PRETRIAL NEGOTIATION, LITIGATION, AND PROCEDURAL RULES

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1 PRETRIAL NEGOTIATION, LITIGATION, AND PROCEDURAL RULES JIONG GONG AND R. PRESTON MCAFEE* We moel the ciil isute resolution rocess as a two-stage game with the arties bargaining to reach a settlement in the first stage an then laying a litigation exeniture game at trial in the secon stage. We fin that the English rule shifts the settlement away from the interim fair an unbiase settlement in most circumstances. Oerall welfare changes are in faor of the arty who makes the offer in the retrial negotiation stage. Lawyers howeer, always benefit from the English rule, because fee shifting increases the stake of the trial an thus intensifies the use of the legal serice. Ž JEL K40. I. INTRODUCTION Newt Gingrich, in the Contract with America 1994, 145, calls for common sense legal reforms, incluing a so-calle loser-ays rule Žin which the unsuccessful arty in a suit ays the attorneys fees of the revailing arty.. The loser-ays rule Žhereafter feeshifting or English rule., Gingrich argues, strongly iscourages the filing of weak cases as well as encourages the ursuit of strong cases, since claimants can get their court costs reimburse if they win 1994, 146. A voluminous literature, iscusse below, suorts Gingrich s view. However, by an large, this literature neglects two imortant consierations in analyzing the English rule. *This aer was comlete while Jiong Gong was oing grauate stuy at the University of Texas at Austin, an the oinions exresse o not reflect those of Telcoria. We are also grateful to Dr. Phil Reny, Davi Sibley, Dale Stahl, Steve Bronars, John Zheng, other faculty members, an grauate stuents of the University of Texas at Austin economics eartment for helful comments. We are also grateful to three anonymous referees for helful suggestions. Gong: Research Scientist, Telcoria Technologies, Morristown, N.J jgong@telcoria.com McAfee: Murray S. Johnson Professor of Economics, Deartment of Economics, University of Texas at Austin, Austin, TX mcafee@ eco.utexas.eu 1. Other names are also use, like the British rule, the Continental rule, an the Euroean rule. In the legal rofession, this kin of ractice is terme inemnity, or fee shifting. We will use fee shifting most of the time in this article.. We have been unable to fin an examle of another nation that uses the American rule for legal fees. Western Euroean nations o not use the American rule. Economic Inquiry Ž ISSN Vol. 38, No., Aril 000, First, the English rule increases the stake of the litigating arties. Thus, although a switch to the English rule may reuce the number of lawsuits, the lawsuits that get to court shoul involve higher legal exenses. Consequently, the overall effect on legal exenses is unclear. Secon, the literature neglects the fairness of the outcome. Do laintiffs obtain an awar justifie by the evience? We rovie some theoretical evience that the English rule istorts outcomes away from a fair outcome. The juicial roceural systems in most nations follow the English rule, which allows for some egree of fee shifting, or the rewaring of legal fees to the winner of legal battles. 1 It is an ol British octrine of the costs follow the event. The Unite States is a remarkable excetion. The American rule eveloe graually. In the early ays of the colonies, the roceural system inherite the English rule, allowing attorneys to recover fees from the losing arty. Graually, attorneys foun it more lucrative to sign rivate contingency agreements with their clients an ai less attention to reemtion from the losing arty. The court in the meantime graually relaxe enforcement of fee shifting an cease monitoring legal costs, until the ABBREVIATIONS F-K: Froeb an Kobayashi Western Economic Association International

2 GONG & MCAFEE: NEGOTIATION, LITIGATION, RULES 19 American rule became art of the statutory law. 3 We evelo a moel that ermits us to aress the question of the effects of fee shifting on fairness an exeniture at trial. Since retrial settlement occurs in a significant ortion of legal isutes, retrial negotiation aears emirically imortant. In environments with symmetrically informe contestants, costly trials will be avoie by settling, because there are no eviating views as to the outcome of the trial. However many isutes o go to trial. Thus, incomlete information also aears to be emirically significant. We allow for incomlete information about the funamental facts of the case as well as the costs of revailing at trial. The actual likelihoo of revailing at trial is itself enogenous. We also enogenize exeniture at trial, which is obviously necessary to answer any question about the effects of fee shifting on legal exenitures. Orley Ashenfelter an Davi Bloom 1993 rovie emirical evience that legal exenitures have a risoner s ilemma nature, thus aearing to be socially wasteful. Since fee shifting works like a subsiy to winning an a tax on losing, one might reasonably exect fee shifting to increase the stake of the trial an thus to encourage greater legal exenitures, than uner the U.S. system. We moel a trial as a isute over a fixe amount of money. This catures a variety of legal contests, incluing tort liability for injuries an eath an rivate antitrust suits. 4 Our moel emloys the enogenous legal exeniture moel of Luke Froeb an Bruce Kobayashi 1993 Ž F.K. as a subgame to the analysis. In this moel, there is a robability P that any given iece of evience favors the laintiff an evience that oes not favor laintiff favors the efenant. At trial, both the laintiff an the efenant choose the amount of evience they collect. It is assume that neither sie resents evience unfavorable to its cause. 5 Thus, a low value of P means that the laintiff must exen a large amount of resources to collect relatively little evience in favor of himself. The ecision of the court is stochastic but is more likely to favor the efenant the more the evience favors the efenant. We moel the isute resolution rocess as a simle two-stage extensive form game with the arties bargaining in the first stage an choosing the amount of trial effort in court in the secon stage. We consier the case of symmetric information first, to rovie a comarison to the more realistic case of asymmetric information. Since the funamental facts of the case are reresente by the robability of samling a unit of favorable evience, it is natural to efine a fair or unbiase settlement to be the rouct of the claim an the robability of evience being favorable to the laintiff. When this robability is commonly known in the negotiation stage as in the case of symmetric information, not surrisingly we fin that both arties refer a settlement. Uner the American rule, the settlement is unbiase, but uner the English rule the settlement favors the arty that has a higher robability of obtaining favorable evience, thus tilting the outcome away from a fair allocation. We then consier the case of asymmetric information, where the equilibrium an the resulting outcome for the two arties een crucially on who makes the offer in the first stage. To hase out the effect of the firstmove avantage, we let each arty make a take-it-or-leave-it offer once an then take the average ayoff of the two cases. 6 Now that the robability of evience being favorable to the laintiff is not observe ex ante, 3. The Sureme Court s first recognition of the American rule was its 1796 ruling on Arcambel. Wiseman. In 1848, the Fiel Coe of Civil Proceure eliminate all rovisions regulating the fees of attorneys an left the measure of fee comensation to the client-attorney agreement. See John Leubsorf 1984, for a etaile account of the history of American rule. 4. The moel is erhas more aroriate for civil cases than criminal cases. It is most alicable to situations where the efenant s behavior is reaily observe, but the legality of the behavior is isute. This situation commonly arises with mergers, reatory ricing, resale rice maintenance, an many liability suits. 5. This assumtion is a reflection of the aversarial system, a efining feature of the Anglo-American roceure. See F-K for a more etaile iscussion. It involves an issue of ethics that is beyon the scoe of this article shoul lawyers resent all the information to the court even though some of the information is not in the interest of their clients? The current governing law requires only that the attorney make accessible all the written ocuments. Although we o not mean to imly that lawyers are not scruulous, casual observation inicate that lawyers selom voluntarily resent information that is etrimental to their clients.

3 0 ECONOMIC INQUIRY the iea of a fair or unbiase settlement is moifie to be the exectation of the rouct of this robability an the claim conitional on the information available. The aroriate terminology in light of Bengt Holmstrom an Roger Myerson 1983 woul be the interim fair settlement, where fairness means as fair as ossible given all the available informationthe signals receive by the laintiff an the efenant. We ientify the situations where the arties reach a settlement or go to trial by constructing a unique searating equilibrium. It is only when both sies receive signals that their case is strong that a trial results. The American rule reners an interim fair allocation of the claim excet when a trial results. In that case, both arties are worse off than the interim fair allocation, with the ifference going towar the attorneys for legal services. With small amount of fee shifting, the legal cost will increase further. When signals for both arties are the same an are inicative of a small P, fee shifting enalizes the laintiff an benefits the efenant comare with the interim fair allocation. When both signals are inicative of a large P, fee shifting oes exactly the oosite. Finally, we investigate the overall ex ante welfare changes in the resence of fee shifting for all the arties involve in the lawsuit, the laintiff, the efenant, an the lawyers. The result much resembles that of the symmetric information caseit tilts towar the arty who has a higher robability of obtaining favorable evience. However, lawyers always benefit from fee shifting, because it increases the stake of the trial an thus intensifies the use of legal reresentation. The rest of the article is arrange as follows. Since the literature is voluminous, iscussion of the aer s relation to the revious stuies is relegate to the next section. Section III resents the moel of equilibrium trial effort uner symmetric information. Section IV analyzes the bargaining an 6. This aroach was mentione in Martin J. Osborne an Ariel Rubinstein Although the aroach is justifie for simlicity, it is quite common that the efenant retains much of the bargaining ower in the first stage. As Sier 199 ointe out, the efenant always refers to ay as late as ossible, an as a result iscounting alone oes not cause the elay to ermit signaling by the efenant. It is likely that the efenant woul reject any offer mae by the laintiff, rovie that the some offer will be accetable later. trial behavior uner the asymmetric information. The welfare imact of fee shifting on the three arties is investigate in Section V. The aer is conclue with a brief iscussion of the olicy imlications. All roofs are relegate to the aenix. II. LITERATURE REVIEW Litigation an retrial negotiation has been stuie extensively by many authors. The voluminous literature on fee shifting is rimarily focuse on the effects of fee shifting on the robability of settlement. Existing theoretical results are controversial an een on how the isute resolution rocess is moele. The early moels of William Lanes 1971, Richar Posner 1973, an John Goul 1973 Ž P-L-G moel. conclue that the risk attitue of litigants lays the major role in etermining settlement. The greater risk associate with the English rule leas risk-averse litigants to settle more cases. However, in a more general version of P-L-G, Steven Shavell 198 showe that uner certain circumstances the English rule leas to fewer out of court settlement. Avery Katz 1987, 1990 further showe that English rule increases litigation exeniture as well. John Hause 1989, on the other han, foun that the English rule is more likely to lea to settlement than the American rule in a moel with assumtions inconsistent with 7 rational beliefs. Kathryn Sier 1994 recently foun that the irect-revelation mechanism that maximizes the robability of settlement is one that resembles Rule 68 of the Feeral Rules of Civil Proceure, a variant of the fee-shift rule. Jennifer Reinganum & Louis wile 1986 argue that the equilibrium robability of trial is ineenent of the extent of fee shifting, because the trial robability is only a function of the total litigation costs, which were exogenous in that stuy. John Donohue 1991, a,b also reicte that both rules woul have ientical effects. Alying the Coase theorem, he argue that the roceural rules are irrelevant as long as the involve arties are free to sign a rivate contract secifying the Pareto otimal rule alicable to the court. 7. Among other things, Hause 1989 assume that the efenant believes the laintiff is less likely to revail than the laintiff believes.

4 GONG & MCAFEE: NEGOTIATION, LITIGATION, RULES 1 Thomas Rowe 1984 an Ronal Braeutigam, Ball Owen, an John Panzar 1984 argue that it is imossible to reict in general whether more or fewer claims woul be settle uner the English rule as oose to the American rule. In a literature review on the subject matter, Robert Cooter an Daniel Rubinfel 1989 wrote that the overall effect of ifferent rules cannot be etermine from theory alone. Emirical stuies have not reache a consensus on the effects of fee shifting on the robability of settlement. Gary Fournier an Thomas Zuehlke 1989 rovie evience that as the stakes in a trial are increase because of fee shifting, arties intensify their efforts to reach a settlement an are more likely to succee. But Ewar Snyer an James Hughes 1990 foun emirical suort for the reiction that fee shifting encourages arties to litigate rather than settle their claims. Finally, it might be worthwhile to oint out that among lawyers it is commonly believe that fewer lawsuits will occur when the loser ays more of the legal costs, although this belief has never been rigorously aresse at a theoretical level ŽCooter an Rubinfel Different theoretical oinions about the effects of retrial negotiation arise from ifferent moels emloye by istinct authors. 8 Extant theory can be usefully classifie into two categories. The first category involves moels with isagreement over the likelihoo of revailing at trial. The secon category imroves on the first by emloying rivate information to create isagreements, thus allowing for rational beliefs. The P-L-G moel an Shavell 198 exlaine costly litigation as the result of eviating views on the likelihoo of revailing at trial. Uner this framework, if the laintiff s execte gain in court is smaller than the 8. The entire litigation rocess is very comlex, involving many ecisions mae by the arties to the isute at several stages. Much of the litigation efforts an exenses are incurre in the retrial iscovery hase where relevant information about facts is iscovere an exert consultants are eose. The arties can still exchange offers for a settlement along the way. An exact moel of this rocess woul be extraorinarily comlex. Therefore, we will follow the framework of the revious stuies all of which assume that the negotiation only takes lace before any litigation exenses are incurre, i.e., before the iscovery rocess. Nalebuff 1987 suggeste that reiscovery negotiation woul be a better terminology. efenant s execte loss, there must exist a value in between that the two arties can agree on for settlement. If the laintiff erceives the robability of winning at trial to be sufficiently greater than the efenant s belief about the same robability, then the laintiff s execte gain coul well be higher than the efenant s execte loss. In this case, settlement terms woul not exist that both arties were willing to accet, an they woul en u in court. This tye of moel has been criticize for suffering from two rawbacks. First, the trial exeniture shoul be an imortant choice variable instea of being treate as fixe, an the arties robabilities of revailing in court ought to be a function of these trial exenitures. Both arties to the isute tyically know much more than the court about the facts, an collecting suorting evience an transmitting the information to the court is costly. In other wors, the trial stage is like a war of attrition, because the arties have ifferent beliefs about revailing in court an hence fail to reach an agreement in the negotiation stage. Recognizing this roblem, Braeutigam, Owen, an Panzar 1984, Charles Plott 1987, Cooter an Rubinfel 1989, an, more recently, Froeb an Kobayashi 1993 imrove on P-L-G moel by incororating the equilibrium level of trial exenitures, shoul a trial take lace. The secon rawback of the moel of eviating views, which none of the above stuies aresse, is the unrealistic assumtion that the robabilities of revailing at trial are both ifferent an common knowlege. At most, one of the arties has correct beliefs. Common knowlege of the beliefs shoul lea to agreement. Rationally eviating assessments shoul be attribute to arties holing rivate information on the merits of the case, rescribing the setting of the moel to be a game of incomlete information where strategic interaction, incluing transmission of rivate information, is taken into account. The secon category of litigation moels emloys rivate information to create isagreements, allowing for rational beliefs. Members of this class inclue I.P.L. P Ng 1983 Lucian Bebchuk 1984, Reinganum an Wile 1986, Barry Nalebuff 1987, Urs Schweizer 1989, an, more recently, Kathryn Sier 199, All but Schweizer

5 ECONOMIC INQUIRY 1989 an Sier 1994 moele the retrial negotiation stage as a stanar bargaining game with one-sie rivate information. The one-sie rivate information stuies may be further istinguishe accoring to whether the informe arty moves first to allow for transmitting of rivate information an accoring to the features of the environment that is rivately observe. 9 However, these stuies o not enogenize exenitures at trial an thus are silent on issues concerning effects of any arameters on the cost of trials. Another feature that istinguishes our work from the existing literature is the way we moel rivate information. Previous stuies usually treat the arties robability estimates of revailing at trial as rivate information, conitional on whether the efenant is guilty or not. In reality, civil lawsuits such as rouct liability, atent infringement, contract isute, antitrust lawsuit, an efamation cases are usually very comlex an often even the efenant oes not know whether he has one anything illegal. Rarely can the jury ientify an action on the art of the efenant that roves liability, nor can it rule out all actions that will exonerate him. Consier a rivate antitrust suit brought to block a roose merger. The law is concerne with the likely harm to cometition of the merger, which is harly a recisely formulate stanar. Consequently, both laintiff an efenant can offer evience, none of which is isutative, an about which reasonable eole might isagree. Similarly, the stanar in liability cases concerns reasonable use of the rouct, an this stanar has not even been constant over time Žsee Anrew Daughety an Jennifer Reinganum 1993, But this uncertainty oes not imly that the two arties ossess no information at all about their resective strength in court. What eter- 9. P Ng, Reinganum an Wile, an Nalebuff ha the informe arty move first. Bebchuck ha the informe arty move last. Reinganum & Wile s rivate information is the laintiff s amount of amage. Bebchuck an Nalebuff treate the efenant s robability assessment of revailing at trial as rivate information. P Ng has the efenant s knowlege of whether he has violate the law or not as rivate information. Schweizer s rivate information is the arties signals that link stochastically to the robabilities of revailing at trial, an thus is closest to our moel. mines whether the arty is likely to win in court is the robability that the funamental facts or evience turn out to favor his sie. The oulation of the otential evience is artly the consequence of the efenant s action. Hence, the robability istribution of evience ought to be conitional on his action, giving the efenant some information about the otential samle of evience. On the other han, the laintiff, after having use the rouct an incurre the amage, shoul also have information as to the likely outcome of evience collection. In light of the above consierations, we let the rivate information be signals the two arties have receive before the negotiation, an these signals are correlate with the funamental facts of the case, which are not observe until trial. Note the subtle ifference between our treatment of rivate information with Schweizer s In his moel, the rivate signals are correlate with the arties estimates of the likelihoo of revailing at trial, while our rivates signals are correlate with the robability that a ranom samle evience from the funamental facts will turn out to be in one arty s favor. Thus, in the resent stuy, rivate information is actually about the costs of revailing at trial, an the robability of revailing is enogenous. III. THE CASE OF SYMMETRIC INFORMATION In this section, we consier the case of symmetric information, where the facts of the suit reresente by the robability of obtaining favorable evience are common knowlege. To cature the imact of fee shifting, we suose that the revailing arty can recover a certain ercentage 0,1. of his own litigation costs from the losing arty. Uner the American rule, 0, an uner the English rule, 0. For ractical uroses, we may think of as quite small. One may question whether a small reresents the true English rule. Werner Pfennigstorf 1984 imlies that a ure form of the English rule is a rarity on the books for the following reasons. In Euroe, not all litigation costs are reimbursable an even the reimbursable costs are limite an regulate. For examle in Englan an Canaa, the costs to be reimburse are efine as inclu-

6 GONG & MCAFEE: NEGOTIATION, LITIGATION, RULES 3 ing only necessary an roer costs. In ractice, the court officials, calle taxing masters, ecie the amount, an it is usually absurly small comare with the total amount charge by the attorney. 10 Suose that the laintiff claims that the laintiff has incurre a loss an emans a comensation of a certain amount. Without loss of generality, the amage is assume to be one ollar. Let P enote the robability that a ranomly samle unit of evience is favorable to the laintiff. Since the efenant has contrary interest against the laintiff, the robability that evience is favorable to the efenant is 1 P. 11 Both the efenant an the laintiff ay the litigation cost c for each unit of evience obtaine. The litigation costs can be thought of as the costs of investigating the case, eosing witnesses, hiring exerts, an attorney fees. Because the roortion of evience favoring the laintiff is P, we efine a fair settlement to be a ayment to the laintiff of P. There are two justifications for this assumtion. One may think of the evience collecte as imlying that laintiffs in ientical circumstances were actually injure with robability P, so the efenant s average liability was P. Alternatively, one coul consier P to be the roortion of the claime amage that the laintiff actually suffere. As the stanar aroach of analyzing a two-stage extensive form game, we first look at the subgame of litigation exenitures at trial. The subgame is essentially a reuce form of Froeb an Kobayashi s 1993 Ž F-K. game of the search for evience, which moels trial exenitures behavior in an economical an tractable manner. Both laintiff an efenant are risk neutral. The choice variable for both arties is the amount of evience to collect. Since P is known when the 10. See Kritzer McAfee ha a ersonal corroboration in Cana. Litigation costs exressly exclue contingent fees. There is substantial agreement in Euroe that contingent fee agreements between the attorney an the client are unethical. Regaring the fact that contingent fee ractice is so wiely accete an involves a substantial amount of money in the Unite States, it is reasonable to exect that only a small ortion of the litigation costs woul be recovere by the revailing arty ha the English rule been written into the American juicial roceural statutes. 11. This is without loss of generality, for if a fraction of the evience favors neither sie, we may ajust the cost to be the cost of obtaining relevant evience by iviing by 1. two arties enter into the trial an the arties only islay to the juge the evience favorable to themselves, it is equivalent to letting the amount of the favorable evience be the choice variable, which is enote by E for the laintiff an E for the efen- ant. The costs of collecting E an E will be ce P an ce Ž 1 P., on average. Each arty s robability assessment of the likelihoo of revailing at trial shoul be an increasing function of the favorable evience resente by himself an a ecreasing function of the unfavorable evience resente by his oonent. We follow F-K an assume the juge or the jury uses a simle rule base entirely on resente evience by both arties to ecie the case, such that the robability of ith arty s revailing at trial is Ž. 1 Ei Ei E j, i j, i, j,. When both terms in the quotient are zero, we assume the robability to be 1, although Ei 0is not equilibrium behavior for at least one arty regarless of the osite robability. 13 Given the environment, the arties face the following objective functions in the subgame Ž. 1 E Ž. max E E E E E c 1 P ž E E / E E c 1 PE E E Ž. min E E E E E c 1 1 Pž E E / E E c P E E Ž. The first equation in 1 is the laintiff s execte gain from the trial. It is the ex- 1. U.S juges tyically instruct juries to consier only the evience resente at trial.this jury technology is a secial form use by Plott The nature of the trial can be broaene to cases where the jury may choose the rewar E Ž E E. for the laintiff, because of the osite risk neutrality.

7 4 ECONOMIC INQUIRY ecte awar by the jury minus his own litigation costs, lus the recovery of a ercentage of it if he wins, an minus a ercentage of the execte litigation costs sent by his oonent if he loses. The secon equation reresents the execte ayment by the efenant. We assume that the two arties lay the unique ure strategy Nash equilibrium to this game. The aenix shows that the solution to Ž. 1 is Proosition 1: The Nash equilibrium of the litigation exeniture game has a unique solution that Ž. 1 E PŽ 1 P. c Ž 1 P. Ž P 1. 1 E PŽ 1 P. c where 1 Ž 1 P., Ž P 1. Ž. Ž. 1 4 P 1 P 0. Several observations immeiately follow. First it is obvious from Ž. 1 that the efenant s equilibrium execte ayment always excees the laintiff s execte gain, resulting in the existence of the cooerative surlus to be istribute between the two arties. Therefore, a settlement is always referre to a trial. How the surlus is slit is also straightforwar. Since both arties are symmetrically informe about P, no strategic action is ossible. Uner this context, no form of bargaining is more natural than the Nash bargaining concet, evenly iviing gains from settlement Žfor a justification, see Ariel Rubinstein Cooter an Rubinfel s Ž moel of symmetric information also uses the Nash bargaining concet. It can be verifie from Ž. that in the secial case of 0, Ž 3. E E P Ž 1 P. c Ž. P 1 P c Ž 4. P P P, while in the limiting case of 1, they een on whether P is less, equal or greater than 1: 0, if P 1 Ž. 18c 1 a,if P 1 Ž 5. E P Ž 1 P. c Ž P 1,. if P 1 P Ž 1 P. cž 1 P., if P 1 E 18c Ž 1.,if P 1 0, if P 1 PŽ 1 P.,if P 1 Ž 1. 4Ž 1., Ž 6. if P 1 1, if P 1 0, if P 1 Ž 3. 4Ž 1., if P 1 P Ž P 1,ifP. 1 Thus, uner the American rule where 0, the execte awar by the jury, E Ž E E. is exactly P, the robability that evi- ence turns out to be favorable to the laintiff. When 1, which is an extreme case of the English rule, the awar goes to 0, 1, or 1, eening on whether P is less than, equal to or greater than 1. Note that the Nash bargaining result uner 0 also yiels a settlement of the amount P, which is an unbiase an fair allocation of the claim. Hence, we conclue that uner symmetric information the American rule generates the fair settlement. Also note that if the arties o go to trial, uner the American rule the laintiff wins the trial

8 GONG & MCAFEE: NEGOTIATION, LITIGATION, RULES 5 with robability P an receives an average comensation of P from the efenant. The simle jury technology secifie thus elivers a fair allocation of the claim uner the American rule, a result rove by F-K. In the case of symmetric information, costly court battles can be avoie by resorting to mutually beneficial negotiations. Trials, however, have the risoner s ilemma nature suggeste by Ashenfelter an Bloom 1993, in that arties exen resources to obtain the commonly known fair ivision. Now consier the imact of switching towar the English rule. Denote the Nash bargaining settlement as a function of by SŽ. Ž. Ž.. First observe that when P 1, in the limiting case of 1, SŽ. PŽ 1 P. is negative. In general, for large enough, a laintiff with a weak case will encounter negative gains from the retrial negotiation, which will certainly revent him from initiating the lawsuit in the first lace. 14 As a result, in the full-information case, fee shifting tens to iscourage frivolous lawsuits, suorting Newt Gingrich s claim. Whether a settlement favors the laintiff or efenant is etermine by the sign of SŽ.. If it is ositive, then the settlement is sai to favor the laintiff. If it is negative, the settlement favors the efenant. First, we establish a lemma. It says that, as increases, the arty that sens more on attorney fees ŽcE P an ce 1 P., e- ens on whether P is greater or less than 1. When P is greater than 1, meaning that the laintiff has a stronger case, his legal exeniture will excee that of the efenant. When P is less than 1, the oosite is true. Furthermore, the ga is an increasing function of. Ž 7. Lemma 1: E E P 1 P ž / 0, if P 1 0, if P 1 0, if P 1 Ž. Now consier the sign of S. Uon ifferentiating with resect to an using lemma, we show in the aenix that 0, if P 1 Ž 8. SŽ. 0, if P 1 0, if P 1 Thus, as eviates away from zero, the amount of the settlement increases if the robability of the ranomly samle evience being favorable to the laintiff is higher than being favorable to the efenant. Figure 1 lots SŽ. against P for sev- eral values of. As increases, SŽ. shifts ownwar for P less than 1 but uwar for P greater than 1. In other wors, as the American rule switches towar the English rule, the settlement rewars more the arty who has a higher robability of obtaining favorable facts suorting the case or, equivalently, a lower cost of resenting a stronger case. We summarize the main results of this section in the following roosition. Proosition : In the case of symmetric information, the arties to the isute always refer the settlement to the trial. Uner the American rule, the Nash bargaining solution generates the fair settlement. As the roceural rule switches towar the English rule, the settlement rewars the arty with a FIGURE 1 Equilibrium Settlement with Symmetric Information 14. For sufficiently small P, SŽ. is negative if an only if 3. That is SŽ. P Ž 3.. P 0

9 6 ECONOMIC INQUIRY higher robability of obtaining suortive evience. Proosition is very intuitive. When the facts of the case are clear to the laintiff an the efenant, an they lay equilibrium strategies of trial exenitures, they shoul have the same robability assessment of the comensation ayment orere by the juge. As increases, the relative gain to winning increases an arties increase exenitures at trial, with the sie more likely to win in equilibrium more heavily subsiize. At the bargaining table before the trial, the arty less likely to win has a isavantage because his oonent knows he is going to ay a large amount at trial. Therefore, as increases, the settlement favors more the arty who has an avantage in terms of fining suortive evience. This oint can be reaily enforce by the revious observation that when 1, the jury awar goes to 0, 1, or 1, eening on whether P is less than, equal to, or greater than 1, suggesting that when P 1, the laintiff s share falls an when P 1, the laintiff s share rises. The symmetric case where settlement is always successful is unrealistic, because no trials occur. Difference of beliefs about revailing at trial can revent the arties from reaching a settlement. If they o not iffer on assessments of revailing at trial as in the case of symmetric information, they shoul reach an agreement easily. To moel the failure of retrial negotiation, we now consier asymmetries of information. VI. THE CASE OF ASYMMETRIC INFORMATION Asymmetries of information may cause the arties to go to trial rather than settle. We examine the simlest moel with twosie asymmetric information. In this moel, the likelihoo that evience favors the laintiff, P, is known to neither arty an is viewe ex ante as a uniform raw from 0,1. Both arties rivately an ineenently observe a single raw from the evience. Evience favors the laintiff with robability P, an we enote this outcome by 1 an evience favoring the efenant by 0. The Bernoulli ranom variables for the laintiff an efenant are enote by X an Y, resectively. 15 The environment is common knowlege to the arties, who are only uncertain about their oonent s information. From the laintiff s ersective, 1 is a strong signal because it brings the goo news that a higher P is likely, whereas 0 is a weak signal. For the efenant, 1 signals a weak case because a higher P is ba news to the efenant, whereas 0 signals a strong case. The retrial negotiation game can take ifferent extensive forms. In our case, the equilibrium eens on who makes the offer in the negotiation stage. We first assume that the efenant makes a take-it-or-leave-it offer an, if the laintiff rejects, a trial results. 16 The case where the laintiff makes the offer is similar in nature. It is quite common that the efenant retains much of the bargaining ower in the first stage. The timing of negotiation is as follows. Both arties receive their rivate signals first an then negotiate. Shoul the arties fail to settle, legal exenses will be incurre as both arties reare for trial. In the rocess, P is reveale, an thus the trial outcome corresons to the full-information case moele in the revious section. This moel has the avantage of simlicity an the isavantage that arties woul actually refer to settle once P is reveale. We consier that our moel catures many of the salient asects of the legal system, incluing the enogenous legal exeniture an interim isagreements about the likelihoo of revailing at trial, in an internally consistent framework. The moel fails to cature the effect of litigants learning about the likelihoo of re- 15. This moel is closely relate to that of Schweizer 1989, although Schweizer s moel i not enogenize effort at trial. It woul obviously be esirable to enogenize the renegotiation evience collection, although the resulting comlexity is aunting. 16. Hause 1989 criticize the iea of one arty making a take-it-or-leave-it offer, arguing that it is inconsistent with general rule that settlements can be offere or accete at any oint in the negotiation rocess. However, as Sier 199 ointe out, the efenant always refers to ay as late as ossible, an as a result iscounting alone oes not cause the elay to ermit signaling by the efenant. That is, the efenant woul reject any offer mae by the laintiff, rovie that the same offer will be accetable later. Therefore cases are likely to be settle in the last minute on the courthouse stes Ž the ealine effect.. In fact, Williams 1983 foun than 70% of all civil cases in Arizona are settle in the last 30 ays before the trial, an 13% were settle on the ay of the trial.

10 GONG & MCAFEE: NEGOTIATION, LITIGATION, RULES 7 vailing at trial through the course of evience collection, an effect rule out by ositing the common knowlege of P at trial. The significance of such learning is left for future research. When the arty ecies what offer to make an similarly when the other arty ecies whether to accet or reject the offer, they must consier the execte equilibrium outcome in the trial subgame. It is straightforwar to comute the execte equilibrium outcome for the two arties uner ifferent signal configurations when is close to or equal to These values are resente in Table I. We first consier the equilibrium uner the American rule. The ayoff uner the American rule corresons to the ayoff in the above table by setting 0. Because P is assume to be uniform, the osterior robability of a arty that the oonent has a ifferent signal is 13 an the robability of same signal is 3. We now consier the equilibrium of this two-stage game. The natural equilibrium concet for games with incomlete information is Davi Kres an Robert Wilson s Ž 198. notion of sequential equilibrium. In the more realistic scenario where the efenant makes the offer, the sequential equilibrium is a set of strategies an beliefs such that, at the beginning of the first stage of the game, the efenant s offer is otimal an, at the beginning of the secon stage, the laintiff reacts otimally given the beliefs of the efenant s signal tye, 17. To erive the ayoffs when 0, it is reasonable to aroximate those ayoffs using the first-orer Taylor exansion EŽ Ž. x, y. EŽ Ž 0. x, y. i i EŽ Ž. x, y. 04 i. Using Mathematica, it can then be shown that Ž. PŽ 0 1 P.Ž1 3P 6P. an Ž. PŽ 0 1 P.Ž 9P 6P.. It is straightforwar to comute the ayoffs using the aroriate conitional ensity functions of P. Note that the conitional ensity functions of P are fž P0, Ž P., fž P0, 1. 6PŽ 1 P. Ž. an f P1, 1 3P. an the beliefs are obtaine from equilibrium strategies an the observe offer using Bayes rule. In other wors, the efenant s offer takes into account the effect of his offer on the laintiff s belief an action, an the laintiff s ecision whether to accet is also otimal given his signal an the osterior belief about the efenant s signal tye. We restrict attention to ure strategy equilibria. With only two tyes of rivate information, ure strategy sequential equilibria will be either fully searating or ooling. A searating equilibrium is an equilibrium in which the two tyes of the efenant choose two ifferent offers in the negotiation stage. In a ooling equilibrium, the efenant s offer will be the same regarless of the signal. The ooling equilibrium is rather simle. Since the efenant s offer oes not transmit any information, it turns out that the ooling equilibrium is for the efenant to offer 1, an the laintiff always accets this offer. 18 We will concentrate on the searating equilibrium where the case will en u in court with ositive robability, which better accounts for actual trial exerience. In a searating equilibrium, for each value y of the two ossible signals, the efenant makes an offer S Ž y.. Several authors have shown that in a bargaining game of two-sie information, the receiver of the offer uses a cut-off strategy Žsee, for examle, Peter Cramton If his cut-off value excees the offer, he will accet it an reject it if otherwise. In the context of our moel, the laintiff aots this cut-off strategy S Ž x, S., which mas his rivate signal an the offer 18. Such an equilibrium where the arties can always reach an agreement is calle the efficient equilibrium by Schweizer He further showe that the efficient ooling equilibrium oes not survive Jeffrey Banks an Joel Sobel s 1987 universal ivinity refinement concet. Reinganum an Wile 1986 rule out the ooling equilibrium in their moel using the same concet. TABLE I Equilibrium Outcome When Is Small ( ( ) ) ( ( ) ) X Y E x, y E x, y

11 8 ECONOMIC INQUIRY by the efenant to 0,1. Because of the first orer stochastic ominance roerty of P an the incentive comatibility constraint, we show in the aenix that both arties strategies are strictly increasing in their receive signals in a searating equilibrium: Lemma : In a searating equilibrium, S Ž1, S Ž y.. S Ž0, s Ž y.., y 0,1, an S Ž. 1 S Ž. 0. Lemma is helful with constructing the searating equilibrium. Since there are only two tyes for the efenant, his signal is fully reveale to the laintiff. Thus, the Bayesian uate for the laintiff is trivial. The laintiff s cut-off value will be the execte ayoff at trial conitional on his signal an the efenant s signal. In the moel, the laintiff s belief after observing an offer is that any offer higher than the equilibrium offer from a strong efenant reveals that the efenant is weak. We will show later that such belief is consistent with the laintiff s otimal strategy. Having observe that, we state the following roosition. Proosition 3: Uner the American rule an when the efenant makes the offer, the searating equilibrium is for the efenant to offer a tough settlement of 110 when he receives a strong signal an offer a generous settlement of 35 when he receives a weak signal. Seeing a tough offer from the efenant, the laintiff accets it if he receives a weak signal, an goes to trial if he also receives a strong signal. Seeing a generous offer, the laintiff always accets the settlement regarless of the signals. Furthermore, the searating equilibrium is unique. Proosition 3 is quite intuitive. In orer for the efenant to searate on tye, the offer from the strong efenant, which is lower, must be rejecte by the strong laintiff. Thus, the strong efenant makes the minimal offer accetable to the weak laintiff. The weak efenant, in contrast, makes an offer just accetable to the strong laintiff so that it is always accete. When the laintiff makes the offer, the equilibrium has the same flavor: Proosition 4: Uner the American rule an when the laintiff makes the offer, the searating equilibrium is for the laintiff to eman a large settlement of 910 when he receives a strong signal an eman a small settlement of 5 when he receives a weak signal. Seeing a large eman from the laintiff, the efenant accets it if he receives a weak signal, an goes to trial if he also receives a strong signal. Seeing a small offer, the efenant always accets the settlement regarless of the signals. Furthermore, the searating equilibrium is unique. Next we consier the equilibrium uner the switch towar the English rule. Here we will resort to exact numerical comutations in our investigation. The ifficulty that arises is that the conitional exectation of the equilibrium ayoff at trial when 0isa comlicate function of Žsee eq... Alying the same logic use in the roof of roosition Ž. 3, it is shown in the aenix that the searating equilibrium of roosition Ž. 3 is entirely robust to the erturbation of the equilibrium ayoff at trial cause by the switch towar the English rule. The maximal level of fee shifting that suorts the searating equilibrium is 0.8, a substantial amount of inemnity that is tyically unrealistic uner most juiciary systems. For exceeing 0.8, the weak laintiff will fin it unrofitable to initiate the lawsuit. The searating equilibrium is without surrise similar to the case uner American rule we have shown in that it is only when the both sies feel they have a strong case that a trial results. Proosition 4 is also robust to large close to 1, when the roceural rule moves towar the English rule. In fact, it is more robust than the case when the efenant makes the offer, because we o not even have to consier the rationality constraint that the laintiff is willing to file a lawsuit. The nature of fee shifting is that it amlifies the first move avantage, so as long as the laintiff is willing to sue for 0, which is obviously true, he will file a lawsuit for all. We ocument the results uner the English rule in the next roosition. Proosition 5: Uner the English rule where the extent of fee shifting is less than 8% an when the efenant

12 GONG & MCAFEE: NEGOTIATION, LITIGATION, RULES 9 makes the offer, the searating equilibrium is for the efenant to offer a tough settlement of EŽ Ž. 0, 0., when he receives a strong signal an offer a generous settlement of EŽ Ž. 1, 1. when he receives a weak signal. Seeing a tough offer from the efenant, the laintiff accets it if he receives a weak signal an goes to trial if he also receives a strong signal. Seeing a generous offer, the laintiff always accets the settlement regarless of the signals. When the laintiff makes the offer, the searating equilibrium is for the laintiff to eman a large settlement of EŽ Ž. 1, 1., when he re- ceives a strong signal an eman a small settlement of EŽ Ž. 0, 0. when he receives a weak signal. Seeing a large eman from the laintiff, the efenant accets it if he receives a weak signal an goes to trial if he also receives a strong signal. Seeing a small eman, the efenant always accets the settlement regarless of the signals. Furthermore the searating equilibrium in each case is unique. V. WELFARE EFFECT OF FEE SHIFTING Base on the results in the revious section, we are now able to investigate the imact of the fee shifting on the attern of the settlement. We comare the results uner the English rule with a small extent of fee shifting to the fair allocation in the following tables. The outcome in the following tables are base on who makes the offer an are interim, meaning that the exectation of the ayoffs shoul be taken conitional on the known signals they receive resectively. A strong signal receive by the laintiff is one reresente by 1, while a strong signal receive by the efenant is one reresente by 0. The last row of each table reresents the ex ante exectation, rior to the realization of the signals. Table IV takes the average of the results in Table II an Table III, with the interretation that half of the time the efenant makes the offer an half of the time the laintiff makes the offer. In terms of the attern of the settlement, we can see from Table IV that uner the American rule where 0, both arties get a fair ayoff for all signal configurations excet when both arties receive signals in- TABLE II Interim Equilibrium Outcome When the Defenant Makes the Offer x y Fair P Plaintiff s Earning Defenant s Payment Legal Costs Average TABLE III Interim Equilibrium Outcome When the Plaintiff Makes the Offer x y Fair P Plaintiff s Earning Defenant s Payment Legal Costs Average

13 30 ECONOMIC INQUIRY TABLE IV Average Interim Equilibrium Outcome x y Fair P Plaintiff s Earning Defenant s Payment Legal Costs Average icative of a strong case, in which instance they go to trial. The imact of the fee shifting is much like that in the symmetric information caseit tens to shift each arty s welfare away from the fair allocation while increasing the legal cost. In two cases where the arties receive the same signals, the English rule shifts the settlement away from the fair allocation. When these same signals all inicate that a small P is likely, fee shifting further enalizes the laintiff an favors the efenant. The oosite is true when both signals inicate otherwise. In one instance where both arties receive strong signals an go to trial, the English rule results in an unambiguous increase in legal fees, further moving both arties welfare in court away from the fair allocation. This is summarize in the next roosition. Proosition 6: The American rule reners an interim fair allocation excet when a trial results. When both arties signals favor the efenant, the infinitesimal move towar the English rule further shifts the settlement away from the interim fair allocation in favor of the efenant. When both arties signals favor the laintiff, the switch towar the English rule shifts the settlement away from the interim fair allocation in favor of the laintiff. When both arties receive strong signals in which case they go to trial, the infinitesimal move towar the English rule increases the legal fees. The likelihoo of both litigants obtaining 0is13, similarly for both obtaining 1, while either a Ž 0, 1. or Ž 1, 0. combination has a robability of 16. This allows us to comute an overall average, resente in the final row of each table. The assignment of the bargaining ower to the efenant as in Table II, in the form of a take-it-or-leave-it offer, results in an avantage, in that, for 0, the efenant on average ays less than the fair allocation. However, when averaging over the two cases where each one has a chance to make the offer, both arties are worse off than the interim fair allocation, because they have to ay for legal services for all signal configuration. The ifference between the two outcomes is slit between the attorneys reresenting each arty. Suose that face a fixe cost Ž which varies across laintiffs. of initiating a lawsuit. This fixe cost is ai after the laintiff learns his signal but before the efenant learns his signal. A small increase in, starting at the American rule, benefits strong Ž X 1. laintiffs. Thus, we woul exect more lawsuits to be brought by laintiffs with strong cases. In aition, the execte ayoff for weak laintiffs, which is 13 0, is ecreasing in. Thus, we fin suort for Gingrich s 1994 claim that the loser-ays system will iscourage frivolous lawsuits while encouraging serious ones. Next we consier the overall ex ante welfare imact of fee-shifting on all the relevant arties in a lawsuit, the laintiff, the efenant an the reresenting attorneys. Again we first consier the case where the efenant makes the offer. Figure is a lot of the ex ante ayoff to the laintiff U Ž, P., the ex ante ayoff to the efenant U Ž, P., an exenitures at trial against P when 0. U Ž, P. an U Ž, P. are comute conitional on P but average over the signals that can arise, using the correct robability Že.g. ProbŽ X 1. P Conitional on P, U Ž,. Ž 1 P. S Ž. 0 Ž. Ž. Ž. Ž Ž.. P 1 PS 1 P 1 P E 0, 1 P S Ž. 1, an U Ž, P. Ž 1 P. S Ž. 0 PŽ 1 PS. Ž. 1 PŽ 1. Ž Ž.. P E 0, 1 SŽ. 1.

14 GONG & MCAFEE: NEGOTIATION, LITIGATION, RULES 31 FIGURE Equilibrium with Asymmetric Information Ž Defenent Makes Offer. In Figure, the two to curves reresent U Ž 0, P. an U Ž 0, P. resectively. Naturally, U Ž 0, P. is above U Ž 0, P.. The ifference between them is the legal costs, which are reresente by the lower arabola. The ifference is wiest as P gets close to 1. This is the best scenario for lawyers as a grou, since no arty has a ominant case an both sies sen heavily on legal service. As P aroaches 0 or 1, the two curves of the litigating arties intersect, inicating the limiting situation of settlement. The reason is straightforwar. When P aroaches 0, the robability that the laintiff receives a strong signal goes to 0. So it is almost sure that the laintiff will accet the settlement. When P aroaches 1, the robability that the efenant receives a strong signal goes to 0. Then it is almost sure that the efenant will always offer a sufficiently high offer to secure a settlement. Recall that the ex ante fair allocation of the claim shoul be P, reresente by the 45 line in Figure. Roughly seaking, the laintiff gets more than he shoul when P is small but less than he eserves when P is large. Similarly, the efenant ays more than he shoul for small P but less than he shoul for large P. In aition, the interval of P in which the laintiff enjoys a surlus over the fair allocation is substantially smaller than that of P in which the efenant enjoys a surlus. Therefore, without fee shifting, the efenant is in a relatively avantageous osition. The avantage stems from the fact that the efenant is assume to be the first mover in offering a settlement in the negotiation stage. As a function of P, our comutation shows that fee shifting benefits the efenant most of the time an increases legal fees while always hurting the laintiff. As argue above, since the efenant makes the offer, he is in an avantageous osition in the absence of fee-shifting. When fee-shifting is introuce, it essentially amlifies this effect, making the laintiff even worse off with the surlus slit between the efenant an the laintiff attorneys. Figure 3 lots the equilibrium outcome changes as increases to 0.1. Figure 3 also resents the effect of fee shifting on the total litigation costs, which increases as increases. Provie that is not increase beyon 0.8 Žat which oint the searating equilibrium breaks own., a move towar the English rule increases legal exenitures. Within the context of the moel, there is no change in the robability of a trial, given the assume iscrete nature of information available to the arties. In the case where the laintiff makes the offer, similar grahs can be rawn. Figure 4 is a lot of the ex ante ayoffs for the three arties uner American rule. Figure 5 recors the changes as a result of fee shifting of 0.1. This time the laintiff gains most of the time, an the efenant incurs loss. The combine legal fees still increase, ai mostly by the efenant. FIGURE 3 Equilibrium Changes When 0.1 Ž Defenent Makes Offer.

15 3 ECONOMIC INQUIRY FIGURE 4 Equilibrium with Asymmetric Information Ž Plaintiff Makes Offer. FIGURE 6 Equilibrium When Plaintiff or Defenant Makes Offer Half of the Time FIGURE 5 Equilibrium Changes When 0.1 Ž Plantiff Makes Offer. FIGURE 7 Equilibrium Changes When 0.1 Ž Each Makes Offer Half of the Time. Finally, we take the average ex ante outcome when half of the time the efenant makes the offer an half of the time the laintiff makes the offer. Figure 6 gives the outcome for each arty as a function of P. Figure 7 reresents the change when moves to 0.1. Clearly, the efenant gains for small P, an the laintiff gains for large P. However, there exists an interval of P in which both the litigating arties incur loss. Attorneys, on the other han, enjoy a win-win situation. The total legal cost will increase as a result of fee shifting. If laintiffs also face a fixe cost of bringing suit that varie across laintiffs, then an increase in fee shifting woul reuce the number of suits, while increasing average legal exenitures on the suits that were brought, resulting in an overall ambiguous effect on the total exeni- tures on legal services an the eman for attorneys. To summarize, we have Proosition 7: A small increase in fee shifting reuces the laintiffs ex ante earnings for small P, increases the efenants execte ayment for large P, an increases the total legal fees for all P. VI. CONCLUDING REMARKS We investigate the effects of charging losers at trial with a ortion of the winner s legal costs, or fee shifting, on the attern of settlement, in a moel with enogenous exeniture at trial. In articular, we comare the American rule, where arties ay their own legal costs, to the English rule, where the loser ays a ortion of the winner s cost. Unlike much of the large literature on this toic, we are concerne with the fairness

16 GONG & MCAFEE: NEGOTIATION, LITIGATION, RULES 33 of the outcome an the exenitures emloye to achieve that outcome. Uner comlete information, the aversarial system without fee shifting rouces a reasonable outcome in the following sense. The fair awar excees the amount that the laintiff receives from trial but is less than the amount that the efenant ays. Moreover, the outcome is such that Nash bargaining with trial as a threat rouces exactly the fair allocation. Consequently, the attern of settlements uner comlete information an without fee shifting is as goo as it can ossibly be. It is no surrise, then, that fee shifting istorts the outcome away from the fair outcome, an that fee shifting favors the arty with the strong case, that is, the arty with the lower cost of roviing evience to the court. Not surrisingly, when the facts of the case are the most uncertain, both arties exen the greatest resources at trial, an these exenitures are increase by fee shifting. Many authors have note that comlete information aears to be inconsistent with the fact that many lawsuits are actually settle in court. Because trials are costly, if there is agreement about the unerlying facts of the case, there shoul always be a mutually avantageous settlement. Consequently, asymmetries of information aear to lay a significant role in legal isutes, with the result that retrial negotiation oes not invariably lea to settlement. Asymmetries of information ten to favor the arty with the weak case so that the laintiff receives more than the fair allocation when ranomly selecte evience favors the efenant an less when ranomly selecte evience favors the laintiff. Thus, the remarkable virtue of the American rule in romoting a fair outcome uner comlete information is lost when the aversaries are asymmetrically informe. In this circumstance, fee shifting can either enhance or etract from the fairness of the allocation, although on balance it makes the arty who makes offers better off at the exense of his oonent. By amlifying the stake of the trial an thus the ayoff of the litigating arties, fee shifting hels the offerer. However, legal exenitures rise with fee shifting, at least until the fee shifting becomes so extreme that the arties settle rather than risk ruinous legal exenitures at trial. Our numerical comutation with a uniform istribution shows such event only occurs when 0.8, a rare level of fee shifting in reality. To balance the first mover avantage, we take the average ayoff of the two cases where each arty has a chance to make the offer. In that case, fee shifting functions much like uner the symmetric information it enalizes the arty with a weak case an subsiizes the arty with a strong case. However, the ifference lies in the fact that, for borer-line cases where P is aroun 0.5, both arties lose with the increase cost of litigation. The moel has two unesirable features. First, the information available to the arties at the time that settlement is iscusse shoul be enogenous. That the information available rior to trial is not enogenous is efensible on the grouns that iscovery is an exensive rocess an that legal costs are alreay sunk at the oint that the arties begin to collect information. However, the assumtion of two signals is clearly inaequate to answer questions about the likelihoo of settlement, as a searating equilibrium will generally entail going to court when both arties erceive themselves to have strong cases. Further refinement of the signal sace woul ermit an analysis of the effect of fee shifting on the likelihoo of settlement. The secon unesirable feature of the moel is the take-it-or-leave-it offer mae by the litigating arties, although a justification of this assumtion is rovie by Sier 199. It woul be referable to have a more comlicate moel in which staning offers are mae by laintiff an efenant as a function of the information they have iscovere to ate. The comlexity of such a moel is aunting, an it woul be reasonable, erhas, to ignore the enogeneity of legal exenitures in a first attemt to solve such a moel, which is inconsistent with our goal of characterizing the effect of fee shifting on the attern of settlement but of interest in its own right. APPENDIX Proof of Proosition ( 1 ). The first-orer conition uner Ž. 1 yiels that the air of equilibrium trial efforts is

17 34 ECONOMIC INQUIRY governe by the following system of equations Ž A1. E c Ž E E. P ž / 1 E 1 1 c P E E P 1 P 0 E c Ž E E. 1 P 0. ž / ž / 1 E 1 1 c 1 P E E 1 P P ž / A few maniulations shoul solve the above system. First, aing u the two equations an multilying both sie by E E, we get ž / 1 1 ž P 1 P / Ž 1. c Ž E E. P 1 P c Ž E E Multilying both sies by PŽ 1 P. c an arranging terms, we have one linear equation Ž A. 1 Ž 1 P. E Ž 1 P. E 1 Ž. P 1 P. c Moving secon terms of the two equations in Ž A1. to the right-han sie an then iviing the first equation of Ž A1. by the secon one, we have 1 Ž E E. E c P 1 1 ce ž P 1 P / 1 Ž E E. E c 1 P 1 1 ce ž 1 P P/ 1 P P. Arranging terms gives Ž A3. 1 PŽ 1 P. PE Ž 1 P. E c Ž 1 P. PŽ E. Ž 1 P.Ž E.. Ž. Ž. Plugging A into A3 yiels Ž A4. Ž 1.Ž 1 P.Ž E. Ž 1. PŽ E. Ž P 1. E E 0. Solving for the ositive E in terms of E in Ž A4. gives another linear equation Ž P 1. Ž A5. E E, 1 Ž.Ž 1 P. where 1 8 P Ž 1 P. Ž. 4 P 1 P. Now Ž A. an Ž A5. constitute a linear system of two equations in two unknowns. Using the usual metho of substitution yiels Ž.. To show the necessary conitions are also sufficient, multily Ž. Ž. E by a ositive term E E to obtain E cpž 1.Ž E E. ce Ž 1P 11 P.. This term is obviously ecreasing in E, an negative if E is sufficiently large. Thus, either Ž Ž. E. 0 for all E where Ž Ž. E. E 0 0, or Ž Ž. E. 0at E 0, an ositive u to some oint where it satisfies the first-orer conition an after which is negative. Thus, Ž. E can only change signs from ositive to negative, an eventually negative. A similar result hols for E. Therefore the solution to the first-orer conitions is an equilibrium. To show it is also the unique equilibrium, we also nee to check ossible bounary solutions where E an E might be zero. But first note that from first-orer conitions only one of them can be zero. Now suose E 0. From the cost function in Ž. 1, we can see that the efenant can sen a minimum amount, say E, where is a arbitrarily small ositive number. But if E, E 0 can not be an equilibrium strategy. This is because lugging E 0 an E back into Ž. E gives Ž 1. cžž 1P. Ž 11 P.., which is always ositive if 1, as is close to 0. Therefore, the laintiff can increase rofits by sening more. So it can not be an equilibrium. The same can be sai about E 0 an E. As a result, Ž. is the unique equilibrium. Using lim Ž 1. 4PŽ 1 P. 0, it is straightforwar to verify Ž. 3 an Ž. 4. Since E can e erive similarly, we only erive E when 1. If P 1, PŽ 1 P. 1 P 1 P lim E 0. 1 cž P 1. 1 P

18 GONG & MCAFEE: NEGOTIATION, LITIGATION, RULES 35 If P 1, PŽ 1 P. P 1 1 P lim E 1 cž P 1. P 1 PŽ 1 P.. cž P 1. If P 1, 1 lim E lim PŽ 1 P.Ž 1 P1 c P1 1 P.Ž P 1. 1 lim Ž 1 P.Ž1 c P1 1 P. P Ž 1 P.Ž Ž.Ž. Ž. P 4 1 P 1 P Ž 1 P. lim c P1 1 PŽ 1 P. lim c P1 1. 8cŽ The secon equality above use L Hoital ˆ rule. E an E are erive, Ž. 6 follows naturally from Ž. 1. () Proof of Lemma 1 : E E c c ž / P 1 P c Ž. P PŽ 1 P. c 1 Ž. P 1 1 PŽ 1 P. c 1 Ž. P 1 Ž. 4Ž 1. Ž. Ž.4 P 1 P 1 4 P 1 P. When P 1, the lemma is true if the last term is not negative. But since 1, an 1 4 PŽ 1 P., we thus have Ž. 4Ž 1. P Ž 1 P. 1 4 PŽ 1 P. 1 4 P Ž 1 P.1 4 P Ž 1 P. 4Ž 1. P Ž 1 P.1 4 P Ž 1 P. 0. When P 1, using L Hoital s ˆ rule, we have Lim c Ž. P1 4Ž 1. P Ž 1 P. 1 4 PŽ 1 P.4 Ž. P 1 3 Lim c½ 1 P1 P ž / 4Ž 1.Ž 1 P. Ž1 4 PŽ 1 P. 4Ž 1 P. PŽ 1 P Ž.Ž P 1. Q.E.D. Proof of Proosition () : We only to rove Ž. 8. By Enveloe theorem, SŽ. ž / 1 Ž. Ž. E c Ž E. Ž. E E P 1 P Ž. Ž. 1 E E E E E c Ž E. Ž. E E P 1 P ½ 1 ce ce E Ž. E 1 P P ce ce E Ž. E 1 P P E c Ž E. Ž. E E P 1 P 1 c E c E. P 1 P

19 36 ECONOMIC INQUIRY Now the sign of the secon art alreay eens on P, as shown in Lemma 1. The first art, E c Ž E. Ž. E E P 1 P c 1 4 E E cž P 1. PŽ 1 P. Ž 1 P. Ž 1 P. P1 Ž 1 P. c 1 E E cž P 1. Ž.Ž. P 1 P 1 P 1 Ž. 4 P 1 P. The first four terms of the above rouct are always ositive, but the last term is nonositive. To see why, substituting the efinition of, it is sufficient to rove Ž. Ž. Ž. 1 4 P 1 P 1 8 P 1 P 4 P 1 P. Since the left sie is always ositive because 1, squaring both sies of the inequality gives 4PŽ 1 P. 1, which certainly hols for all P. Then the sign of SŽ. is ecie entirely by 1 P. Q.E.D. Proof of Lemma () : The cut-off value for the laintiff is simly the execte equilibrium ayoff at trail conitional on his signal an the laintiff s signal. So S Ž1, S Ž y.. EŽ Ž 0. 1, y., an S Ž0, S Ž y.. EŽ Ž 0. 0, y.. Then the first art is rove follow- ing Table 1. Let u Ž y. Ž 13. EŽ Ž 0. y, x y. Ž 3. EŽ Ž 0. y, x y. enote the efenant s ex- ecte ayment at trial conitional on his signal being y. Let VŽ,. enote the efenant s execte ayoff when his signal is an he uses strategy *. Now suose S Ž. 1 S Ž. 0. Consier a tye 0 efenant who uses S Ž. 1 instea of S Ž. 0. Then Ž Ž.. V S 1,0 1 PrŽS Ž 1. rejecte.s Ž 1. Ž Ž.. Ž. Ž. Pr S 1 rejecte u 0 S 0 Ž. Ž Ž. 13 Pr S 1 Ž Ž.. Ž. Ž Ž. S 0, S 1 3 Pr S 1 Ž Ž.. Ž. Ž. Ž. S 1, S 1 u 0 S 0 S 0 Ž. Ž Ž. Ž Ž.. 13 Pr S 0 S 1, S 0 Ž. Ž Ž. Ž Ž.. 3 Pr S 0 S 0, S 0 u Ž 0. S Ž 0. 1 PrŽS Ž 0. rejecte.s Ž 0. PrŽS Ž 0. rejecte. u Ž 0. Ž Ž.. V S 0,0, which violates the efinition of equilibrium strategy when the efenant s signal is 0. Note that the first inequality follows from hyothesis. The secon inequality hols because S Ž1, S Ž 1.. S Ž0, S Ž 1.. S Ž1, S Ž 0.. S Ž0, S Ž 0.. following the first art of the roof, an u Ž. 0 S Ž. 0, that is, the efenant never offers anything higher than his execte ayment at trial. Therefore, the efenant s strategy must be at least weakly increasing. But since we are consiering searating equilibrium, it must further be strictly increasing. Q.E.D. Proof of Proosition () 3 an () 4 : First consier the efenant s strategy of roosition Ž. when his signal is 1. Given the belief of the laintiff that anything higher than the equilibrium offer from a strong efenant reveals that the efenant is weak, he coul either offer EŽ Ž 0. 0,0. 110, which is only going to be accete by the weak laintiff, or EŽ Ž 0. 1,1. 35, which assures a settlement regarless of the laintiff s signal. Now suose 110 is offere. It has a 13 robability of being accete an 3 robability of being rejecte, resulting in an execte ayment of Ž 13.Ž 110. Ž 3. EŽ Ž 0. 1, , which is greater than 35. Thus, offering 35 to secure a settlement is the only otimal strategy for the efenant when his signal is 1. When the efenant receives 0, he coul still offer EŽ Ž 0. 0, or EŽ Ž 0. 1, If he offers 110, his execte ayment will be Ž 3.Ž 110. Ž 13. EŽ Ž 0. 0,1. 310, which is less than EŽ Ž 0. 1,1. 35, the offer than can always secure a settlement. Therefore, we have verifie that S Ž , S Ž is the only otimal strategy for the efenant, given the belief of the laintiff. When these offers are mae, the actual tye of the efenant is exactly the laintiff s belief, showing consistency with his cut-off strategy. The roof of roosition Ž 4.3. where the laintiff makes the offer is similar, using exactly the same logic. Q.E.D. Proof of Proosition () 5 : Accoring to roosition Ž. 3, the searating equilibrium essentially nees to satisfy a series of inequalities. First there are the willingness to offer an accet conitions: Ž A6. S Ž 0. EŽ Ž. 0,0. Ž. Ž Ž.. Ž. Ž Ž.. A7 E 0,0 S 0 E 0,1, an Ž A8. EŽ Ž. 1,1. S Ž 1. Ž. Ž Ž.. 13 E 0,1 Ž. Ž Ž.. 3 E 1,1. The incentive constraints require that Ž A9. Ž 13. S Ž 0. Ž 3. EŽ Ž. 1,1. Ž Ž.. E 1,1

20 GONG & MCAFEE: NEGOTIATION, LITIGATION, RULES 37 an Ž A10. Ž 3. S Ž 0. Ž 13. EŽ Ž. 0,1. Ž Ž.. E 1,1. By the nature of the equilibrium, Ž A6., Ž A7., an the first art of Ž A8. automatically hol. Note that the secon art of Ž A8. is imlie by Ž A9., which fortunately hols for all as Figure 8 shows. Figure 9 is a lot of the right-han sie minus the left-han sie of Ž A10.. It is ositive when Finally we nee to assure that the laintiff s execte ayoff is ositive at least for the weak tye so that he is willing to file the lawsuit. That is, Ž 3. S Ž. 0 Ž 13. S Ž The numerical comuta- tion shows that this inequality hols for all less than 0.8. The roof of the secon half is very similar. Let S Ž. 0 an S Ž. 1 enote the laintiff s offer when he gets signal 0 an 1, resectively. We nee to verify the range of for which the following inequalities hol: Ž A11. S Ž 0. EŽ Ž. 1,1., Ž. Ž Ž.. Ž. Ž Ž.. A1 E 1,1 S 1 E 0,1, Ž. Ž Ž.. A13 E 0,0 S Ž 0. Ž 13. EŽ Ž. 0,1. Ž. Ž Ž.. 3 E 0,0, Ž A14. Ž 13. S Ž 1. Ž 3. EŽ Ž. 0,0. an Ž Ž.. E 0,0, Ž A15. Ž 3. S Ž 1. Ž 13. EŽ Ž. 0,1. Ž Ž.. E 0,0. Ž A11., Ž A1., an the first ar of Ž A13. are straightforwar, accoring to the efinition of the equilibrium. Figure 10 lots the right-han sie minus the left-han sie of Ž A14.. Note that Ž A14. also imlies the secon art of Ž A13.. A similar lot in Figure 11 shows that Ž A15. hols for almost all excet when it is very close to 1. Q.E.D. FIGURE 8 FIGURE 10 FIGURE 9 FIGURE 11

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