  Value factorisation is a useful technique for multi-agent reinforcement
learning (MARL) in global reward game, however its underlying mechanism is not
yet fully understood. This paper studies a theoretical framework for value
factorisation with interpretability via Shapley value theory. We generalise
Shapley value to Markov convex game called Markov Shapley value (MSV) and apply
it as a value factorisation method in global reward game, which is obtained by
the equivalence between the two games. Based on the properties of MSV, we
derive Shapley-Bellman optimality equation (SBOE) to evaluate the optimal MSV,
which corresponds to an optimal joint deterministic policy. Furthermore, we
propose Shapley-Bellman operator (SBO) that is proved to solve SBOE. With a
stochastic approximation and some transformations, a new MARL algorithm called
Shapley Q-learning (SHAQ) is established, the implementation of which is guided
by the theoretical results of SBO and MSV. We also discuss the relationship
between SHAQ and relevant value factorisation methods. In the experiments, SHAQ
exhibits not only superior performances on all tasks but also the
interpretability that agrees with the theoretical analysis. The implementation
of this paper is on https://github.com/hsvgbkhgbv/shapley-q-learning.
