Summary: This article addresses the problem of non-zero sum games for Markov jump multiplayer systems (MJMSs) using the reinforcement \(Q\)-learning method. Firstly, the \(Q\)-functions for each player are derived from the system states and the control inputs. On this basis, by incorporating the integral reinforcement learning scheme and the actor-critic neural networks structure, we design a novel reinforcement learning approach for MJMSs. It should be noted that the designed algorithm does not require any information about the system dynamics and transition probabilities. Furthermore, the stochastic stability and Nash equilibrium of MJMSs can be ensured by the designed algorithm. Finally, a simulation example is presented to illustrate the effectiveness of the designed approach.