Yazar "Perc, Matjaz" seçeneğine göre listele

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    A holistic matrix norm-based alternative solution method for Markov reward games
    (Elsevier Science Inc, 2025) İzgi, Burhaneddin; Özkaya, Murat; Üre, Nazım Kemal; Perc, Matjaz
    In this study, we focus on examining single-agent stochastic games, especially Markov reward games represented in the form of a decision tree. We propose an alternative solution method based on the matrix norms for these games. In contrast to the existing methods such as value iteration, policy iteration, and dynamic programming, which are state-and-action-based approaches, the proposed matrix norm-based method considers the relevant stages and their actions as a whole and solves it holistically for each stage without computing the effects of each action on each state's reward individually. The new method involves a distinct transformation of the decision tree into a payoff matrix for each stage and the utilization of the matrix norm of the obtained payoff matrix. Additionally, the concept of the moving matrix is integrated into the proposed method to incorporate the impacts of all actions on the stage simultaneously, rendering the method holistic. Moreover, we present an explanatory algorithm for the implementation of the method and also provide a comprehensive solution diagram explaining the method figuratively. As a result, we offer a new and alternative perspective for solving the games with the help of the proposed method due to the simplicity of utilization of the matrix norms in addition to the existing methods. For clarification of the matrix norm-based method, we demonstrate the figurative application of the method on a benchmark Markov reward game with 2-stages and 2-actions and a comprehensive implementation of the method on a game consisting of 3-stages and 3-actions.
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    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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    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.
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    Matrix norm based hybrid Shapley and iterative methods for the solution of stochastic matrix games
    (Elsevier Science Inc, 2024) İzgi, Burhaneddin; Özkaya, Murat; Üre, Nazım Kemal; Perc, Matjaz
    In this paper, we present four alternative solution methods to Shapley iteration for the solution of stochastic matrix games. We first combine the extended matrix norm method for stochastic matrix games with Shapley iteration and then state and prove the weak and strong hybrid versions of Shapley iterations. Then, we present the semi-extended matrix norm and iterative semi-extended matrix norm methods, which are analytic-solution-free methods, for finding the approximate solution of stochastic matrix games without determining the strategy sets. We illustrate comparisons between the Shapley iteration, weak and strong hybrid Shapley iterations, semi-extended matrix norm method, and iterative semi-extended matrix norm method with several examples. The results reveal that the strong and weak hybrid Shapley iterations improve the Shapley iteration and decrease the number of iterations, and the strong hybrid Shapley iteration outperforms all the other proposed methods. Finally, we compare these methods and present their performance analyses for large-scale stochastic matrix games as well.