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A Random Rule Model

Avner Seror

TL;DR

Applied to 10,000 binary lottery problems, rule-gating substantially outperforms structured neural benchmarks based on expected utility and prospect theory, approaching the most flexible benchmark while remaining highly restrictive under permutation-fit tests, and retains predictive content on an independent dataset.

Abstract

We propose a Random Rule Model (RRM) in which behavior is generated by switching among a small library of transparent, parameter-free decision rules. A differentiable gate learns environment-dependent rule propensities, producing an interpretable mixture over named procedures. We develop a global identification theory based on two verifiable conditions on the observed support. Applied to 10,000 binary lottery problems, rule-gating substantially outperforms structured neural benchmarks based on expected utility and prospect theory, approaching the most flexible benchmark while remaining highly restrictive under permutation-fit tests, and retains predictive content on an independent dataset. Mechanism diagnostics reveal that extreme-outcome screening, salience, and attention rules carry the largest responsibility weights, with systematic shifts along tradeoff complexity and dispersion asymmetry. Robustness checks confirm that the findings are not driven by the ex-ante library choice, marginal dominance relationships, or the availability of additional regressors.

A Random Rule Model

TL;DR

Applied to 10,000 binary lottery problems, rule-gating substantially outperforms structured neural benchmarks based on expected utility and prospect theory, approaching the most flexible benchmark while remaining highly restrictive under permutation-fit tests, and retains predictive content on an independent dataset.

Abstract

We propose a Random Rule Model (RRM) in which behavior is generated by switching among a small library of transparent, parameter-free decision rules. A differentiable gate learns environment-dependent rule propensities, producing an interpretable mixture over named procedures. We develop a global identification theory based on two verifiable conditions on the observed support. Applied to 10,000 binary lottery problems, rule-gating substantially outperforms structured neural benchmarks based on expected utility and prospect theory, approaching the most flexible benchmark while remaining highly restrictive under permutation-fit tests, and retains predictive content on an independent dataset. Mechanism diagnostics reveal that extreme-outcome screening, salience, and attention rules carry the largest responsibility weights, with systematic shifts along tradeoff complexity and dispersion asymmetry. Robustness checks confirm that the findings are not driven by the ex-ante library choice, marginal dominance relationships, or the availability of additional regressors.
Paper Structure (116 sections, 4 theorems, 79 equations, 13 figures, 15 tables)

This paper contains 116 sections, 4 theorems, 79 equations, 13 figures, 15 tables.

Table of Contents

  1. Introduction
  2. Model
  3. Setup
  4. Rule library and rule-induced perceived lotteries
  5. Conditional on Activity Random Rule Model (CA-RRM)
  6. Rule-gating parameterization
  7. Identification via Rule Switching
  8. From CA-RRM to a collapsed binary logit index (identity, not a new model).
  9. Observed object.
  10. Normalization and parameter vector.
  11. Decisive-side indicators and the $H$ matrix.
  12. Global identification theorem.
  13. Effective dimension.
  14. Menu encoding.
  15. D1 (Two-sidedness).
  16. ...and 101 more sections

Key Result

Theorem 1

Consider the CA-RRM with affine indices $s_f(A)=\alpha_f+\beta_f^\top\psi(A)$, $f\in\mathcal{F}$. Assume $p(A)\in(0,1)$ for all menus in the support, and impose the normalization $(\alpha_{f_0},\beta_{f_0})=(0,0)$. Let $d=\dim(\psi)$. Suppose there exist $(d+1)$ feature values $x^{(0)},\ldots,x^{(d) Then the full parameter vector $\{(\alpha_f,\beta_f)\}_{f\in\mathcal{F}}$ is globally identified fr

Figures (13)

  • Figure 1: Predictive performance. Panel (a) compares out-of-sample MSE across models. Panel (b) reports the learning curve, plotting test-set MSE as a function of the percent training data used.
  • Figure 2: Completeness and Restrictiveness.
  • Figure 3: Completeness--restrictiveness plot. The horizontal axis is ML completeness (Equation \ref{['eq:ml_completeness']}); the vertical axis is permutation-fit restrictiveness (Equation \ref{['eq:restrictiveness']}).
  • Figure 4: Rule responsibility weights $w_f$ for the full 12-rule library. Each $w_f$ is the average share of predicted choice probability attributed to rule $f$ among decisive rules.
  • Figure 5: Rule diagnostics: ablation of each behavioral rule from the full 12-rule library. Error bars show split-variability intervals ($\pm 1.96 \times \mathrm{SE}$) from fold-to-fold variability across 50 CV splits.
  • ...and 8 more figures

Theorems & Definitions (12)

  • Definition 1: First-Order Stochastic Dominance
  • Definition 2: Activity
  • Theorem 1: Global identification under within-feature activity variation
  • Theorem 2: Global identification of the general CA-RRM
  • proof
  • Remark 1
  • Remark 2
  • Theorem 3: Local identification of the general CA-RRM
  • proof
  • Proposition 4: Jacobian entries
  • ...and 2 more