Abstarct
In the preceding Chaps. 7–10, we discussed modeling and solving methods of several kinds of matrix games with intuitionistic fuzzy sets. Obviously, these matrix games are a special case of noncooperative games, i.e., two-person zero-sum finite games. In other words, they are a kind of games in which two players are completely antagonistic, i.e., one player wins the other player loses. In a reality, however, it is not always true that players are completely antagonistic. Thus, it is important and useful to study two-person nonzero-sum noncooperative games in normal form. Bi-matrix games are one of important kinds of the above two-person nonzero-sum noncooperative finite games [1, 2]. In this chapter, we will focus on studying bi-matrix games in which the payoffs of players are expressed with intuitionistic fuzzy sets, which are called bi-matrix games with payoffs of intuitionistic fuzzy sets for short. Specifically, we will propose a total order relation (or ranking method) of intuitionistic fuzzy sets based on the equivalent relation between intuitionistic fuzzy sets and interval-valued fuzzy sets and hereby introduce the concepts of solutions of bi-matrix games with payoffs of intuitionistic fuzzy sets and parametric bi-matrix games. It is proven that any bi-matrix game with payoffs of intuitionistic fuzzy sets has at least one satisfying Nash equilibrium solution, which is equivalent to a Nash equilibrium solution of the corresponding parametric bi-matrix game. The latter can be obtained through solving an auxiliary parametric bilinear programming model. Clearly, bi-matrix games with payoffs of intuitionistic fuzzy sets are a general form of the matrix games with payoffs of intuitionistic fuzzy sets as discussed in Chap. 7.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
Owen, G.: Game Theory, 2nd edn. Academic Press, New York (1982)
Nash, J.F.: Equilibrium points in n-person games. Proc. Natl. Acad. Sci. U.S.A. 36(1), 48–49 (1950)
Yager, R.R.: Some aspects of intuitionistic fuzzy sets. Fuzzy Optim. Decis. Making 8(1), 67–90 (2009)
Mangasarian, O.L., Stone, H.: Two-person nonzero-sum games and quadratic programming. J. Math. Anal. Appl. 9(3), 348–355 (1964)
Lmeke, C.E., Howson, J.T.: Equilibrium points of bi-matrix games. SIAM J. Appl. Math. 12(4), 413–423 (1964)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2014 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Li, DF. (2014). Bi-matrix Games with Payoffs of Intuitionistic Fuzzy Sets and Bilinear Programming Method. In: Decision and Game Theory in Management With Intuitionistic Fuzzy Sets. Studies in Fuzziness and Soft Computing, vol 308. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-40712-3_11
Download citation
DOI: https://doi.org/10.1007/978-3-642-40712-3_11
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-40711-6
Online ISBN: 978-3-642-40712-3
eBook Packages: EngineeringEngineering (R0)