Geodesic
Interval valued bimatrix games
Kybernetika, Tome 46 (2010) no. 3, pp. 435-446 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

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.
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.
Classification : 90C11, 91A05, 91A15
Keywords: bimatrix game; interval matrix; interval analysis 
@article{KYB_2010_46_3_a8,
     author = {Hlad{\'\i}k, Milan},
     title = {Interval valued bimatrix games},
     journal = {Kybernetika},
     pages = {435--446},
     year = {2010},
     volume = {46},
     number = {3},
     mrnumber = {2676081},
     zbl = {1202.91021},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/KYB_2010_46_3_a8/}
}
TY  - JOUR
AU  - Hladík, Milan
TI  - Interval valued bimatrix games
JO  - Kybernetika
PY  - 2010
SP  - 435
EP  - 446
VL  - 46
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/KYB_2010_46_3_a8/
LA  - en
ID  - KYB_2010_46_3_a8
ER  - 
%0 Journal Article
%A Hladík, Milan
%T Interval valued bimatrix games
%J Kybernetika
%D 2010
%P 435-446
%V 46
%N 3
%U http://geodesic.mathdoc.fr/item/KYB_2010_46_3_a8/
%G en
%F KYB_2010_46_3_a8
Hladík, Milan. Interval valued bimatrix games. Kybernetika, Tome 46 (2010) no. 3, pp. 435-446. http://geodesic.mathdoc.fr/item/KYB_2010_46_3_a8/

[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 | MR

[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 | MR | Zbl

[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 | Zbl

[7] Levin, V. I.: Antagonistic games with interval parameters. Cybern. Syst. Anal. 35 (1999), 4, 644–652. | DOI | MR | Zbl

[8] Liu, S.-T., Kao, C.: Matrix games with interval data. Computers & Industrial Engineering 56 (2009), 4, 1697–1700. | DOI

[9] Nash, J. F.: Equilibrium points in $n$-person games. Proc. Natl. Acad. Sci. USA 36 (1950), 48–49. | DOI | MR | Zbl

[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 | MR | Zbl

[12] Thomas, L. C.: Games, Theory and Applications. Reprint of the 1986 edition. Dover Publications, Mineola, NY 2003. | MR | Zbl

[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

[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 | MR | Zbl