Journal of Industrial and Management Optimization (JIMO)

Linear programming technique for solving interval-valued constraint matrix games

Pages: 1059 - 1070, Volume 10, Issue 4, October 2014      doi:10.3934/jimo.2014.10.1059

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Jiang-Xia Nan - School of Mathematics and Computing Sciences, Guilin University of Electronic Technology, Guilin 541004, China (email)
Deng-Feng Li - School of Management, Fuzhou University, Fujian 350108, China (email)

Abstract: The purpose of this paper is to propose an effective linear programming technique for solving matrix games in which the payoffs are expressed with intervals and the choice of strategies for players is constrained, i.e., interval-valued constraint matrix games. Because the payoffs of the interval-valued constraint matrix game are intervals, its value is an interval as well. In this methodology, the value of the interval-valued constraint matrix game is regarded as a function of values in the payoff intervals, which is proven to be monotonous and non-decreasing. By the duality theorem of linear programming, it is proven that both players always have the identical interval-type value and hereby the interval-valued constraint matrix game has an interval-type value. A pair of auxiliary linear programming models is derived to compute the upper bound and the lower bound of the value of the interval-valued constraint matrix game by using the upper bounds and the lower bounds of the payoff intervals, respectively. Validity and applicability of the linear programming technique proposed in this paper is demonstrated with a numerical example of the market share game problem.

Keywords:  Game theory, constraint matrix game, linear programming, interval numbers, economics.
Mathematics Subject Classification:  Primary: 91A05, 91A40; Secondary: 90C05.

Received: January 2012;      Revised: September 2013;      Available Online: February 2014.