Research Article Toward the Nash Equilibrium Solutions for Large-Scale Pentagonal Fuzzy Continuous Static Games Hamiden Abd El-Wahed Khalifa , 1,2 S. A. Edalatpanah , 3 and Alhanouf Alburaikan 2 1 Department of Operations Research, Faculty of Graduate Studies for Statistical Research, Cairo University, Giza 12613, Egypt 2 Department of Mathematics, College of Science and Arts, Qassim University, Al-Badaya 51951, Saudi Arabia 3 Department of Applied Mathematics, Ayandegan Institute of Higher Education, Tonekabon, Iran Correspondence should be addressed to S. A. Edalatpanah; saedalatpanah@gmail.com Received 22 January 2022; Accepted 10 March 2022; Published 4 April 2022 Academic Editor: Muhammad Gulzar Copyright © 2022 Hamiden Abd El-Wahed Khalifa et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. This study aims to characterize and clarify a pentagonal fuzzy continuous static game (PF-CSG) that constraints and cost functions are fuzzy numbers. Pentagonal fuzzy numbers characterize their fuzzy parameters. The α − Pareto optimal solution concept has specified. The decomposition approach has applied to decompose the problem into subproblems each of them having smaller and independent subproblems. In addition, the Nash equilibrium solution concept was used to obtain the solutions of these subproblems. The advantages of this study are the players independently without collaboration with any of the others and that each player seeks to minimize the cost function. Also, the information available to each player consists of the cost function and constraints. An illustrated numerical example has discussed for proper understanding and interpretation of the proposed concept. 1. Introduction Game theory plays a vital role in economics, engineering, biology, and other computational cum mathematical sciences with wide range of applications in real-world prob- lems. Differential games, continuous static games, and matrix games are three major types of games. Matrix games derive their name from a discrete relationship between a finite/countable number of possible decisions and the corre- sponding costs. The relationship is conveniently represented in terms of a matrix (or two-player games) in which the decision of one player relates to the choice of a row and the decision of other player is corresponding to the choice of a column, with the corresponding entries denoting the costs. It is vivid that decision probabilities are not manda- tory in the cooperative games. In addition, there is no time in the relationship between costs and decisions in static games. Differential games are categorized by varying costs along with a dynamic system administrated by ODE. For continuous static games, there are several solution concepts. How a player uses these concepts depends not only on information concerning the nature of the other players, but also on his/her own personality. A given player may or may not play rationally, cheat, cooperate, and bargain. A player in making the ultimate choice of his/her control vector must consider all of these factors. The three basic solution concepts for these games (Vincent and Grantham [1]) are (1) Nash equilibrium solution (2) Min-Max solutions (3) Pareto minimal solutions Early, several researchers worked in fuzzy set theory; Zadeh [2] introduced the notion of a fuzzy set in an attempt to develop the ideology of fuzzy set and mathematical frame- work in which to treat systems or phenomena, which is due to intrinsic indefiniteness as distinguished from a mere statistical variation, cannot themselves be characterized pre- cisely. Dubois and Prade [3] developed the view of using algebraic operations on fuzzy numbers using a fuzzification principle. Decisions in a fuzzy situation were first proposed Hindawi Journal of Function Spaces Volume 2022, Article ID 3709186, 11 pages https://doi.org/10.1155/2022/3709186