Matrix games under a Pythagorean fuzzy environment with self-confidence levels: formulation and solution approach

Over the past few years, there has been an increasing demand for enhanced and efficient tools capable of managing ambiguous and uncertain data. An example of such a tool is the Pythagorean fuzzy set, which was initially presented by Yager (in: Proceedings of joint IFSA world congress and NAFIPS annual meeting, June 24–28, Edmonton, Canada, pp 57–61, 2013). On the other hand, game theory has proved to be a useful framework for analyzing competitive situations involving individuals or organizations across multiple fields. Nevertheless, the conventional matrix game models face limitations in addressing issues under Pythagorean fuzzy circumstances. Furthermore, prior research on matrix games has overlooked the importance of considering the self-confidence levels of the involved experts. To overcome these limitations, this contribution presents a new approach for solving two-player zero-sum matrix games with payoffs represented by Pythagorean fuzzy numbers that include self-confidence levels. First, we introduce a novel aggregation operator called the generalized sine trigonometric Pythagorean fuzzy confidence-weighted average (GST-PFCWA) operator. This operator combines PFNs with self-confidence levels, and its mathematical properties and special cases are explored in detail. Next, we develop basic concepts and mathematical models for matrix games with payoffs represented by PFNs with self-confidence levels. In this context, we derive a pair of Pythagorean fuzzy auxiliary linear/nonlinear-programming optimization models that can be used to solve this class of game problems. Finally, the paper presents a numerical example illustrating the proposed solution approach. In summary, this work presents a novel framework that integrates Pythagorean fuzzy sets and game theory to provide a more comprehensive approach for dealing with competitive situations under uncertain and vague information environments.