The concept of Z-equilibrium has been introduced by Zhuk-ovskii (Mathematical Methods in Operations Research. Bulgarian Academy of Sciences, Sofia (1985) 103–195) for games in normal form. This concept is always Pareto optimal and individually rational for the players. Moreover, Pareto optimal Nash equilibria are Z-equilibria. We consider a bi-matrix game whose payoffs are uncertain variables. By appropriate ranking criteria of Liu uncertainty theory, we introduce some concepts of equilibrium based on Z-equilibrium for such games. We provide sufficient conditions for the existence of the introduced concepts. Moreover, using mathematical programming, we present a procedure for their computation. A numerical example is provided for illustration.
Mots-clés : Bi-matrix game, Pareto optimal, uncertainty theory, $$-equilibrium
@article{RO_2020__54_2_393_0,
author = {Achemine, Farida and Merakeb, Abdelkader and Larbani, Moussa and Marthon, Philippe},
title = {$Z$-equilibria in {Bi-matrix} games with uncertain payoffs},
journal = {RAIRO - Operations Research - Recherche Op\'erationnelle},
pages = {393--412},
publisher = {EDP-Sciences},
volume = {54},
number = {2},
year = {2020},
doi = {10.1051/ro/2019007},
mrnumber = {4069297},
zbl = {1434.91003},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/ro/2019007/}
}
TY - JOUR AU - Achemine, Farida AU - Merakeb, Abdelkader AU - Larbani, Moussa AU - Marthon, Philippe TI - $Z$-equilibria in Bi-matrix games with uncertain payoffs JO - RAIRO - Operations Research - Recherche Opérationnelle PY - 2020 SP - 393 EP - 412 VL - 54 IS - 2 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/ro/2019007/ DO - 10.1051/ro/2019007 LA - en ID - RO_2020__54_2_393_0 ER -
%0 Journal Article %A Achemine, Farida %A Merakeb, Abdelkader %A Larbani, Moussa %A Marthon, Philippe %T $Z$-equilibria in Bi-matrix games with uncertain payoffs %J RAIRO - Operations Research - Recherche Opérationnelle %D 2020 %P 393-412 %V 54 %N 2 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/ro/2019007/ %R 10.1051/ro/2019007 %G en %F RO_2020__54_2_393_0
Achemine, Farida; Merakeb, Abdelkader; Larbani, Moussa; Marthon, Philippe. $Z$-equilibria in Bi-matrix games with uncertain payoffs. RAIRO - Operations Research - Recherche Opérationnelle, Tome 54 (2020) no. 2, pp. 393-412. doi : 10.1051/ro/2019007. http://archive.numdam.org/articles/10.1051/ro/2019007/
[1] and , Robust game theory. Math. Program. Ser. 107 (2006) 231–273. | DOI | MR | Zbl
[2] , Ambiguous act equilibria. Games. Econ. Behav. 71 (2010) 246–260. | DOI | MR | Zbl
[3] , and , Nash equilibrium solution in trapezoidal fuzzy environment. IOSR J. Eng. (IOSRJEN) 3 (2013) 7–14. | DOI
[4] , and , Z-equilibrium for a CSP game. International Symposium on Artificial Intelligence and Mathematics. Fort Lauderdale, FL (2016).
[5] , Fuzzy games: a description of the concept. Fuzzy Sets Syst. 1 (1978) 181–192. | DOI | MR | Zbl
[6] , Advances in Fuzzy Set Theory and Applications, edited by , and . In: Advances in Fuzzy Set Theory and Applications. Kluwer, Boston, MA (1979) 339–359. | Zbl
[7] and , Generalized two-person zero-sun games with fuzzy strategies and fuzzy payoffs. Fuzzy Syst. Math. 20 (2006) 95–101. | Zbl
[8] and , Fuzzy based GA for entropy bimatrix goal game. Int. J. Uncertain. Fuzziness Knowledge-Based Syst. 18 (2010) 779–799. | DOI | MR | Zbl
[9] and , Fuzzy based GA to multi-objective entropy bimatrix game. Opsearch 50 (2013) 125–140. | DOI | MR | Zbl
[10] , Risk, ambiguity and the Savage axiom. Quat. J. Econ. 75 (1961) 643–669. | DOI | MR | Zbl
[11] , , Z-Equilibrium for a Mixed Strategic Multicriteria Game. EURO 25, Vilnius (2012).
[12] , Uncertain bi-matrix game with applications. Fuzzy Optim. Decis. Mak. 12 (2013) 65–78. | DOI | MR | Zbl
[13] , Games with incomplete information played by bayesian players. The basic model. Management Sci. 14 (1967) 317–334. | MR | Zbl
[14] and , A General Theory of Equilibrium Selection in Games. MIT Press, Cambridge, MA (1988). | Zbl
[15] , , and , Communication and equilibrium in discontinuous games of incomplete information. Econometrica 70 (2002) 1711–1740. | DOI | MR | Zbl
[16] , Uncertainty, decision and normal form games. Manuscript (1996).
[17] , Non cooperative fuzzy games in normal form: a survey. Fuzzy Sets Syst. 160 (2009) 3184–3210. | DOI | MR | Zbl
[18] and , A Concept of equilibrium for a game under uncertainty. Euro. J. Oper. Res. 1 (1999) 145–156. | DOI | Zbl
[19] and , Hybrid logic and uncertain logic. J. Uncertain Syst. 3 (2009) 83–94.
[20] , Uncertainty Theory, 2nd edition. Springer-Verlag, Berlin (2007). | MR | Zbl
[21] , Some research problems in uncertainty theory. J. Uncertain Syst. 3 (2009) 3–10.
[22] , Uncertainty Theory: A Branch of Mathematics for Modeling Human Uncertainty. Springer-Verlag, Berlin (2010). | DOI
[23] , Why is there a need for uncertainty theory?. J. Uncertain Syst. 6 (2012) 3–10.
[24] , Uncertainty Theory, 4th edition. Springer-Verlag, Berlin (2015). | MR | Zbl
[25] , Characterization of the equilibrium strategy of the bimatrix game with fuzzy payoff. J. Math. Anal. Appl. 251 (2000) 885–896. | DOI | MR | Zbl
[26] , Deterministic global optimization using interval constraint propagation technique. RAIRO-Rech. Oper. 38 (2004) 277–293. | DOI | Numdam | MR | Zbl
[27] , and , A reliable affine relaxation method for global optimization. 4OR-Q. J Oper. Res. 13 (2015) 247–277. | DOI | MR | Zbl
[28] , A deterministic global optimization algorithm for design problems, edited by , and . In: Chapter in Essays and Surveys in Global Optimization (2005) 267–294. | DOI | MR | Zbl
[29] , and , Birough programming approach for solving bi-matrix games with birough payoff elements. J. Intel. Fuzzy Syst. 29 (2015) 863–875. | DOI | MR | Zbl
[30] , Non-cooperative games. Ann. Math. 54 (1951) 286–295. | DOI | MR | Zbl
[31] , and , Coalitional -Equilibrium in games and its Existence. Int. Game Theory Rev. 17 (2015). | DOI | MR | Zbl
[32] and , Equilibrium solutions in multiobjective bi-matrix games with fuzzy payoffs and fuzzy goals. Fuzzy Sets Syst. 111 (2000) 99–116. | DOI | MR | Zbl
[33] and , A sufficient and necessary condition of uncertainty distribution. J. Interdisciplinary Math. 13 (2010) 277–285. | DOI | MR | Zbl
[34] , A note on robust Nash equilibria in games with uncertainties. RAIRO-REch. Oper. 48 (2014) 365–371. | DOI | Numdam | MR | Zbl
[35] , Fuzzy programming approach to two-person multicriteria bimatrix games. J. Fuzzy Math. 15 (2007) 141–153. | MR | Zbl
[36] and , Bi-matrix game in bifuzzy environment. J. Uncertainty Anal. App. 1 (2013) 1–11.
[37] and , Rough set approach to bi-matrix game. Int. J. Oper. Res. 23 (2015) 229–244. | DOI | MR | Zbl
[38] and , Characterizing the Nash equilibria of three-player Bayesian quantum games. SPIE, forthcoming (2017). | MR | Zbl
[39] , A mathematical theory of evidence. Princeton University Press, Princeton, NJ (1976). | DOI | MR | Zbl
[40] and , Multiagent Systems: Algorithmic, Game-Theoretic, and Logical Foundations. Cambridge University Press, New York, NY (2009). | Zbl
[41] , and , Computing approximate Bayes-Nash equilibria in treegames of incomplete information. In: EC: Proceedings of the ACM Conference on Electronic Commerce (2004) 81–90.
[42] , Multiple Criteria Optimization: Theory, Computation and Application. John Wiley and Sons, New York, NY (1986). | MR | Zbl
[43] and , Games with Ambiguous Payoffs and played by Ambiguity and regret minimising players, edited by and . In: Advances in Artificial Intelligence. AI 2012. Lecture notes in Computer Science. Springer-Verlag Berlin, Heidelberg 7691 (2012) 409–420. | MR
[44] , Fuzzy sets. Informa. Control 8 (1965) 338–353. | DOI | MR | Zbl
[45] , Some problems of non-antagonistic differential games, edited by . In: Matematiceskie metody versus issledovanii operacij [Mathematical Methods in Operations Research]. Bulgarian Academy of Sciences, Sofia (1985) 103–195.
[46] and , Linear-quadratic Differential Games. Naoukova Doumka, Kiev (1994).
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