Subjective Expected Utility With a Spectral State Space

Abstract

An agent faces a decision under uncertainty with the following structure. There is a set $${\mathcal {A}}$$ of “acts”; each will yield an unknown real-valued payoff. Linear combinations of acts are feasible; thus, $${\mathcal {A}}$$ is a vector space. But there is no pre-specified set of states of nature. Instead, there is a Boolean algebra $${\mathfrak {I}}$$ describing information the agent could acquire. For each element of $${\mathfrak {I}}$$, she has a conditional preference order on $${\mathcal {A}}$$. I show that if these conditional preferences satisfy certain axioms, then there is a unique compact Hausdorff space $${\mathcal {S}}$$ such that elements of $${\mathcal {A}}$$ correspond to continuous real-valued functions on $${\mathcal {S}}$$, elements of $${\mathfrak {I}}$$ correspond to regular closed subsets of $${\mathcal {S}}$$, and the conditional preferences have a subjective expected utility (SEU) representation given by a Borel probability measure on $${\mathcal {S}}$$ and a continuous utility function. I consider two settings; in one, $${\mathcal {A}}$$ has a partial order making it a Riesz space or Banach lattice, and $${\mathfrak {I}}$$ is the Boolean algebra of bands in $${\mathcal {A}}$$. In the other, $${\mathcal {A}}$$ has a multiplication operator making it a commutative Banach algebra, and $${\mathfrak {I}}$$ is the Boolean algebra of regular ideals in $${\mathcal {A}}$$. Finally, given two such vector spaces $${\mathcal {A}}_1$$ and $${\mathcal {A}}_2$$ with SEU representations on topological spaces $${\mathcal {S}}_1$$ and $${\mathcal {S}}_2$$, I show that a preference-preserving homomorphism $${\mathcal {A}}_2{{\longrightarrow }}{\mathcal {A}}_1$$ corresponds to a probability-preserving continuous function $${\mathcal {S}}_1{{\longrightarrow }}{\mathcal {S}}_2$$. I interpret this as a model of changing awareness.