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Econ-571 · economics · Economics

Risk preferences versus complexity errors

LLM status Open unverified

Statement

Original subject group: Behavioral and experimental economics

Audit classification: Published research agenda. Full Oprea 2024 article now retrieved. §§III–IV, pp.3806–3807, explicitly proposes further investigation of heuristic behavior and risk-induced complexity. The identification/field decomposition is editorial.

Definitions and precise research target

Model and research target. Consider individuals $i$ choosing between a sure payment and a binary lottery $L=(p,x,y)$, where $p\in[0,1]$ is the probability of receiving $x$, and $y$ is received otherwise. Let $C\in\{0,1\}$ randomly assign a simple or complex representation while preserving $L$, payment rules, and information. Let $a_i(L,C)$ be the reported certainty equivalent. A preference model specifies utility $u_i$, probability weighting $w_i$, and a reference point $r_i$; its latent certainty equivalent $v_i(L)$ satisfies $u_i(v_i)=w_i(p)u_i(x)+(1-w_i(p))u_i(y)$. Observations obey $a_i(L,C)=v_i(L)+e_i(L,C)$, where $e_i$ is a decision or reporting error whose conditional mean need not vanish. The observable causal complexity effect is \[\tau(L)=\mathbb E[a_i(L,1)-a_i(L,0)].\] Identify which restrictions, auxiliary tasks, or repeated measurements permit separation of $v_i$ from systematic $e_i$, rather than attributing every pattern to preferences. Random assignment identifies $\tau$ but does not identify the error-free preference distribution without further assumptions. A field extension should specify a target population $P$, comparable lotteries, and consequential implementation; estimate the corresponding $\tau_P(L)$ and the predictive performance of recovered preferences for held-out decisions.

Definitions and maintained assumptions (editorial unless a source definition is named). For a binary gain lottery with $x\ge y$, strictly increasing utility $u_i$ and weighting $w_i:[0,1]\to[0,1]$, $w_i(0)=0,w_i(1)=1$, the specified model defines a unique certainty equivalent $v_i(L)\in[y,x]$ by the displayed equation. A reference-dependent version must explicitly replace $u_i(x)$ by $u_i(x-r_i)$ and specify treatment-invariance of $r_i$; merely naming a reference point does not use it. Measurements are either monetary certainty equivalents or choices converted by a declared elicitation procedure and error kernel. Potential responses $a_i(L,C)$ exist for both equivalent representations, with randomized $C$. To infer preferences, assume representations preserve utility, beliefs and reference points; otherwise $\tau(L)$ is a representation effect with several channels. Nonzero systematic error permits arbitrary offsets between $v_i$ and $e_i$, so point identification needs anchors, repeated-treatment restrictions or a parametrized error law. A deterministic mirror task is a separate economic task and its validity as an error proxy must be tested.

Variants and assumption changes

  1. Repeated representation (editorial). Repeatedly elicit identical lotteries under randomized equivalent descriptions, with calibrated payment procedures. Identify a maintained stable utility/weighting model under explicit error restrictions, and report failure when systematic errors are unrestricted.
  2. Risk and mirrors (source-related editorial). Jointly vary risk presence and representation complexity using matched tasks. Test whether mirror errors predict lottery errors and bound decomposition sensitivity to risk-dependent error or reference shifts.

References and source audit

  1. §§III–IV pp.3806–3807 in published article. Supports the stated research area; exact displayed specification is editorial. https://www.aeaweb.org/articles?id=10.1257/aer.20221227
  2. §§III–IV pp.3806–3807 in published article. Supports the stated research area; exact displayed specification is editorial. https://ryanoprea.com/papers/decisions-under-risk.pdf
  1. Ryan Oprea. 2024. Decisions under Risk Are Decisions under Complexity. §§III–IV, AER114(12), pp.3806–3807; full article retrieved from author: https://ryanoprea.com/papers/decisions-under-risk.pdf. https://www.aeaweb.org/articles?id=10.1257/aer.20221227. DOI: https://doi.org/10.1257/aer.20221227.
  2. Ryan Oprea. 2024. Decisions under Risk Are Decisions under Complexity. §§III–IV, pp.3806–3807. https://ryanoprea.com/papers/decisions-under-risk.pdf. DOI: https://doi.org/10.1257/aer.20221227.

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-571-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-571

Risk preferences versus complexity errors: What calibrated repeated mirrors can identify

The separation problem

For a consequential binary lottery $L=(p,x,y)$ and a randomized equivalent representation $C\in\{0,1\}$, the decomposition of interest is \[ a_i(L,C)=v_i(L)+e_i(L,C),\qquad u_i(v_i)=w_i(p)u_i(x)+(1-w_i(p))u_i(y). \] The causal representation effect is $\tau(L)=E[a_i(L,1)-a_i(L,0)]$. Errors can be systematic. Oprea's experiments compare lottery choices with deterministic tasks that reproduce aspects of their complexity [oprea]. Such comparisons motivate an auxiliary measurement strategy; they do not, by themselves, prove that two tasks have identical individual errors. We isolate the restriction needed for that strategy and quantify the consequences of imperfect calibration.

An exact observational obstruction

Proposition: Random assignment of representations, arbitrarily many repeated reports, and exact knowledge of their joint distribution need not identify the distribution of preference certainty equivalents, even among expected-utility preferences with the same normalized utility at the prize endpoints.
Proof: Consider $L=(1/2,4,0)$ and deterministic reports $a_i(L,C)=2+C$ for every subject and repetition. One model has $u(z)=z/4$ on $[0,4]$, $w(p)=p$, certainty equivalent $v=2$, and errors $e(C)=C$. A second has $u(z)=\sqrt z/2$, the same probability weights, certainty equivalent $v=1$, and errors $e(C)=1+C$. Both utilities are strictly increasing and satisfy $u(0)=0,u(4)=1$. Both produce identical observations under every randomized representation sequence, but their latent certainty equivalents differ. Their common representation effect is one. On any larger lottery collection the construction extends by defining each error as the observed report minus that model's own certainty equivalent. Repeated observation changes nothing because these errors are systematic. ∎
Thus neither a zero representation effect nor a stable individual ranking across representations establishes an absence of complexity error. The obstruction survives correlated reports and rich panels unless the model restricts the level, rather than only the difference, of systematic errors.

A calibrated mirror panel

Fix one lottery and a baseline population whose latent $v_i$ is in the known prize interval $[y,x]$, where $y<x$. Each subject completes $J$ paired baseline-lottery and auxiliary-mirror trials. Let $D_{ij}$ be the lottery report minus the mirror report. Assume \[ D_{ij}=v_i+d_i+\epsilon_{ij},\quad |d_i|\le\eta, \quad E[\epsilon_{ij}\mid v_i,d_i]=0, \quad |\epsilon_{ij}|\le B. \tag{1} \] Conditional on $(v_i,d_i)$, the errors are independent across $j$. The constant $d_i$ is the mismatch between the tasks' systematic errors; the mirror's known normative value has already been subtracted from its raw report. Different subjects are independently sampled. Neither independence of $v_i$ and $d_i$ nor mean-zero mismatch is imposed. The representation, payment, reference point, and carryover conditions under which (1) is credible are substantive design requirements.
Theorem: Let $\mu_i=E[D_{ij}\mid v_i,d_i]=v_i+d_i$. At the population law of $\mu$, the sharp identified set for the distribution of $v$ consists of all distributions admitting a coupling $(\mu,v)$ with the observed $\mu$ marginal and \[ \max(y,\mu-\eta)\le v\le\min(x,\mu+\eta)\quad\text{almost surely}. \tag{2} \] Assume these intervals are nonempty almost surely. In particular the sharp marginal bounds, for $y\le t<x$, are \[ \Pr(\mu+\eta\le t)\le F_v(t)\le\Pr(\mu-\eta\le t). \tag{3} \] For a panel of $N$ subjects, with probability at least $1-\alpha$, simultaneously for every $i$, \[ |\overline D_i-v_i|\le \eta+B\sqrt{\frac{2\log(2N/\alpha)}{J}}. \tag{4} \]
Proof: Equation (1) implies (2). Conversely, for any coupling satisfying (2), set $d=\mu-v$ and choose trial errors identically zero. This realizes the coupling, satisfies (1), and preserves the specified observed law of $\mu$. It proves sharpness at that population object, without claiming that every finite-panel joint law has the same identified set. For (3), the largest feasible $v$ is $\min(x,\mu+\eta)$ and the smallest is $\max(y,\mu-\eta)$; assigning either endpoint to every subject attains the corresponding CDF bound. The restriction $t<x$ avoids rewriting the trivial endpoint value one. Since each centered error lies in an interval of length $2B$, Hoeffding's inequality gives $\Pr(|\overline D_i-\mu_i|>b\mid v_i,d_i) \le2\exp(-Jb^2/(2B^2))$. Integrate, use a union bound over subjects, and then add $|d_i|\le\eta$ to obtain (4). ∎
The coupling formulation is essential: CDF bands alone do not automatically specify a jointly attainable collection of lottery certainty equivalents. At $\eta=0$, repeated trials identify each $v_i$ as $J\to\infty$ by the conditional strong law, and independent subjects then identify its population distribution. At fixed $J$, the theorem gives an error bound, not nonparametric deconvolution from an unspecified error distribution. At positive $\eta$, more repetitions remove trial noise but cannot remove the calibration interval.

Recovering probability weights under an additional restriction

Suppose utility $u_i$ is independently known, strictly increasing, and continuous on $[y,x]$. For a recovered certainty-equivalent interval $[l_i,r_i]\subseteq[y,x]$, the compatible weights for this probability are exactly \[ \left[\frac{u_i(l_i)-u_i(y)}{u_i(x)-u_i(y)}, \frac{u_i(r_i)-u_i(y)}{u_i(x)-u_i(y)}\right]. \tag{5} \] Indeed the certainty-equivalent equation can be solved for the weight; continuity and strict monotonicity map the entire interval onto (5). If the same $p$ is tested with several prize pairs, a common weight must belong to all the resulting intervals. Empty intersection rejects that combination of calibration and preference restrictions. Unknown utility curvature and unknown weighting are not separately identified by simply applying this formula with a convenient guessed utility.

What remains unresolved

The results identify the exact ambiguity created by bounded mirror mismatch and give a finite-panel calibration guarantee. They do not establish that a particular mirror satisfies (1), identify arbitrary utility and weighting jointly, or transport laboratory preferences to field choices with different stakes and institutions. Those are central parts of the original agenda. The bounds make the missing restriction explicit and permit reporting how conclusions change as its tolerance $\eta$ varies.

Completion Estimate

45%

This is a post hoc, heuristic estimate for the full catalogue target. The attempt establishes observational nonidentification with systematic errors and derives sharp certainty-equivalent bounds and finite-panel guarantees under calibrated repeated mirrors. It still must validate mirror calibration, separate unknown utility from probability weighting, and establish transport and predictive performance for consequential field decisions.

References

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