Transition-state replicator dynamics

•Additional reward parameter is added to the learning algorithm.•The replicator equation is extended from single-state to transition-state.•The algorithm and its dynamic are demonstrated on a 2 states battle of sexes game. Agent-based evolutionary game theory studies the dynamics of the autonomous agents. It is important for application that relies on the agents to perform the automated tasks. Since the agents make their own decision, therefore the stability of the interaction needs to be comprehended. The current state of the art in agent-based replicator dynamics are piecewise and state-coupled replicator dynamics which focus on joint-action single-state reward. This paper introduces additional reward parameter to the learning algorithm, extends the replicator dynamics to joint-action transition-state reward and shows that it can be changed to single-state reward and independent-action reward. The replicator equation is expressed based on the tree diagram approach and is verified with the numerical simulation in a two states battle of sexes coordination game for various types of rewards. The numerical results are consistent with the phase portraits generated by the replicator equation and are able to provide some general insights to the coordination game such as the number of convergence points, the rate of convergence and the effect of initial points on the convergence.