Interval valued bimatrix games. (English) Zbl 1202.91021

Hladík, Milan

Kybernetika 46, No. 3, 435-446 (2010).

This paper deals with a two-player simultaneous game with finite strategy sets. Instead of precise payoffs for the players, the upper and lower values of there payoffs are known. It is shown that the existence of an equilibrium for such an interval game is equivalent to the solvability of a certain linear integer system of equations and inequalities. Moreover, the set of all possible equilibria can be characterized by means of a linear integer system.

Reviewer: Anna Jaskiewicz (Wrocław)

MSC:

91A15	Stochastic games, stochastic differential games
91A05	2-person games
90C11	Mixed integer programming

Keywords:

bimatrix game; interval matrix

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. · Zbl 1236.91019
[2]	Alparslan-Gök, S. Z., Miquel, S., Tijs, S. H.: Cooperation under interval uncertainty. Math. Meth. Oper. Res. 69 (2009), 1, 99-109. · Zbl 1159.91310
[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. · Zbl 1122.91009
[4]	Collins, W. D., Hu, C.: Fuzzily determined interval matrix games. Proc. BISCSE’05, University of California, Berkeley 2005.
[5]	Collins, W. D., Hu, C.: Interval matrix games. Knowledge Processing with Interval and Soft Computing (C. Hu et al., Chapter 7, Springer, London 2008, pp. 1-19. · Zbl 1152.91312
[6]	Collins, W. D., Hu, C.: Studying interval valued matrix games with fuzzy logic. Soft Comput. 12 (2008), 2, 147-155. · Zbl 1152.91312
[7]	Levin, V. I.: Antagonistic games with interval parameters. Cybern. Syst. Anal. 35 (1999), 4, 644-652. · Zbl 0964.91005
[8]	Liu, S.-T., Kao, C.: Matrix games with interval data. Computers & Industrial Engineering 56 (2009), 4, 1697-1700.
[9]	Nash, J. F.: Equilibrium points in \(n\)-person games. Proc. Natl. Acad. Sci. USA 36 (1950), 48-49. · Zbl 0036.01104
[10]	Rohn, J.: Solvability of systems of interval linear equations and inequalities. Linear Optimization Problems with Inexact Data (M. Fiedler et al., 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. · Zbl 1132.91329
[12]	Thomas, L. C.: Games, Theory and Applications. Reprint of the 1986 edition. Dover Publications, Mineola, NY 2003. · 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. · Zbl 1112.91002
[14]	Stengel, B. von: Computing equilibria for two-person games. Handbook of Game Theory with Economic Applications (R. J. Aumann and S. Hart, 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. · Zbl 1113.68542