Willful ignorance in legal contexts: A mechanism design approach

Dong, Xiaoge Working Paper Willful ignorance in legal contexts: A mechanism design approach Center for Mathematical Economics Working Papers, No. 752 Provided in Cooperation with: Center for Mathematical Economics (IMW), Bielefeld University Suggested Citation: Dong, Xiaoge (2025) : Willful ignorance in legal contexts: A mechanism design approach, Center for Mathematical Economics Working Papers, No. 752, Bielefeld University, Center for Mathematical Economics (IMW), Bielefeld, https://nbn-resolving.de/urn:nbn:de:0070-pub-30070577 This Version is available at: https://hdl.handle.net/10419/333505 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ 752 September 2025 Willful Ignorance in Legal Contexts: A Mechanism Design Approach Xiaoge Dong Center for Mathematical Economics (IMW) Bielefeld University Universit¨atsstraße 25 D-33615 Bielefeld ·Germany e-mail: [email protected] uni-bielefeld.de/zwe/imw/research/working-papers ISSN: 0931-6558 Unless otherwise noted, this work is licensed under a Creative Commons Attribution 4.0 International (CC BY) license. Further information: https://creativecommons.org/licenses/by/4.0/deed.en https://creativecommons.org/licenses/by/4.0/legalcode.en Willful Ignorance in Legal Contexts: A Mechanism Design Approach Xiaoge Dong∗ September 24, 2025 Abstract We study “willful ignorance” - choosing not to learn whether a task is illegal - in a lawmaker-principal-agent game and characterize the penalty policies that implement welfare-maximizing behavior. The model delivers an implementability frontier: which equilibrium behaviors can exist and be selected by penalties. With perfect inquiry, this frontier is aligned with the welfare ordering, so the lawmaker can make the welfare-maximizing behavior both exist and be preferred by all parties. With imperfect inquiry, noise breaks that alignment and produces two failures: inquiry that is socially desirable may be infeasible at any penalty, and inquiry that is socially undesirable may persist because it cannot be switched off. We compare harm-based, compliance-based, and dual-penalty rules: harm-based rules preserve control but tightens feasibility; compliance-based rules relax feasibility but sacrifices control; dual penalty rules recover both levers subject to simple bounds. The framework yields practical guidance for calibrating penalties to harm, inquiry accuracy, and inquiry costs. It also implies that ignorance cannot serve as a shield: the absence of knowing crime in equilibrium is driven by incentives rather than morality, making non-inquiry the true strategic margin of liability design. JEL Codes: K14, K42, D82, D86, H23. Keywords: willful ignorance, ostrich instruction, law and economics, asymmetric information. ∗Zeppelin Universität Friedrichshafen, Fallenbrunnen 3, 88045 Germany. I thank Niels Boissonnet, Yves Breitmoser, Herbert David, Manuel Förster, Ruveyda Gozen, Martina Miotto, Frank Riedel, and Gerald Willmann for valuable comments and suggestions. Earlier versions of this paper were presented at the BiGSEM colloquium at Bielefeld University, and I thank the participants for their helpful feedback. Financial support from the DFG through the project RUTHLESS is gratefully acknowledged. All remaining errors are my own. 1 1 Introduction Willful ignorance - also called “willful blindness” or “deliberate ignorance” - arises when a defendant intentionally avoids learning facts that would trigger legal liability (Kirfel and Hannikainen 2023). Since United States v. Jewell (1976), courts have widely adopted the willful-ignorance doctrine, treating failure to inquire as tantamount to knowledge (Charlow 1991; A. Sarch 2018; Hellman 2009; Luban 1998). Yet debate persists about its foundations and the proper design of penalties; definitions vary across jurisdictions and enforcement practice is uneven. Simons (2021) urges caution until clearer, broadly accepted standards emerge. Against this background, the policy question we take up is: how should penalties be structured so that equilibrium behavior aligns with social welfare when actors may strategically avoid information? We study willful ignorance in a lawmaker-principal-agent game. The lawmaker sets a penalty rule ex ante. Nature then draws the principal’s type (good or bad). The principal’s type determines the kind of task he offers: a good principal offers a legal task, while a bad principal offers an illegal one. The principal strategically proposes a transfer for the task to the agent to induce performance, anticipating the agent’s responses and the legal rule. The agent decides whether to inquire into the legality of the task (at a cost) and whether to perform it; if an illegal task is performed, a penalty is imposed. This framework applies directly to compliance domains such as anti-money-laundering due diligence, export controls, and product-safety regulation, where inquiry duties are central and penalties vary in structure. Our objective is to characterize the “implementability frontier”-which equilibrium behaviors can both exist and be selected by penalties-and to map that frontier into welfare and policy guidance. Our main results can be summarized as follows. 1) With perfect inquiry, the lawmaker can always implement the welfare-maximizing behavior. Penalties rule out uninformed action; whether the market remains active then depends only on whether the good type’s surplus covers the cost of inquiry. If so, the agent performs only legal tasks; if not, the market shuts down. In this setting, the agent never knowingly performs an illegal task, so transfers cannot signal types and no separating equilibrium arises. Penalties determine market composition, while inquiry cost determines whether valuable activity survives-an alignment that guarantees welfare optimization. The absence of knowing crime is thus not evidence of higher morality but an endogenous outcome of incentives: all harmful conduct flows through deliberate ignorance, underscoring that ignorance cannot serve as a shield in liability design. 2) With imperfect inquiry, welfare and implementability can diverge. Noise generates false positives and negatives, lowering the value of screening and producing two failures: (i) inquiry may be welfare-maximizing but infeasible, if good types cannot bear inquiry costs plus residual risk; (ii) screening may persist even when shut-down would be better, since penalties cannot fully switch it off. How false positives are treated becomes pivotal. Under harm-based rules, agents remain liable after inquiry, preserving leverage to deter screening when it is welfare dominated, but this can also make desirable screening infeasible. Under compliance-based rules, meeting the inquiry standard eliminates residual risk, sustaining screening, but removes the lawmaker’s off-switch. A dual-penalty scheme sets one penalty for non-inquiry and another for post-inquiry exposure, combining the advantages of both approaches. Equalizing the two recovers harm-based rules; setting the post-inquiry penalty to zero recovers compliance-based rules. Dual penalties thus restore control, allowing the lawmaker both to shut off screening when exclusion is optimal and 2 to sustain it when inquiry is welfare-maximizing. Relation to the literature. This paper contributes to the legal debate on willful ignorance and the “ostrich instruction” by providing a tractable economic model that complements normative and jurisprudential analyses. Additionally, it relates to the literature on strategic ignorance in economics, but differs by focusing on penalty design and implementability rather than on social preferences or self-image. In the end, it speaks to the optimal-deterrence tradition in law and economics, extending standard prescriptions to environments where ignorance itself is the strategic margin. A full discussion of related work appears in Section 2. Contribution. This paper makes three contributions. First, it develops the first tractable lawmaker-principal-agent framework for willful ignorance, embedding the inquiry decision in a mechanism-design setting where penalties are chosen ex ante. In doing so, it formalizes the doctrine that deliberate ignorance may be treated like knowledge (the “ostrich instruction”) and shows how this affects implementability and welfare. Second, it characterizes the lawmaker’s implementability frontier, identifying when the welfare-maximizing behavior can be both sustained and selected, and when feasibility and desirability diverge. This extends the literature on strategic ignorance by showing how liability rules-harm-based, compliance-based, and dual-map into equilibrium outcomes. Third, it translates the analysis into operational guidance: calibrate penalties directly to the cutoff conditions that sustain the socially desirable behavior; use harm-based rules when inquiry cannot be verified; reserve compliance-based exemptions for environments with auditable inquiry and tolerance for persistent screening; and deploy dual penalties where flexibility is needed to deter non-inquiry while keeping screening feasible. When enforcement absorbs real resources, implement the target behavior with the minimal expected penalty and avoid on-path sanctions. Together, these contributions extend the optimal-deterrence tradition to environments where ignorance itself is the strategic margin, and offer a practical menu for legal design. Roadmap. Section 2 reviews the literature. Section 3 presents the model. Section 4 develops the results for perfect and imperfect inquiry and compares liability designs. Section 5 discusses applications in legal practice. Section 6 collects extensions. Section 7 concludes. All proofs are presented in the Appendix. 2 Related Literature The willful-ignorance doctrine addresses a practical tactic: defendants who strategically avoid learning incriminating facts. Since United States v. Jewell, 532 F.2d 697 (9th Cir. 1976), courts have permitted juries to treat deliberate ignorance as knowledge-the socalled “ostrich instruction.” Legal scholarship has debated the legitimacy of this doctrine (Sarch 2014; Simons 2021; Hellman 2009; Luban 1998), but existing analyses remain largely normative. Our model provides the first tractable mechanism-design framework to analyze willful ignorance as a policy instrument, showing how treating ignorance as culpable affects implementability and welfare, and thereby extending the jurisprudential debate into a formal economic analysis. 3 Closest to our analysis is Yaffe (2018), who offers a normative defense of willful ignorance and shows that failure to inquire indicates disregard for others’ interests. Yaffe’s model builds in social-preference concerns, treating ignorance as culpable because it reflects deficient regard for others. Our approach differs in primitives and question. We keep players fully rational and self-interested, and show that willful ignorance arises as an equilibrium response to incentives—transfers, penalties, and inquiry costs—rather than from diminished concern for others. This shift allows us to characterize implementability and to analyze how liability rules map into welfare. Social concern enters in an extension in Section 6, as a parameter that substitutes for legal penalties under pooling. Empirical work speaks to perceptions rather than design. Kirfel and Hannikainen (2023) show that willfully ignorant actors are judged more antisocial than unsuspecting ones but less than knowing violators (see also Alter et al. 2007). These findings inform the legitimacy and likely acceptance of ostrich instructions, but they are orthogonal to the penalty-design problem we study. Our analysis takes incentives as primary and, in extensions, allows for non-material inquiry burdens and social concern as distinct welfare primitives. Our paper is also related to economic models of strategic ignorance. Kartik et al. (2007) develop a theoretical model of motivated information avoidance, where agents prefer to remain uninformed in order to preserve plausible deniability in communication. Grossman and Van Der Weele (2017) study willful ignorance in social decisions, showing that individuals avoid information about the consequences of their actions to protect their self-image. Related experimental work by Dana et al. (2007) demonstrates how principals deliberately remain ignorant about payoffs to excuse exploitative choices. These papers highlight the behavioral logic of ignorance but do not study optimal penalty design. We differ in focusing squarely on legal enforcement: the lawmaker sets penalties ex ante, and ignorance is treated as a strategic margin in equilibrium. This mechanism-design perspective allows us to derive implementability conditions and to compare alternative liability rules, which is absent in the existing information-acquisition literature. Finally, our enforcement results connect to the optimal-deterrence tradition (see Polinsky and Shavell 2000). When severity is resource costly, welfare favors the minimal penalty that implements the desired behavior; when enforcement intensity is costly, welfare favors the minimal intensity. Under imperfect inquiry, these prescriptions interact with residual legal risk after inquiry, explaining why parity can render inquiry infeasible when it is desirable, why exemption can make inquiry hard to switch off when exclusion is better, and how dual penalties can restore control by separating the levers for uninformed action and post-inquiry exposure. 3 Model The game. We study a market environment with three parties: the lawmaker (L), the principal (P), and the agent (A). The lawmaker drafts and announces a penalty rule: if an illegal task is performed, the agent will be convicted and penalized at level T∈R+. Nature (N) then draws the principal’s type ω∈ {G, B}, with Pr(G) = θ∈(0,1) common knowledge.1The principal privately observes ωand offers the agent a corresponding task 1. We assume 0< θ < 1, so that both good and bad types lie in the support of beliefs. This captures the doctrinal idea of “substantial suspicion”: the agent attaches positive probability to illegality, so that non-inquiry is a strategic choice rather than innocent ignorance. Legal scholarship diverges on 4 with transfer X∈R+. By a slight abuse of notation, we use G(resp. B) to denote both the good (resp. bad) principal and the legal (resp. illegal) task he offers. The agent does not directly observe legality but can conduct an inquiry at cost k > 0. We denote her belief that the task is legal by µ∈[0,1]. After observing Xand T, she chooses two strategies in sequence: i) an inquiry strategy σi A(X)∈ {i, ni}, where idenotes inquiry and ni denotes no inquiry; and ii) an action strategy σa A(X, ·)∈ {a, na}, where adenotes acceptance of the task and na non-acceptance. In the baseline game, belief is fully determined by the information structure: without inquiry, µ=θ, reflecting the prior probability that the task is legal; with perfect inquiry, µ∈ {0,1}, as the inquiry reveals type with certainty. Timing. The sequence of moves is summarized in Figure 1. t= 0 t= 1 t= 2 t= 3 t= 4 Lchooses T;N draws ω∈ {G, B} Plearns T, ω; offers Xto A Achooses inquiry σi A∈ {i, ni} If i, she learns ω; then chooses action σa A∈ {a, na} If illegal and accepted, conviction at penalty T Figure 1: Timing of the game Notes: The figure summarizes the sequence of moves in the lawmaker-principal-agent game. At t= 0, the lawmaker sets the penalty Tand nature chooses the principal’s type. At t= 1, the principal offers a task; at t= 2, the agent decides whether to inquire; at t= 3, the agent decides whether to perform the task; and at t= 4, conviction occurs if the task is illegal. Payoffs. If the legal task is performed, the good principal gets yG−Xand the agent gets X−k·1{i}. If the illegal task is performed and illegal, the bad principal gets yB−Xand the agent gets X−T−k·1{i}; society bears harm H. If the task is not performed, both principal and agent get 0, and the agent pays the inquiry cost iff she inquired. Transfers and penalties are pure redistributions between principal and agent. As reference, we formalize the interaction in the game illustrated in Figure A.1, which we refer to as the LPA game. This game models offenses involving inculpatory propositions of the form: “...where the underlying action would not be independently wrongful absent the defendant’s knowledge of the inculpatory proposition.” The delegated task-such as transporting a substance or developing software-is not inherently illicit but becomes so when performed with knowledge that the substance is contraband or the software facilitates money laundering. We assume perfect detection of illegal acts, but only the agent bears the legal consequences. We relax these assumptions in section 6. how demanding this threshold should be. Some argue that any positive probability suffices to ground culpability (e.g. Luban 1998; Hellman 2009), while others emphasize awareness of a high probability of wrongdoing as the correct doctrinal test (e.g. Charlow 1991; A. F. Sarch 2014; Simons 2021). Our baseline collapses the threshold to its minimal value, effectively s= 0. One could alternatively introduce a suspicion threshold s∈(0,1) and treat ignorance as culpable only if 1−˜ θ≥sgiven the agent’s subjective belief ˜ θ. A higher senlarges the range in which ignorance is excused, narrowing implementability, but the core logic of penalty design remains unchanged. 5 Equilibrium concept. Given a fixed penalty T, we analyze the continuation game induced by the lawmaker’s move. We refer to any perfect Bayesian equilibrium of this subgame as a continuation equilibrium. Such an equilibrium satisfies sequential rationality and Bayesian consistency. In the analysis that follows, we assume that if the agent receives an off-equilibrium offer X′, she infers it comes from a bad type principal, refrains from inquiry, and accepts the task only if the legal penalty is fully offset. For off-equilibrium offers X′, we denote her belief by ˆµ. Formally, this implies: ˆµ(G|X′) = 0, σi A(X′) = ni, and σa A(X′, ni) = a⇐⇒ X′≥T. This specification of off-path beliefs maximizes the equilibrium set and is adopted without loss of generality.2More optimistic conventions (ˆµ > 0) shrink the existence windows but do not alter the qualitative welfare rankings, so our main results are robust to nearby belief specifications. We adopt standard tie-breaking in favor of the preceding mover: indifferent agents accept; indifferent principals offer the smallest transfer that induces acceptance. Alternative tie-breaking rules shift only knife-edge boundaries and do not affect the welfare rankings reported below. Given a continuation equilibrium, we classify the structure of play into four types: i. Pooling: Both types of principals offer the same transfer; the agent does not inquire and performs the task. Beliefs are not updated. Formally put: XB=XG,σi A(·) = ni,σa A(·) = a. ii. Semi-pooling: Both types offer the same transfer; the agent inquires before deciding. Beliefs are updated. Formally put: XB=XG,σi A(·) = i,σa A(·)∈ {a, na}. Because the key feature of this equilibrium is that the agent pays the inquiry cost and conditions her action on the inquiry result, we will refer to this equilibrium type as screening in the analysis that follows. “Semi-pooling” and “screening” are thus interchangeable terms in what follows. iii. Separating: Types separate via transfer; the agent does not inquire and decides based on the offer. Beliefs are not updated. Formally put: XB> XG,σi A(·) = ni, σa A(·)∈ {a, na}. iv. Inactive: Both types offer below-penalty transfer; the agent does not inquire and refuses the task. Beliefs are not updated. Formally put: XB, XG∈[0, T),σi A(·) = ni,σa A(·) = na. Nontrivial mixed-strategy equilibria do not exist in our environment; equilibrium behavior is exhausted by the pure types we analyze.3 Social welfare and benchmarks. We use a utilitarian welfare measure: the sum of expected payoffs across players minus expected social harm. We include the bad principal’s payoff with a normative weight τ∈[0,1] to capture different policy views: τ near 0fits predatory or cross-border crimes where the offender’s gain is not valued; τnear 1fits local externalities (e.g., pollution abatement failures) where the principal’s output still counts. Once a behavior is fixed, transfers and penalties are pure redistributions and do not affect SW (penalties are resource-costless; see Section 6 for costly enforcement). 2. Formal bounds for general ˆµ∈[0,1] are provided in Appendix C.2. 3. See Appendix C.3 for a formal argument. 6 We will compare our results to two benchmarks: (i) the Perfect Information Equilibrium (PIE), where the principal’s type is common knowledge; and (ii) the No Inquiry Equilibrium (NIE), where type is unknown and players never inquire. Lemma 3.1 (Benchmark welfare under PIE and NIE).Under PIE, SWPIE =(θ yG+ (1 −θ)(τyB−H),if τyB−H≥0, θ yG,if τyB−H < 0, and under NIE, SWNIE =(θ yG+ (1 −θ)(τyB−H),if θyG+ (1 −θ)(τyB−H)≥0, 0,otherwise. Moreover, SWNIE ≤SWPIE, with strict inequality whenever τyB−H < 0. Proof. All proofs are relegated to the Appendix. Intuition. If the bad type’s net contribution is nonnegative (τyB−H≥0), both PIE and NIE feature both types, so welfare coincides. If it is negative, PIE can exclude the bad type while NIE shuts the market down (welfare 0< θyG= SWPIE). A full proof appears in Appendix B.1. Implementation objective. Let E={pooling,semi-pooling,inactive}denote the set of equilibrium behaviors, and write E(T)for the behaviors sustained under penalty T. For each e∈ E, let SW(e)denote the associated level of social welfare. We call an equilibrium behavior socially desirable if it maximizes welfare, e⋆∈arg max e∈E SW(e). The lawmaker chooses Tto implement the socially desirable behavior, i.e. to ensure that e⋆∈ E(T). When E(T)contains multiple behaviors, we adopt a minimal implementation convention: whenever possible, we pick Tin the interior of the penalty region that sustains e⋆to secure uniqueness; if uniqueness cannot be forced, we credit e⋆as implemented only when it is uniformly preferred (Pareto-dominant) among coexisting behaviors (every player weakly prefers e⋆to any alternative and at least one player strictly prefers it). This convention is not an additional instrument for the lawmaker, but merely a tie-breaking device for exposition. When Pareto dominance does not hold, we adopt the conservative stance and treat the desirable behavior as non-implementable. 4 Analysis Equilibrium preliminaries. Two observations will be used throughout. First, with perfect inquiry (α= 1), no equilibrium separates in transfers: if XB=XG, the agent would skip inquiry, infer type from the offer, and the bad type would profitably mimic the good type. Second, whenever an illegal task is performed with positive probability on the path, it occurs only without inquiry (willful ignorance): if the agent inquired and still performed the illegal task, she would also perform the legal one after inquiry, so inquiry 7 feasibility by limiting residual exposure after inquiry. Proposition 4.4 (Imperfect inquiry with dual penalties).Consider imperfect inquiry and a dual-penalty rule differentiating willful ignorance and knowing crime. The lawmaker can implement the socially desirable continuation behavior in all cases, except when inquiry is desirable but the good type cannot finance the expected inquiry cost. Relative to single-penalty regimes, dual penalties weakly expand implementability and weakly raise the maximal attainable welfare; welfare remains (weakly) below the perfectinquiry benchmark. Intuition. With two levers the lawmaker separates the two margins. The no-inquiry penalty Tnprices uninformed action: raising Tnmakes acceptance without inquiry expensive and can always shut down pooling; lowering Tnpreserves it when desired. The after-inquiry penalty Tiprices the agent’s residual legal exposure from false positives; lowering Tireduces the expected transfer she needs to be willing to inquire, and raising Timakes inquiry unattractive. Under a semi-pooling offer (both types post the same transfer and the agent inquires), her minimal per-performance payment equals tsemi =k+(1−θ)(1−α)Ti φ,where φ:= θα + (1 −θ)(1 −α) is the probability that inquiry leads to performance. The good principal pays tsemi only when his task is actually performed (probability α), so semi-pooling is privately feasible iff yG≥tsemi, i.e. φyG≥k+ (1 −θ)(1 −α)Ti. By setting Tias low as needed (down to 0), the lawmaker makes feasibility easiest; thus inquiry is implementable iff φyG> k. Conversely, if inquiry is not desired, choosing Tilarge pushes tsemi > yGso no one funds screening; Tnthen selects between pooling and inactivity. Hence dual penalties can always “switch screening on or off” except in the single obstruction φyG≤k. Because (Tn, Ti) nest harm-based (Ti=Tn)and compliance-based (Ti= 0) rules, the maximal welfare attainable with dual penalties weakly dominates what either single-penalty regime can achieve, and remains (weakly) below the perfect-inquiry benchmark when α < 1. Numerical Example (dual-penalty). Under dual penalties, the realized welfare at a given equilibrium is as under the other regimes (transfers and penalties cancel in welfare); the gain comes from which equilibrium is implementable and selected. The lawmaker calibrates penalties by setting Tnhigh enough to deter uninformed action while keeping Tilow enough to maintain inquiry feasibility. 4.3 Calibration and comparative analysis under imperfect inquiry Relative to the perfect-inquiry benchmark, the placement of penalties changes in two ways. First, the lower bound that makes inquiry attractive rises as accuracy falls: with noisier signals, the agent requires higher compensation to cover her expected exposure to penalties when she investigates. Formally, the indifference cutoff k/[θ(1 −θ)] is replaced by L(α) = k θ(1−θ)(2α−1), α ∈(1/2,1), so L′(α)<0: higher accuracy reduces the penalty needed to trigger inquiry. Second, screening is now subject to a feasibility constraint, U(α) = φ(α)yG−k (1−θ)(1−α), φ(α) = θα + (1 −θ)(1 −α), 14 0.0 0.5 1.0 1.5 2.0 −4 −3 −2 −1 0 1 Net social harm (τ * yB − H) Social welfare Perfect Information Equilibrium No Inquiry Equilibrium Perfect Inquiry Imperfect Inquiry (dual penalty) Figure 5: Social welfare under dual-penalty imperfect inquiry Notes: Parameters are θ= 0.4,yG= 4,k= 0.5,α= 0.8, and τ= 1. The x-axis plots net social harm, τyB−H; the y-axis plots social welfare. Curves compare Perfect Information Equilibrium (PIE), No Inquiry Equilibrium (NIE), Perfect Inquiry (PIq), and Imperfect Inquiry (dual-penalty), where the lawmaker can set separate penalties for action without inquiry and for post-inquiry violations. For τyB−H≥0, pooling is efficient and all curves coincide. As net harm turns negative, PIq switches from pooling to screening. The dual-penalty curve also selects screening where feasible but lies weakly below PIq because noisy inquiry both lets some illegal tasks slip through and blocks some legal ones. When τyB−His sufficiently negative, screening ceases to be profitable under noise and the dual-penalty curve falls to zero (inactive), while PIq can still sustain positive welfare by screening. so that if accuracy is too low or inquiry costs too high, this constraint binds and the inquiry window collapses even when welfare would favor screening. By contrast, the cutoff for acceptance without inquiry, T≤yG/(1 −θ), and the requirement that Tbe at least as large as the bad type’s output yBremain the same as under perfect inquiry. These thresholds deliver transparent comparative statics. A higher αenlarges the inquiry window (lower L, higher U), making screening easier to implement. A higher k raises Land lowers U, squeezing the inquiry window and pushing the economy toward pooling or inactivity. Greater θor higher yGexpand feasibility by relaxing the acceptance cap and raising U, while a larger yBmakes deterrence harder under harm-based rules and raises the cutoff for inactivity. Net harm (τyB−H)affects only welfare rankings, not feasibility. Across regimes, the calibration logic is unified. With a single penalty, the lawmaker calibrates Tdirectly into the range that sustains the socially desirable outcome: raising it to the indifference cutoff when inquiry must be induced, or lowering it to the feasibility bound when inquiry must be deterred. Under harm-based rules, inquiry can always be deterred by raising Tbut not always sustained when desirable because the feasibility cap may bind. Under compliance-based rules, both problems appear: if φ(α)yG≤k, inquiry is desirable but infeasible; conversely, once uninformed action is deterred and φ(α)yG> k, inquiry persists for all higher T, so exclusion cannot be restored by penalties alone. Dual penalties separate the margins: Tnregulates inquiry versus no-inquiry, while Tiprices residual false-positive risk. This flexibility allows the lawmaker to sustain inquiry when15 ever φ(α)yG> k and to switch it off when exclusion is optimal, though at the practical cost of specifying and enforcing two distinct penalties. All explicit cutoff expressions and comparative statics are collected in Appendix B.4. 5 Applications The preceding analysis derived welfare comparisons and illustrated them with diagrams based on stylized parameter values. These figures already showed how different penalties map into pooling, screening, or exclusion, and how welfare rankings depend on θ,α,k, yG, and yB. What remains is to connect these predictions to real-world settings. This section illustrates the model with legal cases where variation in parameters such as harm H, economic contribution τ, or inquiry accuracy αplayed a decisive role. Environmental enforcement: Reserve Mining (1975) vs. HF Sinclair Navajo (2025) Reserve Mining (D. Minn. 1974; 8th Cir. 1975). Reserve Mining discharged taconite tailings into Lake Superior, releasing asbestos-like fibers into the drinking water of Duluth and nearby communities. The district court, after reviewing epidemiological evidence, concluded that the cancer and respiratory risks were intolerable even under uncertainty, and ordered an immediate shutdown, stressing that “human health must come first.”5On appeal, the Eighth Circuit softened this stance, holding that “no harm to the public health has been shown to have occurred to this date and the danger to health is not imminent.”6Instead of closure, the company was ordered to prepare onland disposal facilities at an estimated cost of $243-300 million (around $1.3-1.6 billion in 2024 dollars).7At the time, toxicological studies placed ingestion risks only slightly above random inference (α≈0.55-0.6), and monitoring costs kwere high because long-term sampling campaigns required millions in expenditure.8Regional economic contribution was substantial: 3,050 direct jobs and 12,000 dependent jobs.9 This sequence illustrates willful ignorance: the company had internal warnings but avoided commissioning comprehensive studies, relying on scientific uncertainty to continue operations. The district court prioritized harm Has catastrophic and imminent, while the appellate court emphasized economic contribution τand downgraded Hto “not imminent.” With αlow and khigh, the feasibility of semi-pooling (screening) was tenuous. In the model, this would place inquiry near the boundary of feasibility; in practice, the courts tolerated continued operation in the short run but coupled it with phased compliance obligations. This resembles present-period pooling combined with an order that compels transformation into compliance in subsequent periods. 5. United States v. Reserve Mining Co., 380 F. Supp. 11, 26-28 (D. Minn. 1974). 6. Reserve Mining Co. v. EPA, 514 F.2d 492, 538 (8th Cir. 1975). EPA = Environmental Protection Agency. 7. Reserve Mining Co. v. Herbst, 262 N.W.2d 596, 605 (Minn. 1977) (“over $300 million” for on-land disposal); 514 F.2d at 505 ($243m estimate for Milepost 7 project). 8. See Gerald Markowitz & David Rosner, Deceit and Denial: The Deadly Politics of Industrial Pollution (2002), ch. 6. 9. TIME, Oct. 22, 1973, “Environment: Crisis in Silver Bay.” 16 HF Sinclair Navajo (D.N.M. 2025). In 2025, HF Sinclair Navajo Refining entered a consent decree requiring a $35 million civil penalty and $137 million in injunctive compliance investments, including flare gas recovery, wastewater upgrades, and advanced monitoring through Continuous Emissions Monitoring Systems (CEMS) and Leak Detection and Repair (LDAR).10 Management had received repeated violation notices but delayed installing available monitoring technology until forced by enforcement. Here, inquiry was feasible but deliberately avoided. Modern monitoring precision was high (α≈0.9), with EPA quality assurance protocols for CEMS targeting relative accuracy within 10-20%.11 Inquiry costs kwere moderate, with EPA manuals estimating CEMS operation and maintenance at $13,000-27,000 per unit-year and LDAR programs costing tens of thousands annually.12 Sectoral compliance was high (θ≈0.8-0.9), as most U.S. refineries had adopted CEMS/LDAR by 2020.13 The harm Hwas well documented: benzene and formaldehyde exposure from refinery emissions carry significant cancer and respiratory risks according to EPA’s National Air Toxics Assessment.14 At these values, harm-based screening was feasible, and compliance-based rules predict that once penalties eliminate the no-inquiry branch, inquiry persists. The scale of the decree easily cleared this threshold. In static terms, the outcome resembles semi-pooling; in practice, the consent decree translated this into a dynamic remedy, combining short-run tolerance with mandated transition into full compliance. A final nuance concerns the difference between our one-shot model and real-world remedies. In the model, the bad type is either tolerated (pooling) or excluded (inactivity). By contrast, environmental enforcement often allows continued operation conditional on transformation—firms are ordered to invest in monitoring and technology that make future violations impossible. This can be understood as the dynamic counterpart of our framework: in the first period, the bad type is punished but tolerated; to remain in the market in subsequent periods, it must transform into a compliant good type. Consent decrees and mitigation orders therefore function as exclusion followed by conditional re-entry, an extension that underscores how our static analysis maps onto repeated regulatory practice.15 5.1 Empirical predictions The case studies illustrate how judicial outcomes align with the model’s logic, but the framework also yields testable comparative statics that go beyond single disputes. These 10. U.S. Department of Justice (DOJ), Press Release, Jan. 17, 2025; EPA Settlement Summary, Apr. 25, 2025, United States v. HF Sinclair Navajo Refining LLC (D.N.M. 2025). EPA = Environmental Protection Agency. 11. EPA, Clean Air Markets: Continuous Emission Monitoring Systems (CEMS), technical guidance. 12. EPA, Technical Support Document for CEMS Costs, 2016 (O&M $13-27k per monitor-year); EPA, Leak Detection and Repair: A Best Practices Guide, 2007. 13. EPA, Air Facility System Compliance Reports (2022). 14. U.S. EPA, National Air Toxics Assessment (2018 cycle). 15. See, e.g., U.S. EPA, Consent Decree: Cummins Inc. (1998, amended 2006) (requiring phased engine monitoring and compliance upgrades over multiple years), available at https://19january2021snapshot. epa.gov/sites/static/files/2013-09/documents/cumminscd.pdf(Last visited: 21st Sep 2025); Joseph A. Hester, “Consent Decrees as Emergent Environmental Law,” 85 Mo. L. Rev. 2020 (discussing how consent decrees often substitute for statutory precision by imposing phased compliance obligations); see also U.S. EPA, Consent Decree: The Williams Companies Inc. (2023) (requiring compliance certification and monitoring within 180 days), available at https://www.epa.gov/system/files/documents/2023-04/ thewilliamscompaniesinc-cd.pdf. (Last visited: 21st Sep 2025) 17 predictions can be taken to data in various legal areas. First, the model predicts that inquiry is more likely when compliant actors are common (θhigh), legal surplus is large (yG), monitoring is accurate (αhigh), and costs are low (k). Sectors with higher baseline compliance and cheaper monitoring technologies should therefore display higher rates of inquiry. Second, liability design leaves distinct empirical footprints. Under harm-based rules, screening should collapse when monitoring costs rise or accuracy falls: for example, facilities may reduce inquiry intensity during periods of equipment failure or tightened QA protocols. By contrast, under compliance-based rules, once the safe-harbor threshold is crossed, inquiry persists even if exclusion would be more efficient. This predicts a divergence: screening rates should be more sensitive to cost shocks under harm-based regimes than under compliance-based ones. Third, the model predicts that observed penalties should align with implementability thresholds. In compliance-based regimes, sanctions that induce inquiry should fall just above the threshold T≥k/[φ(1 −θ)], since higher penalties no longer affect post-inquiry incentives. In harm-based regimes, penalties that allow pooling to persist despite violations should remain at or below T≤yG/(1 −θ), while penalties chosen to induce inquiry should exceed this cutoff. In dual-penalty regimes, front-end penalties Tnshould be set just above yG/(1 −θ)to eliminate pooling, while back-end penalties Tiare tuned to govern inquiry quality. Empirically, one would therefore expect clustering of penalties near these cutoffs: regulators and courts tend to impose the minimum sufficient sanction to induce the desired equilibrium, or to hold penalties below the level that would destabilize an equilibrium they wish to tolerate. Together, these predictions imply cross-sectional correlations between compliance rates and inquiry, differences in sensitivity to cost shocks across liability regimes, and penalty magnitudes that line up with theoretical thresholds. This provides a bridge from the theoretical model to enforcement data, enabling systematic tests beyond qualitative case studies. 6 Further Discussion 6.1 Endogenous inquiry precision. Now assume the agent chooses the precision of inquiry, α∈(1/2,1) as an additional strategy, facing an increasing cost k(α)(k(1/2) = 0,k(1) = ∞,k′>0, k′′ ≥0). In semi-pooling, the agent’s residual legal exposure after inquiry is (1 −θ)(1 −α)multiplied by the penalty that applies after inquiry: under harm-based rules this is T, under compliance-based rules it is 0, and under dual penalties it is Ti. The agent’s private choice of precision therefore solves a simple trade-off between the marginal saving in expected residual exposure and the marginal cost k′(α). Two implications follow. First, precision is increasing in the penalty that prices residual exposure (none under compliance, Tunder harm-based, Tiunder dual): compliance yields the lowest privately chosen precision; dual penalties allow the lawmaker to raise precision by targeting Tiwithout simultaneously inflating the no-inquiry margin. Second, the socially optimal precision balances the social benefit of better screening (fewer false positives and false negatives) against k′(α); with dual penalties the lawmaker can implement this target (subject to the good type’s profitability) by setting Tito match that first-order condition, while using Tnonly to regulate the no-inquiry branch. This arrangement preserves the selection lever 18 (via Tn) and aligns inquiry quality (via Ti). However, verifiability is strictly required: α (or a sufficient proxy) and due-diligence effort must be auditable so that transfers and liability can condition on the chosen precision. Formal statements and proofs appear in Appendix C.1. A close analogue arises in compliance law, where regulators adjust standards upward as inquiry can be made more accurate at similar cost. For example, U.S. environmental rules now mandate continuous emissions monitoring once the technology became feasible, replacing earlier self-reporting schemes. 6.2 Psychological inquiry costs and social preferences Beyond tangible effort, inquiry often carries non-material burdens-embarrassment, fear of social disapproval, and reputational loss (Hellman 2009; Alexander and Ferzan 2009; Grossman and Van Der Weele 2017). Some agents also internalize others’ welfare to varying degrees (Yaffe 2018). Psychological (non-material) costs of inquiry. Let the agent’s private inquiry cost remain k, but let the lawmaker place weight σ∈[0,1] on its non-material component in welfare. Private incentives to inquire are unchanged (they depend on k, not on σ); the social valuation of inquiry is attenuated: the semi-pooling welfare term is θyG−σk rather than θyG−k. Hence, as σfalls, the region where inquiry is socially preferred expands, yet implementability is unaffected-inducing inquiry still requires ruling out action without inquiry and ensuring the legal surplus covers k. Under imperfect inquiry, the same logic holds; with dual penalties one simply reads the no-inquiry side with Tnand the inquiry side with the agent’s residual exposure governed by Ti.16 Social preference (moral) concern under pooling. Suppose an agent who performs an illegal task suffers an internal moral cost m≥0. Under pooling (no inquiry), acceptance depends on expected disutility, so the minimal transfer that induces acceptance is X∗= (1 −θ)T+m (with a dual scheme, replace Tby the no-inquiry penalty Tn). Thus macts like an additive expected penalty: when the lawmaker aims to deter action without inquiry, a larger m permits a lower legal penalty to achieve the same deterrence; when the lawmaker instead prefers to preserve pooling (e.g., when the bad type’s net contribution is non-negative or only mildly negative), a large mcan make pooling infeasible by pushing X∗above a type’s payoff. Under imperfect inquiry, these feasibility effects are unchanged because m bites only on the no-inquiry margin. Interaction and robustness. Psychological inquiry costs and social preferences pull on different levers. Discounting non-material inquiry burdens in welfare (σ < 1) shifts social desirability toward inquiry but does not change the agent’s private cutoff to inquire; moral concern (m > 0) tightens the feasibility of pooling by raising the transfer needed for no-inquiry acceptance. With imperfect inquiry (α < 1), these directions are intact: noise lowers the welfare of inquiry-based screening and narrows its existence window, but σand maffect the same margins as under perfect inquiry. The comparative statics 16. Formal thresholds with (σ, T)under a single penalty and (σ, Tn, Ti)under dual penalties are collected in Appendix C.4. 19 above do not depend on adopting any particular liability scheme; they describe how σ and menter the lawmaker’s trade-offs under each regime. Formal windows and welfare comparisons are in the appendix C.4. 6.3 Enforcement intensity vs. penalty severity. Assume that conviction is not necessarily perfect, but with probability p∈(0,1]. If p is exogenous and enforcement is costless, we can simply read Tas the expected penalty pT throughout; under dual penalties read (Tn, Ti)as (pTn, pTi). Now let us relax the assumption of “costless” conviction and/or enforcement, and introduce cost into the penalty system. Case A: pcostless (p= 1), Tcostly. When enforcement is costless (thus p= 1) but penalty severity Tabsorbs real resources (e.g., incarceration), only equilibria that impose penalties on the path reduce welfare by expected resource cost. Under perfect inquiry, pooling bears (1 −θ)C(T)while semi-pooling (screening) and inactive bear none. Under imperfect inquiry with harm-based liability, semi-pooling additionally bears (1 −θ)(1 − α)C(T)(false positives); under compliance-based liability, inquiring agents are exempt and semi-pooling again bears no resource cost; under imperfect inquiry with dual-penalty liability, semi-pooling additionally bears (1 −θ)(1 −α)C(Ti)(false positives). Hence the lawmaker should always choose the minimal Tthat implements the desired continuation behavior and avoid on-path penalties when C(·)is large. Case B: Tcostless, pcostly. If instead intensity pis resource-costly (with convex K(p)) and severity Tis free, the conclusions are symmetric: equilibria that impose penalties on the path now bear expected Cp(p); the welfare-maximizing choice is the minimal p that implements the desired equilibrium; and designs that avoid on-path penalties (semi under compliance-based rules; inactive) are strictly more attractive when Cp(p)is large. Case C: both pand Tcostly. When both instruments are costly, choose the cheapest mix of (p, T)(or (p, Tn, Ti)under dual penalties) that satisfies the relevant incentive/existence constraints. With convex costs CT(·)and Cp(p), an interior solution equalizes marginal resource cost per unit of expected penalty; corner solutions arise when one instrument is much cheaper. Under dual penalties, Tnprices the no-inquiry margin and Tiprices residual risk after inquiry, so it is efficient to set the unused component to zero whenever it does not bind. 6.4 Market structure and liability sharing Our baseline assumes a single principal makes a take-it-or-leave-it offer to a single agent. The main comparative statics and implementability logic extend to alternative market structures and liability sharings with only cosmetic changes to the feasibility thresholds. We summarize four useful variants and refer to Appendix C.5 for formal statements and proofs. Single principal with uncertainty (one side, one payoff). If neither party knows legality ex ante, set yG=yB≡yand interpret type uncertainty as uncertainty about the legal state. The taxonomy and welfare ranking mirror the baseline. Under perfect inquiry, pooling (no inquiry) is feasible iff y≥(1 −θ)Tn;screening (inquiry) is feasible iff θy ≥k. Under imperfect inquiry with dual penalties, replace the screening threshold by 20 φ y ≥k+ (1 −θ)(1 −α)Ti, where φ:= θα + (1 −θ)(1 −α). Hence the lawmaker’s design problem and the penalty levers carry over verbatim. No distinct principal (self-solicitation). In settings like soliciting contraband services, the “principal” and “agent” collapse into the same decision-maker; transfers vanish and only the decision to inquire (at cost k) versus act without inquiry remains. The screening condition becomes the agent’s private surplus test (as above with y), and the pooling condition becomes her acceptance of expected legal risk. Qualitatively, this is the single-principal case with zero rents: inquiry occurs iff the legal-surplus term clears k, and willful ignorance arises iff expected penalties are (privately) coverable. The lawmaker’s penalty levers and welfare comparison are unchanged. Agent bargaining power (or competition among principals). If the agent can extract more than the minimal acceptance transfer (e.g., Nash bargaining, or many principals bidding), pooling feasibility shrinks (principals must cover higher transfers) while screening feasibility expands (the agent can be funded to cover kand any residual exposure). Implementability therefore tilts toward inquiry. When inquiry is socially desirable, bargaining power helps the lawmaker (lower penalty suffices to rule out no-inquiry action); when inclusion of both task types via pooling is desirable, strong agent power can make pooling infeasible. The lawmaker then trades off desirability against feasibility as in the baseline, but with the windows shifted toward screening. Liability allocation as “just transfers.” The same Coase-style logic behind bargaining power carries over to who bears legal penalties. If parties are risk-neutral, penalties are (fines) costless to society, and indemnity promises are enforceable, then shifting liability shares between the principal (he) and the agent (she)-or allowing the principal to warranty the agent’s penalties-simply reassigns transfers without changing welfare. What can change is implementability: principal-side liability can restore a lever to deter no-inquiry behavior when compliance-based rules would otherwise make screening hard to switch off, whereas full indemnity to the agent effectively replicates an exemption and removes that lever. In short, the logic that “negotiation power translates into pure transfers” extends to shared-liability and warranty arrangements as well; when penalties or enforcement are costly, or indemnity is non-contractible, this equivalence breaks and design trade-offs reappear (outside our baseline). 6.5 Robustness to unknown θ. If the agent does not know the true θ(share of good principals), let her act on a perceived value ˜ θ(posterior mean under Bayes; worst case θunder ambiguity aversion). Then all private thresholds in our analysis hold with θreplaced by ˜ θ(e.g., the no-inquiry acceptance transfer becomes (1−˜ θ)Tn, and screening is feasible under imperfect inquiry iff ˜φ yG≥kwith ˜φ:= ˜ θα+(1−˜ θ)(1−α)). Social desirability still evaluates welfare at the true θ. Ambiguity (lower ˜ θ) simultaneously makes no-inquiry less attractive (higher expected penalty) and reduces the expected upside from inquiry (lower ˜φ), so implementability can shift either way; our comparative statics otherwise carry through verbatim. Dual penalties remain the most robust instrument: set Tnhigh to deter no-inquiry across the relevant ˜ θrange, and keep Tilow to avoid overburdening inquiry when feasible. 21 This formulation also aligns with the doctrine’s “sufficient suspicion” requirement: a court can impose a minimum suspicion threshold on 1−˜ θwithout altering our equilibrium taxonomy. It also clarifies how different mens rea categories map into the model. In our baseline, non-inquiry is a fully rational choice and thus corresponds to deliberate ignorance. By contrast, negligence could be represented by mistaken beliefs (systematically misperceiving θ), and recklessness by agents who recognize risk but underweight it relative to incentives. These cases lie outside the rational baseline but illustrate how the model can accommodate the doctrinal distinction between negligent, reckless, and deliberate ignorance. 6.6 Jurisprudential concerns. Several doctrinal debates around willful ignorance can be mapped into our framework. First, the parity question asks whether deliberate ignorance should be punished like knowledge. In equilibrium, knowing crime is strictly dominated by non-inquiry once ignorance is punished at parity or above, so the equal-culpability debate has no bite except in the implausible case where ignorance is punished more severely than knowledge. Relatedly, some argue culpability should depend on whether knowledge would have changed behavior. Our model shows this criterion is moot: inquiry followed by knowing violation is never optimal, so all harmful conduct flows through ignorance. Second, the legality principle (nulla poena sine lege) cautions against punishing under vague or shifting standards. In our framework, vagueness is captured by reduced accuracy α: inquiry may not yield a clear answer. Compliance-based or dual-penalty rules accommodate this by lowering residual risk once inquiry is documented in good faith. Third, concerns about overbreadth and chilling effects correspond directly to our implementability frontier: excessively high penalties on non-inquiry can make screening infeasible and collapse valuable markets. Finally, doctrine distinguishes negligence, recklessness, and deliberate ignorance. Our baseline assumes rational agents, so non-inquiry is deliberate; negligence and recklessness could be modeled as systematic misperception of θor underweighting of recognized risk. Some courts and commentators also place willful ignorance between recklessness and knowledge, on the view that the actor “in fact” does not know. We treat deliberate ignorance at parity with knowledge, reflecting its strategic nature. This choice has no effect on equilibrium outcomes, since knowing crime is strictly dominated and never arises. 7 Conclusion This paper offers a tractable framework for analyzing willful ignorance under asymmetric information and legal penalties. The central message is an implementability one. With perfect inquiry, penalties can be used to screen out harmful behavior while preserving valuable activity, so the welfare-maximizing behavior can always be made to exist and be selected. With imperfect inquiry, noise simultaneously lowers the value of screening and decouples feasibility from desirability; for non-empty regions of primitives, inquiry is either unattainable when desirable or difficult to switch off when exclusion is preferred. We show how liability design mediates that trade-off. Harm-based rules keep a strong selection lever but can choke off desirable screening; compliance-based rules relax feasibility but risk coexistence and misselection; dual-penalty rules separate these roles and 22 weakly dominate single-penalty rules in implementability, failing only when the legal surplus cannot cover expected inquiry cost. These insights translate into simple guidance: calibrate penalties directly to the cutoff conditions implied by accuracy and inquiry costs. When accuracy is imperfect, use separate penalties for uninformed action and for residual wrongdoing after inquiry. In this sense, ignorance is not a shield: the absence of knowing crime in equilibrium is an endogenous consequence of incentives rather than evidence of a higher moral standard, and liability rules must therefore target the strategic margin of non-inquiry. The framework’s strengths are its clarity about what penalties can and cannot accomplish and its portability: it provides a template for studying inquiry technologies, evidentiary standards, and enforcement frictions. Future work could extend the framework to dynamic and heterogeneous environments-e.g., repeated interactions with reputation, heterogeneity in inquiry costs, subsidies for inquiry and information disclosure regulation. Together, these can deepen the welfare foundations for regulating willful ignorance in practice. References Alexander, Larry, and Kimberly Kessler Ferzan. 2009. Crime and culpability: A theory of criminal law. Cambridge University Press. Alter, Adam L, Julia Kernochan, and John M Darley. 2007. “Morality influences how people apply the ignorance of the law defense.” Law & Society Review 41 (4): 819– 864. Charlow, Robin. 1991. “Wilful ignorance and criminal culpability.” Tex. L. Rev. 70:1351. Dana, Jason, Roberto A Weber, and Jason Xi Kuang. 2007. “Exploiting moral wiggle room: experiments demonstrating an illusory preference for fairness.” Economic Theory, 67–80. Grossman, Zachary, and Joel J Van Der Weele. 2017. “Self-image and willful ignorance in social decisions.” Journal of the European Economic Association 15 (1): 173–217. Hellman, Deborah. 2009. “Willfully blind for good reason.” Criminal Law and Philosophy 3:301–316. Kartik, Navin, Marco Ottaviani, and Francesco Squintani. 2007. “Credulity, lies, and costly talk.” Journal of Economic theory 134 (1): 93–116. Kirfel, Lara, and Ivar R Hannikainen. 2023. “Why blame the ostrich? Understanding culpability for willful ignorance.” K., Prochownik, S. Magen,(Eds.), Advances in experimental philosophy of law, 75–98. Luban, David. 1998. “Contrived ignorance.” Geo. LJ 87:957. Polinsky, A Mitchell, and Steven Shavell. 2000. “The economic theory of public enforcement of law.” Journal of economic literature 38 (1): 45–76. Sarch, Alexander. 2018. “Willful ignorance in law and morality.” Philosophy Compass 13 (5): e12490. 23 B.4 Proof of Proposition 4.2 Lemma B.3 (Threshold geometry under harm-based liability with imperfect inquiry). Let φ:= θα + (1 −θ)(1 −α), p3:= (1−θ)α (1−θ)α+θ(1−α), p4:= (1−θ)(1−α) φ, and define the bad-type incentive-compatibility threshold TIC B:= φαyB+(1−α)k α(1−α+αθ). Then under harm-based liability with α∈(1 2,1), the continuation equilibrium existence conditions are: Pool 1 (insurance-style pooling) exists for k θ(1−θ)(2α−1) ≤T≤yG p3+k (1−θ)α, Pool 2 (minimal-acceptance pooling) exists for T≤minnyG 1−θ,k θ(1−θ)(2α−1)o, Semi (inquiry; act only on g)exists for T > maxnk θ(1−θ)(2α−1), TIC B,yG 1−θoand T≤φ yG−k (1−θ)(1−α), Inactive exists for T > maxnyB,yG 1−θoand (1 −θ)(1 −α)T+k > φ yG. Consequently, multiplicity arises only in: maxnyB,yG 1−θo< T ≤yG p3+k (1−θ)α,(1−θ)(1−α)T+k > φyG⇒Pool 1 and Inactive coexist. In particular, whenever Semi is feasible (i.e. k < θ(2α−1)yG), its lower bound exceeds Pool 1’s upper bound, so Semi and Pool 1 cannot coexist. Proof of Proposition 4.2 (implementation under harm-based rules, imperfect inquiry). Proof. Definitions. Minimal transfers: Xpool min = maxn(1 −θ)T, p3T−k 1−φo, Xsemi =(1−θ)(1−α)T+k φ=p4T+k φ. As in the main text, transfers/penalties are redistributive; welfare is T-invariant within a branch: SWpool =θyG+(1−θ)(τyB−H), SWsemi =θα yG+(1−θ)(1−α)(τyB−H)−k, SWinact = 0. Let eSW ∈ {Pool,Semi,Inactive}denote a welfare-maximizing type (ignoring implementability). Case 1: e⋆=Pool (all trades desirable). (a) Insurance-style pooling (Pool 1). Choose Tstrictly inside the region: T∈hk θ(1−θ)(2α−1),yG p3+k (1−θ)αi. We do not need to rule out Inactive: under our implementation convention, Inactive is weakly Pareto-dominated by Pool 1 and therefore not credited when both 30 coexist.17 (b) Minimal-acceptance pooling (Pool 2). Pick any T≤minyG 1−θ,k θ(1−θ)(2α−1), Semi fails by T≤k θ(1−θ)(2α−1) and we do not need to rule out Inactive. Case 2: e⋆=Inactive (no trade desirable). Choose Tstrictly inside the inactivity region and outside both pooling regions, e.g. T > M := max(yB,yG 1−θ,φ yG−k (1−θ)(1−α),yG p3+k (1−θ)α). Then Pool 2 fails by T > yG 1−θ, Pool 1 fails by T > yG p3+k (1−θ)α, and Semi fails by (1 −θ)(1 −α)T+k > φ yG. Hence Inactive is unique. Case 3: e⋆=Semi (inquiry desirable). Semi must be feasible, i.e. maxnk θ(1−θ)(2α−1), TIC B,yG 1−θo<φ yG−k (1−θ)(1−α). Since Semi’s region is strictly above yG 1−θwhile Pool 1’s upper bound is below yG 1−θwhenever Semi is feasible, Semi and Pool 1 cannot coexist. Also Semi and Inactive cannot coexist because their feasibility inequalities for (1 −θ)(1 −α)T+kare mutually exclusive. Therefore, selecting any T∈maxnk θ(1−θ)(2α−1), TIC B,yG 1−θo,φ yG−k (1−θ)(1−α) puts Tin the interior of Semi’s region and outside all competitors. Hence Semi is unique if feasible. Welfare comparison and conclusion. Within each equilibrium type, SW is independent of T; hence the lawmaker’s choice of Tonly selects between types. Pooling and Inactive equilibria deliver exactly the same welfare as in the No-Inquiry benchmark, while Semi yields SWsemi =θα yG+ (1 −θ)(1 −α)(τyB−H)−k, which is strictly below its perfect-inquiry counterpart θyG−kbut strictly above the NoInquiry outcome whenever Semi is feasible. Thus the lawmaker can always implement pooling or inactivity to deter inquiry when it is not socially desirable. However, there exist non-empty parameter ranges (lower αor higher k) in which Semi is welfare-maximizing yet infeasible, so inquiry cannot be induced by any penalty. In all cases, maximal welfare under imperfect inquiry is weakly below that under perfect inquiry and weakly above that under No Inquiry. B.5 Proof of Proposition 4.3 Proof. Definitions. Let φ:= θα + (1 −θ)(1 −α), 17. Non-emptiness: the interval k θ(1−θ)(2α−1) ,yG p3+k (1−θ)αis non-empty iff k≥θ(2α−1)yG. 31 Minimal transfers: Xsemi =k φ, Xpool = (1 −θ)T. Agent utilities: Ui=φX −k, Uni =X−(1 −θ)T. Good-type feasibility for semi: φyG≥k. Welfare is T-invariant within branches: SWsemi =θαyG+(1−θ)(1−α)(τyB−H)−k, SWpool =θyG+(1−θ)(τyB−H), SWinact = 0. Case 1: e⋆=Semi (inquiry desirable). Semi requires the agent to strictly prefer inquiry. At X=k/φ, Ui= 0, Uni =k φ−(1 −θ)T. Thus inquiry is optimal whenever T≥k φ(1−θ).(C1) Good-type feasibility requires φyG> k. (C2) Together, (C1)–(C2) define the region in which inquiry is implementable. In this region, Semi is unique: Pool fails by (C1), and Inactive fails by good-type deviations. Hence Semi is implemented whenever it is feasible. Case 2: e⋆=Inactive (no trade desirable). Suppose welfare would prefer no action. If φyG≤k, Semi is infeasible by (C2), and the lawmaker can set T > yBto rule out Pool, implementing Inactive uniquely. If φyG> k, however, Semi remains feasible at X=k/φ for any T≥k/[φ(1 −θ)]. Because compliance-based liability exempts inquiry from penalties on false positives, no choice of Tcan remove the inquiry equilibrium. Thus when exclusion is desirable but (C2) holds, inquiry persists in the equilibrium set. Case 3: e⋆=Pool (no inquiry desirable). Without inquiry, the agent accepts if Uni ≥0, i.e. X≥(1 −θ)T. Minimal transfer is X= (1 −θ)T. The lawmaker can always set Tsmall enough that X≤yG, yB; hence Pool can be implemented whenever pooling is welfare-maximizing. Welfare comparison and conclusion. Pooling and Inactive equilibria yield exactly the same welfare as the No-Inquiry benchmark. Semi yields SWsemi =θαyG+ (1 −θ)(1 −α)(τyB−H)−k, which converges to the perfect-inquiry benchmark θyG−kas α→1, and is strictly below it for α < 1. Thus, under compliance-based liability, inquiry can always be sustained when it is feasible but cannot always be deterred when socially undesirable. Maximal welfare is therefore weakly below perfect inquiry and may in case (ii) fall strictly below the No-Inquiry benchmark. 32 B.6 Proof of Proposition 4.4 Proof. Primitives and notation. Let φ:= θα + (1 −θ)(1 −α),1−φ=θ(1 −α) + (1 −θ)α, the probability (under pooling types) that inquiry yields a “good” signal and the task is performed. Under dual penalties we denote by Tnthe penalty if the agent did not inquire and the task is illegal, and by Tithe penalty if the agent did inquire yet an illegal task is performed (false positive). Step 1 (Minimal transfers). If both types offer the same per-performance transfer t and the agent inquires and performs only on a good signal, her expected utility is Ui=φ t −k−(1 −θ)(1 −α)Ti. By tie-breaking, the minimal transfer that induces inquiry is tsemi =k+(1−θ)(1−α)Ti φ.(B.1) If instead she accepts without inquiry, the minimal lump-sum transfer that induces acceptance equals her expected penalty: Xpool = (1 −θ)Tn.(B.2) Step 2 (Turning inquiry off when it is not desirable). Make semi infeasible by choosing Tilarge so that the good type cannot fund tsemi: tsemi > yG⇐⇒ Ti>φ yG−k (1−θ)(1−α). Since α > 1/2, the denominator is positive, so such Tiexists whenever desired. With inquiry infeasible, use Tnto select the no-inquiry branch: implement pooling by keeping Xpool ≤min{yG, yB}(the agent then weakly prefers no inquiry at the minimal transfer), or implement inactivity by taking Tn>max{yG, yB}/(1 −θ). Step 3 (Turning inquiry on when it is desirable). Minimize the semi transfer by lowering Ti(down to 0if needed), so tsemi becomes as small as possible. Sustaining semi requires three conditions: (i) Good-type feasibility. tsemi ≤yG⇐⇒ (1 −θ)(1 −α)Ti+k≤φ yG.(F) (ii) Inquiry optimal for the agent at tsemi.At tsemi we have Ui= 0 by construction, while Uni =tsemi −(1 −θ)Tn. Hence Ui≥Uni iff (1 −θ)Tn≥tsemi ⇐⇒ Tn≥k+(1−θ)(1−α)Ti φ(1−θ).(AO) (iii) Bad-type IC against the deviation X′=Tn.Off path the agent accepts any X′≥Tnwithout inquiry, yielding yB−Tnto type B. On-path under semi, Bearns (1 −α)(yB−tsemi). Deterring the deviation requires yB−Tn≤(1 −α) (yB−tsemi)⇐⇒ Tn≥α yB+ (1 −α)tsemi.(BIC) 33 If φyG> k, take Ti= 0 to minimize tsemi =k/φ, and then choose Tn≥max nk φ(1−θ), αyB+ (1 −α)k φo, which satisfies (AO) and (BIC) with slack; (F) holds since φyG> k. Pooling is then irrelevant because the agent strictly prefers inquiry. If φyG≤k, even Ti= 0 gives tsemi ≥yG, so semi is privately infeasible. Step 4 (Welfare and dominance over single-penalty regimes). The dual-penalty pair (Tn, Ti)nests harm-based (Tn=Ti)and compliance-based (Ti= 0) rules. Therefore the maximal welfare attainable under dual penalties weakly dominates the maximal welfare attainable under either single-penalty design. Since α < 1implies false positives/negatives under inquiry, the maximal welfare under dual penalties is (weakly) below the perfect-inquiry benchmark. The implementability frontier stated in the proposition follows from Step 2 (deterring inquiry) and Step 3 (sustaining inquiry) via the linear bounds (F), (AO), and (BIC). Imperfect inquiry: calibration and comparative statics This appendix collects the full cutoff expressions and comparative statics that underlie the brief discussion in Section 4. They make precise how penalty placement depends on signal accuracy α, inquiry cost k, the prior θ, and the type-dependent outputs yG, yB. Assume imperfect inquiry with precision α∈(1/2,1) and define φ(α) := θα + (1 −θ)(1 −α). Lemma B.4 (Calibration under imperfect inquiry by liability rule).Let e⋆∈ {pooling,semi,inactive} be the socially desirable behavior. (a) Harm-based single penalty T. (i) Semi-pooling. Implementable if maxnyB,k θ(1−θ)(2α−1)o< T < φ(α)yG−k (1−θ)(1−α)and φ(α)yG> k. (ii) Pooling. Implementable if T≤minnyB,yG 1−θowhen θyG≤k,T=k θ(1−θ)(2α−1), T ≤yBwhen θyG> k. (iii) Inactive. Implementable if T > maxnyB,yG 1−θ,φ(α)yG−k (1−θ)(1−α)o. (b) Compliance-based single penalty T(safe harbor after inquiry). (i) Semi-pooling. Implementable if T≥k φ(α)(1−θ)and φ(α)yG> k. 34 (ii) Pooling. Implementable if T≤minnyB,yG 1−θowhen θyG≤k,T < yG 1−θwhen θyG> k. (iii) Inactive. Implementable if φ(α)yG≤kand T > yB. (c) Dual penalties (Tn, Ti). (i) Semi-pooling. Implementable if Tn>yG 1−θand (1 −θ)(1 −α)Ti+k≤φ(α)yG. (ii) Pooling. Implementable if Tn≤minnyB,yG 1−θo, or, if θyG> k, Tn=k θ(1−θ)(2α−1), Tn≤yB,and (1 −θ)(1 −α)Ti+k > φ(α)yG. (iii) Inactive. Implementable if Tn> yBand (1 −θ)(1 −α)Ti+k > φ(α)yG. Proposition B.1 (Comparative statics of imperfect-inquiry cutoffs).For the harm-based rule (and the Ti-arm of the dual-penalty rule), define L(α) := k θ(1−θ)(2α−1), U(α) := φ(α)yG−k (1−θ)(1−α). 1. L′(α)<0: higher accuracy lowers the penalty required to induce inquiry. 2. U′(α)>0if θyG> k;U′(α)=0if θyG=k;U′(α)<0if θyG< k. 3. ∂L/∂k > 0,∂U/∂k < 0;∂U/∂yG>0while Ldoes not depend on yG. 4. L(θ)is minimized at θ= 1/2;U(α)rises in θwhenever αyG> k. 5. Limits: as α→1,L(α)→k/[θ(1 −θ)] and, if θyG> k,U(α)→+∞; as α↓1/2, the interval (L(α), U(α)) shrinks and may vanish unless yGis large relative to k. 6. Under compliance-based rules, once uninformed action is deterred, screening is feasible iff φ(α)yG> k. Pooling and inactivity cutoffs in yG/(1 −θ)and yBremain unchanged. Appendix C Extensions-Proofs and Technical Notes C.1 Endogenous Inquiry Precision Setup and assumptions. In semi-pooling, after agreeing to inquire the agent chooses precision α∈(1/2,1) at cost k(α). Assume: 35 A1.α∈(1/2,1) and k: (1/2,1) →R+is C1, strictly increasing and strictly convex, with k(1/2) = 0 and limα↑1k(α) = ∞. A2. Transfers and liability can condition on (auditable) inquiry and its recorded precision (or a verifiable proxy). A3. Tie-breaking: indifferent agents accept; indifferent principals offer the smallest transfer that induces acceptance. Let λdenote the penalty that prices the agent’s residual exposure after inquiry: λ≡     T(harm-based single penalty), 0(compliance-based exemption), Ti(dual-penalty: after-inquiry arm). In semi-pooling, residual exposure occurs with probability (1 −θ)(1 −α). With verifiability, the good principal (he) pays the minimal transfer that covers the agent’s private inquiry bill k(α)plus expected residual exposure (1 −θ)(1 −α)λ. Lemma C.1 (Agent’s privately optimal precision).Fix a semi-pooling outcome and a residual-exposure price λ≥0. The agent’s privately optimal precision α∗(λ)is the unique solution to k′(α) = (1 −θ)λ, and satisfies α∗(λ)∈(1/2,1),α∗′(λ)>0,limλ↓0α∗(λ) = 1/2, and limλ↑∞ α∗(λ)=1. Proof. In semi-pooling the agent minimizes k(α) + (1 −θ)(1 −α)λon (1/2,1). Strict convexity of kgives a unique minimizer; the FOC is k′(α) = (1 −θ)λ. Monotonicity and limits follow from A1. Welfare with semi-pooling. If inquiry is undertaken with precision α, the welfare contribution (transfers cancel) is SWsemi(α) = θ α yG+ (1 −θ)(1 −α) (τyB−H)−k(α).(C.3) The derivative is d dα SWsemi(α) = θ yG−(1 −θ)(τyB−H)−k′(α), hence the socially optimal precision bα(when semi-pooling is the target behavior) solves k′(bα) = θ yG−(1 −θ)(τyB−H).(C.4) By A1,bα∈(1/2,1) is unique whenever the right-hand side is positive.18 Proposition C.1 (Instrument power across liability regimes).Let α∗(λ)be as in Lemma C.1, and bαsatisfy (C.4). Then: 18. If θ yG≤(1 −θ)(τyB−H)(net benefit of accuracy non-positive), the lawmaker prefers the lowest precision in the admissible set; under A1 this is 1/2. The interesting case for screening has τyB−H < 0, making the RHS strictly larger than θyG. 36 (a) compliance-based: λ= 0 implies α∗= 1/2. The lawmaker cannot raise precision via penalties; inquiry accuracy is privately minimized. (b) harm-based: λ=Tso α∗is increasing in T. Raising Talso tightens the noinquiry margin, potentially eliminating pooling even when inclusion is desirable. (c) Dual: λ=Tiso α∗is increasing in Tiwhile the no-inquiry margin is governed by Tn. Thus Titargets inquiry quality without collateral effects on pooling, and Tn controls selection on the no-inquiry branch. Proof. Immediate from Lemma C.1 and the mapping of λto the penalty by regime. Proposition C.2 (Implementing the socially optimal precision under dual penalties). Suppose semi-pooling is (socially) the target behavior and is feasible at bα, i.e., θbα yG≥k(bα) + (1 −θ)1−bαTi.(C.5) Under dual penalties, setting T⋆ i=k′(bα) 1−θand choosing any Tnthat preserves the desired selection on the no-inquiry branch induces α∗=T⋆ i7→ bαand implements the socially optimal precision. Proof. By Lemma C.1, α∗solves k′(α) = (1 −θ)Ti. With T⋆ ias above the agent’s private FOC coincides with (C.4), hence α∗=bα. Feasibility is guaranteed by (C.5). Tncan be tuned (independently) to preserve or eliminate pooling as desired. Corollary C.1 (Limits under single-penalty regimes).Under compliance-based rules, α∗= 1/2and the lawmaker cannot implement bα > 1/2via penalties. Under harm-based rules, the lawmaker can raise α∗by increasing T, but doing so simultaneously affects the no-inquiry branch, creating selection trade-offs that dual penalties avoid. Existence and welfare. When semi-pooling is socially desirable, feasibility with endogenous αrequires that the good type’s expected legal surplus at α∗(λ)covers the privately chosen inquiry bill: θ α∗(λ)yG≥kα∗(λ)+ (1 −θ)1−α∗(λ)λ, with λmapped to T,0, or Tiby regime. Welfare at the induced precision follows from (C.3). Dual penalties allow the lawmaker to (i) select behavior on the no-inquiry branch via Tnand (ii) align inquiry quality with bαvia Ti, subject only to the feasibility condition above. C.2 Off-path beliefs and implementability Let ˆµ∈[0,1] denote the agent’s belief that the task is legal if she accepts an unexpected (off-path) offer without inquiry. Under a dual-penalty rule, let Tnbe the penalty if she did not inquire, and Tithe penalty if she did inquire but still performs an illegal task (e.g., a false positive). Under single-penalty rule, both Tnand Timerge into T. Let α∈(1/2,1) denote inquiry precision, θ∈(0,1) the prior probability of a good principal (he), and k > 0the inquiry cost. 37 Lemma C.2 (Off-path thresholds).(i) If the agent accepts an off-path offer X′without inquiry, her expected utility is X′−(1−ˆµ)Tn, so (by tie-breaking) she accepts the smallest such offer X′ min = (1 −ˆµ)Tn. (ii) A deviating bad principal earns yB−(1 −ˆµ)Tnat X′ min; he is deterred iff Tn>yB 1−ˆµ. (iii) Suppose the on-path outcome is screening (semi-pooling): the good principal funds inquiry and the agent acts only on a “legal” signal. The minimal on-path transfer that induces inquiry equals Xsemi =k+ (1 −θ)(1 −α)Ti, i.e., the inquiry cost plus the agent’s residual expected exposure under imperfect inquiry. To deter a good-type off-path deviation to “no inquiry” acceptance, it suffices that (1 −ˆµ)Tn≥Xsemi ⇐⇒ Tn≥k+(1−θ)(1−α)Ti 1−ˆµ. Under perfect inquiry (α= 1), this reduces to Tn≥k/(1 −ˆµ). Proof. Part (i) follows from the agent’s off-path acceptance condition with no inquiry: X′−(1 −ˆµ)Tn≥0, minimized at equality. Part (ii) plugs X′min into the deviating bad type’s payoff yB−X′min and requires it <0. Part (iii) equates the off-path acceptance threshold to the on-path payment that exactly covers private inquiry cost kplus residual expected penalty (1 −θ)(1 −α)Ti. The perfect-inquiry simplification is immediate when (1 −α) = 0. The deterrence bound for the bad type, yB/(1−ˆµ), is (weakly) increasing in ˆµ; the nodeviation bound for the good type, k+(1−θ)(1−α)Ti/(1−ˆµ), is also (weakly) increasing in ˆµ. Hence moving from the pessimistic benchmark ˆµ= 0 to more optimistic beliefs (ˆµ > 0)weakly shrinks the penalty windows that support either pooling or screening. This is the sense in which the pessimistic convention maximizes implementability. C.3 Mixed strategies by the principal Fix any liabilities (single-penalty or dual-penalty), α∈(1/2,1], and k > 0. Consider a candidate profile where the bad type randomizes over two transfers, Xℓ< Xh, while the good type plays a single transfer. Let ˆµ(X)denote the agent’s posterior that the task is legal upon observing X(Bayes-consistent on-path; arbitrary off-path subject to the convention above). Proposition C.3 (Extreme-point best reply for the bad type).For any fixed off-path beliefs and the agent’s best response, the bad type’s expected payoff as a function of his own transfer Xis piecewise linear with at most one kink (the point at which the agent switches from “inquire/reject when illegal” to “accept without inquiry”). Hence his best reply is attained at an extreme point: either the lowest Xthat still induces acceptance without inquiry, or the highest feasible Xthat yields acceptance. Generically, the maximizer is unique and pure. Proof. For each observed X, the agent’s best response is threshold in Xand ˆµ(X): below a cutoff, she inquires (and rejects when illegal); above it, she accepts without inquiry. Therefore the acceptance probability as a function of Xis a step function with 38 a single jump. The bad type’s expected payoff is yB−Xwhen acceptance occurs and 0 otherwise, so it is piecewise linear in Xwith at most one kink at the jump. A piecewiselinear function with one kink attains its maximum at an endpoint unless parameters are knife-edge; thus the bad type’s best reply is pure except on a measure-zero set. Implication. Apart from the trivial inactive case (rejection regardless of offers), equilibrium behavior is exhausted by the pure types analyzed in the main text (pooling or semi-pooling). Allowing the bad type to “mix” between legal and illegal tasks with some probability qsimply rescales posteriors ˆµ(X)and leaves the threshold logic unchanged; it does not generate new equilibrium types. C.4 Psychological Inquiry Costs and Social Preferences Let the agent’s private inquiry cost be k, but the lawmaker place weight σ∈[0,1] on its non-material component in welfare. Let m≥0be an internal moral cost the agent suffers when she performs an illegal task without inquiry. Under dual penalties, Tnapplies to no-inquiry acts and Tito post-inquiry illegal acts (false positives). Lemma C.3 (Welfare and feasibility with (σ, m)).(i) The private inquiry condition is unchanged by σand m; thresholds shift only with k,Tn,Ti, and α. (ii) The semi-pooling welfare term becomes θyG−σk (perfect inquiry) and θαyG+(1−θ)(1−α)(τyB−H)−σk (imperfect inquiry). (iii) Under pooling (no inquiry), the minimal acceptable transfer is X∗= (1 −θ)(Tn+m)(replace Tnby Twith a single penalty). Feasibility of pooling requires yG≥X∗and yB≥X∗. Proof. (i) σis a lawmaker’s weight and does not enter the agent’s private problem. mis borne only when the agent performs an illegal task under no inquiry; it does not affect the inquiry branch. (ii) Transfers cancel in welfare; the only change from σis the scaled k. Under imperfect inquiry, the standard welfare term for semi-pooling is reduced by noise; σ scales kidentically. (iii) Under pooling, the agent’s expected disutility is (1−θ)(Tn+m); tie-breaking gives the threshold X∗and the feasibility conditions stated. Implications. Lower σexpands the region where inquiry is socially preferred but leaves implementability unchanged. Higher msubstitutes for Tnwhen deterring no-inquiry action is the objective, but it also makes sustaining pooling harder when inclusion of both types is desirable. Dual penalties keep the levers distinct: tune Tn(with m) for the no-inquiry margin and Tifor residual exposure after inquiry. C.5 Additional notes on market structure Throughout, “screening” refers to the semi-pooling equilibrium in which the agent inquires and performs only when the signal (or belief) indicates legality. Penalty notation under imperfect inquiry: Tnapplies when the agent acts without inquiry; Tiapplies to postinquiry illegal performance (false positives). Precision α∈(1/2,1) and φ:= θα + (1 − θ)(1 −α). A. Single principal with uncertainty about legality 39