  Versatile Video Coding (VVC) has set a new milestone in high-efficiency video
coding. In the standard encoder, the $\lambda$-domain rate control is
incorporated for its high accuracy and good Rate-Distortion (RD) performance.
In this paper, we formulate this task as a Nash equilibrium problem that
effectively bargains between multiple agents, {\it i.e.}, Coding Tree Units
(CTUs) in the frame. After that, we calculate the optimal $\lambda$ value with
a two-step strategy: a Newton method to iteratively obtain an intermediate
variable, and a solution of Nash equilibrium to obtain the optimal $\lambda$.
Finally, we propose an effective CTU-level rate allocation with the optimal
$\lambda$ value. To the best of our knowledge, we are the first to combine game
theory with $\lambda$-domain rate control. Experimental results with Common
Test Conditions (CTC) demonstrate the efficiency of the proposed method, which
outperforms the state-of-the-art CTU-level rate allocation algorithms.
