Theoretical considerations of potential-based reward shaping for multi-agent systems

Potential-based reward shaping has previously been proven to both be equivalent to Q-table initialisation and guarantee policy invariance in single-agent reinforcement learning. The method has since been used in multi-agent reinforcement learning without consideration of whether the theoretical equivalence and guarantees hold. This paper extends the existing proofs to similar results in multi-agent systems, providing the theoretical background to explain the success of previous empirical studies. Specifically, it is proven that the equivalence to Q-table initialisation remains and the Nash Equilibria of the underlying stochastic game are not modified. Furthermore, we demonstrate empirically that potential-based reward shaping affects exploration and, consequentially, can alter the joint policy converged upon.