A Theory of Pre-litigation Settlement and Patent Assertion Entities

Transcription

1 A Theory of re-itigation Settement and atent Assertion Entities Leshui He 1 Department of Economics, Bates Coege January 21, Address: ettengi Ha, Bates Coege, Lewiston, Maine 424. he@bates.edu.

2 Abstract An informed (potentia) paintiff can demand a payment from an uninformed (potentia) defendant with the threat of a aw suit that may not be credibe. The communication takes pace before the aw suit is fied and is costess to both sides. A unique equiibrium that satisfies the intuitive criterion eists and ehibits partia pooing cases beow a cutoff of ega damage reach pre-itigation settement, whie those above are fied for suit. This equiibrium offers an epanation for the distinctive behaviors of itigation AEs who aggressivey fie aw suits for settements and portfoio AEs who obtain icensing revenues through pre-itigation settements. Among other resuts, we shows that, in the US ega system, increasing a paintiff s ega costs or adopting the fee-shifting rue reduces the number of fied aw suits whie risking driving more cases toward pre-itigation settements. Surprisingy, reducing a defendant s ega costs or increasing tria costs has the opposite effects.

3 1 Introduction Individuas and entities resove countess disputes in the shadow of itigation. Most of these resoutions invove a settement even before the case is formay fied for aw suit a fact beckoning a pair of intriguing questions. On the one hand, if cases can be resoved through pre-itigation settement without incurring any ega cost, why are they ever fied for aw suit? On the other hand, if the itigants have to epend etraordinary resources to reach a settement post-itigation, why did they fai to resove the dispute earier? atent assertion entities (AEs) or patent tros, who derive most of their revenues by asserting patent rights against aeged infringers instead of appying patents for productive uses, are a case in point. Their entire business mode hinges on ega settements, which are frequenty caed e-post icensing fees. Two distinct practices stand out among AEs in a case study of the wireess chipset industry by the Federa Trade Commission (216). Some AEs predominanty send demand etters with itigation threats to etract preitigation settements, or so-caed icensing fees, from other producing entities. Others fie many aw suits directy, and reach e-post icensing settements quicky after fiing. Most surprising is the fact that the former type, which may appear more tro-ike, owns much arger patent poos, fies occasiona suits, settes much ater in the ega process, and receives much higher pre-tria (or post-fiing) settements. On the contrary, the atter type accounts for the majority of cases fied for suit and, surprisingy, derives much ess tota revenue and smaer pre-tria settements. Indeed, Federa Trade Commission (216) refers to the former type as portfoio AEs and the atter as itigation AEs. 1 If portfoio AEs can etract settements before itigation, why bother fiing suits at a? Moreover, why do the itigation AEs not imitate the practices of the portfoio AEs to fie fewer suits and etract more settements? A framework anayzing poicy impications reated to AE activities must pass a test by epaining these observations consistenty. This paper presents a mode of pre-itigation settement. A risk-neutra paintiff sustains some privatey observed ega damage from a (potentia) defendant. The risk neutra defendant has a commony known beief over the distribution of the damage, but not its eact vaue. Before formay fiing a aw suit, the paintiff may choose to demand a settement payment from the defendant in echange for reease from further itigation. If the defendant pays a desirabe settement, the dispute is resoved. Otherwise, if the case has sufficient merit, the paintiff may choose to fie suit, which incurs further ega costs for both parties. After the suit is fied and some ega costs incurred, the defendant earns the true damage through discovery. Then, both sides may have another opportunity to sette before going to tria for the court decision. The mode epores a contet in which an informed paintiff makes a demand to an uninformed defendant, whie (1) the threat of a aw suit may not be credibe; (2) the right to fie suit resides with the informed paintiff; and (3) entry to the settement bargaining game is free. These three features are defining characteristics in contets invoving itigation threats such as NE-reated aw suits. The second feature sets the current mode apart from previous works concerning non-credibe cases or negative epected vaue (NEV) cases, whose epected court judgment is dwarfed by the ega costs invoved in pursuing itigation, such that they woud be automaticay dropped before proceeding to the net stage of the itigation process (Naebuff, 1987; Bebchuk, 1988; Meurer, 1989). The third feature highights the distinction between the pre-itigation settement contet and the we-studied pre-tria bargaining contet, in which the paintiff has to pay an initia itigation cost to enter the bargaining game (Reinganum and Wide, 1986; Farmer and ecorino, 27). We characterize a equiibria with pre-itigation settement under continuous and differentiabe probabi- 1 Lemey and Meamed (213) appies three categories for NEs, but with consistent properties. 1

4 ity distributions over the ega damage. In particuar, among a equiibria, there is a unique equiibrium that satisfies the intuitive criterion (Cho and Kreps, 1987), which is aso the equiibrium that maimizes the paintiff s pre-itigation settement and minimizes ega costs. A equiibria with pre-itigation settements ehibit a partia pooing structure, in which paintiffs with case vaues beow a cutoff send out identica demands, inducing the defendants to pay a pre-itigation settement, whie paintiffs with case vaues above the cutoff fie aw suits. In these equiibria, a defendants with NEV cases receive a demand, but not a receiving a demand face NEV cases. With the correct beief of the paintiff s strategy, the defendant forms a correct beief about the chance that the case wi actuay be itigated if he or she defauts on the received demand. The defendant cacuates the epected oss by weighing potentia ega damages pus ega costs with the chance that the paintiff wi fie the case, and is therefore wiing to pay this amount pre-itigation to avoid a aw suit. However, because the defendant s highest wiingness to pay before itigation is weighed down by ow vaue cases and the possibiity of empty threats, paintiffs with high enough case vaues woud rather incur etra ega costs to itigate and force the defendant to discover the case vaue. Once the high case vaue is crediby reveaed to the defendant through discovery, the pre-tria settement net of the ega costs can sti eceed the pre-itigation settement amount eading to partia separation in the equiibria. The eistence of such equiibria with pre-itigation settements reies on severa key ingredients (1) the defendant s perception of a sizabe chance of invovement in high damage cases; (2) non-trivia ega costs for the defendant to wait unti discovery in itigation; (3) the ow cost of itigation for the paintiff reative to the potentia damages, such that the threat of itigation remains credibe. Among other resuts, we show that paintiffs with a higher potentia ega damage can reach pre-itigation settements whie paintiffs with ower damages cannot. They earn higher pre-itigation settements, and receive higher settements pre-tria a key features that fit the distinction between portfoio AEs and itigation AEs. On the poicy side, our resuts suggest that decreasing the defendant s cost to respond to itigation woud increase the number of fied suits, but reduce cases of pre-itigation settements. Moreover, if the eary-stage itigation cost through discovery is ower than the ater-stage costs of going to tria, then increasing the paintiff s eary-stage cost to fie itigation reduces the number of cases fied for itigation, but drives more cases toward pre-itigation settement; surprisingy, increasing the ater-stage court cost wi have the opposite effect by rendering itigation threats ess credibe. Resoving disputes through pre-itigation settements saves substantia ega costs, athough it woud create a mismatch between the settement compensation and the paintiff s rea damages a feature of pooing equiibrium. One the one hand, it aows the paintiffs to receive compensation from ow-damage cases that woud have been too epensive to itigate. On the other hand, it eads some defendants with itte faut to overpay for their true responsibiity. The wefare trade-off is not obvious without imposing some preference for ega costs versus inefficiencies from mismatched transfers, a subject quite beyond the scope of the current artice. However, the mode advocates for caution toward poicies ecusivey aiming either at reducing pre-itigation settement or at reducing the number of fied cases. The rest of the artice proceeds as foows. Section 2 summarizes the most reevant iterature. Section 3 sets up of the mode and anayzes it under the uniform distribution assumption to highight the key intuitions of our resuts. Section 4 reports resuts under genera distribution assumptions. Section 5 etends the baseine mode to discuss the impact of adopting a fee-shifting rue. Section 6 discusses other etensions of the mode and robustness, and Section 7 concudes. 2

5 2 Reated Literature A key feature of the pre-itigation settement phase is the credibiity of the threat to sue. Any entity can in principe send out demands at itte cost; thus, at face vaue, their credibiity in terms of threats of further ega actions is sub-par. Consequenty, these cases introduce a credibiity issue why woud the defendant pay a demand if the paintiff may drop the case him or hersef? Naebuff (1987); Bebchuk (1988) and Meurer (1989) epore this topic in the presence of asymmetric information a setting most cosey reated to ours. Naebuff (1987) and Bebchuk (1988) study screening modes where the uninformed party makes an offer to an informed party. Meurer (1989) anayzes a signaing setting where the rights to fie a aw suit reside with the uninformed defendant. The paintiff in our setting has both the information advantage and the abiity to threaten the defendant with a aw suit. Consequenty, our mode produces different empirica impications. 2 Another key feature of the pre-itigation settement phase is free entry to sending a demand anyone with a patent can send boier pate demands with itte additiona costs. Reinganum and Wide (1986) studies a mode of pre-tria settement in which an informed paintiff sends a demand to an uninformed defendant. They found that, when paintiffs of a types have credibe cases, there eists a mied strategy separating equiibrium in which the defendant pays a mied strategy that rejects higher demands at a higher probabiity, thus inducing a paintiffs to report their types truthfuy. Farmer and ecorino (27) generaizes the mode to incude NEV cases, and shows that the paintiffs with NEV cases have strong incentives to mimic offers from other types, therefore breaks down the proposed mied strategy equiibrium when there is no entry cost of sending the demand. They aso show that, if paintiffs of a types have to pay a fied fee before sending the demand, a mied strategy separating equiibrium can be restored because, in the equiibrium, a paintiffs with NEV cases wi not enter. To our best knowedge, there is itte known about whether itigants can reach settement in the equiibrium and what cases move on to itigation when the paintiff can send demands for free, i.e. some demands aways come from paintiffs with NEV cases a characterizing feature of pre-itigation settement. Our mode fis this void by showing that partia pooing is sti possibe when there is free entry to the settement bargaining game. One strand of the ega studies iterature focuses on the roe of commitment and cost aocation how strategicay arranging the timing of ega costs may either enhance or deter the threat of frivoous aw suits. On the one hand, the paintiff can profitaby etract a settement by strategicay front-oading ega costs (Bebchuk, 1996; Farmer and ecorino, 1999; Chen, 26). On the other hand, the defendant can aso deter frivoous aw suits by sinking cost through different ega contracts (Hubbard, 216) or by estabishing an aggressive reputation for rejecting settements (Micei, 1993; Bone, 1997). In the contet of pre-itigation settement, however, reputation or commitment aone is neither sufficient nor necessary to support or deter AE activities our theory proves the non-necessity and the behavior of many AEs demonstrates insufficiency. Ewing and Fedman (212) and Federa Trade Commission (216) report a distinctive behavior of itigating AEs, who intentionay group their patents into smaer poos and hide behind different she companies before fiing aw suits. If paying tough or sinking ega costs was sufficient to etract settements, we shoud not observe such behaviors. Recent iterature focuses specificay on the rise of the AE phenomenon. Among theoretica anaysis, Hovenkamp (215) studies a screening mode where a short-ived defendant may be a behaviora type that responds to aggressive paintiffs by paying settements. Thus, the AE aggressivey itigates to buid a tough reputation to etract rents from future defendants. Choi and Gerach (215, 217) study modes where the 2 We defer the detaied comparison of the predictions to Section 6. 3

6 eistence of mutipe defendants improves the epected return from individuay non-credibe aw suits, thus providing incentives for the paintiff to pursue itigation. The current mode, however, sheds ight on this topic from the perspective of pre-itigation settement. In a sharp contrast, these modes predict that AEs with weaker patents fie fewer aw suits, whie our mode predicts the opposite a resut driven by the credibiity of the threat to sue. One key ingredient of our mode is that the informed party may send a costess and unverifiabe message to the uninformed party before incurring a cost to verify his or her own type, thus avoiding costs for both parties. Beyond aw and economics, this feature renders our mode a hybrid of a cheap tak game (Crawford and Sobe, 1982), where verification is impossibe, and a emon mode with costy up-front vountary discosure (Verrecchia, 1983; Suijs, 1999), where costess communication before discosure is impossibe. 3 Mode 3.1 Setup A paintiff ( or he) and an aeged defendant (D or she), both risk neutra, are invoved in a dispute with an epected court judgment of, which is randomy distributed on [, ], where may be infinity. has a cumuative distribution function F() and probabiity distribution function f (), which we assume to be continuous and differentiabe everywhere. Before deciding to fie a aw suit, privatey observes the true vaue of, to which we sometimes refer as s type. A of the above information is common knowedge to D, ecept for the reaization of. Before fiing suit, may choose to send a costess message to D demanding a payment of m = m() doars, whie cannot be crediby reveaed to D. In other words, D or any third party cannot verify the vaidity of the demand at this stage. Without incurring any cost, D can choose to either respond by paying the demand (sette) or ignoring it (defaut). If D pays an amount acceptabe to, they reach a pre-itigation settement agreement that reeases D from further ega iabiities pertaining to this dispute. If and D fai to reach an agreement, may choose to incur a constant ega cost of to fie suit, which consequenty requires D to incur a constant ega cost of D to respond. By spending D, D aso acquires more information about. For simpicity, we assume that the reaization of is perfecty reveaed to D after incurring D. Then, foowing a standard bargaining mode under symmetric information, and D may bargain and reach a pre-court settement before incurring additiona cost k on each side to go to tria, where both parties epect the court to mandate that D pay an amount equa to. 3 There is no discounting between any of these stages in the mode, and a ega costs are pubic knowedge. To avoid the trivia scenario where none of the disputes has sufficient merit to pursue itigation, we assume that the upper bound of court vaue eceeds the higher vaue between itigation cost and tria cost, > ma(, k). 4 Subgame after D responds to the demand At tria, the epected payoffs are k D for D and k for. Thus, before tria, chooses to go to tria rather than dropping the case if and ony if k p > p, i.e. > k. 3 Our resuts sti hod when the ega costs are increasing in the judgment vaue, see Appendi C for detais. 4 Under this assumption, it is sti possibe that a cases are frivoous suits, i.e., they a have a negative epected vaue, < + k. 4

7 The pre-tria settement bargaining happens ony if going to court is credibe ( > k); thus, the payoffs after bargaining, assuming equa bargaining power, are v D T = D D if > k if k and v T = if > k, if k where subscript T indicates pre-tria. If D defauts on the demand, decides to fie a aw suit if and ony if v T, which is equivaent to > ma(, k). We refer to these as aw-suit-credibe cases, and others ( ma(, k)) as non-credibe cases. 5 We define the shorthand = ma(, k) to indicate the aw suit-credibiity threshod. In such a continuation game, s pre-itigation demand strategies and D s responses constitute the core aspects of the equiibria with substantive demand. The rest of this section discusses potentia equiibria in detai. Fu separating equiibrium does not eist We first consider the possibiity of a fuy separating equiibrium, where s with different vaues send distinct pre-itigation demands m(), and show that this strategy does not constitute an equiibrium. If adopts this strategy, D is updated about the true perfecty. A D who earns s foresees that wi never crediby fie the case, and thus refuses to pay the demand. This scenario yieds a payoff of for with non-credibe cases ( s ). On the other hand, a D who earns > s foresees the case being fied, and is thus wiing to pay up to + D as a pre-itigation settement. This atter scenario yieds a payoff of + D for with credibe aw suits ( > s ), higher than the payoff from itigation,. This outcome cannot be an equiibrium because s with non-credibe cases have an incentive to deviate and mimic the strategies of s with credibe cases to induce pre-itigation settements from D. Indeed, Farmer and ecorino (27) shows that the same force from NEV cases aso breaks down the mied strategy separating equiibrium à a Reinganum and Wide (1986). Because fu separation is impossibe, it does not oss generaity to focus on pure strategy equiibria. 3.2 Equiibrium definition Definition. A demand m() > in the state of the word is substantive if, in equiibrium, it induces payment m() from D, that is, the best response for any D who receives a demand is to pay the demanded amount rather than to defaut. We adopt a standard pure-strategy perfect Bayesian Nash equiibrium (BE) concept with one modification a pre-itigation demands are substantive. We refer to this soution concept as the equiibrium with substantive demand. Note that there is aways a no-demand equiibrium or babbing equiibrium, where never sends any substantive demand, and D never responds to pre-itigation demands. 6 However, we omit this type of equiibrium, and focus the anaysis on the eistence and properties of the equiibria with substantive demand. Definition. A pure-strategy BE with substantive demand is described by s demand strategy m() R + that maimizes his tota payoff; 5 We avoid using the term frivoous aw suit here (Spier, 27) because some cases may be aw-suit credibe, but sti have a negative epected vaue if carried through to tria, i.e., ma(, k) < + k. 6 One beief for D that woud support such an equiibrium is that any demand indicates the owest type of. 5

8 D s beief of the state of the word according to Bayes rue and to the correct beief of s equiibrium demand strategy, f () = f ( m()) whenever possibe; D s binary response to demand R(m()) {pay, defaut} that maimizes her tota epected payoff; s binary response to itigate L(, R) {itigate, not itigate} that maimizes his continuation payoff; and D s payoff-maimizing settement offers and responses given each partys strategies after the case is fied for suit; a pre-itigation demands are substantive. We do not ose generaity by focusing on equiibria with substantive demand. This additiona requirement on the BE frees the anaysis from considering any additiona hagging process eading up to the pre-itigation settement or other fruitess communications before the suit is fied. Because D has no private information and is unverifiabe, any signa besides m() has no additiona vaue. Suppose an equiibrium with demand m() that induces D to pay m() and sette with pre-itigation; we can then construct an equiibrium with substantive demand to repicate the eact equiibrium outcome demands m() directy and induces the eact pre-itigation settement. Moreover, if, in equiibrium, of a certain sends ecessive demands that fai to induce payments from D, then we can aso construct an equiibrium with substantive demand where such types of do not send any demand to repicate the eact equiibrium path roperties of equiibria Cutoff-pooing strategy in the equiibrium The discussion on separating equiibrium reveas that with non-credibe cases has strong incentives to mimic the strategies of those with aw-suit-credibe cases. Indeed, we show that any equiibrium with substantive demand must ehibit a feature of partia pooing, where a cases beow a threshod reach the same pre-itigation settement. Lemma 1. In any equiibrium with substantive demand, a s who send demands wi send the same demand m(). Specificay, for 1 2, if m( 1 ), m( 2 ), then m( 1 ) = m( 2 ). roof. Suppose an equiibrium where with 1 and 2 send two distinct messages m( 1 ) < m( 2 ). If both demands are substantive, then the of 1 wi deviate and send m( 2 ), a contradiction. Furthermore, if a of particuar type is wiing to accept a given pre-itigation settement, then a s with ower vaue cases must be wiing to sette for the same amount. Lemma 2. In any equiibrium with substantive demand, if a of type sends a substantive demand m( ), then a with any < must aso send the same demand. roof. Without reaching pre-itigation settement, s of type < receive ess continuation payoff, if any, compared to s of type. Specificay, between dropping the case before fiing and moving on to reach pre-tria settement, s aternative payoff without pre-itigation settement is v L () = ma(, v T ()) = ma(, ). Thus, v L () v L ( ) for a <, where subscript L represents pre-itigation. 7 If we assume that bears an infinitesima cost to send a demand it is presumaby costy to make the demand ook remotey credibe, or a potentia ega iabiity for sending demands in bad faith, or risking having the patent invaidated by court it woud be in s best interest to avoid such fruitess attempts and pursue itigation directy. 6

9 If a of type sends substantive demand m( ) in the equiibrium, the demand m( ) must eceed the aternative payoff for a of type ; thus, we have, for any <, m( ) v L ( ) v L (), which impies that a s of type have incentives to mimic type to send m( ) rather than bypassing preitigation settement. Foowing from Lemma 1 and 2, must pay a cutoff strategy in any equiibrium with substantive demand. This impies a simpe structure to update beiefs on D s part. If D received a pre-itigation demand, then hoding the correct beief about s strategy described by the cutoff in the equiibrium, D updates her beief about the case s tria judgment to the truncated distribution with p.d.f. f ( ) = f (). Her epected continuation payoff in choosing to defaut is F( ) E [1( > )vd T ] = 1( > )vd T f ( )d = ( + D ) f ( )d, (1) where 1( ) is an indicator function. The epression aways takes a negative vaue given our assumption that >, indicating a rea risk that the case may be fied for aw suit if she refuses to pay the demand.8 The upper bound of the resuting integra refects the cutoff structure that ony s with case vaues beow woud send a demand. Its ower bound s refects D s reasoning that ony aw-suit-credibe cases wi be fied for suit, eading to a continuation payoff of + D pre-tria, whereas a non-credibe cases wi be automaticay dropped. This epression captures D s cacuation that weighs the potentia ega iabiity against the probabiity that the case wi be fied for suit. Given D s epected payoff from defaut, she is wiing to pay up to m( ) = E [1( > s )vd S ] to as a pre-itigation settement. If a s with are wiing to accept this payment rather than fiing the suit, i.e., m( ), then summarizes an equiibrium with pre-itigation settement, where a cases with a vaue beow sette before being fied for suit for m( ), and a higher vaue cases proceed to aw suit. To compete the equiibrium description, we must impose requirements for off-equiibrium beiefs shoud D receive an ecessive demand deviating from m( ). In such cases, as ong as D updates her beief in such a way that owers her wiingness to pay for pre-itigation settement, the equiibrium sustains. We summarize these key observations in the foowing proposition. roposition 1. A cutoff demand strategy ( s, ] summarizes any equiibrium with substantive demand. In other words, without osing generaity, we can focus on a cutoff strategy by and the corresponding responses from D in the foowing form sends demand m( ) = ( + D ) f ( )d for a and fies suit for a > ; D, who receives the demand m( ), updates her beief about to f () = f ( ); D is wiing to pay m( ) in echange for reease from possibe ega responsibiities; with accepts pre-itigation settement payments m( ); with > fies a aw suit directy; and 8 paying < s is weaky dominated because it induces zero settement payments from D. 7

10 D who receives any demand other than m( ) updates her beief to f () m m( ), such that ( + D ) f ( )d ( + D ) f () m m( ) d. roof. Impied by the previous discussion. For epositiona purposes, we refer to any equiibrium with substantive demand by its cutoff strategy, whie impicity assuming that roposition 1 describes a other aspects of the equiibrium. To highight the intuition of our key resuts, we proceed through the rest of this section with an eampe of a uniformy distributed on [, ]. With the uniform assumption, we can reach cosed-form soutions to the cutoff strategy, and the eampe ehibits amost a of our key resuts. We defer the discussion of the resuts under genera distribution assumptions unti the net section. Fu pooing equiibrium when is uniformy distributed We first consider the possibiity of a fu pooing equiibrium, where any type of sends the same demand m() = m and reaches pre-itigation settement with D. If a types of send identica demands, then receiving the message is competey uninformative; thus, D does not update at a. This is a specia case where the cutoff strategy equas the upper bound of the distribution, =. If D refuses to pay, her continuation payoff is, according to (1), E [1( > )vd S ] = ( + D ) 1 d = s ( + s 2 + D ), where the first term measures the probabiity that the case wi be fied for suit, and the second term measures the average pre-tria ega iabiity pus Ds ega costs to reach that stage. The absoute vaue of this epression, E [1( > s )vd S ] measures D s wiingness to pay for pre-itigation settement. Note that this epression is increasing and concave in the upper cutoff of the integra, in this case an essentia property that shapes the properties of the equiibrium. Given the consideration above, can etract an amount up to m = E [1( > s )vd S ] from D as preitigation settement. It is incentive compatibe for to forego the aw suit if m eceeds his continuation payoff from fiing the suit,. Thus, the fu pooing equiibrium hods if s of the highest type,, are wiing to accept the settement m pre-itigation. Specificay, this requires ( + s 2 which, after some agebra, is equivaent to a quadratic epression in with roots + D ), (2) L = ( D + ) ( D + ) 2 s (2 D + s ) H = ( D + ) + ( D + ) 2 s (2 D + s ). That is, if [ L, H ], D s wiingness to pay is greater than the continuation payoff from fiing aw suit for s of type, m(). Note that L, H are rea if the foowing condition on the ega costs hods D ( s ) + 2( s ) s, (condition D) 8

11 aw-suit-credibe cases setted pre-itigation L H Figure 1: Fu pre-itigation settement ( [ L, H ]) which we can interpret as meaning that the defendant s pre-court itigation cost is reativey high compared to the paintiff s pre-court itigation cost, and the cost to go to court (Figure 1). roposition 2. Let be uniformy distributed on [, ]; then, there eists a fu pooing equiibrium of substantive demand =, where a cases reach pre-itigation settement at the same demand, if condition D hods; and the maimum court judgment ies within an intermediate range, i.e., [ L, H ]. roof. Suppose D beieves that s of a types send identica demands; then, D s wiingness to pay for preitigation settement is m() = s ( +s 2 + D ). By the definition of L, H, we have m() for a [, ], which impies that a types of s have incentives to accept settement m() and forgo fiing the suit. Therefore, s best response, given D s beief and response strategy, is to demand m() for a types. Hence, D s beief of a fu pooing strategy is correct, and thus = is an equiibrium with substantive demand. Figure 1 iustrates the stated equiibrium in the proposition. The paraboa in the figure is defined by (2). In this equiibrium, of a types make an identica pre-itigation demand to D; athough ony a fraction of cases are aw-suit-credibe, D is wiing to pay the demand. Therefore, a cases reach pre-itigation settement and no case is fied for suit. The first condition in the proposition is rather intuitive and echoes the arguments in the ega studies iterature high enough ega costs for the defendant reative to those of paintiff ead to pre-itigation settement. This insight generaizes beyond the uniform distribution assumption and the boundedness of (see roposition 8). By reaching pre-itigation settement, gains on two fronts: avoiding additiona ega cost 9

12 whie etracting D from D. However, s with high case vaues aso ose from foregoing itigation because the pre-itigation settement is diuted by two forces once by pooing with ower-vaue cases, and again by the credibiity of the suit actuay being fied. To make s incentive compatibe to forego itigation, D must sufficienty mitigate the gap between the diuted m and his true. Therefore, a reativey high D is necessary to support the pooing outcome as an equiibrium. The subtety reveaed by the first condition is that can pay two different roes. On the one hand, an increase in can raise the aw suit-credibiity threshod s, rendering a given case ess aw-suit-credibe for D, and owering the settement vaue m, thus making pre-itigation settement ess paatabe for. On the other hand, a higher makes pre-itigation settement more enticing for as it saves etra epenses by setting upfront. When is smaer than k, an increase in ony has the second effect, which eads to a weaker condition to support the equiibrium above. However, when is greater than k, an increase in carries both effects, which counteract each other and can ead to ambiguous resuts. In the uniform eampe, these two forces eacty offset each other; thus, an increase in has no impact on the necessary condition for the equiibrium to hod. 9 Note, however, that these changes in ega costs do not change the pooing equiibrium outcome, =, defined by the upper bound of the distribution. 1 A coser ook at the second condition in roposition 2 raises two questions. First, why is there a possibe interva (, L) in which the pooing equiibrium fais to hod if fas into it? The intuition is that when gets too cose to s, the ikeihood of a aw suit becomes too ow to support a settement vaue m attractive enough for any with aw-suit credibe cases to forego the aw suit, which consequenty breaks the pooing equiibrium. In fact, if < L, there is no equiibrium with substantive demand a cases above s are fied for suit, and others are automaticay dropped without settement (see roposition 3). The counter-intuitive observation in this case is that, affected by the ow credibiity of fiing the suit, wi pursue itigation, athough the vaue of the settement is in fact reativey ow a resut consistent with the behavior of itigation AEs. The second question is what happens if eceeds H? Reca the key property that m is increasing and concave in. When eceeds H, the highest possibe pre-itigation settement m from the fu pooing outcome can no onger keep up with ineary-increasing case vaues at the higher end beyond H. This fact incentivizes with such high vaue cases to deviate bypassing the pre-itigation stage and going directy to a aw suit. In other words, s with high-vaue cases woud prefer going to itigation and waiting for D to discover, thereby etracting a higher settement pre-tria. In such cases, incurring itigation costs is worthwhie, thus bypassing demand stage is optima. An ensuing question is whether there is sti an equiibrium with substantive demand if is greater than H. The answer is yes, and it takes the form of a partia pooing equiibrium indicated by roposition 1. The rest of this section characterizes these equiibria and discusses their properties. artia pooing equiibria when is uniformy distributed One important observation from the anaysis above is that as the cutoff strategy ( in the fu pooing scenario) decreases, D s wiingness to pay may decrease sower than due to the concavity of m( ). Thus, even when the fu pooing equiibrium is unsustainabe, there may sti eist a partia pooing equiibrium with a cutoff strategy <. 9 Specificay, if > k, the first condition in roposition 2 reduces to D, where changes in have no effect. 1 That is, uness fas out of the interva [ L, H ]. 1

13 aw-suit-credibe cases aw-suit-credibe cases setted pre-itigation case fied case case droppedfied L H L H (a) artia pre-itigation settement ( > H ) (b) No pre-itigation settement ( < L ) Figure 2: Equiibria under uniform distribution roposition 3. Let be uniformy distributed on [, ] and suppose that condition- D hods; then, the equiibria with substantive demand are characterized by the foowing cases if > H, = H is an equiibrium (partia pooing); if L H, is an equiibrium (fu pooing); if < L, no equiibrium with substantive demand eists. roof. First, suppose > H ; we then check that = H is an equiibrium. By the definition of H, m( H ) = H for a H. Suppose D beieves that = H ; then, D is wiing to pay m( H ) as pre-itigation settement. Given m( H ), s of type H are wiing to sette. Moreover, m( H ) < for a > H ; thus, s of type > H are unwiing to sette with payment m( H ). Therefore, = H is s best response, hence D s beief that = H is correct, and therefore = H is an equiibrium with substantive demand. The second point reiterates proposition 2. The third point foows from the fact that if < L, by (2), there is no [ s, ] such that m( ). This impies that the most D is wiing to pay for pre-itigation settement is insufficient to encourage s with case vaues above s to give up itigation, regardess of what pays. Figure 2 iustrates the two scenarios beyond the fu pooing equiibrium. If > H, a partia pooing equiibrium eists in which a cases with H reach pre-itigation settements (incuding cases that are not aw suit-credibe), and a cases above H are fied for suit. If < L, no case can reach pre-itigation settement, and ony aw suit-credibe cases are fied for suit. It is important to note that these are not a equiibria with substantive demand when is uniformy distributed. A simiar argument as the proof above verifies that when L, = L is aso an equiibrium with substantive demand (more generay, see roposition 4). Therefore, the equiibrium stated above may not be unique in each scenario. 11

14 However, we wi show that, regardess of the distribution assumptions for, the equiibrium with substantive demand that satisfies the intuitive criterion (Cho and Kreps, 1987) is aways unique (roposition 6). This equiibrium aso maimizes the proportion of cases reaching pre-itigation settement and s net payoff among a equiibria. In the uniform case, they are the equiibria we stated in roposition 3. Compare the two equiibria with substantive demand L against H in the case of > H. Suppose the equiibrium is = L, and D ony updates her beiefs accordingy and is wiing to pay up to m( L ). If D suddeny receives a higher demand m( H ) a deviation from her epected equiibrium strategy shoud she naivey and stubborny deem this an ecessive demand and refuses to pay, or shoud she deem this a reasonabe demand based on a credibe higher-vaue case? The intuitive criterion imposes the additiona requirement on the off-equiibrium beief such that upon receiving such a deviation, D is sophisticated she is forced to concude that m( H ) is in fact a credibe threat of a higher-vaue case that ies between [, H ], and is thus wiing to pay such a higher settement. In other words, equiibria such as = L are ony sustainabe due to the impicit assumption that D can crediby commit to ony paying ower demands, even when a higher demand is credibe. Between the assumption of strong commitment power for D versus the assumption that D is highy sophisticated in choosing her strategy to respond to the demand, we choose the atter. Therefore, we favor the equiibrium that satisfies the intuitive criterion. For its uniqueness, we ca it the equiibrium, on which we focus our comparative statics anaysis. The ast note on the equiibrium characterization under the uniform case is that, if eist, L and H, or L and if < H, are a the equiibria with substantive demand. In fact, there eist at most two equiibria with substantive demand if f () is og-concave, which incudes uniform, norma, truncated norma, eponentia, ogistic, Weibu, Chi-squared, etc. (roposition 5). Comparative Statics when is uniformy distributed The comparative statics hinge on the properties of L and H as the ega costs D, and k change. If =, the equiibrium does not respond to changes in the ega costs, athough changes in ega costs may affect its eistence condition stated in roposition 2. The more interesting case ies in the partia pooing equiibrium, and we focus on anayzing the properties of H rather than L because it satisfies the intuitive criterion. It is readiy verifiabe that H is increasing in D. 11 The intuition reiterates our discussion of the fu pooing equiibrium regarding the effect of D on s incentives a higher D raises D s wiingness to pay and therefore attracts higher-vaue cases to sette without fiing suit. The effects of and k depend on whether or k is greater, which determines the aw suit-credibiity threshod. If < k, an increase in owers the right-hand-side of (2) without changing the eft-hand-side, making the condition easier to satisfy. Intuitivey, it makes a given pre-itigation settement more compeing for because it avoids the higher cost of fiing the suit. Therefore, in this case, an increase in aways increases the equiibrium cutoff strategy H. On the other hand, an increase in k does eacty the opposite it owers the eft-hand-side of (2) without affecting its right-hand-side. That is, an increase in k ony increases the aw suit-credibiity threshod, s, which renders any given case ess aw-suit-credibe, and consequenty owers D s wiingness to pay for settement. Hence, in the equiibrium, it drives more high-vaue cases toward aw suit, if such an equiibrium sti eists. These effects of ega costs on the equiibrium cutoff strategy aso generaizes 11 An increase in D increases both the center and the width of the interva [ L, H ]. If < k, the width is 2 D ; otherwise, it is (D + ) 2 k(2 D + k), which increases in D given condition- D. 12

15 to the amount of pre-itigation settement m( ) and the average pre-tria settements more easiy observed from data they are weaky increasing in and D, and decreasing in k. Moreover, these observations are fuy generaizabe to a distribution assumptions (see roposition 9 and 1, and Tabe 1). If > k, however, an increase in now carries both effects owering both sides of (2). In the uniform case, its effect in pushing to sette dominates its effect in owering D s wiingness to pay, and thus H is increasing in. However, this unambiguous resut can ony generaize to distributions that are non-decreasing in (see ropositions 11, 12, and 13, and Tabe 2). In this case, an increase in k has no effect on the equiibrium it does not change s trade-off between setting pre-itigation versus fiing suit, nor does it affect the aw-suit credibiity of any case. Athough the comparative statics resuts differ depending on the reationship between and k, in reaity, scenarios under < k are ikey to be more reevant because tria costs usuay far outweigh the ega costs to fie the suit and to go through pre-tria discovery. The interpay between D and in the eistence of the equiibrium remains argey true, as the condition- D highights. As D becomes sufficienty high, a cases can reach pre-itigation settement (see roposition 7). On the contrary, if we fi D and k, then, raising sufficienty high can eradicate a equiibria with pre-itigation settements, whie, surprisingy, aso eaving neary no cases fied for suit a dreamand for D and devastation for (roposition 8). In addition to the effects of changes in ega costs, we aso anayze how changes in the distribution of affects the equiibrium behavior. We can imagine that if D epects that a higher ega iabiity is more ikey, her wiingness to pay for pre-itigation settement woud be higher and thus ead to a higher equiibrium cutoff strategy. In the uniform case, changing the distributions upper bound ony affects the equiibrium when it moves beow H. Generaizing beyond the uniform distribution, if one distribution ikeihood ratio dominates another, it has a weaky higher equiibrium cutoff strategy (see ropositions 14 and 15). This impies that s with potentia for higher ega damages sette higher-vaue cases pre-itigation, and potentiay bring fewer suits a resut consistent with both the intuition and the distinctive behaviors of itigation and portfoio AEs. 4 Genera Resuts In this section, we formay characterize the equiibria with substantive demand and anayze their properties without resorting to the uniform assumption for the probabiity distribution. 4.1 Equiibrium characterization roposition 1 impies that, without oss of generaity, we can focus on a cutoff strategy,, by. Therefore, s pre-itigation demand probem is to choose and a corresponding demand m() to maimize his tota payoff from pre-itigation settement and pre-tria return. s.t. ma 1[ ]m() + 1[ > ]( ) m(), m() ˆ ( + D ) f ( ˆ)d, 13

16 where ˆ is D s beief about, and according to Bayes rue, f ( ˆ) = f () is the truncated distribution of F( ˆ) f () on [, ˆ]. This program is equivaent to s.t. ma m() m(), m() ˆ ( + D ) f ( ˆ)d IC for D m() IC for. In equiibrium, D hods the correct beiefs about s strategy, ˆ =. Then, these constraints jointy impy the foowing necessary condition for the equiibrium cutoff strategy ( + D ) f ( )d. Note that s demand in equiibrium, m( ) = ( + D ) f ( )d, is monotone increasing in. 12 By this necessary condition, the equiibria must beong to the foowing set X { In particuar, one of the candidate equiibria maimizes s payoff ma { ( + D ) f ( )d }. (3) ( + D ) f ( )d }. In severa occasions during our anaysis, it is more convenient to work with an aternative form of X, which we can write as foows. Lemma 3. The set X can be rewritten as X = F()d F( ) + C F( ) Φ( ) + D, (4) where C = s f ()d + D F( s ) is constant with respect to and is non-negative. roof. See appendi. 12 To see this, integrating by parts, we have of which we take derivative with respect to, yieding ( + D ) f ( )d = ( D + )F( ) ( s + D)F( s ) + F() d, F( ) More generay, see Lemma 7. [ ( + D ) f ( )d] = s f ( ) (( D + s )F(s ) + s F() d) F( ) 2. 14

17 Φ() Φ() + D + D = = + 2 = (a) < (b) = Figure 3: Iustration of set X By definition, we can write X as a union of cosed and conve subsets, X = i X i. We denote the greatest and smaest eements of these subsets by i + and i, respectivey. Figure 3 iustrates X as defined by (4), as we as the associated i + s and i s. roposition 4 shows that a + i s and i s constitute a equiibria with substantive demand in this mode. In the case iustrated by Figure 3, suppose s < + 1 ; then, there eist three equiibria with substantive demand. roposition 4. A equiibria with substantive demand are characterized by + i and i in every conve subset of X, provided that X is non-empty. roof. See appendi. In addition, if f () is og-concave, the set of potentia equiibria is greaty simpified. Eampes of such distributions incude uniform, norma, truncated norma, eponentia, ogistic, Weibu, Chi-squared, and so on. (Bagnoi and Bergstrom, 24). Lemma 4. The set X is conve if f () is og-concave. roof. See appendi. roposition 5. There eist at most two equiibria with substantive demand if f () is og-concave. roof. By Lemma 4, X is conve; thus, + i and i are unique, if they eist. The resut then foows from roposition 4. roposition 6. If X is non-empty, then its greatest eement characterizes the unique equiibrium with substantive demand that satisfies the intuitive criterion. Moreover, among a equiibria with substantive demand, maimizes the number of cases reaching pre-itigation settement, the settement amount receives, and s payoff. roof. In any other equiibrium <, consider the foowing out-of-equiibrium deviation of, whose type ies in [, ] to send demand m( ) > m( ). 15

18 Such a deviation is equiibrium dominated for a types above, but profitabe for a s beow. Therefore, by the intuitive criterion, D must update her beief that ies in [, ], and thus pays m( ), greater than the equiibrium demand m( ). The equiibrium thus fais to meet the intuitive criterion. The same argument aso shows that an equiibrium with cutoff strategy satisfies the intuitive criterion; thus the uniqueness. The fact that maimizes m( ) foows from the fact that m() is increasing in (footnote 12). The foowing resuts highight the properties of the equiibrium for etreme cases of the parameters. roposition 7. Given, when D is sufficienty arge, a cases reach pre-itigation settement in the equiibrium. roof. See appendi. roposition 8. If is infinity, f () is monotone decreasing for a arge enough ; then, for fied D and sufficienty arge or k, there eists no equiibrium with pre-itigation settements, whie the probabiity of a case being fied for itigation, 1 F( s ), goes to. roof. See appendi. ropositions 7 and 8 provide a stark contrast against each other. When the cost of using the ega system is etremey high, parties invoved in disputes are driven out of the forma itigation process a common resut foowing both the high cost for both and D. However, the outcome under the shadow of high ega costs is competey different when the ega cost for or tria cost is eceptionay high, no pre-itigation settement can be reached, and there is acquiesce. However, if the ega cost for D is eceptionay high, amost a cases are setted outside of the ega system. 4.2 Comparative statics In the rest of the anaysis, we focus on the unique equiibrium with substantive demand that satisfies the intuitive criterion, or the equiibrium for short. We anayze the effects of changes in the ega costs and the distribution of case vaues whie assuming the eistence of the equiibrium. roposition 9. If s itigation cost is ess than the tria cost, < k, then in the equiibrium, is non-decreasing if D increases; increases; or k decreases. roof. See appendi. Figure 4 iustrates the idea behind our comparative statics anaysis. Suppose a change in a parameter, say, owers the function Φ( ) D ; it woud then resut in an epansion of the set X, and thus an increase in in a weak sense. Therefore, the response of Φ( ) D with respect to changes in each parameter is sufficient in determining the responses of. Given the fied distribution F(), a higher cutoff strategy in the equiibrium is aso equivaent to a higher percentage of cases reaching pre-itigation settement. Moreover, these resuts aso etend to the pre-itigation settement amount and the average pre-tria settement amount in the equiibrium. 16

19 Φ() D 1 2 Figure 4: Comparative statics iustration roposition 1. If s itigation cost is ess than the tria cost, < k, then in the equiibrium, the pre-itigation settement amount, m( ), and the average pre-tria settement amount, f ( > )d, are non-decreasing if D increases; increases; or k decreases. roof. See appendi. We summarize these resuts in Tabe 1. The particuary surprising finding here, again, is the opposing effects of and k. The intuition, as we mentioned in the uniform eampe, ies in the fact that when < k, an increase in attracts with a given case to reach pre-itigation settement. On the contrary, an increase in k raises the aw suit-credibiity threshod s and owers D s wiingness to pay for pre-itigation settement, therefore driving more high-vaue cases toward aw suit. We summarize the comparative statics resuts under the ess reaistic assumption of > k beow. roposition 11. If s itigation cost is greater than the tria cost, > k, then in the equiibrium, is non-decreasing if D increases; ambiguous if increases; and unresponsive to changes in k. roof. Impied by the proof of roposition 9. Simiary, these resuts etend to the settement amounts, both pre-itigation and pre-tria. roposition 12. If s itigation cost is higher than the tria cost, > k, then in the equiibrium, the pre-itigation settement amount m( ), and the observed average pre-tria settement amount, f ( > )d, are non-decreasing if D increases; 17

20 Tabe 1: oicy Impications of Changes in Lega Cost (If < k) oicy # of cases fied for itigation # of cases reach pre-itigation settement re-itigation settement amount Average pre-tria settement amount decrease Defendant s cost to respond to itigation, D increase aintiff s cost to fie itigation, increase Tria cost, k are ambiguous if increases; and do not respond to changes in k. roof. Omitted for its simiarity to roposition 1. To iustrate the second point, in which the prediction is ambiguous, we consider an eampe where f () is eponentiay distributed with a parameter of 1. In this case, we can verify that the effect of an increase in can go either way in the equiibrium. 13 Simiary, we summarize the interpretations of the proposition in Tabe 2. Tabe 2: oicy Impications of Changes in Lega Cost (If > k) oicy # of Cases fie for itigation # of Cases reach pre-itigation settement re-itigation settement amount Average pre-tria settement amount decrease Defendant s cost to respond to itigation, D increase aintiff s cost to???? fie itigation, increase Tria cost, k We compement the previous statement with a technica resut, where the effect of an increase in on is definitey non-decreasing. We do not think the premise of the resut is necessariy reaistic; however, we 13 Specificay, et f () = ep( ) and F() = 1 ep( ); then, for =.2, D = 1, we have 1.52, and =.2, D =.2, we have.48, and.4 >..39 <. Whereas if 18