On the Choice of Liability Rules

1 Introduction

Courts from across the world are routinely required to decide on matters relating to apportionment of losses resulting from accidents. A variety of rules are used by courts for the assignment of liabilities for such losses. Which of these rules, by providing appropriate incentives to parties involved, always results in efficient outcomes is a key question addressed in economic analysis of law.^[1] The attempt here is to examine whether an efficiency based explanation can be provided for why some rules are (ought to be) chosen over others.

In the standard framework,^[2] the question is analysed in the context of interactions between two risk-neutral parties who are strangers to one another. It is assumed that the loss, in case of accident, falls on one of the parties called the victim. The other party is referred to as the injurer. The probability of accident and the actual loss in case of accident are assumed to depend on the care levels of the two parties. It is also assumed that, the social objective is to minimize the total social costs which are defined as the sum of costs of care of the parties involved and the expected accidental loss. The assignment of liabilities for accidental losses by courts is usually based on the levels of nonnegligence of the victim and the injurer where the level of nonnegligence of a party is either 0 (indicating that the party is negligent) or 1 (indicating that the party is nonnegligent). A rule for the assignment of liabilities specifies the proportions of the loss, in case of occurrence of accident, that the victim and the injurer have to bear for every possible combination of their levels of nonnegligence.

It is assumed that, parties have common knowledge about their interaction (the rule, the possible care levels for both with their corresponding costs and the expected accident losses) and choose their respective care levels simultaneously to minimize their expected costs where the expected cost of a party is the sum of cost of her chosen care level and the expected value of her share of the loss. Thus, an interaction between the parties is modeled as a static game of complete information. A rule for assignment of liabilities is said to be efficient if and only if it always induces both parties to choose care levels that minimize total social costs.

Since the apportionment of accidental losses by courts is usually based on who is negligent and who is not, determination of negligence of a party involved in an accident is a very crucial element of the process of liability assignment and can have important implications. In the standard literature on the efficiency of rules for the assignment of liabilities for accidental losses, negligence is usually defined as a shortfall from a court-specified total social cost minimizing due care level.^[3] Thus, an injurer (victim) is considered negligent if and only if she is found to have taken less than the care due from her. Negligence has, alternatively, been defined as the failure to take a cost-justified precaution.^[4] Given the care levels of the two parties, any other care level of a party (which is higher than her actual care) is cost-justified if and only if it involves an additional cost which is less than the reduction in expected loss it would have brought about. Thus, according to this notion, injurer (victim) is considered negligent if and only if the other party can demonstrate a failure on her part to take some cost-justified precaution.

It has been established in the literature that, if negligence is defined as shortfall from legally specified total social cost minimizing due care then there are liability rules which provide appropriate incentives for taking efficient levels of care to involved parties.^[5] It has also been established that no rule is efficient when negligence is defined as failure to take some cost-justified precaution.^[6] Thus, while the notion of negligence as shortfall from due care is consistent with the objective of efficiency, the notion of negligence as failure to take some cost-justified precaution is not.

The existence of cost-justified untaken precaution approach to determination of negligence, though inconsistent with the objective of efficiency, has several advantages to its credit. The determination of negligence according to the shortfall from due care approach requires the court to play an active role in collecting and processing information on costs of care of parties and expected accident losses. In the other approach, while the injurer (victim) has to demonstrate the existence of some cost-justified untaken precaution of the victim (injurer) to establish her negligence, the court plays the role of a neutral referee. Thus, in comparison to the shortfall from due care approach, this approach is more in harmony with the adversarial system of law. It has been argued that, the cost of determining efficient level of care by courts can be significantly more than the costs that parties might have to incur in establishing negligence of each other. It has also been argued that, in determining negligence, courts actually do not try to fix due care levels but look for evidence of failure to take a cost-justified precaution.^[7] It is, therefore, important to explore the possibility of retaining efficiency considerations in the standard model while using the notion of negligence as failure to take some precaution which is cost-justified.

In this paper, we pose the following question: if negligence is defined as the existence of some cost-justified untaken precaution then is it possible to choose a rule from a given set on the basis of efficiency considerations? To answer the above question we define a binary relation on a set of rules as follows: a rule is at least as efficient as another if and only if the set of applications for which it is inefficient is a subset of the set of applications for which the other one is inefficient. Accordingly, a rule is said to be more efficient than another if and only if the set of applications for which it is inefficient is a proper subset of the set of applications for which the other one is inefficient. A rule in the set is best if and only if it is at least as efficient as every other rule in the set. A rule in the set is maximal if and only if there is no other rule in the set which is more efficient. If there exists a rule which is best in the set with respect to the above relation then such a rule is an obvious choice. If no rule is best but some rule is maximal in the set then such a rule is socially desirable. In this paper, we focus on the set of the following five rules which are among the most widely used and also the ones most analysed in the literature: no liability, strict liability, negligence, negligence with the defense of contributory negligence and strict liability with the defense of contributory negligence.^[8] It turns out that, none of these rules is comparable to any of the others and therefore none is best but every rule is maximal with respect to the above relation. Thus, it follows that, when negligence is defined as the existence of some cost-justified untaken precaution, it is not possible to make a meaningful choice from these five rules on the basis of efficiency considerations as embedded in the relation discussed above.

The paper is organized as follows: The model is presented in Section 2. All definitions and assumptions are stated here and are illustrated with appropriate examples. Section 3 states an impossibility result. Section 4 presents the main results of the paper in the form of Theorems 1 and 2 and also contains the intermediate results (Lemmas 1–5) which are used to prove the two theorems. Section 5 concludes the paper with a discussion on the implications of the results.

2 Model

We consider interactions between two parties (generically called party i, where i ∈ { 1 ,   2 } ), assumed to be strangers to each other, which can result in an accidental harm falling on party 1. We'll refer to party 1 as the victim and party 2 as the injurer. It is assumed that, the probability of accident and the magnitude of harm in case of an accident depend on the levels of non-negative care that the parties might choose to take. Let a i ≥ 0 be the index of the level of care taken by party i and let A i = { a i | a i ≥ 0 be the index of some feasible level of care which can be taken by party i } . We assume that

We denote by c i ( a i ) the cost to party i of care level a i . Let C i = { c i ( a i ) | a i ∈ A i } . We assume:

We also assume that

In view of (A2) and (A3) it follows that ( ∀ c i ∈ C i ) ( c i ≥ 0 ) . A consequence of (A3) is that c i itself can be taken to be an index of the level of care taken by party i. Let π : C 1 × C 2 ↦ [ 0 ,   1 ] denote the probability of occurrence of accident and H : C 1 × C 2 ↦ R + denote the loss in case of occurrence of accident.

Let L : C 1 × C 2 ↦ R + be defined as: L ( c 1 ,   c 2 ) = π ( c 1 ,   c 2 ) H ( c 1 ,   c 2 ) for all ( c 1 , c 2 ) ∈ C 1 × C 2 . L, thus, is the expected loss due to accident.

We assume:

(A4) implies that L is non-increasing in c 1 and c 2 .

We define the total social cost of the interaction between the two parties, T : C 1 × C 2 ↦ R + , as: T ( c 1 ,   c 2 ) = c 1 + c 2 + L ( c 1 ,   c 2 ) for all ( c 1 ,   c 2 ) ∈ C 1 × C 2 . Let M = { ( c ′ 1 ,   c ′ 2 ) ∈ C 1 × C 2 | ( ∀ ( c 1 ,   c 2 ) ∈ C 1 × C 2 ) [ T ( c ′ 1 ,   c ′ 2 ) ≤ T ( c 1 ,   c 2 ) ] } . Thus, M is the set of all costs of care profiles ( c ′ 1 ,   c ′ 2 ) which are total social cost minimizing. It will be assumed that:

Let p i : C 1 × C 2 ↦ { 0 ,   1 } denote the level of nonnegligence of party i with p i ( c 1 ,   c 2 ) = 0 indicating that party i is negligent and p i ( c 1 ,   c 2 ) = 1 indicating that party i is nonnegligent.^[9] The exact form of the function would depend on the definition of negligence.

2.3 Application of a Rule

An application, ω, is a specification of C 1 ,   C 2 ,   π and H. Let Ω denote the set of all applications which satisfy assumptions (A1)-(A5).

Let g be any simple liability rule and ω ∈ Ω be any application of g. If the victim chooses c 1 , the injurer chooses c 2 and the accident occurs then the actual loss will be H ( c 1 ,   c 2 ) . According to g, party i will be made to bear x i ( p 1 ( c 1 ) ,   p 2 ( c 2 ) ) H ( c 1 ,   c 2 ) . E i : C 1 × C 2 ↦ R + , defined as: E i ( c 1 ,   c 2 ) = c i + x i ( p 1 ,   p 2 ) L ( c 1 ,   c 2 ) for all ( c 1 ,   c 2 ) ∈ C 1 × C 2 , is the expected cost to party i. We assume that for all ( c 1 ,   c 2 ) , ( c ′ 1 ,   c ′ 2 ) ∈ C 1 × C 2 , party i considers ( c 1 ,   c 2 ) to be at least as good as ( c ′ 1 ,   c ′ 2 ) if and only if E i ( c 1 ,   c 2 ) ≤ E i ( c ′ 1 ,   c ′ 2 ) . Thus, a simple liability rule induces a two-player simultaneous move game in every application with party 1 and party 2 as the players, C 1 , C 2 as the sets of strategies and E 1 , E 2 as the payoffs.^[12] We shall denote the game induced by simple liability rule g in application ω by ( g ,   ω ) and, whenever possible, represent it by the corresponding payoff matrix. An application of a liability rule f is defined similarly.

Example 3.

Let C 1 = { 0 ,   4 ,   8 } , C 2 = { 0 ,   2 ,   4 } and let the expected loss function, L ( c 1 ,   c 2 ) , be as given in 2.1.

Table 2.1:

Application ω 5 .

Table 2.2:

( p 1 ,   p 2 ) matrix for ω 5 .

Note that C 1 , C 2 given above and L ( c 1 ,   c 2 ) specified as in Table 2.1 constitute an application in Ω . We refer to this application by ω 5 .^[13] Table 2.2 shows the negligence level for the victim and the injurer for every possible configuration of costs of care. If both parties take no care then both would be negligent. Given that the injurer takes no care, the victim could have spent 4 on care to reduce expected loss by 5. Given that the victim takes no care, the injurer could have spent 2 on care to reduce expected loss by 3. If victim chooses 4 and the injurer chooses 2 then both are nonnegligent. This follows from the fact that (4, 2) minimizes T. Similarly, at (8, 2) the victim is nonnegligent while the injurer is negligent. The victim is already taking the highest level of care possible. Given the victim's care, the injurer could have spent an additional 2 on care to reduce expected loss by three more units.

3 Negligence as Failure to Take Some Cost-justified Precaution and the Efficiency of Simple Liability Rules: An Impossibility

In this section we analyse efficiency properties of simple liability rules with negligence defined as the existence of some cost-justified untaken precaution. The main result here establishes that, if negligence is defined as failure to take some cost-justified precaution then no simple liability rule can always achieve an efficient outcome. The result is stated as Proposition 1 given below:

4 Choice of Rules when Negligence Is Defined as Failure to Take Some Cost-justified Precaution

In this section we focus on, G ′ , the set of five of the most widely analysed rules for the apportionment of accidental losses, and find out the best and the maximal rules in G ′ with respect to the binary relation R. The main results of the paper are presented as Theorems 1 and 2. Theorem 1 states that no rule in G ′ is best with respect to the binary relation R and Theorem 2 states that every rule in G ′ is maximal with respect to the binary relation R. We state and prove five intermediate results, Lemmas 1–5, to prove the two theorems.

Lemma 1 states that there is no rule in G ′ which is at least as efficient as g 1 (the no liability rule). We prove Lemma 1 by providing an example of an application in Ω for which g 1 is efficient but none of the other four rules in G ′ is efficient. This amounts to showing that the set of applications for which any of the other rules is inefficient is not a subset of the set of applications for which g 1 is inefficient. Lemmas 2, 3, 4 and 5 state that there is no rule in G ′ which is at least as efficient as g 2 (the rule of strict liability), g 3 (the negligence rule), g 4 (the negligence rule with the defense of contributory negligence) and g 5 (the rule of strict liability with the defense of contributory negligence) respectively. Proofs of Lemmas 2-5 are similar to that of Lemma 1. Thus, Lemmas 1-5 together establish that none of the five rules is comparable, according to R, to any of the others and, therefore, while the set of best rules in G ′ is empty the set of maximal rules in G ′ is G ′ .

Proof.

Let ω 1 be the application given in Table 4.1. Note that M = { ( 4 ,   0 ) } .

Table 4.1:

Application ω 1 .

Table 4.2:

( p 1 ,   p 2 ) matrix for ω 1 .

Table 4.2 is the ( p 1 ,   p 2 ) matrix for ω 1 . Now we consider ω 1 as an application of each of the five rules in G ′ .

Table 4.3:

Payoff matrix for ( g 1 ,   ω 1 )

Table 4.4:

Payoff matrix for ( g 3 ,   ω 1 ) and ( g 4 ,   ω 1 )

• g 1 is efficient for ω 1 as (4, 0) ∈ M is the unique Nash equilibrium of ( g 1 ,   ω 1 ) . (1.1)

• g 2 is inefficient for ω 1 because 0 is the dominant strategy for the victim in ( g 2 ,   ω 1 ) . (1.2)

• (6, 4) ∉ M is the unique Nash equilibrium of ( g 3 ,   ω 1 ) and, therefore, g 3 is inefficient for ω 1 . (1.3)

• As there is no configuration of costs at which the victim and the injurer are both negligent, ( g 4 ,   ω 1 ) is the same as ( g 3 ,   ω 1 ) . Thus, g 4 is also inefficient for ω 1 . (1.4)

• (6, 4) is also the unique Nash equilibrium of ( g 5 ,   ω 1 ) and, therefore, g 5 is inefficient for ω 1 . (1.5)

Table 4.5:

Payoff matrix for ( g 5 ,   ω 1 )

(1.1) and (1.2) imply that ( g 2 , g 1 ) ∉ R , (1.1) and (1.3) imply that ( g 3 , g 1 ) ∉ R , (1.1) and (1.4) imply that ( g 4 ,   g 1 ) ∉ R , and (1.1) and (1.5) imply that ( g 5 ,   g 1 ) ∉ R .□

Proof.

Let ω 2 be the application given in Table 4.6. Note that M = { ( 0 ,   4 ) } .

Table 4.6:

Application ω 2 .

Table 4.7:

( p 1 ,   p 2 ) matrix for ω 2 .

Table 4.7 is the ( p 1 ,   p 2 ) matrix for ω 2 . We consider ω 2 as an application of each of the rules in G ′ .

Table 4.8:

Payoff matrix for ( g 2 ,   ω 2 ) .

Table 4.9:

Payoff matrix for ( g 3 ,   ω 2 ) and ( g 4 ,   ω 2 ) .

Table 4.10:

Payoff matrix for ( g 5 ,   ω 2 ) .

• g 2 is efficient for ω 2 as (0, 4) ∈ M is the unique Nash equilibrium of ( g 2 ,   ω 2 ) . (2.1)

• g 1 is inefficient for ω 2 because 0 is the dominant strategy for the injurer in ( g 2 ,   ω 2 ) . (2.2)

• (4, 6) ∉ M is the unique Nash equilibrium of ( g 3 ,   ω 2 ) and, therefore, g 3 is inefficient for ω 2 . (2.3)

• As there is no configuration of costs at which the victim and the injurer are both negligent, ( g 4 ,   ω 2 ) is the same as ( g 3 ,   ω 2 ) . Thus, g 4 is also inefficient for ω 2 . (2.4)

• (4, 6) is the unique Nash equilibrium of ( g 5 ,   ω 2 ) also and, therefore, g 5 is inefficient for ω 2 . (2.5)

(2.1) and (2.2) imply that ( g 1 ,   g 2 ) ∉ R , (2.1) and (2.3) imply that ( g 3 ,   g 2 ) ∉ R , (2.1) and (2.4) imply that ( g 4 ,   g 2 ) ∉ R , and (2.1) and (2.5) imply that ( g 5 ,   g 2 ) ∉ R .□

Proof.

Let ω 3 be the application given in Table 4.11. Note that M = { ( 2 ,   4 ) } . Table 4.12 is the ( p 1 ,   p 2 ) matrix for ω 3 . We consider ω 3 as an application of each of the rules in G ′ .

Table 4.11:

Application ω 3 .

Table 4.12:

( p 1 ,   p 2 ) matrix for ω 3 .

• g 3 is efficient for ω 3 as (2, 4) ∈   M is the unique Nash Equilibrium of ( g 3 ,   ω 3 ) . (3.1)

• g 1 is inefficient for ω 3 because 0 is the dominant strategy for the injurer in ( g 1 ,   ω 3 ) . (3.2)

• g 2 is inefficient for ω 3 because 0 is the dominant strategy for the victim in ( g 2 ,   ω 3 ) . (3.3)

• There is no Nash equilibrium in ( g 4 ,   ω 3 ) and, therefore, g 4 is inefficient for ω 3 . (3.4)

• There is no Nash equilibrium in ( g 5 ,   ω 3 ) and, therefore, g 5 is inefficient for ω 3 . (3.5)

Table 4.13:

Payoff matrix for ( g 3 ,   ω 3 ) .

Table 4.14:

Payoff matrix for ( g 4 ,   ω 3 ) .

Table 4.15:

Payoff matrix for ( g 5 ,   ω 3 ) .

(3.1) and (3.2) imply that ( g 1 ,   g 3 ) ∉ R , (3.1) and (3.3) imply that ( g 2 ,   g 3 ) ∉ R , (3.1) and (3.4) imply that ( g 4 ,   g 3 ) ∉ R , and (3.1) and (3.5) imply that ( g 5 ,   g 3 ) ∉ R .□

Proof.

Let ω 4 be the application given in Table 4.16. Note that M = { ( 2 ,   4 ) } . Table 4.17 is the ( p 1 ,   p 2 ) matrix for ω 4 . We consider ω 4 as an application of each of the rules in G ′ .

Table 4.16:

Application ω 4 .

Table 4.17:

( p 1 ,   p 2 ) matrix for ω 4 .

Table 4.18:

Payoff matrix for ( g 4 ,   ω 4 ) .

Table 4.19:

Payoff matrix for ( g 3 ,   ω 4 ) .

Table 4.20:

Payoff matrix for ( g 5 ,   ω 4 ) .

• g 4 is efficient for ω 4 as (2, 4) ∈   M is the unique Nash equilibrium of ( g 4 ,   ω 4 ) . (4.1)

• g 1 is inefficient for ω 4 because 0 is the dominant strategy for the injurer in ( g 1 ,   ω 4 ) . (4.2)

• g 2 is inefficient for ω 4 because 0 is the dominant strategy for the victim in ( g 2 ,   ω 4 ) . (4.3)

• g 3 is inefficient for ω 4 as (0, 8) ∉   M is the unique Nash equilibrium of ( g 3 ,   ω 4 ) . (4.4)

• g 5 is also inefficient for ω 4 as there is no Nash equilibrium in ( g 5 ,   ω 4 ) . (4.5)

(4.1) and (4.2) imply that ( g 1 ,   g 4 ) ∉ R , (4.1) and (4.3) imply that ( g 2 ,   g 4 ) ∉ R , (4.1) and (4.4) imply that ( g 3 ,   g 4 ) ∉ R , and (4.1) and (4.5) imply that ( g 5 ,   g 4 ) ∉ R .□

Proof.

Let ω 5 be the application given in Table 2.1. Note that M = { ( 4 ,   2 ) } . Table 2.2 is the ( p 1 ,   p 2 ) matrix for ω 5 . We consider ω 5 as an application of each of the rules in G ′ .

Table 4.21:

Payoff matrix for ( g 5 ,   ω 5 ) .

Table 4.22:

Payoff matrix for ( g 3 ,   ω 5 ) .

Table 4.23:

Payoff matrix for ( g 4 ,   ω 5 ) .

• g 5 is efficient for ω 5 as (4, 2) ∈   M is the unique Nash equilibrium of ( g 5 ,   ω 5 ) . (5.1)

• g 1 is inefficient for ω 5 because 0 is the dominant strategy for the injurer in ( g 1 ,   ω 5 ) . (5.2)

• g 2 is inefficient for ω 5 because 0 is the dominant strategy for the victim in ( g 2 ,   ω 5 ) . (5.3)

• There is no Nash equilibrium in ( g 3 ,   ω 5 ) and, therefore, g 3 is inefficient for ω 5 . (5.4)

• There is no Nash equilibrium in ( g 4 ,   ω 5 ) and, therefore, g 4 is inefficient for ω 5 . (5.5)

(5.1) and (5.2) imply that ( g 1 ,   g 5 ) ∉ R , (5.1) and (5.3) imply that ( g 2 ,   g 5 ) ∉ R , (5.1) and (5.4) imply that ( g 3 ,   g 5 ) ∉ R , and (5.1) and (5.5) imply that ( g 4 ,   g 5 ) ∉ R .□

5 Concluding Remarks

Law and economics as a discipline tries to explain and evaluate laws in terms of economic efficiency. In the context of rules for the assignment of liabilities for accidental losses, while the positive of the law and economics approach has tried to give an efficiency based explanation for adoption of some rules to the exclusion of others, the normative has tried to determine the desirability or otherwise of such rules on the basis of their efficiency properties. In this paper we focus on five of the most widely analysed rules and demonstrate that, if negligence is defined as failure to take some cost-justified precaution then it is not possible to make pairwise comparisons between these rules based on the notion of efficiency to be able to explain why some rules are (ought to be) chosen over the others.

It has to be noted that the results obtained here are restricted to a set of five rules only and it would be interesting to see if the results hold when we extend our analysis to the set of all possible simple liability rules. Further, it has to be noted that the results of the paper are due to the fact that for any pair of rules g ,   g ′ ∈ G ′ there exist two applications ω ,   ω ′ ∈ Ω such that while g is efficient for ω and inefficient for ω ′ , g ′ is efficient for ω ′ and inefficient for ω. Thus, pairwise comparisons between rules are not possible if the set of permissible applications is Ω . It is not immediately clear, if the results hold for a set of permissible applications which is a subset of Ω. Therefore, it appears that, the possibility of an efficiency based choice of rules for the assignment of liabilities for a restricted class of applications is worth exploring.