Abstract
This paper studies a class of perturbations of a game matrix that alters each row by a different amount. We find that completely mixed optimal strategies are stable under these perturbations provided the norm of the vector of additive amounts is sufficiently small. Using this concept we give a new characterization of completely mixed grames. We also obtain a sensitivity result for a class of perturbations of the technological coefficient matrix of positive linear programs. The stability of an optimal strategy holds throughout at least a spherical neighborhood of the zero perturbation. We give a computational formula and equivalent programming formulations for the radius of this neighborhood.
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References
G. Dantzig, “A proof of the equivalence of the programming problem and the game problem,” in T. Koopmans, ed.,Activity Analysis of Production and Allocation, Cowles Commission Monograph 13, John Wiley & Sons, New York, 1951, 330–335.
G.H. Golub and C.F. Van Loan,Matrix Computations (The Johns Hopkins University Press, Baltimore, Maryland, 1983).
I. Kaplansky, “A contribution to Von Neumann’s theory of games”,Annals of Mathematics, 46 (1945) 474–479.
S. Karlin,Mathematical Methods and Theory in Games, Programming, and Economics (two volumes) (Addison-Wesley Publishing Company, Reading, Massachusetts, 1959).
R. Luce and H. Raiffa,Games and Decisions (John Wiley & Sons, Inc., New York, 1957).
H. Mills, “Marginal values of matrix games and linear programs”, in: H. Kuhn and A. Tucker, eds.,Contributions to the Theory of Linear Inequalities, Volume 38 (Princeton University Press, 1956) 183–193.
T. Parthasarthy and T.E.S. Raghavan,Some Topics in Two Person Games (American Elsevier Publishing Co. Inc., New York, 1971).
M.J. Todd, “Dual families of linear programs,” Technical Report No. 197, Department of Operations Research, College of Engineering, Cornell University, Ithaca, New York, September 1973.
M.J. Todd, “Extensions of Lemke’s Algorithm for the Linear Complementarity Problem,”Journal of Optimization Theory and Applications, 20 (1976) 397–416.
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Troutt, M.D. A stability concept for matrix game optimal strategies and its application to linear programming sensitivity analysis. Mathematical Programming 36, 353–361 (1986). https://doi.org/10.1007/BF02592066
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DOI: https://doi.org/10.1007/BF02592066