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Autor principal:	Fathi, Kevin
Formato:	Recurso digital
Lenguaje:	inglés
Publicado:	Zenodo 2026
Materias:	scalar utility rank obstruction constant-rank theorem Hermann–Krener observability misspecified learning cumulative prospect theory salience theory rational inattention multidimensional screening Harsanyi aggregation CRRA Euler equations reference-dependent preferences loss aversion lexicographic preferences participation ratio falsification protocol behavioral econometrics
Acceso en línea:	https://doi.org/10.5281/zenodo.19794428
Etiquetas:	Agregar Etiqueta Sin Etiquetas, Sea el primero en etiquetar este registro!

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This paper develops a rank-theoretic obstruction to scalar utility in decision theory, mechanism design, and games. The central observation is logical: the absence of an Allais, Ellsberg, preference-reversal, or framing pattern is not evidence for scalar utility, only evidence that a finite paradox detector failed to fire. Scalar utility is a stronger local compression claim — that the entire observable map from hidden behavioral states to choice, pricing, matching, dynamic, welfare, and game behavior factors through a single real number. For a C¹ audit map G, that claim implies rank DG ≤ 1. The implication is one-way: a paradox can serve as a sufficient witness in regular smooth designs, while paradox absence carries no rank information at all. The paper proves the rank obstruction and its converses on regular sets. If G factors as H ∘ K through a scalar K, then DG is an outer product and has rank at most one; if DG has constant rank one on a neighborhood, the constant-rank theorem yields a local scalar factorization. The same machinery extends to k-parameter behavioral repairs (rank ≤ k), to Banach-valued hidden states (continuum reference points, density-valued attributes), to dynamic systems via the Hermann–Krener observability codistribution, and to stochastic populations through the derivative-information operator I(μ) = E[DG⊤Σ⁻¹DG]. A quantitative obstruction index κ_G = σ₂(DG) gives the least first-order scalar misspecification error, with a second-order local approximation bound via Eckart–Young. Several extensions give the framework dynamic and operational content. A rank-flow theorem differentiates the second eigenvalue of I(μ_t) along a Fokker–Planck flow and identifies the exact generator term controlling its movement; a qualified contraction result follows under a checkable matrix dissipation condition (no unconditional contraction is asserted). A capacity-constrained tournament sharpens the comparison with two-parameter cumulative prospect theory, salience theory, and rational inattention, with a binary-separated rank bound for capacity-normalized rational-inattention specifications. A rank-completion theory formalizes the fact that observable dimension is a joint property of the agent and the audit bundle, not an intrinsic number attached to preferences. A local-polynomial estimation section proves uniform consistency and gives finite-sample sufficient conditions for discriminating rank-one from rank-two regions. A stratified extension treats piecewise-C¹ scalar rules and shows that kinks can protect a scalar representation only on lower-dimensional boundary sets; in particular, lexicographic order admits no piecewise-C¹ scalar representation on any open set. The worked examples are model-derived rather than empirical claims: a two-parameter lottery family with no tested Allais or Ellsberg pattern but rank-two certainty-equivalent derivatives at the expected-utility point; a Köbberling–Wakker reference-dependent WTP–WTA rank computation; a CRRA Euler-equation audit with rank two across two assets; a smooth Harsanyi welfare-margin obstruction under varying ethical weights; a habit-formation model that passes the static rank-one test and fails the dynamic one; a multidimensional-screening witness; and a logarithmic public-goods game with rank-two comparative statics across match rates. A universal compression theorem shows the same logic applies beyond utility — to scalar capital aggregation and to scalar-value readouts of multidimensional neural populations. The empirical implication is narrow and testable. Pre-register an audit bundle, estimate its local Jacobian, bootstrap the relevant singular value, and reject a scalar (or k-dimensional) model only when a lower confidence bound on σ_{k+1} remains positive. A companion notebook, rank_audit_demonstration.ipynb, implements the protocol for synthetic rank-one and rank-two agents, with a behavioral-data branch tied to the Choi–Fisman–Gale–Kariv replication package. The claims throughout are conditional: smooth or piecewise-smooth scalar reductions are ruled out where the relevant derivative rank exceeds one on the relevant stratum; arbitrary discontinuous rules are not addressed, and approximate scalar utility can remain locally adequate when σ₂(DG) is small.

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