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Subjective expected utility with a spectral state space

  • Marcus PivatoEmail author
Research Article
  • 6 Downloads

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.

Keywords

Subjective expected utility Awareness Subjective state space Riesz space Banach lattice Commutative Banach algebra 

JEL Classification

D81 

Notes

Supplementary material

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.THEMAUniversité de Cergy-PontoiseCergy-Pontoise CedexFrance

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