GAMELPP: A PASCAL program for solving two-person zero-sum games on a microcomputer

Introduction to mathematical programming: N. K. Kwak and Marc J. Schniederjans: Krieger, Malabar

Operations Research: Applications and Algorithms

Operations Research: Applications and Algorithms

Objectives: To overcome the shortcoming of being limited-time interval derivative based game theory in dealing with the issue of uncertain payoffs and to suggest an optimization-based model of deriving plausible forms of mixed strategies in zero-sum games between two players. Methods: A hybrid interval-fuzzy payoff representation is proposed to describe uncertainty in more impressive way. The value of standard norm-based bounds is computed on approximate game values. There is a formed strategy selection process employed that is based on optimization where the derived bounds are used to derive the upper and lower thresholds of mixed strategies (psup and pinf). These are realistic interval matrices and synthetic simulations. Findings: The suggested approach does not require solving linear programming problems and it has a competitive set of game values as compared to conventional approaches. The strategy on the 3 × 3 matrix games provided consistent approximate values [2.666, 5.000]. The algorithm is scalable and analyzable like bigger games for 10 × 10 matrix. Novelty: The paper presents the first intervalfuzzy payoff approach which maintains both the interval width and the fuzzy confidence at the same time in the zero-sum games. A bounding mechanism that is norm based is created to estimate game values without the use of linear programming, allowing analysis of large matrices on a scale. Theoretical results are now determined new to interval fuzzy saddle-point conditions, mixedstrategy interval convexity and mixed-strategy interval duality-gap control. Keywords: Two-Person Zero-Sum Game; Interval-Fuzzy Matrix; Mixed Strategy; Norm-based Estimation; Saddle Point; Game Theory

Modified fictitious play for solving matrix games and linear-programming problems

A Canonical Game -- Nearly 75 Years in the Making -- Showing the Equivalence of Matrix Games and Linear Programming

Linear fractional programming has been an important planning tool for the past four decades. The main contribution of this study is to show, under some assumptions, for a linear programming problem, that there are two different dual problems (one linear programming and one linear fractional functional programming) that are equivalent. In other words, we formulate a linear programming problem that is equivalent to the general linear fractional functional programming problem. These equivalent models have some interesting properties which help us to prove the related duality theorems in an easy manner. A traditional data envelopment analysis (DEA) model is taken, as an instance, to illustrate the applicability of the proposed approach.

In this and the next chapter we present an application of the learning algorithms developed in the previous chapters to two person zero sum games: Let A and B be the two players. Both are allowed to use mixed strategies. At any instant each player picks a pure strategy as a sample realization from his mixed strategy. As a result of their joint action they receive a random outcome which is either a success or failure. Since the game is a zero-sum game A’s success is B’s failure and vice-versa. The following assumptions are fundamental to our analysis: Either player has no knowledge of the set of pure strategies available to the other player or the pure strategy actually chosen by the other player at any stage of the game or the distribution of the random outcome as a function of the pure strategies chosen by them. Just based on the pure strategy chosen by him and the random outcome he receives both the players individually update their mixed strategies using a learning algorithm. This cycle continues and thus the game is played sequentially. In short we consider a zero-sum game between two players in which the players are totally decentralized, there is no communication or transfer of information between them either before or during the course of the play of the game and in fact they may not even know that they are involved in a game situation at all. In this set-up our aim is to find conditions on the learning algorithms such that both the players in the long run will receive an expected payoff as close to the well established game theoretic solutions (Von Neumann value) as desired.

Aim. The work aimed to analyze the relationship between the optimization competition indicator introduced earlier in the author’s articles and the game models widely used in economics, in particular, zero-sum matrix games.Objectives. The work seeks to determine the quantitative relationship between the solutions of game models and optimization competition, which allows for a new interpretation of the results of game models in economics; as well as to correlate the optimal strategies in game models with the optimization competition indicator.Methods. The analysis was used to perform a study of the relationship between the previously introduced competition indicator (optimization competition) and game models. Examples were applied to establish a quantitative relationship between optimization competition and the results of zero-sum matrix games in pure and mixed strategies.Results. A number of game models involve the use of optimization methods, i.e. linear programming methods, in matrix games with mixed strategies. Since the previously introduced optimization competition indicator was developed specifically for optimization problems, it becomes appropriate to study the relationship between it and the solution of game problems. The idea consists in comparing the calculations of the optimization competition indicator with the results of game models. The examples present the patterns of changes in the optimization competition indicator depending on various types of game models, in particular, matrix games. The differences and features in the cases of matrix games in pure and mixed strategies were determined. Conclusions. The work presents the relationship between optimization competition and the results of zero-sum matrix games. “Pure gain” (or “pure strategies” called in game economic models) is possible only with non-zero optimization competition, and the average one, expected, with probability, depending on the payoff matrix, can be accompanied by both zero and non-zero competition. In other words, pure gain requires that competition be greatest, all other things being equal. This approach allows a new approach to interpreting the results of game models, in particular, zero-sum matrix games. Nowadays, the game result is only its price determined in pure or mixed strategies. But the results given indicate that it is advisable to compare the value of the game price with the value of optimization competition, which provides additional information for analysis in game economic models.

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.

In Operations Research, linear-fractional programming is considered as the generalization of linear programming problem. While in a linear programming the objective function is a linear function, and in a linear-fractional programming the objective function is the ratio of two linear or non-linear functions. The majority of the algorithm used for solving the linear fractional programming problem relies upon the classical simplex method. In this paper, we have proposed a new algorithm for solving a linear fractional programming problem in which the objective function is a combination of linear fractional function, while constraint functions are in the form of linear inequalities. Our proposed algorithm is based on the extension of the method, which is used to solve linear programming problems with linear constraints. The primary intent behind developing this method is that we did not need to transform the linear fractional programming problem into linear programming problem, and also it helps in finding out the feasible region via a sequence of points in the direction that improves the feasibility of the fractional objective function. Numerical examples are given to illustrate the use of these proposed methods. Lastly, to demonstrate the efficacy of the proposed algorithm, we have compared the findings obtained with other approaches to display our algorithm's efficacy.

Linear bilevel programming with upper level constraints depending on the lower level solution

The revised harmonious fuzzy technique (RHFT) is a method used to solve fuzzy optimization problems. It was capitalized as an extension of the classical linear programming technique to handle constraints and objectives that are fuzzy. The harmonious fuzzy technique HFT aims to find a solution that satisfies the uncertain restraints and optimizes the uncertain objectives while taking into account the uncertainty or fuzziness of the problem parameters. This work demonstrates how the RHFT can be utilized to dexterously solve “fully fuzzy multi-goal linear fractional programming (FFMOLFP) problems”. Initially, the FFMOLFP problem can be converted to “single goal linear fractional programming (SOLFP) problems” consuming the modified brittle linear technique. Second, the RHFT is applied to converted brittle problems into linear programming problem, which follow, “the single-goal problem” is made on so on applied the revised harmonious fuzzy for apiece level. at the end, the obtained LPP will be solved by applied the simplex algorithm. To illustrate the application of this method, two examples will be provided. Also, the numerical results are simulated by comparing between proposed method and efficient ranking function methods for fully fuzzy linear fractional programming problems FFLFPP

We apply linear and non-linear programming to find the solutions for Nash equilibriums and Nash arbitration in game theory problems. Linear programming was shown as a viable method for solving mixed strategy zero-sum games. We review this methodology and suggest a class of zero-sum game theory problems that are well suited for linear programming. We applied this theory of linear programming to non-zero sum games. We suggest and apply a separate formulation for a maximising linear programming problem for each player. We move on the Nash arbitration method and remodel this problem as a non-linear optimisation problem. We take the game's payoff matrix and we form a convex polygon. Having found the status quo point (x*, y*), we maximise the product (x-x*)(y-y*) over the convex polygon using KTC non-linear optimisation techniques. The results give additional insights into game theory analysis.

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.