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

Burhaneddin İzgi*, Murat Özkaya, Nazım Kemal Üre, Matjaž Perc

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

5 Citations (Scopus)


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 approaches. 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 matrix 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 addition to these, we refine the boundaries of the game value for the zero/nonzero sum matrix 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 Extended Matrix Norm (EMN) methods work well to obtain a better-estimated game value in view of the obtained convergence results.

Original languageEnglish
Article number127553
JournalApplied Mathematics and Computation
Publication statusPublished - 1 Feb 2023

Bibliographical note

Publisher Copyright:
© 2022


This work is supported by the Scientific and Technological Research Council of Turkey (in Turkish: TÜBİTAK) under grant agreement 121E394. M.P. is supported by the Slovenian Research Agency (Grant Nos. P1-0403 and J1-2457). The authors would like to thank the anonymous referees and the editor for their valuable suggestions and comments that helped to improve the content of the article.

FundersFunder number
Javna Agencija za Raziskovalno Dejavnost RSP1-0403, J1-2457
Türkiye Bilimsel ve Teknolojik Araştırma Kurumu121E394


    • Battle of sexes
    • Bimatrix game
    • Convergence
    • Game theory
    • Matrix norms
    • Nonzero sum game


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