Solving Multiobjective Game in Multiconflict Situation Based on Adaptive Differential Evolution Algorithm with Simulated Annealing

In this paper, we study the multiobjective game in a multiconflict situation. First, the feasible strategy set and synthetic strategy space are constructed in the multiconflict situation. Meanwhile, the value of payoff function under multiobjective is determined, and an integrated multiobjective game model is established in a multiconflict situation. Second, the multiobjective game model is transformed into the single-objective game model by the Entropy Weight Method. /en, in order to solve this multiobjective game, an adaptive differential evolution algorithm based on simulated annealing (ADESA) is proposed to solve this game, which is to improve the mutation factor and crossover operator of the differential evolution (DE) algorithm adaptively, and the Metropolis rule with probability mutation ability of the simulated annealing (SA) algorithm is used. Finally, the practicability and effectiveness of the algorithm are illustrated by a military example.


Introduction
As we all know, in many fields such as optimal control, engineering design, economy, and arms race, deciders usually have not merely considered one objective [1][2][3][4][5][6] but integrated many objectives in decision-making. So multiobjective decisions are more consistent with reality than a single-objective decision. Multiobjective game [7] is an effective method to solve reciprocity among many deciders in the real society. erefore, it is significant to study the multiobjective game. For example, in the military, armed forces will fight with the enemy on multiple battlefields at the same time. Due to the military strength, equipment, and other constraints, its multiple battlefield situations are interrelated and mutually constrained. According to the game theory, a conflict situation can be described as a game environment. In this way, the multiconflict situation can be expressed as multiple interconnected games, and the constraints can be regarded as objective criteria of the game. In the game, the player's strategy choice and the payoff function are the keys to analyzing the game decision. erefore, studying the synthesis of strategies and synthetic payoff function is of great significance for analyzing the multiobjective game in a multiconflict situation.
For the game in a multiconflict situation, Inohara et al. [8] discussed the relationship between strategy sets and studied the synthesis of finite strategies. However, they did not provide a specific strategic integrated model. Under the single objective, Yanjie et al. [9] established an integrated game model for the conflict situation described by multiple bimatrix games [10,11] and gave an application in the military field. Yexin et al. [12] established a game-integrated model in the multiobjective and multiconflict situation, and they solved the game by using the equivalent relationship between quadratic programming and equilibrium solution of the game. When solving these game models in the above literature, the numerical calculation methods and Lingo software were generally used, and less attention has been paid to intelligent algorithms to solve these problems. e paper aims to fulfill this gap, that is, the multiobjective game in multiconflict situation is transformed into a simpler optimization problem, and then an efficient intelligent algorithm is proposed to solve it.
In this paper, in order to solve the integrated game model better, the integrated multiobjective game model is transformed into a single-objective game model by using the Entropy Weight Method [13,14]. en, the game is transformed into an equivalent optimization problem with constraints [15][16][17][18]. Finally, an adaptive differential evolution algorithm based on simulated annealing (ADESA) is proposed to solve this game. e main operation steps of this hybrid algorithm can be divided as follows: first, the mutation and crossover operator of the differential evolution (DE) algorithm are adaptively improved [19,20]. en, the Metropolis process of the simulated annealing (SA) algorithm [21] is applied to the DE algorithm, which has the characteristic that could accept not only the good solutions but also the inferior solutions with a certain probability. It not only improves the convergence speed and accuracy of the DE but also improves the diversity of the population.
us, the hybrid algorithm has a strong global search capability and avoids the occurrence of premature phenomena. At the end of this paper, we use a military game example to demonstrate the practicability and effectiveness of the algorithm in solving the multiobjective game in a multiconflict situation.

Multiobjective Game Model in a Multiconflict Situation
} is the set of players and n is the number of players.
(2) S i � s i1 , . . . , s im i , ∀i ∈ N is the pure strategy set of player i, m i represents the number of strategies available to player i, S � n i�1 S i , and each pure strategy profile meets (s 1m 1 , s 2m 2 , . . . , s ij , . . . , , ∀i ∈ N is the set of mixed strategies of player i. X � n i�1 X i and each mixed strategy profile meets (x 1 , x 2 , . . . , x n ) ∈ X.
(5) f i : X ⟶ R, ∀i ∈ N represents the expected payoff function of player i: where f i (x 1 , . . . , x n ) represents the expected payoff value of player i when he chooses a mixed strategy . . , s nk n ) represents the payoff value of player of i when each player chooses pure strategy s ik i ∈ S i , i � 1, . . . , n. Denote by where is a pure strategy of player i, (x‖s ik i ) represents s ik i of player i instead of x i , and the other players do not change their own mixed strategy.
, ∀i ∈ N, then x * is the Nash equilibrium of n-person finite noncooperative game, where i ∧ � N\i, ∀i ∈ N.

Conclusion 1.
A mixed strategy x * is the Nash equilibrium point of a game if and only if every pure strategy Theorem 1 (see [18]). A mixed strategy x * ∈ X is the Nash equilibrium point of the game Γ if and only if x * is the optimal solution to the following optimization problem, and the optimal value is 0: Especially, for the two-player matrix game, it can be seen from eorem 1 that finding the Nash equilibrium (x * 1 , x * 2 ) of the game is equivalent: where A and B are payoff matrices of players, A i is the ith row of matrix A, and B j is the jth column of matrix B.
where m k and n k are the number of strategies of player 1 and player 2, respectively.

Definition 4.
If player 1 chooses a pure strategy α k i k in the kth (k � 1, . . . , K) game G k and combines pure strategies in all conflict situations, the feasible strategy string of player 1 in the multiconflict situation is obtained, which is recorded as Similarly, the feasible strategy string of player 2 in a multiconflict situation Definition 5. e feasible strategy sets of player 1 and player 2 in a multiconflict situation are recorded as S 1 � α 1 , α 2 , . . . , α t and S 2 � β 1 , β 2 , . . . , β r , where t and r represent the number of feasible strategy strings of player 1 and player 2, respectively. Definition 6. A synthetic strategy in a multiconflict situation is (α i , β j ), where α i is a feasible strategy string selected by player 1 from S 1 , and β j is a feasible strategy string selected by player 2 from S 2 . e synthetic strategy space consists of a set of all synthetic strategies of player 1 and player 2, which is recorded as S � S 1 × S 2 .
Definition 7. In a synthetic strategy, a strategic combination of player 1 and player 2 in each game is called the substrategy of the synthetic strategy. For example, the substrategies of Definition 8. e integrated model of the multiobjective bimatrix game in a multiconflict situation is In order to solve the integrated game model in a multiconflict situation, the Entropy Weight Method is introduced below.

Integrated Game Model Based on the Entropy Weight Method
(1) e payoff matrix of each objective in the integrated model is transformed into a standardized matrix by Mathematical Problems in Engineering using the extreme difference method. For payoff matrices C l and D l of the lth (l � 1, 2, . . . , L) objective, take θ l � max 1≤i≤t,1≤j≤r c l ij , d l ij , θ l � min 1≤i≤t,1≤j≤r c l ij , d l ij . For the negative objective, For the positive objective, Let E l and F l be the normalized matrices corresponding to C l and D l , respectively, and E l � (e l ij ) t×r , F l � (f l ij ) t×r . (2) For E l and F l , the weight of each objective is calculated by the Entropy Weight Method [13,14].
(i) Calculate the entropy value of the lth (l � 1, 2 . . . , L) objective: (iii) Calculate the weight of the lth objective: (3) Weighting and summing E l and F l of all objectives in the synthetic model: where e ij � L l�1 w l e l ij and f ij � L l�1 w l f l ij . By weighted summation, the above-integrated model of a multiobjective bimatrix game in a multiconflict situation is transformed into a single-objective matrix game model G * � S 1 , S 2 , E, F . In order to solve the game model more conveniently, the ADESA algorithm is proposed.

Adaptive Differential Evolution Algorithm
Based on Simulated Annealing (ADESA) In this section, we outline a novel DE algorithm, ADESA algorithm, and explain the steps of the algorithm in detail.

Differential Evolution (DE) Algorithm.
e DE algorithm is first introduced by Storm and Price [22]. Due to its outstanding characteristics, such as simple structure, robustness and speediness, being easy to understand and implement, and fewer control parameters, it has become more and more popular and has been extended to handle a variety of optimization problems. e DE algorithm mainly consists of four operations: initialization population, mutation operation, crossover operation, and selection operation.

Initialization.
In the study of the DE algorithm, it is generally assumed that the initial population conform to a uniform probability distribution. Each individual X � (x i1 , . . . , x ij , . . . , x iD ), i � 1, . . . , N, j � 1, . . . , D can be expressed as where N and D denote population size and space dimension, respectively, and x L i,j and x U i,j , respectively, are the lower and upper bound of the search space.

Mutation.
e mutation operation is mainly executed to distinguish DE from other evolutionary algorithms. e mutation individual V � (v i1 , . . . , v ij , . . . , v iD ) is generated by the following equation: where r 1 r 2 and r 3 are randomly generated integers within [1, N], r 1 ≠ r 2 ≠ r 3 ≠ i, F is a constriction factor to control the size of difference of two individuals, and t is the current generation.

Crossover.
We use the crossover between the parent and offspring with the given probability to generate a new individual U � (u i1 , . . . , u ij , . . . , u iD ): where rand(j) ∈ [0, 1] is random values, CR ∈ [0, 1] is crossover operator, rnbr(i) is a randomly selected integer on [1, D], which ensures at least one component of new individual is inherited from the mutant vector.

Selection.
In the problem with boundary constraints, it is necessary to ensure that the parameter values of the new individuals are in the feasible region. ere is a simple method is boundary treatment, in which the new individuals beyond the bounds are replaced by the parameter vectors randomly generated in the feasible region. en, the offspring X t+1 i is generated by selecting the individual and parent according to the following formula (for a minimization problem): where f(·) is the fitness function. e pseudocode of the standard DE algorithm is shown in Algorithm 1.

Adaptive Differential Evolution Algorithm (ADE).
Although the DE algorithm is widely used in optimization problems, with the increase of the complexity of solving problems, the DE algorithm also has some disadvantages, such as slow convergence, low accuracy, and weak stability. erefore, in order to solve the multiobjective game better, the DE algorithm is improved.
Since the DE algorithm mainly performs the genetic operation through differential mutation operator, the performance of the algorithm mainly depends on the selection of mutation and crossover operations and related parameters. Many scholars have verified that the mutation factor F and crossover operator CR directly affect the searching capability and solving efficiency of the DE [23][24][25]. In order to make the algorithm have better global search capability and convergence speed, adaptive mutation and crossover operator are adopted: where t is the current iterate time, T is the maximum number of iteration, and F 0 and CR 0 are the initial mutation factor and crossover operator, respectively. ere are two main ways of traditional mutation operations: where x t best represents the best individual in the current generation, that is, the optimal position searched by this individual so far [26]. e first mutation method has a strong global search capability but a slow convergence speed. e second method has a fast convergence speed, but it is easy to fall into local optimum values [27]. In order to overcome these shortcomings, many researchers have improved the mutation strategy [28,29], and a new mutation operation is proposed in this paper: e new mutation strategy has strong global search ability and fast convergence speed and can find better solutions.

Simulated Annealing Algorithm (SA).
SA is not only a statistical method, but also a global optimization algorithm. It is first proposed by Metropolis in 1953 [21], and Kirkpatrick first used SA to solve combinatorial optimization problems in 1983 [30]. SA is derived from the simulation of the solid annealing cooling process. e main feature is to accept inferior solutions with a certain probability according to the Metropolis rule, which can avoid the algorithm falling into the local optimum and "premature" phenomenon. e Metropolis rule defines the internal energy probability P M ij of an object state i transferring to state j at a certain temperature M, it can be expressed as follows: where E(i) and E(j) represent internal energy of solid in states i and j, respectively. K is attenuation parameter, and ΔE � E(j) − E(i) represents increment of internal energy. When the combined optimization problem is simulated by solid annealing, the internal energy E is simulated as the objective function value, and the temperature M becomes a parameter. at is, where f(X i ) is the value of the function and M is the temperature at the ith iteration. When f(X i+1 ) ≤ f(X i ), we select f(X i+1 ) with probability 1; otherwise, we select the inferior solution f(X i ) with probability P M i+1 . In this paper, the SA is applied to the DE to enhance its global optimization ability.

e ADESA Algorithm Experimental Steps.
e pseudocode of the ADESA algorithm is shown in Algorithm 2, and the specific steps are described in detail as follows: Step 1: set the parameters of the ADESA, such as N, D, F 0 , CR 0 , K, M, T, ε.
Step 2: initialize the population and each individual satisfies

Mathematical Problems in Engineering
Step 3: calculate the fitness function value f(x) of each individual in population P(t) and determine x t best and f(x t best ).
Step 4: the next generation population P 1 (t) is generated by mutation of formula (20), and population P 2 (t) is generated by crossover of formula (16).
Step 5: the offspring population P(t+1) is selected from the P(t) and P 2 (t) populations according to formulas (21) and (22), and calculate the fitness function value of population P(t+1).
Step 6: determine whether to end this procedure according to the accuracy and the maximum number of iterations and output the optimal value; otherwise, turn to step 3.
According to Algorithm 1, it can be judged that the experimental steps of the ADE algorithm are the same as the DE algorithm, so the time complexity of the ADE algorithm does not change. Comparing the implementation process of Algorithms 1 and 2, we can see that the ADESA algorithm has only one more judgment in the selection operation than the DE algorithm. erefore, the time complexity remains unchanged. In conclusion, the computational complexity provides a guarantee for the performance of the ADESA algorithm proposed in this paper. In the next section, the superiority of the proposed algorithm is verified by calculating an example of the multiobjective military game.

Military Example.
Suppose that there are conflicts between Red and Blue armies on islands A and B. Under the dual objectives of attack time and damage effectiveness, Red's strategies on the two islands are no-attack and attack, and Blue's strategies are retreat and defend. Two parties have different strategic purposes on the two islands; therefore, the degree of preference is different under the same objective. In addition, due to the limitation of military equipment, it is assumed that the Red army cannot choose to attack on both islands at the same time, and the Blue army cannot choose to defend on both islands at the same time. In this way, the available strategies for both Red and Blue on A island are On island B, the available strategies for both Red and Blue are defend}. Aiming at the attack time (l � 1), the payoff matrices of the Red and Blue on the islands A and B are, respectively (the payoff values are expressed in the preference order of − 2, − 1, 0, 1, 2), as follows: Aiming at the damage effectiveness (l � 2), the payoff matrices of the Red and Blue on the two islands A and B are, respectively, as follows: Input: Parameters N, D, T, F, CR, ε Output: e best vector (Solution) · · · Δ (1) t←0 (Initialization) (2) end for (6) end for (7) while |f(Δ)| ≥ ε or t ≤ T do (8) for i � 1 to N do (9) (Mutation and Crossover) (10) for end for (14) (Selection) end if (20) end for (21) t � t + 1 (22) end while (23) return the best vector Δ ALGORITHM 1: DE. 6 Mathematical Problems in Engineering Obviously, considering real situations of the two islands A and B, the available strategy sets of Red and Blue are Using formula (6), the synthetic payoff matrices of the Red and Blue under the two objectives are, respectively, as follows: Using formulas (8) and (9), the above synthetic payoff matrices can be transformed into the standardized matrices as follows: Input: Parameters N, D, T, K, M, F 0 , CR 0 , ε Output: e best vector (Solution) · · · Δ (1) t←0 (Initialization) (2) end for (6) end for (7) while |f(Δ)| ≥ ε or t ≤ T do (8) for i � 1 to N do (9) (Mutation and Crossover) (10) for end if (22) end for (23) M � K · M; t � t + 1 (24) end while (25) return the best vector Δ By formulas (10)- (12), the entropy values of the lth (l � 1,2) objective are h 1 � 0.9581, h 2 � 0.9311. e difference indices are g 1 � 0.0419, g 2 � 0.0689. e weights are ω 1 � 0.3779, ω 2 � 0.6221. Finally, by formula (13) Use formula (4) to solve the following optimization problem: where f 1 � 0.3433x 11 + 0.5622x 12 + 0.4977x 13 x 21 + 0.3779x 11 + 0.4078x 12 + 0.5323x 13

Results and Discussion.
In order to solve the game, the DE, ADE, and ADESA algorithms are used to calculate the above optimization problem. According to [24,31,32] and experimental experiences, the parameters are set to N � 20, D � 6, CR � 0.8, F � 0.6, CR 0 � 1, F 0 � 0.4, K � 0.998, M � 100, T � 60, ε � 10 − 8 . e calculation results and the corresponding Nash equilibria are shown in Table 1. Figure 1 is a comparison of solving this problem with DE, ADE, and ADESA algorithms, and Figure 2 is the result of the first ten iterations of Figure 1.
It can be intuitively seen from Table 1 that when we solve this military game by using these three algorithms, two Nash equilibria are obtained. One is (α 3 , β 3 ), the other is (α 2 , β 1 ), and their corresponding feasible strategy choices are (α A 2 ; means the Red attacks island B and the Blue retreats on both islands. Given the fact that the Blue is unlikely to retreat on both islands, this solution will be abandoned. e other solution (α A 2 ; α B 1 ), (β A 2 ; β B 1 ) represents that the Red attacks island A and the Blue defend on it, which is consistent with the real battle situation, so the final result of this game is (α 2 , β 1 ).
Since finding the Nash equilibria of the integrated game is equivalent to solving the optimal solution of the 8 Mathematical Problems in Engineering optimization problem. It can be clearly seen from Figures 1  and 2 that the ADESA algorithm has the best performance in the calculation process, followed by the ADE algorithm, and DE algorithm at the end. On the one hand, we found that the ADE algorithm and the ADESA algorithm generally have similar performance. It seems rational because the ADESA algorithm and ADE algorithm both have self-adaptively improved the parameters compared with the DE algorithm. As the number of generations increases, it is difficult to find the optimal solution, so the DE algorithm has pauses in the local optimal solution. At this time, the self-adaptive operations of the ADESA and ADE algorithms play a role in avoiding the local optimal solution and accelerating the convergence speed. On the other hand, with the iteration process, there are more and more noninferior solutions. e ADE algorithm only selects the optimal solution according to the greedy criterion, so it will ignore some excellent solutions, while the ADESA algorithm retains some noninferior solutions because it adds the Metropolis rule in the selection operation. erefore, the speed of searching for the global optimal solution is accelerated. It can be clearly observed in Figure 2. By analyzing the results obtained above, it can be seen that the algorithm ADESA proposed in this paper can well find the equilibrium solutions under the two objectives. And comparing this algorithm with DE and ADE algorithms, we find that this algorithm has a faster convergence speed and avoids falling into local optimum. It provides a reference for the study of solving more complex multiobjective games in the future.

Conclusions
In this paper, we study the multiobjective game in a multiconflict situation. Firstly, an integrated multiobjective game model is established in a multiconflict situation; then the multiobjective game is transformed into a single-objective game according to the Entropy Weight Method. Finally, using the equivalence theorem of Nash equilibrium, finding the Nash equilibria of the game is equivalent to solving the optimal solution of an optimization problem. According to the excellent performances of the DE algorithm solving the optimization problems, we choose the DE algorithm to solve this problem. Because the game itself is a complex decision-making problem, and this paper studies the multiobjective game in a multiconflict situation, therefore, the DE algorithm is improved, namely, the ADESA algorithm, which applies the Metropolis rule of the SA algorithm to the selection operation of the ADE algorithm. Moreover, the ADE algorithm is a self-adaptive improvement to the control parameters of the DE algorithm. At the end of the paper, the computational results of a military example show that our proposed ADESA algorithm solves the multiobjective game more quickly and effectively, which provides a reference for future research on related issues. In the future, people may be in the situation of multiobjective mutual transfer in reality, so we will further consider the more complex dynamic multiobjective game in a multiconflict situation. In addition, it could be interesting  Data Availability e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare no conflicts of interest.