With　a　significant　Rate-Distortion　(RD)　improvement　than　H.265/HEVC,　Versatile　Video　Coding　(VVC)　has　set　a　new　milestone　in　lossy　video　compression.　It　also　incorporates　the　emerging　?-domain　rate　control　technique,　aiming　at　a　higher　visual　quality　under　a　fixed　bit　constraint.　However,　the　challenge　remains　how　to　efficiently　allocate　bits　to　all　frames　and　Coding　Tree　Units　(CTUs).　In　this　paper,　we　propose　an　effective　solution　by　formulating　the　above　task　as　a　Nash　equilibrium　problem,　where　all　CTUs　are　treated　as　players　that　bargains　with　each　other.　By　introducing　?-domain　RD　models,　a　constrained　optimization　is　derived　with　no　closed-form　solution.　We　then　propose　a　two-step　strategy　to　address　this　issue:　a　Newton　method　to　iteratively　calculate　an　intermediate　variable,　and　a　final　solution　of　Nash　equilibrium　to　obtain　an　approximately　optimal　?.　Finally,　we　utilize　the　derived　1　to　perform　an　effective　CTU-level　bit　allocation,　which　is　the　very　first　attempt　to　introduce　Nash　equilibrium　in　?-domain　rate　control.　Experimental　results　with　Common　Test　Conditions　(CTC)　demonstrate　the　effectiveness　and　superiority　of　our　method,　which　outperforms　the　state-of-the-art　CTU-level　rate　allocation　algorithms　for　VVC.