Mathematical Problems in Engineering

Mathematical Problems in Engineering / 2021 / Article

Research Article | Open Access

Volume 2021 |Article ID 9957279 | https://doi.org/10.1155/2021/9957279

Huimin Li, Shuwen Xiang, Wensheng Jia, Yanlong Yang, Shiguo Huang, "Solving Multiobjective Game in Multiconflict Situation Based on Adaptive Differential Evolution Algorithm with Simulated Annealing", Mathematical Problems in Engineering, vol. 2021, Article ID 9957279, 11 pages, 2021. https://doi.org/10.1155/2021/9957279

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

Academic Editor: Hao Gao
Received24 Mar 2021
Revised18 May 2021
Accepted04 Jun 2021
Published22 Jul 2021

Abstract

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. Then, 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.

1. 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 [16] 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. Therefore, 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. Therefore, 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. The 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]. Then, the game is transformed into an equivalent optimization problem with constraints [1518]. Finally, an adaptive differential evolution algorithm based on simulated annealing (ADESA) is proposed to solve this game. The 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]. Then, 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. Thus, 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.

2. Game Synthetic Model

2.1. Multiobjective Game Model in a Multiconflict Situation

Definition 1. The N-person noncooperative game with a single objective denoted by , where(1) is the set of players and is the number of players.(2) is the pure strategy set of player , represents the number of strategies available to player , , and each pure strategy profile meets .(3) represents the payoff function of player .(4) is the set of mixed strategies of player . and each mixed strategy profile meets .(5) represents the expected payoff function of player :where represents the expected payoff value of player when he chooses a mixed strategy . represents the payoff value of player of when each player chooses pure strategy .
Denote bywhere is a pure strategy of player , represents of player instead of , and the other players do not change their own mixed strategy.

Definition 2. If there is , , then is the Nash equilibrium of n-person finite noncooperative game, where .

Conclusion 1. A mixed strategy is the Nash equilibrium point of a game if and only if every pure strategy of each player satisfies .

Theorem 1 (see [18]). A mixed strategy is the Nash equilibrium point of the game if and only if 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 Theorem 1 that finding the Nash equilibrium of the game is equivalent:where and are payoff matrices of players, is the row of matrix , and is the column of matrix .

Definition 3. The multiconflict situation of the two persons with multiobjective refers to the situation in which player 1 and player 2 have conflict under the objectives in the situations. Each conflict situation can be described by a multiobjective bimatrix game model. Suppose that in the conflict situation, the payoff matrices of player 1 and player 2 under the objective are and , respectively:where and are the number of strategies of player 1 and player 2, respectively.

Definition 4. If player 1 chooses a pure strategy in the game 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 is .

Definition 5. The feasible strategy sets of player 1 and player 2 in a multiconflict situation are recorded as and , where and represent the number of feasible strategy strings of player 1 and player 2, respectively.

Definition 6. A synthetic strategy in a multiconflict situation is , where is a feasible strategy string selected by player 1 from , and is a feasible strategy string selected by player 2 from . The synthetic strategy space consists of a set of all synthetic strategies of player 1 and player 2, which is recorded as .

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 are , .
The synthetic payoff value of the player in a synthetic strategy is the sum of the payoff values of all substrategies in the same objective. Let the synthetic payoff function values of the players 1 and 2 be and under the objective when players choose the synthetic strategy , and

Definition 8. The integrated model of the multiobjective bimatrix game in a multiconflict situation is , whereIn order to solve the integrated game model in a multiconflict situation, the Entropy Weight Method is introduced below.

2.2. Integrated Game Model Based on the Entropy Weight Method

(1)The payoff matrix of each objective in the integrated model is transformed into a standardized matrix by using the extreme difference method. For payoff matrices and of the objective, take , .For the negative objective,For the positive objective,Let and be the normalized matrices corresponding to and , respectively, and , .(2)For and , the weight of each objective is calculated by the Entropy Weight Method [13, 14].(i)Calculate the entropy value of the objective:where , . If or , then or .(ii)Calculate the difference index of the objective:(iii)Calculate the weight of the objective:(3)Weighting and summing and of all objectives in the synthetic model:where and .

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 . In order to solve the game model more conveniently, the ADESA algorithm is proposed.

3. 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.

3.1. Differential Evolution (DE) Algorithm

The 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. The DE algorithm mainly consists of four operations: initialization population, mutation operation, crossover operation, and selection operation.

3.1.1. Initialization

In the study of the DE algorithm, it is generally assumed that the initial population conform to a uniform probability distribution. Each individual can be expressed aswhere and denote population size and space dimension, respectively, and and , respectively, are the lower and upper bound of the search space.

3.1.2. Mutation

The mutation operation is mainly executed to distinguish DE from other evolutionary algorithms. The mutation individual is generated by the following equation:where and are randomly generated integers within , , is a constriction factor to control the size of difference of two individuals, and is the current generation.

3.1.3. Crossover

We use the crossover between the parent and offspring with the given probability to generate a new individual :where is random values, is crossover operator, is a randomly selected integer on , which ensures at least one component of new individual is inherited from the mutant vector.

3.1.4. 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. There 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. Then, the offspring is generated by selecting the individual and parent according to the following formula (for a minimization problem):where is the fitness function. The pseudocode of the standard DE algorithm is shown in Algorithm 1.

Input: Parameters , , , , ,
Output: The best vector (Solution)
(1) (Initialization)
(2)for to do
(3)for to do
(4)  
(5)end for
(6)end for
(7)while or do
(8)  for to do
(9)   (Mutation and Crossover)
(10)   for to do
(11)     (formula (15))
(12)    
(13)   end for
(14)   (Selection)
(15)   ifthen
(16)    
(17)   else
(18)    
(19)   end if
(20)  end for
(21)  
(22)end while
(23)return
3.2. 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. Therefore, 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 and crossover operator directly affect the searching capability and solving efficiency of the DE [2325]. In order to make the algorithm have better global search capability and convergence speed, adaptive mutation and crossover operator are adopted:where is the current iterate time, is the maximum number of iteration, and and are the initial mutation factor and crossover operator, respectively.

There are two main ways of traditional mutation operations:where represents the best individual in the current generation, that is, the optimal position searched by this individual so far [26]. The first mutation method has a strong global search capability but a slow convergence speed. The 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:

The new mutation strategy has strong global search ability and fast convergence speed and can find better solutions.

3.3. 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. The 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.

The Metropolis rule defines the internal energy probability of an object state transferring to state at a certain temperature , it can be expressed as follows:where and represent internal energy of solid in states and , respectively. is attenuation parameter, and represents increment of internal energy.

When the combined optimization problem is simulated by solid annealing, the internal energy is simulated as the objective function value, and the temperature becomes a parameter. That is,where is the value of the function and is the temperature at the iteration. When , we select with probability 1; otherwise, we select the inferior solution with probability . In this paper, the SA is applied to the DE to enhance its global optimization ability.

3.4. The ADESA Algorithm Experimental Steps

The 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 , , , , , , , .Step 2: initialize the population and each individual satisfiesStep 3: calculate the fitness function value of each individual in population and determine and .Step 4: the next generation population P1(t) is generated by mutation of formula (20), and population P2(t) is generated by crossover of formula (16).Step 5: the offspring population P(t+1) is selected from the P(t) and P2(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.

Input: Parameters , , , , , , ,
Output: The best vector (Solution)
(1) (Initialization)
(2)for to do
(3)  for to do
(4)   
(5)  end for
(6)end for
(7)while or do
(8)  for to do
(9)   (Mutation and Crossover)
(10)   for to do
(11)     (formula (20))
(12)    
(13)   end for
(14)   (Metropolis Selection)
(15)   if then
(16)     (accept the new solutions)
    Let
(17)   else {}
(18)     (accept the inferior solutions)
(19)   else {}
(20)    
(21)   end if
(22)  end for
(23)  ;
(24)end while
(25)return

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. Therefore, 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.

4. Experimental Design and Results

4.1. 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 , .

Aiming at the attack time , 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 , the payoff matrices of the Red and Blue on the two islands A and B are, respectively, as follows:

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:

By formulas (10)–(12), the entropy values of the (l = 1,2) objective are h1 = 0.9581, h2 = 0.9311. The difference indices are , . The weights are , . Finally, by formula (13), we can get

Use formula (4) to solve the following optimization problem:where

4.2. 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 , , , , , , , , , . The 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.


NumberRedBlueSynthetic strategyStrategy string

Nash equilibrium 1(0, 0, 1)(0, 0, 1)(, )(; ), (; )
Nash equilibrium 2(0, 1, 0)(1, 0, 0)(, )(; ), (; )

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 , the other is , and their corresponding feasible strategy choices are (; ), (; ) and (; ), (; ). The feasible strategy 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. The other solution (; ), (; ) 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 (, ).

Since finding the Nash equilibria of the integrated game is equivalent to solving the optimal solution of the 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. The 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. Therefore, 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.

5. 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 to develop other algorithms and compare them with the ADESA algorithm.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare no conflicts of interest.

Acknowledgments

This study was supported by the National Natural Science Foundation of China (Grant nos. 71961003 and 12061020), the Qian Jiaohe YJSCXJH ([2019]029), and the Scientific Research Foundation of Guizhou University ([2019]49).

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