The rock-paper-scissors game, commonly played in East Asia, gives a simple model to understand physical, biological, psychological and other problems. The interacting rock-paper-scissors particle system is a point of contact between the kinetic theory of gases by Maxwell and Boltzmann ( collision model) and the coagulation theory by Smoluchowski ( coalescence model). A $2s+1$ types extended rock-paper-scissors collision model naturally introduces a nonlinear integrable system. The time evolution of the $2s+1$ types extended rock-paper-scissors coalescence model is obtained from the logarithmic time change of the nonlinear integrable system. We also discuss the behavior of a discrete rock-paper-scissors coalescence model.