The　intuitionistic　fuzzy　set　has　been　applied　to　game　theory　very　rarely　since　it　was　introduced　by　Atanassov　in　1983.　The,aim　of　this　paper　is　to　develop　an　effective　methodology　for　solving　matrix　games　with　payoffs　of　Atanassov＇s　triangular　intuitionistic　fuzzy　numbers　(TIFNs).　In　this　methodology;　the　concepts　and　ranking　order　relations　of　Atanassov＇s　TIFNs　are　defined.　A　pair　of　bi-objective　linear　programming　models　for　matrix　games　with　payoffs　of　Alanassov＇s　TIFNs　is　derived　from　two　auxiliary　Atanassov＇s　intuitionistic　fuzzy　programming　models　based　on　the　ranking　order　relations　of　Atanassov＇s　TIFNs　defined　in　this　paper.　An　effective　methodology　based　on　the　weighted　average　method　is　developed　to　determine　optimal　strategies　for　two　players.　The　proposed　method　in　this　paper　is　illustrated　with　a　numerical　example　of　the　market　share　competition　problem.