## Abstract

We present a novel approach to solve and create a two person zero sum matrix game by using matrix norms. Especially, we show how to obtain approximated game value for any zero sum matrix game without solving any equations using our approaches. We firstly, give the results of the lemmas for the game value depend on the matrix norms of the payoff matrix and some constants k containing the game value v. Then, we introduce row-wise and column-wise induced matrix for the payoff matrix. Moreover, we improve our approaches and present some new theorems for the game value to obtain some inequalities which depend on only the 1−norm and ∞−norm of the payoff matrix. Furthermore, we state the min–max theorem for p_{max} and p_{min} which are the maximum and minimum elements of the mixed strategy set, respectively. Finally, we 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 is used the matrix norms in game theory.

Original language | English |
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Pages (from-to) | 148-159 |

Number of pages | 12 |

Journal | Applied Mathematics and Computation |

Volume | 341 |

DOIs | |

Publication status | Published - 15 Jan 2019 |

### Bibliographical note

Publisher Copyright:© 2018 Elsevier Inc.

## Keywords

- Game theory
- Matrix norms
- Min–max theorem
- Zero sum matrix game