Matrix Games Research Articles

Marginal Values of a Stochastic Game

Zero-sum stochastic games are parameterized by payoffs, transitions, and possibly a discount rate. In this article, we study how the main solution concepts, the discounted and undiscounted values, vary when these parameters are perturbed. We focus on the marginal values, introduced by Mills in 1956 in the context of matrix games—that is, the directional derivatives of the value along any fixed perturbation. We provide a formula for the marginal values of a discounted stochastic game. Further, under mild assumptions on the perturbation, we provide a formula for their limit as the discount rate vanishes and for the marginal values of an undiscounted stochastic game. We also show, via an example, that the two latter differ in general. Funding: This work was supported by Fondation CFM pour la Recherche; the European Research Council [Grant ERC-CoG-863818 (ForM-SMArt)]; and Agence Nationale de la Recherche [Grant ANR-21-CE40-0020].

A New Method to Solve Matrix Game with Interval Payoffs and its MATLAB Code

Game theory is of substantial significance in diverse domains, acting as a potent instrument to comprehend and assess strategic engagements among rational decision-makers. It formulates mathematical models to represent strategic interactions among rational decision-makers in the competitive world. Due to ambiguity in the real-world problems, acquiring the precise payoff values of a matrix game proves challenging. However, in numerous scenarios, these payoffs fluctuate within specific ranges, making them suitable for consideration as interval numbers. This leads to the formulation of a special form of game problem known as the interval valued game problem (IVGP). Some methodologies exist in the literature to find the optimal strategies as well as the value of game for IVGP, but most of them possess some limitations, resulting in the need for proposing a new methodology to find the optimal strategies and value of game. Thus, in this paper, a new solution method for game problems with payoffs represented as interval numbers is presented, utilizing the fuzzy concept. The process begins by transforming the interval payoffs into fuzzy numbers using a ranking function. Subsequently, these fuzzy payoffs are converted into crisp values, leading to the formulation of the crisp matrix game. The resulting crisp matrix game is then solved using linear programming approach. Additionally, MATLAB code for the proposed method is developed and proposed to streamline the computation process, enhancing comparison and decision-making efficiency, particularly when dealing with large payoff matrices. Furthermore, three numerical examples are provided to illustrate the validity of the proposed approach as well as its MATLAB code. A real-life example of IVGP in the realm of tourism industry is also provided. Finally, a comparative analysis is conducted, comparing the proposed method with some existing methods.

Data-driven urban waterlogging risk management approach considering efficiency-equity trade-offs and risk mitigation capability evaluation

Because disaster reduction resources are always constrained, risk mitigation measures and policies need to be weighted toward efficiency or equity. Greater efficiency focuses on high-risk areas, and greater equity focuses on identifying additional potential risk areas. To address this efficiency-equity tradeoff decision, a data-driven urban waterlogging risk mitigation capability evaluation and improving framework was proposed for nonprofit operations and government bailouts. First, a novel evaluation index was defined, the F(α,β) score, to consider both precision and recall in the binary classification tasks and to seek an effective efficiency-equity trade-off, after which a two-person, zero-sum matrix game model was developed to calculate the α and β parameters and select the highest valued optimal model. Then, using this model, a risk probability map was generated for Wuhan, China, and data envelopment analysis was used to conduct a risk mitigation capability evaluation and identify the needed policy improvements. It was found that: (1) vulnerability was the primary risk-related factor, followed by adaptability and restorability; (2) more importance should be attached to equity than efficiency when implementing risk reduction capability evaluations, that is, authorities should reduce resource investments in extremely high-risk DEA-inefficient areas and identify a greater number of potential risk areas; (3) there were around 1.55% of extremely high-risk areas, which were mainly concentrated in the Jianghan and Wuchang districts and around 9% of high-risk areas, mainly located in the Hongshan district; and (4) northeastern and southwestern Jianghan district could be a valuable reference for urban waterlogging management and disaster reduction. However, excessive investment in transportation infrastructure and construction in DEA inefficient areas should be appropriately reduced.

The implementation of the Simplex and Optimization packages in the process of solving matrix games

The implementation of the Simplex and Optimization packages in the process of solving matrix games

Optimal scheduling method of low-carbon energy supply chain based on machine learning and game theory

Aiming at the quality and stability of a low-carbon new energy supply chain, this paper puts forward an optimal scheduling method for a low-carbon energy supply chain based on machine learning and game theory. By establishing an index and preprocessing keywords, the similarity between the index and the data in the platform database is calculated, and the data-sharing result of the low-carbon energy supply chain is obtained. This paper analyzes the game benefits of each subject in the optimization of the low-carbon energy supply chain, obtains the game matrix, and realizes the optimal scheduling of the low-carbon energy supply chain. The experimental results show that the scheduling convergence error of the low-carbon energy supply chain is low, the accuracy is high, and the load supply equipment can run stably.

Computational Intelligence Supporting the Safe Control of Autonomous Multi-Objects

The essence of this work, which is an extension of the author’s previous research, is an analysis of computational intelligence algorithms that the support safe control of an autonomous object moving in a large group of other autonomous objects. Linear and dynamic programming methods with neural constraints on the process state, as well as positional and matrix game methods, were used to synthesize computational algorithms for the safe trajectory of one’s own object. The aim of the comparative analysis of intelligent computational methods for the safe trajectory of an object was to show, through their use, the possibility of taking into account the risk of collision resulting from both the degree of cooperation of objects while observing traffic laws and the impact of the environment in the form of visibility and the complexity of the situation. Simulation tests of the algorithms were carried out on the example of a real navigation situation of several dozen objects passing each other at sea.

Cooperation transitions in social games induced by aspiration-driven players.

Cooperation and defection are social traits whose evolutionary origin is still unresolved. Recent behavioral experiments with humans suggested that strategy changes are driven mainly by the individuals' expectations and not by imitation. This work theoretically analyzes and numerically explores an aspiration-driven strategy updating in a well-mixed population playing games. The payoffs of the game matrix and the aspiration are condensed into just two parameters that allow a comprehensive description of the dynamics. We find continuous and abrupt transitions in the cooperation density with excellent agreement between theory and the Gillespie simulations. Under strong selection, the system can display several levels of steady cooperation or get trapped into absorbing states. These states are still relevant for experiments even when irrational choices are made due to their prolonged relaxation times. Finally, we show that for the particular case of the prisoner dilemma, where defection is the dominant strategy under imitation mechanisms, the self-evaluation update instead favors cooperation nonlinearly with the level of aspiration. Thus, our work provides insights into the distinct role between imitation and self-evaluation with no learning dynamics.

Type-2 interval-valued intuitionstic fuzzy matrix game and application to energy vehicle industry development

Game theory is a powerful tool in modeling actual decision-making problems. In many cases, it is difficult for players to give a crisp value of payoff. To conquer this issue, various extensions of fuzzy sets have been introduced to represent the payoffs. Although type-2 intuitionistic fuzzy (T2IF) set (T2IFS) can solve the game problem effectively, sometimes players prefer to give the primary (secondary) membership and non-membership degrees with some approximate ranges rather than crisp values. T2IFS cannot express such uncertain payoffs. Thus, this paper initiates a new concept called type-2 interval-valued intuitionistic fuzzy (T2IVIF) set (T2IVIFS) which has stronger ability than T2IFS and interval-valued intuitionistic fuzzy set (IVIFS) in representing uncertain information. Hamacher operations of T2IVIFSs are given. Hamming and Euclidean distances between T2IVIFSs are defined respectively. Some Hamacher aggregation operators for T2IVIFSs are developed and some desirable properties are analyzed. Afterwards, an effective method is proposed for solving T2IVIF matrix game, which can make game results more accurate and reliable. An example of new energy vehicle industry development is provided to validate the proposed T2IVIF matrix game. Comparative analyses reveal that the matrix game in T2IVIF environment has better flexibility and reliability than that in T2IF and IVIF environments.

DVF:Multi-agent Q-learning with difference value factorization

In cooperative multi-agent games, agents are required to learn effective cooperative behaviors within complex action spaces. An effective approach is to utilize the Individual-Global-Max (IGM) principle to decompose the global reward signal into individual contributions from each agent. However, existing methods confront two significant challenges: ensuring the accuracy of the value function decomposition process and guaranteeing monotonic improvement in joint policies. These challenges become exacerbated in scenarios involving non-monotonic matrices. To address these challenges, we introduce a novel and flexible value factorization method called Difference Value Factorization (DVF). The key idea of our method is to transform the IGM principle into a new form DVF-IGM, which addresses non-monotonic constraints by ensuring the consistency of the IGM process between the joint difference value and the complex non-linear sum of independent difference values. A centralized evaluator is employed to estimate global Q-values, which not only enhances expressiveness but also constructs difference values for updating individual value functions. We demonstrate that DVF-IGM is an equivalent transformation of IGM and that DVF has the monotonic improvement property. Empirically, our method has been shown to maintain and recover the optimal policy in non-monotonic matrix games and achieve state-of-the-art performance in cooperative tasks within the StarCraft Multi-Agent Challenge (SMAC).

Linear programming problems and matrix games

In this article, the relationship between linear optimization and matrix games is discussed, and in particular, the transformation of a linear program to a matrix game is considered.

On the solution of two-person zero-sum matrix games

In this article, 0-sum 2-person matrix games are considered. Some basic concepts and results are recalled. The numerical solution of matrix games by dual problems of linear optimization is discussed. Illustrative numerical examples are also included.

Solving constrained matrix games with fuzzy random linear constraints

In real-world games, players may face an uncertain environment where fuzziness and randomness coexist. The main difficulty in dealing with games involving fuzziness and randomness arises when comparing the payoffs. The purpose of this paper is to introduce a new approach to deal with constrained matrix games where the entries of the constraint matrices are LR-fuzzy random variables. Our methodology is based on constructing a new matrix game using the chance constraint method adapted to the probability-possibility measures. First, a specific type of saddle point is defined as an equilibrium solution. Then, conditions for the existence of the proposed solution are established. Further, a technique based on second-order programming for computing the saddle point is presented. Finally, a numerical illustration of the approach is provided.

Interval-valued Fermatean Neutrosophic Graph with Grey Wolf Optimization for Sarcasm Recognition on Microblogging Data

Game theory is more popular in competitive situations due to its importance in decision making. Several kinds of fuzzy sets can manage uncertainty in matrix games. Neutrosophic set theory has been instrumental in investigating ambiguity, complexity, inconsistency, and incompleteness in real-time issues. Nowadays, sarcastic comments on social media have become a general tendency. Sarcasm is frequently used by individuals to pester or taunt others. It is often conveyed via inflection, tonal stress in speech, or lexical, hyperbolic, and pragmatic features existing in the text. Sentiment Analysis (SA) is regarded as the data mining targets of sentiment organization of the client's criticisms obtainable in textual form. Sarcasm is a form of speech that states an individual's downside feeling through a positive term. Labeling sarcasm in characters is a dynamic task for Natural Language Processing to evade the misconception of sarcastic speeches as a verbatim declaration. The outcome of these kinds of sarcastic speeches is hard for the people and machines. Sarcasm has a considerable influence on the efficacy of SA techniques that are impacted by mendacious sentiments that frequently belong to sarcastic classes. This study introduces an Interval-valued Fermatean Neutrosophic Graph with Grey Wolf Optimization for Sentiment Analysis (IFeNG-GWOSA) on Microblogging Data. The IFeNG-GWOSA technique includes a sarcasm detection technique that categorizes words in sarcastic or non-sarcastic form. The initial phase is preprocessing, where the tokenization and stop word removal are implemented. Then, the preprocessed data is subjected to feature extraction, where the BERT word embedding is applied. The IFeNG model is used for sarcasm detection, and the grey wolf optimizer (GWO) generates its parameter selection technique. Lastly, the efficiency of the presented technique is compared with existing approaches under different measures

Management of the Traffic Networks Transportation in Supplier-Customers Services by Strategic Games

This paper presents an innovative approach to optimize traffic networks in a supplier-customer system based on specific strategies of game theory. The traffic network is represented as a routing configuration in which the arcs represent various possible roads between supplier and customer entities, each route having an associated transport cost for the product unit. The game structure is defined by a matrix of transport costs between the players, supplier and customer, with their corresponding strategies. Based on these costs, the game matrix is built to which the existing quantities in warehouses are added, respectively the quantities requested by the final customers. By solving the game associated to the transportation traffic network, the different strategies are proposed for choosing the optimal solution. The simulation results confirm the effectiveness of the proposed approach and recommend some performant procedures for solving the game, such as Hurwicz, Laplace and Savage, close to optimal solution. This research opens new directions for management optimization in other domains such as comunication, electrical and administrativ networks.

Solving matrix game using rough interval payoffs

Solving matrix game using rough interval payoffs

A non-linear mathematical approach for solving matrix games with picture fuzzy payoffs with application to cyberterrorism attacks

Numerous expansions of fuzzy sets have been proposed to manage uncertain information in real-world problems. Unlike fuzzy sets and intuitionistic fuzzy sets, picture fuzzy sets independently consider membership, non-membership, and indeterminacy. They have been extensively employed to delineate decision-makers’ perspectives when knowledge is lacking and have found application in numerous domains. However, little research has been conducted on uncertain matrix games in picture-fuzzy environments. This paper proposes an approach to resolving a matrix game with picture-fuzzy payoffs. Two non-linear mathematical programming models equipped with multi-objective functions are formulated and converted into crisp linear programming models by applying the weighted average approach. The reduced problems are solved using the LINGO platform, yielding optimal solutions. An illustrative example demonstrates how an information security agency can select the most effective strategies to combat cyberterrorism. Finally, comparing the presented methodology’s outcomes and existing work establishes its dependability and efficacy.

BATTLE PLANNING AND ORGANIZATION LOGISTICS SUPPORT IN THE FORMS AND METHODS OF NATO STANDARDS MILITARY MANAGEMENT TOOLS BASED ON THE USE OF THE GAME THEORY MATHEMATICAL APPARATUS

The subject matter of the article is the analysis of the decision-making options for combat operations in the sphere of operation of the S-4 section of the headquarters of the military unit. The goal of the study is the development of the methodology of the military raffle according to NATO standards for the analysis of decision-making options for combat operations by applying the mathematical apparatus of game theory. The tasks to be solved are: to conduct an analysis of the methods recommended by NATO standards for a military draw; from the point of view their viability objectivity assessing to formulate a matrix game based on an a posteriori analysis of action options to make a decision on planning and organizing a battle in the field of the S-4 section operation. The methodological basis of the research was general scientific and special methods of scientific knowledge. General scientific and special methods of scientific knowledge are used. The following results were obtained: The adequacy of the mathematical apparatus of game theory for the formal description of the decision-making process for combat operations, taking into account all possible risks, has been clarified. Conclusions The method of analyzing options for making a decision on combat operations using the mathematical apparatus of game theory is one that is devoid of subjectivity. The solution of the matrix game compiled on the basis of data on combat operations in the sphere of operation of the S-4 headquarters section of the military unit is the basis of the methodology of logistical support for the planning and organization of the battle in the forms and methods of the military management toolkit according to NATO standards.

ALGORITHMS FOR SOLVING MATRIX GAMES IN THE SPACE OF MIXED STRATEGIES

A number of game issues solutions is performed based on the payment matrix. In the respective model, conditionally, the first player attempts to win, while the second player strives not to lose. It is known that the analysis of this game is carried out based on the saddle element in the payment matrix. The element aij is called the saddle element, if it is the minimum of row number i and the maximum of column number j. In this case, the pure strategy of the first player corresponds to the i-th row of the matrix, and the pure strategy of the second player corresponds to the j-th column of the matrix. When there is no saddle element in the matrix, the analysis of the game is carried out in the space of mixed strategy. Experience shows that it is more efficient to perform this work by removing surplus rows and columns from the matrix. Keywords: matrix games, saddle element, pure strategies, mixed strategies, Braun-Robinson method.

A matrix game‐based coordinated optimisation method of distribution networks with multiple flexible resources

AbstractThe ‘Electric energy substitution’ project in rural areas of northern China aims to replace coal‐fired heating with electric heating. However, this could potentially lead to congestion problems on the local distribution networks (DNs) due to an increase in heating loads. Instead of expanding the lines, flexible resources such as microgrids (MGs) and heat pumps (HPs) could provide auxiliary services in a more cost‐effective manner. A bi‐level optimisation model that coordinates DNs, MGs, and HP clusters is proposed to address this issue. In the lower level, MGs and HP clusters provide auxiliary services through competitive bidding to maximise their own income over multiple periods, considering a series of technical operational constraints and comfort constraints. In the upper level, DNs clear the market based on the bidding information to minimise its operational costs while guaranteeing network constraints. The bi‐level optimisation model is formulated as a multi‐agent matrix game problem, and the Win or Learn Fast‐Policy Hill‐climbing algorithm is used to achieve fast market equilibrium in a decentralised manner. Simulation results demonstrate that the proposed method can improve the revenue of MGs by up to 2.6 times compared to the single period matrix game method, and reduce the convergence time by up to 81.3% compared to the multi‐agent Q‐learning method.

On a value of a matrix game with fuzzy sets of player strategies

The present paper investigates a fuzzy matrix game with fuzzy sets of player strategies. We construct a game value with the help of Zadeh's extension principle in combination with an approach to fuzzy matrix games worked out by Liu and Kao. We prove that the FSs of players strategies in a fuzzy matrix game generate a game value in the form of a type-2 fuzzy set (T2FS) on the real line. Furthermore, the corresponding type-2 membership function is given. It is shown that the T2FS of the game value can be decomposed according to the secondary membership grades into a finite collection of fuzzy numbers. Each of them is the value of the corresponding fuzzy matrix game for crisp sets of player strategies. These sets are the corresponding cuts of the original fuzzy sets of player strategies. Illustrative examples are given.