Subjective Expected Utility With a Spectral State Space

Abstract

An agent faces a decision under uncertainty with the following structure. There is a set A of “acts”; each will yield an unknown real-valued payoff. Linear combinations of acts are feasible; thus, A is a vector space. But there is no pre-specified set of states of nature. Instead, there is a Boolean algebra J describing information the agent could acquire. For each element of J, she has a conditional preference order on A. I show that if these conditional preferences satisfy certain axioms, then there is a unique compact Hausdorff space S such that elements of A correspond to continuous real-valued functions on S, elements of J correspond to regular closed subsets of S, and the conditional preferences have a subjective expected utility (SEU) representation given by a Borel probability measure on S and a continuous utility function. I consider two settings; in one, A has a partial order making it a Riesz space or Banach lattice, and J is the Boolean algebra of bands in A. In the other, A has a multiplication operator making it a commutative Banach algebra, and J is the Boolean algebra of regular ideals in A. Finally, given two such vector spaces A1 and A2 with SEU representations on topological spaces S1 and S2, I show that a preference-preserving homomorphism A2 --> A1 corresponds to a probability-preserving continuous function S1 --> S2. I interpret this as a model of changing awareness.

Keywords: Subjective Expected Utility, Awareness, Subjective State Space, Riesz Space, Banach Lattice, Kakutani, Banach Algebra, Gelfand, Continuous Utility

JEL Classification: D81