In real-life decision-making challenges, experts quite frequently have a preference for expressing their perspective in natural linguistic terms rather than definite numerical format. These linguistic representation has been utilized to resolve plenty of decision-making problems. This paper displays the thorough study of matrix games where in the payoffs are characterized through linguistic interval-valued intuitionistic fuzzy sets (LIVIFSs). Solution of these matrix games are attained by resolving a duo of linear or nonlinear programming problems, originated through non-linear bi-objective programming problems. Finally, a numerical example is used to demonstrate the applicability of the suggested approach.