Probabilistic values on convex geometries

A game on a convex geometry is a real-valued function defined on the family ℒ of the closed sets of a closure operator which satisfies the finite Minkowski–Krein–Milman property. If ℒ is the Boolean algebra 2^N, then we obtain an n-person cooperative game. We will extend the work of Weber on probabilistic values to games on convex geometries. As a result, we obtain a family of axioms that give rise to several probabilistic values and a unique Shapley value for games on convex geometries.