we-solve (beta)

Econ-575 · economics · Economics

Human-aware model training

LLM status Open unverified

Statement

Original subject group: Innovation and AI economics

Audit classification: Resolved or misstated baseline. The paper exists, with Kevin A. Bryan as coauthor. The original proposed heterogeneity extension overlaps analyses already in Sections 4 and 5; only the additional robust/identification specification is new and its open status is unverified.

Definitions and precise research target

Let economic states $\omega$ follow a specified distribution $P$, and let a model trained with parameter $\theta\in\Theta$ generate signal $S_\theta$ according to a known signal experiment. Users have types $v$ drawn from distribution $F$, where type determines verification cost $k_v$, available corrective information and action set $\mathcal A_v$. Let $u_v(a,\omega)$ be a specified bounded economic payoff. After observing the signal, a user may verify, override or accept the suggested action. Let $a_v^\ast$ and $e_v^\ast\in\{0,1\}$ be the realized action and verification indicator under an optimal contingent response policy after observing $S_\theta$, with actions allowed to depend on verification results and ties resolved by a stated rule. Training incurs real cost $C(\theta)$. The proposed training objective is \[ \max_{\theta\in\Theta}E_Q[u_v(a_v^\ast,\omega)-k_v e_v^\ast]-C(\theta). \] The joint law $Q$ has the specified state and user-type marginals; all expectations integrate states, model randomness and user types. Extend this objective to a declared class $\mathcal Q$ of plausible state and user distributions where $\mathcal Q$ is a given finite set of candidate distributions, by maximizing its minimum over $\mathcal Q$, subject to a stated harm constraint. Determine when economically optimal training differs from minimizing average prediction error, and which user-response data identify the relevant payoff and verification functions. Existing work already solves stylized human-aware training problems; heterogeneous users are also analyzed in that work; the additional distribution-robust identification specification here is an editorial extension whose unresolved status is not independently verified.

Definitions, assumptions and mathematical qualifications. Let finite state, user-type and action sets have joint law $Q$; independence of state and type is imposed only when desired. A training choice $\theta$ specifies signal kernel $K_\theta(s\mid\omega,v)$ and verification kernel $K_v^{\rm ver}(h\mid\omega,s)$. A user policy maps its observed type and signal to verification and then maps the available information to an action. Define $V_Q(\theta)$ as maximum expected bounded payoff less verification cost over these policies. For a nonempty finite uncertainty set $\mathcal Q$, robust training is $\sup_{\theta\in\Theta}\min_{Q\in\mathcal Q}\{V_Q(\theta)-C(\theta)\}$, with $E_Q[h(a,\omega)]\leq\bar h$ for all $Q$. Finite $\Theta$, or compact $\Theta$ with suitable continuity, is required for an attained maximum. Specify whether users know $Q$, learn it, or optimize their own worst-case criterion; those produce different $V_Q$.

Variants and assumption changes

  1. Sourced solved baseline. Bryan and Gans analyze a coverage--accuracy frontier and human continuation decisions, including heterogeneous agents and teams. These ingredients alone must not be relabelled unresolved.
  2. Editorial extension. Unknown response primitives are estimated from randomized signal/verification data; optimize distribution-robust training with a uniform harm constraint and quantify policy regret over compatible user models.

References and source audit

Checked source. arXiv:2608.12538v1, submitted 12 August 2026, Sections 2--5 and 6. The paper analyzes optimal training, heterogeneous users, verification, monopoly and teams. Full primary HTML retrieved. https://arxiv.org/html/2608.12538v1.
  1. Kevin A. Bryan; Joshua S. Gans. 2026. Training AI for When Humans Will Use It. arXiv:2608.12538v1, submitted 12 August 2026, Sections 2--5 and 6. https://arxiv.org/html/2608.12538v1. DOI: https://doi.org/10.48550/arXiv.2608.12538.

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-575-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-575

Human-aware model training: Safe robust training under finite response margins

User behavior is part of the objective

For finite states, user types, signals, verification outcomes, and actions, let $K_\theta$ be a trained signal experiment and let users know their type and the true joint state/type law $q$. They may verify at a stated cost before acting. The target is \[ \max_{\theta\in\Theta}\min_{q\in\mathcal Q} \{V_q(\theta)-C(\theta)\}, \qquad H_q(\theta)\le\overline h\quad(q\in\mathcal Q), \tag{1} \] where $H$ is harm under the selected utility-maximizing user response. Bryan and Gans already analyze heterogeneous users and several extensions [bg, Sections 4--5]. Here the additional task is robust training when the state/type law is uncertain and safety must hold for the induced response. We establish a finite-menu guarantee that deals explicitly with response changes. All kernels, payoffs, costs, and tie-breaking rules are known; uncertainty below concerns only $q$.

Reducing verification to policies

A deterministic contingent policy specifies, for every type and signal, either a decision to decline verification with one action, or a decision to verify with an action for every verification outcome. Unused branches are omitted, so they cannot create duplicate policies and artificial zero response margins. There are finitely many such policies, denoted $\Pi_\theta$. Type-specific policies can be combined into one mapping; this does not let a user observe other users' types. For a state/type pair $s$ let $f_{\theta\pi}(s)$ be its expected payoff net of verification cost, integrating signal and verification randomness, and let $g_{\theta\pi}(s)$ be its expected harm. Assume \[ |f_{\theta\pi}(s)|\le B, \qquad 0\le g_{\theta\pi}(s)\le H. \tag{2} \] Maximizing ex ante over these policies implements posterior optimality at every positive-probability information set, because the decisions at different information sets have independent feasibility constraints. Randomization cannot improve the maximum of their expected utilities. Fix a total priority order on policies to resolve any maximizing ties. Thus \[ V_q(\theta)=\max_{\pi\in\Pi_\theta}q^{\mathsf T}f_{\theta\pi}, \qquad H_q(\theta)=q^{\mathsf T}g_{\theta,\pi(q,\theta)}. \tag{3} \] For a finite training menu and finite nonempty uncertainty set, enumeration of (3) solves (1) whenever its feasible set is nonempty. No continuity or equilibrium-existence assertion is needed for this finite calculation.

A margin certificate for estimated environments

Let $q_1,\ldots,q_m$ be the true candidate laws and $\widehat q_1,\ldots,\widehat q_m$ approximations satisfying $\|q_j-\widehat q_j\|_1\le\eta$. For each $\theta,j$ let $\widehat\pi_{\theta j}$ maximize estimated utility. Define its margin as its utility advantage over the best other deterministic policy; the margin is $+\infty$ when only one policy exists. Let $\mathcal M$ be the subset of training choices for which every estimated margin is strictly greater than $2B\eta$. Define the certified set \[ \mathcal F_c=\{\theta\in\mathcal M: \widehat q_j^{\mathsf T}g_{\theta,\widehat\pi_{\theta j}} \le\overline h-H\eta\ \text{for every }j\}. \tag{4} \]
Theorem: Suppose $\Theta$ is finite and $\mathcal F_c$ is nonempty. Choose $\widehat\theta$ maximizing $\widehat W(\theta)=\min_j V_{\widehat q_j}(\theta)-C(\theta)$ over $\mathcal F_c$. Then $\widehat\theta$ is feasible for every true law. For every $\theta^*\in\mathcal M$ whose true harms satisfy $H_{q_j}(\theta^*)\le\overline h-2H\eta$ for all $j$, \[ \min_j V_{q_j}(\widehat\theta)-C(\widehat\theta) \ge \min_j V_{q_j}(\theta^*)-C(\theta^*)-2B\eta. \tag{5} \]
Proof: For every policy, (2) implies $|q_j^{\mathsf T}f-\widehat q_j^{\mathsf T}f|\le B\eta$ and $|q_j^{\mathsf T}g-\widehat q_j^{\mathsf T}g|\le H\eta$. A difference between two policy utilities therefore changes by at most $2B\eta$. The strict estimated margin guarantees that the estimated maximizer is also the unique true maximizer. This holds for every $\theta\in\mathcal M$ and every $j$, so (4) implies true feasibility. It also shows that the stipulated comparator belongs to $\mathcal F_c$: its estimated harm is at most $\overline h-H\eta$. Taking maxima over policies preserves the uniform utility bound $|V_{q_j}-V_{\widehat q_j}|\le B\eta$; taking the minimum over $j$ preserves it again. Therefore $W(\widehat\theta)\ge\widehat W(\widehat\theta)-B\eta \ge\widehat W(\theta^*)-B\eta\ge W(\theta^*)-2B\eta$, which proves (5). ∎
The theorem compares with choices that have both a behavioral margin and a safety margin. It does not silently compare with every feasible optimum. Policies that differ only at zero-probability events can create zero margins even when realized behavior is unique. One may either retain the conservative certificate above or first quotient policies that agree under every candidate law, using the same equivalence for harm.

Why value accuracy alone does not certify safety

Example: There are two states $\omega=0,1$, no signals, no useful verification, and actions safe and risky. Safe utility and harm are zero. Risky utility is $2\omega-1$ and risky harm is one in both states. Write $q(\omega=1)=p$ and select risky at a utility tie. Then \[ V_p=\max(0,2p-1),\qquad H_p=\mathbf1\{p\ge1/2\}. \] For arbitrarily small $\varepsilon>0$, the laws $p=1/2-\varepsilon$ and $p=1/2+\varepsilon$ have arbitrarily close values and distributions but harms differing by one. This proves that a small distribution or value-estimation error cannot give a uniformly small harm error without a condition controlling the selected response.
The same issue explains why compact training parameters and continuous utility primitives alone are insufficient for a constrained maximum: selected harm can jump at a response tie. The finite-menu theorem avoids an attainment claim outside its assumptions.

Sharper uncertainty calculations and the remaining gap

For a fixed experiment and a nonempty compact polytope of possible laws, the worst utility value can also be computed by a linear program: minimize $t$ subject to $q$ belonging to that polytope and $t\ge q^{\mathsf T}f_{\theta\pi}$ for every policy. This is exact because at the optimum $t$ equals the maximum in (3); compactness and finite continuous policy values ensure attainment. Selected harm requires additional policy-region constraints and the declared tie rule, so the utility LP alone is not a safety calculation.

The unresolved part is a data-grounded identified set for verification costs, heterogeneous utilities, and response kernels, together with useful safety guarantees near genuine behavioral ties. Randomized signals can help estimate those objects, but cannot replace the measurement and incentive assumptions. The certificate gives an implementable restricted answer and specifies exactly which comparators its regret guarantee covers.

Completion Estimate

55%

This is a post hoc, heuristic estimate for the full catalogue target. The attempt reduces finite robust training to contingent user policies and proves safety and regret guarantees under behavioral and harm margins, including a discontinuity counterexample. The requested identification of heterogeneous verification costs, utilities and response kernels remains missing, as do useful guarantees near response ties and broader training choices.

References

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