Proportional coalition values for monotonic games on convex geometries with a coalition structure

A new model called games on convex geometries with a coalition structure is introduced, where all players and the coalition structure form a convex geometry, respectively. An explicit form of the proportional coalition solidarity value is given, which endows each union with the solidarity value in quotient games, and then allocates this value to players in a union following the proportions of their solidarity values in original games on convex geometries (i.e., without unions). Meanwhile, an axiomatization of this value is offered. Furthermore, the proportional coalition Shapley value is introduced. Its explicit form is offered, and an axiomatic system is studied.