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Econ-522 · economics · Economics

Welfare from inconsistent choices

LLM status Open unverified

Statement

Original subject group: Economic theory, equilibrium and social choice

Audit classification: Published research agenda. Verified welfare discussion in §6.4 and enriched-data agenda in §7. The robust relation is editorial, not a unique welfare criterion endorsed by the survey.

Definitions and precise research target

Model and research target. Let $X$ be a finite set of economic alternatives, $B\subseteq X$ a feasible menu, and $f\in F$ an observed frame. For each menu and frame, observe choice probabilities $P(x\mid B,f)$ and optional process measurements $L$, such as response time or attention. Fix a publicly specified class $\mathcal M$ of behavioral models. A model $m\in\mathcal M$ consists of a latent welfare ordering $\succeq_m$, a choice procedure, and a distribution of $L$. State explicitly the normative assumption connecting that ordering to welfare; inconsistent choices alone cannot supply it. Let $\mathcal M(P,L)$ contain all models fitting the population observables. Define robust welfare comparison by \[ x\succeq^R y\quad\Longleftrightarrow\quad x\succeq_m y\text{ for every }m\in\mathcal M(P,L). \] Determine the sharp identified relation $\succeq^R$, which additional experimentally manipulated frames or process measurements enlarge it, and which comparisons remain intrinsically ambiguous within the maintained model class. Construct uncertainty regions accounting for sampling error and evaluate policy rankings by their robust welfare implications. A successful solution must report how conclusions change when competing attention or preference-construction models are admitted.

Definitions and maintained assumptions (editorial unless a source definition is named). Menus are nonempty members of a declared $\mathcal B\subseteq2^X$; $P(x\mid B,f)=0$ for $x\notin B$, and probabilities sum to one. A welfare ordering is a complete transitive binary relation on $X$, linked to well-being by an explicitly defended normative assumption. Specify a measurable process space $\mathcal L$, the population joint laws $P^{B,f}$ of $(x,L)$, and a model-generated law $P_m^{B,f}$. Define $\mathcal M(P)=\{m:P_m^{B,f}=P^{B,f}\ \forall(B,f)\}$, and require it to be nonempty before taking its intersection of welfare relations. Otherwise universal quantification over the empty class would misleadingly rank every pair. A comparison is sharp when it holds in every fitting model and each excluded comparison has a fitting countermodel. With sampling, replace exact laws by a simultaneous $(1-\delta)$ confidence region for laws before forming the robust relation. Experiments shrink the fitting class only when their exclusion restrictions preserve the normative welfare ordering.

Variants and assumption changes

  1. One procedure (editorial). Specify one attention mechanism and its admissible parameters. Characterize the identified welfare relation and the smallest additional menu family needed to identify each remaining pair.
  2. Model union (editorial). Allow attention, satisficing and context-dependent preference construction simultaneously. Compute comparisons surviving their union and quantify how results change under an explicit alternative normative interpretation of constructed preferences.

References and source audit

  1. §6.4 pp.51–52; §7 pp.53–54 in February 2023 manuscript. Supports the stated research area; exact displayed specification is editorial. DOI publication is2024; full inspected manuscript is February2023. https://geoffroydeclippel.net/Working%20Papers%20PDFs/JEL.pdf
  1. Geoffroy de Clippel; Kareen Rozen. 2024. Bounded Rationality in Choice Theory: A Survey. Published JEL 62(3),995–1039 (2024); inspected February2023 author manuscript §§6.4–7, printed pp.51–54. https://geoffroydeclippel.net/Working%20Papers%20PDFs/JEL.pdf. DOI: https://doi.org/10.1257/jel.20231592.

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-522-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-522

Welfare from inconsistent choices: attention certificates and sharp orders

Problem ID: D60-518. Status: Unresolved. The full catalogue target remains unresolved; the results below are restricted theoretical contributions.

The model-class target

The catalogue observes choices and process measurements across menus and frames. A fitting model includes a complete transitive normative ranking, a choice procedure and a process-data law. The target intersects the welfare comparisons of all fitting models. To claim sharpness one must exhibit a fitting countermodel for every excluded comparison. Neither a selected behavioral explanation nor a single rationalization establishes that target.

The de Clippel--Rozen review treats welfare inference as sensitive to the maintained bounded-rationality model. This note supplies two exact finite-alternative results: complete nonidentification under unrestricted attention and an efficiently computable sharp order when actual consideration sets are validly observed. It then shows why replacing measured consideration by a noisy proxy can break that conclusion. These results concern explicit subclasses rather than the catalogue's entire union of behavioral theories.

Unrestricted attention gives no welfare comparison

Let $X$ be a finite set with $m\ge2$ alternatives. Menus $B$ are nonempty subsets of $X$. A frame is $f$, the chosen alternative is $C\in B$, and a process vector $L$ is observed. Suppose the procedure draws a nonempty attention set $A\subseteq B$ and chooses its uniquely best member according to a strict normative ranking $\succ$. Permit an arbitrary joint conditional law of $(A,L)$ for each $(B,f)$.
Proposition: Every observable law of $(C,L)\mid(B,f)$ supported on $C\in B$ is consistent with every strict ranking of $X$. Hence no strict welfare comparison is identified in this attention class.
Proof: Fix any ranking. Draw a pair $(c,l)$ from the given observed conditional law, set $A=\{c\}$ and $L=l$, and choose the best member of $A$. The choice is necessarily $c$, independent of the ranking, and the full joint distribution of choice and process data is reproduced. Repeat for every menu and frame. For any two alternatives there are rankings in both directions, each fitting the same law. ∎
The construction does not assert that people literally consider only one option. It proves that a model class allowing that explanation cannot exclude it. Rich eye movements, response times or self-reports have no identifying content without restrictions connecting their distributions to attention or welfare. Correlation between a process measure and choices alone is compatible with the construction because the pair is reproduced jointly.

Verified consideration sets yield a graph problem

Now impose a stronger measurement model: $A$ is observed without error, every considered alternative is feasible, and the chooser always selects the strictly best member of $A$ according to a frame-invariant strict ranking. Choice inconsistencies across menus may arise from varying $A$. Let the population support of $(B,f,A,C)$ be known exactly. Build a directed graph $G$ on $X$ by adding \[ C\longrightarrow x\qquad\text{for every }x\in A\setminus\{C\} \] for each support observation. An edge means that the normative ranking must place its tail above its head.
Theorem: The model fits if and only if $G$ is acyclic. If it fits, its admissible strict rankings are exactly the topological orderings of $G$, and the sharp robust strict welfare relation is reachability in $G$.
Proof: Every fitting ranking respects each edge. A directed cycle would require an alternative to be strictly above itself by transitivity, so acyclicity is necessary. If $G$ is acyclic, a topological ordering exists: repeatedly remove a vertex with no incoming edges; such a vertex exists because following incoming edges forever in a finite graph would create a cycle. Under this ordering every recorded choice is above all other considered alternatives and therefore is the best in its attention set. Keeping the observed law of $(B,f,A)$ reproduces the choices and gives a fitting model. Thus the fitting rankings are precisely the topological orders.

A directed path from $x$ to $y$ forces $x\succ y$ in every such ranking. Conversely suppose there is no path from $x$ to $y$. Add the edge $y\to x$. A new cycle could only be formed from this edge and an old path from $x$ to $y$, which does not exist. The enlarged graph remains acyclic and has a topological ordering with $y\succ x$. It respects the original graph and supplies a fitting countermodel. This proves sharpness of reachability, including when the original graph already contains a path in the reverse direction. ∎

Acyclicity can be checked in $O(m+e)$ operations for $e$ edges, and performing a graph search from every vertex computes the robust order in $O(m(m+e))$ operations. These are finite explicit-input bounds, not claims about obtaining the exact population support from a small survey. They also rely on strict rankings. Allowing welfare indifference with an unrestricted tie-breaking rule changes the inequalities and requires a separate weak-order analysis.

A worked inconsistency with useful welfare information

Take $X=\{a,b,c,d\}$. One observation considers $\{a,b\}$ and chooses $a$; another considers $\{b,c\}$ and chooses $b$; a third considers only $\{c\}$ and chooses $c$ from the full menu. The edges are $a\to b$ and $b\to c$, so the sharp order includes $a\succ b\succ c$ and $a\succ c$. Alternative $d$ remains incomparable to all three. Choosing $c$ from the full menu does not contradict the ranking because the verified consideration set excludes $a$ and $b$.

If a fourth observation considers $\{a,c\}$ and chooses $c$, the new edge $c\to a$ creates a cycle and rejects this model. It would be wrong to repair the fit silently by declaring a different normative ranking in each frame: frame invariance was the restriction giving the earlier comparison meaning. One may enlarge the model to stochastic errors, preference construction or another process, but must then recompute the identified set for that announced class.

An experimental implication and its limits

Suppose an intervention can genuinely force both members of a selected pair to be considered while preserving the same welfare ranking. The selected winner then gives a valid comparison edge. A standard comparison sorting algorithm identifies the entire strict ranking in at most $m\lceil\log_2m\rceil$ pair interventions: recursively sort two halves and merge them using at most the combined size minus one comparisons. The recurrence gives the stated bound by induction. In the worst case any deterministic binary-comparison procedure needs at least $\lceil\log_2(m!)\rceil$ comparisons because its binary decision tree must distinguish $m!$ possible rankings.

These operation counts do not establish practical ability to force consideration. A compulsory display can alter fatigue, tastes or interpretation rather than merely add an option to a fixed attention set. The intervention assumption is therefore stronger than displaying both alternatives on a screen. With choice noise, repeated experiments and a margin assumption are needed before a comparison is reliably assigned.

Likewise, if $L$ is only a proxy for $A$, interpreting every visually fixated item as considered is an extra measurement exclusion. Even an arbitrarily small allowed contamination probability can explain a rare apparent edge by error. Exact logical reachability from population support is then too aggressive. One possible conservative finite-sample approach is to retain all joint latent-attention models in a valid confidence region subject to the specified error restrictions, and intersect their welfare orders. It is informative only when those restrictions exclude singleton-attention rationalizations; this note does not claim a closed-form characterization for that larger problem.

Remaining gap

The graph theorem supplies sharp countermodels and a complete algorithm in a clear restricted model. The nonidentification theorem explains why unstructured process data do not suffice. The full catalogue asks for defensible inference across attention, satisficing and preference-construction models, including sampling uncertainty and valid measurement links. Those empirical and model-selection requirements remain open here. Selecting the graph subclass because it gives attractive welfare comparisons would assume the conclusion rather than validate it.

Completion Estimate

50%

This is a post hoc, heuristic estimate for the full catalogue target. The attempt proves unrestricted-attention welfare nonidentification and a sharp reachability characterization when consideration sets and strict invariant rankings are verified. Full model-union comparisons still lack noisy-process measurement characterization, simultaneous sampling uncertainty, and validated restrictions across attention, satisficing, and preference construction.

References

  • [dcr] G. de Clippel and K. Rozen (2024), Bounded Rationality in Choice Theory: A Survey, Journal of Economic Literature 62(3), 995--1039. Author manuscript, February 2023, discussion in Sections 6.4--7: https://geoffroydeclippel.net/Working%20Papers%20PDFs/JEL.pdf. The primary review provides context for model-dependent welfare inference; the graph and singleton-attention arguments are proved independently above.

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