The persistent complex value

We propose an extension of the classical Shapley value for cooperative games by introducing a new solution concept defined over filtered simplicial complexes. We define a restricted game based on a simplicial complex and a filtration. A filtration is a sequence of simplicial sub-complexes representing the process of simplicial complex formation. The value of a coalition is determined by the values and the persistence of its constituent simplices, where the persistence quantifies their occurrences as maximal simplices in the filtration. A new cooperative solution concept, called persistent complex value, is defined as the Shapley value of this restricted game. Two axiomatic characterizations of this value are provided.