Procedural Mixture Sets

TL;DR

This paper introduces procedural mixture sets, a novel decision-theoretic framework where mixing has intrinsic value, and shows that standard axioms lead to entropy as a measure of procedural value, impacting models of decision processes.

Contribution

It defines procedural mixture sets and proves that von Neumann-Morgenstern axioms imply entropy as a measure of procedural value, extending decision theory.

Findings

Procedural mixture sets generalize standard mixture sets.

Von Neumann-Morgenstern axioms imply entropy as a representation.

Implications for decision processes and choice modeling.

Abstract

The paper characterizes the Shannon (1948) and Tsallis (1988) entropies in a standard framework of decision theory, mixture sets. Procedural mixture sets are introduced as a variant of mixture sets in which it is not necessarily true that a mixture of two identical elements yields the same element. This allows the process of mixing itself to have an intrinsic value. The paper proves the surprising result that simply imposing the standard axioms of von Neumann-Morgenstern on preferences on a procedural mixture set yields the entropy as a representation of procedural value. An application of the theorem to decision processes and the relation between choice probabilities and decision times elucidates the difficulty of extending the drift-diffusion model to multi-alternative choice.