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    Öğe
    Axioms of Decision Criteria for 3D Matrix Games and Their Applications
    (Mdpi, 2022) Ozkaya, Murat; Izgi, Burhaneddin; Perc, Matjaz
    In this paper, we define characteristic axioms for 3D matrix games and extend the definitions of the decision criteria under uncertainty to three dimensions in order to investigate the simultaneous effect of two different states on the decision process. We first redefine the Laplace, Wald, Hurwicz, and Savage criteria in 3D. We present a new definition depending on only the infinity-norm of the 3D payoff matrix for the Laplace criterion in 3D. Then, we demonstrate that the Laplace criterion in 3D explicitly satisfies all the proposed axioms, as well as the other three criteria. Moreover, we illustrate a fundamental example for a three-dimensional matrix with 3D figures and show the usage of each criterion in detail. In the second example, we model a decision process during the COVID-19 pandemic for South Korea to show the applicability of the 3D decision criteria using real data with two different states of nature for individuals' actions for the quarantine. Additionally, we present an agricultural insurance problem and analyze the effects of the hailstorm and different speeds of wind on the harvest by the 3D criteria. To the best of our knowledge, this is the first study that brings 3D matrices in decision and game theories together.
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    Öğe
    Extended matrix norm method: Applications to bimatrix games and convergence results
    (Elsevier Science Inc, 2023) Izgi, Burhaneddin; Ozkaya, Murat; Ure, Nazim Kemal; Perc, Matjaz
    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.