Extended matrix norm method: Applications to bimatrix games and convergence results

In this paper, we extend and apply the Matrix Norm (MN) approach to the nonzero-sum bimatrix games. We present preliminary results regarding the convergence of the MN ap-proaches. We provide a notation for expressing nonzero-sum bimatrix games in terms of two matrix games using the idea of separation of a bimatrix game into two different ma-trix games. Next, we prove theorems regarding boundaries of the game value depending on only norms of the payoff matrix for each player of the nonzero-sum bimatrix game. In ad-dition to these, we refine the boundaries of the game value for the zero/nonzero sum ma-trix games. Therefore, we succeed to find an improved interval for the game value, which is a crucial improvement for both nonzero and zero-sum matrix games. As a consequence, we can solve a nonzero-sum bimatrix game for each player approximately without solving any equations. Moreover, we modify the inequalities for the extrema of the strategy set for the nonzero-sum bimatrix games. Furthermore, we adapt the min-max theorem of the MN approach for the nonzero-sum bimatrix games. Finally, we consider various bimatrix game examples from the literature, including the famous battle of sexes, to demonstrate the consistency of our approaches. We also show that the repeated applications of Ex-tended Matrix Norm (EMN) methods work well to obtain a better-estimated game value in view of the obtained convergence results.(c) 2022 Elsevier Inc. All rights reserved.

Automated summary

This paper extends and applies the Matrix Norm (MN) approach to nonzero-sum bimatrix games, providing preliminary results for the convergence of the MN approach. The authors introduce a notation for expressing nonzero-sum bimatrix games in terms of two matrix games and prove theorems regarding the boundaries of the game value. They also refine the boundaries for zero/nonzero-sum matrix games, successfully improving the game value interval. The paper demonstrates the consistency of the approaches through various bimatrix game examples.