The apportionment of damages is generally determined by the trier of fact in a nontransparent way. The Uniform Comparative Fault Act, Sec. 2.b states that “In determining the percentages of fault, the trier of fact shall consider both the nature of the conduct of each party at fault and the extent of the causal relation between the conduct and the damages claimed.” See also Watson v. State Farm Fire & Cas. Ins. Co., 469 So. 2d 967 (La. 1985) and Restatement (third) of Torts: Apportionment of Liability §8 (2000) detailing the factors that should bear on the determination of the parties’ comparative fault. Cfr. Parisi and Fon (2004) and Parisi and Singh (2010) dealing with the sharing of the loss between two nonnegligent parties.

CV990081115S, 2003 WL 716658 (Conn. Super. Ct. Feb. 19, 2003). Different numbers emerge from litigated cases: the plaintiff’s comparative fault was assessed at 20% in Allen v. Perry, 722 S.W.2d 98, 100 (Mo. Ct. App. 1986), at 33% in Griffin v. LeCompte, 471 So. 2d 1382, 1389 (La. 1985), at 40% in Jensen v. ARA Services, Inc., 719 S.W.2d 121, 122 (Mo. Ct. App. 1986), at 60% in Vincent v. Pabst Brewing Co., 47 Wis. 2d 120, 123, 177 N.W.2d 513, 514 (1970), and at 66% in Griffin v. LeCompte, 471 So. 2d 1382, 1389 (La. 1985).

Dobson v. Louisiana Power and Light Co., 567 So.2d 569 (La. 1990).

The assumption that unilateral care does not affect the accident loss is innocuous and is made only to simplify the examples. The model that we present in Section 4 considers the general case where unilateral care may or may not reduce the accident loss.

United States v. Carroll Towing Co., 159 F.2d 169 (2d. Cir. 1947).

The notion of efficient negligence has been introduced by Grady (1998) in the context of inadvertent violations of the due-care standard. See also Grady (1990). In the conclusion, we will explain how our theory relates to Grady’s theory.

The equilibrium where both parties take care persists but is unlikely to be chosen because it entails higher costs for each party. Mixed strategy equilibria are unstable. In contrast, the negligence equilibrium is stable and is the Pareto efficient equilibrium (and hence a focal point).

567 So.2d 569 (La. 1990).

Note that the notion of “relative fault” could be given a different interpretation: one could argue that the party with the lower costs of care is more at fault than the other party. We have discussed this case in footnotes 3 and 8 and accompanying text. One of the points we make in this paper is that this interpretation is misleading and that relative fault should be defined according to the parties’ relative departures from due care.

Cf. Barnes and Baeverstad (1982:284) using this apportionment rule to make a different point.

Consider the following example: An accident can be prevented if both Xavier (the injurer) and Yvonne (the victim) spend $40 on care. Care taken by only one party has no effect. If an accident occurs, damages amount to $100. This is clearly an accident that should be prevented. However, if an accident occurs and the court erroneously concludes that the injurer’s care cost was not justified (sets the due-care level too low) and thereby finds the injurer nonnegligent, parties will not have an incentive to take care. The injurer prefers spending nothing rather than taking care at the cost of $40. In turn, since the injurer can be expected not to take care and the accident cannot be prevented unilaterally by the victim, she has no reason to take care. Note that this result does not depend on how the loss is shared if both parties act negligently. Therefore, it is irrelevant which negligence rule is implemented.

See Kamin and Rachlinski (1995) showing that, if people are asked to assess the probability of accidents after the fact, their estimates are lager if the accident has actually occurred.

Note that in fact in this case the optimal due-care standards are zero for both parties.

See Shavell (1984:359), explaining that when parties have superior knowledge about factors such as the benefit of the activity, the cost of reducing risk, and the probability and magnitude of losses, liability should be preferred over regulation; otherwise, safety regulation is a better way of alleviating risks. Therefore, the context in which parties are better informed than courts and regulators coincides with the case in which liability is used, while the alternative scenario occurs when regulation is used and hence is not directly relevant for our analysis.

532 P.2d 1226 (Cal. 1975). See Grady (1998:416) for a discussion of this case in the context of comparative negligence.

If instead courts have superior information, the due-care standard should not be based on information that was not available to the parties ex ante, because the parties will not be able to predict the court decision. Thus, superior information by the court will in most cases not be used in trial (see Dari-Mattiacci and Garoupa, 2009 and references therein). Hence, this case reduces to the case in which parties and courts have the same information. If instead regulators have superior information, then regulation of safety should be preferred over liability (Shavell, 1984). Hence this case falls outside the scope of this paper, which focuses on tort liability.

Curran (1992); Calabresi (1997:2206); Best (2007); Robinette and Sherland (2003); van Dam (2006:334–335); Artigot i Golobardes and Gomez Pomar (2009:48–52).

Brown (1973); Posner (1977); cf. Posner (2010:222) for a different view.

Haddock and Curran (1995) are usually credited for what is known as the “allocative equivalence theorem” or the “efficiency equivalence theorem”; however, an earlier proof of the allocative equivalence of negligence rules can be found in Landes and Posner (1980:539, fn. 51). See Jain and Singh (2002) for a general characterization of liability rules and Jain (2009) for an analysis of incremental liability rules.

Posner (2010:222); cf. De Mot (2013) showing that comparative negligence might result in lower litigation costs. We will examine litigation costs in Section 5.5.

See Bar-Gill and Ben-Shahar (2003) and Artigot i Golobardes and Gomez Pomar (2009) for two excellent surveys of the literature.

See Cooter and Ulen (1986), Bar-Gill and Ben-Shahar (2003) and Dari-Mattiacci and De Geest (2005) for references.

See Grady (1990) and further Grady (1998) on lapses and their relation to comparative negligence, see further Section 6.

See Orr (1991) and, for a critique, Chung (1993:404) (“there is no efficiency motivation for favoring one rule over the other”).

The theory that we present in the following relates to lapses in a specific way. We do not address the strategic problem of corrective sequential care discussed above. Rather lapses are relevant to our theory insofar as the presence of lapses is known to the parties but is difficult to verify in court; that is, parties are aware of the possibility of lapses but the court is not. If this is the case, courts might make errors in setting the due-care standards while parties know the socially optimal levels of care. To the extent that lapses can be rationalized as a determinant of court errors our theory applies to lapses.

See also Edlin (1994) submitting that, under evidentiary uncertainty, due-care standards should be more lenient under contributory negligence than under comparative negligence. In our analysis, we assume that the due-care standards are given, so that they are not instrumental toward achieving efficiency.

In a recent paper, Stremitzer and Tabbach (2009) also study the case of erroneous due-care standards in a setting different from ours; in their setting, precaution is unilateral, there is insolvency, and liability is proportional to the probability that the harm was caused by the injurer’s negligence. Leshem and Miller (2009) investigate the performance of all-or-nothing rules versus damages proportionate to the degree of uncertainty in the defendant’s liability. In their paper, erroneous due-care standards play no role and precaution is unilateral.

Haddock and Curran (1995:63–64) discuss a number of errors that the parties or the court could make, including errors concerning the probability of accidents, but conclude that to offset these errors courts would have to first understand the nature and the magnitude of the error and this is unlikely to happen. Therefore, no negligence rule is likely to be unambiguously superior. Our proposal does not require the court to assess the magnitude of errors.

For instance, Posner (2010:223).

Dari-Mattiacci and De Geest (2005:217).

See Orr (1991) and Cooter and Ulen (1986:1092). Rubinfeld (1987:390–391) provides an example with a linear sharing rule based on a fixed sharing of the loss but does not expand on how a court could determine the exact sharing and what information this task requires.

McIntyre v. Balentine, 833 S.W.2d 52 (Tenn. 1992). See Grehan (1981), Barnes and Baeverstad (1982), Cooter and Ulen (1986:1074–1079) and Artigot i Golobardes and Gomez Pomar (2009:52–53).

Stanford v. Chevrolet Division of General Motors, 642 P.2d 624 (or. 1982); Watson v. State Farm Fire & Cas. Ins. Co., 469 So. 2d 967 (La. 1985); Restatement (third) of Torts: Apportionment of Liability §8 (2000). See also Prosser (1953:481), Parisi and Fon (2004) and Edelman (2007) (distinguishing between absolute negligence and relative negligence; what we call relative fault corresponds to absolute negligence in Edelman’s terminology).

Note that if the parties will be found nonnegligent by the court regardless of whether they took care. Hence, an accident may occur even if both parties are nonnegligent in the eyes of the court and the loss will be borne by the victim. In contrast, if , the court will find the parties nonnegligent only if they took care. Hence, if both parties are nonnegligent, there will be no accident and hence no damage to bear.

See footnote 19 above and accompanying text.

In the next section we show that this result applies generally, hence also to unilateral negligence.

Technical details about the equilibria of this game are provided in the Appendix.

Mixed-strategy equilibria are unstable.

See Artigot i Golobardes and Gomez Pomar (2009:52–53) for a classification of the possible sharing rules under comparative negligence. Note that this sharing rule has the desired properties: , , , and . This implies that the second-order conditions for a minimum are satisfied for both parties:

This function has the desired properties: , , , , and . The necessary condition for a minimum is , which is satisfied because . The partial derivatives are not defined at x0 or y0 (or both, for the cross partial derivative). For the first derivatives, the singularity can be removed by defining and . Given that the second and cross-partial derivatives are well-defined in any neighborhood of 0, this does not create problems for the validity of the internal solutions.

The first-order conditions are , which yields the result.

The two equilibria in which a party is unilaterally negligent are ruled out by the Corollary. The case in which both parties are negligent can be ruled out by noting that a necessary condition for this equilibrium is x^dy^d S (x*, y*)S (x^s, y^s). This condition derives from summing up the inequalities in eqs. [7] and [8]. However, note that x^dy^d S (x^s, y^s)99.90. Thus, the outcome in which both parties are negligent cannot be an equilibrium.

The two equilibria in which a party is unilaterally negligent are ruled out by the Corollary. The case in which both parties are negligent can be ruled out as follows. Under simple negligence, if both parties are negligent, the injurer pays the full accident loss and his cost of care, thus this outcome could be an equilibrium only if , where y* is equal to 0, because the victim faces no liability. Therefore the condition becomes , where also x* is equal to 0. However, , hence the outcome where both parties violate the standards of due care is not an equilibrium under simple negligence, because the injurer has an incentive to deviate. An analogous argument proves that this outcome is not an equilibrium under contributory negligence.

The first-order conditions are , which yields . Substituting, we have and

These values are obtained by means of a numerical simulation in the Mathematica 6.0 environment. The code is available with the authors. The exact values returned in the simulation are , , and . Given these equilibrium values, the total costs for each of the parties when both are negligent are lower than if one party unilaterally deviates and takes due care: after rounding up, for the injurer, we have

and, for the victim, we have

In the model developed by Shavell (1983) inefficiencies arise under some liability rules because the first mover’s choice of care affects not only the expected accident loss but also the probability that care by the second mover will be necessary. We do not consider this scenario here.

If due-care standards are too low, all rules perform in a similar way.

This result is true beyond the example. Assume that the parties’ costs of care are X and Y and that the court’s estimates are aX and aY, where a can be greater or less than 1. Xavier’s share of liability is , which is unaffected by the court’s error. The same applies to Yvonne’s share.

This result is true beyond the example. Assume that the parties’ costs of care are X and Y, harm to the victim is L, and harm to a third party is H. Assume further that damages plus punitive damages are equal to LH. If Yvonne is negligent, Xavier will prefer to be negligent if , that is, if his liability share is less than his cost of care. The latter inequality can be rewritten as , which guarantees that the injurer will choose negligence if and only if this outcome is socially desirable (the sum of the costs of care exceeds total harm). Likewise, if Xavier is negligent, Yvonne will prefer to be negligent if , that is, if the harm she suffers minus the damages paid by Xavier when both parties are negligent is less than her cost of care plus the harm she suffers minus the damages paid by Xavier when he is unilaterally negligent. The latter inequality can be rewritten again as , which guarantees that also the victim will choose to be negligent if and only if this outcome is socially desirable.

See also Grady (1984, 1990).

As we have explained in Section 2, our model does not relate to the fact that a party might notice that the other party has failed to take care.

Consider the function and note that this function is strictly convex and that it has first-order conditions identical to those in the text. Convexity implies that there exist levels of x and y such that these conditions are simultaneously satisfied. In order to focus on the interesting cases, we assume such levels of care to be positive.