Summary: The basic aim of this paper is to an application of triangular intuitionistic fuzzy numbers (TIFNs) to a single non-cooperative bi-matrix games. Firstly, the concept of TIFNs and their cut sets are introduced as well as the inequality relations between two TIFNs. Secondly, a bi-matrix game with payoffs of TIFNs is considered and an attempt is made to conceptualize the meaning of a Nash-equilibrium solution for such games. Practical investigations have been discussed for selling a product in a market share for two different companies.