The　intuitionistic　fuzzy　set　(IF-set)　has　not　been　applied　to　matrix　game　problems　yet　since　it　was　introduced　by　K.　T.　Atanassov.　The　aim　of　this　paper　is　to　develop　a　methodology　for　solving　matrix　games　with　payoffs　of　triangular　intuitionistic　fuzzy　numbers　(TIFNs).　Firstly　the　concept　of　TIFNs　and　their　arithmetic　operations　and　cut　sets　are　introduced　as　well　as　the　ranking　order　relations.　Secondly　the　concept　of　solutions　for　matrix　games　with　payoffs　of　TIFNs　is　defined.　A　lexicographic　methodology　is　developed　to　determine　the　solutions　of　matrix　games　with　payoffs　of　TIFNs　for　both　Players　through　solving　a　pair　of　bi-objective　linear　programming　models　derived　from　two　new　auxiliary　intuitionistic　fuzzy　programming　models.　The　proposed　method　is　illustrated　with　a　numerical　example.