Imprecise constrained matrix games (such as fuzzy constrained matrix games, interval-valued constrained matrix games, and rough constrained matrix games) have attracted considerable research interest. This article is concerned with developing an effective fuzzy multi-objective programming algorithm to solve constraint matrix games with payoffs of fuzzy rough numbers (FRNs). For simplicity, we refer to this problem as fuzzy rough constrained matrix games. To the best of our knowledge, there are no previous studies that solve the fuzzy rough constrained matrix games. In the proposed algorithm, it is proven that a constrained matrix game with fuzzy rough payoffs has a fuzzy rough-type game value. Moreover, this article constructs four multi-objective linear programming problems. These problems are used to obtain the lower and upper bounds of the fuzzy rough game value and the corresponding optimal strategies of each player in any fuzzy rough constrained matrix games. Finally, a real example of the market share game problem demonstrates the effectiveness and reasonableness of the proposed algorithm. Additionally, the results of the numerical example are compared with the GAMS software results. The significant contribution of this article is that it deals with constraint matrix games using two types of uncertainties, and, thus, the process of decision-making is more flexible.

The intuitionistic fuzzy set (IFS) is applied in various decision-making problems to express vagueness and showed great success in realizing the day-to-day problems. The principal aim of this article is to develop an approach for solving multi-criteria matrix game with intuitionistic fuzzy (I-fuzzy) goals. The proposed approach introduces the indeterminacy resolving functions of I-fuzzy numbers and discusses the I-fuzzy inequalities concept. Then, an effective algorithm based on the indeterminacy resolving algorithm is developed to obtain Pareto optimal security strategies for both players through solving a pair of multi-objective linear programming problems constructed from two auxiliary I-fuzzy programming problems. It is shown that this multi-criteria matrix game with I-fuzzy goals is an extension of the multi-criteria matrix game with fuzzy goals. Moreover, two numerical simulations are conducted to demonstrate the applicability and implementation process of the proposed algorithm. Finally, the achieved numerical results are compared with the existing algorithms to show the advantages of our algorithm.

Uncertainty plays a crucial role in decision-making problems, particularly in game theory. Various forms of uncertainty have been explored in the literature, including fuzzy, soft, rough, and interval-based approaches. Game theory has been extensively studied under these uncertainty models, with researchers addressing vagueness, and imprecision from multiple perspectives. More recently, neutrosophic sets have emerged as an alternative framework for handling uncertainty. Neutrosophic numbers effectively incorporate indeterminacy in decision-making by considering factors such as intuition, assumptions, judgment, behavior, evaluation, and preferences of decision-makers. This paper presents a novel approach to solving a new class of n-player continuous differential games within a neutrosophic framework. In the proposed methodology, the neutrosophic n-player continuous games are redefined into two separate crisp problems: the lower problem and the upper problem. The study further establishes the necessary and sufficient conditions for determining equilibrium strategies in neutrosophic continuous differential games. To demonstrate its effectiveness and practical applicability, the proposed method is validated through a numerical example.