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An axiomatic formalization of bounded rationality based on a utility-information equivalence
Classic decision-theory is based on the maximum expected utility (MEU)
principle, but crucially ignores the resource costs incurred when determining
optimal decisions. Here we propose an axiomatic framework for bounded
decision-making that considers resource costs. Agents are formalized as
probability measures over input-output streams. We postulate that any such
probability measure can be assigned a corresponding conjugate utility function
based on three axioms: utilities should be real-valued, additive and monotonic
mappings of probabilities. We show that these axioms enforce a unique
conversion law between utility and probability (and thereby, information).
Moreover, we show that this relation can be characterized as a variational
principle: given a utility function, its conjugate probability measure
maximizes a free utility functional. Transformations of probability measures
can then be formalized as a change in free utility due to the addition of new
constraints expressed by a target utility function. Accordingly, one obtains a
criterion to choose a probability measure that trades off the maximization of a
target utility function and the cost of the deviation from a reference
distribution. We show that optimal control, adaptive estimation and adaptive
control problems can be solved this way in a resource-efficient way. When
resource costs are ignored, the MEU principle is recovered. Our formalization
might thus provide a principled approach to bounded rationality that
establishes a close link to information theory.Comment: 22 pages, 4 figures, 1 tabl