A dynamic boundary value problem of the level-set mean curvature flow equation is discussed. We first give a unique existence result of viscosity solutions for more general singular degenerate parabolic equations. The comparison principle is established by employing a so-called flattening argument to avoid a singularity of the equation, while we prove existence of solutions by Perron's method. We also provide a deterministic discrete game interpretation for this problem. The original version of this game was introduced by Kohn and Serfaty (2006) for the case with no boundary, and we propose a modified game including a kind of reflection near the boundary so that the corresponding value functions converge to a viscosity sub-/supersolution satisfying a dynamic boundary condition. This talk is based on a joint work with Y. Giga (The University of Tokyo) and Q. Liu (Fukuoka University).