Risk of Litigation under Legal Expenses Insurance

Transcription

1 Risk of Litigation under Legal Expenses Insurance Myriam Doriat-Duban and Bruno Lovat June 1, 2016 Abstract This article analyzes conflict resolution by introducing the legal expenses insurance and risk aversion (neutrality included). We consider an informational asymmetry on the plainti s insurance policy. We first show that the defendant can have interest to make no o er to the plainti, even in complete information. Under the conditions he makes an o er, we determine the formal conditions under which he can have interest to make an o er rather than another, sometimes with the risk to go to court, and we determine the conflict resolution: judgement or agreement. We show that for the defendant the comparison between the potential o ers can depend on the plainti s probability of winning but not in all situations, especially relative to the plainti s risk aversion. BETA UMR 7522 and University of Lorraine, France ; myriam.duban@univ-lorraine.fr BETA UMR 7522 and University of Lorraine, France ; bruno.lovat@univ-lorraine.fr 1

2 1 Introduction In the law and economic litterature, the conflict is usually considered as a situation of interaction between a plainti (she) and a defendant (he). The aim of the litigation is the recognition of a right to obtain damages in the case of an accident or to enforce a contract for example. Thus, the aim of the economic analysis of conflicts is to determine the way the litigation is solved, by judgment or settlement. In other words, the main question is how the plainti obtains the compensation for his/her damage. Our approach is quite di erent because we consider the litigation as a risk and not only as a mean to obtain compensation. There are several risks in a litigation. The first one is the risk to lose the case and to obtain no damages for the plaintif, to be ordered to pay damages for the defendant. Another risk is to be obliged to pay the costs of the litigation if the case is lost, in particular in legal european systems (in the american system, each party pays is own costs of litigation).the litigants try to protect themselves against these risks. Thus, they can protect themselves against the risk of losing the case by concluding an agreement. But the plainti does not accept any o er as the defendant does not make any o er: the plainti determines an amount below which she refuses any aggreement as the defendant sets the highest o er he accepts to pay to avoid the judgment. The litigants can protect themselves against the risk to pay the litigation costs by another mean. They shift the litigation costs to an insurer through a legal expenses insurance 1. This system prevails in Europe where contingent fees are prohibited. Few articles deal with issues relevant to legal expenses insurance. Most of the authors study the consequences of legal expenses insurance on conflict resolution. Among them, Kirstein [2000] demonstrates that a legal expenses insurance for the plainti has a positive e ect on her strategic position by making her threat to sue more credible, especially for cases with a negative expected value. Heyes, Rickman and Tzavara [2004] study the insurance decision of a risk averse plainti and the e ects of this insurance on settlement amounts, settlement probabilities, volume of accidents and trials. Ancelot, Doriat-Duban and Lovat [2012] extend their analysis by studying the consequences of a substitution between the legal expenses insurance and the legal aid on conflict resolution. More recently, Qiao [2013] takes into account the role of the insurer in the negociations and studies the three-way relationship between the client, the lawyer and the insurer and determines the optimal legal expenses insurance. Some other authors have a more larger perspective, beyond the litigants. Thus, van Velthoven and van Wijck [2001] show that the e ect of legal expenses insurance on the social welfare is uncertain because the deterrence is improved but in the same time, the number of trials increases. In the perspective to choose between two alternative systems, Baik and Kim [2007] compare the American practice of contingent fees with the European practice of legal expenses insurance. They show that the plainti s expected payments are higher under a legal 1 We consider only "before the event insurance" which is an insurance bought before any litigation and whose aim is to cover the costs of the claim. 2

3 insurance system than under a contingent fee system but the conclusion is opposite for litigants legal expenditures. Friehe [2010] generalizes the results of this model by integrating the defendant s degree of fault 2. The aim of this article is to further develop the analysis of the consequences of legal expenses insurance on the way litigation is solved. We propose to apply the tools of the decision theory to determine how the defendant decides of his o er when he is faced to a insured plainti (who is protected against the risk of litigation costs) who wants to avoid the risk to lose the case. By this way, our approach is focused on the litigants and the way they make their decision, not only on the result of the negociations. From an analytical point of view, we introduce an informational asymmetry about the contract of insurance chosen by the plainti (and not about the fact that the plainti is insured or not as in Heyes, Rickman and Tzavara [2004] or Baik and Kim [2007]). More precisely, the plainti can choose between two contracts of legal expenses insurance which di er in their prices and compensations. The defendant does not know the contract chosen by the plainti when he makes his/her o er to settle the case. This informational asymmetry has important consequences on the defendant s decision. In particular, on the one hand, we show that the defendant could have interest to make no o er, especially when the plainti s probability of success is low enough ; on the other hand, if the defendant has interest to make a positive o er, this last can depend or not on the plainti s probability of success. More precisely, we show that it depends on the attitude of the plainti relative to the risk. Beyond the contribution at a very wide literature about conflict resolution in the continuity of Bebchuk [1984] and legal expenses insurance, this article has a methodological interest. While the insurance market operates with the assumption that the insured are risk-averse (even if risk-neutral agents can benefit from being insured 3 ), studies about conflicts resolution often assume risk-neutral parties. We propose to extend the analysis at risk aversion (neutrality included). In this sense, our results can be extended to earlier models that our analysis generalizes. Taking into risk aversion has for consequence to obtain more general results, which are very interesting both in complete and incomplete information. In complete information, the relevant question is to determine if it is better for the defendant to be opposed to one type of plainti s rather than the other. In incomplete information, the question is to determine the better o er because the defendant has to balance the expected gain to make a lower o er with the higher risk to go to court. To study the decisions of the plainti (who has to decide between to accept or to refuse the defendant s o er) and the defendant (who has to set his o er without knowing the plainti s insurance policy), we proceed in three stages. In the second part, we present the conflict situation opposing the plainti and the 2 For a short summary of the economic litterature about legal expenses insurance : Visscher and Schepens [2010]. 3 Skogh [1999], Kirstein [2000]. 3

4 defendant and we determine the minimum o er accepted by the plainti but also the condition to have a positive o er from the defendant. In the third part, we study the negotiations between the parties when the plainti is risk neutral, first in complete and then in incomplete information. In complete information, we show that the defendant can have interest to make no o er. When he makes an o er, it is ever better for him to be opposed to one type of plainti s rather than to another. In incomplete information, we show that the o er of the defendant depends not only on the plainti s probability of success but also on the proportion of each type of plainti s. Above a specified proportion, we show it exists a threshold for the plainti s probability of winning which is decisive for the defendant s o er. In all cases, we determine the issue of the litigation: judgement or agreement. In the fourth part, we consider that plainti is risk averse. We show that in one specific configuration, the results are di erent because it is not possible to compare the o ers for each type of plainti s. Once again, we determine the conditions under which the defendant prefers one o er rather than another (in particular in terms of plainti s probability of success and proportion of each type of plainti s) and the issue of the litigation. Finally, we conclude by discussing the implications of our findings. 2 A situation of litigation We suppose a litigation between a risk-neutral defendant (he) and a risk-averse plainti (she) who has taken out a legal expenses insurance policy. Given: : the plainti s initial wealth (before the accident) : the amount of the injury, equal to the damages granted to the plainti by the court; is known by both litigants. : the aggregated legal costs (i.e.the legal costs both of the plainti and of the defendant), known by both parties. Where no insurance has been taken out, the losing party bears the full legal costs or a part of these costs. These costs are partially covered by the legal expenses insurance, which is more or less extensive depending on the contract. In other words, the insurance covers a part of the legal costs paid by the plainti if she loses (thus also including the defendant s costs). Conversely, if the plainti wins, he has no legal costs as these are paid by the defendant. : the plainti s probability of winning the case ( 2 [0; 1]). The plainti can choose between two policies of legal expenses insurance: - Insurance policy 1 ensures a cover rate 1 2 [0 1] against payment of a premium 1. In these conditions, the plainti accepts any out-of-court settlement 1 2 [0 +1[ to forego the trial ( 1 is the certainty equivalent of the trial for a plainti who chooses insurance policy 1). - Insurance policy 2 ensures a cover rate 2 2 [0 1] against payment of a premium 2. In these conditions, the plainti accepts any out-of-court 4

5 settlement 2 2 [0 +1[ to forego the trial ( 2 is the certainty equivalent of the trial for a plainti who chooses insurance policy 2). We assume that and (consequently, 1 2 [0 1[). In other words, the second policy is more expensive but o ers a better cover than the first 4. There are two types of plainti s who di er from their utility function and consequently by the insurance policy they choose. Thus, 1 is the plainti s utility function of the plainti who chooses the insurance policy 1 and 2 the plainti s utility function of the plainti who chooses the insurance policy 2. The functions 1 and 2 are increasing and concave on the plainti s wealth. For each level of wealth, it is possible to calculate the absolute aversion coe cient ( ) (Arrow-Pratt coe cient) 5. We assume that the type-2 plainti is more risk averse than the plainti 1: 2 ( ) 1 ( ) The informational asymmetry relates to the plainti s insurance policy, unknown to the defendant. Because it determines the level of cover of the plainti s costs if she loses, the insurance policy is expected to influence the latter s response to the defendant s settlement o er. In this context, we determine the minimum amount required by the plainti to accept an agreement and the conditions for a positive o er from the defendant. 2.1 Minimum settlement o er accepted by the plainti The certainty equivalent of the trial, ( = 1 2) is the amount which with absolute certainty provides the plainti with the same utility as the trial. It therefore corresponds to the minimum settlement amount accepted by the plainti to forego the trial. For the plainti, the lotteries [ + ; 1] and [ (1 ) ; 1 ] for = 1 2 are therefore equivalent. As a result, the certainty equivalent of the trial,, is defined by: ( + ) = ( ) + (1 ) [ (1 ) ] (1) with = 1 2. depends on the plainti s probability of winning the case, which does not depends on the plainti s type 6.In the cases of agreement, it can then be interesting to compare the defendant s o ers, depending on whether he is faced with a type-1 or type-2 plainti. At this stage, nothing allows us to conclude that 1 2 or conversely. In other words, we cannot conclude that an o er 4 The value of our approach partly lies in the fact that it does not specify the extent to which the price di erence is compensated for by the di erence in cover. This allows us to develop an analysis that is as general as possible, with emphasis on the balancing of higher insurance costs with better cover. ³ 5 The Arrow-Pratt coe cient 0 is positive when is increasing concave (risk aversion). 6 We consider there is no relation between the legal expenses insurance contract and the plainti s probability of success at trial. 5

6 corresponding to an insurance policy will always be greater than one relative to the other.yet this comparison is important, for if the o ers di er under the two contracts, then for a same injury the out-of-court compensation will not be the same, and will depend on the insurance contract chosen. According to (1), 1 and 2 are respective solutions of the equations: 1 ( ) = 1 ( 1 ) + (1 ) 1 [ 1 (1 1 ) ] 2 ( ) = 2 ( 2 ) + (1 ) 2 [ 2 (1 2 ) ] Let ( ) = ( ) = 2 ( 2 ) = 2 [ 2 (1 2 ) ] 0 = 2 [ 2 (1 1 ) ] = - : utility derived by a type-2 plainti from her initial income reduced by the insurance premium associated with a type-2 policy. - : utility derived by a type-2 plainti from her initial income reduced by the insurance premium associated with a type-2 policy, the injury amount, and the court costs she must bear if she loses, and for which she is covered under the terms of a type-2 policy. - 0 : utility derived by a type-2 plainti from her initial income reduced by the insurance premium associated with a type-2 policy, the injury amount and the court costs she must bear if he loses, and for which she is covered under the terms of a type-1 policy. - : di erence in utility derived by a type-2 plainti between the situation where she benefits from her initial income reduced by the insurance premium of policy 2 and her utility if she loses the case (she bears the damage and the court costs not covered by the insurance). The properties of the function are established in the lemma 2 of the appendix A2. We can then explicitly find the condition on which allows for a comparison of the out-of-court o ers under the two insurance policies. Proposition 1 (Appendix A1): 6

7 The minimum amount required by a type-2 plainti to settle out of court is inferior to that of a type-1 plainti if and only if the following condition is satisfied: 2 1 = [ + (1 ) ] ( ) + (1 ) ( 0 ) (2) 2.2 Conditions for a positive o er from the defendant to the plainti We determine the conditions under which the defendant always makes an o er. The question is justified because the defendant s probability of winning the case is positive ((1 ) 2 [0; 1]) and so it can be in his interest to go to court rather than to settle the case. The defendant never makes an o er higher than his maximum expected losses: ( + ) (3) We first look at two extreme situations in which the defendant is either certain to win the case, or sure of losing. Thus, if = 0, the defendant has no incentive to make an o er because he is certain to win the case. Conversely, if = 1, his o er is equal to the injury As these two situations are of little interest for our analysis, we focus on the situations in between. We therefore assume that 2 ]0 1[. The following proposition can then be made: Proposition 2 (Appendix A2): The defendant makes a strictly positive o er to the plainti only if the plainti s probability of winning is high enough. There exists 0 2 ]0 1[ such that : 0 = 0 0 = = 0 We therefore obtain a plainti s probability of winning the case (which di ers depending on the contract type), 0, below which the defendant makes no o er. We determine the plainti s probability of winning the case, for which the defendant s o er is strictly positive with certainty, 0. This simply requires that min ( 1 2 ) 0 and therefore that the plainti s probability of winning verifies: 0 = ( ) with 01 (respectively 02 ) the probability of winning for a type-1 (respectively type-2) plainti under which the defendant does not make an o er. 7

8 If the plainti is risk neutral, Indeed in this specific case, ( ) =. Hence, according to (*) in appendix A2: = (1 2 ) + (1 2 ) (1 1) + (1 1 ) = 1 2 ( + 1 ) ( + 2 ) 0 and therefore, because 0 = ( ): 0 = 01 = (1 1) + (1 1 ) Equality (4) establishes that faced with a risk-neutral plainti, the minimum probability above which the defendant always makes an o er corresponds to the share of the legal costs he will have to pay if he loses, in the total costs he will have to bear if he loses (including the injury which is not compensated for and the legal costs). Similarly, it exists a minimum probability above which the defendant always makes an o er if the plainti s risk averse. In the rest of our analysis, we assume that 0 1 to study only the situations where the defendant makes an o er. 3 The plainti is risk neutral Our aim is to determine the issue of the conflict in complete and incomplete information when the plainti is risk neutral. Then, we compare the o ers in case of agreement relative to the type of the plainti and show that the defendant has interest to be opposed to a type-1 plainti because Conflict resolution in complete information If the plainti is risk neutral, we have shown that If the plainti s probability of winning the case is between 02 and 01, the defendant makes a positive o er only if the plainti is type-2. But if the plainti s probability of winning the case is higher than 01, the defendant makes an o er corresponding to the type of the plainti and the case is settled out of court. The issue of the conflict depending on the value of is described in the following table. Table 1 : Conflict resolution in complete information when the plainti is risk neutral In the rest of our analysis we eliminate cases where 0 and focus on strictly positive o ers by the defendant. (4) 8

9 According to (1), when the plainti is risk neutral, 1 and 2 are respective solutions of the equations: = ( 1 ) + (1 ) [ 1 (1 1 ) ] and :, 1 = (1 1 )(1 ) = ( 2 ) + (1 ) [ 2 (1 2 ) ], 2 = (1 2 )(1 ) Because by asssumption 1 2, 1 2 Proposition 3 : If the plainti is risk-neutral, then irrespective of the plainti s probability of winning the case: 1 2 Then we show that the plainti s probability of winning the case does not come into play: the o er made to a type-1 plainti is always lower than the o er made to a type-2 plainti. Thus, for a similar case, the defendant has interest to be opposed to a type 1 plainti rather than to a type-2 because the cost of the agreement is lower. That means that the insurance policy has an impact on the agreement. The type-2 contract o ers a better cover than type-1 contract and even if the price of the insurance is higher, the higher price is compensated by the higher cover. 3.2 Conflict resolution in incomplete information The defendant has no reason to make the plainti an o er greater than max( 1 2 ), as it would be too costly, nor inferior to min( 1 2 ), as it would never be accepted, nor any o er in between. All these o ers would be strictly dominated by 1 or 2. After eliminating strictly dominated strategies, the plainti therefore has two possible out-of-court o ers : 2 { 1 2 } The defendant does not know the type of the plainti. He only knows the proportion of type-1 among plainti s ( 1 2 ]0 1[) and the complementary proportion of type-2 ( 2 = 1 1 ). If the defendant chooses to o er = 1, he knows that he has a probability 1 2 ]0 1[ of meeting a type-1 plainti with whom he always reaches an out-ofcourt agreement 1. 9

10 He also has a probability 2 = 1 1 of meeting a plainti who has taken out contract 2, and therefore of going to court. On average this will cost him ( + ). For this defendant, resolving the litigation at this stage of the game costs on average: 1 = (1 1 )( + ) If the defendant chooses to o er = 2, the conflict is solved by an out-ofcourt agreement which costs him 2 = 2. If the plainti s probability of winning the case is very low, the o er is expected to be as low as possible. Conversely, if the plainti s probability of winning the case is very high, the o er must be as high as possible. In all other cases an in-between out-of-court settlement is expected. According to the condition (3) and proposition 3: 1 2 ( + ) The defendant has to choose between 1 et 2. His choice depends on a threshold relative to plainti s probability of winning the case given by the proposition 4. Proposition 4 (Appendix A4): When the plainti is risk neutral and the defendant does not know her type, if ( 0 ) the defendant makes the o er = 2 and if ( 0) , it exists 2 [ 0 1] such that: - the defendant is indi erent between = 1 and = 2 when = - the defendant makes the o er = 2 when - the defendant makes the o er = 1 when. Thus we have demonstrated that if the defendant makes an o er, this o er depends first on the proportion of type-1 plainti s and second, if this proportion exceed a threshold, on the plainti s probability of winning the case. Thus, for these cases in which matters, the defendant balances on the one hand the risk to be opposed to a type-1 plainti and on the other hand the risk to ( 0) lose the case at trial by making a too small o er. Thus, 1 means that the proportion of type-1 plainti exceeds the threshold. In other words, the defendant can consider there is a lot of type-1 plainti s. So, he determines his o er relative to the plainti s probability of winning the case. If this probability is below the threshold, he makes the lower o er 1 and takes the risk that his o er be refused if the plaintif is type-2. But if the 10

11 plainti s probability of winning the case is high enough ( ) then he does not take the risk to go to court and make the highest o er 2 which is accepted ( 0) by all the plainti s. Conversely, if , the risk to be opposed to a type-2 plainti is high and so the defendant always makes the highest o er 2 to avoid the trial with certainty. In this case there is no balance between the propotion of type-1 plainti s and the probability to lose the trial for the defendant. Finally, the conflict resolution is shown in the table 2: Table 2 : Conflict resolution in incomplete information when the plainti is risk neutral (we suppose 0 ): We know that if the plainti is risk neutral then the plainti s certainty equivalent is smaller if he is covered by contract 1 than if he were covered by contract 2 ( 1 2 ). To illustrate this table, we consider the case where the plainti s probability of winning is greater than ( ) and the proportion ( 0) of type-1 plainti s is In this case, the defendant, who is not aware of the plainti s contract type, makes the o er that guarantees him settlement = 2. Thus, if he is opposed to a type 1 plainti, he makes a to high o er and supports all the cost of the informational asymmetry. Conversely, if he is opposed to a type 2 plainti, he makes just a su cient o er to conclude an agreement. Now, we suppose, then he o ers = 1. This o er is rejected by the type-2 plainti and the case is judged; conversely, this o er is accepted by the type-1 plainti. In this case, the defendant accept to take the risk to go to trial because the plainti s probability of winning is low enough. Thus he balances the lower agreement ( 1 2 ) with a higher risk to go to court but compensated by a lower probability of plainti s of winning the case ( ). We have determined the conflict resolution between a defendant and a risk neutral plainti in complete and incomplete information. Then we propose to extend our analysis to risk averse plainti s. 4 The plainti is risk averse We introduce the risk aversion of the plainti and study its e ect on the defendant s o er and on the conflict resolution. We show that the introduction of risk aversion is not always enough to modify the results we obtain with neutral plainti. More precisely, risk aversion matters only in one configuration of insurance policies, a configuration in which the two policies are not too di erent in their compensation. 4.1 Risk aversion does not matter There is one case similar to the neutral case. It occurs when the absolute risk aversion index ( ) decreases when the wealth increases and ( ) = ( ) simultaneously meets condition (***) of lemma 1. 11

12 In this case, function is concave and ( ) decreases with. It corresponds to a classical situation of risk aversion. But in this case, condition (***) of lemma 1 plays a crutial role. It means that and 0 are not too close, i.e. there must be a su ciently big di erence in the type-2 plainti s utilities when she loses the trial, depending on whether she is covered by a type-2 contract ( = 2 [ 2 (1 2 ) ]), or by a type-1 ( 0 = 2 [ 2 (1 1 ) ]). This implies that the two contracts cover rates must be su ciently di erent. Proposition 5 (Appendix A5): If we suppose that the absolute risk aversion index decreases when the wealth increases and if ( ) = ( ) meets condition (***) of lemma 1, then irrespective of the plainti s probability of winning: 1 2. According to (2), this case can be graphically illustrated (Figure 1). If the absolute risk aversion index decreases when the wealth increases and if ( ) = ( ) meets condition (***) of lemma 1, around 0, increasing and convex function! [ + (1 ) ] is situated above the straight line! ( ) + (1 ) ( 0 ) which is also increasing. The two curves only intersect at = 1. Consequently, the greater the plainti s probability of winning, the closer the two o ers will be to each other; however the o er corresponding to insurance policy 1 will always be inferior to the o er under policy 2. Figure 1: Illustration of the proposition 5 Consequently, in this situation, the analysis relative to the conflict resolution is the same as for the risk neutral analysis, both in complete and incomplete information. So we propose to examine the situation of risk aversion in which the results di er because of the relative configuration of the two possible insurance policies. 4.2 Risk aversion matters We study here the configurations in which the results are not the same as in neutral case. Then, we show that the comparison between the two possible o ers, 1 and 2, is not obvious with some consequences on the defendant s behavior. The defendant can take into account both the proportion of each type of plainti s and the plainti s probability of winning the case In complete information Condition (2) allowing for the comparison of the defendant s o er under both contracts can be simplified if we assume that the absolute risk aversion index ( ) decreases when the wealth increases and if ( ) = ( )

13 meets condition (**) of lemma 1 presented in appendix A3 7. Condition (**) of lemma 1 means that the elements concerned, and 0, are close enough, i.e. there must not be a too big di erence in the utilities of the plainti paying the premium of contract 2 when he loses the trial, between on the one hand the case where she is covered by a type-2 contract ( = 2 [ 2 (1 2 ) ]), and that where she is covered by a type-1 contract ( 0 = 2 [ 2 (1 1 ) ]). This implies that the two contracts cover rates must not be too di erent. Under these conditions, a plainti s threshold probability of winning exists, below which the o er made if the plainti is covered by contract 1 is inferior to the o er made if the plainti is covered by contract 2, and vice versa. We find that: 8 2 [0 [ 1 2 and 8 2 [ 1] 1 2 (5) More precisely, we can formulate the following proposition: Proposition 6 (Appendix A6): If we assume that the absolute risk aversion index ( ) decreases when the wealth increases, and if ( ) = ( ) meets condition (**) of lemma 1, then there exists 2 [0 1[ such that: [0 [ 1 2 and [ 1] 1 2 Proposition 6 (which is a refinement of proposition 1) can be graphically illustrated. Under the assumptions used, around 0, the increasing and convex function! [ + (1 ) ] is situated above the straight line! ( )+ (1 ) ( 0 ) which is also increasing. According to lemma 1 and 2, the two curves then intersect at = and = 1. Figure 2 : Illustration of the proposition 6 We thus show that the comparison between the two o ers depends on the plainti s probability of winning the case. When inequality (2) is verified, the defendant s o er is higher if the plainti is covered by contract 1 than if she is covered by contract 2 and vice versa if inequality (2) is not verified. To determine, we consider the special case where 1 = 2. In this case, the trial s certainty equivalent remains the same irrespective of the contract chosen by the plainti, such that the contract has no influence. We then show that a single probability of the plainti winning the trial 2 ]0 1[ exists and is unique for which the defendant makes the same o er to the plainti covered by contract 1 and to the plainti covered by contract 2. 7 Condition (2) does not depend on the risk aversion. 13

14 Thus, in this configuration relative to the two legal expenses insurance contracts o ered to the risk averse plainti, we show that the plainti s probability of success,, plays a determining role not only in the defendant s decision to make an o er but also in the comparison of o ers according to the contract type chosen by the plainti (here assumed to be known by the defendant) In incomplete information We have to distinguish two cases, depending on the relative positioning of 1 and 2 which depends on the value of relative to defined in proposition 6. We show that below, there is no di erence with the neutral case but if the plainti s probability of winning the case exceeds this threshold, then the results di er from the neutral case. Defendant s o er if 1 2 This case occurs if and only if 2 [0 [. In this situation, the analysis is exactly the same as for the neutral plainti. More precisely, the defendant always makes the highest o er 2 if the proportion of type-2 plainti s is high ( 0) enough (i.e ). Conversely, when the proportion of type-1 plainti s is high enough ( ( 0 ) ), if the plainti s probability of winning is higher than a threshold, the defendant does not take the risk to go to court and make the highest o er. But if the plainti s probability of winning is lower than the threshold, then the defendant prefers to make the lowest o er 1 with the risk to go to court if finally the plainti is a type-2. Then, the conflict resolution is given in table 2. The more interesting situation because it di ers from the neutral case occurs when 2 1. Defendant s o er if 2 1 This case occurs when the absolute risk aversion index decreases, meets condition (**) of lemma 1 and. We know that: 2 ( ) 1 ( ) ( + ) As above, if the defendant o ers 2, there is agreement with type-2 plainti s, who represent share 2 of the population, and a trial with type-1 plainti s. If he o ers 1 there is an agreement irrespective of the type of plainti because 2 1. We then search for the condition under which the two policies are equivalent for the defendant: (1 2 )( + ) = 1 Let us consider ª : [ ( + ) 2( ) 1] R given by ª( ) = ( + ) 1( ). We have ª( ) = ª(1) = 1 so there is a critical value in ] 1[ such that ª 0 ( ) = 0 14

15 We assume that is the unique critical value of ª 8 and we have necessarily. Then, we can establish the proposition 7. Proposition 7 (Appendix A7): Under the conditions where 2 1 and when the defendant does not know the type of the plainti, if ( ) the defendent makes the o er + 2( ) + 1( ) = 1 and if 2 +, it exists 2( ) f 1 f 2 in [ ] such that: n o - the defendant is indi erent between = 1 and = 2 when 2 f 1 f 2 - the defendant makes the o er = 1 when 2 - the defendant makes the o er = 2 when 2 h i h i f 1 [ f 2 1 i f 1 f 2 h. Ultimately, as in the case of complete information, we have shown that the comparison of the defendant s o ers depends on the plainti s probability of winning the case. We then show that, depending on the plainti s probability of winning, the defendant does not necessarily choose the o er that guarantees an out-of-court settlement. For each type of plainti and depending on the plainti s probability of winning, Table 3 determines the way in which the litigation is resolved: agreement or judgement. Table 3 : Conflict resolution in incomplete information, in the specific case where 1 2 (i.e. 2 [ 1]) We consider the case where the plainti s certainty equivalent is lower for the contract 2 than for the contract 1 ( 2 1 ). The defendant takes the risk to go to court and o ers the lower o er (in this case 2 ) when he has managed to strike the balance between the proportion of type-2 plainti s 2 and the level of plainti s probability h i of winning We determined two thresholds f 1 f 2 such that : 2 f 1 f 2 = = 2. The higher is the proportion of type-2 h f 1 f i 2 plainti s, the longer is the subinterval for which the defendant prefers i f h 1 and to o er the lowest o er 2 and the shorter are the two subintervals i f 2 1h for which the defendant o ers 1 Finally, it appears that judgment occurs only on two situations. More precisely, the case is judged when simultaneously the certain equivalent of the type- 1 plainti is lower than the type-2 plainti s one and the proportion of type-1 plainti s is high enough and the plainti s winning probability low enough. This occurs because the defendant takes the risk to make the lowest o er 1 because 8 With this assumption, ª( ) is a nonmonotonic (becauseª( ) = ª(1) = 0) quasiconcave function on [ 1]. 15

16 he has a high probability to be confronted to a type-1 plainti with a low probability of winning the case. But this o er is too low if the plainti is finally a type-2. The other situation where trial occurs corresponds to the symetric situation where simultaneously the certain equivalent of the type-2 plainti is lower than the type-1 plainti s one and the defendant has to find a compromise between the proportion of type-2 plainti s and the level of plainti s winning probability. The trial occurs because the defendant takes the risk to make the lowest o er but this o er is too low if the plainti is finally a type-1. In all other situations, the conflict is settled out-of-court. 5 Conclusion This article analyzes conflict as a risk for the litigants. We focus on the way the litigants make their decisions to avoid the risk to lose the case or to pay the litigation costs. We introduce the legal expenses insurance in a context of risk aversion (neutrality included). Relative to the previous literature, we take into account an informational asymmetry on the type of insurance policy chosen by the plainti. We first show that the defendant can have interest to make no o er to the plainti, even in complete information. Under the conditions he makes an o er, we determine the formal conditions under which he can have interest to make an o er rather than another, sometimes with the risk to go to court, and then we determine the conflict resolution: judgement or agreement. In particular, we show that the comparison between the o ers corresponding to two insurance policies which di er in their price and their compensation is not obvious. The consequences are important for the defendant because the comparison between the two potential o ers can depend on the plainti s probability of winning but not in all situations, especially relative to the risk aversion of the plainti (and some other conditions about paiements o ered by the two legal insurance policies). More precisely, in neutral case and in some risk aversion cases, in complete information, the defendant is always better when he is opposed to a type-1 plainti than to a type-2. In incomplete information, if the proportion of type-1 plainti is lower than a threshold, the o er of the defendant is always the highest o er ( 2 because we shown that 2 1 ). But if this proportion is higher than this threshold, then the defendant s o er depends on the plainti s probability of success. In the other case ( 2 [ 1]), in complete information, we can not assert that the defendant is better when he is oppposed to a type-1 or a type-2 plainti because the relative value of the o ers 1 and 2 depends on the plainti s probability of success. It is a first di erence with the neutral case. In incomplete information, the analysis is similar but the thresholds about the plainti s proportion of type-1 (or 2) and about the plainti s probability of winning are di erent. But in all cases, we show that there is a balance between the lowest o er and the risk to be opposed to a more costly plainti (in the sens her certainty equivalent is the highest) and to go to court. The result of this balance 16

17 determine the conflict resolution. Thus we determine the situations where the defendant does not make the good o er to the plainti and consequently goes to court. But the main result of our analysis is probably to show that in the negociations between a risk neutral defendant and a plainti who can be risk neutral or risk averse, four elements determine the litigants decisions and by this way the conflict resolution: the proportion of each type of plainti s, the plainti s probability of success, the plainti s risk attitude and the configuration of the legal expenses insurance. The combinaison of these four elements is central with some threshold e ects that we have determined. In this sense, our analysis is complementary to previous articles about legal expenses insurance because it goes further in the analysis of the interaction between the relevant elements which determine the resolution of a conflict. Nevertheless, our analysis does not take into account the role of the insurer and we know that it could be crucial especially when a problem of moral hazard occurs. The insurer could have incentive to accept agreement that the client would refuse because he avoids the risk to pay the trial costs if the trial is lost. Taking into account the role of the insurer would improve the analysis of conflict resolution. 6 References Ancelot L., Doriat-Duban M. and Lovat B., Aide juridictionnelle et assurance de protection juridique : coexistence ou substitution dans l accès au droit, Revue Française d Economie, vol. XXVII, (2012), pp Baik K.H. and Kim I., Contingent fees versus legal expenses insurance, International Review of Law and Economics, vol. 27, (2007), pp Bebchuck L. A., Litigation and Settlement Under Imperfect Information, RAND Journal of Economics, n ± 15, (1984), pp Friehe T., Contingent Fees and Legal expenses insurance: comparison for varying defendant fault, International Review of Law and Economics, vol. 30, (2010), pp Heyes A., Rickman N. and Tzavara D., Legal expenses insurance, risk aversion and litigation, International Review of Law and Economics, n ± 24, (2004), pp Kirstein R., Risk neutrality and strategic insurance, The Geneva Papers on Risk and Insurance, vol. 25, n ± 2, (2000), pp Qiao Y., Legal E ort and Optimal Expenses Insurance, Economic Modelling, vol. 32, (2013), pp Skogh G., Mandatory Insurance, in Bouckaert B. and Ge Geest G. (eds), Encyclopedia of Law and Economics, Cheltenham: E. Elgar, entry n ± 2400 ( (1998). Tuil M. and Visscher L., New Trends in Financing Civil Litigation in Europe, a Legal, Empirical, and Economic Analysis, New Horizons in Law and Economics; Edward Elgar, Cheltenham, (2010), 200 p. 17

18 Van Velthoven B. and Klein Haarhuis C., Legal Aid and Legal Expenses Insurance, Complements or Substitutes? The Case of the Netherlands, Journal of Empirical Legal Studies, 8 (n ± 3), (2011), pp Van Velthoven B. and Van Wijck P., Legal Cost Insurance and Social Welfare, Economics Letters, 72, (2001), pp Visscher L. and Schepens T., A Law and Economics Approach to Cost Shifting, Fee Arrangements and Legal Expense Insurance, in New Trends in Financing Civil Litigation in Europe, a Legal, Empirical, and Economic Analysis, Tuil M. and Visscher L. eds., chap. 2, Cheltenham : Edward Elgar, (2010), pp

19 Appendix A1. Proof of the proposition 1 Given: ( ) = ( ) = 2 ( 2 ) = 2 [ 2 (1 2 ) ] 0 = 2 [ 2 (1 1 ) ] = We show that 2 1 has the sign: [ + (1 ) ] [ ( ) + (1 ) ( 0 )] Indeed, by definition: 2 1 = { 2 ( 2 ) + (1 ) 2 [ 2 (1 2 ) ]} 1 1 { 1 ( 1 ) + (1 ) 1 ( 1 (1 1 ) )} which can also be written, given ( ) = : 1 2 [ + (1 ) ] 1 1 { 1 ( 2 ) + (1 ) 1 [ 2 (1 1 ) ]} because: 1 ( 1 ) = 1 ( 2 ) 1 [ 1 (1 1 ) ] = 1 [ 2 (1 1 ) ] If we note that 2 = 1 2 ( ) and 2 (1 1 ) = 1 2 ( 0 ), 2 1 can also be written as: 1 2 [ + (1 ) ] ( ) + (1 ) ( 0 ) But because 1 is increasing, for any ( ) has the same sign as 1 ( ) 1 ( ). Therefore, positing that = 1 1 2, the sign of 2 1 is also that of: [ + (1 ) ] [ ( ) + (1 ) ( 0 )] A2. Proof of the proposition 2 When = 0 let 0 the plainti s probability of winning such that the lotteries [ 1] and [ (1 ) ] are equivalent according to (1): 0 = ( ) [ (1 ) ] ( ) [ (1 ) ] 2 ]0 1[ (*) 19

20 For any value 0, = 0 because the defendant has no reason to make the plainti a greater o er but when 0 : ( ) = 0 ( ) + (1 0 ) [ (1 ) ] = 0 [ ( ) ( (1 ) )] + [ (1 ) ] therefore according to (1): [ ( ) ( (1 ) )] + [ (1 ) ] = ( ) + (1 ) [ (1 ) ] and because of the increase in : ( ) ( + ) 0 A3. Proofs of the lemma 1 and 2 Lemma 1. Let be a convex and increasing function. If for 0, we have: then there exists 2 ]0 1[ satisfying: and ( ) 0 ( ) ( ) ( 0 ) (**) 8 2 ]0 [ [ + (1 ) ] ( ) + (1 ) ( 0 ) 8 2 [ 1] [ + (1 ) ] ( ) + (1 ) ( 0 ) If for 0, we have: then: ( ) 0 ( ) ( ) ( 0 ) (***) 8 2 [0 1] : [ + (1 ) ] ( ) + (1 ) ( 0 ) Proof: ( ) = [ ( ) + ] [ ( ) + (1 ) ( 0 )] is a convex function in 2 [0 1] and : (0) = ( ) ( 0 ) 0 because is an increasing function (1) = 0 Figure 3: Illustration 20

21 Let us consider the slope of the tangent for = 1 If 0 (1) = ( ) 0 ( ) [ ( ) ( 0 )] 0 then there exists 2 ]0 1[ satisfying : 8 2 ]0 [ ( ) 0 = [ + (1 ) ] ( ) + (1 ) ( 0 ) and 8 2 [ 1] ( ) 0 = [ + (1 ) ] ( ) + (1 ) ( 0 ) and if 0 (1) 0 : Figure 4: Illustration 8 2 [0 1] : ( ) 0 = [ + (1 ) ] ( ) + (1 ) ( 0 ) Lemma 2. If we suppose that ( ) decreases with then: 1. Function 0 ( ) is convex and decreasing. 2. If 2 ( ) 1 ( ) then for any 0, the function ( ) = ( ) + is convex increasing. Proof: 1. 0 ( ) is decreasing since is concave and, moreover: 0 ( ) = ( ( ) 0 ( ) )0 = ( ) ( ) 0 ( ) 0 =) ( ) ( ) 2 ( ) 0 ( ) ( ) decreases with the fortune, and 2 ( ) 1 ( ), therefore 1 ( + ) 2 ( ) and: 1 ( + ) 0 1 ( + ) 2 ( ) 0 2 ( ) then: 1 ( + ) 0 2( ) 2 ( ) 0 1( + ) 0 which shows that 0 1 ( + ) 0 2 ( ) is increasing. Because 1 2 is also increasing, this is also the case for function 0 1 ( 1 2 ( ) + ) 0 2 ( 1 2 ( )) = 1 ( 1 2 ( ) + ) 0 Finally, for any 0, ( ) = ( ) + is increasing and convex as its derivative is positive and increasing. 21

22 A4. Proof of the proposition 4 The defendant seeks the best possible amicable o er, i.e. that which minimizes his total expected cost: ( ) = min [ (1 1 )( + ) ; 2 ] According to equality (1), 1 and 2 are respective solutions of the equations: 1 = [ 1 ( 1 ) + (1 ) 1 ( 1 (1 1 ) )] 2 = [ 2 ( 2 ) + (1 ) 2 ( 2 (1 2 ) )] As above, if the defendant o ers 1, there is agreement with type-1 plainti s, who represent share 1 of the population, and a trial with type-2 plainti s. If he o ers 2, because 1 2, there is agreement irrespective of the type of plainti. We then search for the condition under which the two contracts are equivalent for the defendant: (1 1 )( + ) = 2 For a fixed value of, if 1 is small enough, then the weighted average (1 1 )( + ) tends to ( + ) which can be greater than 2 and there will be no solution to this equation. Let us consider the general case where indeed the equation has no solution. Because ª( ) = + 2( ) + 1 ( ) = + [( + ) ( )] + [( + ) ( )] = ( ) 1 is an increasing function, if we have : ( 0 ) = ( 0 ) ( + ) ( 0 ) we have also for 0 : 1 + 2( ) + 1 ( ) The equation (1 1 )( + ) = 2 has no solution in [ 0 1] and: (1 1 )( + ) 2 = [ 1 + ( ) ] 1 + ( + ) 2 [ 1 + ( ) ] ( + ) 2 = 0 22

23 So: (1 1 )( + ) 2 and the defendant chooses to o er = ( 0) If 1 ( + ) 0 now there is a solution of the equation (1 1 )( + ) = 2 in [ 0 1] because: implies : 1 = ( + ) 2( ) ( + ) 1 ( ) = ª( ) ª(1) = + 2( ) + 1 ( ) = + + = 1 1 ª( 0 ) = ( 0 ) ( + ) 0 1 and so: ª( ) 1 0 for = 1 and ª( ) 1 0 for = 0. According to the intermediate value theorem, knowing that ª is a continuous and increasing function, we can deduce that a solution = exists and is unique in [ 0 1]. If = the defendant is indi erent between = 1 and = 2 if then : = 2 and if then : = 1 A5. Proof of the proposition has the sign of: [ + (1 ) ] [ ( ) + (1 ) ( 0 )] meets condition (***), so according to lemma 1 and 2, the latter expression is positive: 1 2 A6. Proof of the proposition has the sign of: [ + (1 ) ] [ ( ) + (1 ) ( 0 )] meets condition (**), so according to lemma 1 and 2, there exists 2 [0 1[ such that: 8 2 [0 [ and 8 2 [ 1] A7. Proof of the proposition 7 If the defendant o ers 2, there is agreement with type-2 plainti s, who represent share 2 of the population, and a trial with type-1 plainti s. If he o ers 1, because 2 1, there is agreement irrespective of the type of plainti. The defendant seeks the best possible amicable o er, i.e. that which minimizes his total expected cost: ( ) = min [ (1 2 )( + ) ; 1 ] As above, we search for the condition under which the two contracts are equivalent for the defendant: 23

24 or (1 2 )( + ) = 1 + 2( ) + 1( ) ª( ) = 1 2 where ª( ) = is a nonmonotonic quasiconcave function on [ 1]. Let us note = arg max ª( ) There is no solution to the equation ª( ) = 1 2 when : 1 ª( ) () 2 ( + ) 1 ( ) 2 ( + ) 2 ( ) This inequality implies 1 2 ª( ), (1 2 )( + ) 1 and the defendant chooses to o er = 1. When 2, according to the intermediate value theorem, ( + ) 1( ) ( + ) 2( ) and the quasiconcavity of ª, we can deduce that the equation ª( ) = 1 2 has two solutions f 1 f 2 ( f 1 = f ( + ) 1( ) 2 when 2 = ( + ) 2( ) ). The first one f 1 when ª( ) increases on the subinterval [ ] the second one f 2 when ª( ) decreases on the subinterval [ 1], then: = 2 () (1 2 )( + ) 1 () ª( ) 1 2 () f 1 f 2 Due to the quasiconcavity of ª the higher is 1 2 = ª(f 1 ) = ª(f 1 ), the closer are f 1 and f 2. 24

25 Table 1: Conflict resolution in complete information when the plaintiff is risk neutral Φ Φ Φ < Φ < Φ Φ Φ Plaintiff s type σ σ σ σ σ σ Defendant s σ σ σ offer Conflict resolution judgment judgment judgment agreement agreement agreement Table 2: Conflict resolution in incomplete information when the plaintiff is risk neutral (σ σ ) p < Φ d + Φ F σ (Φ ) p Φ d + Φ F Φ d + Φ F σ (Φ ) Φ d + Φ F Plaintiff s σ σ σ σ type Plaintiff s - - Φ < Φ Φ = Φ Φ > Φ Φ < Φ Φ = Φ Φ > Φ probability of winning Defendant s σ σ σ σ σ σ σ σ σ σ offer Conflict resolution a a a a a j a a a = agreement ; j = judgment

26 Table 3: Conflict resolution in incomplete information,in the specific case where σ > σ (i.e. Φ [Φ, 1]) p < Φ d + Φ F σ (Φ ) p Φ d + Φ F σ (Φ ) Φ d + Φ F σ (Φ ) Φ d + Φ F σ (Φ ) Plaintiff s type σ σ σ σ Plaintiff s - - Φ, Φ Φ, Φ, Φ, Φ Φ, Φ Φ, Φ, Φ, Φ probability Φ, 1 Φ, 1 of winningφ Defendant s σ σ σ σ σ σ σ σ σ σ offer Conflict resolution a a j a a a a a a = agreement ; j = judgment

27 y y = g[φ(v) + (1 Φ)w] 0 y = Φg(v) + (1 Φ)g(w ) 1 Φ Φ [0,1], σ σ Figure 1 : Illustration of the proposition 5

28 y y = Φg(v) + (1 Φ)g(w ) y = g[φ(v) + (1 Φ)w] 0 Φ 1 Φ Φ [0, Φ [, σ < σ and Φ [Φ, 1], σ σ Figure 2 : Illustration of the proposition 6

29 G(0) G(Φ) 0 Φ 1 Figure 3: Illustration Lemma 1 (first part) G(0) G(Φ) 0 1 Figure 4: Illustration Lemma 1 (second part)