Static Bayesian games with finite fuzzy types and the existence of equilibrium. (English) Zbl 1156.91317
Summary: This paper studies a special game with incomplete information, in which the payoffs of the players are both random and fuzzy. Such a game is considered in the context of a Bayesian game with the uncertain types characterized as fuzzy variables. A static fuzzy Bayesian game is then introduced and the decision rules for players are given based on credibility theory. We further prove the existence of the equilibrium of the game. Finally, a Cournot competition model with fuzzy efficiency under asymmetric information is investigated as an application and some results are presented.
MSC:
| 91A15 | Stochastic games, stochastic differential games |
| 91B06 | Decision theory |
| 90C70 | Fuzzy and other nonstochastic uncertainty mathematical programming |
Keywords:
game theory; decision analysis; static fuzzy Bayesian game; uncertainty modeling; fuzzy variablesReferences:
| [1] | Arikan, F.; Gungor, Z., A two-phase approach for multi-objective programming problems with fuzzy coefficients, Information Science, 177, 5191-5202 (2007) · Zbl 1121.90134 |
| [2] | Aubin, J., Optima and Equilibria: An Introduction to Nonlinear Analysis (1993), Springer-Verlag · Zbl 0781.90012 |
| [3] | Aumann, R., Agreeing to disagree, The Annals of Statistics, 4, 1236-1239 (1976) · Zbl 0379.62003 |
| [4] | Aumann, R., Correlated equilibrium as an expression of Bayesian rationality, Econometrica, 55, 1-18 (1987) · Zbl 0633.90094 |
| [5] | Berg, J., Statistical mechanics of random two-player games, Physics Review E, 61, 2327-2339 (2000) |
| [6] | Blau, A., Random-payoff two-person zero-sum games, Operations Research, 22, 1243-1251 (1974) · Zbl 0306.90087 |
| [7] | Buckley, J., Multiple goal non-cooperative conflicts under uncertainty: a fuzzy set approach, Fuzzy Sets and Systems, 13, 107-124 (1984) · Zbl 0549.90095 |
| [8] | Butnariu, D.; games, Fuzzy, a description of the concept, Fuzzy Sets and Systems, 1, 181-192 (1978) · Zbl 0389.90100 |
| [9] | Butnariu, D., Solution concepts for \(n\)-person fuzzy games, (Gupta, M. M.; Ragde, R. K.; Yager, R. R., Advances in Fuzzy Set Theory and Applications (1979), Kluwer: Kluwer Boston) · Zbl 0623.90099 |
| [10] | Campos, L., Fuzzy linear programming models to solve fuzzy matrix games, Fuzzy Sets and Systems, 32, 275-289 (1989) · Zbl 0675.90098 |
| [11] | Cassidy, G.; Field, A.; Kirby, L., Solution of a satisfying model for random payoff games, Management Science, 19, 266-271 (1972) · Zbl 0262.90073 |
| [12] | Charnes, A.; Kirbyand, M.; Raike, W., Zero-zero chance-constrained games, Theory of Probability and its Applications, 13, 628-646 (1968) · Zbl 0209.52008 |
| [13] | Ein-Dor, L.; Kanter, I., Matrix games with nonuniform payoff distributions, Physica A, 302, 80-88 (2001) · Zbl 0993.91001 |
| [14] | Fudenberg, D.; Jean, T., Game Theory (1991), MIT Press: MIT Press Cambridge · Zbl 1339.91001 |
| [15] | Garagic, D.; Cruz, J., An approach to fuzzy noncooperative Nash games, Journal of Optimization Theory and Applications, 118, 475-491 (2003) · Zbl 1073.91002 |
| [16] | Gao, J., Credibilistic game with fuzzy information, Journal of Uncertain Systems, 1, 74-80 (2007) |
| [17] | J. Harsanyi, Games with incomplete information played by ‘Bayesian’ players, I, II, III, Management Science 14 (1967-1968) 14, 159-182, 320-334, 486-502.; J. Harsanyi, Games with incomplete information played by ‘Bayesian’ players, I, II, III, Management Science 14 (1967-1968) 14, 159-182, 320-334, 486-502. · Zbl 0207.51102 |
| [18] | Harsanyi, C., Games with incomplete information, The American Economic Review, 85, 291-303 (1995) |
| [19] | Huang, X., Credibility-based chance-constrained integer programming models for capital budgeting with fuzzy parameters, Information Sciences, 176, 2698-2712 (2006) · Zbl 1149.91315 |
| [20] | Kacher, F.; Larbani, M., Existence of equilibrium solution for a non-cooperative game with fuzzy goals and parameters, Fuzzy Sets and Systems, 159, 164-176 (2008) · Zbl 1168.91308 |
| [21] | Kandel, A.; Zhang, Y., Fuzzy moves, Fuzzy Sets and Systems, 99, 159-177 (1998) · Zbl 1083.91502 |
| [22] | Liu, B., A survey of credibility theory, Fuzzy Optimization and Decision Making, 5, 387-408 (2006) · Zbl 1133.90426 |
| [23] | Liu, B., A survey of entropy of fuzzy variables, Journal of Uncertain Systems, 1, 4-13 (2007) |
| [24] | Liu, B.; Liu, Y., Expected value of fuzzy variable and fuzzy expected value models, IEEE Transactions on Fuzzy Systems, 10, 445-450 (2002) |
| [25] | Liu, Y.; Liu, B., Expected value operator of random fuzzy variable and random fuzzy expected value models, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 11, 195-215 (2003) · Zbl 1074.90056 |
| [26] | Liu, Y.; Gao, J., The independence of fuzzy variables with applications to fuzzy random optimization, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 15, 1-19 (2007) |
| [27] | Maeda, T., Characterization of the equilibrium strategy of the bimatrix game with fuzzy payoff, Journal of Mathematical Analysis and Applications, 251, 885-896 (2000) · Zbl 0974.91004 |
| [28] | Maeda, T., On characterization of equilibrium strategy of two-person zero-sum games with fuzzy payoffs, Fuzzy Sets and Systems, 139, 283-296 (2003) · Zbl 1065.91004 |
| [29] | Myerson, R., Refinement of the Nash equilibrium concept, International Journal of Game Theory, 7, 73-80 (1978) · Zbl 0392.90093 |
| [30] | Narukawa, Y.; Torra, V., Fuzzy measures and integrals in evaluation of strategies, Information Sciences, 177, 4686-4695 (2007) · Zbl 1284.91066 |
| [31] | Nash, J., Equilibrium points in \(n\)-person games, Proceedings of the National Academy of Sciences, 36, 48-49 (1950) · Zbl 0036.01104 |
| [32] | von Neumann, J.; Morgenstern, O., Theory of Games and Economic Behavior (1944), Wiley: Wiley New York · Zbl 0063.05930 |
| [33] | Nishizaki, I.; Sakawa, M., Fuzzy and Multiobjective Games for Conflict Resolution (2001), Physica-Verleg: Physica-Verleg Heidelberg · Zbl 0973.91001 |
| [34] | Oh, S.-K.; Pedrycz, W.; Roh, S.-B., Genetically optimized fuzzy polynomial neural networks with fuzzy set-based polynomial neurons, Information Sciences, 176, 3490-3519 (2006) · Zbl 1119.68159 |
| [35] | Roberts, P., Nash equilibria of Cauchy-random zero-sum and coordination, International Journal of Game Theory, 34, 167-184 (2006) · Zbl 1108.91012 |
| [36] | Vijay, V.; Chandra, S.; Bector, C., Matrix games with fuzzy goals and fuzzy payoffs, Omega - International Journal of Management Science, 33, 425-429 (2005) |
| [37] | Wang, C.; Tang, W.; Zhao, R., On the continuity and convexity analysis of the expected value function of a fuzzy mapping, Journal of Uncertain Systems, 1, 148-160 (2007) |
| [38] | Wilde, P., Fuzzy utility and equilibria, IEEE Transactions on Systems Man and Cybernetics Part B - Cybernetics, 34, 1774-1785 (2004) |
| [39] | Xu, L.; Zhao, R.; Shu, T., Three equilibrium strategies for two-person zero-sum game with fuzzy payoffs, Lecture Notes in Artificial Intelligence, 3613, 350-354 (2005) |
| [40] | Xu, L.; Zhao, R.; Ning, Y., Two-person zero-sum matrix game with fuzzy random payoffs, Lecture Notes in Artificial Intelligence, 4114, 809-818 (2006) |
| [41] | Zadeh, L. A., Fuzzy sets, Information and Control, 8, 338-353 (1965) · Zbl 0139.24606 |
| [42] | Zadeh, L. A., Fuzzy sets as a basis for a theory of possibility, Fuzzy Sets and Systems, 1, 3-28 (1978) · Zbl 0377.04002 |
| [43] | Zadeh, L. A., Toward a generalized theory of uncertainty (GTU) - an outline, Information Sciences, 172, 1-40 (2005) · Zbl 1074.94021 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
