Different Techniques for The Study of Some Game Models

Abstract

Game Theory is defined as a means of mathematical analysis when interests collide with each other to reach the best possible decision-making options taking into consideration the given circumstances to get the desired results. Game Theory is related to well-known games such as checkers, XO, and poker. In this dissertation. We will tackle one of the Combinatorial Games in detail, and link another game to the graph, the topic may suggest lots of fun since games are widely recognized as amusing. But the reader should keep in mind that this is a collection of mathematical results and as such has a moderate fun impact. This dissertation falls into four chapters as following. Chapter One: In this chapter, we will present introduction to basic concepts of game theory. These include basic definitions of game, classifications of games [4], discussion the Combinatorial Games in detail and explain some examples of these games [7], describing strategic games and Nash Equilibrium, games in extensive and in normal form [5-9], the relationship of game theory to other sciences [9]. Chapter Two: In this chapter, we will discuss in detail the famous game of Tic-Tac-Toe and some different names for this game [10-13], the basic rules for playing, explain some similar games [18], history of the game and how to win this game from the viewpoint of one of the researchers. Chapter Three: In chapter Three we deduce a new mathematical technique to define the winning game Tic-Tac-Toe. The results were placed in a 3x3 matrix and initial conversions were performed on the rows to find all possible win states. Programming languages were used to find the matrix to determine the diagonal wins. A simulation algorithm is presented to predict the win, or draw of a game by knowing the first move of a player. The game of Tic-Tac-Toe is simulated by using a Min-max algorithm. The results in this chapter have been published in Journal of Game Theory in 2020 entitled “Shahd H. Alkaraz, Essam El-Seidy and Neveen S. Morcos. Tic-Tac-Toe: Understanding the Minimax Algorithm. Journal of Game Theory (2020), 9(1): 1-7.”.