Matrix Games Research Articles

Stochastic Evolutionary Stability in Matrix Game with Three Pure Strategies Models

Stochastic Evolutionary Stability in Matrix Game with Three Pure Strategies Models

Python Development of Algorithm to find Solution in Pure Strategies of Antagonistic Matrix Game

The article provides an algorithm for finding the optimal solution to the matrix antagonistic game in pure strategies. The algorithm implements the search for the upper and lower prices of the matrix game by implementing the maximal and minimax strategies of the players. The algorithm is encoded using the functionality of the Python programming language. The algorithm is tested using a numerical example.

Applying opponent and environment modelling in decentralised multi-agent reinforcement learning

Multi-agent reinforcement learning (MARL) has recently gained popularity and achieved much success in different kind of games such as zero-sum, cooperative or general-sum games. Nevertheless, the vast majority of modern algorithms assume information sharing during training and, hence, could not be utilised in decentralised applications as well as leverage high-dimensional scenarios and be applied to applications with general or sophisticated reward structure. Thus, due to collecting expenses and sparsity of data in real-world applications it becomes necessary to use world models to model the environment dynamics, using latent variables — i.e. use world model to generate synthetic data for training of MARL algorithms. Therefore, focusing on the paradigm of decentralised training and decentralised execution, we propose an extension to the model-based reinforcement learning approaches leveraging fully decentralised training with planning conditioned on neighbouring co-players’ latent representations. Our approach is inspired by the idea of opponent modelling. The method makes the agent learn in joint latent space without need to interact with the environment. We suggest the approach as proof of concept that decentralised model-based algorithms are able to emerge collective behaviour with limited communication during planning, and demonstrate its necessity on iterated matrix games and modified versions of StarCraft Multi-Agent Challenge (SMAC).

Replicator dynamics generalized for evolutionary matrix games under time constraints.

One of the central results of evolutionary matrix games is that a state corresponding to an evolutionarily stable strategy (ESS) is an asymptotically stable equilibrium point of the standard replicator dynamics. This relationship is crucial because it simplifies the analysis of dynamic phenomena through static inequalities. Recently, as an extension of classical evolutionary matrix games, matrix games under time constraints have been introduced (Garay et al. in J Theor Biol 415:1-12, 2017; Křivan and Cressman in J Theor Biol 416:199-207, 2017). In this model, after an interaction, players do not only receive a payoff but must also wait a certain time depending on their strategy before engaging in another interaction. This waiting period can significantly impact evolutionary outcomes. We found that while the aforementioned classical relationship holds for two-dimensional strategies in this model (Varga et al. in J Math Biol 80:743-774, 2020), it generally does not apply for three-dimensional strategies (Varga and Garay in Dyn Games Appl, 2024). To resolve this problem, we propose a generalization of the replicator dynamics that considers only individuals in active state, i.e., those not waiting, can interact and gain resources. We prove that using this generalized dynamics, the classical relationship holds true for matrix games under time constraints in any dimension: a state corresponding to an ESS is asymptotically stable. We believe this generalized replicator dynamics is more naturally aligned with the game theoretical model under time constraints than the classical form. It is important to note that this generalization reduces to the original replicator dynamics for classical matrix games.

Density amplifiers of cooperation for spatial games

Spatial games provide a simple and elegant mathematical model to study the evolution of cooperation in networks. In spatial games, individuals reside in vertices, adopt simple strategies, and interact with neighbors to receive a payoff. Depending on their own and neighbors' payoffs, individuals can change their strategy. The payoff is determined by the Prisoners' Dilemma, a classical matrix game, where players cooperate or defect. While cooperation is the desired behavior, defection provides a higher payoff for a selfish individual. There are many theoretical and empirical studies related to the role of the network in the evolution of cooperation. However, the fundamental question of whether there exist networks that for low initial cooperation rate ensure a high chance of fixation, i.e., cooperation spreads across the whole population, has remained elusive for spatial games with strong selection. In this work, we answer this fundamental question in the affirmative by presenting network structures that ensure high fixation probability for cooperators in the strong selection regime. Besides, our structures have many desirable properties: (a) they ensure the spread of cooperation even for a low initial density of cooperation and high temptation of defection, (b) they have constant degrees, and (c) the number of steps, until cooperation spreads, is at most quadratic in the size of the network.

Partial sums of antagonistic games and their applications

An abstract algebraic approach to the formation of high-dimensional matrix games is proposed. The considered class of games includes, in particular, statistical games. A general method for constructing randomized decision rules for the problem of estimating the parameter of a discrete random variable is discussed. For economically significant tasks, the cost of additional observation is calculated.

Supervision mechanism of integrated construction consultancy services in China: tripartite game analysis

PurposeMegaprojects contribute greatly to China’s socio-economic development. However, the diverse stakeholders of megaprojects tend to be prone to conflict and plunge the project into lower performance than expected. This study aims to investigate (1) the optimal supervision mechanism under direct supervision of the owner and (2) the optimal indirect supervision mechanism of the owner, with involvement of integrated construction consultancy (ICC) in supervision and considerations into the impact of collusion between the ICC and the general contractor on supervision efficiency.Design/methodology/approachThis study applies the principal–agent theory and game theory to design effective incentive and punishment mechanisms for direct and indirect supervision by the owner. Through the theoretical framework of game theory, it analyses the strategic interactions between the owner, the ICC and the general contractor, and evaluates the effectiveness of various supervision mechanisms based on Nash equilibrium theory.FindingsThe results showed adopting the direct supervision mechanism when the probability of the general contractor’s opportunistic behavior being detected was higher or the occurrence had a greater impact on the owner. However, the increase in supervision cost would reduce the motivation of owner’s direct supervision and prompt the owner to choose the indirect supervision mechanism. Additionally, the indirect supervision mechanism was more likely to inhibit the collusion between the ICC and the general contractor. Thus, increasing penalties for ICC would result in improved supervision of ICC, thereby improving the benefits to the owner.Originality/valueThis study examines the dynamic interaction between the owner, the general contractor and the ICC. The study establishes a game tree and game matrix based on principal–agent theory, and analyses direct and indirect supervision models to determine the Nash equilibrium and optimal supervision strategy that is in the best interest of the owner. The study advocates for improved management and supervision mechanisms and argues that tailored supervision mechanisms can significantly improve the performance of megaprojects, which is well validated in megaprojects in China.

Credible Negotiation for Multi-agent Reinforcement Learning in Long-term Coordination

The coordination of multi-agent is one of the critical problems in Multi-Agent Reinforcement Learning (MARL). The traditional methods of MARL focus on finding a stochastically acceptable solution called Nash Equilibrium (NE) for all agents from the Markov game in which multiple equilibria exist. However, learning a fair equilibrium is crucial for the sustainability and stability of collaboration in the long-term coordination game, especially when the leadership competition exists. In this paper, we propose the bi-level reinforcement learning method N-Bi-AC, whose solution is a Pareto improvement for traditional NE, to choose a fair Equilibrium. There are two parts in our method, the first is that we propose the Negotiator to determine the leader in stage game, and the other is to update the Q-value of agents in the game by using a bi-level actor-critic learning method based on the Joint Mixed Strategy Equilibrium Q-learning algorithm (JMSE Q-learning). The convergence proof is given, and the learning algorithm is compared with the state-of-the-art algorithms. We found that the proposed N-Bi-AC method successfully converged to a fair Nash Equilibrium, which guarantees the fairness of agents in different matrix game environments.

GDT: Multi-agent reinforcement learning framework based on adaptive grouping dynamic topological space

In many real-world scenarios, tasks involve coordinating multiple agents, such as managing robot clusters, drone swarms, and autonomous vehicles. These tasks are commonly addressed using Multi-Agent Reinforcement Learning (MARL). However, existing MARL algorithms often lack foresight regarding the number and types of agents involved, requiring agents to generalize across various task configurations. This may lead to suboptimal performance due to underestimated action values and the selection of less effective joint policies. To address these challenges, we propose a novel multi-agent deep reinforcement learning framework, called multi-agent reinforcement learning framework based on adaptive grouping dynamic topological space (GDT). GDT utilizes a group mesh topology to interconnect the local action value functions of each agent, enabling effective coordination and knowledge sharing among agents. By computing three different interpretations of action value functions, GDT overcomes monotonicity constraints and derives more effective overall action value functions. Additionally, GDT groups agents with high similarity to facilitate parameter sharing, thereby enhancing knowledge transfer and generalization across different scenarios. Furthermore, GDT introduces a strategy regularization method for optimal exploration of multiple action spaces. This method assigns each agent an independent entropy temperature during exploration, enabling agents to efficiently explore potential actions and approximate total state values. Experimental results demonstrate that our approach, termed GDT, significantly outperforms state-of-the-art algorithms on Google Research Football (GRF) and the StarCraft Multi-Agent Challenge (SMAC). Particularly in SMAC tasks, GDT achieves a success rate of nearly 100% across almost all Hard Map and Super Hard Map scenarios. Additionally, we validate the effectiveness of our algorithm on Non-monotonic Matrix Games.

Dynamic Operational Planning in Warfare: A Stochastic Game Approach to Military Campaigns

ABSTRACTWe study a two‐player discounted zero‐sum stochastic game model for dynamic operational planning in military campaigns. At each stage, the players manage multiple commanders who order military actions on objectives that have an open line of control. When a battle over the control of an objective occurs, its stochastic outcome depends on the actions and the enabling support provided by the control of other objectives. Each player aims to maximize the cumulative number of objectives they control, weighted by their criticality. To solve this large‐scale stochastic game, we derive properties of its Markov perfect equilibria by leveraging the logistics and military operational command and control structure. We show the consequential isotonicity of the optimal value function with respect to the partially ordered state space, which in turn leads to a significant reduction of the state and action spaces. We also accelerate Shapley's value iteration algorithm by eliminating dominated actions and investigating pure equilibria of the matrix game solved at each iteration. We demonstrate the computational value of our equilibrium results on a case study that reflects representative operational‐level military campaigns with geopolitical implications. Our analysis reveals a complex interplay between the game's parameters and dynamics in equilibrium, resulting in new military insights for campaign analysts.

Game model of high-speed railway delay and passenger choice behavior under incomplete information

The arrival status of high-speed trains is an important factor affecting the choice behavior of railway passengers. In order to specifically analyze the influence of the arrival state of high-speed trains on the choice behavior of railway passengers, evolutionary game theory was used to analyze the arrival state of high-speed trains and the choice behavior of railway passengers, and the game matrix of the arrival state of high-speed trains and the choice behavior of railway passengers was constructed. The location of the equilibrium point and the system evolution mechanism are deeply analyzed. Matlab simulation results show that the evolutionary game behavior finally has two evolutionary stable states. When high-speed trains delay time is less than 30 min , the train tends to arrive on time, and the railway passengers tend to choose the ride strategy. When the delay time is greater than 30min, the train tends to arrive on time, and the railway passengers tend to choose the refund strategy. By improving the facilities in the high-speed railway station, the railway passengers can choose the transfer strategy. Changing the value of the game parameters can adjust the evolution direction of both sides of the game and effectively improve the proportion of railway passengers when the high-speed train is delayed.

Value-Positivity for Matrix Games

Matrix games are the most basic model in game theory, and yet robustness with respect to small perturbations of the matrix entries is not fully understood. In this paper, we introduce value positivity and uniform value positivity, two properties that refine the notion of optimality in the context of polynomially perturbed matrix games. The first concept captures how the value depends on the perturbation parameter, and the second consists of the existence of a fixed strategy that guarantees the value of the unperturbed matrix game for every sufficiently small positive parameter. We provide polynomial-time algorithms to check whether a polynomially perturbed matrix game satisfies these properties. We further provide the functional form for a parameterized optimal strategy and the value function. Finally, we translate our results to linear programming and stochastic games, where value positivity is related to the existence of robust solutions. Funding: This research was supported by Fondation CFM pour la Recherche, the H2020 European Research Council [Grant ERC-CoG-863818 (ForM-SMArt)], the Austrian Science Fund [Grant 10.55776/COE12], ANID Chile [Grant ACT210005], and Agence Nationale de la Recherche [Grant ANR-21-CE40-0020].

Solving MADM Problems based on Matrix Games with a New Defuzzification Method for Triangular Intuitionistic Fuzzy Numbers

Abstract: The paper discusses about a new defuzzification method called the Comprehensive Li-Samuel ranking for Triangular Intuitionistic Fuzzy Numbers (TrIFNs) which is improved from the Li-Samuel ranking method associated with membership function and Li-Samuel ranking method associated with non-membership function. The Comprehensive Li-Samuel ranking for TrIFNs is utilised to derive the weights for the Multiple Attribute Decision Making (MADM) problems under TrIFN. The MADM problem is solved based on two-person zero-sum game which is converted into a pair of LPP by using Comprehensive Li-Samuel ranking. The process of transforming decision matrix into a matrix game and deriving decision maker’s and attribute weights from the matrix game are discussed. Numerical illustration is given to justify the viability and effectiveness of the proposed method and comparisons are made with existing ranking methods. The comparison study reveals that the proposed method provides the best maximum optimal solution for the decision problem than the existing methods

Проблемы поиска и преследования беспилотных летательных аппаратов с использованием теоретико-игрового подхода

Currently, the application of mathematical modeling theory is widely used in various fields and, among other things, can be used to prevent illegal actions. This study examines scenarios involving a single pursuer tracking a single evader, as well as situations involving multiple pursuers pursuing multiple evaders. The authors formulate this problem as a search and pursuit problem for four-rotor unmanned aerial vehicles (UAVs) or quadcopters. The solution to the problem is based on game theory, as it provides a mathematical framework for modeling and studying strategic interactions involving multiple decision makers. The authors consider a game where the set of strategies of the runner is the set of possible combinations of speeds and directions of his movement, and the set of strategies of the pursuer is the set of all possible permutations of the elements of his speeds. The matrix of the resulting game consists of elements that are the time of capture. A mathematical problem on the optimal assignments of searching and pursuing unmanned aerial vehicles is formulated and solved. Purpose: optimizing the effectiveness of strategies for detecting and capturing four-rotor unmanned aerial vehicles (quadcopters). Methods: methods of mathematical modeling, game theory apparatus, Hungarian method of solving the problem of assignments, decision theory, the principle of dynamic programming, the Maple package for solving examples were used. Results: In order to fulfill the conditions for solving the problem of optimal assignments, it is necessary and sufficient that it is balanced. This assignment problem can be balanced by entering the required number of fictitious boats or escaping boats. After that, it is possible to formulate and solve the dual problem of optimal appointments. The resulting game can be solved by any method of solving matrix games. In this way, it is possible to determine the policy of harassment and search between drones. Practical importance: All escaped quadcopter UAVs can be chased and successfully intercepted using the developed models. Examples of the study of mathematical models using the Maple software package are given.

An expert system of recommendations for combating cyber threats using CVSS metrics and game theory

This study is focused on the creation of an expert system for generating recommendations on cyber security. The developed expert system uses a game-theoretic model as a inference engine to transform expert knowledge into recommendations for end-users, who may be chief IT security officers (CISOs), system administrators, or cyber security engineers. Expert knowledge is presented in the form of an estimate of the base group of CVSS metrics - Common Vulnerability Score System, for each type of attack and adjusted values of CVSS in the case that the counterattack strategy is applied. Given a set of attacks and a base of expert attack knowledge, the system generates a game matrix of zero-sum game with a cybercriminal and a cyberdefense expert as players. The inference engine of the expert system is a game-theoretic model responsible for solving the game using the Brown-Robinson iterative method and generating cyber protection recommendations. An experiment was conducted on the convergence of the BrownRobinson algorithm on the 2022 vulnerability dataset from the Cybersecurity and Infrastructure Security Agency database, as a result of which it was determined that the convergence of the algorithm for solving the matrix game is achieved at a number of iterations of 1000. As a result of the work, expert system was designed and implemented along with the Web interface, which provides input by experts of CVSS level assessments of collected threats, threats countermeasures and output of recommendations for combating cyber threats

Dynamically Unstable ESS in Matrix Games Under Time Constraints

Dynamically Unstable ESS in Matrix Games Under Time Constraints

A probabilistic distributed digital twins approach for short-term stability and islanding of smart grid

The simultaneous evaluation of voltage, frequency, and transient stability enhances the reliability of the power systems, due to the transient effect of stability aspects on each other. However, different quantitative and qualitative parameters in the aspect and nonlinear structural analysis of the system make simultaneous and accurate predictions difficult. For this reason, the simultaneous probabilistic prediction of the stability state of three aspects with high certainty can be considered the most accurate method. This study presents a simultaneous prediction method for the stability aspect states by means of the decision theory and the game theory matrix on the basis of distributed digital twins (DDTs). The decision matrix weighs the aspect states in each bus using the distributed structure measurement and prioritizes the weak buses. In addition, the stability of different bus aspects is predicted using the game matrix. Furthermore, the changes in the system stability are predicted with high probability and certainty in the digital twins (DTs) graph, and if required, islanding is performed on the basis of the weak buses' priorities. The effects of the three aspects on each other, and the use of real-time data of the DDTs, have contributed to the high accuracy and speed of the proposed method. Simulations were performed on IEEE 14-bus, 39-bus, and 118-bus systems by means of MATLAB and DIgSILENT. The results show the advantages of the proposed method over other methods.

Using business games in administration teaching: A case study of the entrepreneurial Matrix Game

Business games, also known as business simulation games, are educational tools used in teaching business administration that simulate real-life business scenarios. They provide a controlled environment for students to practice decision-making, problem-solving, and critical thinking skills in a risk-free environment. In this article we applied the Entrepreneurial Matrix Company Game to a sample group of undergraduate business students. This game is a learning methodology that simulates the step-by-step creation of a new business model in a controlled environment. It is an interactive tool that allows participants to experience different situations and challenges that may arise when opening a new business. Our objective was to identify the importance of such a tool for entrepreneurial motivation, and at the end of this research we identified that the company game Matriz Empreendedora can help identify potential obstacles and opportunities, allowing student-entrepreneurs to better prepare themselves to launch a new business.

Оценка эффективности землепользования в городских агломерациях на основе регрессионно-игрового моделирования

Land use processes are the basic economic processes that determine the efficiency of the functioning of socio-economic systems. This is especially true for large urban agglomerations, for which these processes directly and indirectly significantly affect the volume of revenues to the budgets of various levels, as well as determine the investment attractiveness of the regions. As a result, the principles of rational land use should be based on the results of system analysis and mathematical modeling of the impact of external and internal environmental factors on all subjects involved in the processes of using land plots as the most important resource of the urban economy. The article is devoted to the development of two-level models that involve the use of regression analysis to determine the characteristics of the game matrix in the presence of a sufficiently large number of players with different interests. This approach is distinguished by the possibility of more complete consideration of factors influencing the efficiency of land use when assessing the intensity of land use in large cities and agglomerations, in conditions of external volatility. As a result, the construction of a multiple regression model describing the influence of factors, and the subsequent matrix modeling of game-theoretic conflict situations in land use makes it possible to support decision-making in the development of investment programs for the development of territories. The article discusses an example of the application of the proposed approach to the analysis of factors that have a significant impact on the efficiency of land use in the Moscow agglomeration. The results of the regression analysis led to the conclusion that the predicate of increasing financial and economic returns in the implementation of land use processes is the level of investment in fixed assets. On the example of the Moscow agglomeration, it is also shown that the obtained results of regression analysis can be used in the construction of game-theoretic models of land use, which, in turn, are expedient to apply to support decisions on state management of the market of land plots for various purposes.

Matrix norm methods for zero-sum fuzzy matrix games with payoffs of triangular fuzzy numbers

In this paper, we mainly consider the solution of two-person zero-sum fuzzy matrix games with payoffs of triangular fuzzy numbers. Contrary to the literature, we focus on developing the methods to solve the game directly based on only the norms of the payoff matrix holistically, without solving a linear programming problem or handling sub-games created by taking the components of fuzzy numbers separately. For this purpose, we first present fuzzy versions of 1-norm and ∞-norm with the help of a ranking function and develop the fuzzy matrix norm method to obtain an approximate solution of the zero-sum fuzzy matrix game. In addition to this approach, we provide the fuzzy extended matrix norm method as an enhanced version of the method, which involves the use of newly defined fuzzy matrix norms. Thus, we managed to avoid the complexity of the optimization process of the linear programming problem via the proposed matrix norm-based methods. Finally, we illustrate the implementation of the methods by considering several benchmark examples and a 3 × 3 fuzzy matrix game.