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INTERNATIONAL ECONOMIC REVIEW Vol. 65, No. 4, November 2024 DOI: 10.1111/iere.12706

TORT LIABILITY AND UNAWARENESS∗

By Surajeet Chakravarty, David Kelsey, and Joshua C. Teitelbaum

University of Exeter, U.K.; University of Nottingham, U.K.; Georgetown University, U.S.A.

We explore the implications of unawareness for tort law. We study cases where injurers and victims initially are unaware that some acts can yield harmful consequences, or that some acts or harmful consequences are even possible, but later become aware. We model unawareness by Reverse Bayesianism. We compare the two basic liability rules of Anglo-American tort law, negligence and strict liability, and argue that negligence has an important advantage over strict liability in a world with unawareness—negligence, through the stipulation of due care standards, spreads awareness about the updated probability of harm.

1. introduction

1.1. Background. Expected utility theory (Savage, 1954) posits a space of mutually exclu- sive and collectively exhaustive states of the world, representing all possible resolutions of un- certainty. It assumes that when a person chooses an act, although she is uncertain about the true state of the world and therefore about the consequences of her chosen act, she neverthe- less has complete knowledge of the state space—she knows all the possible acts and all the possible consequences of each and every act. New information contracts the state space, and the person’s beliefs update according to Bayes’ rule.

In reality, people often lack complete knowledge of the state space. This is known as un- awareness. A person may be unaware of some acts, some consequences, or that a known act can cause a known consequence. Unawareness creates the possibility of growing awareness— the expansion of the state space upon the discovery of a new act, consequence, or act– consequence link. Examples include the discovery of a new product or technology (new act), the discovery of a new disease or injury (new consequence), or the discovery of a new link be- tween a known product and a known injury (new act–consequence link).

Unawareness plays an important role in many economic and legal spheres. For example, people may write incomplete contracts due to unforeseen contingencies. They may be un- aware that an act can result in criminal liability. They may not be aware of every objection they can raise to a witness’ testimony in a trial.

In this article, we study the implications of unawareness for tort law, the branch of the com- mon law that governs liability for civil wrongs. Technological progress, scientific discovery, and other unforeseen events can create new possibilities for civil harms. We address the issue of how the common law can deal with this type of change. More specifically, we explore how

∗Manuscript received July 2022; revised February 2024. We thank the editor and the referees for their very helpful comments. We also thank Jonah Gelbach, Tim Friehe,

John Mikhail, David Reinstein, Joshua Rozenberg, Steve Salop, Steve Shavell, Larry Solum, and participants at the following conferences and seminars: the Law and Ambiguity Workshop at the University of Exeter, the Decisions, Markets, and Networks conference at Cornell University, the 2017 conference of the European Association of Law and Economics, the 2018 meeting of the AALS Section on Law and Economics, the 2018 meeting of the Ameri- can Law and Economics Association, the 2018 Foundations of Utility and Risk Conference, the 2018 Law and Eco- nomic Theory Conference, Australian National University, University of Auckland, Bielefeld University, University of Bonn, Heidelberg University, Heilbronn University, University of Marburg, University of Nottingham, Paris Nan- terre University, and Towson University. Please address correspondence to: Joshua C. Teitelbaum, Georgetown Uni- versity, 600 New Jersey Avenue NW, Washington, DC 20001, USA. E-mail: [email protected].

1851 © 2024 the Economics Department of the University of Pennsylvania and the Osaka University Institute of Social and Economic Research Association.

1852 tort liability and unawareness

legal standards and people’s behavior react to the occurrence of such events, and compare how different legal rules perform in the wake of such change.

1.2. Tort Liability Rules and Unawareness. A central question in law and economics is whether negligence or strict liability is the more efficient tort liability rule. Under negligence, a victim can recover damages for harm caused by an injurer who failed to take reasonable care. Under strict liability, by contrast, a victim can recover damages for harm caused by the activity of an injurer irrespective of the injurer’s level of care. The relative efficiency of the two rules is customarily measured by the Kaldor–Hicks criterion.

A bedrock result in the economic analysis of tort law is that, in the case of unilateral accidents with fixed activity levels, negligence and strict liability are equally efficient, pro- vided that, in the case of negligence, the court properly sets the due care standard (the legal standard for what constitutes reasonable care) (Shavell, 1987).1 This equivalence re- sult, however, presents something of a puzzle in light of two facts about negligence. First, negligence is the dominant rule in Anglo-American law.2 Second, negligence is the more costly rule to administer, because the court must determine the due care standard and adjudicate whether it was met. The puzzle is that if negligence and strict liability are equally efficient but negligence is more costly to administer, why is negligence the dominant rule?

The negligence puzzle has led researchers to revisit the equivalence result by exploring de- partures from the standard accident model, which is based on the expected utility framework and the Bayesian paradigm. For instance, Teitelbaum (2007) and Chakravarty and Kelsey (2017) study ambiguity (Knightian uncertainty). They assume the relevant parties have neo- additive preferences and find that this breaks the equivalence in favor of negligence.3

We examine the implications of growing awareness for tort law, and specifically for the negligence versus strict liability debate. We study cases where the parties initially are un- aware that some acts can yield harmful consequences, or that some acts or harmful conse- quences are even possible, but later become aware. Examples of growing awareness that are relevant to tort law include new acts such as fracking4 and cyberbullying5; new consequences such as HIV/AIDS6 and mad cow disease7; and new act-consequence links such as those

1 In unilateral accidents, the injurer, but not the victim, can take care to reduce expected harm. If the activity level is fixed, the injurer affects expected harm only through her level of care (and not through her level of activity). The equivalence result also holds in the case of bilateral accidents with fixed activity levels, provided that strict liability is coupled with the defense of contributory negligence.

2 In modern Anglo-American law, strict liability applies only in a handful of accident cases, including cases involv- ing abnormally dangerous activities or products with manufacturing defects (Dobbs et al., 2011, section 2). Indeed, certain accident cases that were traditionally governed by strict liability are now governed by negligence, including cases involving products with a design or warning defect (Dobbs et al., 2011, section 450).

3 The neo-additive model was developed by Chateauneuf et al. (2007). Franzoni (2017) models ambiguity according to the smooth model of Klibanoff et al. (2005). He finds that strict liability dominates negligence when the injurer has lower degrees of uncertainty aversion than the victim and can formulate more precise estimates of the probability of harm, but that negligence dominates strict liability when harm is dispersed over a very large number of victims, irre- spective of the parties’ respective degrees of uncertainty aversion.

4 In Ely v. Cabot Oil and Gas Corporation, 2017 WL 1196510 (M.D. Pa. 2017), a Pennsylvania jury found in favor of nine plaintiffs on claims that the defendant’s fracking activity negligently permitted methane to flow into under- ground aquifers that wound up polluting the plaintiffs’ water wells.

5 In D.C. v. R.R., 182 Cal. App. 4th 1190 (2010), a Los Angeles high school student brought an action against sev- eral of his fellow students—who had posted messages at his Web site making derogatory comments about his per- ceived sexual orientation and threatening him with bodily harm—claiming, inter alia, defamation and intentional in- fliction of emotional distress.

6 In Quintana v. United Blood Services, No. 86 Civ. 11750 (Colo. Dist. Ct. 1991), a Denver jury held the defendant liable to the plaintiff for negligently supplying her with HIV-contaminated blood.

7 Mad cow disease is formally known as bovine spongiform encephalopathy (BSE). In 2008, a Winnipeg cattle feed supplier settled a class action by Canadian cattle farmers claiming that the defendant negligently supplied them with BSE contaminated feed in the early 1990s (Dowd, 2008).

tort liability and unawareness 1853

between Agent Orange and cancer8 and between American football and chronic traumatic encephalopathy (CTE).9

In order to model growing awareness, which requires a theory of how beliefs update as the state space expands, we adopt the Reverse Bayesian approach of Karni and Vierø (2013). Re- verse Bayesianism posits that as a person becomes aware of new possibilities, she updates her beliefs in a way that preserves the relative likelihoods of events in the original state space. More specifically, it postulates that (i) in the case of a new act or consequence, probability mass shifts proportionally away from the events in the original state space to the new events in the expanded state space, and (ii) in the case of a new act–consequence link, null events in the original state space become nonnull, and probability mass shifts proportionally away from the original nonnull events to these events.10

Using the Reverse Bayesian approach, we present a model of learning in the presence of unawareness in which injurers, victims, and courts update their beliefs about accident risks in the wake of unforeseen events.11 We argue that negligence has an important advantage over strict liability when there is unawareness because negligence is better at spreading growing awareness about newly discovered risks. For example, consider an injurer who uses a new technology that has unforeseen and harmful consequences to a victim. Initially only the par- ties involved—the injurer, the victim, and the court—are aware of this information. Under strict liability, the rest of society learns that an accident has occurred and that the injurer is liable for the harm. This informs the wider world about the new act and that it is potentially dangerous, but it does not allow outsiders to deduce the probability of harm. This is insuffi- cient information to enable future users of the technology to take efficient care. Under neg- ligence, by contrast, the court’s decision serves to transmit more information to the wider world. This is because the court announces a due care standard for the new technology. We ar- gue that Reverse Bayesianism in combination with act independence enables outsiders to de- duce the probability of harm from the court’s announcement.12 This allows future users of the technology to take efficient care. Negligence thus reveals more valuable information to the wider world than does strict liability. This tends to increase awareness and hence efficiency in the legal system and in the economy more widely.

1.3. Structure of the Article. Subsection 2.1 introduces the accident model—a unilateral accident model featuring multiple activities with fixed levels—and presents the equivalence result. Subsection 2.2 describes the unawareness model. Section 3 compares and contrasts negligence and strict liability in a world with unawareness. It considers a simple model with two acts, two consequences, quadratic care costs, and linear expected harm reduction, and separately analyzes the cases of a new act, a new consequence, and a new act–consequence link. Section 4 extends the analysis to a more general model with arbitrary numbers of acts and consequences, convex care costs, and convex expected harm reduction. In Section 5, we relate our results to the existing literature. We offer concluding remarks in Section 6. The Appendix collects the proofs of all results.

8 In In re Agent Orange Product Liability Litigation, 597 F. Supp. 740 (E.D.N.Y. 1984), Vietnam veterans brought a class action against the manufacturers of Agent Orange alleging, inter alia, that their exposure to the chemical re- sulted in a variety of cancers and other diseases.

9 In In re National Football League Players Concussion Injury Litigation, 821 F.3d 410 (3d Cir. 2016), retired NFL players brought a class action against the NFL alleging that the league had failed to take reasonable actions to protect the players from CTE.

10 A null event is an event believed to have zero probability. 11 This has some relation to the literature on learning from misspecified models; see, for example, Heidhues et al.

(2018, 2021) and the sources cited therein. 12 A similar result may be obtained if acts are not independent but their correlation is known; see Chakravarty

et al. (2022).

1854 tort liability and unawareness

2. framework

2.1. The Accident Model. There are two agents: an injurer and a victim. Both are risk- neutral expected utility maximizers. The agents are strangers and not in a contractual relation- ship. Transaction costs are sufficiently high to preclude Coasean bargaining.

The injurer has available m ≥ 2 activities, f1, . . . , fm, which have the potential to cause harm to the victim. The activities are assumed to satisfy an independence assumption, which we call Act Independence and define below in Subsection 2.2.13

There are n ≥ 2 potential degrees of harm, z1, . . . , zn, where z j ≥ 0 for all j = 1, . . . , n. Ac- tivity fi causes harm z j with probability πi j, where

∑n j=1 πi j = 1 for all i = 1, . . . , m. Thus, ac-

tivity fi’s expected harm is ∑n

j=1 πi jz j. In the absence of unawareness, the agents have correct beliefs about each harm probability πi j.

The injurer engages in each available activity. For each activity fi, the injurer, but not the victim, can take care to reduce the expected harm. The injurer chooses a level of care xi ≥ 0 having cost c(xi). Being careless is costless, c(0) = 0, and the marginal cost of care is posi- tive and increasing: c′(xi) > 0 and c′′(xi) > 0 for all xi. Taking care reduces the activity’s ex- pected harm at a nonincreasing rate: hi(xi) ≡ ∑n

j=1 πi jz jτ (xi), where τ (xi) ∈ (0, 1] for all xi

with τ (0) = 1 and where τ ′(xi) < 0 and τ ′′(xi) ≥ 0 for all xi. Care reduces expected harm by decreasing the probability of harm, the magnitude of harm, or both. We assume that c(·) and τ (·) are known to all parties and are the same for all activities. We make the latter assumption for simplicity. It is without loss of generality given the former assumption.

If activity fi causes harm, then the victim may be entitled to damages, depending on the applicable tort liability rule. Under negligence, the victim is entitled to damages equal to the harm if the injurer’s level of care was below the due care standard stipulated by the court.14

Under strict liability, by contrast, the victim is entitled to damages equal to the harm irrespec- tive of the injurer’s level of care. We assume the injurer has the ability to pay any and all dam- ages to which the victim may be entitled.

The social goal is to minimize the total social costs of the injurer’s activities (the sum of the costs of care and the expected harms):

min x1,...,xm≥0

m∑ i=1

c(xi) + hi(xi).

The solution x̃ = (x̃1, . . . , x̃m) is given implicitly by the first-order conditions c′(x̃i) = −h′

i(x̃i), i = 1, . . . , m, and is given explicitly by

x̃i = ξ−1 (∑n

j=1πi jz j

) , i = 1, . . . , m,

where ξ−1 denotes the inverse of ξ (xi) ≡ −c′(xi)/τ ′(xi).15 We refer to x̃i as the efficient level of care for activity fi. It is the level of care at which the marginal cost of care equals the marginal benefit (the marginal reduction in expected harm).

13 Although Act Independence is a reasonable assumption in many settings, there undoubtedly are settings in which it is not. We explore the implications for our analysis of relaxing Act Independence in Subsection 3.4. For a thorough treatment of Reserve Bayesianism and Act Independence, see Chakravarty et al. (2022).

14 Following in the tradition of the tort law and economics literature, we model the due care standard as a precise stipulation. In reality, the due care standard may be less specific. For a discussion on the specificity of the due care standard at common law, see Dobbs et al. (2011, section 145).

15 Note that ξ ′(xi) = [c′(xi)τ ′′(xi) − τ ′(xi)c′′(xi)] / [τ ′(xi)]2 > 0, for all xi; hence ξ is invertible.

tort liability and unawareness 1855

Under strict liability, the injurer’s problem is identical to the social goal. This is be- cause strict liability forces the injurer to internalize the total social costs of her activities. Hence, she will take efficient care in each activity. Under negligence, the injurer’s problem is

min x1,...,xm≥0

m∑ i=1

c(xi) + hi(xi)χ (xi < xi),

where xi is the due care standard for activity fi and χ (xi < xi) is the indicator function de- fined by χ (xi < xi) = 1 if xi < xi and χ (xi < xi) = 0 otherwise. If the court sets xi = x̃i for all i, then the injurer takes efficient care in each activity. The reason is twofold. First, the injurer will not take more than the efficient level of care, because she faces no liability if her level of care equals or exceeds the efficient level. Second, the injurer will not take less than the ef- ficient level of care, because then she faces strict liability, which induces her to take efficient care.

The equivalence result follows immediately from the foregoing.

Theorem 1 (Equivalence Result). The injurer will take efficient care in each activity under either negligence or strict liability, provided that, in the case of negligence, the court sets the due care standard for each activity equal to the efficient level of care for that activity.

2.2. Unawareness. We model unawareness following Karni and Vierø (2013). The primi- tives of the model are a finite set F of feasible acts and a finite set Z of feasible consequences. In our setting, the feasible acts are the injurer’s available activities and the feasible conse- quences are the potential harms to the victim. States are functions from the set of acts to the set of consequences. A state assigns a consequence to each act. The set of all possible states, ZF , defines the conceivable state space. With m acts and n consequences, there are nm conceiv- able states.

The agents and the court (collectively, the parties) originally conceive the sets of acts and consequences to be F = { f1, . . . , fm} and Z = {z1, . . . , zn}. The conceivable state space is ZF = {s1, . . . , snm}, where each state s ∈ ZF is a vector of length m, the ith element of which, si, is the consequence z j ∈ Z produced by act fi ∈ F in that state of the world.

An act–consequence link, or link, is a relationship between an act and a consequence. It indicates that an act will lead to a given consequence in a particular state. The conceivable state space admits all conceivable links. However, the parties may perceive one or more links as infeasible, which brings them to nullify the states that admit such links. We refer to these as null states and denote them by N ⊂ ZF . Taking only the nonnull states defines the feasible state space, S ≡ ZF \N. When N �= ∅, there are

∏m i=1(n − νi) feasible states, where νi denotes

the number of nullified links involving act fi. The parties have common beliefs represented by a probability distribution p on the con-

ceivable state space, ZF . The support set of p is the feasible state space, S. That is, the parties assign nonzero probability to each nonnull state.

Example 1. Consider the case with two acts and two consequences: F = { f1, f2} and Z = {z1, z2}. As illustrated in the following table, the conceivable state space, ZF , comprises four states—s1 = (z1, z1), s2 = (z1, z2), s3 = (z2, z1), and s4 = (z2, z2)—and the parties beliefs are p = (p1, . . . , p4) where pk ≡ p(sk) for all k.

1856 tort liability and unawareness

Suppose the parties believe that act f1 cannot yield consequence z2. That is, suppose they perceive the event {s3, s4} as infeasible (null), which implies p3 = p4 = 0. Then, we can depict the feasible state space, S ⊂ ZF , and the parties’ beliefs, p, as follows:

In general, the parties may initially fail to conceive one or more acts or consequences or to perceive as feasible one or more conceivable links. We refer to such failures of conception or perception as unawareness. However, the parties may later discover a new act or consequence, which expands both the feasible state space and the conceivable state space, or a new link, which expands the feasible state space but not the conceivable state space.16 We refer to such discoveries and expansions as growing awareness.

In the wake of growing awareness, the parties’ beliefs update in a way that preserves the relative likelihoods of the events in the original feasible state space. In each case, probability mass shifts proportionally away from the events in the original feasible state space to the new events in the expanded feasible state space. In the case of a new act or consequence, the new events in the expanded feasible state space are also new events in the expanded conceivable state space. In the case of a new link, the new events in the expanded feasible state space are formerly null events in the original conceivable state space.

Karni and Vierø (2013) refer to this updating as Reverse Bayesianism. Let Ŝ denote the expanded feasible state space and p̂ denote the parties’ updated beliefs. Formally, Reverse Bayesianism implies: (i) in the case of a new consequence or link, p(s)/p(t) = p̂(s)/p̂(t) for all s, t ∈ S; and (ii) in the case of a new act, p(s)/p(t) = p̂(E(s))/p̂(E(t)) for all s, t ∈ S, where E(s) denotes the event in Ŝ that corresponds to state s in S; that is, E(s) ≡ {t ∈ Ŝ : ti = si,∀ i �= m + 1} (assuming the new act is fm+1).

Remark 1. The key point of comparison between the standard Bayesian and Reverse Bayesian models is the conceivable state space. In the Bayesian model, the conceivable state space is given and does not change. The arrival of new information can only render some states null. In the Reverse Bayesian model, by contrast, the conceivable state space is not given and can change. The arrival of new information can add new states.

16 To be clear, by “new,” we mean “not previously conceived” in the case of acts and consequences, and “previously conceived but perceived as infeasible” in the case of links.

tort liability and unawareness 1857

In Chakravarty et al. (2022), we show how strengthening Reverse Bayesianism by assuming Act Independence pins down p̂.17 We define Act Independence formally below.

Assumption 1 (Act Independence). Let Ai(z j ) ⊂ Ŝ denote the event that act fi yields conse- quence z j; that is, Ai(z j ) ≡ {t ∈ Ŝ : ti = z j} is the collection of states in which act fi yields conse- quence z j. We refer to events of this type as act events, and for each act event Ai(z j ) we refer to the act fi as the predicate act. Act Independence holds if for every collection of act events in Ŝ such that no two act events have the same predicate act, the act events in the collection are mutu- ally independent.

Act Independence is the natural extension of the notion of statistical independence to our framework. It says that the outcome of act fi yields no information about the outcome of an- other act fl . Consider an injurer who has two available acts, f1 and f2. Suppose the injurer dis- covers that f1 is linked to a new harmful consequence. This is not informative about the out- come of f2. Thus, it is not necessary to increase precautions when performing f2. However, extra care is required when performing f1.

Act Independence implies additional restrictions on p̂. Observe that we can express each state s = (s1, . . . , sm) as the intersection of a unique collection of act events: s = ⋂

i Ai(si). We refer to this collection as the constituent act events for state s. Act Independence im- plies that the probability of state s equals the product of the probabilities of its con- stituent act events: p̂(s) = ∏

i p̂ ( Ai(si)

) . Act Independence also implies that for any act

event Ai(z j ), its conditional probability given any collection of other act events equals its unconditional probability. In other words, Act Independence implies that the probability that act fi yields consequence z j is independent of the outcomes of any and all other acts fl , i �= l.

2.3. Unawareness and Accidents. In the next two sections (Sections 3 and 4), we present our analysis of tort law in a world with unawareness. In order to preview our analysis, con- sider an example of a new consequence. Suppose the injurer has two available acts, f1 and f2, and the parties initially believe that each act has two potential consequences, no harm z1 = 0 or small harm z2 > 0. We can depict the original feasible state space, S = ZF , as follows:

17 We leverage certain results from Chakravarty et al. (2022) in our analysis below. The proofs of our results specify the extent to which they follow from results in Chakravarty et al. (2022).

1858 tort liability and unawareness

Suppose the parties discover a new consequence, large harm z3 > z2, which they link to f1 and f2. Then, the feasible state space expands from four to nine states:

The expanded feasible state space is characterized by three events, one in which f1 results in no harm, one in which f1 results in small harm z2, and one in which f1 results in large harm z3. Each event contains three states, one in which f2 results in no harm, one in which f2 results in small harm z2, and one in which f2 results in large harm z3.18

Reverse Bayesianism implies that relative likelihoods of the four original states remain un- changed. So, for instance, p1/p2 = p̂1/p̂2. In our analysis, we show that with knowledge of the probabilities of the new act events A1(z3) = {s5, s6, s9} and A2(z3) = {s7, s8, s9}, which are the events that f1 yields z3 and f2 yields z3, Reverse Bayesianism and Act Independence fully determine the updated probability distribution p̂. This enables the court to stipulate new due care standards for f1 and f2, which in turn enables outsiders to learn the new expected harms of f1 and f2, which is the information they need to take efficient care.

3. illustrative results

In this section and the next, we compare and contrast negligence and strict liability in a world with unawareness. Throughout, we assume the parties are fully rational apart from un- awareness. We further assume that when the parties are unaware of an act, consequence, or link, their beliefs, although incorrect with respect to the absolute likelihoods of events, are nevertheless correct with respect to the relative likelihoods of nonnull events. Without this assumption, the parties would not have correct beliefs when they become fully aware, which would be inconsistent with the standard accident model.

In this section, we analyze a simple model, which illustrates our main ideas. We show that these results hold more generally in the next section. In both sections, for simplic- ity, we maintain the assumption that c(·) and τ (·) are known to all parties and are the same for all activities. The latter assumption is without loss of generality given the former assumption.

The simple model has two activities, F = { f1, f2}; two consequences, Z = {z1, z2}, where z1 = 0 (no harm) and z2 > 0 (positive harm); quadratic care costs, c(xi) = (xi)2; and linear expected harm reduction, τ (xi) = (1 − xi).19 The conceivable state space, ZF , comprises four states: s1 = (0, 0), s2 = (0, z2), s3 = (z2, 0), and s4 = (z2, z2). Let pk ≡ p(sk), k = 1, . . . , 4, de- note the parties’ common beliefs on ZF .

3.1. New Link. We start with the case of a new link. We assume the parties initially per- ceive activity f1 as safe (i.e., incapable of causing harm) and activity f2 as risky (i.e., capable of causing harm). That is, we assume they initially perceive the event A1(z2) = {s3, s4} as infeasi-

18 Note that the conceivable state space also expands from four to nine states, so Ŝ = ẐF . 19 In order to preserve the condition τ (xi) > 0 for all xi, we assume xi ∈ [0, 1) in this section.

tort liability and unawareness 1859

ble (null). This implies p3 = p4 = 0. We can depict the original feasible state space, S ⊂ ZF , as follows:

Given S and p, the efficient levels of care are x̃1 = 0 and x̃2 = p2z2/2. Under negligence, the court stipulates x1 = x̃1 and x2 = x̃2 as the due care standards for f1 and f2, respectively.

Suppose the parties discover that activity f1 is risky. In particular, suppose that the injurer engages in f1, that it results in harm z2, and that the victim brings a tort suit against the in- jurer. The feasible state space expands from two to four states to reflect the discovery that f1

can yield z2,20 and the parties update their beliefs to p̂:

We assume that, by virtue of the suit, the parties learn that activity f1 yields harm z2 with probability δ > 0. The fact is the parties have the incentive to expend resources to develop this knowledge. As the Hand formula makes plain,21 the probability of harm is an essential component of a negligence case. Even in a strict liability case, the probability of harm is rel- evant to the issues of foreseeability and proximate cause.22

Note that δ = p̂(A1(z2)) = p̂3 + p̂4 is the total probability of the new states in the expanded state space. It is a measure of the likelihood of the act event of which the parties were previ- ously unaware. Thus, we interpret δ as the degree of unawareness.

Reverse Bayesianism implies that the relative likelihood of states s1 and s2 remains the same after updating, hence

p1

p2 = p̂1

p̂2 .

Note that p1 + p2 = 1. Because p̂1 + p̂2 = 1 − δ, it follows that p̂1 = (1 − δ)p1 and p̂2 = (1 − δ)p2. Act Independence implies that the odds that activity f2 results in harm is the same whether or not activity f1 results in harm, hence

p̂1

p̂2 = p̂3

p̂4 .

20 The feasible state space now coincides with the conceivable state space, that is, Ŝ = ZF . 21 See United States v. Carroll Towing Co., 159 F.2d 169 (2d Cir. 1947); Dobbs et al. (2011, section 161). 22 Alternatively, we could assume the court learns δ by virtue of a sequence of suits (cf. Ott and Schäfer, 1997; Feess

and Wohlschlegel, 2006).

1860 tort liability and unawareness

Because p̂3 + p̂4 = δ, it follows that p̂3 = δp1 and p̂4 = δp2. Thus, Reverse Bayesianism (and knowledge of δ) pins down p̂. The following proposition recaps the foregoing results.

Proposition 1. Assume there are two acts and two consequences. Suppose the parties dis- cover a new link between an act and a consequence and learn that the corresponding new act event has probability δ. Under Reverse Bayesianism and Act Independence, the updated proba- bilities are p̂1 = (1 − δ)p1, p̂2 = (1 − δ)p2, p̂3 = δp1, and p̂4 = δp2.

Remark 2. Note that p is the Bayesian update of p̂ conditional on the event S = {s1, s2} (i.e., the original feasible state space), hence the term Reverse Bayesianism. In the Bayesian model, δ represents the total probability of the nullified states.

Remark 3. Chakravarty et al. (2022) show that if Act Independence does not hold, the up- dated probabilities are p̂1 = (1 − δ) p1, p̂2 = (1 − δ) p2,

p̂3 = δp1 + δρ2

2(1−δ+δρ2 )

[ 1 − 2p1 + √

1 + (4p1(1 − δ)(1 − p1)/δρ2) ] ,

and p̂4 = δp2 − δρ2

2(1−δ+δρ2 )

[ 1 − 2p1 + √

1 + (4p1(1 − δ)(1 − p1)/δρ2) ] ,

where ρ = ρ (A1(z2), A2(z1)) denotes the correlation between the new event A1(z2) = {s3, s4} where f1 yields z2 and the event A2(z1) = {s1, s3} where f2 yields z1. In the case of p̂3 and p̂4, the first term is the updated probability under Reverse Bayesianism if Act Independence holds and the second term corrects for possible correlation between the act events. Thus, Re- verse Bayesianism (and knowledge of δ) still pins down p̂, given knowledge of ρ. The results with Act Independence can be viewed as a special case of this more general result.

Given Ŝ and p̂, the efficient levels of care are

̂̃x1 = ( p̂3 + p̂4)z2

2 = δz2

2 and ̂̃x2 = ( p̂2 + p̂4)z2

2 = p2z2

2 .

Note that ̂̃x1 > x̃1 but ̂̃x2 = x̃2. Thus, the discovery that activity f1 is risky necessitates the stip- ulation of a new due care standard for f1 but not for f2.

Under negligence, the court stipulates x̂1 = ̂̃x1 as the new due care standard for f1 and holds the injurer liable to pay damages of z2 to the victim.23 This makes outsiders aware that f1 is risky. Moreover, they can deduce δ from x̂1; specifically, δ = 2̂x̃1/z2. As a result, they can learn p̂ and ĥ1(x1) = δz2τ (x1), without expending additional resources to learn about δ. This is the information other injurers need in order to take efficient care.

Under strict liability, the court simply holds the injurer liable to pay damages of z2 to the victim. This makes outsiders aware that f1 can potentially cause harm. However, they cannot deduce δ or learn p̂ or ĥ1(x1). Hence, other injurers lack sufficient information to take effi- cient care.

Remark 4. If c(·) and τ (·) are the same for all injurers, then knowledge of ĥ1(x1) is not strictly necessary for them to take efficient care under negligence. They can just blindly adopt x̂1 as their level of care, without bothering to deduce δ from x̂1 and learn ĥ1(x1). If, however, either c(·) or τ (·) varies across injurers, then they need to deduce δ from x̂1 in order to learn

23 Recall that x1 = x̃1 = 0 before the parties discover that f1 is potentially harmful. Under negligence, therefore, the injurer will have exercised no care in conjunction with f1. However, even if the court does not hold the injurer li- able and award damages (perhaps by recognizing a civil ex post facto doctrine which prohibits retroactive application of a due care standard in a negligence suit), our results below would not change, because the world already knows the set of potential harms.

tort liability and unawareness 1861

their own ĥ1(x1), which is necessary for them to take efficient care. Moreover, injurers always need to know ĥ1(x1) in order to take efficient care under strict liability, because the court does not stipulate a due care standard in this case.

3.2. New Act. We next consider the case of a new act. Assume the original feasible state space coincides with the conceivable state space, that is, S = ZF :

The efficient levels of care are x̃1 = [(p3 + p4)z2] /2 and x̃2 = [(p2 + p4)z2] /2. Under negli- gence, the court stipulates x1 = x̃1 and x2 = x̃2 as the due care standards for f1 and f2, respec- tively.

Suppose the parties discover a new activity, f3, which has the potential to cause harm. In particular, suppose that the injurer discovers and engages in f3, that it results in z2, and that the victim brings a tort suit against the injurer before the court. The feasible state space ex- pands from four to eight states:

The expanded feasible state space contains two copies of the original feasible state space, one in which f3 results in no harm and one in which f3 results in harm z2. Stated differently, the expanded space splits each of the original states into two depending on whether or not f3

yields positive harm. For each original state, there is a corresponding event in the expanded feasible state space. For instance, the event {s1, s5} ∈ Ŝ corresponds to state s1 ∈ S, the event {s2, s6} ∈ Ŝ corresponds to state s2 ∈ S, and so forth.24

As before, we assume that, by virtue of the suit, the parties learn that activity f3 yields harm z2 with probability δ > 0. By Reverse Bayesiansim, p1/p2 = ( p̂1 + p̂5) / ( p̂2 + p̂6). By Act In- dependence, p̂1/p̂2 = p̂5/p̂6. Substituting,

p1

p2 =

p̂1 + p̂1 p̂6 p̂2

p̂2 + p̂6 = p̂1

p̂2 = p̂5

p̂6 .

By similar reasoning, p2/p3 = p̂2/p̂3 = p̂6/p̂7 and p3/p4 = p̂3/p̂4 = p̂7/p̂8. Because p1 + p2 + p3 + p4 = 1, p̂1 + p̂2 + p̂3 + p̂4 = 1 − δ, and p̂5 + p̂6 + p̂7 + p̂8 = δ, we have the following re- sult.

24 Note that the conceivable state space also expands from four to eight states, so Ŝ = ZF̂ .

1862 tort liability and unawareness

Proposition 2. Assume there are two acts and two consequences. Suppose the parties dis- cover a new act, which they link to both consequences, and learn that the corresponding new act events have probabilities 1 − δ and δ. Under Reverse Bayesiansm and Act Independence, the updated probabilities are p̂1 = (1 − δ)p1, p̂2 = (1 − δ)p2, p̂3 = (1 − δ)p3, p̂4 = (1 − δ)p4, p̂5 = δp1, p̂6 = δp2, p̂7 = δp3, and p̂8 = δp4.

Given Ŝ and p̂, the efficient levels of care are

̂̃x1 = ( p̂3+p̂4+p̂7+p̂8 )z2

2 = (p3+p4 )z2 2 ,̂̃x2 = ( p̂2+p̂4+p̂6+p̂8 )z2

2 = (p2+p4 )z2 2 ,

and ̂̃x3 = ( p̂5+p̂6+p̂7+p̂8 )z2

2 = δz2 2 .

Thus, the discovery of f3 necessitates the stipulation of a new due care standard, x̂3, while the due care standards for f1 and f2 are unchanged.

Under negligence, the court stipulates x̂3 = ̂̃x3 as the due care standard for the new activ- ity f3 and holds the injurer liable to pay damages of z2 to the victim.25 This makes outsiders aware of f3 (and that it is risky). Moreover, they can deduce δ from x̂3; specifically, δ = 2̂x̃3/z2. As a result, they can learn p̂ and ĥ3(x3) = δz2τ (x3). This is sufficient information for other in- jurers to take efficient care.

As before, however, strict liability does not reveal sufficient information to other injurers. Under strict liability, the court simply holds the injurer liable to pay damages of z2 to the vic- tim. This makes outsiders aware of f3 (and that it is risky), but they cannot deduce δ or learn p̂ or ĥ3(x3).

3.3. New Consequence. This case has already been introduced in Subsection 2.3. As be- fore, we assume S = ZF :

The efficient levels of care are x̃1 = [(p3 + p4)z2] /2 and x̃2 = [(p2 + p4)z2] /2. Under negli- gence, the court stipulates x1 = x̃1 and x2 = x̃2 as the due care standards for f1 and f2, respec- tively.

Suppose that the injurer engages in f1 and f2, that each results in a new harm z3 > z2, and that the victim brings a tort suit against the injurer before the court. The feasible state space expands from four to nine states as illustrated in Subsection 2.3.

We assume that, by virtue of the suit, the parties learn that activity f1 yields z3 with proba- bility α1 > 0 and that activity f2 yields z3 with probability α2 > 0. (This is analogous to assum- ing that the parties learn the probabilities of the new act event(s) in the previous cases.) As before, the degree of unawareness is the total probability of the new states, that is, δ = p̂5 + · · · + p̂9. Under Act Independence, 1 − δ = p̂(Ac

1(z3) ⋂

Ac 2(z3)) = (1 − α1) (1 − α2).26 The en-

suing result follows.

25 Again, because the world already knows the set of potential harms, our results below would not change if the court does not hold the injurer liable and award damages.

26 For the avoidance of doubt, Ac i (z j ) denotes the complement of the act event Ai(z j ).

tort liability and unawareness 1863

Proposition 3. Assume there are two acts and two consequences. Suppose the parties dis- cover a new consequence, which they link to both acts, and learn that the corresponding new act events have probabilities α1 and α2. Under Reverse Bayesianism and Act Indepen- dence, the updated probabilities are p̂1 = (1 − δ)p1, p̂2 = (1 − δ)p2, p̂3 = (1 − δ)p3, p̂4 = (1 − δ)p4, p̂5 = α1(1 − α2)(p1 + p3), p̂6 = α1(1 − α2)(p2 + p4), p̂7 = (1 − α1)α2(p1 + p2), p̂8 = (1 − α1)α2(p3 + p4), and p̂9 = α1α2.

Given Ŝ and p̂, the efficient levels of care are

̂̃x1 = ( p̂3+p̂4+p̂8 )z2+( p̂5+p̂6+p̂9 )z3 2 = (1−α1 )(p3+p4 )z2+α1z3

2 ,

and ̂̃x2 = ( p̂2+p̂4+p̂6 )z2+( p̂7+p̂8+p̂9 )z3 2 = (1−α2 )(p2+p4 )z2+α2z3

2 .

Note that ̂̃x1 > x̃1, and ̂̃x2 > x̃2. Thus, the discovery of z3 necessitates the stipulation of new due care standards for both f1 and f2.27

Under negligence, the court stipulates x̂1 = ̂̃x1 and x̂2 = ̂̃x2 as the new due care standards for f1 and f2, respectively. The court holds the injurer liable to pay damages of z3 to the victim with respect to each of f1 and f2. This makes outsiders aware of z3 (and that it is linked to f1

and f2).28 Moreover, they can deduce α1 and α2 (and hence δ) from x̂1 and x̂2:

α1 = p3z2 + p4z2 − 2̂x̃1

p3z2 + p4z2 − z3 and α2 = p2z2 + p4z2 − 2̂x̃2

p2z2 + p4z2 − z3 .

As a result, they can learn p̂, ĥ1(x1) = [(1 − α1)(p3 + p4)z2 + α1z3]τ (x1), and ĥ2(x2) = [(1 − α2)(p2 + p4)z2 + α2z3]τ (x2), without expending additional resources to learn α1 and α2. Thus, negligence reveals sufficient information for other injurers to take efficient care.

Under strict liability, the court simply holds the injurer liable to pay damages of z3 for each instance of harm. This makes outsiders aware of z3 (and that it is linked to f1 and f2). How- ever, they cannot deduce α1 or α2 or learn p̂, ĥ1(x1), or ĥ2(x2). As before, strict liability yields insufficient information for other injurers to take efficient care.

3.4. Act Independence. Before turning to the general results, we conclude this section with a few remarks about Act Independence. In short, we argue that it is a useful simplifying as- sumption, but that it is not crucial. Even without Act Independence, negligence would reveal valuable information.

Reverse Bayesianism alone does not fully determine the updated probability distribution p̂ in the wake of growing awareness. The reason is that Reverse Bayesianism prescribes how probability mass shifts away from nonnull states in the original state space to the correspond- ing states or events in the expanded state space, but it does not dictate how this mass is dis- tributed among the new states. This is where Act Independence comes in. It determines how the shifted probability mass is apportioned among the new states. There is a unique updated probability p̂ that satisfies Reverse Bayesianism and Act Independence.

How realistic is Act Independence? The answer depends on the nature of the specific activi- ties in question. For instance, the risk that fracking for natural gas results in groundwater con- tamination is likely to be independent of the risk that importing liquefied natural gas results in a fire or explosion. By contrast, the risk of contracting HIV from sharing drug injection nee- dles is likely to be correlated with the risk of contracting HIV from having unprotected sex, as both depend on the prevalence of HIV in the population.

27 One might wonder if the results would be different if only one act, say f1, were linked to the new consequence. This would imply α2 = 0. With this substitution the above results would remain valid.

28 Even if the court does not hold the injurer liable and award damages, the victim’s claims make outsiders aware of z3 (and that it is linked to f1 and f2).

1864 tort liability and unawareness

Because there exist activities whose outcomes are not independent, it is useful to investi- gate the importance of Act Independence for our results. As above, we consider the simple case of two acts and two consequences.

New link: In the case of a new link, Reverse Bayesianism alone implies p̂1 = (1 − δ)p1, p̂2 = (1 − δ)p2, and p̂3 + p̂4 = δ. Importantly, Reverse Bayesianism alone does not separately determine p̂3 and p̂4. As it turns out, this does not create an issue with respect to activity f1. Recall that, by assumption, the parties learn δ (the probability that f1 yields z2). Because the efficient level of care for f1 is a function of the sum p̂3 + p̂4, the court can stipulate a new due care standard for f1 in terms of δ. This is sufficient to make outsiders aware that f1 is poten- tially harmful. Moreover, they can deduce δ from the new due care standard for f1 and, in turn, learn ĥ1(x1) = δz2τ (x1).

Relaxing Act Independence, however, creates ambiguity with respect to the updated risk of activity f2. Because the efficient level of care for f2 is a function of the sum p̂2 + p̂4, without Act Independence (or another assumption that separately determines p̂3 and p̂4), the court cannot stipulate a precise new due care standard for f2. The best the court can do is specify lower and upper bounds, using the knowledge that p̂4 ∈ (0, δ). Given these bounds, the best outsiders can do is infer bounds on ĥ2(x2).

Of course, the ambiguity can be resolved if the parties learn more about p̂. For instance, if the parties learn not only δ but also either p̂2 + p̂4 (the updated probability that f2 yields z2) or p̂4 (the joint probability that f1 and f2 yield z2), this is sufficient to separately determine p̂3

and p̂4. With this, the court can stipulate a precise new due care standard for f2, from which outsiders can deduce p̂2 + p̂4 and, in turn, learn ĥ2(x2) = ( p̂2 + p̂4)z2τ (x2).

New act: In the case of a new act, Reverse Bayesianism alone implies p̂1 + p̂5 = p1, p̂2 + p̂6 = p2, p̂3 + p̂7 = p3, p̂4 + p̂8 = p4, and p̂5 + p̂6 + p̂7 + p̂8 = δ.29 Recall that the efficient level of care for f1 is a function of the sum p̂3 + p̂4 + p̂7 + p̂8; the efficient level of care for f2

is a function of the sum p̂2 + p̂4 + p̂6 + p̂8; and the efficient level of care for f3 is a function of the sum p̂5 + p̂6 + p̂7 + p̂8. Hence, even without Act Independence, the court’s information is sufficiently precise (i) to know that it need not stipulate new due care standards for activities f1 and f2 and (ii) to stipulate a due care standard for the new activity f3. This makes outsiders aware of f3 (and that it is risky). Moreover, they can deduce δ from the due care standard for f3 and, in turn, learn ĥ3(x3) = δz2τ (x3).

New consequence: In the case of a new consequence, Reverse Bayesianism alone implies p̂1 = (1 − δ)p1, p̂2 = (1 − δ)p2, p̂3 = (1 − δ)p3, p̂4 = (1 − δ)p4, and p̂5 + · · · + p̂9 = δ.30 By assumption, the parties learn p̂5 + p̂6 + p̂9 = α1 (the probability that f1 yields z3) and p̂7 + p̂8 + p̂9 = α2 (the probability that f2 yields z3). Assume the parties also learn p̂9 (the joint probability that f1 and f2 yield z3), and let p̂9 = γ . Note that δ = α1 + α2 − γ .

Recall that the efficient level of care for f1 is a function of α1 and the sum p̂3 + p̂4 + p̂8

(the updated probability that f1 yields z2), and the efficient level of care for f2 is a function of α2 and the sum p̂2 + p̂4 + p̂6 (the updated probability that f2 yields z2). Without Act In- dependence, these sums are not separately determined (because p̂6 and p̂8 are not separately determined), creating ambiguity with respect to the updated risks of both activities. As a re- sult, the court cannot stipulate precise new due care standards for either activity. The best the court can do is specify lower and upper bounds. Given these bounds, and given that the vic- tim’s claims make the world aware of z3 (and its links to f1 and f2), outsiders can deduce α1, α2, and δ; however, the best they can do is infer bounds on ĥ1(x1) and ĥ2(x2).

As before, the ambiguity can be resolved if the parties learn more about p̂. For instance, if the parties learn not only δ and γ but also either p̂3 + p̂4 + p̂8 or p̂2 + p̂4 + p̂6, this is suffi- cient to separately determine p̂5, p̂6, p̂7, and p̂8. With this, the court can stipulate precise new due care standards for f1 and f2, from which outsiders can learn ĥ1(x2) and ĥ2(x2).

29 See the proof of Proposition 2 in the Appendix. 30 See the proof of Proposition 3 in the Appendix.

tort liability and unawareness 1865

In summary, without Act Independence, Reverse Bayesianism only partially determines p̂. This does not create an issue in the case of a new act—the court’s information is suffi- ciently precise to stipulate a due care standard for each activity. In the case of a new link or consequence, however, it creates ambiguity with respect to the updated risk of one or both activities, leading to imprecise due care standards. In short, we might say that, without Act In- dependence, negligence achieves only “boundedly” optimal tort deterrence. That said, negli- gence still has a partial advantage over strict liability. What’s more, the ambiguity in any case can be resolved if the parties learn more about p̂. In other words, the more the parties learn about the updated probability of harm, the less important is Act Independence.

4. general model

In this section, we extend the examples from the previous section to a more general model with m acts and n consequences. We also relax the shape restrictions on the care cost and ex- pected harm reduction functions and assume only that each is convex.

Let F = { f1, . . . , fm} be the set of activities and Z = {z1, . . . , zn} be the set of harms, where 0 ≤ z1 < z2 < · · · < zn. For each activity fi, the cost of taking care xi ≥ 0 is c(xi), where c(0) = 0, c′(xi) > 0, and c′′(xi) > 0 for all xi ≥ 0. Activity fi’s expected harm is hi(xi) ≡∑n

j=1 πi jz jτ (xi), where (i) πi j is the probability that fi causes z j and (ii) τ (xi) ∈ (0, 1], τ (0) = 1, τ ′(xi) < 0, and τ ′′(xi) ≥ 0 for all xi ≥ 0.31

Given F and Z, the conceivable state space is ZF , where each state s ∈ ZF is a vector of length m, the ith element of which, si, is the harm z j ∈ Z caused by activity fi ∈ F in that state. The feasible state space is S ≡ ZF \N, where N ⊂ ZF is the set of null states. Each state in N is induced by a nullified link between an activity fi and a harm z j.

Let p represent the parties’ common beliefs on ZF . The support set of p is S. That is, p(s) >

0 for all s ∈ S and p(s) = 0 for all s ∈ N.

Given S and p, the efficient levels of care are x̃i = ξ−1 (∑n

j=1πi jz j

) , i = 1, . . . , m, where

(i) ξ−1 denotes the inverse of ξ (xi) ≡ −c′(xi)/τ ′(xi) and (ii) πi j = ∑ s∈S:si=z j

p(s). Under neg- ligence, the court stipulates xi = x̃i as the due care standard for each activity fi.

4.1. New Link. Assume S ⊂ ZF . Suppose the parties discover a new link from fl to zk for some l ∈ {1, . . . , m} and k ∈ {1, . . . , n}. Let Ŝ denote the expanded feasible state space and p̂ denote the parties’ updated beliefs on Ŝ. Observe that Ŝ = S ∪ �, where � = Al (zk) is the newly discovered event that fl yields zk. Intuitively, � is a copy of any one of the act events Al (z j ) in S, except that fl yields zk (instead of z j) in every state in �. By virtue of a tort litiga- tion, the parties learn that fl yields zk with probability δ > 0. By definition, δ = p̂(�).

For each state s ∈ �, let L(s) ≡ {t ∈ S : ti = si, ∀ i �= l} denote the event in S that corre- sponds to the state s ∈ �. In other words, L(s) comprises the states in S in which every activity (other than fl) yields the same consequence that it yields in state s ∈ �.

By Reverse Bayesianism, the relative likelihoods of the states in S are preserved: p(s)/p(t) = p̂(s)/p̂(t) for all s, t ∈ S. By Act Independence, the probability of each state in Ŝ equals the product of the probabilities of its constituent act events in Ŝ: p̂(s) =∏m

i=1 p̂ ( Ai(si)

) for all s = (s1, . . . , sm) ∈ Ŝ. Given Ŝ and p̂, the efficient levels of care are ̂̃xi =

ξ−1 (∑n

j=1 π̂i jz j

) , i = 1, . . . , m, where π̂i j = ∑

s∈Ŝ:si=z j p̂(s). It follows that:

Proposition 4. Assume Reverse Bayesianism and Act Independence. If the parties discover a new link from fl to zk, then:

(a) p̂(s) = (1 − δ)p(s) for all s ∈ S. (b) p̂(s) = δp (L(s)) for all s ∈ �.

31 For example, we could have τ (xi) = e−xi .

1866 tort liability and unawareness

(c) ̂̃xl = ξ−1 (∑n

j=1(1 − δ)πl jz j + δzk

) .

(d) ̂̃xi = x̃i for all i �= l.

Because activity fl is newly linked to harm zk, the due care standard for fl changes. If zk is less than the activity’s prior expected harm, the standard is reduced. Otherwise it is increased. The standard is unchanged only in the knife-edge case where the newly linked harm exactly equals the activity’s prior expected harm. This is stated formally below.

Corollary 1. The due care standard ̂̃xl increases (resp. decreases)—that is, ̂̃xl > x̃l (resp. ̂̃xl <

x̃l )—if and only if zk > ∑n

j=1πl jz j (resp. zk < ∑n

j=1πl jz j).

Under negligence, the court stipulates x̂l = ̂̃xl as the new due care standard for fl and holds the injurer liable to pay damages of zk if x̂l > xl . This, along with the victim’s claim, makes outsiders aware that fl can yield zk. Moreover, they can deduce δ from x̂l .32

Proposition 5. In the case of a new link from fl to zk,

δ = c′ (̂xl ) + ∑n j=1πl jz jτ

′ (̂xl )∑n j=1πl jz jτ ′ (̂xl ) − zkτ ′ (̂xl )

.

As a result, outsiders can learn p̂ and ĥl (xl ) = ∑n j=1 [(1 − δ)πi jz j + δzk] τ (xl ). This is the in-

formation that other injurers need to take efficient care. Under strict liability, the court simply holds the injurer liable to pay damages of zk. This

makes outsiders aware that fl can yield zk, but they cannot deduce δ or learn p̂ or ĥl (xl ). Strict liability does not reveal sufficient information for other injurers to take efficient care.

4.2. New Act. Assume S ⊆ ZF . Suppose the parties discover a new activity, fm+1. Again, let Ŝ denote the expanded feasible state space and p̂ denote the parties’ updated beliefs on Ŝ. Observe that Ŝ = ⋃n

j=1 � j where � j = Am+1(z j ) is the newly discovered event that fm+1

yields z j. Intuitively, each � j is an augmented copy of S in which fm+1 yields z j in every state. By virtue of a tort litigation, the parties learn that fm+1 yields z j with probability δ j > 0 for all j = 1, . . . , n.33 Note that δ j = p̂(� j ) and

∑n j=1 δ j = 1.

For each state s ∈ S, let E(s) ≡ {t ∈ Ŝ : ti = si, ∀i �= m + 1} denote the event in Ŝ that cor- responds to the state s ∈ S. In other words, E(s) comprises the states in Ŝ in which every act (other than fm+1) yields the same consequence that it yields in state s ∈ S. Observe that Ŝ =⋃

s∈S E(s) where E(s) comprises n states, one in which fm+1 yields z1, one in which fm+1 yields z2, and so forth. Index the states in each E(s) by j = 1, . . . , n such that s j ∈ E(s) is the state in E(s) in which fm+1 yields z j. The connection between the sets of events {E(s) : s ∈ S} and {� j : j = 1, . . . , n}, both of which partition Ŝ, is that � j collects the jth state from each E(s).

By Reverse Bayesianism, p(s)/p(t) = p̂(E(s))/p̂(E(t)) for all s, t ∈ S. By Act Indepen- dence, p̂(s) = ∏m+1

i=1 p̂ ( Ai(si)

) for all s = (s1, . . . , sm+1) ∈ Ŝ. Given Ŝ and p̂, the efficient levels

of care are ̂̃xi = ξ−1 (∑n

j=1π̂i jz j

) , i = 1, . . . , m + 1, where π̂i j = ∑

s∈Ŝ:si=z j p̂(s). It follows that:

Proposition 6. Assume Act Independence and Reverse Bayesianism. If the parties discover a new act fm+1, then:

(a) For all s ∈ S and corresponding E(s) ⊂ Ŝ, p̂(s j ) = δ j p(s), ∀ s j ∈ E(s), j = 1, . . . , n.

32 The formula in Proposition 5 is not necessarily as complex as it seems. For instance, δ = 2̂xl/zk in the example from Subsection 3.1.

33 Assuming δ j > 0 for all j is without loss of generality. We can deal with the case where δ j = 0 for some j by as- suming δ j > 0 for the first k < n and changing n to k as necessary in the statements below.

tort liability and unawareness 1867

(b) ̂̃xi = x̃i for all i �= m + 1.

(c) ̂̃xm+1 = ξ−1 (∑n

j=1δ jz j

) .

Under negligence, the court stipulates x̂m+1 = ̂̃xm+1 as the due care standard for the new ac- tivity fm+1 and holds the injurer liable. (The due care standards for f1, . . . , fm are unchanged.) This makes outsiders aware of fm+1 (and that it is risky). Although they cannot separately de- duce each δ j from x̂m+1, they nevertheless can infer ĥm+1(xm+1) from x̂m+1.34

Proposition 7. In the case of a new act fm+1,

ĥm+1(xm+1) = − c′ (̂xm+1)

τ ′ (̂xm+1) τ (xm+1).

Thus, negligence reveals sufficient information for others to take efficient care. Under strict li- ability, by contrast, the court simply holds the injurer liable to pay damages to the victim. This makes outsiders aware of fm+1 (and that it is risky), but they do not learn ĥm+1(xm+1). Again, strict liability does not reveal enough information to induce efficient care.

4.3. New Consequence. Assume S ⊆ ZF . Suppose the parties discover a new consequence, zn+1. Once again, let Ŝ denote the expanded feasible state space and p̂ denote the parties’ updated beliefs on Ŝ. Observe that Ŝ = S ∪ � where � = ⋃m

i=1 Ai(zn+1) is the union of the newly discovered events that fi yields zn+1 for all i = 1, . . . , m. By virtue of a tort litigation, the parties learn that fi yields zn+1 with probability αi > 0 for all i = 1, . . . , m.35 That is, αi = p̂(Ai(zn+1)). Let δ = p̂(�).

For each state s ∈ �, let I(s) ≡ {i ∈ {1, . . . , m} : si = zn+1} denote the indices of the acts that yield zn+1 in that state of the world, and let I(s) ≡ {i ∈ {1, . . . , m} : si �= zn+1} denote the in- dices of the acts that do not yield zn+1 in that state of the world. In addition, for each s ∈ �, let C(s) ≡ {t ∈ S : ti = si, ∀ i ∈ I(s)} denote the event in S that corresponds to s ∈ � on I(s). In other words, C(s) comprises the states in S in which every act (other than the acts that yield zn+1) yields the same consequence that it yields in state s ∈ �.

By Reverse Bayesianism, p(s)/p(t) = p̂(s)/p̂(t) for all s, t ∈ S. By Act Independence, p̂(s) = ∏m

i=1 p̂ ( Ai(si)

) for all s = (s1, . . . , sm) ∈ Ŝ. In particular, the probability of the event

that no activity yields zn+1 is 1 − δ = ∏m i=1(1 − αi). It follows that:

Proposition 8. Assume Reverse Bayesianism and Act Independence. If the parties discover a new consequence zn+1, then:

(a) p̂(s) = (∏m i=1(1 − αi)

) p(s) = (1 − δ)p(s) for all s ∈ S.

(b) p̂(s) = (∏

i∈I(s)αi

) (∏ i∈I(s)(1 − αi)

) p (C(s)) for all s ∈ � such that I(s) ⊂ {1, . . . , m}.

(c) p̂(s) = ∏m i=1αi for the s ∈ � such that I(s) = {1, . . . , m}.

Part (a) is dictated by Reverse Bayesianism. The relative likelihoods of the states in S are preserved. Parts (b) and (c) are dictated by Act Independence. Part (c) is a direct implica- tion of Act Independence. The probability of the state in which every activity yields zn+1 is∏m

i=1αi. Part (b) applies when some, but not all, states yield the new consequence. It says that the probabilities of the other states are proportionate to the probabilities of the correspond- ing events in S.

34 Note, however, that if each z j is a different type of harm that requires a different type of care, then the court would stipulate a different due care standard x̂m+1, j with respect to each z j , in which case outsiders could separately deduce each δ j .

35 Assuming αi > 0 for all i is without loss of generality. We can deal with the case where αi = 0 for some i by as- suming αi > 0 for the first l < m and changing m to l as necessary in the statements below.

1868 tort liability and unawareness

Given Ŝ and p̂, the efficient levels of care are ̂̃xi = ξ−1 (∑n+1

j=1 π̂i jz j

) , i = 1, . . . , m, where

π̂i j = ∑ s∈Ŝ:si=z j

p̂(s). Specifically:

Proposition 9. Assume Reverse Bayesianism and Act Independence. If the parties discover a

new consequence zn+1, then ̂̃xi = ξ−1 (∑n

j=1(1 − αi)πi jz j + αizn+1

) , i = 1, . . . , m.

It follows that the due care standard for activity fi, i = 1, . . . , m, is unchanged after the dis- covery of zn+1 if and only if zn+1 equals the activity’s prior expected harm.

Corollary 2. The due care standard ̂̃xi, i = 1, . . . , m, increases (resp. decreases)—that is,̂̃xi > x̃i (resp. ̂̃xi < x̃i)—if and only if zn+1 > ∑n

j=1πi jz j, (resp. zn+1 < ∑n

j=1πi jz j).

Thus, the discovery of zn+1 necessitates the stipulation of new due care standards for each ac- tivity fi such that zn+1 �= ∑n

j=1πi jz j. Under negligence, the court stipulates x̂i = ̂̃xi, i = 1, . . . , m, as the new due care standards

for f1, . . . , fm and holds the injurer liable to pay damages of zn+1 to the victim with re- spect to each activity fi such that x̂i > xi. This, along with the victim’s claims, makes outsiders aware of zn+1 (and that it is linked to f1, . . . , fm). Moreover, they can deduce α1, . . . , αm from x̂1, . . . , x̂m.

Proposition 10. In the case of a new consequence zn+1,

αi = c′ (̂xi) + ∑n j=1πi jz jτ

′ (̂xi)∑n j=1πi jz jτ ′ (̂xi) − zn+1τ ′ (̂xi)

for all i = 1, . . . , m.

As a result, outsiders can learn p̂ and ĥ1(x1), . . . , ĥm(xm). This is sufficient information for other injurers to take efficient care.

As before, strict liability reveals too little information. It makes outsiders aware of zn+1

(and that it is linked to f1, . . . , fm), but they cannot deduce α1, . . . , αm and hence cannot learn p̂ and ĥ1(x1), . . . , ĥm(xm).

5. related literature and our contributions

To our knowledge, this article is the first to incorporate unawareness into the economic analysis of tort law. As such, we contribute to the tort law and economics literature and to the unawareness literature, both of which are too vast to review here.36

A handful of papers apply unawareness models to study other legal topics. The bulk of these focus on contracts. For example, Zhao (2011) argues that unawareness may explain the existence of force majeure clauses in contracts; Filiz-Ozbay (2012) posits asymmetric aware- ness as a reason for the incompleteness of contracts; Grant et al. (2012) study aspects of dif- ferential awareness that give rise to contractual disputes; von Thadden and Zhao (2012, 2014) study the properties of optimal contracts under moral hazard when the agent may be par- tially unaware of her action space; Auster (2013) introduces asymmetric unawareness into the

36 The tort law and economics literature was pioneered by Brown (1973). Surveys of this literature include Shavell (2007), Schäfer and Müller-Langer (2009), and more recently Arlen (2017). The unawareness literature was pio- neered by Fagin and Halpern (1988). Surveys of this literature include Schipper (2014) (which offers a “gentle intro- duction”) and Schipper (2015) (which provides an extended review). Karni and Vierø (2013) were among the first to use the choice-theoretic approach (i.e., the state-space approach) to modeling unawareness. Karni and Vierø (2013, 2017) and Dominiak and Tserenjigmid (2022) survey the papers that take this approach.

tort liability and unawareness 1869

canonical moral hazard model and analyzes the properties of the optimal contract; and Board and Chung (2022) argue that asymmetric unawareness provides a justification for the con- tra proferentem doctrine of contract interpretation, which provides that ambiguous terms in a contract should be construed against the drafter.

Within the law and economics literature, the papers closest to ours include Teitelbaum (2007), Chakravarty and Kelsey (2017), and Franzoni (2017), which explore the implications of ambiguity for tort law.37 Although ambiguity and unawareness are distinct phenomena, both are types of uncertainty that the standard accident model does not admit. Hence, we share a common enterprise with the papers on tort law and ambiguity. We enrich the standard accident model to allow the parties to face not just risk but rather a more profound and re- alistic type of uncertainty, and we explore the implications of such uncertainty for the debate over tort liability rules.

We also share connections with Currie and MacLeod (2014), who develop an alternative to the standard accident model that makes use of state-space representations, dubbed “Savage Tables,” to model the decision problems faced by an injurer (who is fully aware of the state space) under different liability rules. They apply their model to argue, inter alia, that negli- gence provides better incentives than strict liability in the case of the Good Samaritan.

Ott and Schäfer (1997) study how the due care standard in negligence develops when the court starts with no information about the efficient level of care and relies on information pro- vided by the parties in litigation. In their model, an efficient standard evolves over time as a result of a learning process based on the information acquired by the court from litigants. In a similar vein, Feess and Wohlschlegel (2006) compare negligence and strict liability when some injurers have better information than others and the court about the efficient level of care and the court does not know which injurers are informed and uninformed. They show that, under certain conditions, the court can learn the efficient level of care by imperfectly observing the injurer’s level of care in a large number of cases, and that under negligence (but not strict li- ability) the uninformed injurers can in turn learn the efficient level of care by observing the court’s due care standard.

Like Ott and Schäfer (1997), we study the evolution of the negligence due care standard in response to knowledge generated by litigants, and like Feess and Wohlschlegel (2006), we argue that negligence has an advantage over strict liability in terms of knowledge transmis- sion. Our motivation and analysis fundamentally differ from theirs, however, as we consider a world with symmetric unawareness whereas they consider a (fully aware) world with asym- metric information. Moreover, we explicitly model the process of belief revision in the wake of growing awareness and of knowledge transmission through the due care standard. In con- trast, Ott and Schäfer (1997) derive transition probabilities from one standard to another and Feess and Wohlschlegel (2006) derive steady-state beliefs in a rational expectations equilib- rium.38

We also contribute to the relatively nascent but rapidly growing behavioral law and eco- nomics literature. Sunstein (1997), Jolls et al. (1998), and Korobkin and Ulen (2000) were early calls for the modification of standard law and economics models to reflect advances in behavioral economics and decision theory. Sunstein (2000) and Parisi and Smith (2005) are edited volumes that collect early papers in the literature. Zamir and Teichman (2014) and Teitelbaum and Zeiler (2018) are more recent volumes. Halbersberg and Guttel (2014) and Luppi and Parisi (2018) provide surveys of behavioral models of tort law.

37 Also related are the papers that explore the implications of risk aversion for tort law. In the seminal paper on the topic, Shavell (1982) shows that strict liability is superior when the injurer is risk neutral and the victim is risk averse, whereas negligence is superior in the opposite case. Franzoni (2017, n. 10) reviews other papers on optimal tort liabil- ity rules under risk aversion and related contributions.

38 Two additional papers that talk about informational advantages of negligence over strict liability are Demougin and Fluet (1999) and Lando (2018).

1870 tort liability and unawareness

6. discussion

This article extends the economic analysis of tort law to incorporate unawareness. We com- pare and contrast negligence and strict liability in a unilateral accident model with unaware- ness and growing awareness, and find that negligence has a key advantage—the due care standard serves as a knowledge transmission mechanism. Under either tort liability rule, a suit involving a newly discovered act, consequence, or link makes the world aware of a new possibility of harm.39 But only negligence, through the stipulation of new due care standards, spreads awareness about the updated probability of harm.40

Knowledge of the determinants of the optimal standard is a public good. The social bene- fit of spreading awareness about the updated probability of harm is that other injurers need not expend additional resources to develop this knowledge. This wastefully duplicative effort would be necessary to achieve optimal deterrence under strict liability. In a sense, negligence is akin to patents; both carry social costs (negligence is more costly to administer; patents cre- ate monopolies and deadweight loss), yet both provide social benefits in terms of knowledge transmission. One should bear in mind, however, that we do not purport to undertake a full welfare analysis. We do not claim that negligence is superior to strict liability in all circum- stances. Rather, we claim that negligence is more robust to unawareness.41

In order to model unawareness and growing awareness, we adopt the Reverse Bayesian ap- proach of Karni and Vierø (2013). This model has (at least) two attractive features. The first is transparency. Karni and Vierø (2013) provide an axiomatic foundation for the model, and so one can judge the theory by the axioms.42 The second is its accessibility. The model is built upon a familiar choice-theoretic framework (expected utility theory), and the upshot is a be- lief revision theory that mirrors the process of Bayesian updating. Becker et al. (2022) present experimental evidence in support of Reverse Bayesianism.

At the same time, the Reverse Bayesian model has its shortcomings. For instance, Cham- bers and Hayashi (2018) criticize its empirical content from a revealed preference perspective. They show that, in the case of a new consequence, the model does not make singular predic- tions about observable choices over feasible acts. A second shortcoming of the model is that it assumes a naive or myopic unawareness—people are unaware that they are unaware. A so- phisticated unawareness, where people are aware that they are unaware, may be more real- istic. In response, Karni and Vierø (2017) extend their model to the case of sophisticated un- awareness. The end result is a generalization that maintains the flavor of Reverse Bayesianism and nests the naive model as a special case.

The importance of unawareness and growing awareness—via technological progress, sci- entific discovery, or otherwise—plainly extends beyond the case of unilateral accidents with fixed activity levels. Natural extensions of this article, therefore, would entail introducing un- awareness into other accident settings. In addition, future research could examine the implica- tions of unawareness for the economic analysis of other areas of law such as contract remedies and criminal law or other topics such as litigation and settlement.

39 It has been suggested that courts may not apply strict liability when consequences are unforeseen. For new con- sequences and links this would reduce the information transmitted by strict liability still further. For new acts there are no unforeseen consequences. Thus our results would be unaffected by this change.

40 Our argument can be extended to bilateral accidents. In this case, it would suggest that negligence with a defense of contributory negligence (which is efficient as long as both due care standards are set correctly) is superior to simple negligence (and to strict liability with a defense of contributory negligence) because the court sets due care standards for both agents and as a result more information is released.

41 One can argue that the court should cut some slack to early injurers. If one views the effect on incentives, the penalty imposed on the first injurer is a lump-sum tax and has no effect on efficiency. The first injurer was unaware of the possibility of harm and therefore was not able to increase care. The spreading of awareness would be the same if the court imposed no penalty but announced a due care standard. The first accident is unfortunate but is beyond the court’s control. However it is common for courts to consider efficient risk sharing between the injurer and the victim. This could be a reason for finding the injurer liable even though doing so has no strong effect on incentives.

42 The key axioms of the model are the “consistency” axioms, which essentially require that preferences conditional on the original state of awareness are not altered by growing awareness.

tort liability and unawareness 1871

appendix

Proof of Proposition 1. By Reverse Bayesianism, the definition of δ, and p̂1 + p̂2 + p̂3 + p̂4 = 1, we have p̂2 = (p2/p1) p̂1 and p̂1 + p̂2 = 1 − δ. Substituting the first equation into the second, we have p̂1 + (p2/p1) p̂1 = 1 − δ, which implies

p̂1 = (1 − δ)p1

p1 + p2 = (1 − δ)p1,

where the last equality follows from p1 + p2 = 1. It follows that

p̂2 = p2

p1 (1 − δ)p1 = (1 − δ)p2.

By Act Independence and the definition of δ, we have p̂3 = δ( p̂1 + p̂3) and p̂4 = δ( p̂2 + p̂4), which imply p̂3 = [δ/(1 − δ)] p̂1 and p̂4 = [δ/(1 − δ)] p̂2. It follows that

p̂3 = δ

1 − δ (1 − δ)p1 = δp1 and p̂4 = δ

1 − δ (1 − δ)p2 = δp2.

�

Proof of Proposition 2. Reverse Bayesianism implies the following conditions:

p2( p̂1 + p̂5) = p1( p̂2 + p̂6), p3( p̂1 + p̂5) = p1( p̂3 + p̂7), and p4( p̂1 + p̂5) = p1( p̂4 + p̂8).

Summing the left- and right-hand sides, and adding p1( p̂1 + p̂5) to each side, yields

(p1 + p2 + p3 + p4)( p̂1 + p̂5) = ( p̂1 + · · · + p̂8)p1.

Because p1 + p2 + p3 + p4 = 1 and p̂1 + · · · + p̂8 = 1, we have p̂1 + p̂5 = p1. Substituting this back into the Reverse Bayesian conditions yields

p̂1 + p̂5 = p1, p̂2 + p̂6 = p2, p̂3 + p̂7 = p3, and p̂4 + p̂8 = p4.

By Act Independence and the definition of δ, we have

p̂5 = ( p̂1 + p̂5)δ, p̂6 = ( p̂2 + p̂6)δ, p̂7 = ( p̂3 + p̂7)δ, and p̂8 = ( p̂4 + p̂8)δ.

These imply

p̂5 = δ

1 − δ p̂1, p̂6 = δ

1 − δ p̂2, p̂7 = δ

1 − δ p̂3, and p̂8 = δ

1 − δ p̂4.

It follows that

p̂1 + δ

1 − δ p̂1 = p1, p̂2 + δ

1 − δ p̂2 = p2, p̂3 + δ

1 − δ p̂3 = p3, and p̂4 + δ

1 − δ p̂4 = p4.

These imply p̂1 = (1 − δ)p1, p̂2 = (1 − δ)p2, p̂3 = (1 − δ)p3, and p̂4 = (1 − δ)p4, which in turn imply p̂5 = δp1, p̂6 = δp2, p̂7 = δp3, and p̂8 = δp4. �

1872 tort liability and unawareness

Proof of Proposition 3. Reverse Bayesianism implies the following conditions:

p2 p̂1 = p1 p̂2, p3 p̂1 = p1 p̂3, and p4 p̂1 = p1 p̂4.

Summing the left- and right-hand sides, and adding p1 p̂1 to each side, yields

(p1 + p2 + p3 + p4)p̂1 = ( p̂1 + p̂2 + p̂3 + p̂4)p1,

which implies p̂1 = (1 − δ)p1.43 Substituting this back in the Reverse Bayesian conditions yields p̂2 = (1 − δ)p2, p̂3 = (1 − δ)p3, and p̂4 = (1 − δ)p4.

By Act Independence, p̂5 = ( p̂5 + p̂6 + p̂9)( p̂1 + p̂3 + p̂5) = α1( p̂1 + p̂3 + p̂5). Hence,

(1 − α1)p̂5 = α1( p̂1 + p̂3) = α1(1 − δ)(p1 + p3),

which implies p̂5 = α1 (1 − α2) (p1 + p3) (using 1 − δ = (1 − α1) (1 − α2)). By similar reason- ing, p̂6 = α1(1 − α2)(p2 + p4), p̂7 = (1 − α1)α2(p1 + p2), and p̂8 = (1 − α1)α2(p3 + p4). Fi- nally, s9 is the event that both acts yield the new consequence. By Act Independence its prob- ability equals the product of the probabilities of its constituent act events: p̂9 = α1α2. �

Remark. Propositions 1–3 are implications of Theorems 1–3 in Chakravarty et al. (2022). The foregoing proofs, however, establish these results directly and are instructive.

Proof of Proposition 4. Parts (a) and (b) follow from Theorem 1 of Chakravarty et al. (2022).

(c) Observe that ∑n

j=1π̂l jz j = ∑ j �=kπ̂l jz j + δzk. By parts (a) and (b),

∑ j �=k

π̂l jz j = ∑ j �=k

[ ∑ s∈Ŝ:sl=z j

p̂(s)

] z j = ∑

j �=k

[ ∑ s∈S:sl=z j

(1 − δ)p(s) + ∑ s∈�:sl=z j

δp(L(s))

] z j.

Observe that sl = zk for all s ∈ �. It follows that, for all j �= k, ∑

s∈�:sl=z j δp (L(s)) = 0. Thus,

∑ j �=k

π̂l jz j = ∑ j �=k

[ ∑ s∈S:sl=z j

(1 − δ)p(s)

] z j = ∑

j �=k (1 − δ)πl jz j = ∑n

j=1(1 − δ)πl jz j,

where the last equality follows from πlk = 0. Hence, ∑n

j=1π̂l jz j = ∑n j=1(1 − δ)πl jz j + δzk.

(d) Take any i �= l and any j. By parts (a) and (b),

π̂i j = ∑ s∈Ŝ:si=z j

p̂(s) = ∑ s∈S:si=z j

(1 − δ)p(s) + ∑ s∈�:si=z j

δp(L(s)).

Observe that L(s) is the union of all t ∈ S such that ti = si for all i �= l. Thus,∑ s∈�:si=z j

p(L(s)) = ∑ t∈S:ti=z j

p(t).

Hence,

π̂i j = ∑ s∈S:si=z j

(1 − δ)p(s) + ∑ s∈S:si=z j

δp(s) = ∑ s∈S:si=z j

p(s) = πi j.

43 Note that p1 + p2 + p3 + p4 = 1 and p̂1 + p̂2 + p̂3 + p̂4 = 1 − δ.

tort liability and unawareness 1873

It follows that ̂̃xi = x̃i for all i �= l. �

Proof of Corollary 1. By Proposition 4, ξ (̂x̃l ) = ∑n j=1(1 − δ)πl jz j + δzk. Observe that

ξ (x̃l ) = ∑n j=1πl jz j. It follows that ξ (̂x̃l ) > ξ (x̃l ) if and only if zk >

∑n j=1πl jz j. Because

ξ ′(xi) > 0 for all xi, we have ̂̃xl = x̃l if and only if zk = ∑n j=1πl jz j. The case where zk <∑n

j=1πl jz j is similar. �

Proof of Proposition 5. By Proposition 4 and x̂l = ̂̃xl , we have ξ (̂xl ) = ∑n j=1(1 − δ)πl jz j +

δzk. It follows that

δ = ξ (̂xl ) − ∑n j=1πl jz j

zk − ∑n j=1πl jz j

.

Observe that ξ (̂xl ) = −c′ (̂xl )/τ ′ (̂xl ). Thus,

δ = c′ (̂xl ) + ∑n j=1πl jz jτ

′ (̂xl )∑n j=1πl jz jτ ′ (̂xl ) − zkτ ′ (̂xl )

.

�

Proof of Proposition 6. Part (a) follows from Theorem 2 of Chakravarty et al. (2022). (b) Recall that {E(s) : s ∈ S} forms a partition of Ŝ. Take any i �= m + 1 and any j. By part

(a),

π̂i j = ∑ s∈Ŝ:si=z j

p̂(s) = ∑ s∈S:si=z j

[ ∑ sl∈E(s)

p̂(sl )

] = ∑

s∈S:si=z j

[ n∑

l=1 δl p(s)

] = ∑

s∈S:si=z j

p(s) [

n∑ l=1

δl

] .

Note that ∑n

l=1 δl = 1. Thus, π̂i j = ∑ s∈S:si=z j

p(s) = πi j. It follows that ̂̃xi = x̃i for all i �= m + 1.

(c) By definition, π̂m+1, j = δ j for all j = 1, . . . , n. Hence, ̂̃xm+1 = ξ−1 (∑n

j=1δ jz j

) . �

Proof of Proposition 7. Observe that ĥm+1(xm+1) = ∑n j=1π̂m+1, jz jτ (xm+1) and x̂m+1 =̂̃xm+1 = ξ−1

(∑n j=1π̂m+1, jz j

) . The latter implies ξ (̂xm+1) = ∑n

j=1π̂m+1, jz j. Thus, ĥm+1(xm+1) = ξ (̂xm+1)τ (xm+1). Recall that ξ (xi) ≡ −c′(xi)/τ ′(xi). Hence,

ĥm+1(xm+1) = − c′ (̂xm+1)

τ ′ (̂xm+1) τ (xm+1).

�

Proof of Proposition 8. This proposition follows from Theorem 3 of Chakravarty et al. (2022). �

Proof of Proposition 9. Take any i ∈ {1, . . . , m}. Observe that

̂̃xi = ξ−1 (∑n+1

j=1 π̂i jz j

) = ξ−1

(∑n j=1π̂i jz j + αizn+1

) .(A.1)

1874 tort liability and unawareness

Let �(αl, s) ≡ (∏

l∈I(s)αl

) (∏ l∈I(s)(1 − αl )

) for all s ∈ �. By Proposition 8,

π̂i j = ∑ s∈Ŝ:si=z j

p̂(s) = ∑ s∈S:si=z j

∏m l=1(1 − αl )p(s) + ∑

s∈�:si=z j

�(αl, s)p(C(s)),

for all j �= n + 1. Observe that∑ s∈S:si=z j

∏m l=1(1 − αl )p(s) = ∏m

l=1(1 − αl ) ∑

s∈S:si=z j

p(s) = (1 − δ)πi j

and that ∑ s∈�:si=z j

�(αl, s)p(C(s)) = ∑ s∈�:si=z j

(∏ l∈I(s)αl

)(∏ l∈I(s)(1 − αl )

) p(C(s))

= ∑ s∈�:si=z j

∏ l∈I(s)αl∏

l∈I(s)(1−αl ) (1 − δ)p(C(s)),

where the last equality follows because 1 − δ = ∏n i=1(1 − αi). Hence,

∑ s∈�:si=z j

�(αl, s)p(C(s)) = ∑ I⊂{{1,...,m}\{i}}

∏ l∈Iαl∏

l∈I (1 − αl ) (1 − δ)πi j = 1 − ∏

l �=i(1 − αl )∏ l �=i(1 − αl )

(1 − δ)πi j.

Thus,

π̂i j = (1 − δ)πi j + 1− ∏

l �=i(1−αl )∏ l �=i(1−αl )

(1 − δ)πi j

= (1 − δ)πi j

( 1∏

l �=i(1−αl )

) = (1 − δ)πi j

( 1−αi 1−δ

) = (1 − αi)πi j.(A.2)

Substituting (A.2) back into (A.1), we have ̂̃xi = ξ−1 (∑n

j=1(1 − αi)πi jz j + αizn+1

) . �

Proof of Corollary 2. Take any i ∈ {1, . . . , m}. By Proposition 9, we have ξ (̂x̃i) =∑n j=1(1 − αi)πi jz j + αizn+1. Observe that ξ (x̃i) = ∑n

j=1πi jz j. It follows that ξ (̂x̃i) > ξ (x̃i) if and only if zn+1 >

∑n j=1πi jz j. Because ξ ′(xi) > 0 for all xi, we have ̂̃xi = x̃i if and only if zn+1 =∑n

j=1πi jz j. The case where zn+1 < ∑n

j=1πi jz j is similar. �

Proof of Proposition 10. Take any i ∈ {1, . . . , m}. By Proposition 9, x̂i = ̂̃xi = ξ−1

(∑n j=1(1 − αi)πi jz j + αizn+1

) , which implies ξ (̂xi) = ∑n

j=1(1 − αi)πi jz j + αizn+1. It fol- lows that

αi = ξ (̂xi) − ∑n j=1πi jz j

zn+1 − ∑n j=1πi jz j

.

Observe that ξ (̂xi) = −c′ (̂xi)/τ ′ (̂xi). Thus,

αi = c′ (̂xi) + ∑n j=1πi jz jτ

′ (̂xi)∑n j=1πi jz jτ ′ (̂xi) − zn+1τ ′ (̂xi)

.

�

tort liability and unawareness 1875

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