Playing with Sets of Lexicographic Probabilities and Sets of Desirable Gambles

Independent natural extension for choice functions

“…For example, Cozman and Seidenfeld [5] explore the notion of layer independence for full conditional probability measures. Cozman [4] shows that the only extant concept of independence for (non-convex) sets of probabilities that has a range of desirable graphoid properties is element-wise layer independence. It is an open question what additional structural constraints on coherent sets of desirable gamble sets are necessary and sufficient to secure the relevant graphoid properties.…”