Abstract
Multiple criteria decision problems with one decision maker have been recognized and discussed in the recent literature in optimization theory, operations research and management science. The corresponding concept with n-decision makers, namely multicriteria n-person games, has not yet been extensively explored.
In this paper we first demonstrate that existing solution concepts for single criterion n-person games in both normal form and characteristic function form induce domination structures (similar to those defined and studied by Yu [39] for multicriteria single decision maker problems) in various spaces, including the payoff space, the imputation space and the coalition space. This discussion provides an understanding of some underlying assumptions of the solution concepts and provides a basis for generalizing and generating new solution concepts not yet defined. Also we illustrate that domination structures may be regarded as a measure of power held by the players.
We then illustrate that a multicriteria problem can naturally arise in decision situations involving (partial) conflict among n-persons. Using our discussion of solution concepts for single criterion games as a basis, various approaches for resolving both normal form and characteristic function form multicriteria n-person games are proposed. For multicriteria games in characteristic function form, we define a multicriteria core and show that there exists a single ‘game point’ whose core is equal to the multicriteria core. If we reduce a multicriteria game to a single criterion game, domination structures which are more general than ‘classical’ ones must be considered, otherwise some crucial information in the game may be lost. Finally, we discuss a parametrization process which, for a given multicriteria game, associates a single criterion game to each point in a parametric space. This parametrization provides a basis for the discussion of solution concepts in multicriteria n-person games.
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Bibliography
Aumann, R. J. and Maschler, M., ‘The Bargaining Set for Cooperative Games’, in Advances in Game Theory (M. Dresher, L. S. Shapley and A. W. Tucker, eds.), Annals of Mathematics Studies, No. 52, Princeton University Press, Princeton, New Jersey, 1964.
Bergstresser, K., Charnes, A., and Yu, P. L., ‘Generalization of Domination Structures and Nondominated Solutions in Multicriteria Decision Making’, Journal of Optimization Theory and Applications 18 (1976), 3–13.
Billera, L. J., ‘Some Recent Results in n-Person Game Theory’, Mathematical Programming 1 (1971), 58–67.
Blackwell, D., ‘An Analog of the Minimax Theorem for Vector Payoffs’, Pacific Journal of Mathematics 6 (1956), 1–8.
Blaquière, A., Girard, F., and Leitman, G., Quantitative and Qualitative Games, Academic Press, New York, 1969.
Blau, R. A., ‘Random-Payoff Two-Person Zero-Sum Games’, Operations Research 22 (1974), 1243–1251.
Cassidy, R., Field, C., and Kirby, M., ‘Solution of a Satisficing Model for Random Payoff Games’, Management Science 19 (1972), 266–271.
Charnes, A. and Cooper, W. W., Management Models and Industrial Applications of Linear Programming, Vols. I and II, John Wiley and Sons, New York, 1961.
Charnes, A. and Granot, D., ‘Coalitional and Chance-Constrained Solutions to N-Person Games I: The Prior Satisficing Nucleolus’, Research Report CCS 123, The Center for Cybernetic Studies, The University of Texas at Austin, Austin, Texas, November, 1973.
Charnes, A. and Granot, D., ‘Prior Solutions: Extensions of Convex Nucleus Solutions to Chance-Constrained Games’, Research Report CCS 148, The Center for Cybernetic Studies, The Unversity of Texas at Austin, Austin, Texas, October 1973.
Charnes, A. and Keane, M., ‘Convex Nuclei and the Shapley Value’, Research Report CCS 12, The Center for Cybernetic Studies, The Unversity of Texas at Austin, Austin, Texas, 1969; also Procedings of the International Congress of Mathematicians, Nice, France, 1970.
Charnes, A., Kirby, M., and Raike, W., ‘Chance-Constrained Games with Partially Controllable Strategies’, Operations Research 16 (1968), 142–149.
Charnes, A., Kirby, M., and Raike, W., ‘Zero-Zero Chance-Constrained Games’, in The Procedings of the Fourth International Conference on Operational Research, Boston, Massachusetts, 1966(D. B. Hertz and J. Melese, eds.) Wiley-Interscience, New York, 1966, pp. 150–169; also Theory of Probability and Its Applications 13 (1968), 663–681.
Charnes, A. and Kortanek, K. O., ‘On Balanced Sets, Cores and Linear Programming’, Cahiers du Centre d'Etudes de Recherche Operationnelle 9 (1967), 32–43.
Charnes, A. and Kortanek, K. O., ‘On Classes of Convex and Preemptive Nuclei for n-Person Games’, in Proceedings of the Princeton Symposium on Mathematical Programming (H. W. Kuhn, ed.), Princeton University Press, Princeton, New Jersey, 1970.
Charnes, A. and Littlechild, S., ‘On the Formation of Unions in n-Person Games’, Journal of Economic Theory 10 (1975), 386–402.
Charnes, A., Littlechild, S., and Sorensen, S., ‘Core-stem Solutions of n-Person Essential Games’, Socio-Economic Planning Sciences 1 (1973), 649–660.
Charnes, A. and Sorensen, S., ‘Constrained n-Person Games’, International Journal of Game Theory 3 (1974), 141–158.
Davis, M. and Maschler, M., ‘The Kernel of a Cooperative Game’, Naval Research Logistics Quarterly 12 (1965), 223–259.
Freimer, M. and Yu, P. L., ‘The Application of Compromise Solutions to Reporting Games’, in Game Theory as a Theory of Conflict Resolution, (A. Rapoport, ed.), D. Reidel Publishing Company, Dordrecht, Holland, 1974.
Freimer, M. and Yu, P. L., ‘Some New Results on Compromise Solutions For Group Decision Problems’, Systems Analysis Program Paper No. F7231, The Graduate School of Management, The University of Rochester, Rochester, New York, June, 1973 (to appear in Management Sciene).
Harsanyi, J. C., ‘A Bargaining Model for the Cooperative n-Person Game’, in Contributions to the Theory of Games, Volume IV (A. W. Tucker and R. D. Luce, eds.), Annals of Mathematics Studies, No. 40, Princeton University Press, Princeton, New Jersey, 1964.
Horowitz, A. D., ‘The Competitive Bargaining Set for Cooperative n-Person Games’, Journal of Mathematical Psychology 10 (1973), 265–289.
Isaacs, R., Differential Games, John Wiley and Sons, New York, 1965.
Keane, M. A., Some Topics in n-Person Game Theory, Ph. D. Dissertation, Mathematics Department, Northwestern University, Evanston, Illinois, June, 1969.
Kohlberg, E., ‘On the Nucleolus of a Characteristic Function Game’, SIAM Journal on Applied Mathematics 20 (1971), 62–66.
Lucas, W. F., ‘Some Recent Developments in N-Person Game Theory’, SIAM Review 13 (1911), 491–523.
Luce, R. D. and Raiffa, H., Games and Decisions, John Wiley and Sons, New York, 1957.
Owen, G., Game Theory, W. B. Saunders Company, Philadelphia, 1968.
Rapoport, A. (ed.), Game Theory as A Theory of Conflict Resolution, D. Reidel Publishing Company, Dordrecht, Holland, 1974.
Rapoport, A., N-Person Game Theory, The University of Michigan Press, Ann Arbor, Michigan, 1969.
Roth, A. E., ‘Subsolutions of Cooperative Games’, Technical Report No. 118, The Economics Series, Institute for Mathematical Studies in the Social Sciences, Stanford University, Stanford, California, December, 1973.
Schmeidler, D., ‘The Nucleolus of a Characteristic Function Game’, SIAM Journal on Applied Mathematics 17 (1969), 1163–1170.
Shapley, L. S., ‘On Balanced Sets and Cores’, Naval Research Logistics Quarterly, 14 (1967), 453–460.
Shapley, L. S., ‘Value for n-Person Games’, in Contributions to the Theory of Games, Volume II (H. W. Kuhn and A. W. Tucker, eds.), Annals of Mathematics Studies, No. 28, Princeton University Press, Princeton, New Jersey, 1953.
Shapley, L. S. and Scarf, H., ‘On Cores and Indivisibility’ Journal of Mathematical Economics 1 (1974), 23–37.
Sorensen, S. W., A Mathematical Theory of Coalitions and Competition in Resource Development, Ph. D. Dissertation, The University of Texas at Austin, Austin, Texas, May, 1972.
Spinetto, R., ‘The Geometry of Solution Concepts for N-Person Cooperative Games’, Management Science 20 (1974), 1292–1299.
Thrall, R. M. and Lucas, W. F., ‘n-Person Games in Partition Function Form’, Naval Research Logistics Quarterly 10 (1963), 281–298.
Von Neumann, J. and Morgenstern, O., Theory of Games and Economic Behavior, John Wiley and Sons, New York, 1944; 2nd ed., 1947; 3rd ed., 1953.
Yu, P. L., ‘A Class of Solutions for Group Decision Problems’, Management Science 19 (1973), 936–946.
Yu, P. L., ‘Introduction to Domination Structures in Multicriteria Decision Problems’, in Multiple Criteria Decision Making (J. L. Cochrane and M. Zeleny, eds.), University of South Carolina Press, Columbia, South Carolina, 1973.
Yu, P. L., ‘Cone Convexity, Cone Extreme Points, and Nondominated Solutions in Decision Problems with Multiobjectives’, Journal of Optimzation Theory and Applications 14 (1974), 319–377.
Yu, P. L., ‘Domination Structures and Nondominated Solutions’, Proceedings of the International Seminar on Multicriteria Decision Making sponsored by UNESCO at CISM, Udine, Italy, June, 1974.
Yu, P. L. and Zeleny, M., ‘The Set of A11 Nondominated Solutions in Linear Cases and a Multicriteria Simplex Method’, Journal of Mathematical Analysis and Applications 49 (1975), 430–468.
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Bergstresser, K., Yu, P.L. Domination structures and multicriteria problems in n-person games. Theor Decis 8, 5–48 (1977). https://doi.org/10.1007/BF00133085
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DOI: https://doi.org/10.1007/BF00133085