Uncertainty Regarding Interpretation of the `Negligence Rule' and Its Implications for the Efficiency of Outcomes

Transcription

1 Jawaharlal Nehru University From the SelectedWorks of Satish K. Jain 2011 Uncertainty Regarding Interpretation of the `Negligence Rule' and Its Implications for the Efficiency of Outcomes Satish K. Jain, Jawaharlal Nehru University Available at:

2 Uncertainty Regarding Interpretation of the Negligence Rule and Its Implications for the Efficiency of Outcomes Satish K. Jain Abstract There are two ways that the negligence rule is interpreted. Under one interpretation a negligent injurer is liable for the entire harm to the victim; and under the other interpretation a negligent injurer is liable only for that part of the harm which can be ascribed to his negligence. Both these versions are efficient. However, if there is uncertainty regarding whether the court will be employing the full liability version or the incremental liability version for determining the liability of a negligent injurer, notwithstanding the fact that both the versions are efficient, inefficiency is possible. In the paper necessary and sufficient conditions for efficiency are derived when there is uncertainty regarding the interpretation of the negligence rule. Keywords: Negligence Rule; Full Liability; Incremental Liability; Standard Version; Incremental Version; Uncertainty Regarding Interpretation of Negligence Rule; Necessary and Sufficient Conditions for Efficiency. JEL Classification: K13 1 Introduction The negligence rule, one of the most important of liability rules, is usually interpreted to mean that the injurer is liable for the entire harm to the victim in case he is negligent; Centre for Economic Studies and Planning, School of Social Sciences, Jawaharlal Nehru University, New Delhi skjain@mail.jnu.ac.in, satishkumarjain@gmail.com 1

3 and he is not at all liable in case he is nonnegligent. There is, however, another way that the negligence rule can be interpreted. Under this interpretation, the injurer, if negligent, is liable for only that part of the harm which can be attributed to his negligence; and as before he is not at all liable if he is nonnegligent. We will use the expressions the standard version and the incremental version respectively for these two ways of defining the negligence rule. Both these versions of the negligence rule are efficient. The efficiency of the standard version was established by Brown (1973). The incremental version has been shown to be efficient for the unilateral care case in Kahan (1989); and for the bilateral care case in Jain (2010). In view of the efficiency of both the versions it follows that from the efficiency perspective it does not really matter which version is used by the courts as long as both the parties have the information as to which version will be used. 1 If courts use one version sometimes and the other version at other times and the parties do not know in advance as to which version will be used in the present instance then notwithstanding the fact that both the versions are efficient it is not at all clear that the outcome will necessarily be efficient. We present an example in this paper where the outcome arrived at is inefficient on account of uncertainty regarding interpretation of the negligence rule. Given that there is uncertainty as to which version of the rule will be used by the courts the two parties must estimate the probabilities with which the two versions could be expected. Let α denote the probability with which the victim expects the standard version to be employed; and let β denote the probability with which the injurer expects the standard version to be employed (Therefore, the victim expects the incremental version to be employed with probability (1 α); and the injurer expects the incremental version to be employed with probability (1 β)). It turns out that, when there is uncertainty regarding interpretation of the negligence rule, a necessary and sufficient condition for efficiency to obtain regardless of the nature of harmful interaction between the two parties is that β be at least as large as α (Theorem 1). That is to say, if β is greater than or equal to α then regardless of which application of the negligence rule obtains the outcome will be efficient; and if β is less than α then efficiency does not obtain for some applications of the negligence rule. For the injurer the incremental version of the rule is clearly better than the standard version; and for the victim it is the other way around. If both parties form their beliefs on the basis of more or less the same information (e.g., the history of the use of the negligence rule by the courts), and if both parties are conservative 1 For illuminating discussions of a large number of court cases, involving both the standard and incremental interpretations, see Landes and Posner (1987) and Grady (1983, 1984). See also Kahan (1989) and the cases cited therein. 2

4 in assessing probabilities, pessimistic in other words, then one would on the whole expect β to be at least as large as α. A careful analysis of the example showing that inefficiency is possible if there is uncertainty regarding the interpretation of the negligence rule shows that the main reason for the failure of efficiency to obtain lies in the complementarity of the cares of the two parties. The natural question that arises then is whether one can always be sure of obtaining efficiency if complementarities in cares are ruled out. It turns out that if complementarities in cares are ruled out then a necessary and sufficient condition for obtaining of efficiency is that β be positive or α be zero (Theorem 2). As long as β is greater than 0 we are assured of efficiency regardless of the value of α. If, however, β is 0 then it is possible that inefficiency might obtain. If both β and α are zero then of course by Theorem 1 we know that inefficiency is impossible; so we must have both β = 0 and α to be greater than 0 for inefficiency to materialize. As the condition is necessary as well, if β = 0 and α > 0 then one can always find an application without complementarities in the cares of the two parties under which efficiency does not obtain. Thus, the assumption of absence of complementarities in the cares of the two parties, although ensuring efficiency in almost all cases, does not completely rule out inefficiency. If in addition to assuming absence of complementarities in the cares of the two parties, it is also assumed that the total social costs are minimized at only one care-configuration, i.e., the minimum is unique, then it turns out that efficiency will obtain regardless of the values of α and β (Theorem 3). In the law and economics literature, in the context of analysis of the efficiency of liability rules, it is generally assumed that there are no complementarities in the cares of the two parties as well as that the care-configuration at which total social costs are minimized is unique. 2 Thus, it follows that under the standard assumptions efficiency will obtain regardless of the subjective probabilities of the two parties. This, indeed, is quite remarkable. Even when the two parties do not which of the two versions is going be used in their case, under the standard assumptions we can be sure that there will be no inefficiency. The paper is divided into four sections, including this introductory section. The second section contains the definitions and assumptions, and spells out the framework of analysis. The third section contains an example which shows that when there is uncertainty regarding as to whether the court will interpret the negligence rule to imply full liability on the part of a negligent injurer or merely the incremental liability inefficiency is possible. The last section discusses the conditions under which the efficiency obtains 2 See Brown (1973), Landes and Posner (1987), and Shavell (1987), among others. 3

5 even when there is uncertainty regarding the interpretation of the negligence rule. All proofs are relegated to the Appendix at the end of the paper. 2 Definitions and Assumptions We consider accidents resulting from interaction of two parties, assumed to be strangers to each other, in which, to begin with, the entire loss falls on one party to be called the victim (plaintiff). The other party would be referred to as the injurer (defendant). At times, the victim would be referred to as individual or party 1 and the injurer individual or party 2. We denote by c 0 the cost of care taken by the victim and by d 0 the cost of care taken by the injurer. Costs of care would be assumed to be strictly increasing functions of indices of care, i.e., care levels; consequently, costs of care themselves can be taken to be indices of care. Let C = {c c is the cost of some feasible level of care which can be taken by the victim} and D = {d d is the cost of some feasible level of care which can be taken by the injurer}. We will identify c = 0 with victim taking no care; and d = 0 with injurer taking no care. We assume: 0 C 0 D. (A1) Assumption (A1) merely says that, for each party, taking no care is always a feasible option. Let π denote the probability of occurrence of accident and H 0 the loss in case of occurrence of accident. Both π and H will be assumed to be functions of c and d; π = π(c, d), H = H(c, d). Let L = πh. L is thus expected loss due to accident. We assume: ( c, c C)( d, d D)[[c > c L(c, d) L(c, d)] [d > d L(c, d) L(c, d )]]. (A2) That is to say: a larger expenditure on care by either party, given the expenditure on care by the other party, does not result in greater expected accident loss. Total social costs (T SC) are defined to be the sum of cost of care by the victim, cost of care by the injurer, and expected loss due to accident; T SC = c + d + L(c, d). Let M = {(c, d ) C D c +d +L(c, d ) is minimum of {c+d+l(c, d) c C d D}}. Thus M is the set of all costs of care configurations (c, d ) which are total social cost minimizing. It will be assumed that: 4

6 C, D and L are such that M is nonempty. (A3) The notion of negligence will be defined as shortfall from a specified level called due care. Let d denote the due care for the injurer. It will be assumed that: ( c C)[(c, d ) M]. That is to say, due care for the injurer is chosen appropriately from the perspective of minimization of total social costs. The injurer is called negligent at (c, d) iff his care level d is less than d ; and nonnegligent iff d is greater than or equal to d. There are two versions of negligence rule which are used in practice. The standard (s) version of negligence rule is defined by: (a) The injurer is liable for the entire loss iff he is negligent; and (b) The injurer is not at all liable iff he is nonnegligent. The incremental (i) version of negligence rule is defined by: (a) The injurer is liable for the loss which can be ascribed to his negligence iff he is negligent; and (b) The injurer is not at all liable iff he is nonnegligent. Let ˆL 2 (c, d) denote expected loss which can be ascribed to injurer s negligence at (c, d). We define ˆL 2 (c, d) as follows: ˆL 2 (c, d) = L(c, d) L(c, d ) if d < d = 0 if d d. Let EC 1 (c, d) and EC 2 (c, d) denote expected costs of the victim and the injurer respectively. Then, under the standard negligence rule the expected costs are given by: EC 1 (c, d) = c EC 2 (c, d) = d + L(c, d) if d < d EC 1 (c, d) = c + L(c, d) EC 2 (c, d) = d if d d. Under the incremental negligence rule the expected costs are given by: EC 1 (c, d) = c + L(c, d ) EC 2 (c, d) = d + L(c, d) L(c, d ) if d < d EC 1 (c, d) = c + L(c, d) EC 2 (c, d) = d if d d. Both parties are assumed to prefer smaller expected costs to larger expected costs and be indifferent between alternatives with equal expected costs. An application of negligence rule, standard or incremental, consists of specification of C, D, π, H and d satisfying (A1)-(A3), where d is such that ( c C)[(c, d ) M]. The set of all applications will be denoted by A. Negligence rule, standard or incremental, is defined to be efficient for a given application belonging to A iff ( (c, d) C D)[(c, d) is a Nash equilibrium (c, d) M] 5

7 and ( (c, d) C D)[(c, d) is a Nash equilibrium]. In other words, negligence rule is efficient for a given application iff (i) every (c, d) C D which is a Nash equilibrium is total social cost minimizing, and (ii) there exists at least one (c, d) C D which is a Nash equilibrium. Negligence rule is defined to be efficient with respect to a class of applications iff it is efficient for every application belonging to that class. We also say that efficiency obtains under a particular a A iff (i) and (ii) hold for a; and that efficiency obtains for a class of applications A A iff efficiency obtains under every application belonging to A. It will be assumed that when courts use negligence rule, they sometimes use the standard version and sometimes the incremental version. The parties do not know in advance which version of the negligence rule will be applied by the court. We assume that the victim expects the standard version with probability α and the incremental version with probability 1 α; and that the injurer expects the standard version with probability β and the incremental version with probability 1 β. 3 An Example Although both versions of the negligence rule are efficient, but if there is uncertainty regarding as to which version of the rule will be applied by the court then there is no guarantee that the outcome will be efficient as the following example shows. Example 1 Let C = D = {0, 1}; and L(c, d), (c, d) C D, be as given in the following array: c d Let α =.9 and β =.1. TSC are uniquely minimized at (1, 1). Let the due care d for the injurer be 1. We obtain (EC 1 (c, d), EC 2 (c, d)), (c, d) C D, as given in the following array: 6

8 c d (0.25, 0.75) (2.50, 1.00) 1 (1.00, 2.50) (1.00, 1.00) (0, 0), which is not TSC-minimizing, is also a Nash equilibrium. Thus efficiency does not obtain under this application. 4 Conditions for Efficiency While from the example considered above it is clear that there are combinations of values of α and β with which efficiency is not guaranteed for every application belonging to A; there do exist combinations of values of α and β with which efficiency obtains under every application belonging to A. In fact, it turns out that, when there is uncertainty regarding interpretation of the negligence rule, a necessary and sufficient condition for obtaining of efficiency under every application belonging to A is that β be greater than or equal to α (Theorem 1). Thus, if β is greater than or equal to α then regardless of which application belonging to A materializes one can be sure that the outcome which will obtain will be efficient. On the other hand, if α is greater than β then there is no guarantee that the outcome will be efficient. There are some applications under which efficiency will obtain even with α > β. But equally there are applications under which efficiency will not obtain if α > β. The proof of the sufficiency part of Theorem 1 is straightforward and is contained in Lemmas 1 and 2. Lemma 1 establishes that the total social costs minimizing configuration involving the due care is a Nash equilibrium; and Lemma 2 establishes that, given β α, all Nash equilibria are total social costs minimizing. The necessity part turns out to be rather complicated; and is established via Lemmas 3-5. Because Lemma 1, which establishes that the TSC-minimizing care-configuration involving the due care is a Nash equilibrium, does not assume anything regarding the values of α and β, it follows that even when α > β (c, d ), where d is the due care for the injurer and c is such that (c, d ) M, is a Nash equilibrium. Thus the inefficiency which is possible when α > β arises not on account of TSC-minimizing care-configuration involving the due care not being a Nash equilibrium but due to some non-tsc-minimizing care-configuration emerging as a Nash equilibrium as well. 7

9 Incremental version of the negligence rule is better for the injurer compared to the standard version; and for the victim it is the other way round. If the liability is going to be determined on the basis of the standard version and if the victim has any reason to believe that the injurer is going to be negligent then the optimal strategy for the victim is to take zero care. On the other hand if the liability is going to be determined on the basis of the incremental version then it is possible, as is the case in Example 1, that given that the victim is taking less than the optimal care the injurer may find the strategy of being negligent to be better than that of being nonnegligent. This is the reason why when α is greater than β it is possible for a care-configuration in which both parties are taking less then the optimal care to emerge as a Nash equilibrium. If both parties are conservative in their expectations, that is to say, have a pessimistic outlook, then, assuming that both the parties form their expectations on the basis of more or less the same objective information, on the whole one can expect the probability with which the injurer would expect the standard version (β) to be at least as large as the probability with which the victim would expect the same to be the case (α). The pessimistic outlook ensures that the injurer s care level is not below the due care and thus eliminating the possibility of non-tsc-minimizing care-configurations emerging as Nash equilibria. The application considered in Example 1 is such that cares of the two parties are complementary. If the injurer increases expenditure on care from 0 to 1 when the victim is taking 0 care then the reduction in expected loss is only.5, but if the victim is spending 1 on care then the same increase by the injurer results in a decrease in expected loss of amount 2.5. As the example is symmetric between the injurer and the victim we have that, if only one party takes care then it is not cost-effective, but if both of them take care then together the cares are. A question which therefore naturally arises is whether inefficiency can be ruled out altogether if complementarities are not there. The answer is a qualified yes. If it is given that β is greater than zero then it can be shown that the absence of complementarities ensures efficiency. The idea of absence of complementarities can be formalized as S-restriction which is defined as follows: An application is S-restricted iff ( c, c C)( d, d D)[[(c > c ) (d > d )] [L(c, d ) L(c, d) L(c, d ) L(c, d)] [L(c, d) L(c, d) L(c, d ) L(c, d )]]. In other words, an application is S-restricted iff it is such that any increase in care from a particular level to a higher level by one party is never more effective with a higher level of care by the other party than with a lower level of care. 8

10 Let the subset of all applications satisfying the S-restriction be denoted by A s. Theorem 2 establishes that if β > 0 then efficiency obtains under every application belonging to A s A regardless of the values of α. If β = 0 then inefficiency is possible for some applications belonging to A s ; α must of course then be greater than 0 in view of Theorem 1. The following example shows inefficiency under an application belonging to A s when β = 0 and α > 0. Example 2 Let C = D = {0, 1}; and L(c, d), (c, d) C D, be as given in the following array: c d Let α = 1 and β = 0. TSC are minimized at (1, 0) and at (1, 1). Let the due care d for the injurer be 1. We obtain (EC 1 (c, d), EC 2 (c, d)), (c, d) C D, as given in the following array: c d (0, 1) (9, 1) 1 (1, 1) (8, 1) (0, 0), which is not TSC-minimizing, is also a Nash equilibrium. Thus efficiency does not obtain under this application. Thus, when β = 0 and α > 0 then efficiency is not guaranteed for every application belonging to A s. In fact, Theorem 2 establishes that given β = 0, for every α > 0 there exists an application for which efficiency does not obtain. In other words, Theorem 2 establishes that a necessary and sufficient condition for obtaining of efficiency under 9

11 every application belonging to A s is that β be positive or α be zero. Thus absence of complementarities rules out inefficiency in almost all cases. The reason why the absence of complementarities ensures efficiency regardless of the values of α and β, as long as β is positive, is as follows: We know that if the victim is taking the optimal care (c = c ) then if the injurer increases his care from a level below the due care (d < d ) to the due care level (d = d ) then the reduction in the expected loss must be at least as large as the increase in care by the injurer. If there are no complementarities then it follows that even if the victim is taking less than the optimal care (c < c ), when the injurer increases his care from a level below the due care level to the due care level the reduction in expected loss must be at least as large as the increase in care. Thus, if there are no complementarities, it can never be advantageous to the injurer to go in for a care level lower than the due care. In case the injurer s care is less than the due care and the victim s care is below the optimal level then the expected loss must be greater than the amount by which the injurer s care falls short of the due care. Consequently, if β is positive then a care-configuration in which the injurer is taking less than the due care and the victim is taking less than the optimal care can never be a Nash equilibrium. The condition that β be positive or α be zero is a very mild condition. Even this condition is not required for efficiency if in addition to ruling out complementarities it is also assumed that there is a unique care-configuration at which total social costs are minimized. Let A m denote the set of applications which are such that there is a unique care-configuration at which total social costs are minimized, i.e., M is a singleton. Then, A s A m is the of applications without complementarities in cares and with a unique TSC-minimizing configuration. Theorem 3 establishes that the efficiency obtains for every application belonging to A s A m regardless of the values of α and β. We have already seen that when there are no complementarities in cares it can never be advantageous for the injurer to take less than the due care. If, TSC are minimized at a unique careconfiguration, then it follows that taking less than the due care will be disadvantageous for the injurer even when β is zero as the uniqueness of the care-configuration coupled with the absence of complementarities implies that regardless of whether the victim is taking the optimal care or less than the optimal care, the lower than the due care by the injurer will result in an increase in expected loss of an amount greater than the amount by which the injurer s care falls short of the due care level. In summary, if there is uncertainty regarding whether the court will apply the standard version or the incremental version then inefficiency is possible. A necessary and sufficient 10

12 condition for efficiency under every application is that β be greater than or equal to α. If complementarities in cares are assumed away then a necessary and sufficient condition for efficiency to obtain under every application without care-complementarities is that β be positive or α be zero. If in addition to assuming that there are no complementarities in cares it is also assumed that TSC are minimized at a unique care-configuration then the efficiency obtains under every application with these characteristics. In the law and economics literature both the absence of complementarities and uniqueness of minimum are generally assumed. Thus under the standard assumptions efficiency obtains notwithstanding uncertainty regarding the interpretation of the negligence rule. References Brown, John Prather (1973), Toward an Economic Theory of Liability, Journal of Legal Studies 2, pp Grady, Mark F. (1983), A New Positive Theory of Negligence, Yale Law Journal 92, pp Grady, Mark F. (1984), Proximate Cause and the Law of Negligence, Iowa Law Review 69, pp Jain, Satish K. (2010), On the Efficiency of the Negligence Rule, Journal of Economic Policy Reform 13, pp Kahan, M. (1989), Causation and Incentives to Take Care under the Negligence Rule, Journal of Legal Studies 18, pp Landes, William M. and Posner, Richard A. (1987), The Economic Structure of Tort Law, Harvard University Press: Cambridge (MA). Shavell, Steven (1987), Economic Analysis of Accident Law, Harvard University Press: Cambridge (MA). Appendix Lemma 1 Let < C, D, π, H, d > be an application belonging to A. Let c be such that (c, d ) M. Then, regardless of the values of α and β, (c, d ) is a Nash equilibrium. Proof: Let < C, D, π, H, d > A; and let (c, d ) M. 11

13 EC 1 (c, d ) = c + L(c, d ) c c EC 1 (c, d ) = c + L(c, d ) Therefore, c c EC 1 (c, d ) EC 1 (c, d ) = [c + L(c, d )] [c + L(c, d )] = [c + d + L(c, d )] [c + d + L(c, d )] = T SC(c, d ) T SC(c, d ) 0 (1.1) EC 2 (c, d ) = d d < d EC 2 (c, d) = d + βl(c, d) + (1 β)[l(c, d) L(c, d )] = d + L(c, d) (1 β)l(c, d ) Therefore: d < d EC 2 (c, d) EC 2 (c, d ) = [d + L(c, d) (1 β)l(c, d )] [d ] = [c + d + L(c, d)] [c + d + L(c, d )] + βl(c, d ) = [T SC(c, d) T SC(c, d )] + βl(c, d ) 0 (1.2) d > d EC 2 (c, d) = d Therefore, d > d EC 2 (c, d) EC 2 (c, d ) = d d > 0 (1.3) (1.1)-(1.3) establish that (c, d ) is a Nash equilibrium. Lemma 2 Let < C, D, π, H, d > be an application belonging to A; and let c be such that (c, d ) M. If β α, then we must have: ( (c, d) C D)[(c, d) is a Nash equilibrium (c, d) M]. Proof: Let < C, D, π, H, d > A. Let c be such that (c, d ) M. Suppose (c, d) is a Nash equilibrium. (c, d) is a Nash equilibrium implies EC 1 (c, d) EC 1 (c, d) (2.1) and EC 2 (c, d) EC 2 (c, d ) (2.2) (2.1) (2.2) EC 1 (c, d) + EC 2 (c, d) EC 1 (c, d) + EC 2 (c, d ) (2.3) 12

14 Now, if d < d we have: EC 1 (c, d) = c + (1 α)l(c, d ); EC 2 (c, d) = d + βl(c, d) + (1 β)[l(c, d) L(c, d )]; EC 1 (c, d) = c + (1 α)l(c, d ); and EC 2 (c, d ) = d. Therefore if d < d, (2.3) reduces to: c + (1 α)l(c, d ) + d + βl(c, d) + (1 β)[l(c, d) L(c, d )] c + (1 α)l(c, d ) + d c + d + L(c, d) c + (1 α)l(c, d ) + d (β α)l(c, d ) c + d + L(c, d) c + d + (1 α)l(c, d ), as (β α) 0 c + d + L(c, d) c + d + L(c, d ), as 0 1 α 1 Therefore, d < d T SC(c, d) T SC(c, d ) (2.4) If d d we have: EC 1 (c, d) = c + L(c, d); EC 2 (c, d) = d; EC 1 (c, d) = c + L(c, d); and EC 2 (c, d ) = d. Therefore, if d d, (2.3) reduces to: c + L(c, d) + d c + L(c, d) + d c + d + L(c, d) c + d + L(c, d ), as L(c, d) L(c, d ), by (A2) Therefore, d d T SC(c, d) T SC(c, d ) (2.5) (2.4) and (2.5) establish that T SC(c, d) T SC(c, d ). As total social costs are minimized at (c, d ), it follows that T SC(c, d) = T SC(c, d ); and consequently we must have (c, d) M. The proposition therefore stands established. Lemma 3 If 1 > α > β > 0, then there exists an application belonging to A under which the efficiency does not obtain. Proof: Let 1 > α > β > 0. Choose positive numbers δ and θ. Let 0 < ɛ 1 < 1 α α β δ; 0 < ɛ 2 < β α β δ. 13

15 Let c 0 = 1 α α β δ + 1 α α θ; d 0 = β α β δ + ɛ 2 + λθ; where 0 < β 1 α 1 β α < λ < 1.3 Now, consider the following application: C = {0, c 0 }; D = {0, d 0 }. L(c, d), (c, d) C D, is as given in the following array: d 0 d 0 c 0 c 0 + d 0 + δ c 0 + d 0 + δ ɛ 2 c 0 c 0 + d 0 + δ ɛ 1 0 As ɛ 1 < c 0 and ɛ 2 < d 0, it follows that (c 0, d 0 ) is the unique TSC-minimizing care-configuration. Let the due care for the injurer d = d 0. We have: EC 1 (0, 0) = (1 α)(c 0 + d 0 + δ ɛ 2 ); EC 2 (0, 0) = β(c 0 + d 0 + δ) + (1 β)ɛ 2 ; EC 1 (c 0, 0) = c 0 ; and EC 2 (0, d 0 ) = d 0. EC 1 (c 0, 0) EC 1 (0, 0) = c 0 (1 α)(c 0 + d 0 + δ ɛ 2 ) = αc 0 (1 α)(d 0 + δ ɛ 2 ) = α[ 1 α α β δ + 1 α α θ] (1 α)[ β α β δ + ɛ 2 + λθ + δ ɛ 2 ] = (1 α)(1 λ)θ > 0. (3.1) EC 2 (0, d 0 ) EC 2 (0, 0) = d 0 β(c 0 + d 0 + δ) (1 β)ɛ 2 = (1 β)d 0 β(c 0 + δ) (1 β)ɛ 2 β = (1 β)[ α β δ + ɛ 2 + λθ] β[ 1 α α β δ + 1 α α θ + δ] (1 β)ɛ 2 = θ[(1 β)λ β 1 α α ] > 0, as λ > β 1 β 1 α α 3 α > β β 1 α 1 β α < 1. > 0. (3.2) 14

16 (3.1) and (3.2) establish that (0,0) is a Nash equilibrium. As (0, 0) / M, it follows that efficiency does not obtain under the application considered here. This establishes the proposition. Lemma 4 If 1 = α > β, then there exists an application belonging to A under which the efficiency does not obtain. Proof: Let 1 = α > β. Choose positive numbers δ, θ, ɛ 2 and c 0. Let 0 < ɛ 1 < c 0 and d 0 = β 1 β (c 0 + δ) + ɛ 2 + θ. Now, consider the following application: C = {0, c 0 }; D = {0, d 0 }. L(c, d), (c, d) C D, is as given in the following array: d 0 d 0 c 0 c 0 + d 0 + δ c 0 + d 0 + δ ɛ 2 c 0 c 0 + d 0 + δ ɛ 1 0 As ɛ 1 < c 0 and ɛ 2 < d 0, it follows that (c 0, d 0 ) is the unique TSC-minimizing care-configuration. Let the due care for the injurer d = d 0. We have: EC 1 (0, 0) = 0; EC 2 (0, 0) = β(c 0 + d 0 + δ) + (1 β)ɛ 2 ; EC 1 (c 0, 0) = c 0 ; and EC 2 (0, d 0 ) = d 0. EC 1 (c 0, 0) EC 1 (0, 0) = c 0 0 > 0. (4.1) 15

17 EC 2 (0, d 0 ) EC 2 (0, 0) = d 0 β(c 0 + d 0 + δ) (1 β)ɛ 2 = (1 β)d 0 β(c 0 + δ) (1 β)ɛ 2 = (1 β)[ β 1 β (c 0 + δ) + ɛ 2 + θ] β(c 0 + δ) (1 β)ɛ 2 = (1 β)θ > 0. (4.2) (4.1) and (4.2) establish that (0,0) is a Nash equilibrium. As (0, 0) / M, it follows that efficiency does not obtain under the application considered here. This establishes the proposition. Lemma 5 If α > β = 0, then there exists an application belonging to A under which the efficiency does not obtain. Proof: Let α > β = 0. Choose positive numbers δ, θ and d 0. Let 0 < ɛ 2 < d 0. Let c 0 = 1 α α (d 0 + δ) + θ; and let 0 < ɛ 1 < c 0. Now, consider the following application: C = {0, c 0 }; D = {0, d 0 }. L(c, d), (c, d) C D, is as given in the following array: d 0 d 0 c 0 c 0 + d 0 + δ c 0 + d 0 + δ ɛ 2 c 0 c 0 + d 0 + δ ɛ 1 0 As ɛ 1 < c 0 and ɛ 2 < d 0, it follows that (c 0, d 0 ) is the unique TSC-minimizing 16

18 care-configuration. Let the due care for the injurer d = d 0. We have: EC 1 (0, 0) = (1 α)(c 0 + d 0 + δ ɛ 2 ); EC 2 (0, 0) = ɛ 2 ; EC 1 (c 0, 0) = c 0 ; and EC 2 (0, d 0 ) = d 0. EC 1 (c 0, 0) EC 1 (0, 0) = c 0 (1 α)(c 0 + d 0 + δ ɛ 2 ) = αc 0 (1 α)(d 0 + δ ɛ 2 ) = α[ 1 α α (d 0 + δ) + θ] (1 α)[d 0 + δ ɛ 2 ] = αθ + (1 α)ɛ 2 > 0. (5.1) EC 2 (0, d 0 ) EC 2 (0, 0) = d 0 ɛ 2 > 0. (5.2) (5.1) and (5.2) establish that (0,0) is a Nash equilibrium. As (0, 0) / M, it follows that efficiency does not obtain under the application considered here. This establishes the proposition. Theorem 1 A necessary and sufficient condition for efficiency under every application belonging to A is that β be at least as large as α. Proof: Let β α. Consider any a =< C, D, π, H, d > application belonging to A. (c, d ) is a Nash equilibrium by Lemma 1. By Lemma 2 we have: ( (c, d) C D)[(c, d) is a Nash equilibrium (c, d) M]. Therefore, it follows that the efficiency obtains under a. Next suppose that α > β. Then by Lemmas 3-5 there exists an application a =< C, D, π, H, d > under which the efficiency does not obtain. This establishes the theorem. Lemma 6 Let < C, D, π, H, d > be an application belonging to A s. Then, if β > 0 we must have: ( (c, d) C D)[(c, d) is a Nash equilibrium (c, d) M]. 17

19 Proof: Let β > 0; and let < C, D, π, H, d > A s. Let c be such that (c, d ) M. Suppose (c, d) is a Nash equilibrium. (c, d) is a Nash equilibrium implies EC 1 (c, d) EC 1 (c, d) (6.1) and EC 2 (c, d) EC 2 (c, d ). (6.2) (6.1) (6.2) EC 1 (c, d) + EC 2 (c, d) EC 1 (c, d) + EC 2 (c, d ). (6.3) If d < d we have: EC 1 (c, d) = c + (1 α)l(c, d ); EC 2 (c, d) = d + βl(c, d) + (1 β)[l(c, d) L(c, d )]; EC 1 (c, d) = c + (1 α)l(c, d ); and EC 2 (c, d ) = d. Suppose c < c d < d. c < c L(c, d) L(c, d ) L(c, d) L(c, d ), (6.4) because the application under consideration belongs to A s. L(c, d) L(c, d ) d d, (6.5) as TSC are minimized at (c, d ). From inequalities (6.4) and (6.5) we obtain: L(c, d) d d + L(c, d ), which in turn implies that: L(c, d) > d d, as L(c, d ) > 0. (6.6) Now, EC 2 (c, d) = d + βl(c, d) + (1 β)[l(c, d) L(c, d )] d + βl(c, d) + (1 β)[l(c, d) L(c, d )], by (6.4) d + βl(c, d) + (1 β)[d d], by (6.5) > d + β[d d] + (1 β)[d d], by (6.6) as β > 0 = d. But by (6.2), EC 1 (c, d) d. This contradiction establishes that if c < c d < d then (c, d) cannot be a Nash equilibrium; and therefore: (c, d) is a Nash equilibrium c c d d. (6.7) 18

20 If d d we have: EC 1 (c, d) = c + L(c, d); EC 2 (c, d) = d; EC 1 (c, d) = c + L(c, d); and EC 2 (c, d ) = d. Therefore, if d d, (6.3) reduces to: c + L(c, d) + d c + L(c, d) + d c + d + L(c, d) c + d + L(c, d ), as L(c, d) L(c, d ), by (A2). Therefore, d d [(c, d) is a Nash equilibrium T SC(c, d) T SC(c, d )]. (6.8) Next suppose d < d c c. If d < d, (6.3) reduces to: c + (1 α)l(c, d ) + d + βl(c, d) + (1 β)[l(c, d) L(c, d )] c + (1 α)l(c, d ) + d (6.9) c + d + L(c, d) c + (1 α)l(c, d ) + d (β α)l(c, d ) c + d + L(c, d) c + d + L(c, d ) αl(c, d ) (β α)l(c, d ) T SC(c, d) T SC(c, d ) α[l(c, d ) L(c, d )] βl(c, d ). (6.10) If c c then L(c, d ) L(c, d ) by (A2). From (6.10), therefore, it follow that: d < d c c T SC(c, d) T SC(c, d ) 0. (6.11) As total social costs are minimized at (c, d ), it follows that: T SC(c, d) T SC(c, d ) 0 T SC(c, d) = T SC(c, d ); and consequently we obtain: T SC(c, d) T SC(c, d ) 0 (c, d) M. (6.7), (6.8) and (6.11), therefore, establish the lemma. Lemma 7 If β = 0 α > 0, then there exists an application belonging to A s under which the efficiency does not obtain. Proof: Let β = 0 α > 0. Choose positive numbers δ and d 0. Let c 0 > 1 α α δ. 19

21 Now, consider the following application: C = {0, c 0 }; D = {0, d 0 }. L(c, d), (c, d) C D, is as given in the following array: d c 0 d 0 0 c 0 + d 0 + δ c 0 + δ c 0 d 0 0 As δ > 0 we obtain: M = {(c 0, 0)(c 0, d 0 )}. Let the due care for the injurer d = d 0. It is clear that this application belongs to A s. We have: EC 1 (0, 0) = (1 α)(c 0 +δ); EC 2 (0, 0) = d 0 ; EC 1 (c 0, 0) = c 0 ; and EC 2 (0, d 0 ) = d 0. EC 1 (c 0, 0) EC 1 (0, 0) = c 0 (1 α)(c 0 +δ) = αc 0 (1 α)δ = α(c 0 1 α α δ) > 0, as α > 0 c 0 > 1 α α δ. (7.1) EC 2 (0, d 0 ) EC 2 (0, 0) = d 0 d 0 = 0 (7.2) (7.1) and (7.2) establish that (0,0) is a Nash equilibrium. As (0, 0) / M, it follows that efficiency does not obtain under the application considered here. This establishes the proposition. Theorem 2 A necessary and sufficient condition for efficiency under every application belonging to A s is that (β > 0 α = 0). Proof: If α = 0 then regardless of the value of β we have: β α; therefore, 20

22 by Theorem 1, efficiency obtains under every application belonging to A s, as A s A. Let β > 0. Consider any a =< C, D, π, H, d > belonging to A s. (c, d ) is a Nash equilibrium by Lemma 1 as A s A. By Lemma 6 we have: ( (c, d) C D)[(c, d) is a Nash equilibrium (c, d) M]. Therefore, it follows that the efficiency obtains under a. This establishes the sufficiency part of the theorem. If (β = 0 α > 0) then by Lemma 7 there exists an application belonging to A s under which efficiency does not obtain; which establishes the necessity part. Lemma 8 Let a = < C, D, π, H, d > be an application belonging to A s A m. Then: ( (c, d) C D)[(c, d) is a Nash equilibrium (c, d) M]. Proof: Let a = < C, D, π, H, d > A s A m. Let c be such that (c, d ) M. Suppose (c, d) is a Nash equilibrium. (c, d) is a Nash equilibrium implies EC 1 (c, d) EC 1 (c, d) (8.1) and EC 2 (c, d) EC 2 (c, d ). (8.2) (8.1) (8.2) EC 1 (c, d) + EC 2 (c, d) EC 1 (c, d) + EC 2 (c, d ). (8.3) If d < d we have: EC 1 (c, d) = c + (1 α)l(c, d ); EC 2 (c, d) = d + βl(c, d) + (1 β)[l(c, d) L(c, d )]; EC 1 (c, d) = c + (1 α)l(c, d ); and EC 2 (c, d ) = d. Suppose c < c d < d. c < c L(c, d) L(c, d ) L(c, d) L(c, d ), as a A s. (8.4) L(c, d) L(c, d ) > d d, (8.5) as TSC are uniquely minimized at (c, d ) because a A m. (8.4) (8.5) L(c, d) > d d + L(c, d ) 21

23 L(c, d) > d d, as L(c, d ) > 0. (8.6) L(c, d) L(c, d ) > d d L(c, d) > d d βl(c, d) + (1 β)[l(c, d) L(c, d )] > β(d d) + (1 β)(d d) = d d (8.7) Now, EC 2 (c, d) = d + βl(c, d) + (1 β)[l(c, d) L(c, d )] > d + (d d), by (8.7) = d. But by (8.2), EC 2 (c, d) d. This contradiction establishes that if c < c d < d then (c, d) cannot be a Nash equilibrium; and therefore: (c, d) is a Nash equilibrium c c d d. (8.8) If d d we have: EC 1 (c, d) = c + L(c, d); EC 2 (c, d) = d; EC 1 (c, d) = c + L(c, d); and EC 2 (c, d ) = d. Therefore, if d d, (8.3) reduces to: c + L(c, d) + d c + L(c, d) + d c + d + L(c, d) c + d + L(c, d ), as L(c, d) L(c, d ), by (A2). Therefore, d d [(c, d) is a Nash equilibrium T SC(c, d) T SC(c, d )]. (8.9) Next suppose d < d c c. If d < d, (8.3) reduces to: c + (1 α)l(c, d ) + d + βl(c, d) + (1 β)[l(c, d) L(c, d )] c + (1 α)l(c, d ) + d (8.10) c + d + L(c, d) c + (1 α)l(c, d ) + d (β α)l(c, d ) c + d + L(c, d) c + d + L(c, d ) αl(c, d ) (β α)l(c, d ) T SC(c, d) T SC(c, d ) α[l(c, d ) L(c, d )] βl(c, d ). (8.11) If c c then L(c, d ) L(c, d ) by (A2). From (8.11), therefore, it follow that: d < d c c T SC(c, d) T SC(c, d ) 0. (8.12) 22

24 As total social costs are minimized at (c, d ), it follows that: T SC(c, d) T SC(c, d ) 0 T SC(c, d) = T SC(c, d ); and consequently we obtain: T SC(c, d) T SC(c, d ) 0 (c, d) M. (8.8), (8.9) and (8.12), therefore, establish the lemma. Theorem 3 Efficiency obtains under every application belonging to A s A m. Proof: Consider any application a =< C, D, π, H, d > belonging to A s A m. (c, d ) is a Nash equilibrium by Lemma 1 as A s A m A. By Lemma 8 we have: ( (c, d) C D)[(c, d) is a Nash equilibrium (c, d) M]. Therefore, it follows that efficiency obtains under a. This establishes the theorem. 23