The　aim　of　this　paper　is　to　develop　a　new　methodology　for　solving　bimatrix　games　with　payoffs　of　triangular　intuitionistic　fuzzy　numbers　(TIFNs),　which　are　called　TIFN　bimatrix　games　for　short.　In　this　methodology,　we　define　the　concepts　of　the　value-index　and　ambiguity-index　and　hereby　develop　a　difference-index　based　ranking　method,　which　is　proven　to　be　a　total　order.　The　parameterized　bilinear　programming　models　are　derived　from　a　pair　of　auxiliary　TIFN　mathematical　programming　models,　which　are　used　to　determine　solutions　of　TIFN　bimatrix　games.　Validity　and　applicability　of　the　models　and　method　proposed　in　this　paper　are　illustrated　with　a　practical　example.