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Keywords:
bimatrix game; interval matrix; interval analysis
bimatrix game; interval matrix; interval analysis
Summary:
Payoffs in (bimatrix) games are usually not known precisely, but it is often possible to determine lower and upper bounds on payoffs. Such interval valued bimatrix games are considered in this paper. There are many questions arising in this context. First, we discuss the problem of existence of an equilibrium being common for all instances of interval values. We show that this property is equivalent to solvability of a certain linear mixed integer system of equations and inequalities. Second, we characterize the set of all possible equilibria by mean of a linear mixed integer system.
References:
[1] Alparslan-Gök, S. Z., Branzei, R., Tijs, S. H.: Cores and stable sets for interval-valued games. Discussion Paper 2008-17, Tilburg University, Center for Economic Research, 2008.
[2] Alparslan-Gök, S. Z., Miquel, S., Tijs, S. H.: Cooperation under interval uncertainty. Math. Meth. Oper. Res. 69 (2009), 1, 99–109. DOI 10.1007/s00186-008-0211-3 | MR 2476050
[3] Audet, C., Belhaiza, S., Hansen, P.: Enumeration of all the extreme equilibria in game theory: bimatrix and polymatrix games. J. Optim. Theory Appl. 129 (2006), 3, 349–372. DOI 10.1007/s10957-006-9070-3 | MR 2281145 | Zbl 1122.91009
[4] Collins, W. D., Hu, C.: Fuzzily determined interval matrix games. In: Proc. BISCSE’05, University of California, Berkeley 2005.
[5] Collins, W. D., Hu, C.: Interval matrix games. In: Knowledge Processing with Interval and Soft Computing (C. Hu et al., eds.), Chapter 7, Springer, London 2008, pp. 1–19.
[6] Collins, W. D., Hu, C.: Studying interval valued matrix games with fuzzy logic. Soft Comput. 12 (2008), 2, 147–155. DOI 10.1007/s00500-007-0207-6 | Zbl 1152.91312
[7] Levin, V. I.: Antagonistic games with interval parameters. Cybern. Syst. Anal. 35 (1999), 4, 644–652. DOI 10.1007/BF02835860 | MR 1729000 | Zbl 0964.91005
[8] Liu, S.-T., Kao, C.: Matrix games with interval data. Computers & Industrial Engineering 56 (2009), 4, 1697–1700. DOI 10.1016/j.cie.2008.06.002
[9] Nash, J. F.: Equilibrium points in $n$-person games. Proc. Natl. Acad. Sci. USA 36 (1950), 48–49. DOI 10.1073/pnas.36.1.48 | MR 0031701 | Zbl 0036.01104
[10] Rohn, J.: Solvability of systems of interval linear equations and inequalities. In: Linear Optimization Problems with Inexact Data (M. Fiedler et al., eds.), Chapter 2, Springer, New York 2006, pp. .35–77.
[11] Shashikhin, V.: Antagonistic game with interval payoff functions. Cybern. Syst. Anal. 40 (2004), 4, 556–564. DOI 10.1023/B:CASA.0000047877.10921.d0 | MR 2136247 | Zbl 1132.91329
[12] Thomas, L. C.: Games, Theory and Applications. Reprint of the 1986 edition. Dover Publications, Mineola, NY 2003. MR 2025526 | Zbl 1140.91023
[13] Neumann, J. von, Morgenstern, O.: Theory of Games and Economic Behavior. With an Introduction by Harold Kuhn and an Afterword by Ariel Rubinstein. Princeton University Press, Princeton, NJ 2007. MR 2316805
[14] Stengel, B. von: Computing equilibria for two-person games. In: Handbook of Game Theory with Economic Applications (R. J. Aumann and S. Hart, eds.), Volume 3, Chapter 45, Elsevier, Amsterdam 2002, pp. 1723–1759.
[15] Yager, R. R., Kreinovich, V.: Fair division under interval uncertainty. Internat. J. Uncertain. Fuzziness Knowledge-Based Systems 8 (2000), 5, 611–618. DOI 10.1142/S0218488500000423 | MR 1784650 | Zbl 1113.68542
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