Subjective probability over a subjective decision tree

Since Savage’s seminal work, a state space has been assumed as a primitive and as a complete description of the world. However, the decision maker may have subjective states in her mind, which are distinct from the primitive states, but relevant for her decisions. Dekel, Lipman and Rustichini (2001) derive a unique subjective state space from preference. In a dynamic setting, a state space S and a filtration {Ft}t=0 over S has been taken as primitives. We derive a triplet (S, {Ft}t=0, μ) from preference, where S is a subjective state space, {Ft}t=0 is a subjective filtration over S, and μ is a subjective probability over S. We also show the uniqueness of the representation.