Recursive matrix games

1972 ◽  
Vol 9 (4) ◽  
pp. 813-820 ◽  
Author(s):  
Michael Orkin
Keyword(s):  
Matrix Game ◽  
Matrix Games ◽  
Inductive Proof ◽  
Stationary Strategies ◽  
A Value ◽  
Recursive Games ◽  
Recursive Matrix ◽  
Optimal Stationary Strategies

“Recursive” games were first defined and studied by Everett. Related results can be found in Gillette, Milnor and Shapley, and Blackwell and Ferguson. In this paper we introduce the notion of a recursive matrix game, which we believe eliminates the vagueness but none of the useful generality of the earlier definition. We then give an inductive proof (different from the proof in [3]) that these games have a value, with ∊-optimal stationary strategies available to each player. We also apply the result and show how a class of games studied in a different framework are games of this type and thus have a value.

Recursive matrix games

1972 ◽  
Vol 9 (04) ◽  
pp. 813-820 ◽  
Author(s):  
Michael Orkin
Keyword(s):  
Matrix Game ◽  
Matrix Games ◽  
Inductive Proof ◽  
Stationary Strategies ◽  
A Value ◽  
Recursive Games ◽  
Recursive Matrix ◽  
Optimal Stationary Strategies

“Recursive” games were first defined and studied by Everett. Related results can be found in Gillette, Milnor and Shapley, and Blackwell and Ferguson. In this paper we introduce the notion of a recursive matrix game, which we believe eliminates the vagueness but none of the useful generality of the earlier definition. We then give an inductive proof (different from the proof in [3]) that these games have a value, with ∊-optimal stationary strategies available to each player. We also apply the result and show how a class of games studied in a different framework are games of this type and thus have a value.

Repeated matrix games

1979 ◽  
Vol 16 (04) ◽  
pp. 830-842
Author(s):  
Dana B. Kamerud
Keyword(s):  
Optimal Strategies ◽  
Matrix Game ◽  
Matrix Games ◽  
Value Of A Game ◽  
A Value ◽  
History Of ◽  
Definition Of

A matrix game is played repeatedly, with the actions taken at each stage determining both a reward paid to Player I and the probability of continuing to the next stage. An infinite history of play determines a sequence (Rn ) of such rewards, to which we assign the payoff lim supn (R 1 + · ·· + Rn ). Using the concept of playable strategies, we slightly generalize the definition of the value of a game. Then we find sufficient conditions for the existence of a value and for the existence of stationary optimal strategies (Theorems 8 and 9). An example shows that the game need not have a value (Example 4).

Repeated matrix games

1979 ◽  
Vol 16 (4) ◽  
pp. 830-842
Author(s):  
Dana B. Kamerud
Keyword(s):  
Optimal Strategies ◽  
Matrix Game ◽  
Matrix Games ◽  
Value Of A Game ◽  
A Value ◽  
History Of ◽  
Definition Of

A matrix game is played repeatedly, with the actions taken at each stage determining both a reward paid to Player I and the probability of continuing to the next stage. An infinite history of play determines a sequence (Rn) of such rewards, to which we assign the payoff lim supn (R1 + · ·· + Rn). Using the concept of playable strategies, we slightly generalize the definition of the value of a game. Then we find sufficient conditions for the existence of a value and for the existence of stationary optimal strategies (Theorems 8 and 9). An example shows that the game need not have a value (Example 4).

Optimal stationary strategies in leavable Markov decision processes

1990 ◽  
Vol 27 (01) ◽  
pp. 134-145
Author(s):  
Matthias Fassbender
Keyword(s):  
Transition Probabilities ◽  
Practical Applications ◽  
Stationary Strategies ◽  
Countable State ◽  
Markov Decision ◽  
Bias Optimality ◽  
The Mean ◽  
Optimal Stationary Strategies ◽  
The Cost ◽  
Reward Criterion

This paper establishes the existence of an optimal stationary strategy in a leavable Markov decision process with countable state space and undiscounted total reward criterion. Besides assumptions of boundedness and continuity, an assumption is imposed on the model which demands the continuity of the mean recurrence times on a subset of the stationary strategies, the so-called ‘good strategies'. For practical applications it is important that this assumption is implied by an assumption about the cost structure and the transition probabilities. In the last part we point out that our results in general cannot be deduced from related works on bias-optimality by Dekker and Hordijk, Wijngaard or Mann.

A Fast Approach to Solve Matrix Games with Payoffs of Trapezoidal Fuzzy Numbers

2016 ◽  
Vol 33 (06) ◽  
pp. 1650047 ◽  
Author(s):  
Sanjiv Kumar ◽  
Ritika Chopra ◽  
Ratnesh R. Saxena
Keyword(s):  
Linear Programming ◽  
Linear Programming Problem ◽  
Fuzzy Numbers ◽  
Optimization Problems ◽  
Fuzzy Linear Programming ◽  
Matrix Game ◽  
Matrix Games ◽  
Trapezoidal Fuzzy Numbers ◽  
Fast Approach ◽  
The Matrix

The aim of this paper is to develop an effective method for solving matrix game with payoffs of trapezoidal fuzzy numbers (TrFNs). The method always assures that players’ gain-floor and loss-ceiling have a common TrFN-type fuzzy value and hereby any matrix game with payoffs of TrFNs has a TrFN-type fuzzy value. The matrix game is first converted to a fuzzy linear programming problem, which is converted to three different optimization problems, which are then solved to get the optimum value of the game. The proposed method has an edge over other method as this focuses only on matrix games with payoff element as symmetric trapezoidal fuzzy number, which might not always be the case. A numerical example is given to illustrate the method.

A Combinatorial method for a class of matrix games

1966 ◽  
Vol 3 (2) ◽  
pp. 495-511
Author(s):  
Rodrigo A. Restrepo
Keyword(s):  
Optimal Strategies ◽  
Matrix Game ◽  
Combinatorial Method ◽  
Matrix Games ◽  
Finite Matrix ◽  
Special Relations ◽  
Class Of Matrices

The optimal strategies of any finite matrix game can be characterized by means of the Snow-Shapley Theorem [1]. However, in order to use this theorem to compute the optimal strategies, it may be necessary to invert a large number of matrices, most of which are not related to the solutions of the game. The present paper will show that when the columns of the pay-off matrix satisfy some special relations, it is possible to enumerate a much smaller class of matrices from which the optimal strategies may be obtained. Furthermore, the maximizing strategies that are determined by these matrices can be written down by inspection as soon as the matrices are specified.

Existence of p-Equilibrium and Optimal Stationary Strategies in Stochastic Games

1976 ◽  
Vol 60 (1) ◽  
pp. 245 ◽  
Author(s):  
C. J. Himmelberg ◽  
T. Parthasarathy ◽  
T. E. S. Raghavan ◽  
F. S. Van Vleck
Keyword(s):  
Stochastic Games ◽  
Stationary Strategies ◽  
Optimal Stationary Strategies

Optimal stationary strategies in leavable Markov decision processes

1990 ◽  
Vol 27 (1) ◽  
pp. 134-145
Author(s):  
Matthias Fassbender
Keyword(s):  
Transition Probabilities ◽  
Practical Applications ◽  
Stationary Strategies ◽  
Countable State ◽  
Markov Decision ◽  
Bias Optimality ◽  
The Mean ◽  
Optimal Stationary Strategies ◽  
The Cost ◽  
Reward Criterion

This paper establishes the existence of an optimal stationary strategy in a leavable Markov decision process with countable state space and undiscounted total reward criterion.Besides assumptions of boundedness and continuity, an assumption is imposed on the model which demands the continuity of the mean recurrence times on a subset of the stationary strategies, the so-called ‘good strategies'. For practical applications it is important that this assumption is implied by an assumption about the cost structure and the transition probabilities. In the last part we point out that our results in general cannot be deduced from related works on bias-optimality by Dekker and Hordijk, Wijngaard or Mann.

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.

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