This paper deals with bi-matrix games with random payoffs. Using probability tools, we propose a solution based on the concept of Z-equilibrium. Then, we give sufficient conditions of its existence. Further, the problem of computation of this solution is transformed into the determination of Pareto optimal solutions of a deterministic bi-criteria minimization problem. Finally, we provide illustrative numerical examples.
Keywords: Bi-matrix game, chance constrained game, Z-equilibrium, normal random variable, Cauchy random variable
@article{RO_2022__56_3_1857_0,
author = {Achemine, Farida and Larbani, Moussa},
title = {$Z$-equilibrium in random bi-matrix games: {Definition} and computation},
journal = {RAIRO. Operations Research},
pages = {1857--1875},
year = {2022},
publisher = {EDP-Sciences},
volume = {56},
number = {3},
doi = {10.1051/ro/2022050},
mrnumber = {4445940},
zbl = {1497.91006},
language = {en},
url = {https://www.numdam.org/articles/10.1051/ro/2022050/}
}
TY - JOUR AU - Achemine, Farida AU - Larbani, Moussa TI - $Z$-equilibrium in random bi-matrix games: Definition and computation JO - RAIRO. Operations Research PY - 2022 SP - 1857 EP - 1875 VL - 56 IS - 3 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/ro/2022050/ DO - 10.1051/ro/2022050 LA - en ID - RO_2022__56_3_1857_0 ER -
%0 Journal Article %A Achemine, Farida %A Larbani, Moussa %T $Z$-equilibrium in random bi-matrix games: Definition and computation %J RAIRO. Operations Research %D 2022 %P 1857-1875 %V 56 %N 3 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/ro/2022050/ %R 10.1051/ro/2022050 %G en %F RO_2022__56_3_1857_0
Achemine, Farida; Larbani, Moussa. $Z$-equilibrium in random bi-matrix games: Definition and computation. RAIRO. Operations Research, Tome 56 (2022) no. 3, pp. 1857-1875. doi: 10.1051/ro/2022050
[1] , and , A new equilibrium for an n-person game with fuzzy parameters. Am. J. Math. Manage. Sci. 32 (2013) 118–132.
[2] , , and , -equilibria in Bi-matrix games with uncertain payoffs. RAIRO: OR 54 (2020) 393–412. | MR | Zbl | Numdam | DOI
[3] , The core of a cooperative game without side payements. Trans. Am. Math. Soc. 96 (1961) 539–552. | MR | Zbl | DOI
[4] , Random-payoff two person zero-sum games. Oper. Res. 22 (1974) 1243–1251. | MR | Zbl | DOI
[5] , and , Z-equilibrium for a CSP game. In: International Symposium on Artificial Intelligence and Mathematics Fort Lauderdale, FL (2016).
[6] , and , Solution of a satisficing model for random payoff games. Manage. Sci. 19 (1972) 266–271. | MR | Zbl | DOI
[7] , and , Zero-zero chanceconstrained games. Theory Probab. App. 13 (1968) 628–646. | MR | Zbl | DOI
[8] , , and , Gaming strategy for electric power with random demand. IEEE Trans. Power Syst. 20 (2005) 1283–1292. | DOI
[9] and , A stochastic version of a Stackelberg-Nash-Cournot equilibrium model. Manage. Sci. 43 (1997) 190–197. | Zbl | DOI
[10] and , A stochastic multiple leader Stackelberg model: Analysis, computation, and application. Oper. Res. 57 (2009) 1220–1235. | MR | Zbl | DOI
[11] and , Z-equilibrium for a mixed strategic multicriteria game. EURO 25 – Vilnius (2012) 8–11.
[12] and , Solving global optimization problems with baron. In: From Local to Global Optimization. Vol. 53 of Nonconvex Optimization and Its Applications, edited by , and . Springer, Boston, MA (2001). | MR | Zbl | DOI
[13] and , Variational inequality approach to stochastic Nash equilibrium problems with an application to Cournot oligopoly. J. Optim. Theory App. 165 (2015) 1050–1070. | MR | Zbl | DOI
[14] , and , Continuous Univariate Distributions, 2nd edition. Vol. 1, John Wiley and Sons Inc. (1994). | MR | Zbl
[15] and , Solving a two person cooperative game under uncertainty. In: Multiple Criteria Decision Making in the New Millennium, edited by and . Springer Verlag (2001) 314–324. | MR | Zbl | DOI
[16] and , A concept of equilibrium for a game under uncertainty. Eur. J. Oper. Res. 1 (1999) 145–156. | Zbl | DOI
[17] , Uncertainty Theory, 2nd edition. Springer-Verlag, Berlin (2007). | MR | Zbl
[18] , , , and , Impact of wind integration on electricity markets: A chance-constrained Nash-Cournot model. Int. Trans. Electr. Energy Syst. 23 (2013) 83–96. | DOI
[19] , Non-cooperative games. Ann. Math. 54 (1951) 286–295. | MR | Zbl | DOI
[20] , and , Coalitional -equilibrium in games and its existence. Int. Game Theory Rev. 17 (2015) 1–18. | MR | Zbl | DOI
[21] and , On the characterization of solution sets of smooth and nonsmooth convex stochastic Nash games. SIAM J. Optim. 21 (2011) 1168–1199. | MR | Zbl | DOI
[22] , Conversation and cooperation in social Dilemma. A meta analysis from 1958 to 1992. Ration. Soc. 71995 (1958) 58–92.
[23] , Administrative Behavior: A Study of Decision-Making Processes in Administrative Organization. Macmillan, New York (1947).
[24] and , A characterization of Nash equilibrium for the games with random payoffs. J. Optim. Theory App. 178 (2018) 998–1013. Springer. | MR | Zbl | DOI
[25] , and , Existence of Nash equilibrium for chance-constrained games. Oper. Res. Lett. 44 (2016) 640–644. | MR | Zbl | DOI
[26] , Random payoff matrix games. In: Systems and Management Science by Extremal Methods. Springer Science + Business Media, LLC (1992) 291–308. | DOI
[27] and , Cournot prices considering generator availability and demand uncertainty. IEEE Trans. Power Syst. 22 (2007) 116–125. | DOI
[28] and , Stochastic Nash equilibrium problems: Sample average approximation and applications. Comput. Optim. App. 55 (2013) 597–645. | MR | Zbl | DOI
[29] , Some problems of non-antagonistic differential games. In: Matematiceskie metody versus issledovanii operacij [Mathematical Methods in Operations Research], edited by . Bulgarian Academy of Sciences, Sofia (1985) 103–195.
[30] and , Linear-Quadratic Differential Games. Naoukova Doumka, Kiev (1994).
Cité par Sources :
