Papers

We study identity testing for high-dimensional covariance matrices against dense alternatives of unknown direction, with $p/n \to \gamma$. Along a globally positive quadratic precision path, mixing Gaussian alternatives over a Gaussian Orthogonal Ensemble direction yields a contiguous experiment whose log likelihood reduces to the corrected Frobenius statistic; its upper-tail test attains the limiting weighted-power envelope at every fixed strength. Fixing the prior's Frobenius radius perturbs the mixture by only $O(p^{-1/2})$ in total variation, and exact whitening carries the experiment, the statistic, and its null law to any known null covariance. Separately, under a product-coordinate null, feasibility needs only $4+\eta$ moments, plus identical distributions over time when means are estimated; studentization and an exact degrees-of-freedom correction preserve the local power. A stability inequality turns near-envelope attainment into null agreement with the Frobenius rule, so uniform noninferiority on the typical dense bulk precludes gains at any contiguous alternative. For trace-matched rank-one alternatives, the corrected statistic is the first likelihood direction when $\vartheta_n \to 0$ and $n\vartheta_n \to \infty$; at fixed strength, the log likelihood ratio in the Onatski-Moreira-Hallin fixed-spike benchmark is governed by a richer linear spectral statistic below the Baik-Ben Arous-Peche threshold, while eigenvalue separation permits cost-free largest-eigenvalue enhancement above it. Simulations illustrate the theory.

Macroeconomic crises are rare, yet they can have large effects on asset prices. Because the probability of entering a crisis from normal times is small relative to the probability of recovery, the dynamics are nearly reducible: setting the crisis-entry probability to zero makes crisis states unreachable from normal states. We develop a graph-theoretic and spectral analysis of long-run pricing under reducible and nearly reducible dynamics. We show that, at the reducible limit, physical and pricing dynamics induce the same classes of economic states and accessibility relations, but rank their long-run importance differently: by recurrence and by class-specific pricing rates. Moreover, a claim's price depends only on its pricing corridor—the classes of states lying on directed paths from the initial state to states where the payoff is positive—and the highest class-specific pricing rate within this corridor determines the claim's long-run pricing rate. When the reducible-limit corridor excludes the globally pricing-dominant class, we characterize how restoring rare transitions introduces a contribution from that class with a weight that vanishes as the transitions become rarer. This contribution becomes dominant only beyond a crossover maturity, which increases with the rarity of the connecting paths and decreases with the dominant class's pricing-rate advantage. Finally, a consumption-based disaster-recovery application shows how preferences, consumption risk, cash-flow exposure, and recovery timing jointly shape the long-run pricing dominance of disaster states and crossover maturities.

The tails of macroeconomic outcomes can respond differently from the centre of their distribution: shocks with modest effects on median growth or inflation can shift downside growth or upside inflation risk. We develop a threshold stochastic-volatility-in-mean VAR with regime-dependent leverage to study their structural drivers. The model allows endogenous interactions between outcomes and volatility, contemporaneous level-volatility dependence, and regime-specific propagation. In nearly 150 years of U.S. data, predictive model selection supports three inflation-defined regimes. We identify business-cycle, financial, macroeconomic-uncertainty, and financial-uncertainty shocks and decompose their contributions to growth- and inflation-at-risk. The structural composition of tail risk differs from that of the predictive median. Business-cycle shocks dominate the median response of GNP growth but account for a substantially smaller share of growth-at-risk. Macroeconomic uncertainty makes a material contribution to both growth- and inflation-at-risk, with its share of growth-at-risk increasing with the magnitude of a positive macroeconomic-uncertainty impulse, despite its limited role at the median. In high-inflation states, the contribution of financial uncertainty to inflation-at-risk rises with the magnitude of positive financial-uncertainty impulses.

This paper develops a model of identifiable smooth ambiguity in a finite-state Anscombe-Aumann framework in which ambiguity perception and the ambiguity-attitude representation may depend on the act's endogenous certainty-equivalent utility level. The framework separates two sources of utility-level dependence that standard smooth-ambiguity models hold fixed. Neutral-transfer experiments within revealed cells recover conditional beliefs, while compensation experiments across cells recover cell weights and locally revealed ambiguity-attitude scales. These objects have distinct behavioral content. After the within-level structure is extended globally, ambiguity perception is identified at each certainty-equivalent level, whereas variation in admissible level-specific global attitude indices need not by itself reflect behaviorally identified variation in ambiguity attitude. Cross-level restrictions determine when the locally revealed attitude scales can instead be represented by one common global ambiguity-attitude index. The baseline characterization yields a doubly implicit smooth-ambiguity representation with level-specific perception and potentially level-specific global attitude indices. Requiring one common ambiguity-attitude index gives an intermediate stable-attitude model in which only perception may vary; imposing perception stability as well yields standard identifiable smooth ambiguity within the maintained full-support, neutrally separated, fixed-partition environment with at least three revealed cells. The contribution is therefore a behavioral decomposition of utility-level dependence in smooth ambiguity into separately revealed perception and attitude components.

Rule-based pricing tools are widespread in digital commerce, yet we know little about how their design shapes market outcomes. In a controlled market experiment, participants use dashboards to build pricing algorithms competing in a sequential Bertrand game over multiple periods. We vary design features commonly found in commercial repricing tools: warnings about price wars, pre-configured strategies, and advice from a large language model. Most treatment variations raise market prices with effects driven by an increase in starting prices and more cooperative algorithm designs. The results matter for competition policy, platform regulation and current discussions on regulating algorithm design tools.

This paper contains new 80-digit positive-weight Gauss-Hermite and positive-weight interior-node Gauss-Legendre quadrature rules for up to five dimensions and varying polynomial degree accuracy (depending on quadrature type and dimension). Some of these rules improve on the best available rules in the literature and some offer rules where none (other than the tensor product) existed. The results were produced by combining methodology developed by previous researchers with new approaches. The full precision rules themselves are at https://doi.org/10.5281/zenodo.22881864. Software in Julia, Python, and R providing the rules is available via a package registry and/or GitHub: Quadriceps.jl (both 64-bit and 128-bit; https://github.com/NittanyLion/Quadriceps.jl), quadriceps-py (64-bit only; https://github.com/NittanyLion/quadriceps-py), and quadriceps-r (64-bit only; https://github.com/NittanyLion/quadriceps-r). The software used to create these rules is available via PositiveWeightQuadratureSolvers.jl (https://github.com/NittanyLion/PositiveWeightQuadratureSolvers.jl). The replication package is at QuadricepsReplicationPackage.jl (https://github.com/NittanyLion/QuadricepsReplicationPackage.jl). A snapshot of all five packages is archived at https://doi.org/10.5281/zenodo.22883240.

Knowledge spillovers have largely been studied as unintended externalities diffusing through geographic proximity, while deliberate interregional networks have been examined mostly through co-inventor ties rather than the scientific layer carried by co-authored publications. This study asks whether spillovers from interregional AI-related co-publications increase regional AI patenting, and whether the ownership composition of regional R&D shapes the capacity to capitalize on them. Using a network-econometric approach across 26 Chinese provincial-level regions from 2006 to 2020, we find that co-publication spillovers foster AI patenting, but the returns concentrate almost entirely in regions where privately owned enterprises dominate R&D, reinforcing regional divergence.

Exponential Second-Order Expected Utility (SOEU) underlies the entropic approach to model uncertainty. This paper explores in what sense that functional form is essential. In the misspecification-robust Smooth Ambiguity criterion, let a single parameter govern both the model-level robustness and the ambiguity-averse aggregation across models Cerreia-Vioglio, Hansen, Maccheroni, and Marinacci (2026). The two-layer criterion then equals Exponential SOEU for every compact set of models and every second-order prior, with the Bayesian predictive measure as the baseline. The main results are converses. On the aggregator side, matched curvature is necessary: at a fixed curvature no other continuous, strictly increasing aggregator delivers the reduction, and with mismatched curvature there are a model set and a prior for which no single-layer entropic value, at any curvature and any baseline, reproduces the criterion. On the cost side, within the power-divergence family, which contains chi-squared and reverse Kullback–Leibler (KL), only KL has a dual of the log-sum-exp form, and the other members first depart from it at the third cumulant. Finally, the value dual to Exponential SOEU is a robust-control value: it is motivated by Rational Inattention, but no Bayes-plausible information-acquisition problem about a fixed act generates it.

Forecasters often score the same units per date against one standardized realized outcome. We show that every standardized forecast splits exactly into a component aligned with this common target and a component uncorrelated with it. Three consequences follow: forecast-error correlation largely mirrors forecast correlation and is therefore a poor measure of diversity; an equally weighted combination beats a no-information forecast only when average alignment is large relative to the combination's dispersion; and the gain from adding a forecaster separates into genuine improvement and mere dilution, which equal-weight admission can mistakenly reward. We develop a cautious selection rule, study it in simulations, and apply it to language-model forecasts of US equity rankings and mechanical signals ranking exchange-traded funds. Selection removes most dilution losses, but no combination beats the no-information forecast.

A collective rule must often choose a public outcome on a constrained boundary. We characterize truthful choice with diagonal unanimity on the entire boundary of any compact full-dimensional strictly convex body in finite dimension at least two. On the full domain of linear support preferences, ordinary individual strategy-proofness and diagonal unanimity force the rule to select one fixed participant's preferred point at every profile. The population is any fixed finite nonempty set. No Pareto condition, boundary smoothness, or continuity or measurability of the rule is assumed. If interior outcomes are allowed instead, two or more participants can use positive fixed weighted averages of preferred points for truthful non-dictatorial compromise on the same type domain. The proof derives regularity from the two agents' incentive inequalities, transports a common support-measure bound across reports, and makes every convexified attainable menu a Minkowski summand of the feasible body. Boundary generation then forces fixed control. The result extends the spherical public-good location question and identifies the role of boundary-valued choice in eliminating truthful compromise.

Causal inference guides operational and managerial decisions but remains challenging in high-dimensional panel or tensor settings, where decisions may depend on the joint conditional distribution of missing control outcomes. We develop Counterfactual Tucker Diffusion (\CFTDiff), which integrates the treatment mask and latent Tucker structure into conditional diffusion to recover this distribution given observed control outcomes through efficient nonlinear score learning in a low-dimensional core. The masked Tucker score preserves dependence across tensor modes while reducing the dimension of nonlinear score learning from the product of mode dimensions to the much smaller product of Tucker ranks. We establish high-probability error bounds for conditional score estimation that depend on the Tucker ranks, largest mode dimension, and the factor-strength-adjusted number of missing outcomes, and show how these bounds translate into recovery guaranties for the conditional distribution of the missing control outcomes. Across missing rates, simulations show more accurate point recovery than common causal panel and matrix/tensor completion methods; comparisons with nested diffusion specifications further demonstrate the gains from masked conditioning and Tucker dimension reduction. In Norway's iFlex experiment, \CFTDiff recovers missing outcomes more accurately than competing methods; when applied to causal analysis, its estimated conditional distributions yield counterfactual prediction intervals and target-attainment probabilities, allowing pricing interventions to be evaluated by demand-reduction magnitude and reliability.

Marketers now deploy generative AI agents as synthetic consumers to pretest visual assets such as logos, packaging, and advertising at a fraction of human-panel cost. However, this procedure assumes that a model seeing a visual cue can also perceive its consumer meaning, which is largely untested. We stress-test the assumption using six canonical visual marketing experiments, varying the two levers managers control: model generation (GPT-4o-mini vs. GPT-5.4-mini) and input format (plain text vs. JSON). Every resulting configuration passed the manipulation checks; however, none of the configurations reproduced more than two of the six human effects, and the remainder were nonsignificant. The one exception was a significant reversal of the human pattern. Providing conceptual or empirical evidence through in-context learning steers average responses toward the human effect. Yet steering has a limit: even when it succeeds, a configuration reproduces less than half of the natural spread of human responses and so understates consumer heterogeneity. We integrate these results into an AI governance protocol (Calibrate, Intervene, Deploy) that delineates when synthetic consumers can responsibly screen creatives and when human panels remain necessary.

This paper studies agency problems when both parties worry that the model linking action to output is misspecified. With observable actions, an optimal contract is linear in output, so performance pay arises solely to share misspecification exposure, the slope reflects the parties' relative robustness concerns, and its allocation is Pareto efficient. With hidden actions, this sharing rule survives and incentives add a nonlinear correction. Misspecification concerns can polarize effort by making intermediate actions impossible to implement. Moreover, ambiguity across competing models has asymmetric effects: uncertainty about desired actions raises the principal's payoff, whereas uncertainty about deviations can lower it.

Consumers can invest resources to understand and avoid their behavioral mistakes, and their incentives to do so depend on the market consequences of remaining naive. We incorporate this feedback between consumers' cognitive states and market outcomes into a general contracting model. Firms face a trade-off between the depth and reach of exploitation: deeper exploitation raises profit from a naive consumer but induces greater cognitive investment, promoting sophistication and shrinking the exploitable consumer base. This trade-off disciplines exploitation and can cause policies that benefit consumers when cognition is fixed to backfire when cognition is endogenous.

We study how timestamps affect dynamic disclosure. At a random date, a sender privately obtains hard evidence about an evolving binary state and chooses when to disclose it. With timestamps, the unique equilibrium features immediate good-evidence disclosure and timestamp-dependent bad-evidence delay. Without timestamps, high priors relative to impatience generate a stock of undisclosed good evidence, which is stochastically purged before permanent transparency. High impatience yields immediate good-evidence disclosure, while bad evidence can be arbitrarily delayed and disclosed in bursts. Timestamps prevent pretending old good evidence is fresh and certify bad evidence is old, accelerating good-evidence disclosure and facilitating bad-evidence disclosure.

Institutions often face constraints richer than a single capacity. Protecting the priority of a student who cannot be accommodated may then block a lower-priority student who fits. We introduce COP-and-Choose, a cumulative offer process followed by a choice step, and show that its greedy version, which skips such blockers, leaves no student worse off than any fair matching or Knapsack Deferred Acceptance (KDA). Greedy is the unique consistent rule preserving this guarantee. With substitutability and size monotonicity at every school, greedy is strategy-proof and Pareto-undominated among strategy-proof mechanisms. If either fails at a school, one additional capacity-one school yields a market where, under every proposal order, greedy is manipulable and no strategy-proof mechanism weakly Pareto dominates it. In a three-resource refugee-resettlement calibration with simulated preferences, 19–33 percent of families gain relative to KDA and none lose, while 21–83 percent strongly envy another family.

We study strategic disclosure of the action space in principal-agent contracting, where an agent selects a disclosed action set to shape the principal's perception of her capabilities before contract design. Unaware of the strategic disclosure, the principal designs a revenue-optimal contract as if the disclosed action set were complete and accurate. We consider two variants distinguished by cost verifiability. When costs are unverifiable, the agent can extract the entire first-best surplus, leaving the principal with zero revenue. When costs are verifiable, we characterize the agent's optimal disclosure strategy in binary-outcome settings and, more generally, when the principal is restricted to linear contracts, reducing the agent's problem to a two-variable convex optimization problem. We prove that the agent can secure utility of at least a $1/e$ fraction of the first-best surplus, which also yields a $1/e$ welfare guarantee under optimal disclosure. While the principal's revenue can be arbitrarily small compared to the first-best surplus, when the ratio of maximum to minimum expected reward among non-null base actions is at most $L$, we establish a revenue guarantee of $\Theta(1/\log L)$ relative to the first-best surplus. We also compare utilities and welfare under strategic disclosure with their counterparts in the canonical model. Finally, we extend the agent's $1/e$ utility guarantee to general outcome spaces without restricting the principal to linear contracts. Our results show how strategic action-space disclosure changes the distribution of surplus while preserving a constant-factor welfare guarantee under the agent's optimal disclosure.

Standard valuation methods, including discounted cash flow, the income approach standard IDW S 1 of the Institute of Public Auditors in Germany, and market multiples, compress milestone probabilities, continuation options, and risk shifts into opaque aggregate parameters; none provides a structured protocol for decomposing AI integration into auditable option-level assumptions. We propose an industry-agnostic taxonomy separating AI Integrators from AI Providers. AI Integrators are further classified by their Integration Depth Level, ranging from no integration to AI at the core of the product or process. A milestone-gated real-options overlay decomposes milestone state value into five components, and an Analytic Hierarchy Process-based Success Readiness Index derives per-option probabilities from structured pairwise comparisons for scenario analysis. Applied to an AI-native energy software-as-a-service firm, the framework yields a coherent valuation band traceable to identifiable option-level assumptions. Risk concentrates in later-stage continuation options, matching the structural prediction for AI Providers. The protocol applies across the firm lifecycle, including mergers and acquisitions due diligence. The case is a single-firm demonstration of protocol coherence, not empirical validation; multi-case testing against realised post-exit valuations is left to future research.

This paper provides a theoretical framework to evaluate the trade-off between the self-regulated peer review system and independent government inspection (PCAOB) in the auditing profession. We model the peer review system as a Judgment Regime, where a stakeholder utilizes professional expertise, captured as a private signal, to make ex-post decisions on verifying audit failures. In contrast, PCAOB inspection is modeled as a Commitment Regime, where the stakeholder lacks private information but can commit ex-ante to a predetermined level of verification. We find that the Judgment Regime benefits from a resource-allocation effect and a deterrence effect driven by informed verification, whereas the Commitment Regime deters audit failures through the first-mover advantage of ex-ante commitment. Our analysis demonstrates that the stakeholder prefers the peer review system if and only if the private signal is sufficiently informative. Furthermore, comparative statics reveal that higher verification costs or stronger audit incentives shift the stakeholder's preference toward PCAOB inspection.

We develop a stochastic nested fixed point (SNFP) estimator for random coefficients logit demand models that updates model parameters using stochastic gradients and performs demand inversion one market at a time. Relative to the conventional nested fixed point (NFP) estimator, SNFP substantially reduces memory requirements and computational cost, making estimation feasible in very large datasets. We establish the large-$T$ (number of markets) asymptotic properties of the estimator under regularity conditions. We also characterize the effect of sharing one block of simulation draws across markets and show how to correct for it. Monte Carlo simulations show that the SNFP estimator achieves statistical accuracy comparable to the NFP estimator, and in our benchmark a single online pass estimates a model with 100 million markets in about 5.5 hours. An empirical application using scanner data further demonstrates the practical advantages of SNFP for large-scale demand estimation.

Organizations invest heavily in internal mechanisms such as incentive systems, governance structures, performance measurement, analytics, and repeated reorganizations. Yet realized performance gains are often weak or short-lived, particularly in large and mature organizations. Standard explanations emphasize allocative inefficiency arising from incentive misalignment, information problems, or bounded rationality. While relevant, these explanations struggle to account for environments in which effort is high, optimization capabilities are sophisticated, and aggregate performance remains far below apparent potential. This paper develops a theory of structural inefficiency inside organizations. The central mechanism is the presence of internal externalities: unpriced cross-effects of organizational mechanisms acting simultaneously on multiple performance objectives. When such effects are mixed in sign, organizational effort is partially canceled internally, creating a structural ceiling on realized performance. The paper introduces a simple representation of organizational primitives and key performance indicators and derives a realized-performance law in which internal conflict enters as a multiplicative discount factor. Under weak and realistic conditions, structural inefficiency dominates allocative inefficiency, explaining why optimization often fails before organizational coherence is restored.

Many structural and dynamic economic models imply that key parameters are identified by conditional quantile restrictions. Building on the exponential-weighting approach of Bierens (1990) and recent advances in penalized maximum statistics for conditional moment restrictions (Chen et al., 2025), we develop a unified inference framework for such parameters. We propose an adaptive $\ell_1$-penalized supremum statistic that transforms the conditional restriction into a continuum of unconditional moment conditions and aggregates evidence across quantile indices. The penalty regularizes the maximization over the weighting direction. Under the stated uniformity conditions, the known-parameter adaptive selector has no lower maximin local power than the unpenalized test and yields a strict maximin local-power gain whenever some positive candidate penalty has a strictly larger population maximin criterion than the zero penalty. We extend the theory to settings with pre-estimated nuisance parameters, characterizing the additional terms induced by the plug-in step in the limiting process. We derive an analytically corrected variance estimator that accounts for plug-in estimation uncertainty and establish the validity of a Gaussian multiplier bootstrap under the null and sequences of local alternatives. Monte Carlo simulations show that the proposed CvM-KS aggregation scheme has rejection rates relatively close to nominal size under pre-estimation in the linear design and approaches the nominal level in nonlinear designs as the sample size grows. The reported power comparisons between the adaptive and unpenalized procedures are design-specific, with higher empirical rejection frequencies for the adaptive procedure along some directions.

An expert may misreport her forecast to influence a decision she cares about. We study contracts that elicit both her belief about a verifiable state and her stake in a binary action. If every belief is possible at two distinct stakes, exact incentive compatibility prevents a belief-only mechanism from letting the action depend on the report. Asking about both belief and stake allows the action to respond, but may require implementation rent. For threshold rules, we show that minimum rent equals the cutoff's positive curvature weighted by the belief distribution. The principal therefore chooses a cutoff by balancing virtual surplus against the implementation rent of truthful reporting. We characterize the best cutoff rule and give sufficient conditions under which it is optimal even among randomized mechanisms.

While corporate sustainability mandates are expanding, the systemic reliance on self-reported emissions data exposes financial markets to pervasive greenwashing. Current literature relies heavily on subjective ESG ratings or textual sentiment analysis, leaving a critical econometric gap in objectively quantifying physical climate realities. To resolve this information asymmetry, we fuse U.S. SEC financial fundamentals with facility-level EPA greenhouse gas registries to establish a mathematically guaranteed baseline of physical corporate emissions. Leveraging a gradient boosting architecture and Mondrian Conformal Prediction, we quantify the shortfall between self-reported data and this algorithmic baseline into a novel Conformal-Weighted Continuous Divergence (CWCD) metric. Evaluating this divergence via a cross-sectional lead-lag econometric design, we uncover a robust mechanism of market discipline: algorithmic emissions divergence exhibits a severe, statistically significant negative relationship with subsequent market valuation (Tobin's Q) and operational profitability (ROA). Providing definitive evidence against the market blindness hypothesis, this study proves that institutional capital actively prices environmental deception not merely as an ethical lapse, but as a leading indicator of fundamental corporate mismanagement. Ultimately, these findings provide the quantitative justification necessary for asset managers and regulators to deploy algorithmic auditing infrastructure at scale.

Day-ahead electricity price forecasts support trading and storage decisions, but for battery arbitrage predicting intraday price spreads is more relevant than predicting individual hourly prices. Here we show that a temporal hierarchy forecasting (THieF) framework that jointly reconciles forecasts of hourly electricity prices and all intraday price spreads consistently improves performance across two major European electricity markets and three different forecasting architectures. Using five years of out-of-sample data from Germany and Spain, we obtain accuracy improvements of up to 19.7% and profit gains of up to 10.4% relative to unreconciled hourly price forecasts. The gains persist even for a highly accurate pretrained TabPFN foundation model. Our results demonstrate that exploiting coherent relationships between economically relevant forecasting targets can improve both predictive accuracy and decision value, and that better statistical forecasts do not necessarily imply better economic decisions.

We provide sufficient conditions for the consistency of penalized least squares procedures that select the order (dimension) of a regression model from a sequence of nested classes, allowing for dependent, martingale-difference errors. The main contribution is to relax the classical identifiability requirement: parameters indexing classes larger than the true order need not be identified, provided the additional, excess directions admit a linear approximation to the truth in a neighbourhood of the true parameter. This relaxation lets the number of candidate models grow with the sample size, removing the usual need for a fixed upper bound. We verify the resulting high-level conditions for two classes of nonlinear regression models used in applied work: multiple-regime smooth transition regression and mixture-of-experts models with generic generalized linear model experts. Under a BIC-type penalty, the resulting order selection rule is consistent in either model class whenever the number of candidate models grows slower than the logarithm of the sample size.

We study randomized experiments involving two interacting populations, such as buyers and sellers in a marketplace. In the two-sided experiments we consider, we randomize the two populations separately and independently. For a pair consisting of one member from each population, the two assignments jointly determine one of four exposure conditions. Under a local interference assumption, we consider a broad class of linear estimands, including total, interaction, and buyer- and seller-side spillover effects. Our first main result establishes that researchers can estimate these effects using ordinary least squares and conduct asymptotically valid design-based inference using the conventional two-way cluster-robust variance estimator, clustered at the buyers' and sellers' levels. Our second main result develops a sharper variance estimator for a single linear estimand that better preserves dependence within the buyer and seller dimensions and is asymptotically less conservative than the two-way clustered estimator and existing alternatives. Our third main result establishes the theory for covariate adjustment and recommends a two-way analysis-of-variance-type covariate representation to ensure efficiency gains.

A government may ask for a high tax because it faces a genuine expense. The same demand can also be made by a government that intends to keep the proceeds. We study this ambiguity in a two-period reputation model with privately observed spending needs and an endogenous tax base. The opportunist chooses between outright confiscation and levies that an honest government might impose. We characterize the three possible regimes through a single equilibrium equation. A mean-preserving spread of legitimate needs weakly raises the opportunist's lifetime value, with a strict increase precisely when the spread changes the upper tail relevant for mimicry. We then give a necessary and sufficient curvature condition for this ordering to extend to a general reputational continuation prize. Verification reduces the value of concealment, but also changes how the opportunist extracts. In the benchmark economy, greater auditing weakly lowers current citizen welfare, even before audit costs, while improving the future allocation through better information. The welfare case for verification consequently depends on the balance between these two effects. Examples yield no auditing, an optimum within a regime, and an optimum at the boundary between regimes.

This paper studies social learning when buying access to previous decisions also determines whose decision enters the record. Individuals first receive private information and then decide whether to pay for access. A positive price selects relatively weak private beliefs. With one founding observation, every buyer copies it, and the record acquires no further information. With a richer initial history, buyers can disagree and the record can improve; nevertheless, a price bounded away from zero prevents complete learning, even with unbounded private beliefs. We then consider a rule under which a buyer makes a preliminary choice before access and a revised choice afterward. A positive probability of implementing the preliminary choice makes it follow the private signal. Retaining both choices restores complete learning under a fixed effective fee or pooled myopic pricing, including with bounded beliefs. The rule reduces willingness to pay and sacrifices some current decision accuracy. A worked example shows how the subsequent improvement in information can outweigh both initial losses. The results show why preserving a judgment made before exposure can change what later users learn from the same selected population.

We study the design of priority pricing systems with heterogeneous agents in environments in which improving quality for some agents reduces the average quality that can be provided. Contrary to the equity-efficiency tradeoff emphasized in public debates, we show that under economically natural conditions priority pricing can Pareto-improve on an equal-allocation benchmark. Three priority tiers suffice for such an improvement, combining higher quality for a fee, lower quality with compensation, and an intermediate tier at the benchmark quality; two tiers are never enough. Our results provide a framework for overcoming equity-efficiency tensions in applications such as lane pricing, waiting-line design, public provision, and insurance.