Patent validity and the timing of settlements

International Journal of Industrial Organization 67 (2019) 102535

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Patent validity and the timing of settlementsR Talia Bar∗, Jesse Kalinowski Department of Economics, University of Connecticut, 365 Fairfield Way, U-1063, Storrs, CT 06269-1063, United States

a r t i c l e

i n f o

Article history: Received 29 January 2018 Revised 8 September 2019 Accepted 13 September 2019 Available online 19 September 2019 JEL classification: D83 O31 O34

a b s t r a c t We study the timing of settlement in patent disputes, accounting for an alleged infringer’s search for evidence to challenge patent validity. Early settlements are more likely reached when the disputed patent has a low quality, and late settlements take place when the alleged infringer finds strong evidence that challenges patent validity. Thus, there is a tendency to settle disputes over patents that would have likely been invalidated in court. Fee shifting induces more early settlements, and when parties did not settle early, a higher expenditure on invalidating the patent. Our model sheds light on disputes involving patent assertion entities. © 2019 Elsevier B.V. All rights reserved.

Keywords: Patent Litigation Infringement Prior art Fee shifting America Invents Act

1. Introduction The United States Patent and Trademark Office (USPTO) receives more than 50 0,0 0 0 patent applications each year. The office examines every application to determine if it satisfies the legal patentability requirements. This process is imperfect and the patent office grants “bad” patents – ones that would not have been granted had the examiner been perfectly informed. Bad patents are believed to impose harm to society (Frakes and Wasserman, 2019), and only some are later invalidated.1 We model settlements and litigation in patent disputes. We aim to better understand the timing and size of settlement and the role of search for evidence of invalidity in determining the outcome of negotiation. Settlement timing plays an important financial role in litigation. Settling shortly after a legal demand will often mean avoiding the majority of the costs. But, it is not uncommon for parties to settle just before trial and incur substantial litigation expenses (Gryphon, 2011). In our symmetric information model, there is a common belief that the patent is good. There are two stages of negotiations. In the first stage, the patentee makes an early settlement demand. If her demand is rejected, then the alleged infringer R We thank Brendan Costello and Lu Cao for excellent research assistance. We are grateful to workshop participants at the Innovation Workshop in Stony Brook and at the University of Connecticut, and to Haim Bar, Tom Miceli, Mark Myers and Mike Shor for helpful comments. ∗ Corresponding author. E-mail addresses: [email protected] (T. Bar), [email protected] (J. Kalinowski). 1 The prevalence of “bad” or “questionable” patents has been widely discussed in the literature. Shapiro (2004) reviews some evidence and argues that “questionable” patents can harm innovation. Allison et al. (2014) use a dataset of cases filed in 2008 and 2009 and find that alleged infringers win validity challenges 42% of the time. Using data on Inter Partes Reviews filed between 9/2012 and 2/2016, Bar and Costello (2019) find that 47% of validity challenges result in cancellation of the first patent claim. Poor quality patents might be more prevalent in some areas such as software and business methods (Merges, 1999).

https://doi.org/10.1016/j.ijindorg.2019.102535 0167-7187/© 2019 Elsevier B.V. All rights reserved.

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will expend resources to search for prior art or other evidence that could challenge patent validity. The strength of evidence is revealed to both parties, and it determines the probability that the patent would be found valid in trial. If the alleged infringer finds strong evidence then the probability that the patentee wins is lower than if the alleged infringer only finds weak evidence. The patentee makes her late settlement demand contingent on the strength of evidence. If her demand is rejected again, then the patentee decides whether to drop the suit or to continue to trial. In many models of pre-trial negotiations, asymmetric information is the reason why parties sometimes fail to settle before trial. In our model, no party has private information, instead trials are driven by asymmetric stakes – the gain to the patentee from winning might be higher than the loss to the alleged infringer. Patent disputes likely exhibit stakes asymmetry; for example, when winning could strengthen a patentee’s position in future disputes.2 This simple modeling approach allows us to combine important characteristics of patent litigation. In particular, in our model, parties can settle early (before the alleged infringer searches for evidence of invalidity), settle late (after evidence is revealed), or go to trial; there is endogenous expenditure on search for evidence; the model addresses both validity and infringement; and it accounts for situations where the patentee would choose to drop the suit if strong evidence of invalidity is found.3 When the patentee’s gain from winning a trial is sufficiently higher than the alleged infringer’s loss (a high-returns-tolitigation range), a trial is credible regardless of whether the evidence of invalidity is weak or strong. In this case, we find a range of parameter values for which the parties fail to settle early. Instead, they either settle late, if the alleged infringer finds strong evidence of invalidity, or continue to trial, if evidence is weak. In a low-returns-to-litigation range, a trial is only credible when evidence is weak. In this case, we find a range of parameter values for which the patentee drops the suit if evidence of invalidity is strong and continues to trial if evidence is weak. Importantly, in our model the alleged infringer chooses the intensity of search for evidence of invalidity. We show that an alleged infringer expends more resources to find evidence of invalidity the lower is the initial probability that the patent is good. The alleged infringer also searches more intensely when finding strong evidence of invalidity would induce the patentee to drop the suit. The patentee chooses the amount of settlement demands. Her early settlement demand increases with the initial probability that the patent is good. When a late settlement is reached, the settlement amount is lower than the highest amount the alleged infringer would have been willing to settle for in the early stage. This is because late settlements take place when the patentee’s position has been weakened by the finding of strong evidence of invalidity and because the pre-trial litigation costs have been incurred earlier. Farrell and Merges (2004) reason that litigation does not reliably fix patent office errors due to skewed incentives to challenge and defend patents. Complementing their arguments, we claim that the courts’ ability to correct the patent office’s mistakes is limited by the parties’ higher incentive to settle disputes over patents that are more likely to be bad. The reason is that at an early stage, the owner of a lower quality patent makes a lower settlement demand, because she anticipates a higher probability that the alleged infringer would find strong evidence of invalidity if they do not settle early. Moreover, if the parties do not settle early, then after the strength of evidence is revealed, the patentee has a higher incentive to settle late or to drop the suit if the alleged infringer revealed strong evidence of invalidity. The tendency to settle weak cases before trial suggests that if all disputed patents had their validity challenged in court, the share of invalidated patents would have been higher than in the cases that reach a trial. Still, even under this pattern of selection, empirical findings suggest that a significant share of litigated patents are found to contain invalid claims (Allison et al., 2014). Asymmetric information models similarly predict selection into trial; typically strong cases of the informed party are more likely to reach a trial. For example, in Lee and Bernhardt (2016), where privately informed defendants signal through the timing and the size of settlement offers, weaker defendants attempt to settle pre-discovery. Our model also highlights a special feature of patent litigation. For the patentee to win an infringement suit, the patent needs to be valid and infringed; while for the alleged infringer to win, the patent should either be invalidated or not infringed. Extending the base model to consider the patentee’s search for evidence of infringement, we find that if she searches for evidence before filing a suit and files only if she finds strong evidence of infringement, then the patentee will have a stronger incentive to search for evidence of infringement when she has a higher quality patent. This is because the patentee expects a higher payoff from filing a suit when her patent is more likely to be good. In a version of the model with simultaneous search for evidence, we show that the patentee would want to spend more resources to show infringement the less resources the alleged infringer expends to show invalidity. However, the alleged infringer would want to spend more resources to invalidate the patent the more resources the patentee spends to find evidence of infringement. In patent cases in the US, each party typically pays its own attorney fees. Under a fee shifting rule, the loser may be required to also pay the legal fees of the winning party. US 35 U.S.C. 285 states that, “The court in exceptional cases may award reasonable attorney fees to the prevailing party.” A 2014 Supreme Court ruling tends toward a more supportive environment for fee-shifting in patent litigation.4 The Innovation Act further proposed changes that would make fee shifting the default.5 In our model, if the alleged infringer only reveals weak evidence of invalidity, under a fee shifting rule the threat

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Siegelman and Waldfogel (1999) provide evidence consistent with asymmetric stakes in intellectual property cases. Much of the early literature on litigation allowed only one round of negotiations before trial. Spier (1992) and Lee and Bernhardt (2016) assume that defendants have private information. They allow for multiple rounds of pre-trial negotiations, but do not model the choice of expenditure on search for evidence, and in their models the plaintiff never drops the suit. 4 Octane Fitness, LLC v. ICON Health & Fitness, Inc., 134 S. Ct. 1749 (2014), see http://www.supremecourt.gov/opinions/13pdf/12-1184_gdhl.pdf. 5 See https://www.congress.gov/bill/114th-congress/house-bill/9. 3

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of trial is credible for a wider range of returns to the patentee compared to the case without fee shifting. This is because when evidence is weak, the patentee is likely to prevail, thus, under fee shifting her expected cost is lower. However, if the alleged infringer finds strong evidence of invalidity, under a fee shifting rule the patentee is more likely to drop the suit, since the cost of losing is higher. This strengthens the alleged infringer’s incentive to search for evidence of invalidity. We consider the behavior of Patent Assertion Entities (PAE) and the alleged infringers they sue. PAEs—firms that do not research or produce the goods for which they own the intellectual property, but assert a patent to extract rent—are a subject of policy debate. According to Chien (2013), PAEs were responsible for 62% of total US patent litigation cases in 2012, up from 19% in 2006. Mazzeo et al. (2013) suggest that PAEs may be more willing to settle litigation before a court decision and that they have lower success rates in terms of validity and infringement. PAEs specialize in patent litigation, and so they likely have lower litigation costs. Our model predicts that patentees with lower litigation costs file infringement suits on patents that are, on average, of lower quality; and if they fail to settle, they might bring to trial even cases for which there is strong evidence of invalidity. 2. Related literature In models of legal disputes, trials can occur when there is information asymmetry, when the parties have different perceptions regarding the probability that the plaintiff will prevail, or with asymmetric stakes – when the patentee would gain sufficiently more if she wins compared to the loss of the alleged infringer (Cooter and Rubinfeld, 1989; Lanjouw and Lerner, 20 0 0; Priest and Klein, 1984). Like Farrell and Shapiro (2008), we assume symmetric information with a common belief about the probability that the patent is good.6 Asymmetric stakes drive late settlements and trial in our model.7 Asymmetric stakes might arise in patent disputes for several reasons. Scholars note that winning a trial against one alleged infringer could strengthen a patentee’s position against future potential infringers (Choi, 1998; Farrell and Shapiro, 2008). With reputation benefits, the patentee may benefit more than the alleged infringer would lose. In markets with homogeneous goods, excluding a competitor from using a patented technology that lowers costs can lead to higher profits than the joint profits from sharing the technology (Lanjouw and Lerner, 20 0 0), and so a trial can create joint surplus. Asymmetry could also arise when the parties are optimistic – the patentee might expect to be awarded higher damages than the alleged infringer expects to pay. Bar-Gill (2006) argues that “lawyers and litigants are systematically optimistic with respect to the outcome at trial.” In a seminal paper on the selection of disputes into litigation, Priest and Klein (1984) suggest that disputes that reach trial are different than disputes that settle before trial. In their model, each dispute is characterized by a measure of the fault of the defendant, and the judge rules in favor of the plaintiff if this measure exceeds a decision standard. The parties observe noisy signals of this measure resulting in asymmetric estimates of the probability of plaintiff win. With symmetric stakes and small errors in parties’ estimates, Priest and Klein show that trials occur when the measure of defendant fault is close to the decision standard. They show that as parties’ error diminishes, litigation rate declines and the proportion of plaintiff victory tends to 50%. With asymmetric stakes in favor of the plaintiff, they argue that “smaller differences between the parties’ expectations of success will be sufficient to generate litigation.” In our paper, the parties’ expectations are equal, trials are driven by asymmetric stakes in favor of the patentee. Consistent with Priest and Klein’s discussion of the asymmetric stakes case, trials are more likely when the probability that the patentee will prevail is higher (in our model, when evidence of invalidity is weak). In many contributions on legal disputes, there is only one settlement stage (see for example Bebchuk (1984) and Reinganum and Wilde (1986) for general litigation, Meurer (1989) and Crampes and Langinier (2002) for patent litigation). Spier (1992) introduces dynamics to a model of pre-trial negotiations, and examines the timing of settlement. In her model, costs incurred preparing for trial and bargaining provide an incentive to settle early, while private knowledge that the defendant has about the outcome of trial provides an incentive for a settlement “on the courthouse steps.”8 Like Spier (1992), we also allow for more than one round of negotiations and consider the timing of settlement. While in her model late settlements and trial are driven by one sided incomplete information and the plaintiff can learn about the defendant’s type when a settlement offer is refused, in our model, the driving force is asymmetric stakes and both parties learn about the quality of the patent if an early settlement is not reached. Importantly, we account for the resources that the alleged infringer expends to invalidate the patent, such as prior art search. Lee and Bernhardt (2016) study the discovery process in civil litigation. In their model, a defendant has private information about the probability that he would be found not liable in trial. The timing of settlements serves as a signal of the defendant type. Our model is similar to theirs in some ways; for example, they have two take-it-or-leave-it settlement offers (in our case, demands) with a discovery process in between, and the timing and size of settlements are endogenous. They find that an informed defendant who has a low probability of prevailing in a trial is more likely to accept an early 6 Patents are publicly available and include the relevant prior art that is known to the examiner at the time the patent was issued. Atal and Bar (2010) argue that patentees likely have little incentive to search intensively for prior art before they apply for a patent. Langinier and Marcoul (2016) examine strategic concealing of prior art by patent applicants. We expect that in a patent dispute, both parties are skilled in the art. 7 We obtain nearly identical results if we formulate a model with differences in belief. 8 Fournier and Zuehlke (1996) simulate Spier’s model to derive comparative statics and empirically explore its predictions. Fenn and Rickman (1999) test predictions from Spier’s model using data from English health care providers on clinical negligence and employee claims.

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settlement offer, since discovery would reveal his low type. In our (symmetric information) model, the patentee would make a low enough early settlement demand when her patent has a low probability of being good, since the alleged infringer will more likely reveal strong evidence of invalidity if they do not settle early.9 Kamenica and Gentzkow (2011) provide a model of Bayesian persuasion, where a Sender (the prosecutor) designs a signal to reveal to a Receiver (the judge) so as to maximize the probability of getting the Receiver to take the Sender’s preferred action (convict). The sender optimally chooses an investigation (a distribution mapping the true state—guilty or innocent—to signals) and must truthfully reveal the realization of the signal. In our model, the alleged infringer’s search for evidence can be thought of as an “investigation” and our set of possible investigations is restricted by a search technology. We do not assume that the court is Bayesian. Froeb et al. (2016) model courts’ decision as a choice between competing interpretations of evidence, offering a model of persuasion with non-Bayesian courts. Lemley and Shapiro (2005) describe patent prosecution and patent litigation in the United States. They offer an excellent review of related empirical and theoretical research. According to the authors, “Modeling patents as probabilistic rights requires us to rethink [...] our approach to patent litigation, the efficacy of litigation as a means of invalidating patents that were improperly issued and antitrust policy toward the settlement of patent lawsuits.” We account for the probabilistic nature of patents and offer insights into invalidation through litigation. A number of studies have examined the economic consequences of remedies for licensing agreements and the incentive to innovate (Shapiro, 2016; 2010; Schankerman and Scotchmer, 2001). Chen and Sappington (2018) show that the design of damages plays an important role in fostering efficient levels of sequential innovation. The type of remedies awarded for infringement and the nature of the market, specifically whether transfer of the patented technology would increase joint profits, have consequences for our model. For example, in a linear homogeneous goods Cournot duopoly, if the patented technology entails a significantly lower marginal cost than that of the other firm, then joints profits are higher when that firm is excluded.10 However, joint profits can be higher if the technology is shared when there is sequential innovation and an infringing innovator can expand the market (Chen and Sappington, 2018). When joint profits are higher if the technology is not shared, Scotchmer (2006) argues that “lost profits” damages are higher than “unjust enrichment”. Our model then predicts that with “lost profits” the settlement amounts will be higher, and if the parties do not settle early, there is more intense search for invalidating evidence. In such market, the possibility of an injunction could lead to asymmetric stakes in favor of the patentee, potentially resulting in late settlements. However, if joint profits are higher when the patented technology is shared, “unjust enrichment” leads to higher settlements and more resources spent on invalidating a patent. Cremers et al. (2016) compare a bifurcated patent system in which infringement and validity are determined independently in separate courts, and a non-bifurcated system in which infringement and validity are decided jointly. The US has a non-bifurcated system. However, this is to some extent changing with a significant increase in administrative challenges of patent validity since the introduction of Inter Partes Review. Our paper also joins an emerging body of literature that examines the behavior of PAEs. Tucker (2013), Mazzeo et al. (2013), and Bessen and Meurer (2014) offer empirical studies. Theoretical work related to PAEs, or of the broader group of non-practicing entities (NPEs), is scarce. He (2018) compares the strategies of litigation PAEs and portfolio PAEs. Cosandier et al. (2014) model competition between NPEs and operating firms to win a patent auction. Choi and Gerlach (2017) examine the incentives of NPEs to acquire patent portfolios and compare them to the incentives of practicing entities. Bergin (2018) studies the impact of NPEs on R&D investment arguing that “in some cases, the NPE is beneficial to investment, in others, detrimental.” Our results are also related to contributions that compared the English rule (in which litigation costs are shifted) and the American rule. Consistent with the earlier findings of Katz (1987), we also find that the alleged infringer expends more resources to invalidate the patent in a system with fee shifting.

3. Model Our model is a sequential game between a patentee and an alleged patent infringer. The legal process we model has potentially two rounds of settlement demands surrounding a discovery stage in which the alleged infringer might reveal new evidence of invalidity. If settlements fail and the patentee does not drop the lawsuit, a trial follows. In a trial the court decides on two issues: was there an act of infringement (e.g., unauthorized use or sale of the patented innovation), and is the patent valid? The patentee wins if the patent is found valid and infringed. Fig. 1 illustrates the timing of actions, and we describe it below in more detail.

9 A discovery stage is also accounted for in other asymmetric information litigation models. For example, Schrag (1999) shows that judges can increase the probability of early settlement by limiting pre-trial discovery; Schwartz and Wickelgren (2009) show that discovery can reduce the probability and size of pre-discovery settlement offers. 10 In a Cournot duopoly with demand P = A − bQ and constant marginal costs Cp for a patentee and Cn > Cp for a firm using a non-patented technology. Excluding the second firm from using the patented technology results in higher joint profit than sharing the patented technology if and only if (Cn − C p ) > 2 A − Cn ). Relatedly, Bernhardt and Dvoracek (2009) show that a multinational firm that has a superior technology compared to that of domestic firms, 3( would pay wage premiums to maintain its technology supremacy when there is a large cost difference.

T. Bar and J. Kalinowski / International Journal of Industrial Organization 67 (2019) 102535

P

Patentee

A

Alleged Infringer

P Don’t file suit

File suit

N

P

5

Nature

Early selement demand A

Accept

Reject P Drop

Connue A

Search for evidence of invalidity

N

Weak evidence

P

Strong evidence

P Late selement demand

A

A

Reject

Accept

Reject

Accept

P Drop

P Trial

Drop

Trial

N

N

Valid and infringed Valid, not infringed α0β α0(1-β)

Valid and infringed Valid, not infringed α1β α1(1-β)

Trial Invalid 1-α0

Invalid 1-α1

Fig. 1. The timing of actions in the game.

The disputed patent can be “good” or “bad” – a good patent is truly novel and non-obvious, had examiners been perfectly informed and unbiased, they would only grant good patents. The parties have symmetric information; initially, they share a common belief that the patent is good with probability θ . We say that a patent has a higher quality when θ is higher. The alleged infringer can expend resources to search for evidence that challenges patent validity.11 The evidence can be either strong (i = 1) or weak (i = 0). The parties also have a common belief β that the court will find that the patent was infringed (if it finds the patent valid).12

11 Challenging the validity of the patent is a common defense for an alleged infringer. In the US, the burden of proof of invalidity is on the challenger. To invalidate a patent in court, an alleged infringer needs to provide “clear and convincing evidence.” (Kesan and Ball, 2006) 12 It is possible that if θ is lower then there is more incentive to infringe, and thus θ and β might be negatively correlated. In our model θ and β are exogenous and known to both parties.

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At the start of the game, given θ and β , the patentee decides whether to file a suit at a cost c0 . If so, she demands a settlement, s1 . The alleged infringer can accept the demand, pay the patentee s1 and then the game ends. If the alleged infringer rejects the demand, the patentee decides if to drop the suit or continue. If the dispute continues, the alleged infringer expends resources ea ∈ [0, ∞ ) to search for new evidence that the patent is invalid. If the patent is bad, then the alleged infringer’s search reveals  strong evidence of invalidity with  probability F(ea ), where F (0 ) = 0. The probability is increasing at a decreasing rate F  (ea ) = f (ea ) > 0, and f  (ea ) < 0 and limea →∞ F (ea ) = 1.13 If the patent is good, the probability of strong evidence of invalidity is smaller, φ F(ea ) where 0 ≤ φ < 1. Thus, when φ is lower, strong evidence is a more accurate signal of invalidity. Overall, the probability that the alleged infringer finds strong evidence is

p(θ , φ , ea ) = (1 − θ )F (ea ) + θ φ F (ea ).

(1)

This probability is decreasing in θ and increasing in φ and in ea . Between negotiation stages, the parties incur pre-trial litigation costs (discovery costs) denoted by dp and da . Before the second round of settlement negotiations, the patentee learns the results of the alleged infringer’s search.14 Given evidence i, the patentee then makes a new settlement demand si2 . If the alleged infringer accepts the patentee’s demand, he pays the patentee si2 and the game ends. Otherwise, the patentee chooses whether to drop her suit or to litigate to trial. The alleged infringer does not challenge the validity of the patent unless the patentee decides to continue to trial. If the parties continue to trial, the probability that the court affirms patent validity is α 0 if evidence is weak and α 1 if evidence is strong; the court is more likely to affirm validity if the evidence is weak, α 0 > α 1 .15 The patentee wins the trial if the court finds that the patent is valid and infringed (with probability α i β ). If the patentee wins, she receives W and the alleged infringer incurs a loss D.16 Trial costs for the alleged infringer and the patentee are ca and cp , respectively. The assumption that the conditional probabilities α i are constant seems reasonable since we expect judges to rule based on knowledge of the law and the evidence presented, but not based on prior information specific to the current case. The initial probability that the patent is good (θ ) affects the intensity of search for evidence of invalidity (ea ) and the probability of finding strong evidence of invalidity, p(θ , φ , ea ). However, once uncertainty about the strength of evidence is resolved, the conditional probabilities that the court will find the patent valid (α i ) do not depends on θ . We discuss this assumption further in Section 6.3, as well as consider a version of the model that partially relaxes it, accounting for the possibility of Bayesian persuasion. 4. Analysis We derive the subgame perfect equilibrium using backwards induction. If the patentee filed a suit and the parties did not reach a settlement agreement, their expected payoffs from trial given evidence i are i πT,p = αi β W − c p − d p for the patentee and

(2)

i πT,a = −αi β D − ca − da − ea for the alleged infringer.

(3)

To understand these payoffs, observe that just before trial, the parties know whether the evidence of invalidity is strong or weak. The patentee wins with a probability α i β , which is the probability that the court affirms validity and finds that the patent was infringed. If the patentee wins, her gain is W and the alleged infringer’s loss is D.17 Either way, each  party  incurs its own costs. If the patentee drops the suit, the parties’ payoffs are the costs they incurred before trial, −d p for the patentee and (−da − ea ) for the alleged infringer. If settlements failed, the patentee will continue to trial if she expects a higher payoff from trial than from dropping the suit. The value of W that makes the patentee indifferent between continuing to trial and dropping the suit given evidence i = 0, 1, solves αi β W − c p − d p = −d p , leading to the thresholds,

WTi =

cp

βαi

.

(4)

Since α 0 > α 1 , the threshold is lower when evidence is weak: WT1 > WT0 > 0. Definition 1. There are high returns to litigation if W > WT1 , and low returns to litigation if W ≤ WT1 . 13 The function F has properties of a cumulative distribution function. But, the interpretation is different. F(e) is the probability of finding strong evidence given e, and not the probability that the expenditure is lower than e. 14 The legal pre-trial discovery process in which a party can obtain documents and request answers from another party provides one justification for why the patentee can learn about the evidence that the alleged infringer has found. 15 Evidence of invalidity could be weak either because the alleged infringer does not look for evidence or because his search was not successful. 16 The payoffs W and D are the incremental benefits if the patentee wins the trial compared to no trial, not accounting for litigation costs. In additional to damages awarded, these payoffs account for the effect that the outcome of the dispute has on the firms’ future profits and reputation. 17 i = αi β W − (1 − αi )L − c p − d p . We can generalize the payoffs to include an additional loss to the patentee if the patent is invalidated, L. In this case, πT,p For brevity, we assume here L = 0.

T. Bar and J. Kalinowski / International Journal of Industrial Organization 67 (2019) 102535

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In the high-returns-to-litigation range, if settlements failed, the patentee litigates to trial regardless of whether the alleged infringer found weak or strong evidence of invalidity. In the low-returns-to-litigation range, if WT1 ≥ W > WT0 and settlements failed, the patentee litigates to trial only when the alleged infringer found weak evidence of invalidity, but when evidence is strong the patentee drops the suit. If W ≤ WT0 the patentee never continues to trial. When a trial is never a credible threat, the patentee does not file the suit. Consider first the high-returns-to-litigation case. 4.1. High returns to litigation Late negotiations start after the strength of evidence is revealed. Given evidence i, a late agreement is reached if there exists a settlement amount si2 that would make both parties at least as well off as they would be if they continued to trial. Using the trial payoffs in (2) and (3), (excluding the costs ea , da and dp , which were incurred earlier), a late settlement can be reached if

αi β D + ca ≥ si2 ≥ αi βW − c p for i = 0, 1.

(5)

Rearranging this inequality, a settlement is reached when total trial costs exceed the total expected benefits from the trial

ca + c p ≥ αi β (W − D ).

(6)

The parties necessarily settle if W ≤ D. However, asymmetric stakes in favor of the patentee (W > D) could lead to trial. In this case, when evidence is weak, the expected benefit from trial is higher than when evidence is strong, but trial costs are the same. So, the parties are more likely to settle when evidence is strong. For comparison with Priest and Klein (1984), we use (6) to write a condition for trial, 0 > ca + c p + αi β (D − W ).18 In their model, the left-hand side has the difference between the defendant’s and the plaintiff’s estimate of the probability of plaintiff win; in our model this difference equals to 0. When stakes are asymmetric in favor of the plaintiff, Priest and Klein predict relatively more plaintiff wins in disputes that reach trial. Likewise, in our model, if D − W < 0, the condition for trial is more likely to hold the higher the probability of plaintiff win (α i β ), and this probability is higher when evidence is weak. 1 > W 0 , which we derive using (6): The thresholds for late settlement are Wlate late i Wlate :=

ca + c p

αi β

+ D for i = 0, 1.

(7)

1 > W 1 . Therefore, there are values of W in the high-returns-to-litigation range for which, if the alleged Note that Wlate T infringer found strong evidence, the parties would settle late.19 By (5) and the assumption that the patentee makes a takei it-or-leave-it demand, when W ≤ Wlate the patentee’s late settlement demand equals

si2 = αi β D + ca .

(8)

If the first settlement demand was rejected, then before entering the second stage of negotiations, the alleged infringer expends resources ea to search for evidence of invalidity. Using πah to denote the alleged infringer’s payoff in the highreturns-to-litigation range, the alleged infringer chooses ea to minimize his cost from the dispute:

−πah = min[(1 − p(θ , φ , ea ) )(α0 β D + ca ) + p(θ , φ , ea )(α1 β D + ca ) + da + ea ],

(9)

ea ≥0

where p(θ , φ , ea ) is given in (1).20 An −(1 − θ + θ φ ) f (ea )(α0 − α1 )β D + 1 = 0. Thus,



e∗ah

=



f −1 (1−θ +θ φ )1(α −α )β D 0 1 0



interior

optimal

choice

of

ea

solves

the

first

order

condition:



if f (0 ) > (1−θ +θ φ )1(α −α )β D  0 1 .  otherwise

(10)

If the parties do not settle early, then in the second period the parties could: (i) go to trial regardless of whether evidence of invalidity is weak or strong (when W is high); (ii) go to trial only if evidence is weak (intermediate W); (iii) settle either way. Considering these options, we find the patentee’s expected payoff if an early settlement is not reached. Writing it as a function of W (fixing all other parameter values) we have:

⎧ 0 1 (1 − p∗h )πT,p (W ) + p∗h πT,p (W ) ⎪ ⎨   0 π ph (W ) = (1 − p∗h )πT,p (W ) + p∗h s12 − d p ⎪     ⎩ (1 − p∗h ) s02 − d p + p∗h s12 − d p

18

1 if W > Wlate , 1 0 if Wlate ≥ W > max{Wlate , WT1 },

(11)

0 if Wlate ≥ W ≥ WT1 ,

In Priest and Klein (1984) Eq. (10) is the following condition p p − pd >

C−S J

+ P J J . Using our notation, p p − pd = 0, C = c p + ca , S = 0, P = αi β , J =

D − W, J = D+2W , and multiplying by J we obtain the condition we just stated. c +c c 19 0 0 and WT1 depends on the parameters of the model. WT1 < Wlate if and only if α pβ < αp βa + D. This holds when the alleged infringer’s The order of Wlate 1 0 trial costs (ca ) and damages (D) are high enough. 20  The function in (9) is convex in ea , because the search technology is such that f (ea ) < 0. The unique optimal expenditure of resources e∗ah is finite because the benefit of search is bounded and so, as ea → ∞, πah → −∞.

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i where πT,p and si2 are defined in Eqs. (2) and (8), p∗h = (1 − θ + θ φ )F (e∗ah ) and e∗ah is given in (10). If the parameters were

such that π ph (W ) < 0, then if no early settlement was reached, the patentee would drop the suit before the discovery stage, and so the alleged infringer would reject any s1 > 0, and the patentee would not have filed a suit to begin with. Thus, if the suit was filed, it must be that π ph (W ) ≥ 0.21 When the suit was filed, an early settlement can be reached if there exists s1 such that π ph (W ) ≤ s1 ≤ −πah . The parties





h h settle early if W ≤ Wearly where π ph Wearly = −πah . In this case, the patentee demands



sh1 = −πah = (1 − p∗h )α0 + p∗h α1

 β D + ca + da + e∗ah .

(12)

Fig. 2 illustrates the following two possible cases: in panel a, the solution to

π ph (W )

=

−πah

falls in the high segment of

  1 π ph (W ), W > Wlate ; and in panel b, it falls in the intermediate segment.22 Lemma’s 1 and 2 in the appendix show that the intersection is in the intermediate range only when there are “high trial costs”. Definition 2. There are “high trial costs” when c p + ca >

(1 − θ + θ φ )F (e∗ah ), otherwise, there are “low trial costs”.23

  α1 e∗ah +da +d p   ∗ (α0 −α1 ) 1−ph

where e∗ah is given in (10) and

p∗h =

When trial costs are high relative to the costs incurred between settlement rounds, there is a range of returns that are high enough so that no early settlement can be reached, yet not high enough for the patentee to litigate to trial if strong evidence is revealed. In this range, if the alleged infringer finds strong evidence, the parties settle late. The outcome of the patent dispute in the high-returns-to-litigation range is as follows:



c



h Proposition 1. If the patentee filed a suit, in the high-returns-to-litigation range W > WT1 = βαp , there are thresholds Wearly ≤ 1 h Wtrial so that: h . (i) The parties reach a trial if W > Wtrial

h (ii) The parties settle late if evidence of invalidity is strong and go to trial if evidence is weak, if Wtrial ≥W >



h max Wearly , WT1 . h (iii) The parties settle early if Wearly ≥ W > WT1 . e∗ +da +d p

∗

c +c

c +c

a p a p h h h h The thresholds are Wearly = βah (1−p∗h )α0 + βα0 + D < Wtrial = α1 β + D when trial costs are high and Wearly = Wtrial =

eah +da +d p +ca +c p  + D, when trial costs are low, where e∗ is given in (10) and p∗ = (1 − θ + θ φ )F (e∗ ).

ah h ah β (1−p∗h )α0 + p∗h α1

The thresholds exceed D, which implies that a late settlement or a trial can arise only when W > D. Thus, in our symmetric information model, stakes asymmetry in favor of the patentee is necessary for late settlements or trial. We provide two simple examples illustrating asymmetric stakes in the online appendix. Fig. 3 illustrates the outcomes of patent dispute for different combinations of W and cp (fixing all other parameters). The solid black line indicates the threshold WT1 that separates the high-returns and the low-returns ranges. Above this line, parameters are in the high-returns-to-litigation range, and the figure shows the three ranges of outcomes in Proposition 1. The late settlement range is non-empty for sufficiently high trial costs (cp ) and intermediate levels of return (W). The intuition for why a late settlement may be reached is that the return to litigation could be high enough for the patentee not to want to settle early, yet after new evidence against patent validity is revealed, the patentee becomes more pessimistic about her chances of prevailing in court. In this case, she might be willing to make a small enough settlement   demand that will be accepted by the alleged infringer just before a trial. For sufficiently large returns, as trial costs ca + c p increase compared to the search and discovery costs, the set of parameters for which late settlements  h occur  is larger. A h h decrease in discovery costs da + dd results in a lower threshold Wearly , and thus, in a wider range Wearly , Wtrial in which a settlement is reached only in the second stage if strong evidence was found. 4.2. Low returns to litigation c

If W ≤ WT0 = βαp , a trial is not a credible threat and the patentee would not have filed a suit. Consider WT1 ≥ W > WT0 , 0 so that after the second stage of negotiations, if the parties did not reach an agreement, the patentee drops the suit if the 21 i The condition π ph (W ) ≥ 0 can be written in terms of exogenous parameters by substituting p∗h , πT,p (W ) and si2 and the thresholds which were all derived in terms of exogenous parameters. It holds when the discovery costs, dp , are not too high. The following sufficient condition is obtained from the low range of (11) using p∗h ≤ (1 − θ + θφ ): [α1 + θ (1 − φ )(a0 − α1 )]β D + ca > d p . 22 It can be verified that (1 − p∗h )(s2,0 − d p ) + p∗h (s2,1 − d p ) < −πah = s1 . The patentee’s payoff is higher with early settlement than if the parties settle late





0 whether the evidence is weak or strong. Thus, there is either a solution to π ph (W ) = −πah for some W > max Wlate , WT1 , or π ph (W ) > −πah , for all W ≥ WT1 . 23 The right-hand side of the “high trial costs” does not depend on cp , ca because e∗ah is independent of these parameters. Therefore, this condition holds for some high enough trial costs.

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−

= Early settlement

Trial

Panel a: Low Trial Costs.

−

= Early settlement

Late if strong Trial if weak

Trial

Panel b: High Trial Costs. Fig. 2. Payoffs and thresholds in the high-returns region, for low or high trial costs.

alleged infringer has strong evidence of invalidity, but litigates to trial if the alleged infringer found only weak evidence. The alleged infringer will reject any positive demand for settlement if he has strong evidence of invalidity. If the evidence is 0 , the parties will agree on a settlement, s0 = α β D + c ; and if W > W 0 , the patentee will litigate weak, then if W ≤ Wlate a 0 2 late to trial. If the initial settlement demand is rejected, and the patentee did not drop the suit, the alleged infringer will expend resources, ea that solves

−πa = min[(1 − (1 − θ + θ φ )F (ea ))(α0 β D + ca ) + da + ea ]. ea ≥0

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Fig. 3. Thresholds and regions of dispute outcomes by patentee trail cost and return. Note: The bold line indicates the threshold W T1 which separates between the high returns to litigation area (above the bold line) and low returns to litigation (shaded area below the line). In the high returns to litigation h h = Wearly = WThT (eah ). range: when cp is in the low trial costs range, the dotted segment represents the threshold below which the parties settle early, Wtrail h 1 When cp is in the high trial costs range: the thin dashed line represents the threshold above which the parities go to trial, Wtrail ; the thick dashed = Wlate h h h line is the threshold below which the parties settle early, Wearly = WThS (eah ). In the region between Wtrail and Wearly , the parties settle late if strong evidence of invalidity is revealed and go to trial if the evidence is weak.

The alleged infringer’s optimal ea in the low-returns range is



e∗a =



f −1 (1−θ +θ φ )1(α β D+ca ) 0 0





if f (0 ) > (1−θ +θ φ )1(α β D+ca )  0 .  otherwise

(13)

Denoting p∗ = (1 − θ + θ φ )F (e∗a ), the patentee’s payoff if a settlement was not reached is:



π p (W ) =

0 (1 − p∗ )πT,p − p∗ d p

0 if WT1 ≥ W > Wlate ,

(1 − p∗ )s02 − d p

0 if WT0 < W ≤ Wlate .

(14)

0 , an early settlement is reached and if W 0 < W ≤ If dp is low enough so that a suit was filed, then if WT0 < W < Wlate late a settlement s1 is reached if

WT1 ,

(1 − p∗ )(α0 β D + ca ) + da + e∗a ≥ s1 ≥ (1 − p∗ )(α0 βW − c p ) − d p .

(15)

The settlement demand is then

s1 = (1 − p∗ )(α0 β D + ca ) + da + e∗a .

(16)

The outcome of the patent dispute in the low-returns-to-litigation range when the patentee sues for infringement is as follows:



c

c



Proposition 2. When there are low returns to litigation and a suit is filed W ∈ (WT0 , WT1 ] = ( βαp , βαp ] , there is a threshold 0 1  Wearly so that  (i) If Wearly < W ≤ WT1 , the parties do not settle; if the evidence of invalidity is weak they go to trial and if it is strong the patentee drops the suit.  (ii) If WT0 < W ≤ min{Wearly , WT1 }, the parties settle early. h Moreover, if in the high-returns-to-litigation range early settlements can occur (Wearly > WT1 ), then in the low-returns-to-

 litigation range there can only be early settlements (Wearly ≥ WT1 ). e∗ +d +d

c +c

a p  The threshold is Wearly = βa(1−pa∗ )αp + βα + D where e∗a is given in (13), and p∗ = (1 − θ + θ φ )F (e∗a ). 0 0 

In Fig. 3, the low-returns-to-litigation outcomes are illustrated below the solid line, WT1 . 4.3. Search for evidence and settlements For given parameter values (β , θ , φ , α 0 , α 1 , D, ca ), the alleged infringer expends more resources in the low-returns range than in the high-returns range, e∗a > e∗ah . Intuitively, this is driven by the difference in the credibility of trial in the

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Trial

Early settlement

Low trial costs

Late settlement or trial

High trial costs

Fig. 4. Thresholds and outcomes in the high returns to litigation range. Note: For low values of the probability there are “low trial costs” relative to the h h h h costs of search for evidence of invalidity. In this range, Wtrial = Wtrial . For θ in the high trail costs range, Wtrial > Wearly , for returns between these thresholds the parties settle late if evidence of invalidity is strong, and go to trial if evidence is weak.

two ranges. With high returns to litigation, a trial is a credible threat even if the alleged infringer has strong evidence of invalidity, while with low returns, a trial is only a credible threat if the alleged infringer has weak evidence. As a result, the benefit from finding strong evidence of invalidity in the low-return range is higher than in the high-returns range. The expenditure e∗a is continuous in all parameters in each range (high or low return returns to litigation). However, due to the differences in the credibility of the threat of trial, there is a discontinuity in e∗a when a change in α 1 , W, cp or β causes a switch from the high to the low-returns-to-litigation range. Nevertheless, e∗a is (weakly) monotone with respect to all the parameters of the model, except for the probability of infringement (β ). As we would expect, e∗a and e∗ah increase with the probability of infringement, because the alleged infringer benefits more from showing invalidity the more likely he is to be found infringing. However, an increase in the probability of infringement that just causes a switch from the low-returns-to-litigation range to the high-returns-to-litigation range, results in a discrete jump down in the expenditure of resources to invalidate the patent. We derive detailed comparative statics in the appendix proposition A1. Here, we focus on the effect of the initial probability that the patent is good. Proposition 3. (i) The alleged infringer expends less resources to find evidence of invalidity as the probability that the patent is good (θ ) increases. (ii) Disputes are more likely to settle early when θ is lower. Fig. 4 illustrates the wider range of W that leads to early settlement when the values of θ are low. As θ increases, the probability of finding invalidating prior art decreases, because strong evidence is less likely to be found when the patent is good, and because the alleged infringer expends less resources when θ is high. Turning to settlement amounts, recall that a late settlement could only be reached in the high-returns range, and only when strong evidence of invalidity was revealed. Proposition 4. (i) The early settlement amount increases with the probability that the patent is good, θ ; (ii) when a late settlement is reached, it is lower than the highest amount the alleged infringer would have been willing to settle for in the first stage, s12 < sh1 . The early settlement amount increases with θ because, the alleged infringer’s probability of finding strong evidence of invalidity decreases with θ allowing the patentee to demand a higher settlement. The second period settlement is lower because the parties settle late only when the patentee’s position has been weakened by new evidence of invalidity. Additionally, the alleged infringer’s search for evidence and additional discovery costs have been sunk at the time of late stage negotiations. In the appendix proposition A2 we derive detailed comparative statics for settlement amounts.

4.4. The decision to file an infringement suit Anticipating the chain of events described in the previous sections, the patentee who identifies an alleged infringer decides whether to file a suit at an initial cost, c0 . This cost captures preliminary legal counsel, time, and administrative fees (but it does not include future litigation costs which we account for separately). It can also reflect expected losses from a possible counter infringement suit. The patentee weighs c0 against her expected payoff from initiating the suit. Let denote all the model parameters save θ .

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Proposition 5. In equilibrium there is a cutoff patent quality, θ ∗ ( ), such that, all else equal, the patentee files an infringement claim if θ > θ ∗ ( ). The threshold θ ∗ ( ) (weakly) increases with the cost of initiating the suit c0 , the patentee’s discovery cost, dp , and trial cost cp , and decreases with the return to successful litigation W. There are parameter values for which a suit is never filed even if the patent is known to be good, θ = 1. This can arise when trial cost is too high so the patentee never continues to trial (βα 0 W < cp ), or when discovery costs are so high so that the patentee would always drop the suit if early settlements fail. In these cases θ ∗ ( ) = 1. 5. Fee shifting In the US, each party in a patent dispute typically pays its own legal fees. The court awards reasonable attorney fees to the prevailing party only in exceptional cases. However, provisions for attorney fee shifting in patent litigation have been proposed and debated. Proponents suggest that this rule can deter low-merit legal suits and eliminate nuisance suits in which the plaintiff is unwilling to litigate to trial but only attempts to extract settlement fees (Chen, 2013). Fee shifting makes the benefit from winning and the costs of losing larger. In our model, the patentee can drop the suit when new evidence against her patent is revealed. Thus, fee shifting could have a positive effect on the patentee’s expected payoffs and encourage suits.24 Assume that if the parties reach a trial, the court shifts litigation fees (dp , cp ) or (da , ca ) to the loser, but not the additional resources that the alleged infringer expends trying to invalidate the patent (ea ). Keep all the other assumptions we made earlier and additionally assume that

βα0 >

da + ca > βα1 where C = da + d p + ca + c p . C

(17)

This inequality implies that α i are such that if the alleged infringer finds  weak  evidence of invalidity, then fee shifting is expected to save the patentee litigation costs, (1 − βα0 )(da + ca ) − βα0 c p + d p < 0; while if he finds strong evidence of





invalidity, fee shifting is expected to save the alleged infringer costs, (1 − βα1 )(da + ca ) − βα1 c p + d p > 0. With fee shifting, the expected payoffs of the patentee and alleged infringer if they go to trial are i T,p π = αi β W − (1 − αi β )C, i T,a π = −αi β D − αi β C − ea .

Since only the court can determine which party prevails and award fees, if the patentee drops the suit, payoffs are the costs already incurred, −d p for the patentee and (−da − ea ) for the alleged infringer. The threshold return for the patentee to continue to trial given evidence i is

i = (1 − βαi )C − d p = c p + (1 − βαi )(da + ca ) − βαi (c p + d p ) . W T

βαi

βαi

βαi

(18)

Comparing with the thresholds obtained in (4) for the system without fee shifting we find: Proposition 6. Given (17), in a system with fee shifting, a trial is credible for a smaller range of returns when evidence is strong, 0 < W 0 . 1 > W 1 , but for a wider range of returns when evidence is weak, W W T T T T The inequalities hold because with fee shifting, the patentee’s expected cost is higher when evidence is strong, and lower when evidence is weak compared to the system without fee shifting. Hence, with fee shifting the patentee is more likely 0 , W 0 ], some patentees will assert the patent even to drop the suit when evidence is strong. In the non-empty range (W T T though they would not have done so in a system without fee shifting. Fee shifting could encourage even a patentee with a low-quality patent to file an infringement suit because it makes going to trial when evidence is weak a credible threat for a wider range of returns than in the system without fee shifting. These cases settle early (see Lemma 6 in the Appendix). If the parties do not settle early the alleged infringer solves:

min{[(1 − (1 − θ + φθ )F (ea ))βα0 + (1 − θ + φθ )F (ea )βα1 ](D + C ) + ea }. ea ≥0

The optimal expenditure on finding evidence of invalidity in the system with fee shifting is therefore,



∗ = f −1 e ah



1

.

(1 − θ + φθ )(α0 − α1 )β (D + C )

∗ > e∗ . That is, for values of W that fall in the high-returns-to-litigation range with or Comparing with (10) we find that e ah ah without fee shifting, the alleged infringer’s expenditure is higher with fee shifting, and it increases with the patentee’s and the alleged infringer’s fees (cp , ca , da , dp ). Similarly, if W is in the low-returns range in the system with fee shifting:



∗ = f −1 e a

24

1

(1 − θ + φθ )[α0 β (D + C ) − da ]



.

A similar intuition emerges in Bernhardt and Lee (2015) who analyze trial incentives in a model with sequential litigation.

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∗ > e∗ > e∗ . Comparing with (13) we find e a a ah If the parties do not settle early, the alleged infringer would expend more resources on finding evidence of invalidity in the system with fee shifting than without fee shifting. This gives the parties an extra incentive to settle early. If they settle late, the settlement amount will be smaller because they settle late only if the alleged infringer found strong evidence of invalidity, and in this case, he expects lower litigation costs in a system with fee shifting than in a system without fee shifting.

Proposition 7. Under condition (17), in a legal system with fee shifting compared to the system without fee shifting: (i) If the parties do not settle early, then the alleged infringer will expend more resources searching for evidence of invalidity; (ii) The parties are less likely to settle late; (iii) In a late settlement, the settlement amount is smaller.25 6. Discussion 6.1. Other policy changes The Leahy–Smith America Invents Act (AIA), enacted on September 16th, 2011, introduced broad reforms to the US patent system. We discuss provisions that fall within the scope of this paper. After March 16th, 2013, pursuant to the AIA, the USPTO deemed several additional forms of prior art admissible, for example, foreign public use technologies. Additionally, the effective dates for admissible prior art shifted from before the date of invention to “before the effective filing date of the claimed invention”.26 The change means that for a given patent, more prior art could potentially be considered invalidating. We interpret this as a decrease in the ex-ante probability that the patent is good, θ . Our model predicts that this change would decrease a patentee’s expected payoff, which makes her less likely to file an infringement suit (Proposition 5). If a suit was filed and the parties did not settle early, we expect more intense search for evidence by the alleged infringer (Proposition 3). A policy change that strengthens the non-obviousness requirement for patentability can also be modeled as a decline in θ and would result in the same qualitative changes as those described above. In 2012, the AIA introduced Inter Partes Reviews (IPR) as a new procedure to challenge patent validity. Compared with challenges of validity in civil court, IPR challenges cost less and require a lower burden of proof to invalidate a patent (Bar and Costello, 2019). In our model, we could view these differences as suggesting lower pre-trial cost (da ) and a lower probability that the patentee prevails in a validity challenge when the alleged infringer finds strong evidence of invalidity (α 1 ).27 Thus, if no early settlement is reached, the alleged infringer will expend more resources to search for evidence of invalidity. Late settlement decreases because α 1 decreases, and early settlement decreases too because of both the decrease in α 1 and the decrease in da (see Proposition A2). However, the effect of IPR on da + e∗a is ambiguous (lower da but higher e∗a ), and therefore the effect on the early settlement threshold, and on the timing of settlement is indeterminate. 6.2. Patent assertion entities PAEs differ from their practicing entity counterparts. First, as they specialize in litigation, PAEs likely face lower litigation costs.28 Second, PAEs may avoid the risk of countersuit and not face product disruptions which are typically seen in patent suits between practicing firms (Chien, 2012). This might suggest that PAEs are more likely to fall into the high-returns-tolitigation range. In this range, if the parties did not settle earlier, the patentee would proceed to trial even if the alleged infringer has strong evidence of invalidity. This observation is consistent with the empirical findings of Allison et al. (2011) that when taking cases to judgement “product-producing entities are far more likely to win their cases than NPEs.” Additionally, all else equal, we expect that with lower costs the threshold θ ∗ for initiating a suit is lower (Proposition 5). Thus, PAEs would be willing to acquire and assert relatively lower quality patents, which might result in PAEs holding lower quality patents overall. 6.3. The probability of validity in trial We assumed so far that the probability that a judge affirms validity depends only on the evidence presented in court, and it is constant conditional on evidence i. This means that the alleged infringer searches for evidence before the trial, and that judges do not know, or do not use the parties’ prior belief that the patent is good (θ ) or the expenditure (e∗a ) 25 If the inequalities in (17) are reversed, (which we find less plausible), the results in Proposition 7 (i) and (ii) continue to hold if W is in the high  < W < W . The results in Proposition 7(iii) and returns to litigation range in both systems, or in the low range in both systems. But not necessarily if W h h in Proposition 6 are reversed. 26 Prior art definitions are provided in 35 U.S.C. 102. More details about these changes and the legal definition of effective filing date are provided in https://www.uspto.gov/web/offices/pac/mpep/s2151.html. 27 With IPR, validity will likely be decided before the judge rules on infringement. Late settlements should now be interpreted as ones taking place after evidence of invalidity is revealed, but before an IPR ruling is made. Settlements during the IPR process are permitted. 28 See for example the 2013 report “Patent Assertion and US Innovation” at http://www.whitehouse.gov/sites/default/files/docs/patent_report.pdf.

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to make decisions. A judge’s decision is likely based on knowledge of the law and on the evidence presented, but not on other information specific to the current case. This allows judges to remain impartial. As law professor Kermit Roosevelt writes, “[Judges] must make factual findings based only on the evidence presented by the parties....”29 The Code of Conduct for United States Judges indicates that “[a] judge shall disqualify himself or herself in a proceeding in which the judge’s impartiality might reasonably be questioned, including but not limited to instances in which: (a) the judge has a personal bias or prejudice concerning a party, or personal knowledge of disputed evidentiary facts concerning the proceeding...”30 While we believe that constant α i is a reasonable simplifying assumption, one can consider relaxing it. For example, suppose one maintains the assumption that α 0 is constant,31 but assumes that if evidence of invalidity is strong then the probability that the patentee wins the validity challenge is α 1 (θ , φ ), where α 1 (θ , φ ) increases in the probability θ that the patent is good and in φ , the search technology parameter. Then the results and proofs of Propositions 1 and 2 do not change. Propositions 3–5 also continue to hold, but the proofs must be revised to account for the effect of θ on α 1 . However, some of the comparative statics results with respect to θ and φ change (Propositions A1 and A2). In our main model, at the time of the late negotiations, uncertainty about the strength of evidence has been resolved, and the amount of late settlement, s12 = α1 β D + ca , does not depend on θ or φ . But if α 1 increases with θ and φ , then s12 increases with these c parameters. Similarly, the trial threshold, WT1 = βα (pθ ,φ ) , decreases with θ and φ , so that if the parties did not settle and 1

evidence is strong, a trial is more likely when θ and φ are higher. Building on Priest and Klein (1984) and Kamenica and Gentzkow (2011), in the online appendix we offer a simple version of our model with Bayesian persuasion that gives rise to the relation α 1 (θ , φ ) described above. 7. Search for evidence of infringement We consider now the possibility that the patentee expends resources to establish infringement. Evidence of infringement can be strong or weak, resulting in probabilities β 1 or β 0 that the court will find the patent infringed. To focus on one c +d

p p simple case, assume β0 < W α0 so that a trial is not credible if evidence of infringement is weak and the patentee only files suit if she finds strong evidence of infringement. Denote by ep the patentee’s expenditure on search for evidence of infringement. Strong evidence is found with a probability ηG(ep ) where η is a prior probability of infringement, and the function G(ep ) is increasing and concave. Assume at first that the patentee searches before filing the initial suit, and she sues only if she finds strong evidence of infringement. Thereafter, the game proceeds exactly as described in our main model. Since the patentee’s expected profit increases with the probability that the patent is good, if θ is higher, the patentee will spend more resources to find evidence of infringement. Alternatively, suppose that if an early settlement was not reached, the patentee and the alleged infringer simultaneously choose expenditures ep and ea to prove infringement and invalidity, respectively. All other earlier assumptions remain unchanged. Assume also parameters are such that there are high returns to litigation, W > WT1 . The alleged infringer maximizes his payoff solving:

min ea ≥0

{ηG(e p )[(1 − p(θ , φ , ea ))(α0 β D + ca ) + p(θ , φ , ea )(α1 β D + ca )] + da + ea },

where p(θ , φ , ea ) is given in (1). In the interior range:



ea (e p ) = f −1

ηG(e p )β (1 − θ



1 . + φθ )(α0 − α1 )D

(19)

This reaction function is increasing in ep . Intuitively, the more the patentee searches, the higher the probability that she would establish infringement, in which case establishing invalidity is more beneficial to the alleged infringer. The patentee solves:

π ph = max{ηG(e p )[(1 − p(θ , φ , ea ))0,1 + p(θ , φ , ea )1,1 ] − d p − e p } e p ≥0

where,

i,1 = max {αi β D + ca , αi βW − c p } for i = 0, 1. In the interior range:



e p (ea ) = g−1

1

η[(1 − (1 − θ + φθ )F (ea ))0,1 + (1 − θ + φθ )F (ea )1,1 ]

 .

(20)

29 Statement made in a 2016 NYT article by Kermit Roosevelt, a professor of law at the University of Pennsylvania, see https://www.nytimes.com/ roomfordebate/2013/11/03/judges- appearance- of- impartiality/ways- a- judge- should- and- should- not- be- impartial. 30 See Code of Conduct: https://www.uscourts.gov/judges-judgeships/code-conduct-united-states-judges#d For an example in which a judge explicitly objected to arguments using Bayes’ rule see https://understandinguncertainty.org/court- appeal- bans- bayesian- probability- and- sherlock- holmes. 31 For example, α0 = 1 could represent a presumption of validity and that weak evidence does not meet the challenger’s burden of proof.

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This reaction function is decreasing in ea , because the patentee’s payoff in the case of weak evidence of invalidity is higher than her payoff with strong evidence of invalidity, 0,1 > 1,1 . The more the alleged infringer searches, the more likely evidence of invalidity will be strong, and the patentee’s benefit from proving infringement is lower the more likely the patent is invalid. Proposition 8 (Reaction functions and equilibrium expenditure searching for evidence). With simultaneous search for evidence, and high returns to litigation, W > WT1 : (i) The alleged infringer’s resource expenditure on finding evidence to prove invalidity (ea ) increases  ∗ ∗  with the patentee’s expenditure to prove infringement (ep ). (ii) ep decreases with ea . (iii) There is a unique equilibrium e p , ea . The reason for the difference in the shape of their reaction functions is that, in patent litigation, the patentee wins if the court finds the patent valid and infringed, and for the alleged infringer to win it suffices if it is either found invalid or not infringed. 8. Concluding remarks We offer a model of patent litigation that accounts for the timing of settlement and for the resources that an alleged infringer expends to invalidate the patent that he is alleged of infringing. Our findings support the view that we cannot rely on the court to correct the imperfections of the patent office. We find that the parties are more likely to settle early when the initial probability that the patent is good is low. If they do not settle early, they are more likely to continue to trial if evidence of invalidity is weak than if it is strong. As a result, many disputes involving “bad” patents would not reach a trial and these patents will remain in the economy. The way courts rule can have implications on the timing and size of settlements, and on the outcomes of patent validity challenges. For example, injunctions might contribute to stakes asymmetry and increase late settlements. Also, higher damages increase an alleged infringer’s expenditure on searching for evidence to invalidate the patent. We assumed that the probabilities of court ruling only depend on evidence. Our findings remain largely unchanged with a simple model of a Bayesian court that allows the pre-trial probability that the patentee will prevail given strong evidence to increase on the initial probability that the patent is good. A more general model of Bayesian courts is left for future work. In light of our model, we argue that PAEs, who likely have lower costs associated with litigation, will tend to assert patents of lower quality. For a given patent quality, PAEs are more likely to fall into a high-returns-to-litigation range, meaning that they will be more inclined to litigate even when there is evidence questioning the validity of their patent. The firms PAEs accuse will expend the same or less resources on searching for evidence of invalidity than if the patentee was a practicing entity. Our model suggests that, in a system with fee shifting, alleged infringers will expend more resources when searching for evidence to invalidate a patent. Parties will be more likely to settle early. Some patentees that hold patents of low returns will have an incentive to file an infringement suit, while absent fee shifting they would not have initiated the dispute. This is true even for patents of low quality. Therefore, fee shifting does not necessarily discourage infringement suits on low quality patents by patentees who intend to benefit from early settlement. Welfare considerations for patent litigation are complex, as the outcome of a dispute often affects parties that are not directly involved in the dispute, such as additional users of the invention, consumers and future innovators. If bad patents impose a social cost, it might be socially desirable to determine the validity of disputed patents that are likely “bad”. Our analysis, however, shows that disputes over low quality patents are more likely to settle early, and if they don’t, late settlements are more likely when evidence of invalidity is strong. IPR, which was introduced following the America Invents Act, offers alleged infringers a way to administratively challenge patent validity. In the current system, the review process often terminates if the parties settle. Given our findings, we suggest that it could be socially beneficial to require that any IPR be continued until a decision on validity is made even if the parties settle. 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