From eXplainable artificial intelligence to game theory with coalitional hodge-shapley value

In cooperative game theory, a set of players or decision makers should negotiate to decide how to allocate the worth gained by the coalition composed by all the players. A value is a solution concept that suggests the negotiation outcome among players. Among the many alternatives, the Shapley value solution concept is very popular: it has the property of being a fair allocation, where the fairness is described by a set of desirable properties, or axioms. The axioms characterize the Shapley value in the sense that it is the unique value satisfying those properties; in addition, axioms allow deriving a simple explicit combinatorial formula to compute the Shapley value. In our approach, coalitions are the main subjects of cooperation, instead of single players. Inspired by the Shapley value, the goal is to derive a fair allocation to coalitions. The methodology uses the Hodge decomposition of the simplicial complex associated to the poset of the subsets of the set of players ordered by inclusion, like the one proposed by [1]. We will motivate this investigation in terms of eXplainable Artificial Intelligence (XAI), and link our work with the SHAP algorithm [2].