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Econ-534 · economics · Economics

A portable model of cognitive biases

LLM status Open unverified

Statement

Original subject group: Behavioral and experimental economics

Audit classification: Published research agenda. Enke–Graeber §8 explicitly discusses cognitive defaults and noise-measure validity. The cross-domain shrinkage and prediction test are editorial.

Definitions and precise research target

Model and research target. Let $d$ index risky choice, ambiguous choice, belief updating, and intertemporal choice. In each domain individual $i$ faces a task with observed attributes $X_{id}$, gives response $Y_{id}$, and reports cognitive uncertainty $Q_{id}\in[0,1]$, defined as subjective doubt that the chosen action maximizes expected utility. Let $m_d(X_{id};\theta_i)$ denote a domain-specific benchmark decision, where $\theta_i$ contains stable preference or belief parameters. A candidate shared mechanism has the form \[Y_{id}=(1-\lambda_{id})m_d(X_{id};\theta_i)+\lambda_{id}b_d(X_{id};\eta)+\varepsilon_{id},\qquad \lambda_{id}=g(Q_{id},X_{id};\gamma).\] Here $b_d$ is a cognitive default, $g$ maps measurements to shrinkage intensity, and $\eta,\gamma$ are parameters shared across specified domains. Define prediction loss in a held-out domain $d$ as $R_d=\mathbb E[\ell(Y_{id},\widehat Y_{id})]$, for a preregistered loss $\ell$; estimate the prediction using parameters trained without that domain. Can measured uncertainty and a low-dimensional account of defaults and task difficulty improve $R_d$ over independently fitted domain models, while preserving calibration across populations and description-versus-experience treatments? Also estimate whether $Q_{id}$ predicts independently measured response noise. This formulation asks for testable portability and calibration rather than asserting that a universal model exists. Utility benchmarks and scaling of responses must be specified separately before the cross-domain comparison has empirical meaning.

Definitions and maintained assumptions (editorial unless a source definition is named). Restrict $Y_{id}$ to scalar responses on a declared domain-specific scale, or replace the convex-combination equation by a model on a shared feasible vector space. $m_d$ and $b_d$ must lie in the same response space; for discrete choices a mixture of probabilities, not a fractional action, is appropriate. Require $g(Q,X;\gamma)\in[0,1]$, integrable disturbance, and a stated conditional mean restriction if regression parameters are to be interpreted. Define measured $Q$ by the exact incentivized or survey question and reporting-error kernel; it is not assumed to equal true cognitive noise. Stable individual parameters $\theta_i$ require independent identification or prespecified pooling. Held-out-domain risk $R_d$ uses training independent of test observations; domain-specific benchmark models must receive comparable training sample sizes and tuning budgets. Calibration is a declared conditional prediction condition or bin-based approximation. The substantive hypothesis is reduction in out-of-domain loss by a restricted mechanism, not the unconstrained ability of a flexible model to fit existing responses.

Variants and assumption changes

  1. Fixed defaults (editorial). Prespecify a default per domain and a shared finite-dimensional shrinkage function. Test whether it predicts held-out risk and belief tasks better than equally constrained baselines.
  2. Noise validation (source agenda, editorial design). Obtain repeated independent responses and uncertainty reports on the same tasks. Identify when reported uncertainty predicts independently measured variance rather than only systematic central tendency.

References and source audit

  1. §8 printed pp.32–34 in March2023 manuscript. Supports the stated research area; exact displayed specification is editorial. https://thomasgraeber.com/assets/papers/EnkeGraeber_CognitiveUncertainty.pdf
  1. Benjamin Enke; Thomas Graeber. 2023. Cognitive Uncertainty. §8 printed pp.32–34 in March2023 manuscript. https://thomasgraeber.com/assets/papers/EnkeGraeber_CognitiveUncertainty.pdf. DOI: https://doi.org/10.1093/qje/qjad025.

Common conventions: Source mathematical conventions

These conventions apply to each entry unless explicitly replaced.

Models and identification. A primitive model $m$ specifies all probability laws, feasible choices, information, equilibrium and measurement rules used by its target. The admissible class $\mathcal M$ is fixed independently of outcomes. If $P_O(m)$ is its observable probability law and $\tau(m)$ the target, its identified set at observable law $P$ is \[ \mathcal I_\tau(P)=\{\tau(m):m\in\mathcal M,\ P_O(m)=P\}. \] Point identification means that this set is a singleton. An empty set signals incompatibility of data and assumptions. Coordinate bounds need not describe the joint set. Bounds are sharp only if every retained target is attainable or arbitrarily approachable by compatible models. Finite bounds require explicit restrictions on unobserved quantities; unrestricted confounding, hidden wealth, extrapolation or arbitrary equilibrium selection can yield uninformative sets.

Causal interventions. A potential outcome $Y_i(p)$ is the outcome under a complete policy regime $p$, including timing, financing, information and market spillovers. The target population and units are fixed before treatment. With interference, use the complete assignment vector or a justified exposure mapping; individual no-interference notation alone cannot identify an economy-wide intervention. A difference of mean potential outcomes does not require the cross-world joint distribution. Conditioning on post-treatment migration, survival, enrollment or employment changes the estimand and needs separate justification.

Statistical statements. An estimator, confidence set, forecast or test specifies a sampling law, model class and loss/coverage criterion. Uniform $(1-\alpha)$-coverage means $\inf_{m\in\mathcal M}\Pr_m\{\tau(m)\in C_n\}\geq1-\alpha$, or an explicitly asymptotic version. The whole parameter space trivially has coverage, so informativeness requires a stated width, risk or power objective. A forecasting improvement is relative to a named benchmark, information set and evaluation population.

Equilibrium and optimization. An equilibrium comprises feasible plans, optimality, beliefs where needed, budget constraints and all market/resource clearing. The primitives or a stated benchmark define each correspondence $\mathcal E(p;m)$. Nonemptiness is assumed or proved, never inferred just from compact policy sets. A selected-equilibrium target uses a declared selection rule; a robust target quantifies over all permitted equilibria. Use suprema or infima unless attainment is established. An $\varepsilon$-optimal policy is feasible and within $\varepsilon$ of the finite optimum value. Report an empty feasible set separately.

Welfare and accounting. Normative utility, interpersonal normalization, social weights, numeraire, discounting and the use of public receipts are primitives. Behavioral utility need not coincide with the welfare criterion. Household welfare includes ownership income and rents once; adding profits again to compensating variation generally double-counts them. A monetary equivalent is defined by an explicit transfer and price convention, with monotonicity and range conditions. Average effects, mechanism contributions, transfers and welfare losses are different targets. Infinite sums require convergence and dynamic feasible plans require terminal or no-Ponzi conditions.

Source versions. A working-paper date, revision date and journal publication year are distinguished. A DOI for a journal version is not automatically the DOI of an earlier manuscript. Locators refer to the stated version and printed pages where specified. A metadata-only check does not verify a claim about a full-text conclusion. Every entry's source subsection narrows the provenance claim accordingly.

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Econ-534-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-534

When a shared shrinkage parameter is identified and portable

Problem ID: D91-530. Status: Unresolved. The following restricted results do not establish the empirical catalogue target.

Target and restrictions

The catalogue asks whether cognitive uncertainty, task difficulty and defaults support a low-dimensional model that predicts across domains. Its displayed mixture is a hypothesis to test, not a universal behavioral law. Enke and Graeber's discussion of cognitive defaults and noise motivates the question [eg]. This attempt isolates three separate requirements: identifying a shrinkage coefficient within a domain, measuring response noise independently, and testing prediction in a genuinely held-out domain. None follows merely from a significant regression coefficient on reported uncertainty.

For a tractable scalar submodel, prespecify benchmarks $m(X)$ and defaults $b(X)$ on the same response scale, observe $Q\in[0,1]$, and posit \[ Y=m(X)+\gamma w(X,Q)+\varepsilon,\qquad w=Q\{b(X)-m(X)\},\quad 0\le\gamma\le1. \tag{1} \] Thus the shrinkage intensity is $\lambda=\gamma Q$. For binary actions, $Y$ can be a response indicator and (1) a probability model whose conditional mean lies in $[0,1]$. An individual preference parameter hidden inside $m$ must have been independently identified or fixed before the following argument. Jointly fitting an unrestricted benchmark would erase the identifying restriction.

Proposition (Identification and a precise failure): If $0<{\mathbb{E}} w^2<\infty$, ${\mathbb{E}}|w(Y-m)|<\infty$, and ${\mathbb{E}}[w\varepsilon]=0$, then \[ \gamma=\frac{{\mathbb{E}}[w(Y-m)]}{{\mathbb{E}} w^2}. \tag{2} \] If instead $Q=q\in(0,1)$ is constant, $b=0$, and the unknown benchmark is $m=\theta$, the conditional mean alone generally identifies only $(1-q\gamma)\theta$.
Proof: Multiply (1) by $w$, take expectations, and divide by its positive second moment. For the second assertion let $c>0$ be the observed mean. Every $\gamma\in[0,1]$ paired with $\theta=c/(1-q\gamma)$ produces that mean; with a common additive error law it produces the same entire response law. For example $q=1/2,c=1/4$ permits $(\gamma,\theta)=(0,1/4)$ and $(1,1/2)$, even with benchmarks restricted to $[0,1]$. ∎

The relevance condition concerns uncertainty times the benchmark--default difference. Variation in reported uncertainty has no identifying power when benchmark and default coincide. The moment restriction also is substantive: endogenous reporting error, changing attention or omitted preferences can correlate $w$ with $\varepsilon$. Random assignment of a description does not automatically randomize subjective $Q$. Equation (2) identifies a predictive coefficient under the stated model; it is not, without more restrictions, the causal effect of increasing uncertainty.

What repeated answers measure

Let $L$ denote stable latent individual parameters and fix the task, report and presentation. Suppose two responses satisfy \[ Y_r=f(X,Q,L)+\varepsilon_r,\qquad r=1,2, \] with errors conditionally independent given $(X,Q,L)$, conditional mean zero and common conditional variance $\sigma^2(X,Q,L)$. Then \[ {\mathbb{E}}\left[\frac{(Y_1-Y_2)^2}{2}\mid X,Q\right] ={\mathbb{E}}[\sigma^2(X,Q,L)\mid X,Q]. \tag{3} \] Indeed the common latent mean cancels before squaring, the two error variances add, and the conditional cross product vanishes; averaging over $L$ proves the identity. This distinguishes within-person response variation from heterogeneity in preference levels. Regressing the squared difference on a preregistered transformation of $Q$ can therefore validate a predictive noise relationship under these assumptions.

The independence requirement cannot be replaced by equal marginal variances. If the errors have conditional covariance $c(X,Q,L)$, the right side of (3) becomes ${\mathbb{E}}[\sigma^2-c\mid X,Q]$. A perfectly persistent common error is invisible to differencing. Conversely, genuine learning or fatigue that changes the latent mean contributes to the squared difference and is misclassified as noise. Repeated reports of $Q$ can themselves change reflection. A useful implementation separates the uncertainty-measurement session from randomized repeat timing, records order, and interprets failures of stationarity explicitly. These are design implications of (3), not empirical claims about any existing sample.

A calculation of the cost of failed portability

Assume for this calculation that the $n$ training covariates are fixed, $S=\sum_{j=1}^n w_j^2>0$, and errors are independent with mean zero and variance $\sigma^2$. The unconstrained least-squares estimator is \[ \widehat\gamma=\frac{\sum_jw_j(Y_j-m_j)}S,\qquad {\operatorname{Var}}(\widehat\gamma)=\sigma^2/S. \tag{4} \] If training observations in domain $d_j$ have true coefficients $\gamma_{d_j}$, its expectation is the weighted average $\gamma_{\rm pool}=\sum_j w_j^2\gamma_{d_j}/S$. In an independent target domain with coefficient $\gamma_T$ and mean-zero noise conditional on covariates, squared prediction loss above the target conditional-mean oracle equals \[ {\mathbb{E}}_T[w^2]\left\{\frac{\sigma^2}{S} +(\gamma_{\rm pool}-\gamma_T)^2\right\}. \tag{5} \] To prove this, write the prediction error as $(\widehat\gamma-\gamma_T)w-\varepsilon_T$. Its cross term has zero expectation. Subtract target noise risk and apply the bias--variance identity to (4). Clipping to $[0,1]$ can only reduce coefficient squared error when $\gamma_T$ lies there, but (5) is the exact formula for the unclipped estimator.

A larger pooled sample reduces the first term and leaves the portability bias intact. For example, if all training tasks have $\gamma=0$ and target tasks have $\gamma_T=1$, even infinite training data leave excess risk ${\mathbb{E}}_Tw^2$. Conversely, with a genuinely shared coefficient and matched response scales, pooling can lower variance. These two completely specified models refute any unconditional claim that the shared specification must outperform separate domain models. Which situation describes people is the unresolved empirical question.

A finite-sample comparison that tests the intended claim

Train a shared predictor and its comparator without the held-out observations, with their tuning rules fixed. On $n_T$ independent target observations let $D_j$ be shared-model loss minus comparator loss. Suppose each loss is scaled to $[0,1]$, so $D_j\in[-1,1]$. Conditional on the training data, Hoeffding gives \[ \left|n_T^{-1}\sum_j D_j-{\mathbb{E}}_T D_j\right| \le\sqrt{\frac{2\log(2/\alpha)}{n_T}} \tag{6} \] with probability at least $1-\alpha$. A negative upper endpoint establishes improvement for that target distribution and those trained rules. Searching across many domains, losses or uncertainty scales requires simultaneous correction or another holdout; it cannot use (6) unchanged. If individuals contribute correlated tasks, the independence unit must instead be the person or an appropriately analyzed cluster.

Calibration is distinct from average loss. Conditional residual means across prespecified uncertainty bins can detect systematic over-shrinkage even when average loss improves. Description and experience treatments require their own target populations and scales. A model can be portable across tasks within one sample yet fail across populations through a changed default or reporting kernel. No data are supplied here to determine these quantities, compare actual prediction risks, or establish a universal model. Equations (2)--(6) provide exact identification restrictions, noise diagnostics and falsifiable comparisons for the specified submodel only.

Completion Estimate

60%

This is a post hoc, heuristic estimate for the full catalogue target. The attempt identifies a restricted shared shrinkage coefficient, derives repeated-response noise diagnostics, and quantifies portability bias and honest held-out loss comparisons. The full cross-domain prediction hypothesis still requires independently identified benchmarks, validated uncertainty-report kernels, population calibration, and empirical comparisons across description and experience treatments.

References

  • [eg] B. Enke and T. Graeber (2023), Cognitive Uncertainty, March manuscript, discussion section 8, printed pp.32--34. Author manuscript. The source motivates defaults and noise validation; (1)--(6) are self-contained restricted calculations, not attributed empirical findings.

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