Approximations and well-posedness in multicriteria games. (English) Zbl 1138.91407

Morgan, Jacqueline

Ann. Oper. Res. 137, 257-268 (2005).

Summary: We address bargaining games where the agents have to take into account different criteria to value the decisions. We propose the class of generalized maximin solutions as the natural extension of the maximin solution concept in conventional bargaining games. In order to refine this solution concept, we define a multicriteria lexicographic partial ordering and present the class of generalized leximin solutions as those that are nondominated with respect to this relation. We establish some properties of these solutions and characterize them as solutions of multicriteria problems.

MSC:

91B26	Auctions, bargaining, bidding and selling, and other market models
91A06	\(n\)-person games, \(n>2\)

References:

[1]	Altman, E., T. Boulogne, R. El-Azouzi, and T. Jimenez. (2000). ”A Survey on Networking Games in Telecomunications.” working paper.
[2]	Aubin, J.P. (1978). Mathematical Methods in Games and Economic Theory. North Holland, Amsterdam.
[3]	Basar, T. and G.J. Olsder. (1995). Dynamic Noncooperative Game Theory. New York: Academic press. · Zbl 0479.90085
[4]	Bednarczuk, E. (1994). ”An Approach to Well-Posedness in Vector Optimization: Consequences to Stability.” Control and Cybernetics 23, 107–122. · Zbl 0811.90092
[5]	Blackwell, O. (1956). ”An Analog of the Minimax Theorem for Vector Payoffs.” Pacific Journal of Mathematics 6, 1–8. · Zbl 0074.34403
[6]	Berge, C. (1959). Espaces Topologiques. Fonctions Multivoques, Dunod, Paris. · Zbl 0088.14703
[7]	Borm, P., F. van Hegen, and S.H. Tijs. (1999). ”A Perfectness Concept for Multicriteria Games.” Math. Methods Oper. Res. 49, 401–412. · Zbl 0942.91006
[8]	Cavazzuti, E. and J. Morgan. (1983). ”Well-Posed Saddle Point Problems.” Lecture Notes in Pure and Applied Mathematics, New York: Marcel Dekker, vol. 86, pp. 61–76. · Zbl 0519.49015
[9]	Corley, H.W. (1985). ”Games with Vector Payoffs.” Journal ofOptimization Theory and Applications 42, 491–498. · Zbl 0556.90095
[10]	Del Prete, I., M.B. Lignola, and J. Morgan. (2003). ”New Concepts of Well-Posedness for Optimization Problems with Variational Inequality Constraints.” Journal of Inequalities in Pure and Applied Mathematics 4, Issue 1. · Zbl 1029.49024
[11]	Dontchev, A.L. and T. Zolezzi. (1993). ”Well-Posed Optimization Problems.” Lecture Notes in Mathematics, Springer Verlag, vol. 1543. · Zbl 0797.49001
[12]	Fernandez, F.R., L. Monroy, and J. Puerto. (1998). ”Games with Vector Payoffs.” Journal of OptimizationTheory and Applications 99, 195–208. · Zbl 0915.90269
[13]	Fernandez, F.R., M.A. Hinojosa, and J. Puerto. (2002). ”Games with Vector Payoffs.” Journal of OptimizationTheory and Applications 112, 331–360. · Zbl 1005.91016
[14]	Ghose, D.B. (1991). ”A Necessary and Sufficient Condition forPareto-Optimal Security Strategies in Multicriteria Matrix Games.” Journal of Optimization Theory and Applications 68, 468–480. · Zbl 0697.90088
[15]	Ghose, D. and U.R. Prasad. (1989). ”Solution Concepts inTwo-Persons Multicriteria Games.” Journal of Optimization Theory andApplications 63, 167–188. · Zbl 0662.90093
[16]	Huang, X.X. (2000). ”Extended Well-Posedness Properties of Vector Optimization Problems.” Journal of Optimization Theoryand Applications 106, 165–182. · Zbl 1028.90067
[17]	Huang, X.X. (2001). ”Well-Posedness of Perturbed Vector Optimization Problems in a Vector-Valued Variational Principle.” Journal of Optimization Theory and Applications 108, 671–686. · Zbl 1017.90098
[18]	Kuratowski, K. (1968). Topology, New York-London: Academic Press.
[19]	Lignola, M.B. and J. Morgan. (1994). ”Semicontinuities and Episemicontinuity: Equivalence and Applications.” Bollettino Unione Matematica Italiana 7(8-B), 1–16. · Zbl 0799.49013
[20]	Lignola, M.B. and J. Morgan. (2000). ”Well-Posedness for Optimization Problems with Constraints Defined by Variational Inequalities Having a Unique Solution.” Journal of Global Optimization 16, 57–67. · Zbl 0960.90079
[21]	Lignola, M.B. and J. Morgan. (2002). ”Approximate Solutions and {\(\alpha\)}-Well-Posedness for Variational Inequalities and Nash Equilibria.” Decision and Control in Management Science. Kluwer Academic Publishers, pp. 367–378.
[22]	Loridan, P. (1995). ”Well-Posedness in Vector Optimization.”In R. Lucchetti and J. Revalski (Eds.), Recent Developments in Well-Posed Variational Problems. Kluwer Academic Publishers, pp. 171–192. · Zbl 0848.49017
[23]	Loridan, P. and J. Morgan. (2000). ”Convergence of Approximate Solutions and Values in Parametric Vector Optimization.” In F. Giannessi (eds.) Vector Variational Inequalities and Vector Equilibria. Kluwer Academic Publishers, pp. 335–350. · Zbl 0989.65066
[24]	Luc, D.T. (1989). ”Theory of Vector Optimization.” Lecture Notes in Economics and Mathematical Systems. Berlin: Springer-Verlag, vol. 319. · Zbl 0688.90051
[25]	Lucchetti, R., F. Patrone, S.H. Tijs, and A. Torre. (1987). ”Continuity Properties of Solution Concepts for Cooperative Games.” OR Spektrum 9, 101–107. · Zbl 0619.90098
[26]	Margiocco, M., F. Patrone, and L. Pusillo Chicco. (1997). ”A New Approach to Tikhonov Well-Posedness for Nash Equilibria.” Optimization 40, 385–400. · Zbl 0881.90136
[27]	Monroy, L. and A. Marmol. (1999). ”Aplicaciones Economicas,Avances en Teoria de Juegos con Aplicaciones Economicas y Sociales.” J.M. Bilbao and F.R. Fernandez (eds.) Universidad de Sevilla, pp. 73– 88.
[28]	Monroy, L. and J. Puerto. (1999). ”Juegos Matriciales Vectoriales.” In J.M. Bilbao and F.R. Fernandez (eds.), Avances en teoria de juegos con aplicaciones economicas y sociales. Universidad de Sevilla, pp. 27–53.
[29]	Morgan, J. (1989). Constrained Well-Posed Two-Level Optimization Problems, In F. Clarke, V. Demianov and F. Giannessi (eds.), Nonsmooth Optimization and Related Topics. Ettore Majorana International Sciences Series, New York: Plenum Press, pp. 307–326. · Zbl 0786.90112
[30]	Owen, G. (1993) (III ed.), Game Theory. New York: Academic Press. · Zbl 0802.90132
[31]	Puerto, J. and F.R. Fernandez. (1995). Solution Concepts Based on Security Levels in Constrained Multicriteria Convex Games. Opsearch vol. 32, pp. 16–30. · Zbl 0843.90136
[32]	Puerto, J. and F.R. Fernandez. (1999). ”A Refinement of the Concept of Equilibrium in Multiple Objective Games.” In W. Takahashi and T. Tanaka (eds.) Nonlinear Analysis and Convex Analysis. World Scientific, Singapore.
[33]	Sawaragi, Y., H. Nakayama, and T. Tanino. (1985). Theory of Multiobjective Optimization. Academic Press Inc. · Zbl 0566.90053
[34]	Selten, R. (1975). ”Reexamination of the Perfectness Concept for Equilibrium Points in Extensive Games.” International Journal on Game Theory 4, 25–55. · Zbl 0312.90072
[35]	Shapley, L.S. (1959). ”Equilibrium Points in Games with Vector Payoffs.” Naval research Logistics Quarterly 6, 57–61.
[36]	Singh, C. and N. Rueda. (1994). ”Constrained Vector Valued Games andMultiobjective Minmax Programming.” Opsearch 31(2), 144–154. · Zbl 0815.90139
[37]	Staib, T. (1988). ”On Two Generalizations of Pareto Minimality.” Journal of Optimization Theory and Applications 59, 289–306. · Zbl 0628.90076
[38]	Szidarovszky, F., M.E. Gershon, and L. Duckstein. (1986). Techniques for Multiobjective Decision Making in Systems Management. Elsevier,Amsterdam. · Zbl 0615.90029
[39]	Tykhonov, A.N. (1966). ”On the Stability of the Functional Optimization Problem.” U.S.S.R. Computational Math. and Math. Phys. 6(4), 26–33.
[40]	van Damme, E. (1987). Stability and Perfection of Nash Equilibria. Springer-Verlag. · Zbl 0696.90087
[41]	Wang, S.Y. (1993). ”Existence of a Pareto Equilibrium.” Journal of Optimization Theory and Applications 79, 373–386. · Zbl 0797.90124
[42]	Yang, H. and J. Yu (2002). ”Essential Components of the Set of Weakly Pareto-Nash Equilibrium Points.” Applied Mathematics Letters 15, 553–560. · Zbl 1016.91008
[43]	Yu, P.L. (1979). ”Second-Order Game Problems: Decision Dynamicsin Gaming Phenomena.” Journal of Optimization Theory and Applications 27, 147–166. · Zbl 0393.90117
[44]	Yu, J. and G.X.-Z. Yuan. (1998). ”The Study of Pareto Equilibria for Multiobjective Games by Fixed Point and Ky Fan Minmax Inequality Methods.” Computers Math. Applic. 35, 17–24. · Zbl 1005.91008
[45]	Yuan, X.Z. and E. Tarafdar. (1996). ”Non-Compact Pareto Equilibria for Multiobjective games.” Journal of Math. Analysis andApplications 204, 156–163. · Zbl 0870.90106
[46]	Zeleny, M. (1976). ”Game with Multiple Payoffs.” Intern.Journal of Game Theory 4, 179–191. · Zbl 0395.90093
[47]	Zolezzi, T. (1995). ”Wellposed Criteria in Optimization with Application to the Calculus of Variations.” Non-linear Analysis TMA 25, 437–453. · Zbl 0841.49005