Three new papers: "Polyvalent decision theory", "SEU on orthomodular lattices", and "Global SEU representations"

[marcus...@gmail.com]

Dear all,

I have recently finished three closely related papers may interest some members of this community. One of these has recently been published in Philosophical Transactions of the Royal Society A.   The other two are still preprints, and are available on SSRN.  Here are the abstracts.

Polyvalent decision theory

Abstract: A single agent may encounter many sources of uncertainty and many menus of outcomes, which can be combined together into many different decision problems. There may be analogies between different uncertainty sources (or different outcome menus). Some uncertainty sources (or outcome menus) may exhibit internal symmetries. The agent may also have different levels of awareness. In some situations, the state spaces and outcome spaces have additional mathematical structure (e.g. a topology or differentiable structure), and feasible acts must respect this structure (i.e. they must be continuous or differentiable functions). In other situations, the agent might only be aware of a set of abstract "acts", and be unable to specify explicit state spaces and outcome spaces. We introduce a new approach to decision theory that addresses these issues. It posits multiple uncertainty sources and outcome menus, linked by the aforementioned analogies, symmetries, and awareness changes. It makes no assumption about the internal structure of these sources and menus, so it is applicable in diverse mathematical environments (i.e. categories). In this framework, we define a suitable notion of subjective expected utility (SEU) representations, and provide conditions under which such SEU representations are unique.

Subjective expected utility on orthomodular lattices  (Philosophical Transactions of the Royal Society A 383: 20240534, 2025) [Preprint version]

Abstract: In recent work, I have developed a general category-theoretic framework for decision theory. This paper applies this to the category of orthomodular lattices. Every Boolean algebra is an orthomodular lattice, so this yields a new ("syntactic") model of decision-making with classical uncertainty. The lattice of closed subspaces of a Hilbert space is also an orthomodular lattice, so this also yields a new model of decision-making with quantum uncertainty.


Global subjective expected utility representations

Abstract:  In recent work, I have developed a new model of decision-making under uncertainty, which can simultaneously accommodate multiple sources of uncertainty and multiple outcome menus, related by analogies and/or changes in awareness. This framework can also accommodate state spaces and outcome spaces with additional structure (e.g. a topology), as well as decision problems where states and outcomes are not explicitly specified. The present paper axiomatically characterizes a subjective expected utility representation that is “global” in two senses. First: it posits probabilistic beliefs for all uncertainty sources and utility functions over all outcome menus, which simultaneously rationalize the agent’s preferences across all possible decision problems, and which are consistent with the aforementioned analogies and awareness levels. Second: it applies in many mathematical environments (i.e. categories), making it unnecessary to develop a separate theory for each one. We illustrate this by applying our representation in several categories (sets, measurable spaces, topological spaces, bounded distributive lattices, Riesz spaces, and Banach algebras), and showing that it delivers utility functions suitable to each one.


I hope that you find these papers interesting, and I would be very happy to receive any comments.  

With my best wishes for the holidays,
  Marcus