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Econ-279 · economics · Economics

Mechanism design for nonstandard agents

LLM status Open unverified

Statement

Original atlas ID: M8. Formulation: editorial specification of a research family.

Definitions and assumptions

Let $\tau=(\theta,\alpha)$ be a participant type, where $\theta$ indexes a declared welfare-relevant preference and $\alpha$ indexes cognitive or behavioral characteristics. Types have law $F$ on a specified space. Mechanism $m\in\mathfrak M$ presents a finite decision protocol with outcomes $x\in X$ and payments; context $z$ has known law $P_Z$. A behavioral response model $p\in\mathcal P$ supplies a joint outcome law $p(\mathrm dx\mid m,z,\tau)$, allowing errors and dependence among participants. A welfare evaluator $w_\theta(x)$ and complexity cost $c_\alpha(m,z,x)$ are bounded and fixed before optimization. Feasibility and the chosen participation condition define $\mathfrak M_F$. Neither $w_\theta$ nor complexity cost is automatically identified from inconsistent observed choices.

Formal research question

Using data from randomized protocols and a specified response-model confidence or identification set $\mathcal P(D)$, characterize mechanisms attaining or approximating \[ \sup_{m\in\mathfrak M_F}\inf_{p\in\mathcal P(D)} \mathbb E_{F,P_Z,p}\!\left[w_\theta(x)-c_\alpha(m,z,x)\right]. \] Determine which experimental variation identifies responses to mechanisms outside the observed menu, and how much institutional simplification improves the objective. When participants optimally choose cognitive effort, model that effort as a strategic action and require an equilibrium selector; when behavior is an empirical policy, state its invariance restrictions instead. Examine welfare sensitivity to alternative defensible evaluators, not just sensitivity to behavioral parameters, and report outcomes common to all admissible specifications.

Interpretation and scope

This question requires both positive behavioral prediction and a normative objective. A participant's displayed choice can be an error relative to a stable preference, or it can reveal a context-sensitive preference; those interpretations imply different welfare comparisons. Complexity costs must specify who pays them and how they enter the evaluation. A redesign can change attention, beliefs, or sophistication, so holding fitted behavioral error probabilities fixed across protocols is a substantive assumption. Obvious strategy-proofness is a particular mechanism property with an interpretation for cognitively limited agents; it does not by itself identify welfare or eliminate every misunderstanding. A useful study randomizes protocol complexity and measures understanding, participation, and outcomes alongside choices.

Source audit and status

Glazer--Rubinstein (1998) is verified as Motives and Implementation: On the Design of Mechanisms to Elicit Opinions; its central variation concerns experts' motives, not a general cognitive-error model. Li (2017) and de Clippel (2014) are verified as Obviously Strategy-Proof Mechanisms and Behavioral Implementation. They supply formal partial answers under their own restrictions. The source atlas overstates the direct cognitive relevance of the first citation and leaves the welfare criterion unspecified. The robust welfare-response program above is editorial, with normative assumptions exposed; it is not an assertion that all behavioral implementation or simple-mechanism design remains unresolved.

References

  1. S1 Jacob Glazer and Ariel Rubinstein (1998). Motives and Implementation: On the Design of Mechanisms to Elicit Opinions. Journal of Economic Theory 79(2), 157--173. https://www.sciencedirect.com/science/article/abs/pii/S0022053197923851. Locator: abstract; opinion-elicitation model. Establishes: Dependence of implementation on experts' motives. Does not establish: a general model of cognitive error or complexity cost.

  2. S2 Shengwu Li (2017). Obviously Strategy-Proof Mechanisms. American Economic Review 107(11), 3257--3287. https://pubs.aeaweb.org/doi/10.1257/aer.20160425. Locator: abstract; obvious-dominance definition. Establishes: A formal strategy-proofness criterion with a behavioral interpretation. Does not establish: welfare optimality for every boundedly rational population.

  3. S3 Geoffroy de Clippel (2014). Behavioral Implementation. American Economic Review 104(10), 2975--3002. https://www.aeaweb.org/articles?id=10.1257/aer.104.10.2975. Locator: abstract; choice-based implementation framework. Establishes: Implementation analysis when context-independent preference maximization is relaxed. Does not establish: a unique normative welfare criterion for inconsistent choices.

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-279-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-279

Mechanism design for nonstandard agents Cognitive effort, robust rewards, and protocol experiments

Question and scope

The broad problem is to choose a feasible interaction protocol that maximizes normative welfare when observed choices may depend on cognitive limitations and context. Neither revealed preference nor strategic truthfulness, by itself, determines that welfare. This attempt solves a reward-design problem with endogenous cognitive effort, and gives a finite experimental reduction for comparing protocols. These are restricted results; they do not establish implementation for arbitrary behavioral agents.

A protocol with endogenous effort

An agent must produce a correct answer to a task. A protocol has two known parameters: automatic success probability $q\in[0,1)$ and the productivity $a\in(0,1-q]$ of effort. Effort $e\in[0,1]$ raises success probability to $q+ae$ and incurs real cognitive cost $\kappa e^2/2$. A correct answer earns a monetary reward $s\ge0$. The agent's utility is expected reward less cognitive cost. Outside utility is zero; rewards are funded externally and agents can decline participation. The designer values a correct answer at $B>0$, includes the cognitive cost, and charges public funds at shadow price $\lambda\ge0$. Thus monetary rewards are not added a second time to social benefit. This is a declared normative convention, not an inference from behavior.

The cost coefficient is unknown and belongs to $[\underline\kappa,\overline\kappa]$, with $0<\underline\kappa\le\overline\kappa$. A reward is fixed before the coefficient is known. Restrict rewards to $0\le s\le\bar s$, where $\bar s\le\underline\kappa/a$. This explicit bound ensures that the effort constraint never binds except possibly at its upper endpoint.

Theorem (Exact robust reward): Put $c=1/2+\lambda$. The agent's unique optimal effort is $e(s,\kappa)=as/\kappa$ and participation is individually rational. A maximizer of worst-case normative welfare is \[ s^*={\operatorname{clip}}_{[0,\bar s]}\left(\frac{a^2B-\lambda q\overline\kappa}{2ca^2}\right). \] Its worst-case welfare equals \[ Bq-\lambda q s^*+ \frac{a^2}{\overline\kappa}\{Bs^*-c(s^*)^2\}. \] In particular, a strictly positive reward is optimal if and only if $\bar s>0$ and $a^2B>\lambda q\overline\kappa$.
Proof: The agent maximizes $s(q+ae)-\kappa e^2/2$, a strictly concave quadratic. Its unconstrained maximizer $as/\kappa$ belongs to $[0,1]$ by the reward bound. Utility there is $sq+a^2s^2/(2\kappa)\ge0$. Substitution into the designer's criterion gives \[ W(s,\kappa)=Bq-\lambda qs+\frac{a^2}{\kappa}(Bs-cs^2). \] For $0\le s\le B/c$, the bracket is nonnegative, so the worst coefficient is $\overline\kappa$. For $s>B/c$, the bracket is negative and $W(s,\kappa)\le Bq=W(0,\kappa)$; such rewards cannot improve the optimum. On $[0,B/c]$ the robust criterion is a strictly concave quadratic whose stationary point is the displayed expression before clipping. That point is at most $B/(2c)$, including when $q=0$ or $\lambda=0$. If it is negative, zero is optimal; otherwise clipping to $\bar s$ cannot move it beyond $B/(2c)$. Hence it also maximizes over the originally admissible interval. Its derivative at zero is $a^2B/\overline\kappa-\lambda q$, proving the last assertion. ∎

This calculation identifies a specific cost of incentives: rewards paid to automatically correct agents consume funds without increasing their effort. It is therefore insufficient to compare protocols solely by their maximal attainable accuracy. For each protocol $m$ with parameters $(q_m,a_m,\underline\kappa_m,\overline\kappa_m,\bar s_m)$, the theorem computes a robust score. Maximizing these finitely many scores solves protocol selection in this model. Different complexity parameters change the effort equilibrium rather than merely subtracting a fixed complexity penalty.

Proposition (Strict preference for a simpler protocol): Suppose $\lambda>0$. A protocol $S$ with $a_S^2B\le\lambda q_S\overline\kappa_S$ has optimal robust value $Bq_S$. A protocol $C$ with $q_C<q_S$ has value at most \[ Bq_C+\frac{a_C^2B^2}{4c\overline\kappa_C}. \] Consequently $S$ is strictly preferred whenever $B(q_S-q_C)>a_C^2B^2/(4c\overline\kappa_C)$, even if $q_C+a_C>q_S+a_S$.
Proof: The first statement is the zero-reward case of the theorem. For the second, discard the nonpositive term $-\lambda q_Cs$ and maximize $Bs-cs^2$ over all $s\ge0$, obtaining $B^2/(4c)$. Rewards outside its nonnegative region never improve on zero, as already proved. The strict inequality then compares the two bounds. The final possibility is consistent: choose $q_C<q_S$, $a_C$ large enough for the accuracy comparison, and then choose $\overline\kappa_C$ sufficiently large. ∎

What protocol experiments can identify

For a separate finite-data formulation, let $m=1,\ldots,M$ index complete protocols, and $j=1,\ldots,J$ index observed pretreatment strata (including measured welfare-relevant type and context). Their population probabilities $f_j$ are known. Outcomes $x$ belong to a finite set. The normative net value $g_{mj}(x)$ is known and lies in $[-G,G]$. Randomized protocol experiments identify the conditional probabilities $p_{mj}(x)$ on their support; a model for unmeasured normative type would be an additional assumption.

Let $\mathcal P$ be a nonempty compact polytope of such probability arrays, specified by probability constraints and linear moment restrictions. It may link responses across protocols; randomization across protocols is announced only after nature selects this one common array. A protocol lottery $z\in\Delta_M$ has worst-case value \[ V(z)=\min_{p\in\mathcal P}\sum_{m,j,x}z_m f_jp_{mj}(x)g_{mj}(x). \] This lottery is feasible only if the original participation and resource constraints hold for each included protocol or are separately imposed on $z$.

Theorem (Finite robust experimental reduction): If $p^1,\ldots,p^K$ are the vertices of $\mathcal P$ and $A_{mk}=\sum_{j,x}f_jp^k_{mj}(x)g_{mj}(x)$, the optimal lottery value is the attained value of the linear program \[ \max_{z,v}v\quad\text{subject to }z\in\Delta_M, \quad v\le\sum_m z_m A_{mk}\quad(k=1,\ldots,K). \] If $|A_{mk}-\widehat A_{mk}|\le\eta$ for every $m,k$, an optimal lottery for $\widehat A$ loses at most $2\eta$ in true robust value.
Proof: For fixed $z$, the criterion is linear in $p$. Every element of a compact polytope is a convex combination of its vertices, so its smallest value is attained at a vertex. This gives exactly the displayed constraints. The resulting minimum of finitely many continuous functions is continuous on the compact simplex, hence has a maximizer. Uniform entrywise perturbation by $\eta$ perturbs every lottery's value against each vertex by at most $\eta$, and therefore its minimum by at most $\eta$. Comparing the estimated maximizer to a true maximizer twice gives the $2\eta$ bound. ∎

The common ambiguity set matters. With two protocols and two admissible response models, a value matrix $A=\left(\begin{smallmatrix}1&0\\0&1\end{smallmatrix}\right)$ gives deterministic robust value zero and lottery value $1/2$. This can be realized by two binary-success protocols: one response model succeeds only under the first, the other only under the second. If nature may independently choose a different response law after observing the realized lottery draw, both protocols instead have worst-case value zero. These are different maintained assumptions about stability of behavior.

Remaining gap and source scope

The effort theorem assumes the agent correctly optimizes a specified perceived reward-cost objective, and that the designer's $B$ and cognitive cost are normatively justified. It leaves open misspecified beliefs, multidimensional types, equilibrium selection among interacting agents, and extrapolation to unseen protocols. The polytope reduction is finite-dimensional optimization, not identification of its restrictions from sparse experiments. In particular, unexplored protocols require response restrictions or new experiments.

Li's obvious strategy-proofness concerns strategic reasoning, and de Clippel studies implementation when choices need not maximize context-independent preferences. They motivate distinct aspects of the agenda, but neither is being cited as proving the reward formula or the experimental ambiguity restrictions used here. Publisher abstracts and bibliographic metadata were checked; no claim about an uninspected full-text theorem is made.

Completion Estimate

40%

This is a post hoc, heuristic estimate for the full catalogue target. The attempt solves robust rewards with endogenous cognitive effort and reduces finite experimentally restricted protocol lotteries to an attained optimization with perturbation guarantees. General behavioral mechanism design still lacks justified welfare evaluators, response identification for unseen protocols, interacting-agent selection and robustness to misspecified beliefs and multidimensional types.

References

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