The two-person and zero-sum matrix game with probabilistic linguistic information

2021 ◽  
Author(s):  
Xiaomei Mi ◽  
Huchang Liao ◽  
Xiao-Jun Zeng ◽  
Zeshui Xu
Keyword(s):  
Matrix Game ◽  
Linguistic Information ◽  
Zero Sum

scholarly journals Max-min solution approach for multi-objective matrix game with fuzzy goals

2016 ◽  
Vol 26 (1) ◽  
pp. 51-60 ◽  
Author(s):  
Sandeep Kumar
Keyword(s):  
Linear Programming ◽  
Linear Programming Problem ◽  
Optimization Problem ◽  
Matrix Game ◽  
Game Model ◽  
Solution Approach ◽  
Multi Objective ◽  
Fuzzy Goals ◽  
Fuzzy Goal ◽  
Zero Sum

In this paper, we consider a multi-objective two person zero-sum matrix game with fuzzy goals, assuming that each player has a fuzzy goal for each of the payoffs. The max-min solution is formulated for this multi-objective game model, in which the optimization problem for each player is a linear programming problem. Every developed model for each player is demonstrated through a numerical example.

Games in Extensive Form

2017 ◽  
Author(s):  
João P. Hespanha
Keyword(s):  
Saddle Point ◽  
Matrix Game ◽  
Extensive Form ◽  
Multi Stage ◽  
Form Representation ◽  
Other Information ◽  
The Matrix ◽  
Recursive Computation ◽  
Zero Sum ◽  
Computation Of Equilibria

This chapter discusses a number of key concepts for extensive form game representation. It first considers a matrix that defines a zero-sum matrix game for which the minimizer has two actions and the maximizer has three actions and shows that the matrix description, by itself, does not capture the information structure of the game and, in fact, other information structures are possible. It then describes an extensive form representation of a zero-sum two-person game, which is a decision tree, the extensive form representation of multi-stage games, and the notions of security policy, security level, and saddle-point equilibrium for a game in extensive form. It also explores the matrix form for games in extensive form, recursive computation of equilibria for single-stage games, feedback games, feedback saddle-point for multi-stage games, and recursive computation of equilibria for multi-stage games. It concludes with a practice exercise with the corresponding solution, along with additional exercises.

scholarly journals An Algorithmic Procedure for Finding Nash Equilibrium

2020 ◽  
Vol 40 (1) ◽  
pp. 71-85
Author(s):  
HK Das ◽  
T Saha
Keyword(s):  
Nash Equilibrium ◽  
Nash Equilibria ◽  
Payoff Matrix ◽  
Matrix Game ◽  
Perfect Equilibrium ◽  
Repeated Use ◽  
Definition Of ◽  
Zero Sum ◽  
Mathematical Construction ◽  
Algorithmic Procedure

This paper proposes a heuristic algorithm for the computation of Nash equilibrium of a bi-matrix game, which extends the idea of a single payoff matrix of two-person zero-sum game problems. As for auxiliary but making the comparison, we also introduce here the well-known definition of Nash equilibrium and a mathematical construction via a set-valued map for finding the Nash equilibrium and illustrates them. An important feature of our algorithm is that it finds a perfect equilibrium when at the start of all actions are played. Furthermore, we can find all Nash equilibria of repeated use of this algorithm. It is found from our illustrative examples and extensive experiment on the current phenomenon that some games have a single Nash equilibrium, some possess no Nash equilibrium, and others had many Nash equilibria. These suggest that our proposed algorithm is capable of solving all types of problems. Finally, we explore the economic behaviour of game theory and its social implications to draw a conclusion stating the privilege of our algorithm. GANIT J. Bangladesh Math. Soc.Vol. 40 (2020) 71-85

scholarly journals Matrix Games with Interval-Valued 2-Tuple Linguistic Information

Games ◽  
2018 ◽  
Vol 9 (3) ◽  
pp. 62 ◽  
Author(s):  
Anjali Singh ◽  
Anjana Gupta
Keyword(s):  
Linear Programming ◽  
Lower Bound ◽  
Real World ◽  
Upper Bound ◽  
Duality Theorem ◽  
Matrix Game ◽  
Matrix Games ◽  
Linguistic Information ◽  
The Real ◽  
Interval Valued

In this paper, a two-player constant-sum interval-valued 2-tuple linguistic matrix game is construed. The value of a linguistic matrix game is proven as a non-decreasing function of the linguistic values in the payoffs, and, hence, a pair of auxiliary linguistic linear programming (LLP) problems is formulated to obtain the linguistic lower bound and the linguistic upper bound of the interval-valued linguistic value of such class of games. The duality theorem of LLP is also adopted to establish the equality of values of the interval linguistic matrix game for players I and II. A flowchart to summarize the proposed algorithm is also given. The methodology is then illustrated via a hypothetical example to demonstrate the applicability of the proposed theory in the real world. The designed algorithm demonstrates acceptable results in the two-player constant-sum game problems with interval-valued 2-tuple linguistic payoffs.

Resolving Indeterminacy Approach to Solve Multi-Criteria Zero-Sum Matrix Games with Intuitionistic Fuzzy Goals

Mathematics ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 305 ◽  
Author(s):  
M. G. Brikaa ◽  
Zhoushun Zheng ◽  
El-Saeed Ammar
Keyword(s):  
Intuitionistic Fuzzy Set ◽  
Implementation Process ◽  
Fuzzy Programming ◽  
Matrix Game ◽  
Great Success ◽  
Matrix Games ◽  
Intuitionistic Fuzzy ◽  
Fuzzy Goals ◽  
Linear Programming Problems ◽  
Zero Sum

The intuitionistic fuzzy set (IFS) is applied in various decision-making problems to express vagueness and showed great success in realizing the day-to-day problems. The principal aim of this article is to develop an approach for solving multi-criteria matrix game with intuitionistic fuzzy (I-fuzzy) goals. The proposed approach introduces the indeterminacy resolving functions of I-fuzzy numbers and discusses the I-fuzzy inequalities concept. Then, an effective algorithm based on the indeterminacy resolving algorithm is developed to obtain Pareto optimal security strategies for both players through solving a pair of multi-objective linear programming problems constructed from two auxiliary I-fuzzy programming problems. It is shown that this multi-criteria matrix game with I-fuzzy goals is an extension of the multi-criteria matrix game with fuzzy goals. Moreover, two numerical simulations are conducted to demonstrate the applicability and implementation process of the proposed algorithm. Finally, the achieved numerical results are compared with the existing algorithms to show the advantages of our algorithm.

COMPACT REPRESENTATIONS OF SEARCH IN COMPLEX DOMAINS

2005 ◽  
Vol 07 (01) ◽  
pp. 73-90
Author(s):  
YOSI BEN-ASHER ◽  
EITAN FARCHI
Keyword(s):  
Nash Equilibrium ◽  
Search Strategy ◽  
Matrix Game ◽  
Set Systems ◽  
Compact Representations ◽  
Mathematical Generalization ◽  
General Search ◽  
Pure Strategies ◽  
Zero Sum ◽  
Search Domain

We introduce a new zero-sum matrix game for modeling search in structured domains. In this game, one player tries to find a "bug" while the other tries to hide it. Both players exploit the structure of the "search" domain. Intuitively, this search game is a mathematical generalization of the well known binary search. The generalization is from searching over totally ordered sets to searching over more complex search domains such as trees, partial orders and general set systems. As there must be one row for every search strategy, and there are exponentially many ways to search even in very simple search domains, the game's matrix has exponential size ("space"). In this work we present two ways to reduce the space required to compute the Nash value (in pure strategies) of this game: • First we show that a Nash equilibrium in pure strategies can be computed by using a backward induction on the matrices of each "part" or sub structure of the search domain. This can significantly reduce the space required to represent the game. • Next, we show when general search domains can be represented as DAGs (Directed Acycliqe Graphs). As a result, the Nash equilibrium can be directly computed using the DAG. Consequently the space needed to compute the desired search strategy is reduced to O(n2) where n is the size of the search domain.

Solving Two-Person Zero-Sum Game by Matlab

2011 ◽  
Vol 50-51 ◽  
pp. 262-265 ◽  
Author(s):  
Yan Mei Yang ◽  
Yan Guo ◽  
Li Chao Feng ◽  
Jian Yong Di
Keyword(s):  
Game Theory ◽  
Linear Programming ◽  
Special Class ◽  
The Other ◽  
Matrix Game ◽  
Matlab Software ◽  
Zero Sum Games ◽  
Programming Techniques ◽  
Zero Sum

In this article we present an overview on two-person zero-sum games, which play a central role in the development of the theory of games. Two-person zero-sum games is a special class of game theory in which one player wins what the other player loses with only two players. It is difficult to solve 2-person matrix game with the order m×n(m≥3,n≥3). The aim of the article is to determine the method on how to solve a 2-person matrix game by linear programming function linprog() in matlab. With linear programming techniques in the Matlab software, we present effective method for solving large zero-sum games problems.

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