Matrix games with proportional linguistic payoffs

Abstract

This paper aims to present a novel approach for computing two-player constant-sum matrix games laid on the notion of a “symbolic proportional” linguistic term set. It is not always possible to lay assessments based on uniformly and symmetrically distributed linguistic term sets; hence, the defined concept motivates the decision-makers to represent their opinions using 2-tuples composed of two proportional linguistic terms. The proportional 2-tuple linguistic representation of payoffs concerns linguistic labels, which do not certainly have to be symmetrically distributed or do not have the conventional prerequisite of having uniform distance among them. This representation confers an opportunity to describe the linguistic payoffs of a matrix game using members of a continuous linguistic scale domain. In our work, we have defined a two-player constant-sum proportional linguistic matrix game and proposed an approach of Proportional Linguistic Linear Programming (PLLP) to evaluate these game problems. The framed PLLP problem is then transformed into a crisp LPP that can be easily solved, decreasing the computation complexities involved in solving the linguistic decision-making problems. This perspective of proportional 2-tuples provides decision-makers an approach to represent their opinions not by just using one label, rather by proportional linguistic labels of the form $(\delta u_i,\,\gamma u_{i+1})$, where $u_i$ and $u_{i+1}$ are two successive linguistic terms, with  $0\le \delta ,\,\gamma \le 1$ and $\delta \,+\,\gamma \,=\,1$. Besides, some test examples are also presented to show the consistency of our designed approach. Further, the PLLP formulation is utilized to solve a Multi-Criteria Decision-Making problem based on actual-time linguistic data.