The roles of matrix norms in the game theory

In this thesis, we make some significant contributions and give a new perspective to the game theory and 3-dimensional matrix theory. We present our contributions and developments in three diffrent chapters as follows: In the first chaper, a brief history of the matrices is given. Some examples are given in order to demonstrate the usage in science for different purposes. In the second chapter, some definitions and properties for the 2-dimensional matrices are extended to the 3-dimensional matrices. The basic concepts of the 3-dimensional matrices are presented by extending the definitions for the 2-dimensional matrices. The 3-dimensional matrix product is defined as it is defined for the 2-dimensional matrices. Moreover, the matrix inversion of a 3-dimensional matrix, determinant vector and some other definitions are made.The condition number vectors for the 3-dimensional matrices is defined. In addition these definitions, the singular and nonsingular 3-dimensional matrices are defined based on the definition of the determinant vector. Furthermore, the definition of ill-conditioned and well-conditioned 3-dimensional matrices are presented by using the definition of the condition number vector. Beside these, Cauchy-Schwarz inequality is represented for the 3-dimensional matrices and proved by inducing the 3-dimensional matrix to 2-D matrix. Additionally, some other important inequalities related to the 3-dimensional matrix norms are demonstrated. Finally, in this chapter, the effects of the third dimension with the new definitions and inequalities by some examples are investigated.\\ In addition to these, the norm inequalities for 3-dimensional matrices are presented and comprehensively proved. The proofs are completed with the similiar methodology being used for the 2-dimensional matrix norm inequalities. That is, we first induce the 3-D matrix to the 2-dimensional matrix. Then, we use the 2-D matrix norm inequalities and necessary vector norm inequalities. Moreover, the relationships between these norms are showed and the coefficients of the 3-dimensional matrix norm inequalities are presented with a table in order to simplify the usage of these norm inequalities. Furthermore, the usefulness of these inequalities is illustrated for 3-dimensional matrices which are obtained from simulations and real data applications.\\ In the third chapter, a novel approach to solve and create a two person zero sum matrix game by using matrix norms is presented. Especially, we show how to obtain approximated game value, $v_{app}$, for any zero sum matrix game without solving any equations using our approaches. Firstly, some lemmas are given and the results of these lemmas for the game value depend on the matrix norms of the payoff matrix and some constants $k$ containing the game value $v$. Then, the row-wise and column-wise induced matrix for the payoff matrix are introduced. Moreover, the proposed approaches are improved and the game value in the constant $k$ is vanished off. Then, some new improved theorems for the game value are presented in order to obtain some inequalities which depend on only the $1-norm$ and $\infty-norm$ of the payoff matrix. Furthermore, the min-max theorem for $p_{max}$ and $p_{min}$ is stated and clearly proved, where $p_{max}$ and $p_{min}$ are the maximum and minimum elements of the mixed strategy set, respectively. The min-max theorem shows the relationship between $p_{max}$ and $p_{min}$. Additionally, this theorem provides an opportunity to obtain more optimal interval for the game value. We also illustrate and show the consistency of our approaches with some test examples. To the best of our knowledge, this is the first study in the literature that the game theory meets the matrix norms.