This paper presents a generalization of the Kostant game, a combinatorial framework originally for generating positive roots in Lie algebras. By introducing an arbitrary multi-vertex modification, we prove that the resulting game configurations naturally biject with the minimal length representatives of parabolic quotients W/W_J. This yields a dynamical and algorithmic perspective on reduced words. Finally, we apply this framework to derive a novel root counting identity, formalize the Coxeter-theoretic foundation for combinatorial approaches to the Mukai conjecture, establish the regularity of reduced word languages via finite state automata, and dynamically construct Standard Young Tableaux.