2020

A New Game Value Approach for Infinite Interval Matrix Games

The purpose of this paper is to determine when and under which conditions the solution and game value of the infinite interval matrix games will exist.Firstly, the concept of a reasonable solution defined in interval matrix games was extended to infinite interval matrix games. Then, the solution and game value were characterized by using sequences of interval numbers (defined by Chiao, 2002) and their concept of convergence of interval numbers. Considering that each row or column of the payoff matrix is a sequence of interval numbers, we assume that each row converges to the same interval number α ̃=[α_l,α_r] and each column to the same interval number β ̃=[β_l,β_r]. In a conclusion, the existence of the solution of G ̃ is shown.

PDF

___

1] E. N. Barron, Game Theory an Introduction, John Wiley & Sons Inc., New Jersey, 1-108, 2008.
[2] A. Cegielski, “Approximation of some zero-sum noncontinuous games by a matrix game”, Comment. Math., 2261-267, 1991.
[3] K.P. Chiao, “Fundamental Properties of Interval Vector Max-Norm”, Tamsui Oxf J Math Sci, 18(2):219-233, 2002.
[4] D.W. Collins and C. Hu, “Studying interval valued matrix games with fuzzy logic” Soft Comput, 12(2):147-155, 2008.
[5] H. Ishibuchi and H. Tanaka, “Multi- objective Programming in Optimization of the Interval Objective Function”, European Journal of Operational Research, 48: 219- 225, 1990.
[6] D.F. Li, J.X. Nan and M.J. Zhang, “Interval programming models for matrix games with interval payoffs”, Optimization Methods and Software, 27(1), 1-16, 2012.
[7] E. Marchi, “On the minimax theorem of the theory of games”, Ann. Mat. Pura Appl. 77, 207-282, 1967.
[8] R.E. Moore, Method and Application of Interval Analysis, SIAM, Philadelphia, 1979.
[9] L.M. Naya, “Zero-sum continuous games with no compact support”, International J. Game Theory, 25(1), 93-111, 1996.
[10] L.M. Naya, “Weak topology and infinite matrix games”, Int J Game Theory, 27, pp:219-229, 1998.
[11] L.M. Naya, “On the Value of Some Infinite Matrix Games”, Mathematics of Operation Research, 26(1): 82-88, 2001.
[12] P.K. Nayak and T.K. Pal, “Linear Programming Technique To Solve Two- Person Matrix Games With Interval Payoffs”, Asia Pacific J of Operation Research, 26(2), 285-305, 2009.
[13] V.J Neuman and O. Morgenstern, “Theory of Games and Economic Behavior”, New York, Science Editions, John Wiley and Sons, Inc. third edition, 85-165, 1944.
[14] G., Owen, Game Theory, Third Edition Academic Press, 1995.
[15] A. Sengupta and T.K. Pal, “A-index for ordering interval numbers”, Presented in Indian Science Congress. Delhi University, January 3-8, 1997.
[16] A. Sengupta and T.K. Pal, “On comparing interval numbers”, European Journal of Operational Research, 27: 28-43, 2000.
[17] A. Sengupta, T.K. Pal and D. Chakraborty, “Interpretation of Inequality Constraints Involving Interval Coefficients and a Solution to Interval Linear Programming”, Fuzzy Sets and Systems, 119: 129-138, 2001.
[18] A. Sengupta and T.K. Pal, “Fuzzy Preference Ordering of Interval Numbers in Decision Problems”, Studies in Fuzziness and Soft Computing 238, Springer, Berlin, 2009.
[19] S.H Tijs, “Semi Infinite and infinite matrix and bimatrix games”, Ph.D. thesis, University of Nijmegan, 1975.