The　purpose　of　this　paper　is　to　develop　the　concept　and　mathematical-programming　methodology　of　matrix　games　with　payoffs　represented　by　Atanassov＇s　interval-valued　intuitionistic　fuzzy　(IVIF)　sets.　In　this　methodology,　the　concept　of　solutions　of　matrix　games　with　payoffs　represented　by　Atanassov＇s　IVIF　sets　is　defined,　and　some　important　properties　are　studied　using　multiobjective-programming　and　duality-programming　theory.　It　is　proven　that　each　matrix　game　with　payoffs　represented　by　Atanassov＇s　IVIF　sets　has　a　solution,　which　can　be　obtained　through　solving　a　pair　of　auxiliary　linear/nonlinear-programming　models　derived　from　a　pair　of　nonlinear　biobjective　interval-programming　models.　Validity　and　applicability　of　the　proposed　methodology　are　illustrated　with　a　numerical　example.