George Mason University SCHOOL of LAW

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1 Geore Mason University SCHOOL of LAW Te Filterin Effet of Sarin Rules Giuseppe Dari Mattiai Gerrit De Geest 04-4 LAW AND ECONOMICS WORKING PAPER SERIES Tis paper an be downloaded witout are from te Soial Siene Resear Network Eletroni Paper Colletion: ttp://ssrn.om/abstrat_id= 5568

2 THE FILTERING EFFECT OF SHARING RULES Giuseppe DARI MATTIACCI a, b, Gerrit DE GEEST, * a Universiteit van Amsterdam, Roetersstraat, 08 WB Amsterdam, Te Neterlands b Geore Mason University Sool of Law, 330 Nort Fairfax Drive, Arlinton, Virinia 0, USA Utret Sool of Eonomis, Vredenbur 38, 35 BG Utret, Te Neterlands ABSTRACT Sarin rules ave a filterin effet on violations: tey prevent te most armful violations and let te least armful ones our. We sow under wat onditions te filterin effet improves soial welfare and arue tat tis may explain wy, in most areas of te law, sarin rules are, in eneral, preferred to rules tat entirely burden one party. Our analysis applies to omparative neliene, ommunal liability, te alloation of polie investiation efforts, ontrat remedies for nonverifiable breaes su as tose tat may our in marriae and employment ontrats, and to te distribution of sares in partnersips. JEL lassifiation: K3 Keywords: omparative neliene, law enforement, divore, employment ontrats, teory of te firm. * Correspondin autor: Pone , Fax s: darimat@mu.edu Giuseppe Dari-Mattiai,.deeest@eon.uu.nl Gerrit De Geest. We would like to tank Oren Bar-Gill, Winand Emons, Carlo Ferrino, Jana MLean, Hans-Bernd Säfer, Ben van Veltoven, te partiipants in te 7 t annual onferene of te European Assoiation of Law and Eonomis, Gent 000, and in te annual meetin of te Italian Eonomists Soiety, Rome 00, for elpful omments on a previous draft of tis artile, irulated as Comparative Neliene as a Filter Meanism Wen Courts Cannot Observe Preaution Costs, written wen bot of us were at Utret University. We are also rateful to Bruno Lovat and an anonymous referee wo enerously provided us wit key indiations on ow to improve te ore of te analysis. Devlin Cooper and Andrea Naylor provided valuable editorial assistane.

3 G. DARI-MATTIACCI & G. DE GEEST THE FILTERING EFFECT OF SHARING RULES. Introdution Sarin rules are endorsed in law but distrusted in aademia. One of te most troublin perplexities but not te sole arises from omparative neliene in tort. Altou tis is te most ommon liability rule aross leal systems, te puzzle of its wide use is still to be resolved, and most previous explanations ave been questioned in a reent artile. Oter rules tat allow for impliit loss sarin, su as no-fault divore systems and te employment-at-will dotrine, are also fierely debated in te literature. In tis artile, we sow tat sarin rules ave a filterin property tat may advane soial welfare beause tey stop te most armful violations wile lettin te least armful our. To illustrate, suppose tat an aident ould ave been avoided if eiter te vitim or te injurer ad taken preaution. Ex post, te ourt annot determine wi party ould ave taken preaution and only knows tat te probability tat injurers are avoiders is 60%, wile it is 40% for vitims. Assume furter tat te arm is $00, and te avoider s preaution may ost eiter $30 or $70 wit equal probabilities. An all-or-notin rule wi fully tarets te injurer su as strit liability, maximizes te ex ante probability of avoidin te aident, and in fat prevents 60% of te aidents all tose in wi te injurer is te avoider. If te ourt instead sares te loss amon te parties 60% to te injurer and 40% to te vitim, ten te avoider will prevent te aident wen is preaution osts less tan is sare in te arm $30, and will let te aident our wen preaution osts more $70. In tis ase, tere are more aidents. In fat, te injurer will prevent te aident in only alf of te ases in wi e is te avoider 60%/ and te vitim will do likewise 40%/. Tus, only 50% instead of 60% of te aidents will be prevented. Neverteless, te net soial ost is lower tan before. In fat, if preaution osts are i $70, ten te net soial ost of lettin an aident our is low $00 $70=$30. On te ontrary, if preaution osts are low $30, te net soial ost is i $00 $30=$70. Under an all-or-notin rule, aidents take plae witout any seletion and teir expeted net soial ost is 60% times te averae net ost of an aident $30+$70/, tat is, te expeted net soial ost is equal to $30. Quite te reverse, te aidents tat take plae under a sarin rule are only tose tat ost $70 See Bar-Gill and Ben-Saar 003. Tat no-fault divore systems and employment-at-will ontrats ave an impliit loss-sarin funtion is explained in setion 5. Dnes 000 review of te literature on marriae ontrats is ily ritial of no-fault divore systems. Te oriinal justifiation of te employment-at-will dotrine by Epstein 984 as been ritiized in several studies. See Buetemann and Walwei 999.

4 G. DARI-MATTIACCI & G. DE GEEST THE FILTERING EFFECT OF SHARING RULES 3 to prevent and ene yield a net soial ost of only $30. Sine only 50% of te aidents our, sarin rules yield an expeted net soial ost equal to only $5. Tus, sarin yields a lower soial ost tan te all-or-notin approa. Tis example illustrates te main two points tat we will elaborate upon in te next setions. First, te filterin effet of sarin rules enerally improves soial welfare, beause fousin te inentive on tose violations tat yield a ier soial ost ultimately redues te ost borne by soiety. Seond, tis redution in te soial ost may be aompanied by an inrease in te probability of violations. In fat, as it is obvious, if te oal of law enforement is to redue te number of violations, ten inentives sould be foused entirely on te party tat is most likely to violate te rule. As te example sows, owever, tis poliy may redue soial welfare instead of enanin it. Te optimal sarin will be sown to depend on tree variables: te type of aident ontext, te information available to te jude, and te distribution of te parties osts of preaution. In unilateral-are ases tat is, wen only one party is able to avoid te aident at reasonable osts we sow tat tere always exists some sarin tat dominates all-or-notin rules for any distribution of te preaution osts. We also sow tat, if te distribution is symmetri, te optimal sare of a party is equal to tat party s probability of bein te avoider. Tis rule is very easy to apply sine it requires no omplex alulation and is valid for a broad lass of distributions, wi inlude, amon oters, te plausibly very ommon ases of uniform distributions and random distributions. If te jude as no information on te distribution funtion, ten it is plausible to attribute te same probability to ea ost, tat is, to assume tat te parties preaution osts are distributed aordin to a uniform distribution 3 and ene tat, as we ave just notied, te optimal sarin for ea party is equal to te probability of bein te avoider. If tis probability is also unknown, ten an equal sarin is evidently desirable. In alternative-are ases tat is, wen bot parties an avoid te aident at reasonable osts but, aain, one party s preaution is suffiient to prevent it sarin is affeted by a free-ridin problem, as te parties tend to externalize te ost of preaution on ea oter. We sow under wat onditions tis problem disappears and sarin is still superior to all-or-notin rules. Also in tis ase, we devote partiular attention to te uniform-distribution ase and in eneral to symmetri distributions and sow tat equal sarin as some advantaes bot in terms of soial welfare and its easiness to apply. 3 On tis point see Savae 954 and Mas-Colell, Winston, and Green 995, pp. 05 ff..

5 G. DARI-MATTIACCI & G. DE GEEST THE FILTERING EFFECT OF SHARING RULES 4 In a nutsell, we arue tat if ourts know te distribution of te parties preaution osts, an optimal sarin an be found tat is better for soial welfare tan all-or-notin rules. If ourts do not know te distribution, ten sarin aordin to te probability of uilt is optimal. If tis bit of information is also missin, sarin equally is optimal. Tis artile is strutured as follows. Setion provides an overview of te intelletual taste and distaste for sarin in te eonomi literature. Setion 3 develops te first part of our formal analysis, presentin te model of unilateral are. Setion 4 ontains te model of alternative are. Setion 5 disusses several appliations of our analysis and onludes.. Te distaste for sarin Wile sarin rules are larely used in different areas of law, te eonomi literature as mainly 4 foused on te relatively reent rise of omparative neliene in tort. 5 Te effiieny reasons for su a rise are, owever, intensely debated. Te opinion of early writers about te effiieny of omparative neliene was larely neative. 6 Later solarsip proved a now enerally aepted effiieny-equivalene teorem, wi states tat all neliene rules provide, in priniple, equivalent inentives to take optimal are. 7 Comparative neliene, owever, is believed to enerate ier administrative osts tan its all-or-notin alternatives 8 and, tus, its pervasive diffusion is puzzlin. Te effiieny equivalene teorem is based on a number of assumptions, inludin tat ourts operate under perfet information and tat due are is set at te optimal level. Terefore, many autors ave tried to develop eonomi aruments for omparative neliene by relaxin some of tese assumptions. Wen tere is evidentiary unertainty, 9 parties mit take an ineffiient level of 4 For a disussion of te literature on impliit sarin rules in oter areas of tort law, law enforement, and in marriae law and employment reulation we refer to setion 5. 5 See Curran 99, Restatement Tird of Torts: Produts Liability se , and Restatement Tird of Torts: Apportionment of Liability 999 for a perspetive on Amerian law. 6 See Brown 973, Green, 976, p. 558, and Diamond 974a, pp Tose studies, owever, did not onsider tat te sarin of te loss under omparative neliene is onneted wit te implementation of a standard of are, and tus, te loss is sared only if bot parties are found nelient. On te first formal approaes to tort liability see also Diamond 974b and Diamond and Mirrlees Aordin to te effiieny-equivalene teorem, all neliene rules simple neliene, ontributory neliene and omparative neliene lead to optimal inentives if te due level of preaution is orretly defined. Landes and Posner 980 proved tis result. See also Haddok and Curran 985. For a survey see Säfer and Sönenberer 000, pp Low and Smit 995 ontend tat omparative neliene ives more inentives to sue and to ire attorneys tan allor-notin rules. Wite s 989 empirial study suests tat omparative neliene ives parties weaker inentives to take preaution tan ontributory neliene. 9 Evidentiary unertainty ours wen te ourts make random errors in omparin te due level of preaution to te level of preaution atually taken by te parties or wen judes err wit respet to te preaution osts atually borne by te parties. See Cooter and Ulen 986 for te former situation and Haddok and Curran 985 for te latter.

6 G. DARI-MATTIACCI & G. DE GEEST THE FILTERING EFFECT OF SHARING RULES 5 preaution in order to minimize teir expeted liability. It as been arued tat omparative neliene, by distributin te loss between te parties, redues te effet of errors and results in less distortion. 0 Comparative neliene as also been defended wen te due level of preaution is uniform for all parties but te individual osts of preaution differ and wen parties make preaution deisions sequentially. In tese ases, omparative neliene as been said to ave a sreenin effet on te parties. A reent artile asts serious doubts on te validity of any previous ase for omparative neliene. In partiular, it is sown tat distributin te loss between te parties does not always improve soial welfare under evidentiary unertainty and tat simple and ontributory neliene may also ave sreenin properties. 3 Consequently, te ase for or aainst omparative neliene is one aain reopened. Our analysis belons to te set of ontributions tat attempt to provide a justifiation for omparative neliene on te basis of informational sortfalls of te judiial system, but it is different from previous ontributions in several respets. Previous studies deal wit situations in wi bot parties need to take preaution in order for te aident to be avoided. Instead, we fous on situations in wi one party s preaution is suffiient to prevent te aident. 4 Te rule we study is not, terefore, te traditional form of omparative neliene under wi te loss is sared between nelient parties. In fat, sine only one party needs to take preaution, it an never our tat bot are nelient. Instead, sarin takes plae in our framework wen it is not possible to 0 Edlin 994 arued instead tat in tese situations omparative and ontributory neliene only differ wit respet to te standard of neliene tat is to be implemented for an effiient outome to result. Savell 987 sowed tat omparative neliene ould be superior in non-equilibrium situations, wen for some reasons vitims or injurers are found nelient. See also Orr 99 and Cun s 993 related omment. See Rubinfeld 987. See also te analysis on aidents wen parties are not idential by Emons 990a and 990b and Emons and Sobel 99. In tis ase, one party an observe te oter party s level of preaution before oosin is. See Rea 987 and Grady Bar-Gill and Ben-Saar 003 ave reexamined te literature on te topi, tou tey did not tou upon all studies mentioned in tis setion. Tey first analyze ontributions based on evidentiary unertainty. Te ommon tenet of tese ontributions is tat all-or-notin rules severely distort one party s preaution deision, wile omparative neliene reates two smaller distortions for bot parties. Tese aruments are sown to be rounded on te wron belief tat two small distortions ave less onsequenes on soial welfare tan one reater distortion. Te autors disprove tis ontention by sowin tat te omparative neliene rule does not enerally indue te most symmetri distortions and tat, in any ase, two small distortions are not neessarily preferable to a sinle reater one in terms of resultin soial welfare. Furter, tey examine aruments based on te ontention tat omparative neliene works as a sreenin meanism wen parties different osts of are are undistinuisable by te ourts. In tis ase, te ounterarument is tat all liability rules ave sreenin properties, wi may or may not enane soial welfare. Tus, sreenin does not provide a eneral ase for omparative neliene 4 Bar-Gill and Ben-Saar 003, p. 438 fous teir analysis on te standard model of bilateral preaution situations, in wi parties preautions are at least to some extent omplementary inputs to aident prevention. In our framework, we fous instead on ases in wi parties preautions are perfet substitutes. Te ase of joint are in wi tey are perfet omplements would be uninterestin as it is obvious tat under all-or-notin rules all aidents would our, wile sarin rules would at least prevent some of tem.

7 G. DARI-MATTIACCI & G. DE GEEST THE FILTERING EFFECT OF SHARING RULES 6 sinle out te only nelient party. 5 For te similar reasons, our analysis also differs from sreenin aruments. In addition, in our framework te problem is tat inentives are not always direted to te rit party te avoider. Sarin te loss maximizes te inentive effet of liability rules by assurin tat te avoider is tareted wen te violation as more armful onsequenes. On te ontrary, sreenin refers to te different problem tat arises wen it is soially desirable for some individuals to obey te neliene standard, and for oters to violate. Wile sreenin may also our under all-or-notin rules, we sow tat te filterin effet only arises under sarin rules. 3. Te model of unilateral are We onsider a simple model wit two risk-neutral, utility-maximizin parties wo are straners to ea oter. Only one of tem may take preaution at a reasonable ost a ost tat is less tan or equal to te arm in order to prevent 6 te aident: 7 we all tis party te avoider. 8 Te oter party is passive. 9 Te avoider may be of various types, dependin on is preaution ost. Ex post, it is not possible to determine wi party was te avoider, but te probability tat eiter party is te avoider is known. A leal rule determines ex ante te sares of ea party in te arm. Let: = te arm, > 0; = te avoider type tat is, te avoider s ost of preaution, 0 ; is distributed 5 Altou te filterin effet may be seen as a form of sreenin meanism, it does not operate trou te settin of a neliene standard but purely trou te sarin of te aident loss. For tis reason, filterin is only proper of sarin rules and is absent in all-or-notin rules. Tus, te laim tat all neliene-based liability rules ave sreenin properties as tey all apply a standard of due are annot be extended to filterin. Filterin ours in situations tat are lose to evidentiary unertainty. However, in all ases belonin to tis roup, te standard of are is erroneously applied wit some positive probability. In our framework, tere is no standard to apply and te jude is ompletely blind wit respet to identifyin te avoider. 6 Our model is based on a binary deision on weter or not to take preaution, and it is, in tis respet, similar to te standard models used for law enforement. In tort law and eonomis, instead, models usually but not always desribe preaution as a ontinuous variable. Tus, our model diretly applies to tose ases in wi preaution involves a binary deision and only approximates tose in wi tere are multiple levels of preaution to be taken into aount. Examples of te former types of situations are not drivin trou a red lit, yieldin for pedestrians, adoptin a new pollutionontrol tenoloy in a emial plant, installin a naviation devise on a boat, and so fort. 7 We are assumin tat, if preaution is taken, no aident ours and tat, onversely, if no preaution is taken an aident ours. Considerin aidents tat only our wit a ertain probability less tan if no preaution is taken or aidents tat our wit a ertain positive probability even if preaution is taken would not ane te substane of our aruments or te results of our model iven tat te oie of weter or not to take preaution is assumed to be binary and parties are assumed to be risk neutral. 8 Note tat te avoider is not neessarily te injurer. Te problem ourts fae in te ase under examination is indeed te identifiation of te avoider between te vitim and te injurer. Note also tat our model is not restrited to unilateral-risk ases, in wi only te vitim suffers arm. Quite te opposite, te model allows for bot parties to be affeted by te aident. We onsider te total loss arisin from te aident and ask wo sould bear it, irrespetive of were te loss oriinally fell. 9 In our model we do not onsider possible distortions in te levels of te parties ativities due to te fat tat te party wo is innoent is sometimes santioned.

8 G. DARI-MATTIACCI & G. DE GEEST THE FILTERING EFFECT OF SHARING RULES 7 aordin to a density funtion f and a umulative distribution F; wit f ontinuously differentiable over 0, positive over te same interval and zero elsewere, F0 = 0 and F = ; π = te probability tat party is te avoider, 0 < π < ; = party s sare in te arm, 0. Let a sarin rule be a rule tat implements 0 < <, and let an all-or-notin rule be a rule tat implements = 0 or =. Te avoider knows te leal rule before e deides weter to take preaution. Te net soial ost funtion S is defined as te differene between te arm and te avoider s ost of preaution if no preaution is taken and an aident ours. S is zero oterwise. 0 Avoiders will take preaution if te ost of preaution is less tan or equal to teir sare in te arm. Terefore, te ex ante net soial ost resultin from a rule tat sares te arm ex post aordin to an be alulated as te weied sum of te net osts of ea parties non-prevented aidents: S = π f d + π f Likewise, te expeted probability of an aident atually ourrin is: P = π [ F ] + π [ F ] Fiure depits te avoider type on te vertial axis and its umulative distribution on te orizontal axis. Te soial ost is te sum of te two saded areas weied by te probabilities π and -π. Te probability of an aident is instead te sum of te projetions of tese areas on te orizontal axis weied by te same probabilities. d Observation. Sarin rules always filter out te relatively most armful aidents; all-or-notin rules do not ave su a filterin effet. Te relative armfulness of aidents is iven by teir net soial ost. In our model, te arm is onstant; ene, aidents wit a lower ost of preaution are te relatively most armful ones. If 0 Note tat tis formulation is equivalent to sayin tat te soial ost is equal to eiter te arm or te preaution ost, iven tat if preaution is taken tere is no arm. Te net soial ost is simply te differene between te soial ost so alulated and te ideal ase in wi preaution is taken in ea individual ase. We adopt te latter perspetive for onveniene. Canin tis assumption would not, owever, impair te results we derive.

9 G. DARI-MATTIACCI & G. DE GEEST THE FILTERING EFFECT OF SHARING RULES 8 = 0, party is always inentivized, wile party will never take preaution. Hene te soial ost under su an all-or-notin rule is: 3 S 0 = π f 0 d Te arument we are makin olds symmetrially for =. By lookin at te lower and te upper boundary of te interal in Exp. 3 wi is alulated on te interval from 0 to it is easy to see tat in tis ase a fration π of te aidents our, irrespetive of teir armfulness. In ontrast, under te sarin rule in Exp., te most armful aidents are prevented preisely tose wose ost of preaution is less tan te party s sare in te arm sine te interals are alulated on te interval from and -, respetively, to. Te same observation an be made by lookin at fiure and notiin tat te most armful aidents are tose to te left of te fiure, for wi te ost of preaution is relatively low if ompared to te arm. [fiure ] Te filterin effet due to sarin enanes soial welfare, as it redues te ost borne in te areate by soiety. Terefore, as te next proposition sows, sarin rules are superior to all-ornotin rules. Proposition. In unilateral-are ases, tere always exists some sarin tat yields a lower soial loss tan all-or-notin rules. Proof. Te first derivative of Exp. yields: 4 S' = πf + π f Te interpretation of Exp. 4 is straitforward: an inrease in improves te inentives for party to take preaution, and, tus, te soial ost dereases te first term is neative. At te same time, te inentives of party are redued, and, ene, te soial ost rises te seond term is positive. Tus, anes in enerate ontrastin effets on te soial ost. Evaluatin Exp. 4 at =0 and at =, we ave, respetively:

10 G. DARI-MATTIACCI & G. DE GEEST THE FILTERING EFFECT OF SHARING RULES 9 5 S' 0 = πf 0 < 0 S' = π f 0 > 0 From Exp. 5 it is evident tat te soial ost dereases if deviates from 0 or towards internal values. Tus, sarin rules must yield a lower soial ost tan all-or-notin rules. To illustrate te loi of tis result, if we move from = 0 to some internal value of, party will take more preaution, wile party will redue is. Tus, party will prevent some aidents tat e would not ave prevented at = 0, and, to te ontrary, party will let some aidents our tat e would ave prevented at = 0. Te aidents tat party now prevents are more armful tan tose tat party lets our, as teir ost of preaution is very small say $ if ompared to te arm $00, and namely below we are onsiderin a lose to zero. Correspondinly, te ost of avoidin te aidents tat party lets our is relatively very i say $99 and namely above -. Terefore, tere is a soial ain in tradin off te former more armful for te latter less armful aidents. Te same symmetrially applies to a redution of from = towards internal values. Tis result proves tat tere is always te possibility to improve soial welfare by switin from an all-or-notin rule to some sarin of te aident loss. In priniple, tis is a rewardin oie beause te aidents we let our by switin rules as a lower net soial ost tan tose we prevent, even if te probabilities for te two types of aidents are different; by finely tunin te sare, it is always possible to apture tis benefit. Te value of te optimal sarin is easy to determine for symmetri distributions tat is, in situations in wi te ost of preaution is distributed symmetrially around its mean. In tis ase, te soial ost is minimized by a sarin equal to te probability of bein te avoider. For example, if te probabilities of bein te avoider are /3 for te injurer and /3 for te vitim, te optimal sarin requires tat te injurer pays /3 of te arm and te vitim bears te remainin /3. Two ompellin examples of symmetri distributions are te uniform distribution, wi we will study in te next setion, and te bell-saped normal distribution. Tese ases are likely to play an important role in reality. Te uniform distribution, as we will arue, may be used to desribe situations is wi tere is a lak of information onernin te atual sape of te distribution funtion. Te normal distribution desribes ases in wi te ost of preaution is randomly distributed. Proposition. In unilateral-are ases, if te distribution of avoider types is symmetri, te optimal sarin rule is te one tat sares te arm between te parties aordin to te probability of

11 G. DARI-MATTIACCI & G. DE GEEST THE FILTERING EFFECT OF SHARING RULES 0 ea party bein te avoider, = π. Proof. Sine te distribution is symmetri, we ave f = f[ ]. Substitutin te latter equivalene into Exp. 4 and rearranin, we an alulate te followin first order ondition for an optimal sarin *: 6 S' = f π = 0 wi is learly satisfied at * = π. Te seond order ondition yields: 7 S' ' = f ' π + f > 0 3 Te first addendum is zero at te optimum and te oter terms are all positive. Terefore, Exp. 7 is satisfied at te stationary point, and, ene, * is a loal minimum. Sine * is te unique solution to 6 and te funtions we employ are differentiable, we an infer tat * is also a lobal minimum. 3. Te uniform-distribution ase Wen te avoider s osts of preautions are uniformly distributed, ea type ours wit te same probability. Te uniform distribution is a speial ase of te more eneral set of symmetri distributions tat we ave studied in te previous setion. Terefore, te optimal sarin of a party is also equal to tat party s probability of bein te avoider in tis ase. Wen te avoider types are distributed uniformly, we ave f = / and F = /. By makin some simple substitutions into Exp., we an write te soial ost as follows: 8 S = π + π or S = π + π In tis ase, te preaution ost funtion depited in fiure takes te form of a linear funtion wit slope equal to ; tus, te soial ost may also be rapially seen as te weied sum of te areas of two trianles of basis and and eit and, respetively as it is lear from te first Exp. in 8. It is redundant to verify tat, as a result of proposition, if te distribution of te avoider types is uniform, te soially optimal rule is te one tat sares te arm between te parties aordin to te probability of ea party bein te avoider, * = π. It an be easily sown tat tis value of uniquely minimizes Exp. 8, wi is stritly onvex. Now, let us derive some additional orollaries.

12 G. DARI-MATTIACCI & G. DE GEEST THE FILTERING EFFECT OF SHARING RULES Corollary.a. In unilateral-are ases, if te distribution of te avoider types is uniform, te differene between te soial loss due te optimal sarin and te soial loss due te optimal all-or-notin rule inreases wen π approaes /; at π = / te soial loss due to te optimal sarin rule is alf te soial loss due to te optimal all-or-notin rule. Proof. Te optimal all-or-notin rule makes te party wit te iest probability bear te arm; ene, te probability tat an aident ours is iven by min{π, π}. Let us assume tat π π, i.e. let us assume π / if te opposite ase obtains, we simply re-label te probabilities. Te soial loss under te optimal all-or-notin rule is ene equal to S0= π/. By substitutin = π into Exp. 8 and rearranin we obtain tat te soial loss due to te optimal sarin rule is equal to Sπ = π/ π /. Te differene between te S0 and Sπ is π /, wi inreases in π at an inreasin rate equal to π and is maximal wen π = /. At tis level of π, S0 = /4, wile Sπ = /8. Corollary.b. In unilateral-are ases, any sarin rule 0 < < yields a lower soial ost tan any all-or-notin rule if te distribution of te avoider types is uniform and if parties ave equal probabilities of bein te avoider π = /. Proof. Sine te soial ost is a onvex funtion of, it is at its maximum wen equals eiter or 0. By substitutin π = / into Exp. 8 we an alulate te soial loss for te two all-or-notin rules, S0 = S = /4. Sine = 0 and = yield te same maximum soial ost, any oter sarin is preferable. It is evident tat te same does not old if π is not equal to /. Corollary.. In unilateral-are ases, if te distribution of te avoider types is uniform, te optimal sarin rule * = π yields if π = / te same number of aidents as or if π / a ier number tan te optimal all-or-notin rule. Proof. By substitutin F = / = and = π into Exp., we obtain tat te probability of an aident under te optimal sarin rule is Pπ = π π. Assumin as above tat π / and re-labelin, oterwise, under te optimal all-or-notin rule te probability tat an aident ours is π. It is easy to sow tat Pπ = π π = / if π = / and tat Pπ = π π > / if π < /. It is evident tat te soial ost is lower if te party tat takes preaution is te one wit te ier probability of bein te avoider, iven tat te type distribution is te same for bot parties. Consider Exp. wen = or =0.

13 G. DARI-MATTIACCI & G. DE GEEST THE FILTERING EFFECT OF SHARING RULES [fiure ] [fiure 3] Our results suest tat, for uniform distributions, te santion sould be apportioned amon potential violators aordin to te probability of uilt. Te soial-loss savin due to te filterin effet may be very relevant, and it depends on te probability of te parties bein te avoider. Wen parties are equally likely to be te avoider te probability is / for ea of tem, te filterin effet saves alf of te soial loss tat would our under an all-or-notin rule. Te manitude of tis advantae tends to derease altou it remains positive wen te parties probabilities divere from ea oter, as sarin tends to ause inreasinly more aidents ompared to all-ornotin rules. Anoter important arateristi of te ase in wi te probability of bein te avoider is / for ea party is tat any sarin is preferable to an all-or-notin rule altou te optimal sarin in tis ase is / as we ave remarked. Tis means tat even if te sarin rule annot be applied wit perfet auray, errors in te settin or te alulation of te sares do not undermine te superiority of su a rule over all-or-notin rules. Conernin te number of aidents or, more in eneral, of violations, sarin rules tend to perform as well as all-or-notin rules only if te probability of bein te avoider is / for ea party. In all oter ases, sarin yields more violations. It does not seem obvious tat an inrease in te number of aidents may improve soial welfare, but it is wort remarkin tat even tou more aidents our, tey are te less armful ones sine te most armful ones are filtered out. Tis point may be exploited to empasize a potential onflit between te oal of maximizin te probability of appreendin te uilty wi suests tat te party wit te ier probability sould be santioned and te oal of maximizin soial welfare wi requires tat te santion sould be sared Te model of alternative are In te previous setion, we analyzed a model in wi only one party an avoid te aident at a reasonable ost a ost less tan te arm. However, in reality, it is often te ase tat bot parties 3 A related point onerns transation osts. An inrease in te number of aidents may inrease te ost of applyin a sarin rule over te ost of implementin an all-or-notin rule. Comparative neliene in tort as lon been said to be more expensive to administer tan its all-or-notin alternative, su as simple or ontributory neliene. Our analysis suests tat omparative neliene may yield ier administrative osts due to more ases to adjudiate rater tan or in addition to a ier ost per ase.

14 G. DARI-MATTIACCI & G. DE GEEST THE FILTERING EFFECT OF SHARING RULES 3 an take preaution at a reasonable tou not te same ost and tat one party s preaution is suffiient to prevent te aident. In te literature, tis situation is referred to as alternative are. Let us onsider te ase in wi bot parties an take preaution, wi an ave a ost tat varies from 0 to. Te value of amounts to a limit to te rane of te parties types, and it an be equal to or ier tan te arm. Te point we will make in tis setion is tat if te parties osts of preautions rane suffiiently beyond te arm, ten te filterin effet also improves soial welfare in alternative-are ases. Te reason is tat sarin in alternative-are ases filters out bot te most armful aidents and te iest preaution osts tus, aidents are prevented at a lower averae ost, but it may reate onditions under wi no party takes preaution. It is also vulnerable to reiproal free ridin by te parties, wo attempt to externalize preaution osts on ea oter. We will sow tat te advantaes of sarin aument and its sortomins weaken wen te rane of te parties osts of preaution expands beyond te arm tat is, wen some avoiders ave a ost of preaution sinifiantly ier tan te arm. In addition to our previous notation, let: 4 i = upper limit for te rane of te parties types, wit i and i =, ; = ea party s type, 0 ; is distributed aordin to a density funtion f i and a umulative distribution F i, wit F i 0=0, F i i =0 note tat F i =0 only if = i, and i =,. It is onvenient to alulate te net soial ost as te sum of te osts of preautions taken by te parties and te aident osts minus te minimum between te parties ost and te arm te ideal situation; 5 sine te latter does not depend on te sare, it an be treated as a onstant K. 6 = 9 S, f d + f d + [ F ][ F ] K 0 0 Te values and are te maximum osts tat parties take iven te leal rule. Te probability tat an aident ours in alternative-are ases is iven by te followin expression. 4 Clearly, tere is no need to define te probability of bein te avoider sine bot parties are in fat avoiders. Impliitly, te parties probabilities of bein te avoider depend upon te distributions of teir types. 5 In te ideal ase if at least one party s ost is below te arm, preaution is only taken by te least-ost avoider, and no arm ours. Oterwise, preaution is not taken and te arm materializes. 6 Te first interal in Exp. 9 may be interpreted as te averae ost of preaution for party times te probability tat party takes preaution. Te averae ost of preaution is to be alulated as te interal divided by F, beause preaution only ranes from 0 to. Te probability tat party takes preaution is also equal to F. Terefore, we an simplify and only onsider te interal. Te same applies to party. Te tird term is te expeted ost of te aidents tat atually our wen no party takes preaution.

15 G. DARI-MATTIACCI & G. DE GEEST THE FILTERING EFFECT OF SHARING RULES 4 0 P, [ F ][ F = ] Parties preaution deisions in alternative are not only depend upon teir sare in te arm but also upon teir expetations onernin ea oter s beavior. To illustrate, if te sare of party is, party will atually bear tis ost only if te oter party does not take preaution and te aident ours. If parties at simultaneously, as we assume, ten teir maximal preaution osts may be alulated as follows: = = [ F ] [ F ] Exp. sows te maximal osts tat parties are willin to bear to prevent te aident. Tis maximal ost is equal to te expeted ost of not takin preaution, wi, in turn, is equal to te probability tat te oter party will not take preaution oterwise tere is no arm, times te party s sare, times te arm. Tus, parties only take preaution if teir ost of preaution is less tan tis value. As in te unilateral-are ase, te filterin effet does, under a suffiient ondition, enane soial welfare. Details onernin te followin proofs are iven in te appendix. Proposition 3. In alternative-are ases, if i >, tere exists some sarin tat yields a lower soial loss tan all-or-notin rules. Proof. Calulatin te total derivative of Exp. 9 wit respet to and evaluatin it at = 0 and at =, we ave, respetively: S' 0 = f0 S' = f 0 [ F ] [ F ] < 0 > 0 Sine te soial ost dereases wen departs from 0 or toward internal values, tere exists some sarin wi yields a lower soial ost tan te all-or-notin rules. It is to be notied tat if i =, ten F i =, and, ene, bot expressions equal zero. In tis ase, te soial ost does not ane by switin rules. Te intuition beind tis result is as follows. Wen and rane beyond, tere will be ases in wi at least one party s ost of preaution is ier tan te arm. On te one and, te inentives to free ride diminis as a party may rely less often on te oter party s preaution. On te oter and, tose ases are in fat of unilateral-are type, in relation to wi te superiority of sarin as already been proven.

16 G. DARI-MATTIACCI & G. DE GEEST THE FILTERING EFFECT OF SHARING RULES 5 Te optimal sarin will depend on te speifi arateristis of te parties types distributions. Neverteless, it is possible to sow tat, under ertain onditions, equal sarin may be preferable to all-or-notin rules, altou tis is not neessarily te optimal sarin. Proposition 4. In alternative-are ases, if te parties types are distributed aordin to te same distribution funtion, tere exists some level of above wi equal sarin yields a lower soial ost tan all-or-notin rules. Proof. If f = f = f, and if = =, ten we an evaluate Exp. 9 at = 0, =, and = /, respetively, as follows: 3 S0 = S = f d + 0 ^ S = + f d 0 [ F ] [ F ^ ] wit ^ = [ F^]/ < /. Te relative performane of te different rules may be expressed as follows: ^ { } 4 S 0 S = f d f d + [ F ] [ F ^ ] 0 0 It is easy to sow tat te first term is surely larer tan te seond beause ^ < /. Te tird term is neative wen = beause F = 0 but approaes 0 as dereases below beause bot F and F^ move toward. Terefore, tere exists some suffiiently larer tan su tat Exp. 4 is positive, and, ene, te soial ost is lower wit = / tan under all-or-notin rules. Tis result may be interpreted in te lit of our previous results onernin symmetri distributions in te unilateral-are ase. In a way, te symmetry ere is iven by te fat tat te parties distributions are te same, and, as for symmetrial distributions, we are advoatin a sarin equal to te probability of bein te avoider. In fat, avin equal distributions also means avin equal parties probabilities of bein te avoider. Equal sarin, owever, only beomes viable wen te rane of te ost of preaution beomes suffiiently lare, for te same reasons iven above.

17 G. DARI-MATTIACCI & G. DE GEEST THE FILTERING EFFECT OF SHARING RULES 6 4. Te uniform-distribution ase In tis setion, we study in more detail te simpler ase in wi parties types are distributed aordin to te same uniform distribution funtion. Terefore, we ave f = f = f = /, F = F = F = /, and = =. Substitutin into Exp. 9 and Exp., and rearranin, we an write te soial ost as a funtion of te sarin, as follows: 5 S = 3 [ ] K As enuniated in Proposition 3, if >, te soial ost is minimized by some internal values of. Proposition 4, instead, implies tat equal sarin, even tou it may not be te optimal sarin, still yields a lower soial ost tan all-or-notin rules if is suffiiently larer tan. Moreover, sine te probability tat an aident ours is iven by te produt between te probabilities tat ea party does not take preaution, tese probabilities will be equal wen te loss is equally sared, and, ene, teir produt will be at te maximal level. If, on te ontrary, te loss is onentrated on one party, at least tat party will take preaution wenever is ost is less tan te arm, and, tus, tere will be fewer aidents. Te followin orollaries proved in te appendix refine tese laims. Corollary 3.a In alternative-are ases, if te parties types are distributed aordin to te same uniform distribution funtion, for < < 3/, te soially optimal sares are some values a < / and b > / or, indifferently, a > / and b < /, subjet to a + b = ; = / is instead a loal maximum. As moves toward 3/, te optimal sares move toward /. Corollary 3.b In alternative-are ases, if te parties types are distributed aordin to te same uniform distribution funtion, for > 3/, ten = / is te optimal sarin rule. Corollary 4.a In alternative-are ases, if te parties types are distributed aordin to te same uniform distribution funtion, for > +3 ½ /, ten = / yields a lower soial loss tan all-or-notin rules. Te orollaries sow tat sarin rules are superior to all-or-notin rules for any >. As beomes even larer tan, te optimal sarin rule anes. Initially, it is optimal to sare te loss unequally amon te parties. As te differene between and inreases, equal sarin, altou it

18 G. DARI-MATTIACCI & G. DE GEEST THE FILTERING EFFECT OF SHARING RULES 7 is not yet te optimal sarin, is still preferable to all-or-notin rules; fiure 4 depits tis ase in wi te soial ost funtion is w-saped. Wen is above a ertain tresold, equal sarin is te optimal sarin. In tis ase, te soial ost funtion assumes a u-saped onvex form. [fiure 4] Corollary 4.b In alternative-are ases, if te parties types are distributed aordin to te same uniform distribution funtion, ten equal sarin yields te iest probability of aidents. Proof. For uniform distributions, Exp. 0 may be rewritten as follows: 6 = P, Tis expression is maximized wen =, and it is at its minimal level wen one of tem is equal to. As we ave already remarked for te unilateral-are ase, tere mit be a potential onflit between reduin te probability of aidents and minimizin teir soial ost. Equal sarin tends to yield many more aidents tan all-or-notin rules even in tose ases in wi it minimizes te soial ost. 5. Conlusions and appliations Wat sould ourts deide wen tey do not know wi of te parties sould ave done more to prevent an aident or te failure of a ontratual relationsip? We ave arued tat sarin rules ave an important advantae over all-or-notin rules: tey filter out te losses tat were relatively less expensive to prevent. Sine te most notorious sarin rule is omparative neliene, te prose of our model refers to an aident ontext. Yet, te filterin effet of sarin is a more eneral priniple, wi applies to a vast array of seeminly different areas of te law ranin from tort and ontrat to law enforement. In a broad sense, tis artile disusses te eneral enforement problem of ow to alloate inentives wen te roup of potential violators and teir probabilities of uilt are known, but it is not possible to identify te atual violator ex post. In an aident ontext, te problem is to identify te avoider between te injurer and te vitim wen teir osts of preaution are unverifiable. In divore ases, tis means sinlin out wi of te partners ould ave saved te relationsip. In

19 G. DARI-MATTIACCI & G. DE GEEST THE FILTERING EFFECT OF SHARING RULES 8 employment ases, te problem may be one of findin wo is responsible for te lak of ooperation between te employer and is employee. In ommunal liability ases, tere may be an unidentifiable member of a tribe wo as aused arm to a neiborin tribe. In law enforement, te issue may be related to te alloation of investiation effort between two riminal roups. If our analysis applies to a ase, te next question is wat uidelines it ould provide for te ourts. It is instrutive to make a distintion between different typoloies of aidents or, in eneral, violations. Pure unilateral-are ases are tose ases in wi te arm ould ave been avoided by one party only. We ave disussed tis ase at lent and sown tat sarin always dominates all-or-notin rules. Pure alternative-are ases are tose ases in wi eiter party ould ave avoided te arm at reasonable even tou different osts. We ave sown tat te ase for sarin is seriously weakened. Mixed alternative-are ases are tose in wi parties osts of preaution rane beyond te arm and tus onsist of a portion of pure alternative-are ases and one of unilateral-are ases. We ave arued tat if te rane is suffiiently lare, sarin reains its advantae. We ave not disussed te obvious ase for sarin under joint are. In our framework, it turns out tat if bot parties were to take preaution, all-or-notin rules prevent no aident at all, wile sarin still yields some positive level of deterrene. Terefore, we suspet tat even beyond te ases in wi our main laims ave been proven sarin may, in reality, be more likely tan not to dominate all-or-notin rules wen ourts ave no information about te aident typoloy unless te aident is learly a pure alternative-are type. Our analysis also suests tat, in eneral, a 50/50 sarin may be reommendable. It is also wortwile remarkin tat if all-or-notin rules are applied in a way tat annot be predited ex ante, ten teir effets are equivalent to a stoasti sarin rule. In tis ase, te only differene between sarin and all-or-notin rules would be te risk involved in te probabilisti attribution of responsibility, wi is absent wen sarin rules are announed ex ante and orretly implemented. 5. Tort As we suested in te introdution, our analysis may furnis a justifiation for omparative neliene. Stritly speakin, omparative neliene sares te loss wen bot te injurer and te vitim are found nelient, wi annot be te ase in our unilateral-are model. Wat we are advoatin is, in fat, te sarin of te loss wen fault annot be proven. Loss sarin in tort as also been advoated under oter eadins. Comparative ausation, for example, refers to te sarin of te loss between non-nelient parties and mirrors te omparative

20 G. DARI-MATTIACCI & G. DE GEEST THE FILTERING EFFECT OF SHARING RULES 9 neliene rule. In tis ase, sarin aims at ontrollin te inentives to take optimal ativity level rater tan are. Wen an aident as ourred, notwitstandin te fat tat parties took optimal are, our analysis suests tat sarin te loss ould improve parties inentives wit respet to te level of ativity. 7 Also, rules of ommunal liability may amount to a sarin of te loss amon several potential tortfeasors. 8 In anient soieties, it was ommon tat if a member of a tribe did arm to anoter tribe, ompensation was to be sout from te tortfeasor s roup. In tese ases, te literature as foused on te reasons for implementin different forms of liability rater tan on teir sarin effets. 9 Current explanations in te literature fous on te need to ive ea individual inentives to monitor te oters. 30 We suest tat sarin may be supported by its filterin properties in tis ase as well. Tat ommunal liability disappeared in more modern soieties is also in line wit our framework. Over time, it mit ave beome eaper and, more often, tenially possible to find out wat individual was responsible. As te non-verifiability problem disappears, te filterin justifiation for sarin rules also disappears. Empirial studies ave onluded tat omparative neliene ives parties weaker inentives to take preaution tan ontributory neliene. 3 Tese results sould be interpreted wit aution. Our analysis sows tat sarin rules are indeed likely to even tou tey do not always inrease te number of aidents. Tis means tat parties will more often be areless. Te soial ains of omparative neliene onsist of te fat tat injurers, on averae, spend less in preaution. Tis means tat injurers wo exerise are ave, on averae, less are osts. In oter words, under omparative neliene, tere are more aidents, and less is spent on preaution, but wat is spent on preaution is relatively well-spent. Terefore, te urrent empirial literature does not allow us to make any statements on weter omparative neliene worsens te parties inentives. A ane in te number of aidents is an evident effet of tort reforms, wile a ane in teir soial 7 See Parisi and Fon fortomin for an analysis of loss sarin between non-nelient parties as a funtion of teir ausal ontribution to te aident. Tey do not tou upon te points we make. Also, Savell 985 makes a point in favor of sarin, wen ausation is unertain for bot te ourts and te parties. Te distintion between are and ativity level, altou already present in earlier studies, was first formalized by Savell Kornauser and Revesz 990 and 993 provide an analysis of loss-sarin amon several atual injurers but make points tat are different from ours. 9 Sykes 98 and Kornauser 98 provided te first breaktrous in te eonomi understandin of rules of viarious and ommunal liability and te standard view on teir desirability. 30 See Posner 980 and Parisi and Dari-Mattiai Wite 989 omparative neliene ives parties weaker inentives to take preaution tan ontributory neliene; Sloan, Reilly, and Senzler 995 used self-reported data on patterns of alool use amon individuals and found tat switin from ontributory to omparative neliene inreased bine drinkin. Flanian, Jonson, Winkler, and Feruson 989 found tat omparative neliene is assoiated wit ier automobile liability insurane premiums, but tey did not investiate weter tis inrease was due to more laims towards injurers or to more aidents. On te omparison between ontributory and omparative neliene see also Swartz 978.

21 G. DARI-MATTIACCI & G. DE GEEST THE FILTERING EFFECT OF SHARING RULES 0 ost may be a rater idden feature, and it is more umbersome to measure. Terefore, te tension between, on one and, reduin te number of aidents or te number of areless parties and, on te oter, reduin te soial ost may be of relevant politial impat beause poliies tat pursue te latter oal at te expense of te former may be favoured by uninformed voters. 5. Law enforement Our analysis also bears on issues of law enforement and partiularly on te question weter investiation effort sould be sared amon te population of potential violators or foused on a subroup. An example is useful for larifiation. Assume tat a robbery as been perpetrated and tat tere are only two riminal roups in te ity wit different probabilities of ostin te tief. If all investiation effort is direted toward one of te two roups, te probability of findin and provin te uilt of te riminal, if e is tere, is equal to ; if e is in te oter roup, te probability is obviously zero. If poliemen are sent to inquire into bot roups, te probability is less tan for bot and proportional to te number of poliemen. Sarin te poliemen between te two roups an be analoized to te sarin of te loss in an aident ontext. Te osen poliy produes ex ante inentives for te two roups, wi an be studied by applyin our model. A reent artile on tis topi fouses on te effets of varyin enforement resoures and onludes tat te optimal distribution of enforement effort depends on te availability of tese resoures. 3 In eneral, te more resoures are available, te more optimal law enforement effort tends to be uniformly distributed under te assumption tat te violators probabilities of uilt are te same. Instead we make a different point. Our fous is on differenes amon parties, bot wit respet to teir probabilities of uilt 33 and to teir varyin osts of preautions. 34 We study ow te 3 Lando and Savell 004 disuss tis under te label fousin law enforement effort. Our study interats wit teirs in several respets and deviates in oters. First, fousin is not te opposite of sarin. Fousin denotes any poliy tat is different from uniformly sarin polie effort amonst te population of violators tat is sarin 50/50. In our framework, sarin denotes any poliy tat is different from fousin te full enforement effort on a subroup of te violators an all-or-notin rule. Tere exists, terefore, a rey zone between uniform sarin 50/50 and full fousin 0/00 or 00/0 in wi law enforement effort is sared between two subroups of violators, but not uniformly. Lando and Savell also all tis ase fousin, stressin te fat tat law effort is not distributed uniformly. We all it sarin, empasizin te fat tat law enforement effort is not onentrated on one subroup only. Seond, wit tese onsiderations in mind, it is easy to see tat our findins are ompatible wit tose by Lando and Savell, as bot of us sow in fat te benefits of distributin law enforement effort between two subroups of potential violators wit varyin proportions. For bot of us, te santion is iven. Tey vary enforement levels by means of redistributin enforement effort and, tus, trou variations in te probability of appreension. We rea te same result by means of sarin a fixed probability times santion mix between te parties. It is easy to see tat, beyond terminoloy and matematis, te two approaes onvere. In our settin owever, te enforement resoures are impliitly iven as fixed, as te probability of appreension is iven and te santion is set equal to te arm. 33 Lando and Savell 004 dismiss tis issue as an obvious arument for fousin. However, te issue is not trivial, and we sow tat omplete fousin all-or-notin rules is not in eneral te optimal response to different probabilities of uilt. 34 It is immaterial tat we determine te parties type aordin to teir osts of preautions, wile Lando and Savell