SHAQ: Incorporating Shapley Value Theory into Multi-Agent Q-Learning.

Value factorisation proves to be a useful technique in multi-agent reinforcement learning (MARL), but the underlying mechanism is not yet fully understood. This paper explores a theoretical framework for value factorisation with interpretability. We generalise Shapley value in coalitional game theory to Markov convex game (MCG) and use it as a value factorisation method for MARL. We show that the generalised Shapley value possesses several features such as (1) efficiency: the sum of optimal generalised Shapley values is equal to the optimal global value, (2) fairness in factorisation of the global value, and (3) sensitiveness to dummy agents. Moreover, we show that MCG with the grand coalition and the generalised Shapley value is within $\epsilon$-core, which means no agents would deviate from the grand coalition. Since MCG with the grand coalition is equivalent to global reward game, it is the first time that Shapley value is rigorously proved to be rationally applied as a value factorisation method for global reward game. Moreover, extending from the Bellman operator we propose Shapley-Q operator that is proved to converge to the optimal generalised Shapley value. With stochastic approximation, a new MARL algorithm called Shapley Q-learning (SHAQ) is yielded. We show the performance of SHAQ on Predator-Prey for modelling relative overgeneralisation and StarCraft Multi-Agent Challenge (SMAC). In experiments, we also demonstrate the interpretability of SHAQ that is lacking in the state-of-the-art baselines.