Two-person Zero-sum Matrix Game Research Articles

A novel technique for solving two-person zero-sum matrix games in a rough fuzzy environment

This study proposes a novel way to deal with uncertainty in a two-person zero-sum matrix game with payoffs expressed as fuzzy rough numbers. Complete and reasonable solutions to these types of games are obtained. In this research we develop two linear programming models with upper and lower approximation intervals of fuzzy rough numbers and handle multi-objective crisp linear programming models by incorporating trapezoidal fuzzy rough numbers as payoffs. To provide each opponent with the optimal strategy and value of the game, the usual simplex approach is applied. Finally, two numerical examples demonstrate the matrix game outcomes using Wolfram Cloud.

On solving two-person zero-sum fuzzy matrix games via linear programming approach

In this paper, a two-person zero-sum matrix game with fuzzy numbers payoff is introduced. Using the fuzzy number comparison introduced by Rouben's method (1991), the fuzzy payoff is converted into the corresponding deterministic payoff. Then, for each player, a linear programming problem is formulated. Also, a solution procedure for solving each problem is proposed. Finally, a numerical example is given for illustration.

Intuitionistic-Fuzzy Goals in Zero-Sum Multi Criteria Matrix Games

The classical matrix theory is deficient to express the vagueness of the real life. The fuzzy set theory has been successfully applied to bridge this gap. Much work has already been done on a two-person zero sum matrix game with fuzzy goals. In continuation, this paper is dedicated to define and study a multi-criteria two-person zero sum game with intuitionistic fuzzy goals. It is shown that solving such games is equivalent to solving two crisp multi object linear programming problems. Our work generalizes the previous study on a multi-criteria game with fuzzy goals by adopting the approach of linear programming with intuitionistic fuzzy sets. Finally, an illustrative numerical example is provided to elaborate the proposed approach.

Optimal strategies for equal-sum dice games

In this paper, we consider a non-cooperative two-person zero-sum matrix game, called dice game. In an ( n , σ ) dice game, two players can independently choose a dice from a collection of hypothetical dice having n faces and with a total of σ eyes distributed over these faces. They independently roll their dice and the player showing the highest number of eyes wins (in case of a tie, none of the players wins). The problem at hand in this paper is the characterization of all optimal strategies for these games. More precisely, we determine the ( n , σ ) dice games for which optimal strategies exist and derive for these games the number of optimal strategies as well as their explicit form.

Classification of Matrix Types for Multicriteria Two-Person Zero-Sum Matrix Games

The paper is concerned with multicriteria games, or games with a vector payoffs. The aim of this paper is to give structure information for the graph of a payoff function in two-person zero-sum matrix game with two 2×2 matrices by using computational program on computer. As a result, six different types of classification on matrices for such multicriteria matrix game is presented. Moreover, complete characterizations for such classification and some examples of illustrations for graphs of payoff functions are given.

Some Further Thoughts on the Minimax Principle

In a recent paper Aumann and Maschler [Aumann, R. J., M. Maschler. 1972. Some thoughts on the minimax principle. Management Sci. 18 (5, Part 2, January) P-54.] (we will subsequently refer to them as A-M) raised some questions about certain matters that had long been considered settled. In their words, “It is generally agreed that the minimax solution to a two-person zero-sum matrix game is intuitively satisfactory. Now in many applications of game theory a game is not described in matrix (or ‘normal’ or ‘strategic’) form, but rather in extensive form, i.e., by its rules. A game described in such a way may be reduced to a matrix game by means of the concept of ‘strategy.’ If, moreover, it is of perfect recall, then all mixed strategies, and in particular the optimal strategies of each player, are equivalent to behavior strategies The usual conclusion from these considerations is that for 2-person 0-sum games in extensive form, the minimax solution is intuitively satisfactory, and that in games of perfect recall, in particular, the players would do well to play in accordance with optimal (minimax) behavior strategies. In this paper we shall discuss some examples that, we believe, cast doubt on these conclusions.”

Game solutions by improved fictitious play algorithm

An improved algorithm is presented for solving the two-person zero-sum matrix game problem.

Some Thoughts on the Minimax Principle

It is generally agreed that the minimax solution to a two-person zero-sum matrix game is intuitively satisfactory. Now in many applications of game theory, a game is not described a priori in matrix (or “normal” or “strategic”) form, but rather in extensive form, i.e., by its rules. A game described in such a way may be reduced to a matrix game by means of the concept of “strategy.” If, moreover, it is of perfect recall, then all mixed strategies, and in particular the optimal strategies of each player, are equivalent to behavior strategies. The usual conclusion from these considerations is that for 2-person 0-sum games in extensive form, the minimax solution is intuitively satisfactory; and that in games of perfect recall, in particular, the players would do well to play in accordance with optimal (minimax) behavior strategies. In this paper we shall discuss some examples that, we believe, cast doubt on these conclusions.