Using a Two-Person Zero-Sum Game to Solve a Decision-Making Problem

Joseph Gogodze

doi:doi:10.11648/j.pamj.20180702.11

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Using a Two-Person Zero-Sum Game to Solve a Decision-Making Problem

Joseph Gogodze

Published in Pure and Applied Mathematics Journal (Volume 7, Issue 2)

Received: 30 May 2018 Accepted: 25 June 2018 Published: 17 July 2018

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Abstract

This study proposes a game-theoretic approach to solve a multiobjective decision-making problem. The essence of the method is that a normalized decision matrix can be considered as a payoff matrix for some zero-sum matrix game, in which the first player chooses an alternative and the second player chooses a criterion. Herein, the solution in mixed strategies of this game is used to construct a weighted sum of the primary criteria that leads to a solution of the primary multiobjective decision-making problem. The proposed method leads to a notionally objective weighting method for multiobjective decision-making and provides “true weights” even in the absence of preliminary subjective evaluations. The proposed new method has a simple application. It can be applied to decision-making problems with any number of alternatives/criteria, and its practical realization is limited only by the capabilities of the solver of the linear programming problem formulated to solve the corresponding zero-sum game. Moreover, as observed from the solutions of the illustrative examples, the results obtained with the proposed method are quite appropriate and competitive.

Published in	Pure and Applied Mathematics Journal (Volume 7, Issue 2)
DOI	10.11648/j.pamj.20180702.11
Page(s)	11-19
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2018. Published by Science Publishing Group

Keywords

Multiobjective Optimization, Decision-Making Problem, Two-Person Zero-Sum Matrix Game

References

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[2]	Neumann, Von J. & Morgenstern O. Theory of Games and Economic Behaviour. Princeton University Press, Princeton, NJ, 1944.
[3]	Marler, R. T. & Arora, J. S., The weighted sum method for multi-objective optimization: new insights. Structural and multidisciplinary optimization, 41(6), 2010, pp. 853-862.
[4]	Farag, M. M. Quantitative methods of materials selection. In: Kutz M, editor. Handbook of materials selection; 2002.
[5]	Chatterjee, P., Athawale, V. M., and Chakraborty, S. Selection of materials using compromise ranking and outranking methods. Materials & Design, 30(10), 2009, 4043-4053.
[6]	Khabbaz, R., Sarfaraz, B., Dehghan Manshadi, A., Abedian, and R. Mahmudi. A simplified fuzzy logic approach for materials selection in mechanical engineering design. Materials & design 30(3), 2009, pp. 687-697.
[7]	Jahan, A., Mustapha, F., Ismail, M. Y., Sapuan, S. M., and Bahraminasab, M. A comprehensive VIKOR method for material selection. Materials & Design, 32(3), 2011, pp. 1215-1221.
[8]	Karande, P., and Chakraborty, S. Application of multi-objective optimization on the basis of ratio analysis (MOORA) method for materials selection. Materials & Design 37, 2012, pp. 317-324.
[9]	Yazdani, M. New approach to select materials using MADM tools. International Journal of Business and Systems Research, 12(1), 2018, pp. 25-42.
[10]	Anyfantis, K., Foteinopoulos, P. and Stavropoulos, P. Design for manufacturing of multi-material mechanical parts: A computational based approach. Procedia CIRP, 66, 2017, pp. 22-26.
[11]	Deng, H., Yeh, C. H., & Willis, R. J. Inter-company comparison using modified TOPSIS with objective weights. Computers & Operations Research, 27(10), 2000, pp. 963-973.
[12]	El Gibari, S., Gómez, T. and Ruiz, F. Building composite indicators using multicriteria methods: a review. Journal of Business Economics, 2018, pp. 1-24.
[13]	Shih, H. S., Shyur, H. J., & Lee, E. S. An extension of TOPSIS for group decision making. Mathematical and Computer Modelling, 45(7-8), 2007, pp. 801-813.
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Joseph Gogodze. (2018). Using a Two-Person Zero-Sum Game to Solve a Decision-Making Problem. Pure and Applied Mathematics Journal, 7(2), 11-19. https://doi.org/10.11648/j.pamj.20180702.11

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ACS Style

Joseph Gogodze. Using a Two-Person Zero-Sum Game to Solve a Decision-Making Problem. Pure Appl. Math. J. 2018, 7(2), 11-19. doi: 10.11648/j.pamj.20180702.11

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AMA Style

Joseph Gogodze. Using a Two-Person Zero-Sum Game to Solve a Decision-Making Problem. Pure Appl Math J. 2018;7(2):11-19. doi: 10.11648/j.pamj.20180702.11

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@article{10.11648/j.pamj.20180702.11,
  author = {Joseph Gogodze},
  title = {Using a Two-Person Zero-Sum Game to Solve a Decision-Making Problem},
  journal = {Pure and Applied Mathematics Journal},
  volume = {7},
  number = {2},
  pages = {11-19},
  doi = {10.11648/j.pamj.20180702.11},
  url = {https://doi.org/10.11648/j.pamj.20180702.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20180702.11},
  abstract = {This study proposes a game-theoretic approach to solve a multiobjective decision-making problem. The essence of the method is that a normalized decision matrix can be considered as a payoff matrix for some zero-sum matrix game, in which the first player chooses an alternative and the second player chooses a criterion. Herein, the solution in mixed strategies of this game is used to construct a weighted sum of the primary criteria that leads to a solution of the primary multiobjective decision-making problem. The proposed method leads to a notionally objective weighting method for multiobjective decision-making and provides “true weights” even in the absence of preliminary subjective evaluations. The proposed new method has a simple application. It can be applied to decision-making problems with any number of alternatives/criteria, and its practical realization is limited only by the capabilities of the solver of the linear programming problem formulated to solve the corresponding zero-sum game. Moreover, as observed from the solutions of the illustrative examples, the results obtained with the proposed method are quite appropriate and competitive.},
 year = {2018}
}

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TY  - JOUR
T1  - Using a Two-Person Zero-Sum Game to Solve a Decision-Making Problem
AU  - Joseph Gogodze
Y1  - 2018/07/17
PY  - 2018
N1  - https://doi.org/10.11648/j.pamj.20180702.11
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JF  - Pure and Applied Mathematics Journal
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AB  - This study proposes a game-theoretic approach to solve a multiobjective decision-making problem. The essence of the method is that a normalized decision matrix can be considered as a payoff matrix for some zero-sum matrix game, in which the first player chooses an alternative and the second player chooses a criterion. Herein, the solution in mixed strategies of this game is used to construct a weighted sum of the primary criteria that leads to a solution of the primary multiobjective decision-making problem. The proposed method leads to a notionally objective weighting method for multiobjective decision-making and provides “true weights” even in the absence of preliminary subjective evaluations. The proposed new method has a simple application. It can be applied to decision-making problems with any number of alternatives/criteria, and its practical realization is limited only by the capabilities of the solver of the linear programming problem formulated to solve the corresponding zero-sum game. Moreover, as observed from the solutions of the illustrative examples, the results obtained with the proposed method are quite appropriate and competitive.
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Author Information

Joseph Gogodze

Institute of Control System, Techinformi, Georgian Technical University, Tbilisi, Georgia

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APA Style

Joseph Gogodze. (2018). Using a Two-Person Zero-Sum Game to Solve a Decision-Making Problem. Pure and Applied Mathematics Journal, 7(2), 11-19. https://doi.org/10.11648/j.pamj.20180702.11

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ACS Style

Joseph Gogodze. Using a Two-Person Zero-Sum Game to Solve a Decision-Making Problem. Pure Appl. Math. J. 2018, 7(2), 11-19. doi: 10.11648/j.pamj.20180702.11

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AMA Style

Joseph Gogodze. Using a Two-Person Zero-Sum Game to Solve a Decision-Making Problem. Pure Appl Math J. 2018;7(2):11-19. doi: 10.11648/j.pamj.20180702.11

Copy | Download

@article{10.11648/j.pamj.20180702.11,
  author = {Joseph Gogodze},
  title = {Using a Two-Person Zero-Sum Game to Solve a Decision-Making Problem},
  journal = {Pure and Applied Mathematics Journal},
  volume = {7},
  number = {2},
  pages = {11-19},
  doi = {10.11648/j.pamj.20180702.11},
  url = {https://doi.org/10.11648/j.pamj.20180702.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20180702.11},
  abstract = {This study proposes a game-theoretic approach to solve a multiobjective decision-making problem. The essence of the method is that a normalized decision matrix can be considered as a payoff matrix for some zero-sum matrix game, in which the first player chooses an alternative and the second player chooses a criterion. Herein, the solution in mixed strategies of this game is used to construct a weighted sum of the primary criteria that leads to a solution of the primary multiobjective decision-making problem. The proposed method leads to a notionally objective weighting method for multiobjective decision-making and provides “true weights” even in the absence of preliminary subjective evaluations. The proposed new method has a simple application. It can be applied to decision-making problems with any number of alternatives/criteria, and its practical realization is limited only by the capabilities of the solver of the linear programming problem formulated to solve the corresponding zero-sum game. Moreover, as observed from the solutions of the illustrative examples, the results obtained with the proposed method are quite appropriate and competitive.},
 year = {2018}
}

Copy | Download

TY  - JOUR
T1  - Using a Two-Person Zero-Sum Game to Solve a Decision-Making Problem
AU  - Joseph Gogodze
Y1  - 2018/07/17
PY  - 2018
N1  - https://doi.org/10.11648/j.pamj.20180702.11
DO  - 10.11648/j.pamj.20180702.11
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 11
EP  - 19
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20180702.11
AB  - This study proposes a game-theoretic approach to solve a multiobjective decision-making problem. The essence of the method is that a normalized decision matrix can be considered as a payoff matrix for some zero-sum matrix game, in which the first player chooses an alternative and the second player chooses a criterion. Herein, the solution in mixed strategies of this game is used to construct a weighted sum of the primary criteria that leads to a solution of the primary multiobjective decision-making problem. The proposed method leads to a notionally objective weighting method for multiobjective decision-making and provides “true weights” even in the absence of preliminary subjective evaluations. The proposed new method has a simple application. It can be applied to decision-making problems with any number of alternatives/criteria, and its practical realization is limited only by the capabilities of the solver of the linear programming problem formulated to solve the corresponding zero-sum game. Moreover, as observed from the solutions of the illustrative examples, the results obtained with the proposed method are quite appropriate and competitive.
VL  - 7
IS  - 2
ER  -

Copy | Download