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This paper investigates the impact of reward shaping in multi-agent reinforcement learning as a way to incorporate domain knowledge about good strategies. In theory, potentialbased reward shaping does not alter the Nash Equilibria of a stochastic game, only the exploration of the shaped agent. We demonstrate empirically the performance of reward shaping in two problem domains within the context of RoboCup KeepAway by designing three reward shaping schemes, encouraging specific behaviour such as keeping a minimum distance from other players on the same team and taking on specific roles. The results illustrate that reward shaping with multiple, simultaneous learning agents can reduce the time needed to learn a suitable policy and can alter the final group performance.

Reinforcement Learning (RL) is a powerful and well-studied Machine Learning paradigm, where an agent learns to improve its performance in an environment by maximising a reward signal. In multi-objective Reinforcement Learning (MORL) the reward signal is a vector, where each component represents the performance on a different objective. Reward shaping is a wellestablished family of techniques that have been successfully used to improve the performance and learning speed of RL agents in single-objective problems. The basic premise of reward shaping is to add an additional shaping reward to the reward naturally received from the environment, to incorporate domain knowledge and guide an agent's exploration. Potential-Based Reward Shaping (PBRS) is a specific form of reward shaping that offers additional guarantees. In this paper, we extend the theoretical guarantees of PBRS to MORL problems. Specifically, we provide theoretical proof that PBRS does not alter the true Pareto front in both single-and multi-agent MORL. We also contribute the first published empirical studies of the effect of PBRS in single-and multi-agent MORL problems.

The majority of multi-agent reinforcement learning (MARL) implementations aim to optimize systems with respect to a single objective, despite the fact that many real-world problems are inherently multi-objective in nature. Research into multi-objective MARL is still in its infancy, and few studies to date have dealt with the issue of credit assignment. Reward shaping has been proposed as a means to address the credit assignment problem in single-objective MARL, however it has been shown to alter the intended goals of a domain if misused, leading to unintended behaviour. Two popular shaping methods are potential-based reward shaping and difference rewards, and both have been repeatedly shown to improve learning speed and the quality of joint policies learned by agents in single-objective MARL domains. This work discusses the theoretical implications of applying these shaping approaches to cooperative multi-objective MARL problems, and evaluates their efficacy using two benchmark domains. Our results constitute the first empirical evidence that agents using these shaping methodologies can sample true Pareto optimal solutions in cooperative multi-objective stochastic games.