Bargaining with Heterogeneous Beliefs: A Structural Analysis of Florida Medical Malpractice Lawsuits. Abstract


 Georgiana Ross
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1 Bargaining wih Heerogeneous Beliefs: A Srucural Analysis of Florida Medical Malpracice Lawsuis Anonio Merlo and Xun Tang 1 Deparmen of Economics Rice Universiy May 16, 2015 Absrac We propose a srucural bargaining model where players hold heerogeneous beliefs abou he nal resoluion if no selemen is reached ouside he cour. We show he disribuion of heir beliefs and he sochasic surplus are nonparamerically ideni ed from he probabiliy for reaching an selemen and he disribuion of nal ransfers beween players. We hen use a Simulaed Maximum Likelihood (SML) approach o esimae he beliefs of docors and paiens in medical malpracice lawsuis in Florida in he 1980s and 1990s. We nd srong evidence ha he beliefs for boh paries vary wih he severiy of he injury and he quali caion of he docors in he lawsuis, even hough hese characerisics are saisically insigni can in explaining wheher he cour rules in favor of he plaini or he defendan. Key words: Bargaining wih heerogeneous beliefs, nonparameric ideni caion, medical malpracice lawsuis 1 We hank paricipans a Brown, OSU and Economeric Sociey Norh America Summer Meeing 2014 for useful feedbacks. This research is funded by NSF Gran # xxxxxx. We hank Devin Reily and Michelle Tyler for capable research assisance. Errors (if any) are our own. 1
2 1 Inroducion A major heme in recen developmen of bargaining heory is o raionalize he delay in reaching an agreemen ha are prevalen in realworld bargaining episodes. (See, for example, Cho (1990), Merlo and Wilson (1995) and Yildiz (2004).) One explanaion for he delay is ha he paries involved in bargaining are oo opimisic abou heir respecive bargaining power in he absence of a common prior belief. Speci cally, consider a bilaeral bargaining episode where players could learn abou each oher s bargaining power hrough hisory of negoiaion. In he presence of opimism, a player i will decide o wai in hopes ha he oher player j will learn abou his (i s) selfperceived srong posiion and agree o his (i s) erms. As ime passes, he learning slows down, and i becomes no longer worhwhile o wai for he oher paries learning. Tha is when hey reach an agreemen. Yildiz (2004) inroduced his model and showed ha here is a deerminisic selemen dae, which is predeermined by he prior and he discoun facor of he players, such ha players will wai unil his dae o reach an agreemen. Since is inroducion, he model of bargaining wih opimism has been applied in a wide range of empirical conex, such as prerial negoiaions in medical malpracice lawsuis (Waanabe (2006)), negoiaion abou marke condiions (Thanassoulis 2010), and crosslicense agreemens (Galasso (2006)). Despie he recen surge in he heory and he applicaion of bargaining wih opimism, we are no aware of any exising work which addresses he ideni caion quesion in such a model formally. Tha is, under wha condiions can he srucural elemens of he model be unambiguously recovered from he hisory of bargaining repored in he daa. One of he objecives of our paper is o ll in his gap beween heory and empirical work by inroducing a framework for he srucural esimaion of bargaining wihou a common prior. In paricular, we propose a model for bilaeral bargaining where players have opimism abou he sochasic nal oucome in case no agreemen is reached. The players have a oneime opporuniy for reaching an agreemen a an exogenously scheduled dae during he bargaining process, and make decisions abou he selemen based on heir beliefs and ime discoun facors. We show ha all srucural elemens in his model are ideni ed nonparamerically from he probabiliy for reaching an agreemen and he disribuion of ransfer in he nal resoluion. The ideni caion sraegy does no rely on any paramerizaion of he srucural primiives such as players beliefs or he surplus disribuion. We hen propose a Simulaed Maximum Likelihood (SML) esimaor based on exible paramerizaion of he join beliefs. The model we inroduce is a simpli caion of he model of bargaining wihou a common prior in Yildiz (2004). We model a oneime selemen opporuniy for he players o reach an agreemen. Consequenly here is no dynamic learning consideraion in players decisions, and he daes of he nal resoluion of he bargaining episodes are deermined by players opimism, heir paience and heir percepion of he surplus o be shared. 2
3 There are boh heoreical and empirical moivaion for such simpli caion and modi  caion. Firs o, in realworld bargaining episodes, daa limiaion prevens researchers from deriving robus (paramerizaionfree) argumens for he ideni caion of srucural elemens in a full edged model of bargaining wih uncommon prior. For insance, he ideniy of proposers and he iming or he size of rejeced o ers in negoiaions are seldom repored in he daa available o researchers. By absracing away from he dynamic learning aspecs in Yildiz (2004), we adop a pragmaic approach o build a model ha is ideni able under less sringen daa requiremens and mild economeric assumpions. Despie his simpli caion, our model capures a key aspec of models wihou a common prior in ha he iming of he agreemen is deermined by he players opimism and heir paience. Thus our work provides a benchmark for undersanding wha addiional daa or economeric assumpions are needed o recover he primiives in more elaborae models of bargaining wihou common priors. Our modeling choices are also moivaed by he empirical quesion addressed in his paper: In medical malpracice lawsuis, wha do he selemen decisions by paiens and docors ell us abou heir respecive percepion of how likely he cour would rule in heir favor in case a cour hearing is necessary? The law of he Sae of Florida requires ha here should be a oneime mandaory selemen conference beween he plaini and he defendan, which is scheduled by he couny cour someime prior o he hearing and is mediaed by courdesignaed legal professionals. Besides, unlike Yildiz (2004) where a player s bargaining power is modeled as his chance for being he proposer, we model bargaining power as a player s percepion of he probabiliy ha he will receive he favored oucome in he nal resoluion in case no agreemen is reached. This modi caion is mean o be a beer approximaion of he acual decision environmen in he legal conex. Our sraegy for idenifying his model builds on a couple of insighs: Firs o, if he lengh of ime beween he scheduled selemen conference and he cour hearing (a.k.a. he waiime ) were repored in he daa, we would be able o recover he disribuion of opimism by observing how he condiional selemen probabiliy varies wih he lengh of waiime. Second, he disribuion of he poenial surplus o be divided beween players can be recovered from he disribuion of oal compensaion awarded o he plaini by he cour decision, provided he surplus disribuion is orhogonal o he beliefs. Third, because he acceped selemen o ers re ecs a plaini s imediscouned expecaion of his share of he oal surplus, we can idenify he disribuion of he plaini s belief condiional on selemen using he disribuion of acceped selemen o ers given he lengh of he waiime. This is done hrough a deconvoluion argumen using he disribuion of surplus recovered above. Las, since opimism is de ned as he sum of boh paries beliefs minus one, he objecs ideni ed from he preceding seps can be used o back ou he join disribuion of he beliefs hrough a sandard Jacobian ransformaion. 3
4 A key challenge for implemening he ideni caion sraegy in he environmen of malpracice lawsuis is ha he lengh of waiime is no direcly repored in he daa. In order o solve his issue of unobserved waiime, we ap ino a branch of recen lieraure ha uses an approach based on eigenvalue decomposiion o idenify nie mixure models or srucural models wih unobserved heerogeneiy. (See for example Hall and Zhou (2003), Hu and Schennach (2008), Kasahara and Shimosu (2009), An, Hu and Shum (2010) and Hu, McAdams and Shum (2013).) To do so, we rs exploi he insiuional deails in our environmen o group lawsuis ino smaller clusers (de ned by he couny and he monh in which a lawsui is o cially led) ha can be plausibly assumed o share he same (albei unobserved) lengh of waiime beween selemen conference and cour hearings. We hen use he cases in he same cluser as insrumens for each oher and apply an eigenvalue decomposiion o he join disribuion of selemen decisions and acceped o ers wihin he cluser. This allows us o recover he selemen probabiliy and he disribuion of acceped o ers condiional on he unobserved waiime. Then he argumens from he preceding paragraph applies o idenify he join disribuion of beliefs. The inference of docors and paiens beliefs in medical malpracice lawsuis is an ineresing empirical quesion in is own righ. In paricular, a cenral issue in he reform of U.S. healh care sysem is how o minimize he liigaion coss in medical malpracice lawsuis, which are known o consiue a large porion of he soaring insurance expenses. Knowing how selemen ouside he cour depends on paiens and docors opimism in he bargaining process could shed lighs on policy design. Using daa from medical malpracice lawsuis in Florida in he 1980s and 1990s, we nd clear evidence in our esimaes ha he beliefs of he docors and paiens vary wih observed characerisics of he lawsuis such as he severiy of he injury and he quali caion of he docors. This conrass wih he realiy ha he cour and jury decisions depend mosly on he naure and he cause of he malpracice and no so much on hese observed case characerisics (which is anoher fac revealed in our esimaes in he applicaion). Our esimaes can be used for answering fuure policy design quesions such as how he selemen probabiliy would change if he disribuion of he waiime is changed or some caps on pu on he poenial compensaion possible. The res of he paper is organized as follows: Secion 2 inroduces he model of bilaeral bargaining wih uncommon beliefs. Secion 3 esablishes he ideni caion of srucural elemens in he model. Secion 4 de nes he Simulaed Maximum Likelihood (SML) esimaor. Secion 5 describes he daa and he insiuional deails in he applicaion of medical malpracice lawsuis in Florida. Secion 6 presens and discusses he esimaion resuls. Proofs and a mone carlo sudy are presened in he appendices. 4
5 2 The Model Consider a lawsui following an incidence of medical malpracice involving a plaini (or paien) and a defendan (or docor). The oal amoun of poenial compensaion C is common knowledge among he plaini and he defendan. (I should be inerpreed as a sunk cos for he defendan, analogous o he money paid by he defendan for bailou.) Afer he ling of a lawsui, he plaini and he defendan are noi ed of a dae for a oneime selemen conference, which is mandaory by he Sae Saues in Florida. The conference requires aendance by boh paries (and heir aorneys), as well as legal professionals designaed by he couny cour where he lawsui is led. Such selemen conferences ake place wihin 120 days afer he ling of he lawsui. 2 During he conference, he defendan has he opporuniy o make an selemen o er of S C o he plaini. If he plaini acceps i, hen he legal process ends wih plaini receiving S and he defendan reclaiming C S. Oherwise he case needs o go hrough a cour hearing process ha culminaes in jury decisions. Boh he defendan and he plaini are aware ha he cour hearing needs o be a leas hree weeks laer han he selemen conference; and he exac dae is deermined by he schedule and he backlogs of all judges available a he coun cour. Le T denoe he lengh of ime beween he selemen conference and he scheduled cour hearing dae. Le A 1 if a selemen is reached a he conference; and A 0 oherwise. In he laer case, a he end of he cour hearing process, he jury makes a binary decision D as o wheher he plaini ges compensaed wih he full amoun C (i.e. D = 1) or he defendan is acquied wih no compensaions o he plaini required (i.e. D = 0). The plaini and he defendan believe heir chances of winning are p and d 2 [0; 1] respecively. These beliefs are common knowledge beween he paries, bu are no repored in daa. The join suppor of beliefs is f( p ; d ) 2 (0; 1] 2 : 1 < p + d 2g. This means excessive opimism always occurs (i.e. p + d > 1 wih probabiliy 1). We mainain he following assumpion hroughou he paper. Assumpion 1 (i) ( p ; d ) and C are independen from he wai ime T ; and he disribuions of ( p ; d ) is coninuous wih posiive densiies over. (ii) Condiional on A = 0, he jury decision D is orhogonal o C and T. Assumpion 1 allows plaini s and defendans beliefs o be correlaed wih each oher and asymmeric wih di eren marginal disribuions. This is empirically relevan because he marginal disribuion of beliefs may well di er beween paiens and docors due o 2 See Secion 108 in Chaper 776 of Florida Saues Web link: hp:// www. senae. gov/ Laws/ Saues/ 2012/
6 facors such as informaional asymmeries (e.g. docors are beer informed abou he cause and severiy of he malpracice) or unobserved individual heerogeneiies. Beliefs of plaini s and defendans are also likely o be correlaed hrough unobserved heerogeneiy of he case of malpracice. For example, hey may boh observe aspecs relaed o severiy or cause of he malpracice ha are no recorded in daa. Such aspecs lead o correlaions beween paiens and plaini s beliefs from an ousider s perspecive. Assumpion 1 also accommodaes correlaion beween ( p ; d ) and C. The independence beween he wai ime T and he beliefs is a plausible condiion, because he wai ime T is mosly deermined by availabiliy of judges and juries in he couny cour during he lawsuis. This depends on he schedule and backlogs of judges, which are idiosyncraic and orhogonal o paries beliefs ( p ; d ). The orhogonaliy of C from D given T and A = 0 in condiion (ii) is also jusi ed. On he one hand, C is a moneary measure of he magniude of he damage in iced on he plaini regardless of is cause; on he oher hand, D capures he jury s judgemen abou he cause of damage based on cour hearings. I is likely ha he jury decision is correlaed wih speci c feaures of he lawsui ha are repored in daa and ha may also a ec he beliefs of boh paries. Neverheless, once condiional on such observable feaures, jury decisions are mos likely o be orhogonal o measure of damage capured by C. A he end of his secion, we discuss how o exend our model o accoun for heerogeneiies across lawsuis repored in daa. We now summarize how he disribuions ha are direcly ideni able from daa are linked o model primiives under he assumpion ha boh paries follow raional sraegies. A he selemen conference, he plaini acceps an o er if and only if S T p C, where is a consan ime discoun facor xed hroughou he daageneraing process and available in daa. The defendan o ers he plaini S = T p C if he remainder of he poenial compensaion C S exceeds T d C. Hence a selemen occurs during he conference if and only if: C T p C T d C, d + p T. The resuled disribuion of selemens, condiional on he wai ime beween he selemen conference and scheduled cour hearing being T =, is: Pr (S s j A = 1; T = ) = Pr p C s j d + p. (1) where lower cases denoe realized values for random variables; and he equaliy follows from par (i) in Assumpion 1. Besides, he disribuion of poenial compensaion, condiional on he absence of selemen in he conference T = periods ahead of he cour hearing and condiional on he jury ruling in favor of he plaini, is: Pr(C c j A = 0; D = 1; T = ) = Pr(C c j d + p > ) (2) 6
7 where he equaliy follows from boh condiions in Assumpion 1. In pracice, he daa repors di erences in he characerisics of plaini s and defendans, such as he professional quali caion of he defendan or he demographics of he plaini. Besides, he daa also repors feaures relaed o he cause and he severiy of malpracice in quesion. Such informaion available in daa (denoed by a vecor X) are correlaed wih oal compensaion C and beliefs ( p ; d ). The simplisic model above can incorporae such observed case heerogeneiies by leing he primiives (i.e. disribuions of beliefs ( p ; d ), compensaions C, jury decisions D and he waiime T ) depend on X. If boh resricions in Assumpion 1 hold condiional on X, hen raional sraegies are characerized in he same way as (1) and (2) excep ha all disribuions needs o be condiioned on X. More imporanly, he ideni caion sraegy proposed in Secion 3 below are applicable when daa repors heerogeneiies across lawsuis. Formally, he resuls in Secion 3 (Lemma 1 and Proposiion 1) hold afer condiioning on X, provided he idenifying condiions (Assumpions 2, 3, 4 and 5) are formulaed as condiional on X. Noneheless, in order o simplify exposiion of he main idea for ideni caion, we choose o suppress dependence on observable case characerisics in Secion 3, and only incorporae hem explicily laer in he esimaion secion. 3 Ideni caion This secion shows how o recover he disribuion of boh paries beliefs from he probabiliy for reaching selemens and he disribuion of acceped selemen o ers. We consider an empirical environmen where for each lawsui he daa repors wheher a selemen occurs during he mandaory conference (A). For cases seled a he conference, he daa repors he amoun paid by he defendan o he plaini (S). For he oher cases ha underwen cour hearings, he daa repors jury decisions (D) and, if he cour rules in favor of he plaini, he amoun of oal compensaions paid by he defendan (C). However, exac daes of selemen conferences and scheduled daes for cour hearings (if necessary) are never repored in daa. 3 Thus he waiime T beween selemen conference and scheduled cour hearings, which is known o boh paries a he conference, is no available in daa. 3 For example, he daa we use in Secion 5 repors Daes of Final Disposiion for each case. However, for cases seled ouside he cour, hese daes are de ned no as he exac dae of he selemen conference, bu as he day when all o cial adminisraive paperwork are concluded. There is a subsanial lengh of ime beween he wo. For insance, for a large proporion of cases ha are caegorized as Seled wihin 90 days of he ling of lawsuis, he repored daes of nal disposiion are acually more han 150 days afer he iniial ling. Similar issues also exis for cases ha underwen cour hearings in ha he repored daes of nal disposiion are no idenical o he acual dae of cour hearings. 7
8 To address his issue wih unrepored waiime, we propose sequenial argumens ha exploi an implici panel srucure of he daa in he curren conex. In paricular, we noe ha lawsuis led wih he same couny cour during he same period (week) pracically share he same waiime T. The reason for such a paern is as follows: Firs, he daes for selemen conferences are mosly deermined by availabiliy of auhorized legal professionals a liaed wih he coun cour, and are assigned on a rscome, rs served basis. Thus selemen conferences for cases led wih he same couny cour a he same ime are pracically scheduled for he period. Besides, he daes for poenial cour hearing are deermined by he schedule and backlog of judges a he couny cour. Hence cases led wih he same couny cour simulaneously can be expeced o be handled in cour in he same period in he fuure. This allows us o e ecively group lawsuis ino clusers wih he same T, despie unobservabiliy of T in daa. We formalize his implici panel srucure as follows. Assumpion 2 Researchers have su cien informaion o divide he daa ino clusers, each of which consiss of a leas hree lawsuis sharing he same waiime T. Across he cases wihin he same cluser, he beliefs ( p ; d ), he oal compensaion C and he poenial jury decision D (if necessary) are independen draws from he same disribuion. This implici panel srucure in our daa allows us o use acceped selemen o ers in he lawsuis wihin he same cluser as insrumens for each oher, and apply eigendecomposiionbased argumens in Hu and Schennach (2008) o recover he join selemen probabiliy and disribuions of acceped selemen o ers condiional on he unobserved T. We hen use hese quaniies o back ou he join disribuion of beliefs using exogenous variaions in T. For he res of his secion, we rs presen argumens for he case where T is discree (i.e. jt j < 1). A he end of his secion, we explain how o generalize hem for ideni caion when T is coninuously disribued. 3.1 Condiional disribuion of selemen o ers An inermediae sep for idenifying he join disribuion of beliefs is o recover he condiional selemen probabiliy and he disribuion of selemen o ers given he waiime before cour hearings T. Le S; T denoe he uncondiional suppors of S; T respecively. Assumpion 3 (i) The suppor of T is nie (jt j < 1) wih a known cardinaliy and inff : 2 T g 1=2. (ii) Given any ( p ; d ), he poenial compensaion C is coninuously disribued wih posiive densiy over a conneced suppor [0; c]. 8
9 We focus on he model wih nie T wih known cardinaliy because of is empirical relevance. Wihou loss of generaliy, denoe elemens in T by f1; 2; :; jt jg. Condiion (i) also rules ou unlikely cases where a cour hearing is scheduled so far in he fuure or he oneperiod discoun facor is so low ha he compounded discoun facor is less han one half. Condiion (i), ogeher wih he nonincreasingness of E[A i j T = ] over 2 T due o Assumpion 1, pin down he index for eigenvalues and eigenvecors in he aforemenioned decomposiion. Par (ii) in Assumpion 3 is a mild condiion on he condiional suppor of poenial compensaion. A su cien condiion for his is ha C is orhogonal from ( p ; d ) wih a bounded suppor. 4 resul below. The role of par (ii) will become clear as we discuss he ideni caion Lemma 1 Under Assumpions 1, 2 and 3, E (A j T = ) and f S (s j A = 1; T = ) are joinly ideni ed for all and s. This inermediae resul uses argumens similar o ha in Hu, McAdams and Shum (2013) for idenifying rsprice sealedbid aucions wih nonseparable aucion heerogeneiies. I explois he panel srucure of he daa and he condiional independence of beliefs across lawsuis in Assumpion 2. These condiions allow us o break down he join disribuion of he incidence of selemen and he size of acceped selemen o ers across muliple lawsuis wihin one cluser ino he composiion of hree linear operaors. More speci cally, le f R1 (r 1, R 2 = r 2 j :) be a PrfR 1 ~r, R 2 = r 2 j :gj ~r=r1 for any discree random vecor R 2 and coninuous random vecor R 1. For any hree lawsuis i; j; k sharing he same waiime T, le A i;k = 1 be a shorhand for A i = A k = 1. By consrucion, f Si ;S k (s; s 0 ; A j = 1 j A i;k = 1) = P f Si (s j S k = s 0 ; A j = 1; T = ; A i;k = 1)E[A j j S k = s 0 ; T = ; A i;k = 1]f T;Sk (; s 0 j A i;k = 1) 2T = P 2T f Si (s j A i = 1; T = )E[A j j T = ]f T;Sk (; s 0 j A i;k = 1). (3) The second equaliies follow from Assumpion 1; from S = T p C whenever A = 1 and A = 1 if and only if p + d T ; and from he fac ha beliefs ( p ; d ) and poenial compensaion C are independen draws across he lawsuis i; j; k according o Assumpion 2. To illusrae he ideni caion argumen, i is useful o adop marix noaions. Le D M denoe a pariion of he uncondiional suppor of acceped selemen o ers S ino 4 I is worh noing ha our ideni caion argumen remains valid even wih c being unbounded, as long as he fullrank condiion in Lemma A1 holds for some pariions of S. 9
10 M inervals. Each of he inervals has a nondegenerae inerior and is denoed by d m. 5 For a given pariion D M, le L Si ;S k probabiliy ha S i 2 d m and S k 2 d m 0 be a MbyM marix whose (m; m 0 )h enry is he condiional on A i;k = 1 (selemens are reached in cases i and k); and le Si ;S k be a MbyM marix wih is (m; m 0 )h enry being f(s i 2 d m ; A j = 1; S k 2 d m 0 j A i;k = 1). Noe ha boh Si ;S k and L Si ;S k are direcly ideni able from daa. Thus a discreized version of (3) is : Si ;S k = L Si jt j L T;Sk (4) where L Si jt be a MbyjT j marix wih (m; )h enry being Pr(S i 2 d m j A i = 1; T = ); j be a jt jbyjt j diagonal marix wih diagonal enries being [E(A j j T = )] =1;:;jT j ; and L T;Sk be a jt jbym marices wih is (; m)h enry being Pr (T = ; S k 2 d m j A i;k = 1). Besides, due o condiional independence in Assumpion 2. L Si ;S k = L Si jt L T;Sk (5) Par (ii) in Assumpion 3 implies he supreme of he condiional suppor of acceped o ers given T = is c and hence decreases in. This, in urn, guaranees here exiss a pariion D jt j such ha L Si jt as well as L Si ;S k are nonsingular (as proved in Lemma A1 in Appendix B). Then (4) and (5) imply Si ;S k (L Si ;S k ) 1 = L Si jt j L Si jt 1 (6) where he L.H.S. consiss of direcly ideni able quaniies. The R.H.S. of (6) akes he form of an eigendecomposiion of a square marix, which is unique up o a scale normalizaion and unknown indexing of he columns in L Si jt and diagonal enries in j (i.e. i remains o pin down a speci c value of 2 T for each diagonal enry in j ). The scale in he eigendecomposiion is implicily xed because he eigenvecors in L Si jt are condiional disribuions and needs o sum up o one. The quesion of unknown indices is solved because in our model E[A j j T = ] is monoonically decreasing in over T provided he paries follow raional sraegies described in Secion 2. This is again due o he independence beween iming and he beliefs in Assumpion 1 and he moderae compounded discouning in Assumpion 3. This esablishes he ideni caion of j and L Si jt, which are used for recovering L T;Sk and hen he condiional densiy of acceped selemen o ers over is full suppor (see proof of Lemma 1 in Appendix B). 3.2 The join belief disribuion We now explain how o idenify he join disribuion of beliefs ( p ; d ) from he quaniies recovered from Lemma 1 under he following orhogonaliy condiion. 5 Tha is, d m [s m ; s m+1 ] for 1 m M, wih (s m : 2 m M) being a vecor of ordered endpoins on S such ha s 1 < s 2 < :: < s M < s M+1 and s 1 inf S, s M+1 sup S. 10
11 Assumpion 4 The join disribuion of beliefs ( p ; d ) is independen from poenial compensaions C. This condiion requires he magniude of poenial compensaion o be independen from plaini and defendans beliefs. This condiion is plausible because C is mean o capure an objecive moneary measure of he severiy of damage in iced upon he paien. On he oher hand, he beliefs ( p ; d ) should depend on he evidence available as o wheher he defendan s neglec is he main cause of such damage. I hen follows from (2) ha he disribuion of C is direcly ideni ed as: Pr(C c) = Pr(C c j A = 0; D = 1). (7) Le S [0; c ] denoe he condiional suppor of acceped selemen o ers S = T p C given A = 1 and T = ; 6 and le ' (s) denoe he probabiliy ha a selemen is reached when he lengh of waiime beween he selemen conference and he dae for cour hearing is and ha he acceped selemen o er is no greaer han s. Tha is, for all (s; ), ' (s) Pr (S s; A = 1 j T = ) = Pr p C s= ; d + p 1= (8) where he equaliy is due o Assumpion 1. The nonnegaiviy of C and ( p ; d ) and an applicaion of he law of oal probabiliy on he righhand side of (8) implies: ' (s) = Z c 0 1 Pr c p s 1 ; f(c)dc = d + p Z c 0 h (c=s) f C (c)dc (9) where f C (c) is he densiy of C and h (v) Prfp 1 v ; ( d + p ) 1 g; and he rs equaliy is due o orhogonaliy beween C and ( p ; d ). Changing variables beween C and V C=S for any xed and s, we can wrie (9) as: ' (s) = Z 1 0 h (v)(v; s)dv (10) where (v; s) sf C (vs)1fv c=sg. Wih he disribuion (and hence densiy) of C recovered from (7), he kernel funcion (v; s) is considered known for all (v; s) hereinafer for ideni caion purposes. Also noe for any s > 0, (:; s) is a wellde ned condiional densiy wih suppor [0; c=s]. 7 Le F V ja=1;t = denoe he disribuion of V given T = and A = 1 (or equivalenly d + p T ), whose suppor is denoed as V. Assumpion 5 For any and g(:) such ha E[g(V ) j A = 1; T = ] < 1, he saemen R 1 0 g(v)(v; s) = 0 for all s 2 S implies he saemen g(v) = 0 a.e. F V ja=1;t =. 6 In general, we could also allow suppors S;T and S o depend on observed heerogeneiies of lawsuis as well. Noneheless, hroughou his secion, we refrain from such generalizaion in order o simplify exposiion. 7 This is because (v; s) > 0 for any v 0, s > 0. Besides R 1 (v; s)dv = R c=s sf 0 0 C (vs)dv = 1 for any s. 11
12 This condiion, known as he compleeness of kernels in inegral operaors, was inroduced in Lehmann (1986) and used in Newey and Powell (2003) for ideni caion of nonparameric regressions wih insrumenal variables. Andrews (2011) and Hu and Shiu (2012) derived su cien condiions for various versions of such compleeness condiions when g(:) is resriced o belong o di erence classes. This condiion is analogous o a fullrank condiion on if he condiional suppors of S and V were nie. 8 Proposiion 1 Under Assumpions 15, Pr( p ; p + d ) is ideni ed for all 2 (0; 1] and 2 T. For he res of his secion, we discuss how o generalize resuls above when T is in nie (T is coninuously disribued over a known inerval). Firs o, he key idea of using eigendecomposiions in Secion 3.1 remains applicable, excep ha L Si jt and L T;Sk become linear inegral operaors, and heir inveribiliy needs o be saed as an assumpion as opposed o being derived from resricions on model primiives and implicaions of raional sraegies (as is he case when T is discree). Under he suppor condiion ha inff : 2 T g 1=2, he eigenvalues in he decomposiion E[A j j T = ] remains sricly monoonic over he inerval suppor T when T is coninuously disribued. On he oher hand, he argumen ha uses monooniciy of he eigenvalues over a nie suppor T o index hem is no longer applicable when T is coninuously disribued. However, raional sraegies in our model imply he supremum of he suppor of acceped selemen o ers given T = mus be c. Wih he supremum of he suppor of compensaions c ideni ed and known, his means can be expressed hrough a known funcional of he eigenvecors f Si (: j A i = 1; T = ) in he eigendecomposiion ideni ed in he rs sep. Thus he issue wih indexing eigenvalues is also solved. The remaining sep of idenifying he join disribuion of ( p ; p + d ) from f S (: j A = 1; T = ) and E[A j T = ] follow from he same argumen above. I is worhy of noe ha an addiional sep based on Jacobian ransformaion leads o ideni caion of he join disribuion of ( p ; d ) when T is coninuously disribued. 8 If he suppor of poenial compensaion is unbounded, here are pleny of examples of parameric families of densiies ha saisfy he compleeness condiions. For example, suppose poenial compensaions follow a Gamma disribuion wih parameers ; > 0. Tha is, f C () = () 1 expf g. Then, wih s > 0, he kernel (v; s) sf C (vs) = [s] () v 1 expf v (s)g is a densiy of a Gamma disribuion wih a shape parameer > 0 and a scale parameer s > 0. Tha is, (v; s) remains a condiional densiy wihin he exponenial family, and sais es he su cien condiions for he compleeness condiion in Theorem 2.2 in Newey and Powell (2003). 12
13 4 Simulaed Maximum Likelihood Esimaion Our ideni caion resuls in Secion 3 lay he foundaion for nonparameric esimaion of he belief disribuion. However, a nonparameric esimaor based on hose argumens would require a large daa se, and he curse of dimensionaliy aggravaes if he daa also repor caselevel variables ha may a ec boh paries beliefs (such as he severiy of injury in iced upon he plaini and he quali caion of he defendan) and herefore should be condiioned on in esimaion. To deal wih caseheerogeneiies in moderaesized daa, we propose in his secion a Maximum Simulaed Likelihood esimaor based on a exible paramerizaion of he join belief disribuion. Consider a panelsrucure daa conain N clusers. Each cluser is indexed by n and consiss of m n 1 cases, each of which is indexed by i = 1; :::; m n. For each case i in cluser n, le A n;i = 1 when here is an agreemen for selemen ouside he cour and A n;i = 0 oherwise. De ne Z n;i S n;i if A n;i = 1; Z n;i C n;i if A n;i = 0 and D n;i = 1; and Z n;i 0 oherwise. Le T n denoe he waiime beween he selemen conference and he scheduled dae for cour decisions. We propose a Maximum Simulaed Likelihood esimaor for he join beliefs ( p ; d ) ha also explois variaion in he heerogeneiy of lawsuis repored in he daa. Throughou his secion, we assume he idenifying condiions also hold once condiional on such observed heerogeneiy of he lawsuis. Le x n;i denoe he vecor of caselevel variables repored in he daa ha a ecs he disribuion of C. (We allow x n;i o conain a consan in he esimaion below.) These include he age, severiy, and couny average/median income (as well as heir ineracion erms). The oal poenial compensaion C in a lawsui wih observed feaures x n;i is drawn from an exponenial disribuion wih he rae parameer given by: (x n;i ; ) expfx n;i g for some unknown consan vecor of parameers. In he rs sep, we pool all observaions where he jury is observed rule in favor of he plaini o esimae : P ^ arg max n;i d n;i(1 a n;i ) [x n;i expfx n;i gc n;i ]. Nex, le w n;i denoe he vecor of caselevel variables in he daa ha a ecs he join belief disribuion. (The wo vecors x n;i and w n;i are allowed o have overlapping elemens.) We suppress he subscrips n; i for simpliciy when here is no confusion. In he second sep, we esimae he belief disribuion condiional on such a vecor of caselevel variables W using ^ above as an inpu in he likelihood. To do so, we adop a exible paramerizaion of he join disribuion of ( p ; d ) condiional on W as follows. For each realized w, le (Y 1 ; Y 2 ; 1 Y 1 Y 2 ) be drawn from a Dirichle disribuion wih concenraion parameers j expfw j g for j = 1; 2; 3 for some consan vecor ( 1 ; 2 ; 3 ). In wha follows, we suppress he dependence of j on w o simplify he noaion. 13
14 Le p = 1 Y 1 and d = Y 1 + Y 2. The suppor of ( p ; d ) is f(; 0 ) 2 [0; 1] 2 : g, which is consisen wih our model wih opimism. (Table C1 and Figure C1 in Appendix C show how exible such a speci caion of he join disribuion of ( p ; d ) is in erms of he range of momens and he locaion of he model i allows.) Also noe Y 2 = p + d 1 by consrucion, so i can be inerpreed as a measure of opimism. Under his speci caion, he marginal disribuion of Y 1 condiional on W = w is Bea( 1 ; ), where of course j s are funcions of w. The condiional disribuion Y 2 j Y 1 = ; W = w is he same as he disribuion of (1 )Bea( 2 ; 3 ). For any y and 2 (0; 1), we can wrie: Y2 PrfY 2 y j Y 1 = ; W = wg = Pr 1 y 1 Y 1 = ; W = w where he righhand side is he c.d.f. of a Bea( 2 ; 3 ) evaluaed a y=(1 Le q n;i Pr(D n;i = 1 j A n;i = 0; W n;i = w n;i ). Recall ha we mainain D is orhogonal o (T; C) once condiional on A = 0 and W. Hence q n;i does no depend on c n;i. This condiional probabiliy is direcly ideni able from he daa. Le h n (; ) denoe densiy of he waiime T n a T n = in cluser n. This densiy may depend on cluserlevel variables repored in he daa, and is speci ed up o an unknown vecor of parameers. The loglikelihood of our model is: L N (; ; ) P N n=1 ln P 2T h n(; ) Q m n i=1 f n;i(; ; ) where f n;i (; ; ) is shorhand for he condiional densiy of Z n;i ; A n;i ; D n;i given T n =, W n;i = w n;i and wih parameer, evaluaed a (z n;i ; a n;i ; d n;i ). Speci cally, where f n;i (; ; ) [g 1;n;i (; ; )] a n;i fg 0;n;i (; ) [1 p n;i (; )] q n;i g (1 a n;i)d n;i f[1 p n;i (; )] (1 q n;i )g (1 a n;i)(1 d n;i ) p n;i (; ) Pr(A n;i = 1 j T n = ; W n;i = w n;i ; ) = Pr( p;n;i + d;n;i j w n;i ; ) = Pr(Y 2 1 = j w n;i ; ); g 0;n;i (; ) g 0 (z n;i ; x n;i ; ; Pr(C n;izja n;i =0;T n=;x n;i =x n;i ). = f C (z n;i j x n;i ;(11) ); Z=zn;i wih f C (: j x n;i ; ) being he condiional densiy of he poenial compensaion given X n;i = x n;i ; and g 1;n;i (; ; ) g 1 (z n;i ; w n;i ; x n;i ; ; ; Pr(S n;iz;a n;i =1jT n=;w n;i ;x n;i Z 0 Pr Y 1 1 Z=(c ); Y 2 1 wn;i ; 14 Z=zn;i f C (c j x n;i ; )dc (12). Z=zn;i
15 In he derivaions above, we have used he condiional independence beween C n;i and D n;i ; T n ; ( p;n;i ; d;n;i ) condiional on W n;i ; X n;i. Under mild regulariy condiions, he order of inegraion and di ereniaion in (12) can be exchanged. Tha is, g 1;n;i (; ; ) equals: Z 1 z n;i n Pr Y 2 1 Y1 = 1 z n;i =c; w n;i ; o f Y1 (1 z n;i =c j w n;i ; ) f C(c j x n;i ; ) c dc where he lower limi is z n;i because he inegrand is nonzero only when 1 z n;i =c 2 (0; 1), c 2 ( z n;i ; +1).Changing variables beween c and 1 z n;i =c for any i; n and xed, we can wrie g 1;n;i (; ; ) as: Z 1 0 Y2 Pr 1 1 (1 ) Y 1 = ; w n;i ; fy1 ( j w n;i ; )f C zn;i (1 (1 ) j x ) n;i; d 1 where he rs condiional probabiliy in he inegrand is a Bea c.d.f. evaluaed a (1 ) and parameers ( 2 (w n;i ; 2 ); 3 (w n;i ; 3 )) and he second erm f Y1 ( j w n;i ; ) is he Bea p.d.f. wih parameers ( 1 (w n;i ); 2 (w n;i ) + 3 (w n;i )). For each n, i, and a xed vecor of parameers (; ), le ^g 1;n;i (; ; ) be an esimaor for g 1;n;i (; ; ) using S > N simulaed draws of. (We experimen wih various forms of densiy for simulaed draws.) I follows from he Law of Large Numbers ha ^g 1;n;i (; ; ) is an unbiased esimaor for each n; i and (; ). sep is Our Maximum Simulaed Likelihood Esimaor for he belief parameers in he second (^; ^) arg max ; ^L N (; ; ^). (13) where ^L N (; ; ) is an esimaor for L N (; ; ) by replacing g 1;n;i (; ; ) wih ^g 1;n;i (; ; ) and replacing q n;i wih a parameric (logi or probi) esimae ^q n;i ; and ^ is he esimaes for he parameers in he disribuion of poenial compensaion in he rs sep. Under regulariy condiions, (^; ^) converge a a p Nrae o a zeromean mulivariae normal disribuion wih some nie covariance as long as N! 1, S! 1 and p N=S! 1. The covariance marix can be consisenly esimaed using he analog principle, which involves he use of simulaed observaions. (See equaion (12.21) in Cameron and Trivedi (2005) for a deailed formula.) 5 Daa Descripion Since 1975 he Sae of Florida has required all medical malpracice insurers o le repors on heir resolved claims o he Florida Deparmen of Financial Services. Using his source, we consruc a sample ha consiss of 13,351 lawsuis led in Florida beween 1984 and Sieg (2000) and Waanabe (2009) also used he same source of daa. Our sample includes 15
16 he cases ha are eiher resolved hrough he mandaory selemen conference or by a jury decision ha followed he cour proceedings. For each lawsui, he daa repors he dae when i is led (Sui_Dae) and he couny cour wih which i is led (Couny_Code), he dae of he nal disposiion (Year_of_Disp) (when he claim was closed wih he insurer), and wheher he case is resolved hrough a selemen conference or by a jury decision in cour (A=1 if seled ouside he cour). The daa also repors he size of he ransfer from he defendan o he plaini upon he resoluion of he lawsui. This equals he amoun of acceped o er o he plaini (S) if a selemen is reached ouside he cour, or he oal compensaion awarded o he plaini according o he cour decision (C ) oherwise. In addiion we also observe caselevel variables ha may be relevan o he disribuion of he join belief or ha of he poenial compensaion. These include he severiy of he injury due o negligence (Severiy), he age (Age) and gender of paiens and wheher he docors responsible are boardceri ed (Board_Code). (Tha is, Board_Code = 1 if he docor is ceri ed by a leas one professional board and = 0 oherwise.) For he lawsuis seled ouside he cour, he daes for selemen conferences are no repored in he daa. The scheduled daes for cour hearings are no repored for he cases resolved by cour decisions eiher. Furhermore, he recorded daes for he nal disposiion only reveal when he claim is closed wih he insurer, which are ypically laer han he acual daes when an agreemen is reached in a selemen conference or when a decision is made by he judge in he cour. Therefore, he lenghs of he ime beween scheduled cour hearings and he selemen conferences are no direcly measured in he daa. Despie hese daa limiaions, we de ne clusers wihin which he cases could be reasonably assumed o share he same lengh of waiime. I is plausible ha he lawsuis led wih he same couny cour in he same monh would be scheduled for cour proceedings in he same monh. This is of course because he schedule for hearings in a couny cour is mosly deermined by he backlog of unresolved cases led wih ha cour, and by he availabiliy of judges and oher legal professional from he cour. By he same oken, he schedule for selemen conferences, which require he presence of cour o cials who have auhoriy o coordinae a selemen, are also mosly deermined by he backlog cases as well as he availabiliy of aorney represening boh paries. Due o hese empirical consideraions, we mainain ha he waiime beween selemen conferences and cour hearings are idenical for he cases led wih he same couny in he same monhs. As explained in Secion 3, he disribuion of selemen decisions and acceped o ers in lawsuis from hese clusers are su cien for recovering he join beliefs of plaini s and defendans. The daa consiss of 3,545 clusers de ned by monhcouny pairs. In oal here are 1,344 clusers which repor a leas hree medical malpracice lawsuis. Abou half of hese clusers (661 clusers) conain a leas six cases. Besides, among hese 1,344 clusers, 1,294 have a leas wo lawsuis ha were seled ouside he cour due o he mandaory conference. 16
17 These numbers con rm ha we can apply our ideni caion sraegy from Secion 3 o recover he join disribuion of paiens and docors beliefs. I is worh noing ha in our SML esimaion he likelihood includes all 3,545 clusers o improve he e ciency of he esimaor, even hough in heory ideni caion only requires he join disribuion of selemen decisions and acceped o ers from he subse of clusers ha have a leas wo selemens ou of hree or more cases. Table 1(a): Selemen probabiliy and acceped o ers Board Cer n Severiy # obs ^p sele s:e:(^p sele ) ^ SjA=1 ($1k) s:e:(^ SjA=1 ) ($1k) ceri ed low 1, medium 2, high 2, unceri ed low 1, medium 2, high 2, Nex, we repor some evidence from he daa ha he belief of he plaini s and he defendans are a eced by cerain observed characerisics in he lawsuis. Table 1(a) summarizes he selemen probabiliy and he average size of acceped o ers in he sample afer conrolling for he docors quali caion and he level of severiy. There is evidence ha boh he selemen probabiliy and he size of acceped selemen o ers di er sysemaically across he subgroups. Table 1(b) repors he pvalues of wosided ess (using he unequal variance formula) for he equaliy of selemen probabiliies in subgroups. We le (u,c) and (l,m,h) be shorhand for he realized values of (unceri ed, ceri ed) in Board_code and (low, medium, high) in Severiy respecively. Wih he excepion of hree pairwise ess, he nulls in he oher ess are all rejeced a he 1% signi cance level. Among he hree excepions, he null for equal selemen probabiliy beween (u,l) and (u,h) is also rejeced a he 10% level. The only wo cases where he null can no be rejeced even a he 10% signi cance level are (u,l) versus (c,m) and (u,l) versus (c,h). This is somewha consisen wih he inuiion ha a plaini may end o be more opimisic ha he jury would rule in his favor when he injury in iced is more severe, or when he docor s quali caion is no suppored by board ceri caion. Our esimaes in he nex secion are also consisen wih his inuiion. The failure o rejec he null of equal selemen probabiliy beween he wo subgroups (u,l) and (c,h) for example may be due o he fac ha he impacs of severiy and of board ceri caion on he plaini s belief o se each oher. Pairwise ess for he equaliy of 17
18 average acceped selemen o ers beween he subgroups de ned severiy and docor quali caion also demonsrae similar paerns. Speci cally, he null of equal average selemen o ers is almos always rejeced a he 1% signi cance level for all pairwise ess using unequal variances, wih he only excepion being he ess comparing (u,l) versus (c,l). Table 1(b): pvalues for ess: selemen probabiliy u,l u,m u,h c,l c,m c,h u,l < < < u,m < < u,h < < < c,l < < c,m c,h The daa also conain some evidence ha he disribuion of oal compensaion may be parly deermined by he age of he plaini and he severiy of he injury. Ou of he oal 2,298 lawsuis which were no resolved hrough selemen, 359 were ruled in favor of he plaini by he cour. The observaions of he realized oal compensaion in hese cases are useful for inference of he disribuion of C. Figure 1(a) and 1(b) in Appendix A repor hisograms of he acceped o ers (S) from he cases seled ouside he cour and he oal compensaion (C ) from he cases where he cour ruled in favor of he plaini, condiioning on he informaion abou he plaini s. The variable Age is discreized ino hree caegories: young (Age < 33), older (Age > 54) and middle, wih he cuo s being he 33rd and he 66h perceniles in he daa. Figure 1(a) suggess he younger plaini s end o receive higher ransfers eiher hrough acceped o ers in selemen or hrough he oal compensaion paid by he defendan when he cour rules in favor of he plaini. Figure 1(b) shows he cases wih more severe injuries in general are associaed wih higher ransfers. Boh paerns are inuiive, and consisen wih our esimaes in he nex secion. To furher compare he disribuion of he acceped o ers wih ha of he oal compensaion ruled by he cour, we compare he perceniles of boh variables condiional on Age and Severiy. We nd ha he 10h, 25h, 50h, 75h and 90h condiional perceniles of he acceped o ers are consisenly lower han hose of he oal compensaion ruled by he cour. This is consisen wih he noion ha he acceped selemen o ers are he discouned expecaion of he oal compensaion o be ruled by he cour. The quali caion of he docors does no seem o have any noiceable e ec on he disribuion of he oal compensaion. Figure 1(c) repors he hisogram of he oal compensaion 18
19 for he cases where he cour ruled in favor of he plaini, condiioning on he board ceri caion of he docors. A es for he equaliy of he average compensaion for he wo subgroups wih and wihou board ceri caion repors an asympoic pvalue of (assuming unequal populaion variance). Besides, a onesided KomolgorovSimirnov es agains he alernaive ha he disribuion of C is sochasically lower when he defendan is boardceri ed yields a es saisic of and an asympoic pvalue of Thus in eiher es he null can no be rejeced even a he 15% signi cance level. On he oher hand, i is reasonable o posulae ha he oal poenial compensaion in a malpracice lawsui is posiively correlaed wih he conemporary income level in he couny where he lawsui is led. In order o conrol for such an income e ec, we collec daa on household income in all counies in Florida beween 1981 and We rs collec he daa on he median household income in each Florida couny in 1989, 93, 95, 97, 98 and 99 from he Small Area Income and Povery Esimaes (SAIPE) produced by he U.S. Census Bureau. 9 We also collec a ime series of saewide median household income in Florida each year beween 1984 and 1999 from U.S. Census Bureau s he Curren Populaion Survey. We hen combine his laer saewide informaion wih he counylevel informaion from SAIPE o exrapolae he median household income in each Florida couny in he years , 92, 94 and We hen incorporae his yearly daa on household income in each couny while esimaing he disribuion of oal compensaion nex year. 6 Esimaion Resuls As he rs sep in esimaion, we use a logi regression o he cour decisions in hose lawsuis ha are resolved hrough cour hearings. The goal is o provide some evidence abou wheher he jury decisions were a eced by case characerisics repored in he daa. Besides, he prediced probabiliy for D = 1 (he jury ruled in favor of he plaini ) from he logi regression will be used in he SML esimaion of he join beliefs of docors and paiens. 9 See hp://www.census.gov/did/www/saipe/daa/saecouny/daa/index.hml 10 The exrapolaion is done based on a mild assumpion ha a couny s growh rae relaive o he saewide growh rae remains seady in adjacen years. For example, if he raio beween he growh rae in Couny A beween 1993 and 1995 and he conemporary saewide growh rae is, hen we mainain he yearly growh raes in Couny A in (and ) are boh equal o p imes he saewide growh raes in (and respecively). Wih he yearly growh rae in Couny A beeen calculaed, we hen exrapolae he median household income in Couny A in 1994 using he daa from he SAIPE source. 19
20 Table 2. Logi Esimaes for Cour Decisions 11 (Response Variable: D. Toal # of observaions: 2,289 cases.) (1) (2) (3) Board_Code (0.120) (0.282) (0.288) Severiy ** (0.023) (0.033) (0.056) Age (0.003) (0.003) 0.021* (0.012) SeveriyBoard_Code (0.046) (0.046) Age (0.011) SeveriyAge (0.011) Consan *** (0.189) *** (0.228) *** (0.391) Log likelihood PseudoR pvalue for L.R.T Noes: Sandard errors are repored in parenheses. (*** signi can a 1%; ** sig. a 5%; * sig. a 10%).Age 2 is repored in unis of 100 yr 2. Table 2 repors he logi regression esimaes under di eren speci caions, using 2,289 lawsuis from he daa ha were no seled ouside he cour and hus had o be resolved hrough scheduled cour hearings. The case heerogeneiy used in he logi regressions include Board_Code, Severiy and he age of he paiens Age. In all hree logi regressions, he consan erm is highly saisically signi can a he 1% level. The severiy is saisically insigni can in he laer wo speci caions. Besides, he age of he paien is only signi can a 10% level in he hird speci caion. The board ceri caion of docors and he ineracion erms in he logi regressions are all insigni can. The pseudo Rsquares are low for all hree speci caions. This suggess ha he paien and case characerisics considered are raher insigni can in explaining he cour decisions. Furhermore he pvalues for he likelihood raio ess of he join signi cance of all slope coe ciens are 0:1676, 0:2533 and 0:2557 in he hree speci caions respecively. Therefore we conclude from Table 2 ha he docor s board ceri caion, he severiy of he malpracice and he age of he plaini do no have signi can impac on jury decisions in he cour. Nex, we esimae he disribuion of oal poenial compensaion using a subse of he observaions of lawsuis above where he cour ruled in favor of he plaini s (A = 0 and D = 1). The descripive saisics in Secion 5 show ha he severiy of he injury and he age of he plaini s have a noiceable impac on he size of he oal poenial compensaion, while he docors board ceri caion does no. In one of he speci caions, we include he 11 Sandard errors are repored in he parenhesis. B.C. is shorhand for Board_Code, and Sev. for Severiy. The variable Age 2 is repored in unis of 100 yr 2. 20
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