Interval Payoffs Research Articles

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.

Solving matrix game using rough interval payoffs

Solving matrix game using rough interval payoffs

Fuzzy Weighted Pareto–Nash Equilibria of Multi-Objective Bi-Matrix Games with Fuzzy Payoffs and Their Applications

Based on our previous research, this paper further discusses the multi-objective bi-matrix game with fuzzy payoffs (MBGFP), which is a special case of the fuzzy constrained multi-objective game with fuzzy payoffs. We first prove that any bi-matrix game with interval payoffs (BGIP) has at least one Pareto–Nash equilibrium. Then, with the help of BGIP, we obtain the necessary and sufficient conditions for the existence of fuzzy Pareto–Nash equilibrium of MBGFP. Secondly, based on the bilinear programming method for calculating Nash equilibrium in crisp bi-matrix games, we established a bilinear programming method with parameters for calculating fuzzy Pareto–Nash equilibrium. By considering the importance of each objective to the players, MBGFP is transformed into a bi-matrix game with fuzzy payoffs (BGFP). Furthermore, we obtained the necessary and sufficient conditions for the existence of fuzzy weighted Pareto–Nash equilibrium and its calculation method. Finally, a practical example is used to illustrate the effectiveness of our proposed calculation method.

Moran process in evolutionary game dynamics with interval payoffs and its application

Moran process is an essential dynamic process of updating in stochastic evolutionary game dynamics for analyzing evolutionary game models with exact payoffs. The payoffs in the game are not crisp numbers because of how complicated and uncertain the environment is. Considering this problem, this work investigates the evolutionary game dynamics of Moran process with interval payoffs by using interval numbers to characterize the uncertainty in evolutionary games. Firstly, the interval fixation probability of the Moran process is derived under weak selection. Secondly, the conditions and the degrees of possibility for the dominant strategy in the population are calculated and the interval evolutionary equilibrium strategy is obtained. Then, the evolutionary game model under the prisoner’s dilemma is studied with the interval Moran process. Finally, the evolutionary process of the enterprise epidemic prevention strategy is studied with an evolutionary game model based on the interval Moran process. In addition, the influence of the uncertainty of local government regulation and public supervision on the choice of enterprise epidemic prevention strategies is analyzed by comparing numerical examples, and the superiority of the method in practical application is verified.

Fuzzy Ranking Approach to Bi-Matrix Games with Interval Payoffs in Marketing-Management Problem

World is full of uncertainties. In decision-making problems, uncertainty occurs in many forms such as fuzzy, rough, interval, soft and researchers use data in one of these forms for presenting uncontrollable factors. This paper aims to develop an effective method for solving bi-matrix game problem with interval payoffs assuming that the players know the upper and lower bounds on payoffs. First, bi-matrix game model with interval payoffs is considered. This model is then transformed to another model with fuzzy payoffs. Finally, ranking approach is used to convert the model to crisp-valued bi-matrix game model. Further, due to correspondence between games and programming problem, the Nash equilibrium of interval bimatrix game is obtained by solving a deterministic nonlinear programming problem with nonlinear objective and linear constraints. Finally, a real-life problem of marketing management is solved to validate, approve and illustrate the effectiveness of the proposed model and its solution method. The results derived are compared with some previously defined methods and conclusions are drawn further.

Nash Equilibrium Points for Generalized Matrix Game Model with Interval Payoffs

Game theory-based models are widely used to solve multiple competitive problems such as oligopolistic competitions, marketing of new products, promotion of existing products competitions, and election presage. The payoffs of these competitive models have been conventionally considered as deterministic. However, these payoffs have ambiguity due to the uncertainty in the data sets. Interval analysis-based approaches are found to be efficient to tackle such uncertainty in data sets. In these approaches, the payoffs of the game model lie in some closed interval, which are estimated by previous information. The present paper considers a multiple player game model in which payoffs are uncertain and varies in a closed intervals. The necessary and sufficient conditions are explained to discuss the existence of Nash equilibrium point of such game models. Moreover, Nash equilibrium point of the model is obtained by solving a crisp bi-linear optimization problem. The developed methodology is further applied for obtaining the possible optimal strategy to win the parliament election presage problem.

Evolutionary game dynamics of death-birth process with interval payoffs on graphs

In this paper, we study the evolutionary game dynamics of the death-birth process with interval payoffs on graphs. First of all, we derive the interval replication dynamic equation. Secondly, we derive the fixation probability of the B-C prisoner’s dilemma game based on the death-birth process under the condition of weak selection, analyze the condition of the strategy fixed in the population, that is the condition of strategy A being dominant is analyzed. So we can judge whether natural selection is beneficial to strategy A in the game process through this condition. Finally, the feasibility of this method is verified by several examples.

A reformulated Shapley-like value for cooperative games with interval payoffs

In this paper, a new value for cooperative interval games is proposed which may remedy the disadvantages of the interval Shapley-like value and of the improved interval Shapley-like value introduced by Han et al. (2012). Moreover, it is shown that the reformulated interval value uniquely satisfies the properties of efficiency, indifference null player, symmetry, and additivity.

On the Banzhaf-like Value for Cooperative Games with Interval Payoffs

By using Moore’s subtraction operator and a total order on the set of closed intervals, we introduce a new variation of the Banzhaf value for cooperative interval games called the interval Banzhaf-like value which may accommodate the shortcomings of the interval Banzhaf value. We first reveal the relation between this introduced value and the interval Banzhaf value. Then, we present two sets of properties that may be used to determine whether an interval value is median-indifferent to the interval Banzhaf-like value. Finally, in order to overcome the disadvantages of the interval Banzhaf-like value, we propose the contracted interval Banzhaf-like value and give an axiomatization of this proposed value.

A Direct Method of Interval Banzhaf Values of Interval Cooperative Games

The goal of this paper is to develop a direct method for computing interval Banzhaf values for a class of interval cooperative games. In this method, it is proven that the Banzhaf values of the associated cooperative games of interval cooperative games are monotonic and non-decreasing functions of coalitions’ payoffs under weaker coalition monotonicity-like conditions. So the interval Banzhaf values can be explicitly obtained through using only the lower and upper bounds of the coalitions’ interval payoffs, respectively. The proposed method does not employ interval subtractions and hereby can effectively avoid the irrational issues such as irreversibility and uncertainty enlargement. We prove some important and desirable properties of the interval Banzhaf values and illustrate the applicability and validity of the proposed method with a numerical example.

On solution of constraint matrix games under rough interval approach

The aim of this article is to propose an effective technique for solving constraint matrix games with rough interval payoffs, which are a class of non-cooperative two-person matrix games with rough interval payoffs and the strategies of the players are constrained. Since the payoffs of the rough constraint matrix games are rough intervals, then its game value is also a rough interval. In this technique, we derived four linear programming problems, which are used to obtain the upper–lower bound, lower–lower bound, lower–upper bound and upper–upper bound of the rough interval game values of the players in rough constraint matrix games. Moreover, the expected value operator and trust measure of rough interval have been used to obtain the α-trust equilibrium strategies and the expected equilibrium strategies of the problem under study. In addition, the different advantages of the proposed technique over those existing are discussed. Finally, a numerical experiment of market share game model is given and solved by the three mentioned methods to illustrate the effectiveness and practicality of the proposed method.

GRAPHICAL METHOD FOR INTERVAL BIMATRIX GAMES

This paper deals with two-person non-zero sum games with interval payoffs. Graphical method to find a mixed strategy equilibrium is adapted to interval bimatrix games. In addition, interval bimatrix games Nash equilibrium is attained by graphical method. Nu- merical examples are also illustrated.

A SATISFACTORY STRATEGY OF MULTI OBJECTIVE TWO PERSON MATRIX GAMES WITH FUZZY PAYOFFS

The multiobjective two person matrix game problem with fuzzy payoffs is considered in this paper. It is assumed that fuzzy payoffs are triangular fuzzy numbers. The problem is converted to several multiobjective matrix game problems with interval payoffs by using the $alpha$-cuts of fuzzy payoffs. By solving these problems some $alpha$-Pareto optimal strategies with some interval outcomes are obtained. An interactive algorithm is presented to obtain a satisfactory strategy of players. Validity and applicability of the method is illustrated by a practical example.

Existence of Equilibrium Points for Bimatrix Game with Interval Payoffs

This paper deals a bimatrix game with payoffs as closed intervals. Existence of equilibrium point of this game is discussed by using suitable interval quadratic programming problem. Further, a methodology is proposed for finding optimal strategies for each player of the game. The methodology is illustrated by numerical example.

An Analytıc Method for Interval Bimatrix Games

This paper deals with ݔ two-person non-zero sum games with interval pay-offs. An analytic method for solving such games is given. A pair of Nash Equilibrium is found by using the method. The analytic method is effective to find at least one Nash Equilibrium (N.E) for two-person bimatrix games. Therefore, the analytic method for two-person bimatrix games is adapted to interval bimatrix games.

随机区间收益金融市场的无强占优分析

文章提出了具有随机区间收益的金融市场，在该市场中，证券价格被描述为区间值随机变量，并将研究的金融市场所在的概率空间框架从有限维拓展为一般概率空间，是传统随机市场的推广。在此模型中，提出了强占优策略和线性定价测度的概念，证明了市场无强占优与存在线性定价测度等价的定理，并且研究了随机区间收益市场无强占优与经典市场无占优的关系。该结论既包含经典随机金融市场的结论，又将金融分析推广到随机区间收益金融市场中。 In this article, a new financial market model, in which all securities have random interval pay- offs, is proposed. In the market, the probability space from finite dimensional expands general probability space. Some concepts, such as robust dominant strategy and linear pricing measures, are given and discussed parallel to those in traditional market analysis. With these new concepts, it is discussed that the requirement of no robust dominant strategy is equivalent to the existence of linear pricing measures and the relations between the analysis of the no robust dominant strategy in the market with random interval payoffs and it’s in traditional market. This conclusion includes the traditional random market analysis conclusions, extending to this new financial market.

Some results on cooperative interval games

Uncertainty is a daily presence in the real world. It affects our decision-making and may have influence on cooperation. On many occasions, uncertainty is so severe that we can only predict some upper and lower bounds for the outcome of our actions, i.e. payoffs lie in some intervals. A suitable game theoretic model to support decision-making in collaborative situations with interval data is that of cooperative interval games. Solution concepts that associate with each cooperative interval game sets of interval allocations with appealing properties provide a natural way to capture the uncertainty of coalition values into the players’ payoffs. In this paper, the relations between some set-valued solution concepts using interval payoffs, namely the interval core, the interval dominance core, the square interval dominance core and the interval stable sets for cooperative interval games, are studied. It is shown that the interval core is the unique stable set on the class of convex interval games.

Cooperative Fuzzy Games with a Coalition Structure and Interval Payoffs

Based on the extension Hukuhara difference between interval numbers, a generalized form of cooperative fuzzy games with a coalition structure and interval payoffs is proposed, which can be seen as an extension of crisp case. The interval Owen value for this kind of fuzzy games is studied, and its explicit form is given. When the fuzzy games are convex, the proposed interval Owen value is an interval population monotonic allocation function (IPMAF), and belongs to the associated core. Furthermore, we discuss a special kind of fuzzy games with a coalition structure and interval payoffs, and study the interval Owen value and the core of it. Some properties are also examined, which are coincided with the crisp case.

Interval programming models for matrix games with interval payoffs

The aim of this paper is to study how to solve a type of matrix game with interval payoffs. In this paper, the interval inequality relations and the concept of solutions of the matrix game with interval payoffs are defined. Based on the fuzzy ranking index defined, the solution of the matrix game with interval payoffs can be obtained through solving a pair of bi-objective linear programming models derived from the constructed auxiliary interval programming models. It is shown that the models proposed in this paper extend those of the classical matrix game. The validity and applicability of the proposed methodology are illustrated with a numerical example.

NOTES ON "LINEAR PROGRAMMING TECHNIQUE TO SOLVE TWO-PERSON MATRIX GAMES WITH INTERVAL PAY-OFFS"

The aim of this note is to point out and correct some vital mistakes in the paper by P K Nayak and M Pal, "Linear programming technique to solve two person matrix (games with interval pay-offs). Asia-Pacific Journal of Operational Research, 26(2), 285–305". Lots of serious mistakes on the definitions, conclusions, models, methods, proofs and computing results have been corrected and modified in this note. We also indicate inappropriate formulations regarding their proposed linear programming models for solving generic matrix games with interval pay-offs and suggest a pair of linear programming models with any minimal acceptance degree of the interval inequality constraints which may be allowed to violate. The lexicographic method is suggested so that a rational and credible solution of the generic matrix game with interval pay-offs can be achieved.