1I would like to thank Professor Polinsky and my colleagues at the University of Chicago Law School, especially Judge Richard Posner, for their comments on earlier drafts of this paper.

2For purposes of simplicity, I assume that gain, loss, and punishment all occur with certainty, although the size of the loss may vary probabilistically. The generalization of my arguments to the more complicated and realistic situation where only a fraction of offenders are detected and punished should be obvious.

3A. Mitchell Polinsky, "Optimal Liability When the Injurer's Information About the Victim's Loss is Imperfect," IRLE (1987), 7, pp. 139-147.

4Polinsky (1987) pp. 139-140.

5Making the argument more precise requires, as Polinsky makes clear, more specific assumptions about the form of h, the probability distribution for l'. The verbal form of the analysis is, however, sufficient to show the logic of the argument.

6For an act that imposes costs but (arguably) is always efficient, consider carbon dioxide emission by people. Every time we exhale we increase the amount of CO2 in the atmosphere, but that does not seem like a good reason to stop breathing.

7Polinsky's analysis of this situation is similar to the argument for the existence of the "winner's curse" in auction theory. If the bidders in an auction have different estimates of the (common) value of what is being auctioned, the bidder with the highest estimate will win. If the estimates are an unbiased measure of the actual value, the highest estimate (and, with more than three bidders, the second highest as well) will usually be higher than the actual value, so the winner will have paid more than the good is worth.

What is wrong with the argument is that a rational bidder knows that he will only win if everyone else bids less, in which case he is probably overestimating the value of what he is bidding for. So the maximum bid he will be willing to make is not simply his estimate of the good's value but his estimate of the good's value conditional on everyone else having a lower estimate than he does. A similar problem arises with Polinsky's analysis. If the activity is socially desirable, a rational actor will realize that if l'>g it is probably an overestimate of l.

8Polinsky denies that his argument depends on individuals consistently overestimating or underestimating the damage, but that is because he identifies such behavior with a biased h(l'). The actors in his model are, however, biased in a different sense, as we will see below.

9This assumes that potential offenders will be aware of the court's rule; in practice few tortfeasors know much about tort law. But that objection applies with at least equal strength to Polinsky's proposed rule. Whatever the legal rule is, it can only affect the decision to commit a tort to the extent that the potential tortfeasor is aware of it. It is surely at least as easy to transmit the message "the penalty for this tort is a \$500 fine" as to transmit the message "the penalty for this tort is a fine equal to the damage done plus \$100."

Polinsky writes that his assumption of superior knowledge by the court "is more realistic in some situations than others. For example, a court may deal with some types of harms-such as automobile accidents-repeatedly, whereas each injurer may only be involved with such harms infrequently, if ever." But if the injurer is involved with such harms "infrequently, if ever," he will be very unlikely to know that the court has set the punishment for that particular sort of harm at damage done + d.

10Polinsky suggests such a possibility by writing that "It will become clear that the results will generalize to a model in which both the probability and the magnitude of the victim's loss may be uncertain." Polinsky (1987), fn 5.

11Strictly speaking, the conditional probability distribution is , shown on the top axis of Figure 1b.

12I describe Tr as rational because he correctly estimates l from the information available to him. It does not follow that Tp is irrational. He might either be someone who has access to [[rho]](l,l') but (irrationally) fails to incorporate the information into his estimate of l, or someone who does not have access to [[rho]] and for whom l' is the correct estimate of l, given his information. The latter corresponds more closely to what Polinsky assumes.

13This is equivalent to g-d<1.5, as the reader can easily check for himself, so the regions defining cases 1 and 2 are the same as under Polinsky's assumption. This is because the lines showing the estimate of l as a function of l' on Figures 3 and 4 (l=l' on Figure 3, <l>=l'/2 +3/4 on Figure 4) happen to have the same value at l'=1.5.

14The alternative would be a command system where the actor is told what he must do and not do, and where the punishment is intended to make sure that he always acts accordingly. In the context of environmental regulation, the corresponding contrast is between an effluent fee and a set of direct regulations telling each firm how much it is permitted to pollute, with severe punishments for violation of the regulations.

15More realistically, one might assume that the court is imperfectly informed about what the actor believes his gain from committing the act will be. Even if the court has better information than the actor about the actual gain, it is not the actual gain but the actor's belief that determines what he will do, which is what the court is trying to control.

16Throughout this discussion, I interpret negligence in the conventional economic sense: an act is negligent if the gain is less than the cost. I will not consider the complicated question of how closely this corresponds to negligence as actually applied in the law. I will consider different ways in which one might define the cost-ex post or ex ante, with the ex ante based either on the actor's or the court's probability distribution.

17This is not true for Tp; as Polinsky correctly argues, he will in some cases make a different and less efficient choice under negligence than under strict liability. This will occur when l'+d>g>l'. Under strict liability, Tp expects to pay damages of l'+d, so does not take the action. Under negligence, he does not expect to be held negligent, since g>l', so he does take the action. The same thing is true if, as Polinsky assumes, negligence is defined in terms of the real loss l rather than the actor's expected loss l', since Tp believes that l=l' and acts accordingly.

This difference between the effects on Tp of negligence and strict liability would be eliminated if the court modified its negligence rule in the same way that, under Polinsky's system, it modifies its damage rule, announcing that an actor would be held negligent if and only if g>l+d.

18A more realistic version of the assumption would be that the court has some but imperfect information about what the actor knew at the time he decided to commit the action.

19The expected value of l is .

20Some readers may be disturbed by a definition of negligence in terms of actual loss, since in the typical case actual loss occurs with low probability and is almost always much larger than the cost of the precaution that would have prevented it. Negligence is then defined by comparing the cost of the precaution with the expected savings-the probability of loss times the amount of loss. In this model I, following Polinsky, have assumed away that part of the calculation by assuming that the probability of loss is one and the only uncertainty is in the amount of loss. A more realistic model would add a probability p of loss, and define the act as negligent if g<pl. Nothing important in the results would be changed.

21For simplicity, I have not included the corresponding set of calculations for l'<3/2.

22The court could, in principle, eliminate this inefficiency by announcing a still more elaborate system in which d depended on g in a complicated fashion which in turn depended on the form of h(l,l')-giving Equation 6 for the particular h I have just been working with. That would, however, carry us very far from anything recognizable as a system of civil damages.

23So long as we are considering the choice to commit or not commit an act, where not committing the act means not being liable for any damages, this implies that a negligence rule will lead to underdeterrence-the act will be committed too often.

The implications are more complicated if we consider an actor choosing a level of precaution from a continuous range. A higher level of precaution produces two benefits -a decrease in the expected loss and a decrease in the probability that the act will be judged negligent. The latter effect increases the benefit from precautions and thus tends to result in an inefficiently high level of precautions; the former effect reduces the benefit from precautions and thus tends to result in an inefficiently low level of precaution; the sign of the net result is indeterminate. The logic of such a situation (with the court's error involving the level of care taken, rather than the ex ante probabilities of loss) is discussed in Steven Shavell, Economic analysis of Accident Law, pp. 80-81, 93-96. His formal analysis is correct but the verbal summary ("... a general consequence of uncertainty over the assessment of true levels of care is that parties will tend to be led to take more than due care..." pp. 80-81) is somewhat misleading, since it suggests that the net effect is predictable overdeterrence rather than being (as the formal analysis implies) indeterminate.

24A closely related point appears in the discussion of Gorris v. Scott in W. Landes and R. Posner, The Economic Structure of Tort Law, pp. 247-248. "... the plaintiff's animals were washed overboard and lost in a storm. The defendant's ship was not equipped with pens, as required by statute in order to prevent diseases from spreading among the animals. Had the ship been so equipped, the plaintiff's animals would not have been washed overboard in the storm; but the court excused the defendant from liability on the ground that the statute did not create a duty to avoid the type of accident that had occurred."

In terms of the argument of this article, the act (not having pens) may produce a loss larger than the gain (disease transmission) or smaller than the gain (loss of animals in the storm). The situation differs from ours in that both the small and the large loss are themselves probabilistic events with low probability-but this, as already pointed out, does not affect the analysis.

Under a negligence rule, the ship owner is only responsible for the large loss, so he ignores the expected cost associated with the small loss in deciding whether to have the pens. So the negligence rule as applied leads to an inefficient result (like Landes, Posner and the court I ignore the effect on the shipowner's incentives of his desire to have a reputation for delivering his customers' animals to their destination alive and well). Landes and Posner justify the court's decision on the basis of administrative costs, writing that:

"there is a practical problem with using this insight as a basis for liability in Gorris-type cases: it requires the court to consider a type of accident not before it in the litigation-in Gorris, to weigh the costs of the pens against their benefits in disease prevention when the only accident being litigated involved the washing overboard of plaintiff's animals in a storm."

They do not, however, note that the administrative problem would disappear under a rule of strict liability. Each court would only have to consider the case before it. The calculation of whether building pens was desirable given the probability distribution of possible outcomes would be made by the ship owner, not the court.

25This problem is analysed in David Friedman, "Reflections on Optimal Punishment or Should the Rich Pay Higher Fines?," Research in Law and Economics, 1981. The analysis there assumes that only a fraction of offenders are caught and convicted, unlike the discussion here, which assumes (for simplicity) that all offenders are punished.

Throughout the present article, I assume that actors are risk neutral. Risk aversion can easily be incorporated in the analysis. The cost of imposing a punishment lottery on a risk averse actor, whether through uncertainty about whether an offender will be caught and convicted (as discussed in Friedman (1981)) or uncertainty about the outcome of an act (as here) is simply one of the elements that goes into punishment cost.

26More precisely, this condition implies that the derivative of NB(f) is zero at f=l. It is thus a necessary but not a sufficient condition for f=l to be the optimum fine. There might be other and higher local maxima.

27We would expect increasing marginal punishment cost if increasing the level of punishment required a shift from relatively inexpensive punishments such as fines to more expensive punishments such as imprisonment or execution. The latter punishments are expensive because nobody gets what the offender loses. See Friedman (1981) for a discussion of this issue in the criminal context. For very low punishments we might expect decreasing marginal cost, since collecting a hundred dollar fine probably does not cost ten times as much as collecting a ten dollar fine.