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

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.

💡 Research Summary

The paper presents a unified information‑geometric framework that brings together two major strands of bounded‑rationality modelling: entropy‑regularised (ER) choice models (including rational inattention) and quantum‑like (QL) models that employ Hilbert‑space amplitudes to capture context and order effects. The authors start from the Fisher–Rao metric on the interior of the probability simplex Δⁿ⁻¹ and introduce a “least‑action rationality” (LAR) principle formulated in amplitude (square‑root) coordinates. By lifting the dynamics to the cotangent bundle N = T*ℝⁿ of unnormalised amplitudes, they obtain a phase‑space equipped with the canonical symplectic form and a neutral (para‑Kähler) structure consisting of a metric, a symplectic form, and a product.

A linear evaluator V̂ is decomposed into a symmetric part Ŝ (the evaluative channel) and a skew‑symmetric part F̂ (the co‑utility or circulatory channel). The symmetric component generates an exact one‑form on the simplex (the gradient of a utility function), while the skew component yields a divergence‑free co‑exact one‑form whose curvature measures the path‑dependence of preferences. The LAR variational principle yields a quadratic Lagrangian L(ρ, ẋ)=½‖ẋ‖²−⟨ρ, V̂ ρ⟩, whose Euler–Lagrange equations lift to a Hamiltonian system on N. On the distinguished zero‑residual Lagrangian leaf (where ⟨ρ, V̂ ρ⟩=0) the dynamics admit a split‑complex (hyperbolic) Schrödinger‑type representation; observable probabilities are obtained via a hyperbolic Born rule q_i = ψ_i².

Projecting the latent flow back onto the simplex produces a preference one‑form α = dU + β, where dU is the exact utility component and β is the co‑utility component with curvature Ω = dβ. Non‑zero Ω generates holonomy: the same start‑end lottery can be reached by different paths, leading to context‑ and order‑dependent choice probabilities, violations of the law of total probability, and interference‑like terms—precisely the empirical signatures of QL models. Thus the apparent “quantum‑like” phenomena emerge as shadows of an underlying rational flow on the Fisher–Rao manifold.

Finally, the authors show that standard complex (elliptic) quantum dynamics arise from the same real symplectic phase space when an additional Kähler polarisation is imposed. This restriction selects a holomorphic Lagrangian subbundle, projecting the full Hamiltonian evolution onto a complex Hilbert space where it becomes unitary. Consequently, unitary quantum evolution is not a primitive postulate but a coherent restriction of the broader least‑action framework.

The paper concludes that ER and QL models are not competing paradigms but different projections of a single geometric structure. The para‑Kähler lift provides a principled foundation for real‑valued “hyperbolic quantum” formulations, while the polarisation step recovers the familiar complex quantum formalism. This unification offers new analytical tools for behavioural economics, cognitive psychology, and quantum cognition, and opens avenues for exploring non‑linear evaluators, boundary effects, and empirical estimation of the curvature Ω as a measure of contextual irrationality.