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Econ-582 · economics · Economics

Empirically disciplined bounded rationality

LLM status Open unverified

Statement

Original atlas ID: M3. Formulation: editorial specification of a research family.

Definitions and assumptions

Let $\mathcal G$ be a declared population of finite games, with utilities normalized by a fixed experimental convention and observable covariates $x$. A structural model $m\in\mathcal M$ maps $(G,x,\alpha,\theta_m)$ into a probability distribution $p_m(\cdot\mid G,x,\alpha,\theta_m)$ on actions, where $\alpha$ is a subject characteristic with population law $F_m$ and $\theta_m$ is shared across games. For example, logit quantal response imposes \[ \sigma_i(a)=\frac{\exp\{\lambda_i\,\mathbb E_{\sigma_{-i}}u_i(a,a_{-i})\}} {\sum_{b\in A_i}\exp\{\lambda_i\,\mathbb E_{\sigma_{-i}}u_i(b,a_{-i})\}},\quad\lambda_i\ge0. \] This is a fixed-point condition and may require an explicit equilibrium selector. Cognitive-hierarchy models instead specify a distribution of reasoning depths and beliefs about lower-depth behavior.

Formal research question

Determine whether a finite-dimensional model family admits identified shared parameters and improves predictive performance across games drawn from a prespecified target distribution $Q$. Define the out-of-game log-score risk \[ \mathcal R_Q(m,\theta_m,F_m) =\mathbb E_{(G,x)\sim Q}\mathbb E_{a\sim P_{G,x}} \left[-\log\int p_m(a\mid G,x,\alpha,\theta_m)\,F_m(\mathrm d\alpha)\right], \] where $P_{G,x}$ is the population action law and candidate probabilities must be positive on its support. Characterize observational equivalence, estimate risk using games withheld from fitting, and test restrictions that prevent game-specific parameters from absorbing every discrepancy. Report the identified set when the shared parameters or depth distribution are not uniquely identified.

Interpretation and scope

The target is transportable prediction, not a theorem that one model describes all humans in all games. Utility normalization matters because payoff rescaling changes the effective logit precision unless its parameter is transformed accordingly. Subject heterogeneity, repeated-game experience, and equilibrium selection within quantal response must be represented in the likelihood rather than silently omitted. A within-subject game battery can distinguish stable subject effects from population compositional changes; assignment and attrition must still be handled. Comparing models by training likelihood alone does not answer the question. The held-out game distribution, score, benchmark, and permitted parameter dimension should be declared before collecting confirmatory outcomes.

Source audit and status

McKelvey--Palfrey (1995), Camerer--Ho--Chong (2004), and Crawford--Costa-Gomes--Iriberri (2013) are verified with their exact titles and journal records below. The first develops quantal-response equilibrium, the second cognitive hierarchy, and the third reviews nonequilibrium strategic thinking. Existing fits and comparisons are substantive partial answers; none establishes universal parameter invariance. This entry is therefore an empirical modeling agenda with an editorial validation and identification target. The absence of a universal cross-game model in the cited papers is not a certified mathematical impossibility or proof that the specified empirical comparison remains unperformed in every relevant population.

References

  1. S1 Richard D. McKelvey and Thomas R. Palfrey (1995). Quantal Response Equilibria for Normal Form Games. Games and Economic Behavior 10(1), 6--38. https://authors.library.caltech.edu/records/4fh4n-4c340. Locator: abstract; fixed-point definition and estimation. Establishes: Existence and empirical estimation of quantal-response equilibria. Does not establish: context-invariant cognitive parameters.

  2. S2 Colin F. Camerer, Teck-Hua Ho, and Juin-Kuan Chong (2004). A Cognitive Hierarchy Model of Games. Quarterly Journal of Economics 119(3), 861--898. https://academic.oup.com/qje/article-abstract/119/3/861/1938841. Locator: abstract; model and experimental applications. Establishes: A parsimonious nonequilibrium reasoning model. Does not establish: uniform prediction across all strategic environments.

  3. S3 Vincent P. Crawford, Miguel A. Costa-Gomes, and Nagore Iriberri (2013). Structural Models of Nonequilibrium Strategic Thinking: Theory, Evidence, and Applications. Journal of Economic Literature 51(1), 5--62. https://www.aeaweb.org/articles?id=10.1257/jel.51.1.5. Locator: abstract and review of level-k evidence. Establishes: Evidence and applications of structural nonequilibrium models. Does not establish: a theorem certifying the atlas prediction problem as open.

Common conventions: Source mathematical conventions

Unless an entry states otherwise, agent and firm sets are finite. State and action spaces are measurable, strategies and outcome maps are measurable, probability laws are specified, and expectations exist for the targets under discussion. Infinite-horizon discount factors lie in $(0,1)$ and discounted objectives are finite. Norms, tolerances and complexity input sizes must be specified for computational comparisons. Constants or primitives that appear locally are inputs, not empirical facts asserted by this catalogue.

Identification. A statistical model $m\in\mathcal M$ implies an observable law $P_m$ and target $\theta(m)$. At law $P$, its identified set is \[ \Theta(P;\mathcal M)=\{\theta(m):m\in\mathcal M,\ P_m=P\}. \] Point identification holds when this set is a singleton, or equivalently when $P_{m_1}=P_{m_2}$ implies $\theta(m_1)=\theta(m_2)$ for all compatible models. Sharp bounds mean bounds attained, or approached when the boundary is not attained, by compatible models. Writing this set is a definition, not a solution: the substantive task is to establish informative restrictions and derive or estimate the set. An unrestricted-confounding model may give only trivial bounds.

Inference. Identification, estimability and valid uncertainty quantification are separate questions. Fix $\alpha\in(0,1)$. A confidence procedure $C_n$ over a declared law class $\mathcal P$ has asymptotic uniform coverage at level $1-\alpha$ when \[ \liminf_{n\to\infty}\inf_{P\in\mathcal P}\Pr_P\{\theta(P)\in C_n\}\geq1-\alpha. \] For set-identified targets the coverage event must be defined separately, for example coverage of the full identified set. If an entry uses identification-indexed models instead of uniquely indexed laws, the same coverage convention is applied over those models. Pointwise limits do not establish uniform validity, and asymptotic validity does not guarantee finite-sample precision. Sample sizes, dependence structures and weak-identification sequences are part of each application.

Counterfactuals. $Y_i(a)$ is a potential outcome under a specified intervention, including its timing, implementation and versions. It is not $Y_i$ conditional on voluntarily choosing $a$. A full intervention vector permits interference; shorthand scalar treatment notation does not silently impose no interference. Assignment, selection, attrition, anticipation and measurement must be specified. A claim about an instrument's local effect is not automatically a population policy effect.

Economic equilibrium. A counterfactual model includes best responses, constraints, feasibility, market clearing, state transitions and expectations consistent with the stated information. When several equilibria exist, either a declared selector is an input or the target is set-valued. Comparisons across policy regimes must use the stated selection convention. Reduced-form relationships are not presumed invariant after an equilibrium-changing intervention.

Optimization and welfare. Optimization is over the stated feasible set. If attainment is not established, $\sup$ or $\inf$ and $\varepsilon$-optimal choices replace an asserted maximizer or minimizer. For example, with value $V=\sup_{a\in\mathcal A}W(a)<\infty$, an $\varepsilon$-optimal set is $\{a\in\mathcal A:W(a)\geq V-\varepsilon\}$ for $\varepsilon>0$. Benefits, costs, risk and health or environmental outcomes can be aggregated only using declared units and conversion weights. Interpersonal utility weights, the treatment of fiscal transfers and foreign profits, discounting, and the behavioral welfare criterion are explicit inputs. A comparative-static derivative additionally requires existence and appropriate differentiability of the selected counterfactual.

Sources. Local labels [S1], [S2], and so forth restart in every question. Locators identify the relevant abstract, model, section or result at the level actually checked; a bibliographic or abstract-level verification is not represented as inspection of an entire proof. Working papers are distinguished from later publications and changing versions. Source summaries are paraphrased. All URL checks and editorial formulations refer to the audit date stated above.

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Econ-582-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-582

Empirically disciplined bounded rationality: Designed payoff gaps identify a finite logit mixture

A disciplined submodel

The general target is a shared behavioral model that survives evaluation on new games. For a logit specification, utility normalization and an equilibrium selector are part of that model. Quantal response equilibrium is a fixed-point concept in [mp]; the results below instead use diagnostic games in which the subject's payoff gap does not depend on the opponent, removing equilibrium multiplicity from this identification step. This restricted battery cannot by itself validate a strategic model.

Normalize the two available payoffs to $d$ and zero, with $0<d<1$. Let an individual's stable response precision be $\lambda\ge0$. In this battery the high-payoff action has probability \[ k_\lambda(d)=\frac{1}{1+\exp(-\lambda d)},\qquad p(d)=\int k_\lambda(d)\,dF(\lambda). \tag{1} \] Any opponent can be given a payoff-independent strict best action; the subject's choice law remains (1). Games are randomly assigned within one baseline population and the mixture $F$ is invariant across them. This fixes the payoff scale: without it, multiplying every payoff by $c>0$ and dividing every precision by $c$ leaves all predictions unchanged.

A finite support can be separated by a finite battery

Lemma: For distinct nonnegative numbers $\lambda_1,\ldots,\lambda_K$, the functions $k_{\lambda_j}$ are linearly independent on $(0,1)$.
Proof: Suppose $\sum_j a_j k_{\lambda_j}(d)=0$ on that interval. The sum is real analytic on the whole real line, hence is identically zero there. Its limit as $d\to+\infty$ is $\sum_{\lambda_j>0}a_j+\frac12\sum_{\lambda_j=0}a_j=0$. Subtracting this limit from the identity gives \[ \sum_{\lambda_j>0}\frac{a_j}{1+\exp(\lambda_j d)}=0. \] If a positive-precision coefficient is nonzero, take its smallest precision $\lambda_*$ and multiply by $\exp(\lambda_*d)$. As $d\to+\infty$, its term tends to $a_*$ and all terms of larger precision tend to zero, a contradiction. Remove these coefficients successively. At most one zero precision remains, whose constant function is $1/2$, forcing its coefficient to vanish too. ∎
Theorem: For any declared set of $K$ distinct nonnegative precision types, there exist $K$ payoff gaps $d_1,\ldots,d_K\in(0,1)$ such that population choice probabilities in those games identify all common mixture weights. For any such invertible design matrix $A_{ij}=k_{\lambda_j}(d_i)$, estimates satisfying $\|\widehat p-p\|_2\le\varepsilon$ obey \[ \|A^{-1}\widehat p-w\|_2\le \frac{\varepsilon}{s_{\min}(A)}, \tag{2} \] where $s_{\min}(A)>0$ is its smallest singular value.
Proof: If the vectors $(k_{\lambda_1}(d),\ldots,k_{\lambda_K}(d))$ failed to span $\mathbb R^K$ as $d$ varies over $(0,1)$, a nonzero vector orthogonal to their span would contradict the lemma. Choose $K$ spanning evaluation vectors. The resulting matrix is invertible, and $p=Aw$ identifies $w$. Equation (2) follows from the operator norm of its inverse. The raw inverse estimate may leave the simplex; its Euclidean projection back onto the simplex cannot increase its distance to the true feasible vector, by the projection inequality. ∎
This is a population-identification result with a conditioning bound, not a claim that all gap designs are informative. Nearly identical precisions or badly chosen gaps can make $s_{\min}(A)$ very small. The design should be fixed using the declared types before comparing predictions, rather than chosen to interpolate the realized choices.

Why unrestricted heterogeneity changes the conclusion

Proposition: No finite battery of $m$ payoff gaps identifies every probability law $F$ on a nondegenerate compact interval $[0,L]$, $L>0$.
Proof: Select $m+2$ distinct precision values in that interval. Their column vectors of $m$ choice probabilities augmented by a row of ones lie in $\mathbb R^{m+1}$ and are linearly dependent. Thus a nonzero vector $a$ has zero sum and annihilates every observed choice row. Its positive and negative parts have the same strictly positive total mass. Normalize them separately to produce different probability distributions $F_+$ and $F_-$ with identical probabilities at all tested gaps. They are different finite-support laws, so identification fails even within this small subclass of the unrestricted model. ∎
For a declared finite type dictionary the sharp feasible set is $\{w\in\Delta_K:Aw=p\}$. The range of a new-game probability $c^{\mathsf T}w$ is obtained by two linear programs. If $c$ is in the row span of $A$ and the normalization row, that new prediction is identified even when the weights are not. Conversely, when the feasible set contains a vector positive on all dictionary types, a prediction outside that row span varies along a feasible null direction. Thus prediction can be better identified than the latent distribution.

From battery identification to an honest prediction test

Assume the finite dictionary also specifies, before evaluation, each type's full choice kernel in strategic games, including any QRE selector. Then identified weights generate an unambiguous predictive distribution. If its probabilities are bounded below by a fixed $0<\epsilon<1$ on the evaluation support, independent held-out groups give bounded log losses in $[0,-\log\epsilon]$. For a fixed fitted predictor, Hoeffding's inequality gives \[ \Pr\{|\widehat R-R_Q|>t\mid\text{training data}\} \le2\exp\!\left(-\frac{2nt^2}{(\log\epsilon)^2}\right). \tag{3} \] Here evaluation games and covariates are independently drawn from the declared target distribution $Q$, and actions follow their actual population law; no correctness of the fitted model is needed for this evaluation bound. It follows by applying the scalar bounded-variable inequality to the independent conditional losses. If one observes only several actions within each interacting group, that group, not each action, is the independent unit.

Remaining gap

The finite-support assumption, stable utility scale, and transport of precision from diagnostic to strategic games remain falsifiable restrictions. Cognitive hierarchy, context-dependent beliefs, and heterogeneous equilibrium selection can be observationally equivalent to parts of this battery and need separate diagnostic variation. The results therefore identify a particular shared-parameter model and its prediction set; they do not settle which bounded-rationality model best describes a population or authorize game-specific refitting.

Completion Estimate

51%

This is a post hoc, heuristic estimate for the full catalogue target. The attempt proves finite-battery identification and conditioning bounds for a declared logit-type dictionary, unrestricted-mixture nonidentification, and valid held-out log-score evaluation. Choosing and validating a transportable bounded-rationality model still requires strategic diagnostic variation, stable payoff scales and beliefs, heterogeneous equilibrium selection, and comparison with cognitive-hierarchy alternatives across new games.

References

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