Mathematical Treatments for Some Game Issues

Abstract

We, humans, cannot survive without interacting with other humans, and ironically, it sometimes seems that we have survived despite those interactions. The subject matter of game theory is exactly those interactions within a group of individuals (or governments, firms, etc.) where the actions of each individual have an effect on the outcome that is of interest to all. In this thesis, we will study the concept of surreal numbers and their relationship to some games and link them in an understandable way, and then put some concepts related to the relationship of the graph to the games and study a game represented on the graphs and we show some properties. This dissertation falls into four chapters as following. Chapter One: In this chapter, we will present a critical introduction to basic concepts of game theory. These include basic definitions of simultaneous game and sequential game, describing strategic games, discussion the combinatorial games in detail and explain some models of combinatorial games. Chapter Two: In this chapter we study construction of the Surreal Numbers, showing it is a class that forms the totally ordered field, and then explore some of new numbers, we present to the reader some algebraic operations related to combinatorial games and gives a detailed outlook of the Surreal Numbers. A fresh outlook to some combinatorial mathematical algebraic operations, through the evaluation of a deduced several algebraic concepts. The findings in this chapter have been published in International Journal of Scientific & Engineering Research in 2020 under the name “Maan T. Alabdullah, Essam El-Seidy and Neveen S. Morcos (2020). On Numbers and Games, International Journal of Scientific & Engineering Research, Volume 11, Issue 2, February -2020.”.