Matrix Games Research Articles

On the Analytics of Infinite Game Theory Problems

We consider zero-sum and a non zero-sum games of two players with generalized, not necessarily linear, utility functions and infinite, compact pure strategy spaces. Emphasis is given to comparisons with results obtained in mathematical theorems. The games chosen make specific points in relation to the conditions of the theorems. The idea of δ functions is exploited to construct mixed strategies. We interpret their significance in joining pure strategies and show the application in confirming NE. Uniqueness of NE is looked at. An issue is also how far an analogy can be drawn from the case of the finite matrix games. The usually discussed game theory problems are easy to analyze but they do not cover the whole range of possibilities.

Solution of interval-valued matrix games using intuitionistic fuzzy optimisation technique: an effective approach

Solution of interval-valued matrix games using intuitionistic fuzzy optimisation technique: an effective approach

НЕЧІТКІ МОДЕЛІ АНТАГОНІСТИЧНИХ ІГОР

The article is a continuation of a series of works on modeling situations in competitive markets at both micro and macro levels and the development of approaches to finding solutions to the obtained models. The paper proposes a method for solving a certain class of game-theoretic models under conditions of uncertainty. It is substantiated that a significant part of the problems of economic competition can be reduced to a finite matrix game of two players with zero sum, the matrix of winnings of the first player which has a specific form. Given the high degree of uncertainty in modern domestic markets and the need to simplify the current situation in its modeling due to the impossibility of including in the developed model of all real multifaceted relationships, the article considers antagonistic games with fuzzy parameters. It is proposed to look for the solution of the considered class of finite matrix games by reducing them to two dual optimization problems of linear programming with flexible limit constraints. The case is considered when the coefficients in the system of constraints of these models of linear programming are approximated by piecewise-linear membership functions, because they do not raise the question of linearity of the studied models. Using certain linear transformations, the optimization models of linear programming obtained in this work are reduced to models of a special kind, the method of solving which has been developed by other scientists. The essence of this method is that according to the Bellman-Zadeh approach, the resulting fuzzy model is reduced to the decision problem described by the multi-purpose optimization model, the solution of which includes only those alternatives, in such problems are called Pareto effective. Using this method, the fuzzy model obtained in the work is reduced to a "clear" problem of linear programming, some parameters of which are rationally determined by the person making managerial decisions, based on certain limitations obtained by solving two "clear" optimization models with known coefficients. By finding the solution to these dual problems and calculating the mixed strategies of the two players, the person making management decisions will be able to make the right choice among a set of alternative solutions.

IsoMaTrix: a framework to visualize the isoclines of matrix games and quantify uncertainty in structured populations.

Evolutionary game theory describes frequency-dependent selection for fixed, heritable strategies in a population of competing individuals using a payoff matrix. We present a software package to aid in the construction, analysis and visualization of three-strategy matrix games. The IsoMaTrix package computes the isoclines (lines of zero growth) of matrix games, and facilitates direct comparison of well-mixed dynamics to structured populations on a lattice grid. IsoMaTrix computes fixed points, phase flow, trajectories, (sub)velocities and uncertainty quantification for stochastic effects in spatial matrix games. We describe a result obtained via IsoMaTrix's spatial games functionality, which shows that the timing of competitive release in a cancer model (under continuous treatment) critically depends on the initial spatial configuration of the tumor. The code is available at: https://github.com/mathonco/isomatrix. Supplementary data are available at Bioinformatics online.

Predicting Nash equilibria for microbial metabolic interactions.

Microbial metabolic interactions impact ecosystems, human health and biotechnology profoundly. However, their determination remains elusive, invoking an urgent need for predictive models seamlessly integrating metabolism with evolutionary principles that shape community interactions. Inspired by the evolutionary game theory, we formulated a bi-level optimization framework termed NECom for which any feasible solutions are Nash equilibria of microbial community metabolic models with/without an outer-level (community) objective function. Distinct from discrete matrix games, NECom models the continuous interdependent strategy space of metabolic fluxes. We showed that NECom successfully predicted several classical games in the context of metabolic interactions that were falsely or incompletely predicted by existing methods, including prisoner's dilemma, snowdrift and cooperation. The improved capability originates from the novel formulation to prevent 'forced altruism' hidden in previous static algorithms while allowing for sensing all potential metabolite exchanges to determine evolutionarily favorable interactions between members, a feature missing in dynamic methods. The results provided insights into why mutualism is favorable despite seemingly costly cross-feeding metabolites and demonstrated similarities and differences between games in the continuous metabolic flux space and matrix games. NECom was then applied to a reported algae-yeast co-culture system that shares typical cross-feeding features of lichen, a model system of mutualism. 488 growth conditions corresponding to 3221 experimental data points were simulated. Without training any parameters using the data, NECom is more predictive of species' growth rates given uptake rates compared with flux balance analysis with an overall 63.5% and 81.7% reduction in root-mean-square error for the two species respectively. Simulation code and data are available at https://github.com/Jingyi-Cai/NECom.git. Supplementary data are available at Bioinformatics online.

Об одном алгоритме поиска седловой точки для непрерывных линейных игр применительно к задачам защиты информации

The paper introduces a game formulation of the problem of two players: the defender determines the security levels of objects, and the attacker determines the objects for attack. Each of them distributes his resources between the objects. The assessment of a possible damage to the defender serves as an indicator of quality. The problem of a continuous zero-sum game under constraints on the resources of the players is formulated so that each player must solve his own linear programming problem with a fixed solution of the other player. The purpose of this research was to develop an algorithm for finding a saddle point. The algorithm is approximate and based on reducing a continuous problem to discrete or matrix games of high dimension, since the optimal solutions are located at the vertices or on the faces of the simplices which determine the sets of players' admissible solutions, and the number of vertices or faces of the simplices is finite. In the proposed algorithm, the optimization problems of the players are sequentially solved with the accumulated averaged solution of the other player, in fact, the ideas of the Brown --- Robinson method are used. An example of solving the problem is also given. The paper studies the dependences of the number of algorithm steps on the relative error of the quality indicator and on the dimension of the problem, i.e., the number of protected objects, for a given relative error. The initial data are generated using pseudo-random number generators

Tarım Sigortası Gerekliliğinin Oyun Teorisi Yardımıyla Gösterilmesi: Matris Norm Yaklaşımı

In this paper, we investigate whether the farmers should get agriculture insurance or not under the uncertainty by the view of the game theory and decision theory. For this purpose, we create a zero-sum matrix game against nature using the real data. We solve the matrix game with Wald maximin criterion and Savage regret criterion, separately. Then, we resolve this game, while the mixed strategies are allowed, by using the well-known method in the literature for the solution of the zero-sum matrix games. Later, we demonstrate the game in the different aspect with the approach only consists of the matrix norms of the payoff matrix, called the matrix norm approach (MNA), under the same condition. Finally, we show the consistency of the results obtained by using MNA and the other methods.

Node Attitude Aware Information Dissemination Model Based on Evolutionary Game in Social Networks

The information dissemination in social networks is affected by many factors. However, node attitude which is an important influence factor of information dissemination in social network have not been fully considered in the previous works. Aiming at the problem of the influence of node attitude on information dissemination, this paper proposes an information propagation model based on evolutionary game. Firstly, from the individual point of view, the node’s attitude update rules are defined according to non-Bayesian social learning rules. Secondly, an inter-node game matrix based on attitude value is established. Based on the evolution analysis paradigm, an information dissemination model based on node attitude is established. The equilibrium solution of dynamic equations is replicated for both positive and negative attitudes, and the corresponding equilibrium points are stabilized. The validity of the proposed model is verified by numerical analysis and simulation experiments. They all show that the different attitudes of nodes play an important role in information dissemination.

Noncooperative game theory in response surface methodology decision of pricing strategy in dual-channel supply chain

ABSTRACT Manufacturers utilize dual-channel supply-chain to cope with customer preferences. This structure allows manufacturer’s online facility and conventional retailer to deliver their products more efficiently. However, this mechanism also brings up anew internal competition between manufacturers and retailers. To observe the price effect within this system, response surface methodology is used to model the customer preference. The independent variables are online and retailer prices, while the dependent variables are price and cross-price sensitivities. To simulate proposed model into real-practice, game theory is employed. The strategies are the priority of each channel price and cross-price sensitivities. Based on this game model, a Nash equilibrium is obtained. This research proposes anovelty by means of the usage of response surface results as game theory payoffs to represent the uncertainty in a competitive environment. The integration of response surface model and game matrix can be used as managerial implication in predicting competitors’ behaviour.

Differential evolution particle swarm optimization algorithm based on good point set for computing Nash equilibrium of finite noncooperative game

In this paper, a hybrid differential evolution particle swarm optimization (PSO) method based on a good point set (GPDEPSO) is proposed to compute a finite noncooperative game among N people. Stochastic functional analysis is used to prove the convergence of this algorithm. First, an ergodic initial population is generated by using a good point set. Second, PSO is proposed and utilized as the variation operator to perform variation crossover selection with differential evolution (DE). Finally, the experimental results show that the proposed algorithm has a better convergence speed, accuracy, and global optimization ability than other existing algorithms in computing the Nash equilibrium of noncooperative games among N people. In particular, the efficiency of the algorithm is higher for determining the Nash equilibrium of a high-dimensional payoff matrix game.

Gaddum’s test for symmetric cones

A real symmetric matrix A is copositive if $$\left\langle {Ax},{x}\right\rangle \ge 0$$ for all x in the nonnegative orthant. Copositive programming gained fame when Burer showed that hard nonconvex problems can be formulated as completely-positive programs. Alas, the power of copositive programming is offset by its difficulty: simple questions like “is this matrix copositive?” have complicated answers. In 1958, Jerry Gaddum proposed a recursive procedure to check if a given matrix is copositive by solving a series of matrix games. It is easy to implement and conceptually simple. Copositivity generalizes to cones other than the nonnegative orthant. If K is a proper cone, then the linear operator L is copositive on K if $$\left\langle {L \left( {x}\right) },{x}\right\rangle \ge 0$$ for all x in K. Little is known about these operators in general. We extend Gaddum’s test to self-dual and symmetric cones, thereby deducing criteria for copositivity in those settings.

New expected impact functions and algorithms for modeling games under soft sets

Soft set is the power tool to deal with uncertainty in a parametric manner. In applications of soft set, one of the most important steps is to define mappings on soft sets. In this study, we model theory of game under theory of soft set which is an effective tool for handling uncertainties events and problems that may exist in a game. To this end, we first define some expected impact functions of players in soft games. Then, we propose three new decision making algorithms to solve the 2.2 × p, 2 . n × p and m . 2 × p soft matrix games, which cannot be settled by the relevant soft methods such as saddle points, lover and upper values, dominated strategies and Nash equilibrium. The proposed soft game algorithms are illustrated by examples.

Multi-objective linguistic-neutrosophic matrix game and its applications to tourism management

Game theory plays an important role in numerous decision-oriented real-life problems. Nowadays, many such problems are basically characterized by various uncertainties. Uncertainties come to happen due to decision makers' collection of data, intuition, assumption, judgement, behaviour, evaluation and lastly, due to the problem itself. Fuzzy concept with membership degree made an initialization towards the treatment of uncertainty, but it was not enough. Intuitionistic fuzzy concept was evolved concerning with both membership and non-membership degrees but failed to express reality more accurately. Then, neutrosophy logic was developed with a new degree in uncertainty, say, indeterminacy degree besides membership and non-membership degrees. Multi-objective optimization is an area of multiple-criteria decision making related with mathematical optimization problems involving more than one objective function to be optimized at the same time. Game theory (matrix game) problems with imprecise, vague information, like neutrosophic, can be formed with multiple objective functions. We develop and analyse a matrix game with multiple objectives, and solve the problem under a single-valued neutrosophic environment in linguistic approach. The main achievement of our study is that we here introduce a problem-oriented example to justify our designed methodologies with a successful real-life implications using linguistic neutrosophic data rather than crisp data as used in previous researches.

Solving matrix games based on Ambika method with hesitant fuzzy information and its application in the counter-terrorism issue

The hesitant fuzzy set has been studied as a powerful tool to describe the decision makers’ judgements under uncertain environment and applied to many domains. For solving the matrix games whose payoffs are expressed by the hesitant fuzzy information, the paper proposes the Ambika method of hesitant fuzzy matrix games (HFMGs). In this paper, firstly, the formal representation of HFMGs is established to meet the conditions of two-person finite zero-sum games. Secondly, after a new method of adding elements to the shorter hesitant fuzzy elements (HFEs), i.e. the hesitant fuzzy elements with possibility, is developed to keep the same length of HFEs, a weighting method based on the position of element in the HFEs is proposed. Then the hesitant fuzzy bi-objective nonlinear programming models for both players are established for HFMGs. Thirdly, according to the proposed value and ambiguity indexes, the Ambika method of HFMGs is developed to find the optimal solutions of mixed strategies by solving the converted linear programming models. Finally, as the illustration of the proposed method, a numerical example about how to choose the optimal solutions for a state security department is given in the counter-terrorism issue.

A game-theoretic approach for stochastic estimation of equilibrium in land use data: stochastic estimation of equilibrium in land use data

This work is focused on the use of linear games in the normal form for evaluation of Nash equilibria probability distribution in spatial data eg land use. Spatial data with varied content represent the observed state and, as a rule, do not automatically offer information about what that state is the result of. However, individual cases can be defined in such data, representing, for example, different sizes or degrees of representation of selected characteristics. The essence of the proposed approach is that such data pertaining to individual cases are regarded as payoff values of multiple interacting entities and can form a matrix of a symmetric linear game in the normal form. For this game, an NE representing a particular distribution of strategies of interacting entities can then be determined. This distribution assigns to each case the share of NE depending on the respective location in the symmetric game matrix. However, spatial data does not provide any information for the specific design of this deployment. The proposed solution is therefore stochastic: all effective permutations of the game matrix or a multidimensional symmetric game configuration leading to a different result for the NE distribution are evaluated. The result are values of the occurrence of NE probability for individual evaluated cases. The properties of the method are tested on the example of a practical application. The results show that the method is applicable for evaluating the spatial distribution of the land use stability.

Expected Worth for 2 2 Matrix Games with Variable Grid Sizes

We offer a detailed examination of a broad class of 2 × 2 matrix games as a first step toward considering measures of resource distribution and efficiency of outcomes. In the present essay, only noncooperative equilibria and entropic outcomes are considered, and a crude measure of efficiency employed. Other solution concepts and the formal construction of an efficiency index will be addressed in a companion paper.

What is a Solution to a Matrix Game

These notes are provided to describe many of the problems encountered concerning both structure and behavior in specifying what is meant by the solution to a game of strategy in matrix or strategic form. In the short term in particular, it is often reasonable for the individual to accept as given, both the context in which decisions are being made and the formal structure of the rules of the game. A solution is usually considered as a complete set of equations of motion that when applied to the game at hand selects a final outcome. There are many different theories and conjectures about how games of strategy are, or should be played. Several of them are noted below. They are especially relevant to the experimental gaming facility noted in the companion paper.

Solving matrix games with hesitant fuzzy pay-offs

The objective of this paper is to develop matrix games with pay-offs of triangular hesitant fuzzy elements (THFEs). To solve such games, a new methodology has been derived based on the notion of weighted average operator and score function of THFEs. Firstly, we formulate two non-linear programming problems with THFEs. Then applying the score function of THFEs, we transform these two problems into two non-linear multi-objective programming problems with triangular fuzzy numbers (TFNs). Finally, the Lexicographic method is used to solve these two multi-objective programming problems. A market share problem is considered to show the validity and applicability of the proposed methodology.

Completely mixed bimatrix games

Kaplansky (Ann. Math. 46(3) (1945) 474–479) introduced the notion of completely mixed games. Fifty years later in 1995, he wrote another beautiful paper where he gave a set of necessary and sufficient conditions for a skew symmetric matrix game to be completely mixed. In this work, we attempt to answer when bimatrix games will be completely mixed. In particular, we give a set of necessary and sufficient condition for a bimatrix game (A, B) to be completely mixed when A and B are skew symmetric matrices of order 3. We give several examples to show the sharpness of our result. We also formulate a conjecture when A and B are skew symmetric matrices of order 5.

The Best and Worst of All Possible Worlds: Some Crude Evaluations

The 2 x 2 matrix game plays a central role in the teaching and exposition of game theory. It is also the source of much experimentation and research in political science, social psychology, biology and other disciplines. This brief paper is addressed to answering one intuitively simple question without going into the many subtle qualifications that are there. How efficient is the non-cooperative equilibrium? This is part of a series of several papers that address many of the qualifications concerning the uses of the 2 x 2 matrix games.