \left(103.4,-76.5\right)& \hfill \left(87.4,-50.1\right)& \hfill \left(111.4,-76.9\right)& \hfill \left(119.6,-85.0\right)& \hfill \left(116.0,-74.1\right)& \hfill \left(83.6,-46.3\right)\\ \hfill \left(92.5,-42.1\right)& \hfill \left(108.0,-47.2\right)& \hfill \left(116.5,-63.4\right)& \hfill \left(97.6,-57.6\right)& \hfill \left(81.6,-31.2\right)& \hfill \left(105.6,-58.0\right)& \hfill \left(113.8,-66.1\right)& \hfill \left(110.2,-55.2\right)& \hfill \left(77.8,-27.4\right)\\ \hfill \left(66.1,-26.1\right)& \hfill \left(81.6,-31.2\right)& \hfill \left(90.1,-47.4\right)& \hfill \left(71.2,-41.6\right)& \hfill \left(55.2,-15.2\right)& \hfill \left(79.2,-42.0\right)& \hfill \left(87.4,-50.1\right)& \hfill \left(83.8,-39.2\right)& \hfill \left(51.4,-11.4\right)\\ \hfill \left(92.9,-50.1\right)& \hfill \left(108.4,-55.2\right)& \hfill \left(116.9,-71.4\right)& \hfill \left(98.0,-65.6\right)& \hfill \left(82.0,-39.2\right)& \hfill \left(106.0,-66.0\right)& \hfill \left(114.2,-74.1\right)& \hfill \left(110.6,-63.2\right)& \hfill \left(78.2,-35.4\right)\\ \hfill \left(101.0,-58.3\right)& \hfill \left(116.5,-63.4\right)& \hfill \left(125.0,-79.6\right)& \hfill \left(106.1,-73.8\right)& \hfill \left(90.1,-47.4\right)& \hfill \left(114.1,-74.2\right)& \hfill \left(122.3,-82.3\right)& \hfill \left(118.7,-71.4\right)& \hfill \left(86.3,-43.6\right)\\ \hfill \left(90.1,-54.7\right)& \hfill \left(105.6,-59.8\right)& \hfill \left(114.1,-76.0\right)& \hfill \left(95.2,-70.2\right)& \hfill \left(79.2,-43.8\right)& \hfill \left(103.2,-70.6\right)& \hfill \left(111.4,-78.7\right)& \hfill \left(107.8,-67.8\right)& \hfill \left(75.4,-40.0\right)\\ \hfill \left(62.3,-22.3\right)& \hfill \left(77.8,-27.4\right)& \hfill \left(86.3,-43.6\right)& \hfill \left(67.4,-37.8\right)& \hfill \left(51.4,-11.4\right)& \hfill \left(75.4,-38.2\right)& \hfill \left(83.6,-46.3\right)& \hfill \left(80.0,-35.4\right)& \hfill \left(47.6,-7.6\right)\end{array}\right]$$

Assuming that the strategy of the projectile side is determined and that the aircraft target side has nine strategies, the damage gains of the confrontation game are calculated under the different numbers of warhead fragments hitting the aircraft target. The results are shown in Figure 4.

From the results of Figure 4, when the number of warhead fragments hitting the target is 1500 and the target adopts the strategy X3, the growth rate of the profit degree of the projectile and the target confrontation game is the fastest, while the profit degree of the projectile and the target confrontation game is the best when the target adopts the strategy X5; when the target adopts the strategy X9, the profit degree of the projectile and the target confrontation game is the worst, and its growth rate is the slowest. When the numbers of warhead fragments hitting the target are 2000 and 2500, respectively, the profit degree of the projectile and the target confrontation game and the change laws of their growth rates are similar to the state when the number of warhead fragments hitting the target is 1500. The difference is that, when different numbers of warhead fragments hitting the target adopt the same strategy, the maximum value of the profit degree is different. The lowest profit degree curve is also the slowest, and the speed of its change is different. Hence, to achieve the best damage profit of the projectile and the target game, the appropriate target strategy is selected according to the profit degree of the warhead fragments.

When the flight velocities of the warhead fragments are 1500 m/s, 2000 m/s and 2500 m/s, respectively, the aircraft target has nine strategies, the damage gain of the confrontation game is shown in

Table 2. The gain degree under different flight velocities of the warhead fragments.

Strategic Factors		Gain Degree
		Less	Moderate	More
Flight velocities of the warhead fragments	1500 m/s	0.2	0.3	0.4
	2000 m/s	0.3	0.5	0.7
	2500 m/s	0.5	0.6	0.8

.

Based on the gain degree of the warhead fragments’ flight velocity, when the flight velocity of the warhead fragment is 1500 m/s, the strategy X1 of the aircraft target indicates a small gain degree, the strategy X2 of the aircraft target indicates a moderate gain degree and the strategy X3 of the aircraft target indicates more of a gain degree. When the flight velocity of the warhead fragment is 2000 m/s, the strategy X4 of the aircraft target indicates a small gain degree, the strategy X5 of the aircraft target indicates a moderate gain degree and the strategy X6 of the aircraft target indicates more of a gain degree. When the flight velocity of the warhead fragment is 2500 m/s, the strategy X7 of the aircraft target indicates a small gain degree, the strategy X8 of the aircraft target indicates a moderate gain degree and the strategy X9 of the aircraft target indicates more of a gain degree. The calculation result of the game gain matrix is:

$$\left[\begin{array}{rrrrrrrrr}\hfill \left(69.0,-29.0\right)& \hfill \left(93.0,-44.5\right)& \hfill \left(96.0,-48.3\right)& \hfill \left(84.5,-53.0\right)& \hfill \left(59.7,-19.7\right)& \hfill \left(75.7,-51.9\right)& \hfill \left(88.3,-56.0\right)& \hfill \left(91.9,-35.7\right)& \hfill \left(64.3,-24.3\right)\\ \hfill \left(84.5,-53.0\right)& \hfill \left(108.5,-68.5\right)& \hfill \left(111.5,-72.3\right)& \hfill \left(100.0,-77.0\right)& \hfill \left(75.2,-43.7\right)& \hfill \left(91.2,-75.9\right)& \hfill \left(103.8,-80.0\right)& \hfill \left(107.4,-59.7\right)& \hfill \left(79.8,-48.3\right)\\ \hfill \left(88.3,-56.0\right)& \hfill \left(112.3,-71.5\right)& \hfill \left(115.3,-75.3\right)& \hfill \left(103.8,-80.0\right)& \hfill \left(79,-46.7\right)& \hfill \left(95.0,-78.9\right)& \hfill \left(107.6,-83.0\right)& \hfill \left(111.2,-62.7\right)& \hfill \left(83.6,-51.3\right)\\ \hfill \left(93.0,-44.5\right)& \hfill \left(117.0,-60.0\right)& \hfill \left(102.0,-63.8\right)& \hfill \left(108.5,-68.5\right)& \hfill \left(83.7,-35.2\right)& \hfill \left(99.7,-67.4\right)& \hfill \left(112.3,-71.5\right)& \hfill \left(115.9,-51.2\right)& \hfill \left(88.3,-39.8\right)\\ \hfill \left(59.7,-10.9\right)& \hfill \left(83.7,-35.2\right)& \hfill \left(86.7,-39.0\right)& \hfill \left(75.2,-43.7\right)& \hfill \left(50.4,-10.4\right)& \hfill \left(66.4,-42.6\right)& \hfill \left(79.0,-46.7\right)& \hfill \left(82.6,-26.4\right)& \hfill \left(55.0,-15.0\right)\\ \hfill \left(91.9,-38.9\right)& \hfill \left(115.9,-51.2\right)& \hfill \left(118.9,-55.0\right)& \hfill \left(107.4,-59.7\right)& \hfill \left(82.6,-26.4\right)& \hfill \left(98.6,-58.6\right)& \hfill \left(111.2,-62.7\right)& \hfill \left(114.8,-42.4\right)& \hfill \left(87.2,-31.0\right)\\ \hfill \left(96.0,-48.3\right)& \hfill \left(120.0,-63.8\right)& \hfill \left(123.0,-67.6\right)& \hfill \left(111.5,-72.3\right)& \hfill \left(86.7,-39.0\right)& \hfill \left(102.7,-71.2\right)& \hfill \left(115.3,-75.3\right)& \hfill \left(118.9,-55.0\right)& \hfill \left(91.3,-43.6\right)\\ \hfill \left(75.7,-51.9\right)& \hfill \left(99.7,-67.4\right)& \hfill \left(102.7,-71.2\right)& \hfill \left(91.2,-75.9\right)& \hfill \left(66.4,-42.6\right)& \hfill \left(82.4,-74.8\right)& \hfill \left(95.0,-78.9\right)& \hfill \left(98.6,-58.6\right)& \hfill \left(71.0,-47.2\right)\\ \hfill \left(64.3,-24.3\right)& \hfill \left(88.3,-39.8\right)& \hfill \left(91.3,-43.6\right)& \hfill \left(79.8,-48.3\right)& \hfill \left(55.0,-15.0\right)& \hfill \left(71.0,-47.2\right)& \hfill \left(83.6,-51.3\right)& \hfill \left(87.2,-31.0\right)& \hfill \left(59.6,-19.6\right)\end{array}\right]$$

Assuming that the strategy of the projectile side is determined and that the aircraft target side has nine strategies, the damage gains of the confrontation game are shown in Figure 5 under the different flight velocities of the warhead fragment.

From the results of Figure 5, when the flight velocity of the warhead fragment is 1500 m/s, the aircraft target adopts the strategy X4, and the growth rate of the damage gain of the confrontation game is the slowest. The growth rate of the damage gain of the confrontation game is the fastest when the aircraft target adopts the strategy X6 or X9. When the flight velocity of the warhead fragment is 2000 m/s, the growth rate of the damage gain of the confrontation game is the slowest when the aircraft target adopts the strategy X4. However, when the aircraft target adopts the strategy X5 or X7, the growth rate of the damage gain of the confrontation game is the fastest and the best. When the flight velocity of the warhead fragment is 2500 m/s, the growth rate of the damage gain of the confrontation game is the slowest when the aircraft target adopts the strategy X1. When the aircraft target adopts the strategy X9, the growth rate of the damage gain of the confrontation game is the fastest and the best. For the different flight velocities of the warhead fragment, the strategy of the aircraft target is also different. Therefore, it is necessary to change the corresponding strategy of the aircraft target according to the change of the flight velocity of the warhead fragment.

When the intersection angles of the projectile and the aircraft target are 30°, 45° and 60°, respectively, the aircraft target has nine strategies. The damage gain of the confrontation game is shown in

Table 3. The game gain degree under different intersection angles of the projectile and the aircraft target.

Strategic Factors		Gain Degree
		Less	Moderate	More
Intersection angle of the projectile and the aircraft target	30°	0.5	0.6	0.8
	45°	0.3	0.4	0.6
	60°	0.2	0.3	0.5

.

Based on the gain degree of the intersection angle of the projectile and the aircraft target, when the intersection angle of the projectile and the aircraft target is 30°, the strategy X1 of the aircraft target indicates a small gain degree, the strategy X2 of the aircraft target indicates a moderate gain degree and the strategy X3 of the aircraft target indicates more of a gain degree. When the intersection angle of the projectile and the aircraft target is 45°, the strategy X4 of the aircraft target indicates a small gain degree, the strategy X5 of the aircraft target indicates a moderate gain degree and the strategy X6 of the target indicates more of a gain degree. When the intersection angle of the projectile and the aircraft target is 60°, the strategy X7 of the aircraft target indicates a small gain degree, the strategy X8 of the aircraft target indicates a moderate gain degree and the strategy X9 of the aircraft target indicates more of a gain degree.

Assuming that the strategy of the projectile side is determined and that the aircraft target side has nine strategies, the damage gains of the confrontation game are shown in Figure 6 under the different intersection angles of the projectile and the target.

From the results of Figure 6, when the intersection angles of the projectile and the aircraft target are 30°, 45°and 60°, respectively, the growth rate of the damage gain of the confrontation game is the fastest when the aircraft target adopts the strategy X9, while the growth rate of the damage gain of the confrontation game is the slowest when the aircraft target adopts the strategy X6 and the intersection angles are 30° and 45°. When the intersection angle of the projectile and the aircraft target is 60°, the growth rate of the damage gain of the confrontation game is the slowest at the aircraft target strategy of X2. With the increase in the intersection angle, the number of warhead fragments effectively penetrating and damaging the aircraft target gradually decreases. Accordingly, the aircraft target will choose different strategies to defend against the projectile damage. Therefore, when the intersection angle is 45° and 60°, respectively, the growth rate of the damage gain of the confrontation game of the aircraft target under different strategies is different. Through the calculation and analysis, it can be seen that, for different quantities of effective warhead fragments, flight velocities of the warhead fragment and intersection angles of the projectile and the aircraft target, the aircraft target adopts corresponding strategies, and the damage gain of the confrontation game under different strategies is different.

To verify that the target damage effectiveness evaluation method proposed in this paper is scientifically effective, this paper regards the target as a static state for comparative analysis. Assuming that the projectile warhead generates warhead fragments by prefabrication, the warhead fragments are evenly dispersed at a certain angle, and the effective damage area of each vulnerable area is the same. In the intersection area between the warhead fragments and the aircraft target, the warhead fragment velocity is greater than the limit penetration velocity, and the projectile explodes at the same position. As shown in Figure 1, in the state of the intersection of the projectile and the aircraft target, the target damage probability of the warhead fragments to the aircraft target is analyzed by using the methods of reference [3], reference [7] and this paper, and the change curves of the target damage probability under different intersection angles are obtained, as shown in Figure 7.

It can be seen from Figure 7 that, under the same intersection angle of the projectile and the aircraft target, the aircraft target damage probability increases gradually with the increase in the number of effective warhead fragments hitting and penetrating the target. At the same time, the larger the intersection angle of the projectile and the aircraft target, the lower the damage probability of the aircraft target. Because when the intersection angle of the projectile and the aircraft target is small, the aircraft target surface area affected by the warhead fragment field is large, the specific kinetic energy of the warhead fragments penetrating the same area of the aircraft target is small, and there are few warhead fragments that fail to hit the aircraft target in the fragment field, that is, the number of invalid fragments is small, which leads to a higher damage probability. Even if the number of warhead fragments is the same, the damage probability of the aircraft target is different under different intersection angles of the projectile and the aircraft target. The reason for this is that the vulnerable characteristics of each cabin compartment of the aircraft target are different, and its vulnerable weight is accordingly also different. With the increase in the intersection angle of the projectile and the aircraft target, the damage caused by the intersection of different segments of the aircraft target and the projectile is also different.

Based on the premise that the aircraft target is static, the aircraft target damage probability obtained by the method proposed in this paper is not significantly different from that obtained by the methods in the other two references, that is, the damage effectiveness evaluation of the aircraft target is basically the same, which shows that the aircraft target damage effectiveness evaluation method proposed in this paper is scientifically effective.

In order to more accurately evaluate the damage effectiveness of warhead fragments against the aircraft target after warhead explosion, it needs to be analyzed in combination with the dynamic characteristics of the aircraft target. The aircraft target on the actual battlefield is not stationary but rather dynamically rendezvous with the projectile at a certain angle ${\theta }_2$ and a certain velocity $\overrightarrow{v_2}$. Because the warhead fragments formed by the projectile explosion also fly away at a certain angle ${\theta }_1$ and a certain speed $\overrightarrow{v_1}$, the kinetic energy of the warhead fragments of the effective warhead penetrating the aircraft target in the actual dynamic rendezvous increases from $m{v}_1{}^2/2$ to $m{(v_1+v_2)}^2/2$. Because the mechanism of the adversarial game strategy can reflect the interests of both sides of the game and obtain the optimal income in the confrontation, this mechanism and method provide a good research idea for the selection and decision of the interests of the intelligent missile developed in the current weapon system in the air and the attacking enemy aircraft. Therefore, this paper introduces the confrontation game strategy mechanism into the target damage effectiveness evaluation and puts forward the damage game strategy. In the game damage, the index of gain is used to characterize the number of warhead fragments effectively penetrating the damaged target. Assuming that the intersection angle of the projectile and the aircraft target is 30°, analyze the damage probability of the warhead fragments to the target by using the methods of reference [3] and reference [7] and this paper. The relationship curves between the damage gain and the target damage probability under different damage game strategies are obtained, as shown in Figure 8.

It can be seen from Figure 8 that the target damage probability is different under different game damage strategies. Under the same gain condition, the target damage probability of the game damage strategy (X9, Y9) is significantly greater than that of strategy (X6, Y6). However, no matter which strategy is adopted, with the increase in gain, the target damage probability increases to the highest value when the gain is a certain value and then decreases slowly. The core idea of this paper is to find the optimal strategy for the maximum damage caused by the projectile. According to the established damage strategy set and the gain function of the projectile and aircraft target intersection, combined with the properties of the optimal hybrid strategy, we solve the linear programming of the hybrid game and obtain the strategy value and the optimal strategy situation in the sense of the Nash equilibrium of the two-person non-cooperative damage game strategy. The strategy value in the sense of the Nash equilibrium is the equilibrium point of the game damage between the projectile and the aircraft target, and the target damage probability is the largest at this point. Once this point is exceeded, the damage probability of the target will gradually decrease.

Compared with the calculation methods in references [3,7], the damage calculation model in this paper considers some actual parameters of the projectile and the target and calculates the damage effect from the dynamic information of both. Therefore, the damage calculation model proposed in this paper can better reflect the real damage effect.

In comparison with the target damage probability when the aircraft target is stationary in Figure 6c, the target damage probability obtained by using the method in this paper has a certain attenuation under the condition of the dynamic intersection of the projectile and the aircraft target. This is not to say that the method proposed in this paper has errors or no advantages, but, because the dynamic parameters of the aircraft target are ignored when the target is regarded as a static state, resulting in incomplete calculation parameters, the results are ideal. This just shows that the method in this paper takes into account the dynamic characteristics of the aircraft target in the actual battlefield environment, which better reflects the damage effectiveness of the actual projectile on the dynamic aircraft target and provides a damage calculation method and idea closer to the actual combat for the weapon damage experimental test.

The advantage of this paper is that, considering the confrontation characteristics of the dynamic rendezvous between the projectile and the aircraft target, the damage effectiveness of the projectile attacking the aircraft target is different from that of the traditional projectile directly attacking the static target, which is relatively more complex. Therefore, this paper regards the intersection confrontation between the projectile and the aircraft target as a two-person non-cooperative game problem and puts forward the damage game strategy method based on the intersection confrontation between the projectile and the aircraft target to evaluate the damage formed under the dynamic intersection conditions of the projectile and the aircraft target, which better reflects the damage effectiveness of the actual projectile on the dynamic target and provides a damage calculation method and idea closer to the actual combat in the weapon damage experiment and test.

6. Conclusions

In order to more comprehensively analyze the damage effect of the projectile and the aircraft target in the state of arbitrary intersection, this paper proposes a numerical calculation method of aircraft target damage game strategy on the intersection confrontation with a two-person non-cooperative game and hybrid game strategy and obtains the following results.

(1) A two-person non-cooperative game can reflect the mechanism of the game against the two sides, effectively avoiding the projectiles for the aircraft target during the execution of the mission and reducing the probability of its own damage.

(2) The projectile is regarded as one of the players in the confrontation damage game, and the aircraft target (damage target) is regarded as the other player in the confrontation damage game. This makes the game damage strategy pay more attention to the benefits of the dynamic parameters of the projectile warhead fragments on the formation of the aircraft targets.

(3) The damage calculation model in this paper considers some actual parameters of the projectile and the aircraft target and calculates the damage effect from the dynamic information of both. Therefore, the damage calculation model proposed in this paper can better reflect the real damage effect.

Due to the target damage about the intersection status of projectiles and aircraft targets, it is difficult to obtain more accurate warhead fragment parameters and the vulnerability characteristics data of the aircraft. The damage game calculation model of the intersection of the projectile and the aircraft target needs further in-depth research. The research of this paper has certain application prospects for the development of new damage calculation methods in the future.