This note provides an additional argument in favour of the use of a negligence rule in tort law. When the probability of suit varies among injurers and is not observable by the judge, the judge will fail to implement the socially optimal level of care using a strict liability rule (for this implies to set damages equal to harm multiplied by the inverse of the probability of suit). However, he can still implement it using a negligence rule, subject to the condition that he chooses the negligence standard properly (for if damages are set large enough, potential injurers will comply with the standard, regardless their probability of suit).Liability, asymmetric information

Sébastien ROUILLON (GREThA)

English

Research Papers in Economics

 
GRETHA UMR CNRS 5113 
Université Montesquieu Bordeaux IV 
Avenue Léon Duguit  -  33608 PESSAC  -  FRANCE 
Tel : +33 (0)5.56.84.25.75  -  Fax : +33 (0)5.56.84.86.47  -  www.gretha.fr 
              
 
 
 
 
Variable Probabilities of Suit and Liability Rules 
 
 
Sébastien ROUILLON 
 
 
GREThA UMR CNRS 5113 
 
 
 
 
 
Cahiers du GREThA 
n° 2008-15 
              Cahier du GREThA 2008 – 15 
GRETHA UMR CNRS 5113 
Université Montesquieu Bordeaux IV 
Avenue Léon Duguit  -  33608 PESSAC  -  FRANCE 
Tel : +33 (0)5.56.84.25.75  -  Fax : +33 (0)5.56.84.86.47  -  www.gretha.fr 
Probabilités de poursuite hétérogènes et règles de responsabilité 
Résumé 
Cette note énonce un argument supplémentaire en faveur d'une utilisation de la 
responsabilité pour faute. Quand la probabilité d'être poursuivi varie d'un 
responsable à l'autre et n'est pas observable par le juge, il est impossible de 
décentraliser le niveau de précaution optimal en utilisant une règle de 
responsabilité stricte (car ceci nécessiterait que les indemnités soient égales au 
dommage multiplié par l'inverse de la probabilité d'être poursuivi). Ceci reste 
toujours possible avec la responsabilité pour faute, à condition de choisir le 
standard de faute correctement (car en réclamant une indemnité suffisamment 
grande, les responsables potentiels auront intérêt à respecter le standard légal).  
Mots-clés : Responsabilité, asymétrie d'information. 
 
Variable Probabilities of Suit and Liability Rules 
Abstract 
This note provides an additional argument in favour of the use of a negligence 
rule in tort law. When the probability of suit varies among injurers and is not 
observable by the judge, the judge will fail to implement the socially optimal level 
of care using a strict liability rule (for this implies to set damages equal to harm 
multiplied by the inverse of the probability of suit). However, he can still 
implement it using a negligence rule, subject to the condition that he chooses the 
negligence standard properly (for if damages are set large enough, potential 
injurers will comply with the standard, regardless their probability of suit). 
Keywords: Liability, asymmetric information. 
 
JEL : D82 ; K13 
 
Reference to this paper:ROUILLON Sébastien, “Variable Probabilities of Suit and 
Liability Rules”, Working Papers of GREThA, n°  2008-15, 
http://ideas.repec.org/p/grt/wpegrt/2008-15.html. 
   1
Variable Probabilities of Suit and Liability Rules 
Sébastien Rouillon 
February 16, 2007 
 
Université de Bordeaux 4 
GRETHA 
Avenue L. Duguit 
33 608 Pessac cedex 
+033 (0)5 56 84 25 85 
rouillon@u-bordeaux4.fr 
 
 
Keywords: liability, asymmetric information. 
JEL: D82, K13. 
 
Summary: This note provides an additional argument in favour of the use of a negligence 
rule in tort law. When the probability of suit varies among injurers and is not observable by 
the judge, the judge will fail to implement the socially optimal level of care using a strict 
liability rule (for this implies to set damages equal to harm multiplied by the inverse of the 
probability of suit). However, he can still implement it using a negligence rule, subject to the 
condition that he chooses the negligence standard properly (for if damages are set large 
enough, potential injurers will comply with the standard, regardless their probability of suit). 
 
 
1. Introduction 
 
  The Law and Economics literature provides valuable insights about the choice of a 
strict liability rule vs. a negligence rule, under various technological and informational 
circumstances (for a review, see Kaplow and Shavell, 2002). 
  This note supplies an additional argument in favour of a negligence rule, assuming that 
the probability of suit varies among injurers and that this information is not observable to the 
judge. It also shows the determination of the optimal damages under strict liability and 
imperfect information about the probability of suit. 
  Although natural, this assumption has not been considered in the economic analysis of 
liability. Bebchuk and Kaplow (1993) use a similar assumption in a model of law 
enforcement, assuming that individuals are not equally easy to apprehend. In contradiction 
with Becker (1968), they show that the penalty should not be maximal, because otherwise it 
would be too dissuasive for individuals with a large probability of being caught. In Section 4, 
a similar result is obtained for the optimal damages under the rule of strict liability. 
  The familiar model of tort law is introduced in Section 2. We build on Schmitz (2000) 
and add the assumption of variable probabilities of suit among the population. In Section 3, 
we assume that both the injurers and the judge have perfect information. In particular, the 
judge can observe the probability of suit and rely on it to set the damages. This assumption 
being standard, Section 3 only restates, as a benchmark, known results (Kaplow and Shavell, 
2002; Polinsky and Shavell, 1998). In Section 4, we consider the case where the judge is 
unable to observe the probability of suit after an accident has occurred and prove that only the 
negligence rule is optimal. Under a strict liability rule, we derive the optimal damages and 
compare with the results in Section 3. In Section 5, the case where both the judge and the 
injurers are unaware of the probability of suit is introduced. 
   2
 
2. The model 
 
  In the model, risk-neutral agents (injurers) exert hazardous activities. They can reduce 
the probability of accident by taking care. Following Schmitz (2000), we assume that if an 
injurer expends c(x) in care, the resulting probability of accident will be 1 – x, and we admit 
the assumptions: c(0) = c’(0) = 0, c(x) is twice differentiable, strictly increasing and strictly 
convex, and c’(x) → ∞ when x → 1. When an injurer provokes an accident, a third party (a 
victim) suffers a harm h and brings the suit to the judge (goes to trial) with a probability q (
1). 
 
  2.1. Distribution of information. The assumptions about the distribution of 
information among the injurers and the judge appear to be central. 
  Since we focus on the effects of the variability of q in the population, we will assume 
everywhere that the technology c(x) and the harm h are known to everyone. From this, the 
injurers and the judge are able to calculate the level of care that best fits their objective. 
  Concerning the information about q, we postulate that it is common knowledge that q 
is distributed in the population of victims on the support [a, b] according to the density g(q) 
(we assume that 0 < a < b ≤ 1 and g(q) > 0, for all q∈[a, b]). For the rest, the following table 
summarizes the cases considered in Sections 3 to 5: 
 
Information available to… :
Case of…: 
…the injurers  …the judge 
…perfect information (§ 3)  Current value of q  Current value of q 
…imperfect information of the judge (§ 4) Current value of q Distribution  of  q 
…imperfect information of injurers (§ 5)  Distribution of q Both  cases 
Table 1. Distribution of information 
 
  2.2. Social optimum. This part borrows much from Shavell (1984) and is thus only 
sketched. The social problem is to minimize the expected social cost of the risky activity: 
c(x) + (1 – x) h. (1) 
The solution to this problem is x = x*(h), where x*(h) is defined, for all h ≥ 0, by: 
c’(x*(h)) = h. (2) 
 
  It is immediate that x*(h) is differentiable and strictly increasing (since x*’(h) = 
1/c’’(x*(h)) > 0, for all h). In other words, optimal care is an increasing function of the harm 
done. 
 
 
3. Perfect information 
 
  We postulate thereafter that the judge seeks to minimize (1), using the liability rules 
(strict or negligence). In this section, we assume that their probability q is kno wn to th e 
injurers when they choose x and is observable by the judge after an accident has occurred 
(implicitly, the judge is allowed to make damages depend on it). 
 
  3.1. Strict liability. Consider first the use of strict liability. The judge fixes damages, 
denoted f. The injurer is liable for f when an accident occurs. Therefore, his problem is to 
choose x to minimize his expected private cost: 
                                                 
(
1) We implicitly assume thereafter that all trials go to an end and implement the correct decision. Therefore, q is 
also the probability that the injurers pay for damages when they ought to (the liability rule being given).   3
c(x) + (1 – x) q f. (3) 
The solution to this problem is x = x*(q f). 
  From this, the judge implements the social optimal level of care if (Polinsky and 
Shavell, 1998): 
q f = h ⇔ f = h/q. (4) 
In other words, the injurers should face, when they choose x, the true cost h of their activity. 
At the time a suit is brought to the judge, damages should thus be multiplied by the inverse of 
the probability of suit (1/q). 
 
  3.2. Fault-based liability. Consider now the use of the negligence rule. The judge 
defines a damage f and a legal standard s. The injurer is liable for f when an accident occurs if 
he is found negligent (i.e., if x < s). Therefore, his problem is to choose x to minimize his 
expected private cost: 
c(x) + (1 – x) q f, if x < s, and c(x), otherwise.  (5) 
 
  It is immediate to show that the judge implements the social optimal level of care if he 
chooses f ≥ h/q and s = x*(h) (Polinsky and Shavell, 1998). Indeed, if the injurer is liable for 
damages (x < s), his private cost (3) is minimized when x = x*(q f) ≥ x*(h) = s. Otherwise, he 
chooses x = s (doing more is costly and useless). And x = s minimizes (5) since c(s) < c(x*(q 
f)) + (1 – x*(q f)) q f < Inf{c(x) + (1 – x) q f ; x < s}. 
 
 
4. Imperfect information of the judge 
 
  Here, we assume that the injurers know their q when they choose x, but that the judge 
can not observe it after an accident has occurred (or, equivalently, that he is not allowed to 
differentiate damages with respect to it). However, he is aware that the distribution of q in the 
population follows the density g(q). 
 
  4.1. Strict liability. Consider the use of strict liability under this new assumption. 
Since the judge cannot observe q, he is bound to charge a uniform liability to the injurers, 
irrespective of their probability of suit. His problem is therefore to choose f to minimize: 
∫a
b
 [c(x*(q f)) + (1 – x*(q f)) h] g(q) dq, (6) 
knowing that an injurer of type q facing a liability f cares x*(q f). 
 
  The first-order condition for this problem is (
2): 
∫a
b
 (q f – h) x*’(q f) g(q) dq = 0.  (7) 
 
 Since  x*’(h) > 0, this implies that the optimal damages, denoted f*, satisfy: 
a f* < h < b f* ⇔ h/b < f* < h/a. 
Hence, f* is bounded below (resp., above) by the damages necessary to induce an injurer of 
type b (resp., of type a) to choose the socially optimal level of care x*(h). 
 
  From this, there exists a type q
0 = h/f*∈]a, b[, such that all types q < q
0 will undertake 
too little care and all types q > q
0 will undertake too much care. As an immediate 
consequence, strict liability fails to implement the social optimum level of care. 
                                                 
(
2) We assume everywhere that the second-order condition holds. It is immediate to verify that it is true when 
c(x) is (locally) quadratic.   4
 
  More can be said about f* if we assume that the cost of care c(x) is (locally) quadratic. 
Indeed, if c(x) = C x
2/2 (with C > 0), we get, for all h: 
x*(h) = h/C and x*’(h) = 1/C. 
Then, eliminating the term x*’(q f) = 1/C in (6) and rearranging, we get: 
f* = m h/(m
2 + σ
2), (8) 
where m = ∫a
b
 q g(q) dq and  σ
2 = ∫a
b
 (q – m)
2 g(q) dq are respectively the mean and the 
variance of q in the population of victims. 
  Equation (8) shows that the economic principle of Section 3.1, namely that injurers 
should always face the true cost of their activity at the moment they choose how much care to 
take, is not the correct one here. Indeed, when q varies among the victims and is not 
observable by the judge, it would a priori imply to set f = h/m (since then m f = h and the 
judge asks the injurers, on average, for the payment of the true cost of their activity). From 
(8), we see that this rule is obtained as a limit case, when σ
2 → 0. Otherwise, smaller damages 
should be asked (Equation (8) implies that f* < h/m). 
 
  4.2. Fault-based liability. Consider now the use of fault-based liability when q varies 
among the population and is not observable by the judge. Using the results obtained in 
Section 3.2, it is immediate to show that the judge can still implement the social optimum 
level of care. 
  Indeed, assume that he chooses f ≥ h/a and s = x*(h). For all q, we have f ≥ h/a ≥ h/q. 
From our results in 3.2, all injurers choose to follow the standard s = x*(h), which is the 
socially optimal level of care. 
 
 
5. Imperfect information of the injurers 
 
  The results above rely on the assumption that the injurers know their probability of 
suit at the moment they choose how much to expend in care. However, in reality, q is partly 
exterior to the injurers (it depends on the incentives to go to trial of the victims, on the 
probability of an erroneous decision by the judge, etc.) Therefore, our assumption will not 
necessary be correct in all cases. 
  This drives us to consider the case where the injurer is unaware of his actual q and 
chooses x knowing its distribution only. Under this assumption, he seeks to minimize over x 
his expected private cost: 
∫a
b
 [c(x) + (1 – x) q f] g(q) dq = c(x) + (1 – x) m f. (9) 
The solution to his problem is x*(m f). 
  As a consequence, under a strict liability rule, the judge can implement the socially 
optimal level of care if (
3): 
m f = h ⇔ f = h/m. (10) 
In other words, there is no need that the judge observes the probability of suit when the 
injurers do not. When the injurers only anticipate the average probability of suit m, uniform 
damages are sufficient and should be set such that parties internalize the true cost of their 
activity, given their common belief. A corollary is that it would be socially costly to provide 
the injurers with finer information about their actual q. 
  Comparing with Section 4 (if c(x) is locally quadratic), our results also imply that 
injurers should be less severely ‘penalized’ when they own and use information to avoid their 
                                                 
(
3) Of course, the results of Section 4.2 apply and the first-best is also possible using a negligence rule.   5
liability than when they do not. Indeed, according to equations (8) and (10) respectively, the 
injurers should pay f* when they know q, h/m when they do not, and we have f* < h/m. 
 
 
6. References 
 
Bebchuk L. and L. Kaplow, 1993, “Optimal Sanctions and Differences in Individuals’ 
Likelihood of Avoiding Detection”, International Review of Law and Economics 13, 217-
224. 
Becker G., 1968, “Crime and Punishment: An Economic Analysis”, Journal of Political 
Economy, 76, 169-217. 
Kaplow L. and S. Shavell, 2002, “Economic analysis of law”, in Handbook of Public 
Economics, Volume 3, Alan J. Auerbach and Martin Feldstein (editors), Elsevier, 2002, pages 
1661-1784. 
Polinsky A.M. and S. Shavell, 1998, “Punitive Damages: An Economic Analysis”, Harvard 
Law Review, 111(4), 869-962. 
Schmitz P.W., 2000, “On the joint use of liability and safety regulation”, International Review 
of Law and Economics, 20, 371-382. 
Shavell S., 1984, “A Model of Optimal Use of Liability and Safety Regulation”, Rand Journal 
of Economics, 15(2), 271-280. 
  
   
 
              
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La coordination scientifique des Cahiers du GREThA est assurée par Sylvie FERRARI et Vincent 
FRIGANT. La mise en page est assurée par Dominique REBOLLO. 

A Model of Optimal Use of Liability and Safety Regulation”,

Crime and Punishment: An Economic Analysis”,

Economic analysis of law”,

On the joint use of liability and safety regulation”,

Optimal Sanctions and Differences in Individuals’ Likelihood of Avoiding Detection”,

Punitive Damages: An Economic Analysis”,

Variable Probabilities of Suit and Liability Rules

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Variable Probabilities of Suit and Liability Rules

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