Information Geometry of Bounded Rationality: Entropy--Regularised Choice with Hyperbolic and Elliptic Quantum Geometries

2026-03-02

AI Summary

This work unifies quantum-inspired models and entropy-regularized models of bounded rationality by revealing their shared Fisher–Rao information geometry on the probability simplex. It introduces a variational principle of “least-action rationality” to formulate decision dynamics in the cotangent phase space of unnormalized amplitudes, integrating utility evaluation with cyclic co-utility channels. The framework innovatively incorporates both hyperbolic (split-complex) and elliptic (standard complex) quantum geometries within a common information-geometric setting, combining symplectic geometry, para-Kähler structures, and Born-like normalization to derive an evolution equation for observable probabilities. This approach naturally accounts for context effects, order effects, and violations of the law of total probability, while showing that standard quantum unitary evolution arises as a special case under coherence constraints.

Technology Category

• Reasoning under Uncertainty: Decision/Utility Theory
• Game Theory and Economic Paradigms: Imperfect Information
• Machine Learning: Probabilistic Circuits and Graphical Models

Application Category

• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Web Mining and Content Analysis: Models for Web evolution
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

Models of bounded rationality include quantum--like (QL) models, which use Hilbert--space amplitudes to represent context and order effects, and entropy--regularised (ER) models, including rational inattention, which smooth expected utility by adding an information cost. We develop a unified information--geometric framework in which both arise from the same structure on the probability simplex. Starting from the Fisher--Rao geometry of the open simplex $Δ^{n-1}$, we formulate \emph{least--action rationality} (LAR) as a variational principle for decision dynamics in amplitude (square--root) coordinates and lift it to the cotangent phase space $N:=T^*\mathbb R^n$ of unnormalised amplitudes. The lift carries its canonical symplectic form and a para--Kähler geometry. For a linear evaluator $\widehat V=\widehat S+\widehat F$ with symmetric part $\widehat S$ and skew part $\widehat F$, the dynamics separate an evaluative channel from a circulatory (co--utility) channel. On a distinguished zero--residual Lagrangian leaf the flow closes as a split--complex (hyperbolic) Schrödinger--type evolution, and observable probabilities follow from a quadratic (Born--type) normalisation. When reduced to the simplex, the induced preference one--form decomposes into an exact utility component and a divergence--free co--utility component whose curvature measures path dependence. Context effects, order effects, and interference--like deviations from the law of total probability emerge as projections of this latent rational flow. Finally, standard complex (elliptic) quantum dynamics arises within this real symplectic phase space by imposing an additional Kähler polarisation that restricts admissible variations. Unitary evolution is thus a coherent restriction of the underlying least--action framework rather than a primitive postulate.

Problem

• bounded rationality
• quantum-like models
• entropy-regularized choice
• information geometry
• context effects

Innovation

• information geometry
• bounded rationality
• least-action rationality
• para-Kähler geometry
• quantum-like decision models