Little Patents and Big Secrets: Managing

Intellectual Property ∗

James J. Anton� and Dennis A. Yao�

August 5, 2000

We examine how an innovator should manage its intellectual property when con-fronted with limited intellectual property rights and possible imitation. Exploita-tion of an innovation commonly requires some disclosure of enabling knowledge toselected Þrms or to the public (e.g. to obtain a patent or induce complementaryinvestment). When property rights offer only limited protection, the value ofthe disclosure is offset by the created threat of imitation. Our model incorporatesthree features critical to understanding this decision: innovation creates asymmet-ric information, innovation often has only limited legal protection, and disclosurefacilitates imitation by transferring enabling knowledge. Imitation depends inpart on inferences the imitator makes about the innovator�s advance. We Þnd anequilibrium in which small inventions are fully disclosed, medium inventions areprotected by both legal property rights and secrecy, and large inventions are pro-tected primarily through secrecy when property rights are weak. Our discussion isframed in terms of a Þrm�s decision of what to patent, what to disclose, and whatto keep secret, but the model is adaptable to other intellectual property settings.

∗The authors thank Tracy Lewis, Marvin Lieberman, Rob Merges, Scott Stern, and seminarparticipants at Berkeley, Florida, FTC, NBER, Penn, and UCLA for helpful comments and theFuqua Business Associates Fund for research support.

�Associate Professor, Fuqua School of Business, Duke University, Durham, NC 27708-0120 and Visiting Scholar, Economics Department, University of North Carolina, Chapel Hill;james.anton@duke.edu

�Associate Professor, Wharton School, University of Pennsylvania, Philadelphia, PA19104-6372 and Visiting Scholar, Haas School, University of California, Berkeley;yao@wharton.upenn.edu

1. Introduction

We examine how an innovator should manage its intellectual property (IP) whenconfronted with limited intellectual property rights and possible imitation. Ex-ploitation of an innovation commonly requires some disclosure of enabling knowl-edge to selected Þrms or to the public (e.g. to obtain a patent, obtain an alliancepartner, or to induce investment in complementary assets). When property rightsoffer only limited protection, however, the value of the disclosure is offset by thecreated threat of imitation.1

Our analysis can be applied to IP settings involving patents, copyrights, con-tractual property rights, trade secrets and conÞdentiality agreements, as well as tothe polar case of no property rights. We focus on the decision of a Þrm concerninghow much of an innovation should be disclosed (with and without legal protec-tion) and how much should be kept secret. A major business concern is thatdisclosure through patenting or voluntary disclosure will provide competitors withusable information. In a survey of U.S. Þrms in twelve industries MansÞeld 1986[14] found that a substantial fraction of patentable inventions were not patented.This Þnding plus Cohen, Nelson, and Walsh�s 2000 [4] Þnding from a generalsurvey of U.S. manufacturing Þrms that secrecy was viewed as more importantthan patenting for appropriability indicate the importance of understanding thesecrecy-patent decision.2 Along these lines we explore, among other issues, whyÞrms (holding legal speciÞcities constant but allowing changes in the strength ofproperty rights) are likely to employ secrecy more heavily as the signiÞcance ofthe invention increases.Three features of the economic environment of innovation are critical for under-1Jolly 1997 [11], for example, argues that an important aspect of a business strategy for

mobilizing interest in a technology is �formulating a communication strategy that balancesinterest creation with secrecy..� (p.83)

2There are, of course, numerous anecdotes supporting the selection of secrecy over patenting.Jackson 1998 [10], p.41, for example, discusses Intel�s decision to keep a portion of a patentable�reßow� manufacturing process secret because they were concerned that a full patent wouldprovide competitors with too much information and also provides an example of a manufacturingtrade secret (�walking out�) that allowed Intel to maintain a complete monopoly over EPROMsfor nearly two years because of Intel�s superior yield from its wafers. Milgrim 1974 [16] gives theexample that a French company kept a valuable process for producing cellophane secret. DuPont spent millions of dollars in an unsuccessful attempt to duplicate this process. Many Þrmsincluding Kodak involve their business managers in the decision of what parts of the underlyingtechnology should be patented and what parts kept secret. (�How Kodak, Fearing Theft ofTrade Secrets, Mounted Its Own Sting� Wall Street Journal 11/25/96 p.1).

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standing the management of IP. First, incomplete information about the extentof innovation is often fundamental. Second, it is common for inventions to haveavailable only limited intellectual property protection. Third, enabling knowledgerevealed through disclosures makes imitation feasible.Asymmetric information gains force when property rights are limited. If an

innovation were fully protected, full disclosure would facilitate appropriation ofthe beneÞts associated with signaling the extent of innovation and would entail norisk of imitation or other unauthorized use.3 With limited protection, however,disclosure risks imitation and incomplete information remains a primary concern.Imitation depends on each of these factors. Disclosure determines what imita-

tion is possible and limited property right protection frequently makes unautho-rized imitation economically attractive despite the possibility of legal damages.4

Asymmetric information also Þgures prominently in the imitation decision. Forinstance, if an innovation were known a priori to be minor, a competitor mightprefer to remain with existing technology rather than imitate and risk legal dam-ages. If an innovation were known to be major, it could render the status quotechnology noncompetitive and trigger imitation or perhaps exit. Under incom-plete information an imitation decision must necessarily be based on an inferenceabout the extent of innovation and an assessment of the downstream competi-tive position relative to the innovator. The extent of disclosure and protectionchosen by the innovator provide an important clue. Thus, signaling concernsare important for managing IP vis-a-vis competitors. For example, Ford MotorCompany�s disclosure of substantial amounts of unprotected technical knowledgeabout its revolutionary moving assembly line system may have been motivated inpart for its value as a signal to competitors of its dominant low-cost manufactur-ing position.5 Inferences are also pivotal when the innovator discloses enablingknowledge to obtain sales (e.g. publications by management consulting Þrms),

3There are a number of beneÞts to making signaling disclosures beyond that of obtainingpatents. Such disclosures can induce weaker competitive responses or discourage rival innovation(e.g. some biotechnology Þrms avoid or abandon projects for which others are perceived to beahead). The disclosures are also valuable for inducing third-party complementary investmentor securing Þnancing.

4MansÞeld, Schwartz, and Wagner 1981[15] found, for example, imitation within four yearsof 60% of the patented successful innovations in their sample of 48 innovations.

5See, e.g., Nevins and Hill 1954 [18] for a discussion of the Ford disclosures. Publication is acommon form of disclosure. Based on extensive interviews with European and Japanese R&Dmanagers, Hicks 1995 [7] argues that publication is used by Þrms in part to gain �a reputationfor possessing useful tacit knowledge� and �signals the area ...and the quality of that work.�(p.420)

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secure third-party complementary investments or Þnancing (e.g. business plansrevealed to venture capitalists, academic research proposals) or in strategic al-liance negotiations (e.g. licensing, contract disputes over rights to IP) becausethe decisions of these third parties depend ultimately on an assessment of the rel-ative advantages of the innovator over its competition.6 Our basic model providesa foundation for understanding such interactions.Thus, the amount of the innovator�s disclosure is critical to the imitation

decision. In economic terms, one can view the innovator�s disclosure choice asa decision of how to �manage� market competition between direct competitorswhen imitation is a central concern. The relative positions of the competitorsare managed by moderating the amount of disclosed knowledge, and competitionoccurs in the shadow of an ongoing legal dispute over IP rights (e.g., Polaroid andKodak in instant photography).7

The strength of intellectual property rights is critical to our story and meritsfurther comment. If patents, copyrights, and trade secret law could fully protectall economically important inventions, circumvention and possible infringementwould be of secondary importance to the management of intellectual property.But clearly this is not the case. With respect to patents, for example, infringe-ment suits are common and surveys of Þrms on the question of appropriability ofIP suggest that Þrms in a majority of industries do not see patents as providingstrong appropriability (Levin, Klevorick, Nelson, and Winter 1987 [13] and Co-hen, Nelson, and Walsh 2000 [4]). Some reasons for these results include the easewith which some patents can be circumvented, the possibility that a patent will beinvalidated if challenged, and the sometimes modest damages awarded in success-ful infringement suits. Especially for process innovations, lack of appropriabilitymay also simply result from the difficulty of detecting infringement.8

6A common form of voluntary disclosure is a conference presentation or other public disclosuremade subsequent to a patent Þling but before the patent is issued and becomes public. (e.g.,�Fina Þled the patent application...and shortly afterword Dr. Ewen gave a speech about it [aplastic] at a symposium organized by Exxon [Fina�s competitor].� �Battle Over Patents PitsTwo Oil Concerns Against One Scientist,� Wall Street Journal 3/1/96, p.1; �[O]n December 14he mailed in his patent application. The next day, a joint article by him and Dr. Srinivasanappeared in the American Journal of Opthalmology.� �Patent Challenges Face Leader in LaserSurgery for Nearsightedness,� Wall Street Journal, 5/26/99).

7Other prominent examples include Intel and AMD over microprocessor technology,Kimberly-Clark and Procter & Gamble in disposable diaper technology, Visx and Nidek invision-correcting laser surgery, and Fonar, Johnson & Johnson and GE in MRI technology.

8An example of the difficulty of detection is provided in Northern Petrochemicals Co. v.Tomlinson, 484 F.2d 1057 (7th Cir.) 1973 in which a trade secret theft was discovered only

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We model these trade-offs in a duopoly competition setting where an innovat-ing Þrm (the �innovator�) has private information regarding an invention, givingit (possibly) lower costs of production than the other Þrm (the �follower�). Thedegree of innovation is a continuously valued variable, ranging from no advanceover the status-quo to major innovations that, under complete information, couldinduce the follower to exit. The innovator makes two choices: whether to patentand how much knowledge to disclose. Having observed the protection and disclo-sure choices of the innovator, and using these as a basis for assessing the inven-tor�s total knowledge, the follower decides whether to use the inventor�s disclosedknowledge and risk infringement or to stay with the old technology. Thus, ouranalysis conforms best to a process innovation setting. The subsequent marketcompetition is modeled as Cournot. The Cournot structure, while special, cap-tures the basic tension between the cost of enabling knowledge transfer and thebeneÞts of signaling toughness to a competitor, which is a central concern in avariety of relevant competitive settings. Market competition takes place underthe shadow of infringement, and legal damages, if any, are assessed after marketcompetition. We assume that damages take the form of payments linear in theimitating Þrm�s sales.9

Preview of Results Our analysis leads to four main results. For small inno-vations we Þnd an exclusion effect under which small innovations are patented andfully disclosed and no imitation occurs. The second result involves a licensing ef-fect where larger innovations are protected both through patents and secrecy and

after the victim acquired the perpetrator. See also Milgrim 1974 [16].Although the advent of the Federal Circuit (and its changes in policy) has resulted in increased

protection for IP in the United States in recent years, effective legal property right protectionin a wide variety of settings and industries should arguably still be characterized as limited.One problem is that the patent office issues patents somewhat leniently, relying, in part, onsubsequent litigation to make the ultimate validity determination on economically meaningfulpatents. The weaknesses in the protection regime are multiplied when one considers IP protec-tion internationally. Copyright protection is limited by its inherent narrowness while provinga trade secret violation is quite difficult.

9It is common for litigation involving infringement to last for many years and sometimes tobe resolved after the effective economic life of the invention has ended. A common remedyinvolves assessing a royalty on an infringer�s past sales that represents a �reasonable royalty�(sometimes augmented with a punitive component) and an injunction of the use into the future.By statute, the reasonable royalty constitutes the ßoor with respect to the damage award. (35U.S.C. § 284) In recent years, the US courts have been more willing to grant preliminaryinjunctions but such injunctions are difficult to obtain because of the burden of proof imposedon the patentee. See, e.g., Rhodes 1997 [20]. The economic implications of disclosure under apreliminary injunction are analyzed in Anton and Yao 1999 [2].

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imitation occurs, leading to an implicit licensing relationship between competi-tors. Third, we Þnd a waiver effect where, for sufficiently weak property rights,very large innovations are not protected via patent but instead through a strongreliance on secrecy. Finally, we have a no-exit result under which innovations thatwould be drastic (force exit) if the follower knew the innovator�s cost (i.e. undercomplete information) become incremental when the follower does not know theinnovator�s cost (i.e. under incomplete information). The common thread in theseresults is that weak property rights imply disclosure incentives that are relativelystronger for smaller innovations and, as a result, larger innovations are protectedmore through secrecy as a response to the problem of imitation by a competitor.Property rights provide protection to the extent that they discourage imita-

tion and create an expectation of a damage payment for the use of protectedknowledge. For a small innovation relative to the status quo technology, even arelatively weak patent will discourage imitation because the gain to using the newknowledge is insufficient to justify a possible infringement payment. This createsa strong incentive to patent and disclose fully since the threat of imitation canbe discounted. As a result, we Þnd that for a range of small innovations, whichexpands as property rights become stronger, that weak patents are economicallyequivalent to strong patents and, in equilibrium, the patent will fully disclose theinnovator�s enabling knowledge because of the downstream competitive beneÞt toappearing to have lower costs.When the disparity between costs associated with old and new technology is

greater, a sufficiently large (protected) disclosure will trigger imitation. A switchto an imitation regime means that expected damages no longer perform the �ex-clusionary� role. Instead, the competitors Þnd themselves in what amounts to alicensing relationship governed by the property rights regime. In this interpreta-tion of imitation and infringement, the innovator chooses the technology transfer(via the disclosure) and the license fee is set exogenously by the court (via ex postexpected per unit damages). Imitation can then be viewed as an exercise of theimplicit option to license.Because the exclusion effect leads high-cost types in the nonimitation range

to disclose fully, greater absolute-sized disclosures are required in equilibrium toachieve separation by innovators with more substantial advances. When imita-tion occurs, the innovator has an incentive for partial disclosure to preserve anadvantage over the rival. The disclosure-secrecy-signaling trade-off consists of thecost of the rival�s use of the disclosed knowledge and the beneÞts of expecteddamage payments (the licensing effect) and an anticipated weaker competitive

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response when low costs are signaled. In equilibrium, the enabling knowledgebehind a large innovation is not completely disclosed and, as the innovation be-comes larger, the gap between disclosed and actual knowledge increases: partialdisclosure and imitation go hand-in-hand.Weak patent protection exacerbates the imitation problem. As the size of the

innovation increases, the relative beneÞt from expected damages decreases becausethe larger gap between disclosed and actual knowledge implies an increasinglyinefficient production allocation for the innovator and the imitator. Togetherwith the substantial amount of disclosure required to separate in equilibrium, thisraises the question of whether a different signal, if it exists, may be preferable andthis leads to our third result.The signal we identify is for the innovator to give up the beneÞts of the patent,

say by eschewing the patent or through licensing the patent rights for a nominallump sum fee. Such an action substantially reduces the amount of disclosure(knowledge transfer) that is needed to separate from higher types and causesthe imitator to infer that the innovator has low costs. Because the knowledgedisclosure is effectively unprotected, though, only big-innovation low-cost Þrms arewilling to take this action. Thus, weak property rights and incomplete informationregarding the innovation lead to an incentive cost for large innovations under whichthe value of protection is outweighed by a combination of secrecy and unprotected,but smaller, disclosures.One prominent example that can be interpreted through the lens of disclosure

to signal low costs involves the actions of the Ford Motor Company in 1913-1915during their implementation of the arguably revolutionary moving assembly lineprocess for mass production of automobiles. During this period Ford alloweda number of journalists to write extensively about its (unpatented) processes.Hounshell 1984 [9] in a historical study of American manufacturing technologiesremarks that Ford �...educated the American technical community in the ways ofmass production.� (p. 261) One reason Ford may have done this was to signal itscompetitors that it had extremely low costs and that a head-to-head competitionwith Ford would have been foolish.10

10One series of articles in Engineering Magazine resulted in a 440 page book on Ford methodsof manufacturing which was published in 1915. Ford also disclosed much about its pre-movingassembly line mass production system. While the systems were described in detail, this wasclearly only a partial disclosure of the knowledge it took to make the system work and adaptthe system to various applications. Further, the efficiency of the system depended in part oneconomies of scale that Ford could exploit given its dominant sales position in the industry.Ford may have also signaled its low costs with aggressive pricing, its $5/day wage (double the

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The Þrst three results of the model (exclusion, licensing, and waiver effects)when taken together lead to an interesting implication for the traditional distinc-tion between incremental and drastic innovations. When the range of potentialinnovation is large relative to the status quo technology, a setting with completeinformation will lead the follower to exit the market when the innovator has asufficiently large success. Now consider a large success under incomplete infor-mation and weak property rights. Disclosure incentives lead to imitation whichmay make it feasible for the follower to remain in the market: here disclosureand protection choices signal a major innovation but simultaneously reduce thecost differential between the innovator and the follower. The distinction betweendrastic and non-drastic is endogenous in our analysis and, ceteris paribus, we Þndthat, in equilibrium, the follower cannot be forced from the market given weakproperty rights. The difference in conclusions under complete versus incompleteinformation highlights the value to studying IP transactions and decisions in anincomplete information setting.Our model implements protection via an ex post expected transfer of some

damage payment back to the innovator. This feature is not speciÞc to patenting,but is a general feature of property rights. Thus, the model allows an explorationof a wide variety of IP settings, including negotiations held under a conÞdentialityagreement between an innovator and a supplier that is threatening entry11, depar-ture of key employees who had access to trade secrets, and the sale of an inventionto a buyer that is skeptical of the invention�s value but is unwilling to talk unlessthe seller waives conÞdentiality rights (e.g. in the toy industry and with venturecapitalists). Further, the approach is helpful for developing intuition relevant tomore complex IP settings such as (potentially) licensing an invention to a Þrmthat does not know the value of the IP. This value depends on the resulting rel-ative competitive position offered by the license, which, in turn, depends on howmuch is licensed and how much remains exclusive to the innovator. We discussvarious applications in the conclusion.Literature Review The decision of whether to patent has been explored

directly in Horstmann, MacDonald, and Slivinski 1985 [8] and is an integral partof the analysis in the models of Scotchmer and Green 1990 [21] and Gallini 1992[5]. The patent decision has two critical components that have not been fully

going rate) and its 1914 sales-volume-triggered rebate policy. (See Nevins and Hill [18] andHoushell 1984 [9] for accounts of Ford actions during this period.)11See, e.g., �Hardball Beans Citrix, but it Recovers,� USA Today June 11, 1997 for a story

about negotiations between software-maker Citrix and Microsoft.

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explored in this previous work. First, what is patented (and/or publicly disclosed)is a decision on the amount of enabling knowledge to transmit to one�s competitorand, second, the amount that is disclosed is a signal of the total knowledge theinnovator possesses.12 Both of these aspects are important for understanding theimitation choice of the noninnovators.Horstmann, MacDonald, and Slivinski (HMS) 1985 [8] model an information

signaling problem but not the strategic choice in the amount of knowledge trans-fer.13 The innovator chooses whether to patent and the follower chooses either tostay out, imitate (without infringing), or directly duplicate (only possible if nopatent). The follower�s choice directly affects the innovator�s payoff. The inno-vator has private information about the competitors�s payoff for each action and,while this information has no direct effect on the innovator�s payoff, the patentchoice can provide a signal that inßuences the competitor. The optimal innovatorstrategy involves mixing between patenting and not patenting; the follower staysout of the market when the innovator patents and imitates when the innovatordoes not. In contrast, we Þnd infringement-risking imitation (closest to HMS�sduplication) in the face of a patent. This contrast underscores some basic dif-ferences between our work and that of HMS. Imitation in our model is moreattractive for a follower as the patent decision and disclosure transmits enablingknowledge as well as information about the innovator�s costs. Second, HMS donot examine cases where safe imitation or staying in the market will always beproÞtable, cases that one would frequently encounter when innovation providessmall cost reductions from current technology. Finally, no duplication is permit-ted against a patent. These are critical differences and we interpret HMS to bemost appropriate for cases where property rights are strong.Scotchmer and Green 1990 [21] and Gallini 1992 [5] focus on the impact of

patent policy on the incentives to innovate. The decision to patent or suppressthe innovation is important to their models, but is handled as a binary choicein which patents disclose fully and there are no incomplete information problemsregarding whether innovation has occurred. These properties make our concernwith the interaction between enabling knowledge transmission and information(cost) signaling essentially moot as the knowledge transmission and information

12The signaling and disclosure elements in our paper explore issues related to those exam-ined in Bhattacharya and Ritter 1983 [3], Milgrom and Roberts 1986 [17], Okuno-Fujiwara,Postlewaite, and Suzumura 1990 [19], and Anton and Yao 1994 [1].13HMS employ a leader-follower equilibrium concept which involves differences in observability

and commitment relative to the equilibrium concept we use.

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signal are inseparable.14

We begin with a discussion of the model. Sections 3 through 5 provide thecore analysis leading to our main propositions which characterize the patenting,disclosure, and infringement-risking imitation decisions under incomplete infor-mation. We end with a discussion of the results in the wider context of managingintellectual property, suggest some testable implications of our model, and in-dicate how the model informs our understanding of licensing, the incentives toinnovate, and disclosure to induce demand.

2. The Model

We examine the choices to protect intellectual property and disclose informationand the resulting market interactions in a model with two Þrms: an innovatingÞrm (innovator) and a competing Þrm (imitator or follower). Each Þrm is risk-neutral and seeks to maximize expected proÞts. The model has three stages.First, in the protection and disclosure stage, the innovator realizes an R&D out-come and decides whether to protect via patent and how much of the invention(�enabling knowledge�) to disclose. Next, after observing these choices, the fol-lower decides whether to use the disclosed knowledge or to stay with the priorstatus-quo technology. For simplicity we assume that the follower has not en-gaged in R&D. Finally, there is a competition stage in which market outcomesare determined and, following the market outcomes, a third party (court) deter-mines if the follower is liable for use of knowledge disclosed by the innovator. Wespecify each stage in turn and then deÞne equilibrium.

2.1. Protection and Disclosure Stage:

The innovator, i, privately observes the realized outcome of i�s prior R&D invest-ment. This outcome involves the discovery of a process innovation which entailsan associated marginal cost of producing (Þxed costs are set to zero). The inno-vation is fully summarized by this marginal cost, c, and we assume that c is drawnfrom a c.d.f. F with support [0, c̄]. The upper bound, c̄, is the cost of the priorstatus-quo technology (an R&D failure is an atom in F at c̄); We set the lower

14Our result that small innovations (in terms of variable cost reduction) are fully disclosedand not imitated in equilibrium is similar in spirit to a result in Gallini 1992 [5] in which a shortpatent life in conjunction with a Þxed cost of imitation implies no imitation.

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bound at zero as we seek to examine a wide range of potential cost innovations(results with a positive lower bound are a minor extension).The innovator chooses whether to protect its innovation with a patent. We use

{P,S} to denote the choice of patent, P, and secrecy (no patent), S. Firm i alsochooses how much enabling knowledge s to disclose. When i chooses to patent,P , the disclosure s can be interpreted as the amount of enabling knowledge thatis patented.15 An alternative interpretation is that s is a disclosure outside thepatent and that the underlying patent can potentially block the free use of thisdisclosed knowledge. Of course, we may also have disclosure when i choosessecrecy, S, although the disclosure is then not protected.A disclosure transfers technological information that makes it feasible for Þrm

j to produce at cost s. We require that

s ≥ c (2.1)

so that the innovator cannot disclose more knowledge than is actually possessed.The innovator can disclose less. We refer to s > c as partial disclosure; j doesnot directly observe how much enabling knowledge remains conÞdential since c isprivate information of i.Our model does not allow licensing a competitor as a strategic option, but

this is not unrealistic for licensing involving direct competitors in a concentratedindustry where antitrust considerations would severely circumscribe licensing pos-sibilities. We discuss licensing in the concluding section.

2.2. Infringement-Risking Imitation Stage

The follower observes whether the innovator patented or not and the disclosedknowledge. Given (s,S), there is no patent in place and the follower is free touse disclosed knowledge without penalty and produce at cost s. Given (s,P), thefollower decides whether to use the disclosed knowledge of s or the old technology.These choices are denoted by I, infringement-risking imitation, and N , no imita-tion. As we specify below, a choice to imitate is actually a choice to risk a legalÞnding of infringement. We assume that the innovator can observe whether thefollower has chosen to imitate or not prior to the subsequent competition stage.16

15It is feasible to patent any invention (c < c̄) and the direct cost of obtaining a patent isassumed to be zero. Because the imitator can always use the prior technology without legalrisk, there is no economic force to a patent with s = c̄.16There are many settings in which the infringement-risking imitation choice is known either

before or shortly after competition commences, e.g., Polaroid-Kodak and Intel-AMD. In other

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2.3. Competition Stage

This stage consists of a duopoly market competition which we model as quantitysetting (Cournot) with linear market demand

p(Q) = α− βQ (2.2)

where Q ≡ qi + qj is the sum of outputs. Market competition takes place underone of three possible regimes, depending on prior moves. First, at (s,S) whereno patent is in place and s was disclosed, the market reduces to pure Cournotcompetition between i at cost c and j at cost s (in which j may remain uncertainabout i�s cost). Second, at (s,P ,N ), the market reduces to pure Cournot betweeni at c and j at c̄ (again, j may remain uncertain about i�s cost). Third, andmost importantly, at (s,P, I) the market competition occurs under the shadow ofinfringement. Firms Þrst choose quantities and this determines price, revenues,and production costs (with i at c and j at s). Then, infringement (court outcome)is determined as follows. With probability γ, j is found to have infringed and isforced to pay damages to i. Infringement damages are assessed at a royalty rateof τ on the realized market price for each unit j produced. Thus, j is required topay τpqj to i with probability γ and nothing with 1-γ. As the expected penaltyis what matters, we deÞne g ≡ γτ where 0 < g < 1. To avoid nuisance divisionby zero, we treat full property rights (g = 1) and no property rights (g = 0) aslimiting cases of the analysis.We assume that c̄ < α so that both Þrms would be active in the absence of

innovation. Cases for α, c̄ and g are introduced below.

2.4. Equilibrium

Strategic options are as follows. A protection and disclosure strategy for theinnovator is a map from [0, c̄] into {P ,S}× [0, c̄]. By feasibility, disclosures mustsatisfy 2.1. We use ϕP(c) to denote the disclosure of an innovator with cost drawc who decides to patent and ϕS(c) for the disclosure when no patent is chosen.An imitation strategy for the follower is a choice from {I,N} based on the

observed protection and disclosure choice of the innovator, which is of the form(s,P) or (s,S). Quantity strategies for i and j at the competition stage are

cases, the imitation choice may not be known to the innovator prior to production and compe-tition. The effect of unobservability is that j will mix between I and N for a subset of the(c, s) points in Figure 4.1.

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choices for output based on the observed prior disclosure, protection and imitationhistory and, for i, the privately observed cost c. Thus, for example, if i is of typec and (s,P, I) is the history, then quantity choices are denoted by qj(s,P ,I) andqi(c, s,P, I). Finally, a belief of j regarding i�s cost conditional on an observedprotection and disclosure choice by i is a c.d.f. on [0, c̄] and Bayes Law impliesthat a belief must put zero probability on all cost types greater than the observeddisclosure.A perfect Bayesian equilibrium (PBE) is a protection and disclosure strategy

for i, an imitation strategy for j, and quantity strategies for i and for j as wellas beliefs for j such that, given these strategies and beliefs, i) quantity choicesare optimal for each history at the competition stage, ii) the imitation strategy isoptimal for j at the second stage, and iii) the disclosure and protection strategyis optimal for i at the Þrst stage. A PBE is separating if, in equilibrium, eachobserved disclosure and protection choice is made by a unique cost type of i. Wefocus on equilibria of this type.

3. Market Competition

Competition between the innovating Þrm i and the follower j depends on theirrelative cost positions and on the property right positions they have chosen. Atthe point when i and j choose quantities, the history of the game consists ofa disclosure, s, and property right choice {P ,S} by i and, given P by i, aninfringement-risking imitation choice, {I,N}, by j. In a separating equilibrium,j infers i�s cost c from the observed disclosure and property right choices.For any given c and s, one of three cases for competition arises: i) if i chose S,

then we have (pure) Cournot competition between i with cost c and j with costs; (ii) if i chose P and j chose N , we have Cournot competition between i at costc and j at cost c; (iii) if i chose P and j chose I, we have Cournot competitionunder the imitation regime with i at c and j at s.We focus on case (iii) since a full understanding of the consequences of infringement-

risking imitation by j is essential for the rest of the analysis. Summary resultsare provided for cases (i) and (ii) which are standard Cournot situations.

3.1. Patent and Infringement-Risking Imitation

Consider the competition stage given that i chose to patent and disclose s andthat j chose to imitate. Thus, the observed history is (s,P , I). In equilibrium,

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j infers that the innovator i has cost c = ϕ−1P (s).Imitation allows j to produce at cost s, an improvement over c, but it also

exposes j to the risk of paying infringement damages. Given imitation, if iproduces qi and j produces qj, the resulting market price of p = α − β(qi + qj)leads to payoffs of πj = (p− s)qj − gpqj and πi = (p− c)qi + gpqj where g is theexpected infringement damages rate.17 The best response for j to qi is given by

qBRj =1

2β

·α− 1

1− gs− βqi¸,

when the interior term is positive (and zero if not). In the strategic response to agiven qi, the damages payment leads j to be more timid and produce as if it hada higher marginal cost of 1

1−gs rather than s. Thus, as property rights weaken,j will produce more aggressively. The best response for i is given by

qBRi =1

2β[α− c− β(1 + g)qj ] ,

when the interior term is positive (and zero if not). Thus, infringement damageslead the innovator to be less aggressive in its response to a given qj. The prospectof an infringement payment provides i with an incentive to keep prices higherthan otherwise as the damages payment is a function of j�s revenue.The resulting competition stage outcome is given by:

Lemma 1. Consider an equilibrium and suppose (s,P , I) is observed at the com-petition stage. Let c = ϕ−1P (s). Then the unique outcome is given by

qj(s,P ,I) =1

β(3− g)·α− 2

1− gs+ c¸

qi(c, s,P ,I) =1

β(3− g)·α(1− g)− 2c+ 1 + g

1− gs¸

if s < 1−g2(α+ c) and by monopoly for i at an output of (α−c)

2βif s ≥ 1−g

2(α+ c).

Note two effects. The Þrst relates to changes in cost differentials betweenthe innovator and follower. If i were to disclose more information (at a given

17In this model g is not a function of the amount of the patented enabling knowledge, soimitation always involves the maximum use possible. When key disclosures are made outsideof the patent, g is not likely to be a function of s.

14

inference of c), then an imitating follower has lower costs as s is lower. Firm jproduces more while i produces less. A higher cost c for i, in equilibrium, alsoleads j to produce more and i to produce less (at a given disclosure of s).Second, weaker property rights (smaller g) lead j to increase output while i

reduces output. When the follower imitates, it is as if i and j have entered ade facto licensing agreement with the royalty rate g set exogenously through thelegal system of intellectual property rights. Due to the infringement payment, ihas a weaker incentive to increase quantity as this reduces the market price and,with it, the (expected) infringement payment from j to i.In equilibrium, the payoffs at the competition stage under imitation are given

by

πj(s,P ,I) =1− g

β(3− g)2·α− 2

1− g s+ c¸2

(3.1)

πi(c, s,P ,I) =1

β(3− g)2 [α− (2− g)c+ s]2

+g

β(3− g)·α+ c− 2

1− gs¸c (3.2)

We see that i�s proÞt increases when i is (correctly) inferred to have lower costs andthat i�s proÞt falls as more information is disclosed. These incentive propertiesform the basis for i to signal low costs through disclosure.

3.2. Patent and No Infringement-Risking Imitation

In this case, j operates at cost c since j did not imitate. Also, in equilibrium,the disclosure s leads j to infer that i has cost c = ϕ−1P (s). Thus, at a historyof (s,P ,N ), equilibrium competition is analogous to a full information Cournotsetting with i at cost c and j at cost c.

Lemma 2. Consider an equilibrium and suppose (s,P ,N ) is observed at thecompetition stage. Let c = ϕ−1P (s). Then the unique outcome is given byqj(s,P,N ) = 1

3β(α− 2c+ c) and qi(c, s,P ,N ) = 1

3β(α− 2c+ c) if c > max{2c−

α, 0} and by monopoly for i with output of (α− c)/2β if c ≤ max{2c− α, 0}.

The payoffs are given by πi(c, s,P ,N ) = βqi(c, s,P ,N )2 and πj(s,P,N ) =βqj(s,P ,N )2.

15

3.3. No Patent

In this case, as i did not patent, j produces using the disclosed information ofs and faces no damages payment. In equilibrium, j infers that i has cost c =ϕ−1S (s). Thus, at a history of (s,S), equilibrium competition is analogous to fullinformation Cournot between i at cost c and j at cost s.

Lemma 3. Consider an equilibrium and suppose (s,S) is observed at the com-petition stage. Let c = ϕ−1S (s). Then the unique outcome is given by qj(s,S) =13β(α− 2s+ c) and qi(c, s,S) = 1

3β(α− 2c+ s) if c > max{2s − α, 0} and by

monopoly for i with output of (α− c)/2β if c ≤ max{2s− α, 0}.

The payoffs are given by πi(c, s,S) = βqi(c, s,S)2 and πj(s,S) = βqj(s,S)2.The key aspect of market competition captured in our model is that there are

proÞt beneÞts to (a) having a differential cost advantage over one�s competitorand (b) making this fact known prior to some downstream competitive interaction.That is, it pays to be (relatively) strong or to appear to be relatively strong atthe start of the market competition phase. Many competitive settings otherthan a straightforward market competition (e.g., a choice of capacity followed bymarket competition or a second-phase R&D competition in which lagging Þrmsreposition the direction of or reduce their innovation efforts) have this strategicsubstitutes feature and, to a Þrst approximation, should create similar incentivesfor disclosure and imitation to those captured here.

4. Infringement-Risking Imitation

Because it is always technically feasible for j to access disclosed knowledge, wemust consider when j will Þnd it proÞtable to imitate rather than stay with thenoninfringing old technology. Suppose that i chose to patent, P, and disclose sand that, based on these actions, j infers that i has cost c = ϕ−1P (s). By notimitating, N , j faces a cost disadvantage of c̄ versus c. By imitating, I, j reducesthe cost disadvantage to s versus c but risks the infringement payment. To decidewhich is better, the anticipated payoffs from the competition stage at an observedhistory of (s,P ,N ) and at (s,P, I) must be compared.The results from Section 3 are used to compare the payoffs for j under N and

I. The choice depends on the relative cost positions of i and j and the strength

16

c

s

D

EP line

c* s*2c −α c

c 45 line! ( )c s=

s g c= − +12

( )α line

imonopoly

I by j

N by j

Figure 4.1: Imitation Choice

of the property right. The essential economic features of the imitation decisionare illustrated in Figure 4.1.18

First, consider the vertical line at 2c̄−α. Under N , j is active in the competi-tion stage and earns a positive proÞt, πNj > 0, provided that c is to the right of thevertical line at 2c̄− α. Since j operates at cost c̄ under N , the disclosure s onlymatters indirectly via the inference c = ϕ−1P (s). To the left, where c < 2c̄− α, jis inactive under N and earns zero as, with i at c and j at c̄, the cost advantageis sufficiently large to force j from the market.Next, consider the upward sloping line, given by s = 1−g

2(α + c), that crosses

the 45◦ line at s∗ ≡ 1−g1+gα. Under I, j is active and earns a positive proÞt, πIj > 0,

provided that the observed disclosure s and inferred c lie below this line. Abovethe line, j is inactive under I and earns zero. In this case, the cost gap betweens and c is large enough, relative to the damages payment implied by g, that j isforced from the market.The I versusN choice has substance in the lower right region (where c > 2c̄−α

and s < 1−g2(α + c)) as this is the only case for which j is active under both I

18For the purposes of Figure 4.1, we have assumed that 0 < 2c̄− α < s∗ ≡ 1−g1+gα < c̄. While

useful for the graph, these conditions are much stronger than necessary; Lemma 4 providesmuch weaker necessary and sufficient conditions on α, c̄, and g for the imitation choice.

17

and N . We compare the payoffs under I and N . In the lower right region ofFigure 4.1, j is active at each of (s,P, I) and (s,P ,N ) against an inferred costof c = ϕ−1P (s) for the innovator, and we have

πIj ≷ πNj ⇐⇒1− g

β(3− g)2·α− 2

1− gs+ c¸2≷ 1

9β[α− 2c+ c]2

⇐⇒·1− g2

− (3− g)√1− g

6

¸(α+ c) +

·(3− g)√1− g

3

¸c ≷ s (4.1)

Equality in 4.1 deÞnes a linear relationship between s and c, denoted by s = e (c),along which j is indifferent between N and I. This equal payoff (EP) line has anegative slope and always passes through the point c = 2c̄− α, s = (1− g)c̄. InFigure 4.1 the EP line begins at point D, which is at the edge of the monopolyregion for i, and falls as c rises, hitting the 45◦ line at the point labelled c∗. Abovethe EP line, j will choose N and, below EP, j will choose I. Thus, as (4.1) andFigure 4.1 suggest, the follower will imitate when the cost disadvantage for j ats with i at c is small enough, given the expected damages payment implied byg under imitation, relative to the larger cost disadvantage for j at c̄ with i at c,under N .The Þnal step is to characterize the infringement-risking imitation decision

formally with respect to the underlying parameters of the model. Two propertiesare necessary for j to have a non-trivial imitation decision. First, j must beactive in the competition stage under N and under I for a nonempty set of cand s values; otherwise, whenever one choice has a positive payoff, the other willhave a zero payoff. Second, given that j is active under N and I at a set of cand s values, one of the choices must not strictly dominate the other across theset. When either of these features is not present, we will necessarily have a trivialimitation choice for j.In terms of the graph, we see that 0 < c∗ < c̄ is the necessary property with

respect to a non-trivial imitation choice. If c∗ > c̄, then N is never chosen by jand, if c∗ < 0, then I is never chosen. Solving for the intersection of EP withthe 45◦ line, we Þnd

c∗(g,α, c̄) =2c̄ + α[(1− g)h(g)− 1]

1 + (1 + g)h(g)(4.2)

where h(g) ≡ 3(3−g)√1−g . For reference, deÞne the set

B ≡½(c, s) | max{0, 2c̄− α} < c ≤ min{c̄, s∗} and c ≤ s < min{c̄, 1− g

2(α+ c)}

¾.

18

B corresponds (in general) to the lower right region in Figure 4.1 where j isactive under I and N . Analysis of the EP line and the N versus I choice yields

Lemma 4. Suppose that c̄ < α1+g. Consider the infringement-risking imitation

decision for j given that i patented and disclosed s. Then

i) the set B is non-empty and for any (c, s) ∈ B the follower j is active in thecompetition stage at (s,P,N ) and at (s,P ,I);

ii) for (c, s) ∈ B, a choice of N is optimal for j when s ≥ e(c) and I is optimalwhen s ≤ e(c);

iii) c∗ < c;

iv) c∗ > 0 and c∗ is strictly decreasing in g iff c̄ > αk(g) ≡ α2[1− (1− g)h(g)] .

The condition c < α1+g

necessarily holds as property rights become weak. Thisimplies B 6= φ, the Þrst necessary feature. When c > α

1+g, the active regions for

j in Figure 4.1 do not intersect, as 2c − α > s∗, and the choice of I versus Nis trivial at any disclosure s and inferred cost c. The second essential feature iscaptured by 0 < c∗ < c. This ensures that the EP line crosses the 45◦ line at apossible innovation cost draw and, hence, that j will be induced to choose I andN in response to the observed disclosure and inferred cost type.An important implication of Lemma 4 is that the innovator will not, in equi-

librium, be able to force the follower to exit the market, no matter how large theinnovation (when c̄ < α

1+g). Refer to Figure 4.1 and note that i can induce j to

exit the market (when c < 2c̄−α) only if a relatively small fraction of the innova-tion is disclosed. But if the equilibrium has j exiting, then the partial disclosurenecessarily implies that innovators with a much smaller innovation will be able tomimic the disclosure, ensuring exit and a monopoly payoff for themselves. Thiscannot be an equilibrium outcome. Referring again to Figure 4.1, we see that thesize of the disclosure required to induce j to exit also necessarily makes it feasibleand desirable for types above c∗ to mimic such a disclosure. However, j will notexit against types above c∗. Formally, we have

19

Corollary 1. Suppose c̄ < α1+g. Then, in equilibrium, the follower is active in

the competition stage (does not exit) for all patent and disclosure choices of theinnovator.

This result is a consequence of incomplete information regarding innovation.If partial disclosure could induce exit, then higher-cost types would have a strongincentive to mimic such a disclosure. In contrast, under complete information, ifj had cost c̄ and knew that i had cost c, Þrm j would exit whenever c < 2c̄− α.Thus, we Þnd that there are no drastic innovations in equilibrium for c < α

1+g.

While other factors not considered here might moderate this result, the analysisdoes highlight how incomplete information impacts the market structure effect ofmajor innovations.In summary, the imitation decision depends critically on the follower�s assess-

ment of the cost position of the innovator. This underscores the focus of thispaper on the signaling aspect of the competitive interaction: an optimal imita-tion decision when there is the option of producing with noninfringing technologynecessarily turns on how the follower assesses the advantage of the innovator.

5. Equilibrium Protection and Disclosure

With the results for the competition and infringement-risking imitation stages inhand, we can examine the incentives of the innovator in the protection and dis-closure stage. The equilibrium involves three distinct regions: i) high-cost typeswho have a relatively small innovation and choose to patent and disclose fully, ii)medium-cost types with a more signiÞcant innovation who, while still choosing topatent, disclose partially and thus rely in part on secrecy, and iii) low-cost typeswith large innovations who eschew a patent entirely, disclose partially, and relymore extensively on secrecy. We develop the economic analysis of each regionand then state our main result, Proposition 1, which establishes existence of thisthree-region equilibrium. Finally, we consider uniqueness issues in Proposition 2.

5.1. Existence of a Three-Region Equilibrium

Small Innovation Region: Patent and Full Disclosure Suppose c ≥ c∗

so that the innovation is relatively small. Suppose further that i patents and,in equilibrium, discloses s = ϕP(c). Then, we know from the imitation analysisthat j will not imitate since, with s ≥ c ≥ c∗, the cost reduction beneÞt of s

20

Large{S}

Medium{P,I}

Small{P,N}

s

ccc∗cL

4 5!

c∗

Innovation Region

jSjP

Figure 5.1: Equilibrium Protection and Disclosure

over c̄ is insufficiently attractive to justify infringement-risking imitation. As aresult, i would earn a payoff of 1

9β(α− 2c+ c)2 . The important point is that this

payoff does not depend directly on s; the disclosure only impacts the inference ofj regarding i�s innovation since j�s cost remains at c absent imitation.The innovator, therefore, has a strong incentive to disclose fully in the small

innovation region. If some type c were to disclose partially, say, at s = ϕP(c) > c,then all higher-cost types �c between c and s would Þnd it feasible and proÞtableto disclose s in order to be perceived as the lower cost type of c. SigniÞcantly,for types above c∗, even a very weak property right (small g) is sufficiently strongto deter imitation when the innovation is relatively small. Thus, we Þnd that iwill fully disclose its innovation under a patent and that ϕP(c) = c for c ≥ c∗. InFigure 5.1, ϕP coincides with the 45

◦ line above c∗.

Medium Innovation Region: Patent and Partial Disclosure The inno-vation is more signiÞcant when c < c∗. We know from above that the followerwill Þnd infringement-risking imitation to be attractive in this range provided thedisclosure s is not too far above the inferred cost type of the innovator (s andc lie below the EP line). Recall that types above c∗ disclose fully. Thus, for adisclosure to signal cost below c∗, it must be that s < c∗. Then, in equilibrium, j

21

will choose to imitate.The payoff for the innovator, from patenting and disclosing s = ϕP(c) is then

given by equation (3.2). Consider the incentive of a different innovator type, say�c, who Þnds it feasible to disclose such an s < c∗ This would lead j to infer thatthe innovator is type c = ϕ−1P (s) and, hence, to imitate and produce qj(s,P, I).By choosing a best response to this quantity, type �c can obtain the deviationpayoff of

1

β(3− g)2·α− 3− g

2�c− 1− g

2c+ s

¸2+

g

β(3− g)·α− 2

1− gs+ c¸�c. (5.1)

In equilibrium, since j imitates and operates at cost s, the innovator no longerhas an incentive to disclose fully. Instead, as s varies, we Þnd a trade-off betweensignalling low costs (a larger innovation) and transferring enabling knowledge toan imitating follower. A simple incentive compatibility argument based on (5.1)then establishes that as c rises ϕP(c) must rise at the rate of

1−g2. As this rate

is below one, we Þnd partial disclosure of innovations when c < c∗. See ϕP inFigure 5.1 in the medium innovation region.

Large Innovation Region: Partial Disclosure Without Patent Protec-tion The Þnal question to consider before presenting the formal equilibrium iswhy an innovator should patent rather than rely exclusively on secrecy. For smallinnovations (c ≥ c∗) and for medium-sized innovations (c < c∗ but not too muchsmaller), we Þnd that the patent incentive is dominant. For large innovations,however, the economic trade-off to signalling via partial disclosure in a patentbecomes less attractive: high-cost types disclose fully, which pushes disclosuredown to c∗, and then medium-cost types disclose partially, which forces still moreinformation knowledge disclosure by the innovator. The innovator, however, hasthe option not to patent. In equilibrium, the choice to give up property rights sig-nals a large innovation and permits less disclosure of valuable enabling knowledge.See ϕS and the jump from ϕP at cL in Figure 5.1.We can assess the economic strength of this incentive in the following way.

Suppose the innovator has achieved total success and that c = 0! Then, ratherthan patenting and disclosing ϕP(0), as implied by extending ϕP , suppose theinnovator does not patent and discloses c∗. Provided that the follower beliefsregarding i are not too unfavorable, the innovator necessarily Þnds it more prof-itable to give up a patent when g is sufficiently small.19 It can be more proÞtable19If c = 0 deviates to (�s,S), then j is free to operate at cost �s. If �c, where �c ≤ �s, is j�s belief,

22

not to patent large innovations than to patent them. This result contrasts withthe value even weak patents have for protecting small innovations.Let cL denote the upper boundary of the large innovation range. The formal

parameter condition on c, α and g that ensures cL > 0 is provided next.

Lemma 5. Suppose αk(g) < c̄ < α1+g. Then, there exists a unique cL that

satisÞesπi(cL, c

∗,S) = πi(cL, 1 + g2c∗ +

1− g2cL,P ,I)

provided that g < 1/3 and c̄ > αm(g) wherem(g) ≡ k(g)+ g[1+(1+g)h(g)]3−5g . Further,

we have c∗ > cL > max{0, 2c∗ − α}. Otherwise, when g > 1/3 or when g < 1/3and c̄ < αm(g), we have πi(c, c∗,S) < πi(c, 1+g2 c∗ + 1−g

2c,P ,I) for all c ∈ [0, c∗].

The type cL is indifferent in equilibrium between patenting at a high level ofdisclosure and secrecy with a lower level of disclosure. Regarding the parameterconditions, a small value for g is important. In particular, as g → 0 both m(g)and k(g) approach zero and, consequently, the jump type exists for any value c̄of the older technology cost. When the existence conditions in Lemma 5 do nothold, we effectively have cL at zero and a special case of our analysis provides thecorresponding equilibrium.For reference, deÞne the parameter set

A ≡½(g, c̄,α) | g ∈ (0, 1/3) and αm(g) < c̄ < α

1 + g

¾.

We now show the equilibrium exists.

Proposition 1. Assume that (g, c,α) ∈ A. Then an equilibrium exists and isgiven by the following strategies:

i) the innovator patents and discloses according to

then calculating j�s optimal production (via Lemma 3) and the best response for the innovatoryields the deviation payoff to the innovator. A comparison with the payoff at ϕP(0) then showsthe deviation will be proÞtable when �s > 1

2�c+g3−gα+

3(1+g)3−g c

∗. When �s = c∗, this reduces to[(3− 5g)c∗ − 2gα] /(3− g) > �c; as g → 0, the left-hand side converges to c̄ and the deviation isproÞtable for a wide range of �c beliefs.

23

a) in the small innovation range, c ≥ c∗, i patents and fully discloses withϕP(c) = c;

b) in the medium innovation range, c∗ > c > cL, i patents and partiallydiscloses with

ϕP(c) =1 + g

2c∗ +

1− g2c

c) in the large innovation range, cL ≥ c, i eschews a patent and partiallydiscloses with

ϕS(c) =³c∗ − cL

2

´+1

2c;

ii) the follower imitates under the risk of infringement according to (a) jchooses not to imitate, N , when (s,P) is observed and s > c∗ and (b) jchooses to imitate, I, when (s,P) is observed and s ≤ c∗.

iii) at each observed history on the equilibrium path, (s,P,N ) for s ≥ c∗,(s,P, I) for s < c∗, and (s,S) for s ≤ c∗, the innovator and follower producein the competition stage according to the implied Cournot outputs.20

The proof of Proposition 1 involves verifying that each of i and j Þnds itoptimal to follow the speciÞed strategies and this entails a set of proÞt comparisonsfor deviations from the equilibrium strategies. With the formal result established,we now discuss three additional important properties: the impact of imitation onmarket structure, the switch to a no patent strategy for large innovations whenpatent protection is weak, and the importance of the strength of property rights.

5.2. Discussion

Boundary Between Small and Medium Innovations: An Implicit Li-censing Interpretation Consider what happens to market competition be-tween the innovator and the follower at c∗, the innovation level that just triggersinfringement-risking imitation. Once c falls below c∗, the follower strictly prefersto imitate because the disclosure s = ϕP(c) allows j to operate at a small cost

20The equilibrium also requires that we specify out-of-equilibrium beliefs and actions. Seethe Appendix.

24

disadvantage relative to i at c while remaining at c now implies a large cost dis-advantage. At c∗, the follower is exactly indifferent between N and I.An intriguing feature of the equilibrium is that the innovator is not indifferent

with respect to j�s imitation choice because there is a discrete jump upwards inproÞt for i at c∗ when j imitates. The innovator beneÞts from imitation by jbecause imitation leads to a qualitative change in the competitive relationship.Fix c = c∗ for purposes of discussion and examine the competition stage as

j switches from N to I. Imitating and producing at cost c∗ leads j to increaseoutput while i is led to reduce output relative to no imitation (qIj > qNj andqIi < q

Ni ). The net effect is that aggregate output falls, the market price rises,

and total proÞts for i and j rise. Thus, imitation has the beneÞcial joint effectof creating a market relationship in which i is effectively �licensing� j. The�royalty rate� is determined implicitly by the strength of patent protection via g,the infringement penalty. The extent of technology transfer is determined by thedisclosure of the innovator.

Boundary Between Medium and Large Innovations: Declining Value ofthe Implicit License Once the innovation size reaches cL (which occurs if g issmall), the innovator chooses not to patent and this functions, in equilibrium, asa strong signal that i has low costs. The associated beneÞt is the discrete jumpto c∗ in disclosure: although the follower can operate at cost c∗ without riskinginfringement, the cost disadvantage jumps to c∗ − cL from ϕP(cL)− cL and, so, jbecomes a weaker competitor. In essence, a large innovation leads i to give upthe beneÞts from the �licensing relationship� under imitation.The innovator sacriÞces patent protection and the associated �licensing� rev-

enues because the value of licensing falls as c falls. Recall that the innovator hastwo sources of proÞt under imitation: �licensing� revenues of gpIqIj , and operatingproÞts of (pI − c)qIi . In turn, the price and the quantities depend on the costdifferential between i and j (s = ϕP(c) and c) measured relative to the strengthof property rights, g. Consider, how the innovator�s proÞt sources change as cfalls from c∗.At c = c∗, where imitation commences, we have ϕP(c

∗) = c∗ and there is nocost differential in absolute terms. As c falls, i is led to disclose more knowledgebut, signiÞcantly, a cost differential opens up since the gap ϕP(c) − c increasesas c falls. It is easy to verify that i increases output as c falls. The follower,however, does not change quantity: the rate at which equilibrium disclosure ϕP(c)falls exactly balances the effect of lower costs for j against that of facing a lower

25

cost opponent. In sum, the effect of lower cost is to increase industry output andreduce price with the cost advantage leading i to assume the more dominant role.These effects shift the innovator�s proÞt source from licensing revenue towards

operating proÞt. With pI falling and qIj steady, licensing revenue of gpIqIj falls.

The price-cost margin for i, pI − c, rises as c falls because of the widening costdifferential relative to j and, consequently operating proÞt, (pI − c)qIi , is drivenup by each of the margin and volume changes.A large innovation leads i to rely more strongly on the cost advantage against

j and less on expected infringement revenues. When g is sufficiently small andc̄ is sufficiently large, the cost advantage effect necessarily becomes dominant.Then, the innovator switches strategy at c = cL and abandons patent protectionto open up a wider cost advantage by disclosing less information. This is whyeschewing the patent �signals� a large innovation. Licensing revenue is high forsmaller innovations and only a large innovation (large cost advantage) makes itproÞtable to give up the patent.Finally, we need to explain why the jump is to c∗ and not higher. Low-cost

innovators prefer the jump to be as large as possible since this would mean less�free� disclosure. What limits the size of the jump is the necessity of maintaininga consistent signal of low costs. A jump above c∗ would make it feasible forinnovator types in the no-imitate region (c > c∗) to mimic this signal. Moreover,these types would Þnd it proÞtable to deviate if the jump point were above c∗.Thus, the jump is capped at c∗.

Impact of Property Rights Consider how the strength of property rights, asmeasured by g, impacts the equilibrium. We focus on the two limiting cases inwhich property rights vanish, g → 0, or become perfect, g → 1.Weaker property rights lead the innovator to rely more heavily on secrecy

(disclose less). As g falls, imitation becomes more attractive for the follower. c∗

rises and the full-disclosure, patent region shrinks. In the medium innovationregion, we Þnd less disclosure as ϕP shifts upwards. At the type cL, the jumpto secrecy involves less disclosure as c∗ is larger. In the limit, as g → 0, thereis no economic distinction between P and S. Both c∗ and cL converge to c anddisclosure under P and S approach a common limit of 1

2(c+ c).

The impact of stronger property rights depends on the status quo technology.If 2c− α > 0, then c is relatively large and a drastic innovation is possible undercomplete information. As g rises, we eventually reach the case of c > α

1+g. At

this point, types above 2c − α disclose fully. Further, there is no overlap in the

26

N versus I choice of the follower, and upon observing ϕP(2c − α) = 2c − α,the follower exits the market. Thus, types below 2c − α achieve the monopolyoutcome. Once c > α

1+gholds, property rights are effectively perfect for the

innovator although equilibrium requires that the disclosure must still be partial(s above the 1+g

2(α+ c) line) in order to deter an imitation choice.

When 2c−α < 0, the ex ante range for process innovation is smaller. Now, asg rises, we eventually cross into the case of c < αk(g) where c∗ has been pushedto zero. This means that the size of g has rendered I unproÞtable relative to Nand the innovator chooses to patent and disclose fully. Due to the small value ofc, however, the follower cannot be forced from the market.21

5.3. Uniqueness of the Separating PBE

In Proposition 2 we characterize the set of equilibria (separating PBE). Forreference, deÞne σ= 1

2(3−g) [2gα+ 3(1 + g)c∗] ; for (g, c,α) ∈ A, we have σ< c∗.

A generalization of Lemma 5 (see Lemma A5 in the Appendix) shows that foreach disclosure σ ∈ [σ, c∗], there is a unique type cσ at which πL(cσ,σ,S) =πi(cσ,

1+g2σ + 1−g

2cσ,P, I). We then have

Proposition 2. Assume (g, c,α) ∈ A. For each disclosure σ ∈ [σ, c∗] and asso-ciated crossing type cσ, the strategies in Proposition 1 constitute an equilibriumwhen we replace cL with cσ and set ϕS(c) = (σ− cσ

2)− c

2. Further, every equilib-

rium (separating PBE) is of this form. Finally, for all types, the equilibrium ofProposition 1 is maximal with respect to the payoff of the innovator.

Thus, equilibrium choices and outcomes for all types above cL are unique.22

The switch from a patent to secrecy strategy can occur, in equilibrium, at any typebelow cL. All innovator types in [0, cL], however, strictly prefer the equilibriumin which secrecy is used to the maximum extent possible.

21As speciÞed in Lemma 5 there is also an intermediate region where α1+g > c > αk(g) and

cL has been pushed to zero.22Consider the possibility of all types pooling at no disclosure (s = c̄) so that the distinction

between P and S is moot. Letting µ denote the mean of the prior F on innovation draws, thepayoff to type c is given by 1

9β (α− 32c− 1

2µ+ c̄)2. Suppose type c deviates to (s,P) and that j

forms the (most) pessimistic belief of �µ = s. For a choice of s > c∗, j optimally chooses N andthe deviation payoff to i is given by 1

9β (α − 32c − 1

2s + c̄)2. Then the deviation is proÞtable if

µ > s. Thus, pooling at c̄ is not an equilibrium if µ > c∗.

27

6. Discussion: Managing Intellectual Property

Our model focuses on three features of innovation and the patent system: inno-vation implies incomplete information, property rights often provide only limitedprotection from imitation, and disclosures make imitation feasible. In settingswhere imitation is a real possibility, the interplay between property rights, disclo-sure and the imitation decision is key to managing IP.In terms of patents and managing IP, we interpret disclosure in our model in

two ways. First, the patent itself may be the primary vehicle for disclosure ofthe enabling knowledge. A second interpretation is that disclosure is separatefrom the patent, though the patent is the means of overall protection againstunauthorized use of the disclosed knowledge.23 We discuss the implications ofdisclosure in this section and note some empirical implications of our theory andsome modeling extensions.

6.1. Information Transmission and Signaling

In the last decade some Þrms have begun to proactively manage their IP.24 Theolder lawyer-driven conventional wisdom that emphasized the value of patentingall that is worth patenting is being supplanted by a more strategic business-drivenlogic that balances the advantages and disadvantages of protection and disclosure.The strategic logic implies as well that the choice of what to patent and what todisclose serves an important signaling role.25 This aspect of the managementchoice is especially salient when patent protection is not strong. Weaker patentprotection makes infringement-risking imitation an economically attractive optionfor the follower. The question of how much to patent then turns on the economic

23One might also consider the IP management as a portfolio of technologies problem (whichof a set of related technologies to patent) instead of as a problem involving a single technology.24See, e.g., Grindley and Teece 1997 [6] for a discussion of these practices in some semicon-

ductor and electronics Þrms.25Hicks 1995 [7] indicates that many Þrms try to signal �credibility� through publishing

technical work and Cohen, Nelson, and Walsh 2000 [4] suggest that some Þrms patent to signalcredibility as a �player.� Also, competitive intelligence experts are now using patents to assess acompetitor�s relative position in developing individual technologies or as indications of progressthat a competitor has made. Some of these experts have gone the next step and urged theirclients to conduct analyses to determine how they appear to others. See, e.g., S. Wilkinson,�Competitors Reveal Own Strengths, Weaknesses,� Chemical & Engineering News, 4/13/98pp. 27-30 or M. Ojala, �A Patently Obvious Source for Competitor Intelligence: The PatentLiterature,� Database, August 1989, pp. 43-49.

28

trade-off between the cost of imitation enablement and the beneÞt of signaling alarge innovation.Small innovations are fully disclosed and protected because imitation through

the use of that disclosed knowledge is not attractive to a competitor. Mediumand large innovations, on the other hand, invite imitation; hence a portion of theinnovation is kept secret so that the innovator can maintain its competitive edge.Somewhat more surprising is the result that with weak property rights an in-

novator will choose to give its competitor a free road to the disclosed technologyby making unprotected disclosures (e.g., Ford Motor Company�s disclosures of itsmoving assembly line processes) or, alternatively, by patenting but licensing theenabling knowledge for a nominal payment. When property rights are weak, theamount of disclosure under a patent needed to signal low costs is increasinglyunattractive for large-innovation Þrms and this makes the patent increasingly lessvaluable relative to reducing disclosure. By sacriÞcing expected infringementdamages the innovator can signal with much less disclosure. This sacriÞce ofexpected damages is the core signal. Eschewing the patent and then makingunprotected disclosures is one implementation of the signal, though this imple-mentation has practical deÞciencies because the choice to not patent is a nonactionand even if noticed might be attributed to a problem with the legal patentabilityof the invention (e.g. too close to prior art). The patent plus nominal lump-sumlicense, on the other hand, has the same expected damages feature, but is bettertailored to creating the desired and ineluctable signal. Actions consistent withsuch signals include public disclosures of enabling knowledge in conferences andpapers and licenses of technology for (surprisingly) low fees.26

Because the management of IP has traditionally been handled as a legal ratherthan business matter, our model can be viewed as partly positive and partly nor-mative. The cost-oriented model does, however, offer some predictions relatinginnovation size, market structure, imitation, and disclosure. To the extent thatÞrms have been treating IP in a business-sophisticated way, the model predictsthat small process innovations will not be imitated, in contrast to medium andlarge process innovations which will be imitated, where imitation will be associ-ated with an infringement lawsuit or perhaps licensing. The size of the innovationmay be estimated by examining the change between pre and post innovation mar-ket share. Large cost differentials�larger innovations�will lead to a larger relative

26This approach also avoids in part the problem associated with having some other Þrminventing and then patenting the technology that the innovator kept secret.

29

market share for the innovator. In industries where property rights are gen-erally considered weak, we further predict very large cost differentials will notbe associated with infringement suits, though some (low royalty) licensing mayoccur. In such cases, for example, our model suggests that an exogenous increasein the strength of property rights might lead a Þrm that had previously relied onnominal license fees and (more) secrecy to try to collect substantial license fees(through the threat of enforcement of the patent) and to increase the fee revenueby disclosing additional enabling knowledge which in turn encourages greater out-put by the competition. The change in the strength of patent protection afterthe advent of the Federal Circuit provides a natural experiment with which tolook for such changes.27

6.2. Investment Incentives

A potentially interesting extension of our model would be to add considerationof R&D incentives. Investment in R&D will reßect the level of appropriabilityexpected for the R&D outputs and this, in turn, depends on the strength ofthe property rights and the endogenous actions of the players post invention.When a Þrm faces pathways offering ex ante different proÞles with respect to theprobability and extent of the resulting innovation, the proÞt outcomes from theanticipated downstream protection, disclosure, and imitation decisions are criticalfor assessing each path�s attractiveness. Along these lines, we speculate thatwhere property rights are weaker (small g in our model), there may exist a biastowards investment in smaller over larger innovations because small innovationshave effectively better IP protection.28

6.3. Licensing

Licensing is a strategic option that is not directly considered in our analysis. Weomit this option because abstracting away from licensing allows a more precise in-vestigation of the information transmission and signaling issues associated with IP

27There is some evidence that supports a change in the policy of some Þrms from a very lowlicensing rate to a much higher rate in the years subsequent to the new patent regime. Thereare, however, a plethora of possible explanations for these changes, of which our theory is onlyone.28Other important elements relating to the management or administration of IP such as the

role of cumulative innovation and blocking (Scotchmer and Green 1990 [21]), reverse engineering,etc. would also be valuable to bring to our framework. See also Katz and Shapiro 1987 [12] foran analysis of how imitation inßuences the incentive to innovate.

30

protection and exploitation and licensing is a legally questionable option betweendirect competitors.Under the assumptions adopted in our model, antitrust laws present an imped-

iment to the use of licensing as the innovator and follower compete in the marketeven when the follower is using the old technology.29 Even where licensing mightbe (legally) permissible, the form of licensing is likely to be constrained in �closecall� situations. For example, per unit licensing fees which appear to have thegreatest potential for interfering with competition tend to be frowned upon byantitrust law. If detection probabilities are low, however, some Þrms may not bedeterred from licensing by the legal risk, so augmenting our analysis with somelicensing possibilities, especially those not involving per unit licensing fees wouldbe a useful extension.The analysis in our �implicit licensing� case, however, does provide a prelimi-

nary result�the fall back or threat points for each party to the license negotiation�helpful for the analysis of fully strategic licensing in the joint presence of incom-plete information and limited property rights.Licensing is a more complex variant of the problem we analyze. The basic

inference and imitation issues remain critical: the potential licensee needs toknow the value of the license before it can strike a deal and as the value is difficultto establish absent disclosure, some amount of partial, but enabling, disclosurewill typically be necessary.

6.4. Signaling to Third Parties and Induced Demand

Innovating Þrms often disclose enabling knowledge to third parties to induce mar-ket demand (e.g., indirectly through the development of complementary productsor facilitating Þnancial backing, or directly to obtain buyers). Such disclosuresare particularly important when the innovation makes a relatively large breakfrom the past, say, as a completely new product or in the form of a process in-novation that substantially decreases costs and makes new applications for theoriginal product economically feasible. These disclosures induce additional mar-ket demand, but often at the expense of transferring enabling knowledge (either

29�[A]ntitrust concerns may arise when a licensing arrangement harms competition amongentities that would have been actual or likely potential competitors in a relevant market in theabsence of the license.� Antitrust Guidelines for the Licensing of Intellectual Property, U.S.Department of Justice and U.S. Federal Trade Commission, 1995.

31

directly from the public disclosures or indirectly through leaks) to competitors.30

The interest of noncompetitors will turn on their assessment of the relative advan-tage of the new technology over the old. That assessment requires noncompetitorsto make inferences based on disclosed knowledge, just as the competitors do in ourbasic model.31 Further, even though the targets of the signal may be noncom-petitors, the signal is likely to reach competitors as well. Thus, information ßowstrategies directed to noncompetitors should not be divorced from issues affectingdirect competitors such as we have analyzed in this paper.

References

[1] Anton, James J. and Yao, Dennis A., �Expropriation and Inventions: Ap-propriable Rents in the Absence of Property Rights,� American EconomicReview 84:1 (March 1994), pp. 190-209.

[2] Anton, James J. and Yao, Dennis A., �Patents, Invalidity, and the StrategicTransmission of Information,� Duke University working paper, August 1995(revised 1999).

[3] Battacharya, Sudipto and Ritter, Jay R., �Innovation and Communication:Signalling with Partial Disclosure,� Review of Economic Studies 50 (1983),pp. 331-346.

[4] Cohen, Wesley M., Nelson, Richard R., and Walsh, John, �Protecting TheirIntellectual Assets: Appropriability Conditions and Why ManufacturingFirms Patent (or Not),� NBER WP 7552, February 2000.

[5] Gallini, Nancy T., �Patent Policy and Costly Imitation,� RAND Journal ofEconomics 23:1 (Spring 1992), pp. 52-63.

30Management consultants, for example, frequently publish their general frameworks in hopesof attracting business. Revelation of conÞdential information to encourage sales is common inmanufacturing as well. For example, Tippens, a steel plant construction company, alleged thatconÞdential knowledge it disclosed to a potential buyer was transferred to competing companythat was hired to do the job. (see, e.g., �Tippens Sues Lukens over Steckel-mill Technology,�Iron Age New Steel 12 (3) March 1996, pp. 10-14.).31Induced demand could be included in our model by allowing the demand function to increase

with disclosure. This would capture the public good aspect of the signal for the follower Þrm.

32

[6] Grindley, Peter C. and Teece, David J., �Managing Intellectual Capital: Li-censing and Cross-Licensing in Semiconductors and Electronics,� CaliforniaManagement Review 39:2 (Winter 1997), pp.8-41.

[7] Hicks, Diana, �Published Papers, Tacit Competencies and Corporate Man-agement of the Public/Private Character of Knowledge,� Industrial and Cor-porate Change 4:2 (1995) pp. 401-424.

[8] Horstmann, I., MacDonald, G.M., and Slivinski, A., �Patents as InformationTransfer Mechanisms: To Patent or (Maybe) Not to Patent,� Journal ofPolitical Economy 93 (1985), pp. 837-856.

[9] Hounshell, David A., From American System to Mass Production 1800-1932:The Development of Manufacturing Technology in the United States, JohnsHopkins University Press, 1984.

[10] Jackson, Tim, Inside Intel: Andy Grove and the Rise of the World�s MostPowerful Chip Company, Dutton 1998.

[11] Jolly, Vijay K., Commercializing New Technologies: Getting from Mind toMarket, HBS Press 1997.

[12] Katz, Michael L. and Shapiro, Carl, �R&D Rivalry with Licensing or Imita-tion,� American Economic Review 77 (June 1987), pp. 402-420.

[13] Levin, Richard C., Klevorick, Alvin K., Nelson, Richard R. and Winter,Sidney G. �Appropriating the Returns from Industrial Research and Devel-opment,� Brookings Papers on Economic Activity 3, (1987), 783-831

[14] MansÞeld, Edwin, �Patents and Innovation: An Empirical Study,� Manage-ment Science 32:2 (February 1986), pp. 173-181.

[15] MansÞeld, Edwin, Schwartz, M., and Wagner, S., �Imitation Costs andPatents: An Empirical Study,� Economic Journal 91 (December 1981) pp.907-918.

[16] Milgrim, Roger M., �Get the Most Out of Your Trade Secrets,� HarvardBusiness Review (November-December 1974), pp. 105-112.

[17] Milgrom, Paul and John Roberts, �Relying on the Information of InterestedParties,� RAND Journal of Economics 17:1 (Spring 1986), pp. 18-32.

33

[18] Nevins, Allan and Hill, Frank Earnest, Ford: The Times, the Man, theCompany, Charles Scribner�s Sons, 1954.

[19] Okuno-Fujiwara, Masahiro, Postlewaite, Andrew and Kotaro Suzamura,�Strategic Information Revelation,� Review of Economic Studies 57:1 (Jan-uary 1990), pp. 25-47.

[20] Rhodes, Glenn W., Patent Law Handbook, 1997-98 Edition.

[21] Scotchmer, Suzanne and Green, Jerry, �Novelty and Disclosure in PatentLaw,� RAND Journal of Economics 21:1 (Spring 1990), pp. 131-146.

34

7. Appendix

A. Proofs of Lemmas 1, 2, and 3 and Related ResultsLemmas 1, 2, and 3 are special cases of a more general Cournot game with

one-sided incomplete information. We establish existence and uniqueness of aBayesian equilibrium for the more general game in Lemma A1 and apply theresult.Suppose payoffs satisfy πj = (p− cj)qj − gpqj and πi = (p − ci)qi + gpqj and

each Þrm chooses quantity.

Lemma A1 Let t ∈ [0, c̄] and g ∈ [0, 1). Suppose Þrm j has cost cj ∈ [t, c̄].Suppose Þrm i�s cost type is private information and takes values in [0, t]with c.d.f. G and let µ ≡ R t

0cidG(ci) be the mean cost type of i. Strategies

are non-negative quantity choices of qj for j and qi(ci) for each ci ∈ [0, t].Then a unique Bayesian equilibrium exists and is given by

q∗j =1

β(3− g)·α− 2

1− gcj + µ¸and

q∗i (ci) =1

β(3− g)·α(1− g)− 3− g

2ci − 1 + g

2µ+

1 + g

1− gcj¸

when (1− g)(α+ µ) > 2cj and by monopoly for i with q∗j = 0 and q∗i (ci) =(α−ci)2β

when (1− g)(α+ µ) ≤ 2cj.

Proof : Consider the best-response for i and j. For a given qj, type ci ofi maximizes proÞt at qBRi (ci, qj) = (α− ci − β(1 + g)qj) /2β when the numera-tor is positive, and at qBRi (ci, qj) = 0 if not. Given qi : [0, t] → [0,∞), letµi ≡ R t

0qi(ci)dG(ci) be the mean output of i. Then, j maximizes expected

proÞt at qBRj (qi) =³α− 1

1−gcj − βµi´/2β when the numerator is positive, and

at qBRj (qi) = 0 if not.It is then straightforward to verify that q∗j and q

∗i (ci) as speciÞed in Lemma

A1 satisfy the best-response conditions and constitute a Bayesian equilibrium.Consider uniqueness. Suppose qj and qj : [0, t] → [0,∞) are a Bayesian

equilibrium. To begin, we show qi(ci) > 0 for all ci ∈ [0, t]. First, supposeα(1 − g) ≤ cj. Then qBRj (qi) = 0 as [α − 1

1−gcj − βµi] ≤ α − 11−gcj ≤ 0 for any

35

µi ≥ 0. Hence, qj = 0 in equilibrium and we must have qi(ci) = qBRi (ci, 0) =(α−ci)2β

> 0. Now suppose α(1 − g) > cj. If µi = 0 in equilibrium, then qj =(α − 1

1−gcj)/2β. We then have qBRi (ci, qj) > 0 ⇐⇒ α − ci − β(1 + g)qj >0 ⇐⇒ 1

2

hα(1− g) + 1+g

1−gcji> ci. With α(1 − g ú) > cj, this last inequality holds

strictly for all ci ≤ t. Hence, qi(ci) > 0 must hold and, in fact, qi(ci) =12β

h12α(1− g)− ci + 1+g

2(1−g)cji> 1

2β

³cj1−g − ci

´. Hence, µi > 1

2β

³cj1−g − µ

´≥ 0, a

contradiction. Thus, qi(ci) > 0 for all ci in any equilibrium.Now, suppose that qj > 0 in equilibrium. Then qj and qi(ci) satisfy the best-

response conditions at equality. Solving simultaneously directly implies that qjand qi(ci) must assume the values in Lemma A1 for j at positive output. Then,a positive output best response for j implies that (1 − g)(α + µ) > 2cj. Now,suppose that (1 − g)(α + µ) > 2cj. If qj = 0, then we have qi(ci) =

(α−ci)2β

in

equilibrium. With µi = (α−µ)2β, however, the best response for j has positive

output ashα− 1

1−gcj − βµii> 0 holds. Hence, we must have qj > 0 when

(1− g)(α+ µ) > 2cj .Next, suppose qj = 0 in equilibrium. Then, qi(ci) =

(α−ci)2β

must hold and

µi = (α−µ)2β. A best response of zero for j then implies that (1− g)(α+ µ) ≤ 2cj.

Going the other way, suppose (1 − g)(α + µ) ≤ 2cj. If qj > 0, then the valuesin Lemma A1 for qj and qi(ci) must apply for j at positive output. A positivebest response for j, however, then implies (1− g)(α + µ) > 2cj, a contradiction.Hence, qj = 0 when (1− g) (α+ µ) ≤ 2cj.Thus, the equilibrium in Lemma A1 is unique when (1− g) (α+ µ) > 2cj and

when (1− g) (α+ µ) ≤ 2cj.¤Lemmas 1, 2, and 3 are special cases of Lemma A1. For Lemma 1 set t = cj =

s and let G be degenerate with an atom at ci = ϕ−1P (s), so that µ = ci holds. ForLemma 2 set t = s, cj = c̄, let G be degenerate at ci = ϕ−1P (s), and set g ≡ 0 asthe infringement damage does not apply when j chooses N . Finally, for Lemma3 set t = cj = s, let G be degenerate at ci = ϕ−1S (s), and set g ≡ 0.¥

Let π∗i (ci) be the payoff to type ci for the game in Lemma A1. We record thefollowing result for future reference.

Lemma A2 π∗i (ci) is strictly decreasing in the mean belief µ whenever Þrm j isactive and it is constant in µ if j is inactive.

36

Proof: Begin with the case where 0 < 21−gcj − α < c̄. By Lemma A1, j is

inactive for µ ≤ 21−gcj −α, and i earns a monopoly proÞt which is independent of

µ. Firm j is active when µ > 21−gcj −α. Then, substituting q∗j and q∗i (ci) into the

deÞnition of πi yields

π∗i (ci) =1

β(3− g)2·α− 3− g

2ci − 1− g

2µ+ cj

¸2+

gciβ(3− g)

·α− 2

1− gcj + µ¸.

Differentiating with respect to µ, ∂∂µπ∗i < 0⇔ (3−g)ci < 2s∗+2

³1−g1+g

´ £cj − 1−g

2µ¤,

where s∗ ≡³1−g1+g

´α. The right-hand side is strictly increasing in cj and strictly

decreasing in µ. As µ ≤ t ≤ cj, the right-hand side is bounded below by thevalue when we set cj = µ = t. Further, since ci ≤ t, it is sufficient to show that(3 − g)t < 2s∗ + (1 − g)t. This reduces to t < s∗ which is valid by j active andcj ≥ t ≥ µ.Finally, for the case of 2

1−gcj ≤ α, Þrm j is active for all µ ∈ [0, t]. For21−gcj ≥ α + c̄, Þrm j is never active for any µ ∈ [0, t]. The result for these casesfollows trivially.¤B. Proof of Lemma 4 and Corollary 1 and Related ResultsProof of Lemma 4: (i) We show that j is active under (s,P, I) and (s,P ,N )

for a disclosure s and inferred type c = ϕ−1P (s) iff (c, s) ∈ B 6= φ. Note thatc̄ < α

1+gimplies (1− g)c̄ > 2c̄− α; in Figure 4.1, the point D thus lies above the

45◦ line. Otherwise, as when c̄ ≥ α1+g, the point D lies below the 45◦ line.

Assume c̄ < α1+g. This implies s∗ > 2c̄ − α. All feasible disclosures satisfy

c̄ ≥ s ≥ c ≥ 0. Now, by Lemma 2, j is active under N at any (c, s) wheremax{0, 2c̄− α} < c ≤ c̄ and c ≤ s, which is a non-empty set. Also, by Lemma 1,j is active under I at any (c, s) where 0 ≤ c < min{c̄, s∗} and c ≤ s < 1−g

2(α+ c),

which is a non-empty set. Note that max{0, 2c̄−α} < min{c̄, s∗} and, as a result,the intersection of these two sets, deÞned as B, is non-empty.(ii) This follows directly from (4.1) in the text.(iii) From (4.2), we have c∗ < c̄ ⇐⇒ c̄ [1− (1 + g)h(g)] < α [1− (1− g)h(g)],

after simplifying terms. Since h(g) > 1 and, therefore, positive, we have [1− (1 + g)h(g)] <[1− (1− g)h(g)]. Then, with c̄ < α, we are done and c∗ < c̄.(iv) The denominator in (4.2) is always positive and, hence, c∗ > 0 iff

c̄ > α2[1− (1− g)h(g)] ≡ αk(g). The function k(g) is easily seen to be strictly

increasing, strictly convex, and rising in value from zero as g ↓ 0 to 1/2 as g ↑ 1.Thus, αk(g) < α

1+gfor all g < 1.

37

To show that c∗ is strictly decreasing in g, note that the numerator in (4.2) ispositive when c̄ > αk(g). Thus, it is sufficient to show that the numerator in (4.2)is strictly decreasing while the denominator is strictly increasing. Differentiationshows that h(g) and, hence, (1 + g)h(g) are strictly increasing, while (1− g)h(g)is strictly decreasing. Hence, c∗ is strictly decreasing in g.When c̄ < αk(g), we have c∗ < 0. As limg↑1 c∗(g, c̄,α) = 0, c∗ cannot be

decreasing for all of this part of the (g, c̄,α) parameter space.¥Table 7.1 summarizes the optimal choice in {I,N} for j. We employ Lemma

A1 to cover off-equilibrium cases. Thus, suppose j has observed a choice of (t,P)by i, where t ∈ [0, c̄], and holds (mean) belief µ ∈ [0, t].In Figure 4.1, consider a horizontal line at vertical height t with µ ranging

from 0 to t. We can divide [0, t] into at most three sub intervals, say KM , KI,andKN . As µ ranges over [0, t], j chooses to be inactive (a monopoly for i) whenµ ∈ KM , j chooses I when µ ∈ KI , and j chooses N when µ ∈ KN . Notethat when t lies on the boundary between cases, one of these intervals typicallycollapses to a single point; we omit these details from the table. Finally, recallthat t = e(µ) is the EP line and e−1 denotes the inverse; the domain for the EPline is µ ∈ [2c̄− α, c∗] when 2c̄− α > 0 and [0, c∗] when 2c̄− α ≤ 0.

Table 7.1: Follower I Versus N ChoiceRange for mean belief µ

Disclosure t KM KI KNCase: 2c̄−α > 0 ___________ ______________ ________t > (1− g)c̄ [0, 2c̄− α] φ (2c̄− α, t](1− g) c̄ > t > M

h0, 2

1−g t− αi

( 21−g t− α, e−1(t)] [e−1(t), t]

Case: M > t > c∗ φ [0, e−1(t)] [e−1(t), t]

Case: M > t > 1−g2α

h0, 2

1−g t− αi

( 21−g t− α, t] φ

m > t φ [0, t] φCase 2c̄− α < 0 ___________ _____________ _______t ≥ e−1(0) φ φ [0, t]e−1(0) > t > c∗ φ [0, e−1(t)] [e−1(t), t]c∗ > t φ [0, t] φ

M ≡ max½1− g2α, c∗

¾,m ≡ min

½1− g2α, c∗

¾

38

With the optimal choice of j between I and N (and inactivity) characterized,we can establish the following result for the payoff to i. Whenever a disclosure t(with P) and a mean belief µ lie on the EP line, j is indifferent between I andN . Firm i, however, strictly prefers that j chooses I in this case. To show this,deÞne πIi (c) to be the payoff to type c for game in Lemma A1 when we set cj = t(i.e., j chose I). DeÞne πNi (i) to be the corresponding payoff for the game whenwe set cj = c̄ and g ≡ 0 (i.e., j chose N ). We have

Lemma A4 Suppose αk(g) < c̄ < α/(1+g), so that c∗ ∈ (0, c̄). For the Bayesiangame of Lemma A1, suppose (µ, t) lies on the EP line. Then πIi (c) > π

Ni (c)

for all c ∈ [0, t].

Proof: We have µ ∈ [max{0, 2c̄ − α}, c∗] and t = e(µ). Let a superscript Idenote a variable in the Bayesian game of Lemma A1 with cj = t and g ∈ (0, 1);let a superscript N refer to the game with cj = c̄ and g ≡ 0. We suppressthe arguments of variables when no confusion arises. Lemma A1 implies qNi =13β(α − 3

2c − 1

2µ + c̄), qNj = 1

3β(α − 2c̄ + µ) and, with QN ≡ qNi + q

Nj , we Þnd

pN = 13(α+ c̄+ 3

2c− 1

2µ). Similarly, Lemma A1 implies

qIi =1

β(3− g)·α(1− g)− 3− g

2c− 1 + g

2µ+

1 + g

1− g t¸,

qIj =1

β(3− g)·α− 2

1− g t+ µ¸

and, with QI = qIi + qIj , we Þnd p

I = 13−g

¡α+ t+ 3−g

2c− 1−g

2µ¢.

We claim pI > pN . Comparing expressions, pI > pN ⇔ (3−g)c̄ < 3t+g(α+µ).Since t = e(µ), we can substitute for t in terms of µ using (4.1). This yieldspI > pN ⇔ 2c̄− α < µ, which is valid on the EP line. Thus, pI > pN .Next, we have as an accounting identity that πIi > π

Ni ⇔ (t−c)qIj −(c̄−c)qNj <

(pI−c)QI−(pN −c)QN , as follows from the deÞnition of proÞts and the fact thatπIj = π

Nj on the EP line. We Þrst show (p

I − c)QI > (pN − c)QN . The monopolyproÞt function deÞned by (p(x)−c)x has unique maximum at x = 1

2β(α−c) and it

is strictly decreasing at larger x values. Since QI < QN , as follows from pI > pN ,we need only show QI > 1

2β(α − c) to establish the result. Comparing, we have

QI > 12β(α− c)⇔ t < 1−g

2(α+ µ), which is valid on the EP line (as qIj > 0).

Next, we show (t−c)qIj − (c̄−c)qNj < 0. Note this is a decreasing function of cprovided that qNj > q

Ij (note that j�s output depends on µ, but not c). Comparing

39

and simplifying via t = e(µ) from (4.1), we Þnd that qNj > qIj ⇔ 2c̄ − α < µ,which is valid. Thus, we are done if we can show the function is negative at c = 0.Thus, we must show tqIj < c̄qNj . Note that t < (1 − g)c̄ holds on the EP line.Thus, it is sufficient to show (1− g)qIj < qNj . Comparing and simplifying, we seethis holds ⇔ (3− g)c̄ < 3t+ g(α+ µ) and, as above, this is valid on the EP line.Combining the above results, we have (t−c)qIj − (c̄−c)qNj < 0 < (pI−c)QI−

(pN − c)QN . By the accounting identity, this implies πIi > πNi .¥Proof of Corollary 1: We must show that qj > 0 holds for any possible

equilibrium path preceding the competition stage. Recall that each path takesone of three forms, namely, (s,P ,I), (s,P ,N ) or (s,S).Case 1: 2c̄− α ≤ 0. For any type c ∈ [0, c̄], we apply Lemmas 1, 2, and 3 and

it is easy to verify that j is active for each path.Case 2: 2c̄ − α > 0. For a type c ∈ (2c̄ − α, c̄], the argument from Case

1 applies. Now consider c ∈ [0, 2c̄ − α]. We Þrst show that if j is inactive inequilibrium, then type c necessarily discloses s ≥ 1−g

2(α + c). There are three

possibilities.First, if i chose (s,S), then Lemma 3 implies j is inactive ⇔ s ≥ (α + c)/2.

Next, if i chose (s,P) and j chose N , then Lemma 2 implies j is inactive, sincec ≤ 2c̄−α holds. Finally, if i chose (s,P) and j chose I, then Lemma 1 implies jis inactive⇔ s ≥ 1−g

2(α+ c). Combining, we see that s ≥ 1−g

2(α+ c) is necessary

for j to be inactive in the competition stage; otherwise, j could always produceproÞtably whether i chose (s,S) or (s,P).Suppose that, in equilibrium, j is inactive following the disclosure and patent

choice of some type c0 ∈ [0, 2c̄ − α]. Then, from above, type c0 must discloses0 ≥ 1−g

2(α+ c0) and, as j is inactive, type c0 earns the monopoly payoff (of type

c0). Let c1 ≡ 1−g2α and note that c1 ≤ s0. Then the disclosure s0 is feasible for

any type c ∈ [0, c1]. Further, since j is inactive at the equilibrium disclosure andpatent choice of type c0, a deviation by type c must yield the monopoly payoff (forc). Hence each type c ∈ [0, c1] must earn the monopoly payoff in equilibrium andj must be inactive following the disclosure and patent choice of type c. Otherwise,if j were active, then by Lemma A2, type c would necessarily earn a payoff belowthe monopoly level.Consider type c1. Since j must be inactive, we know from the above analysis

that c1 discloses s1 ≥ 1−g2(α + c1) ≡ c2. Then all types in [c1, c2] must earn a

monopoly payoff and j must be inactive, in equilibrium, as the disclosure s1 isfeasible. Repeating this argument, we construct a sequence via cn =

1−g2(α+cn−1).

By induction, cn = ραPn−1

k=1 ρk, where ρ ≡ 1−g

2∈ (0, 1). Hence, cn → αρ

1−ρ = s∗.

40

However, 2c̄− α < s∗ by c̄ < α1+g. Then, j must be active for sufficiently large n

since, eventually, we have cn > 2c̄ − α. Thus, there can be no type c0 ≤ 2c̄ − αfor which j is inactive in equilibrium.¥C. Proof of Lemma 5We generalize Lemma 5 (to Lemma A5). Consider the payoff πi(c,σ,S) for

type c ∈ [0,σ] and a disclosure σ ∈ [0, c∗]; Consider πi(c, r(c),P ,I) for c ∈ [0, c∗]and the disclosure r(c) ≡ 1+g

2c∗ + 1−g

2c. Lemma A5 establishes when there exists

a type-disclosure pair (cσ, σ) such that these payoff functions cross at cσ.For these disclosures, the payoff functions are as follows. At (r(c),P) by type

c, we have r(c) < 1−g2(α + c) ⇔ c∗ < s∗, which is valid, and Lemma 1 implies j

is active. Further, Lemma 4 and Table 7.1 imply that I is optimal for j. Then,substitution with s = r(c) in (3.2) of the text implies

v(c) ≡ πi(c, s,P, I) = 1

β(3− g)2·α− 3− g

2c+

1 + g

2c∗¸2+

gc

β(3− g)·α− 1 + g

1− gc∗¸,

for any c ∈ [0, c∗]. Next, at (σ,S) by a type c, where c ∈ [0, σ] and σ ∈ [0, c∗], weapply Lemma 3 to Þnd j is active ⇔ c > 2σ− α. Substituting into the payoff fori, we have

w(c, σ) ≡ πi(c, σ,S) =½ 1

9β(α− 2c+ σ)2 for c > M ≡ max{2σ − α, 0},

14β(α− c)2 for c ≤ 2σ − α,

where the lower branch irrelevant if 2σ − α ≤ 0; w is deÞned for types c ∈ [0,σ]and σ ∈ [0, c∗]. Note that the interval [M,σ] is non-empty asM < σ always holds;the cases of 2σ − α ≷ 0 both arise (as 2c∗ − α ≷ 0 occurs across (g, c̄,α) values)and must be dealt with.DeÞne ∆(c,σ) = w(c, σ)− v(c) for c ∈ [0, σ],σ ∈ [0, c∗]. It is easy to verify the

following properties. First, ∆ is strictly convex for c ∈ [M, σ]. Next, if 2σ−α > 0,then i) ∆ is linear in c over [0, 2σ−α], ii) ∆ is continuous at c = 2σ−α, iii) thereis a kink in ∆ at c = 2σ−α and the partial derivative of w satisÞes 0 > w−c > w+cat (2σ − α,σ).We now show∆(σ,σ) < 0. Since c = σ > 2σ−α, we apply the upper branch of

w. Note that the second term in v(σ) is always positive as gσβ(3−g)

³α− 1+g

1−gc∗´>

0 ⇔ 1−g1+gα ≡ s∗ > c∗, which holds. Hence, it is sufficient for ∆(σ, σ) < 0 to

show w(σ, σ) is less than the Þrst term in v(σ). Comparing, this reduces to0 < gα+ 3

2(1 + g)c∗ − 1

2(3− g)σ. This expression is linear decreasing in σ, so we

41

need only show it is positive at σ = c∗ and this reduces to 0 < g(α+ 2c∗), whichis valid. Hence, ∆(σ, σ) < 0 for σ ∈ [0, c∗].Now consider the value of ∆(0, σ). First, suppose 2σ − α ≤ 0. Then, upon

simplifying, we have ∆(0,σ) > 0 ⇔ σ > σ where σ ≡ 12(3−g) [2gα+ 3(1 + g)c

∗].We also note, for later use, σ ≷ c∗ ⇔ 2gα ≷ (3− 5g)c∗.Now suppose 2σ − α > 0. We claim ∆(c, σ) > 0 holds for c ∈ [0, 2σ − α].

We employ a monopoly versus duopoly payoff argument. By Lemma 3, j isinactive at (σ,S) for c ≤ 2σ − α and i earns the monopoly payoff of w(c, σ) =14β(α− c)2 = Max

q≥0(p− c)q. From above, j chooses I and is active at (r(c),P) by

i and j earns πIj ≡ πj(r(c),P, I) from Lemma 1 and (3.1). Letting superscriptI denote values at (r(c),P , I) for j when i is type c and similarly for i, we haveπIi < π

Ii +π

Ij = (p

I−c)qIi +gpIqIj +(pI−cj)qIj −gpIqIj = (pI−c)qIi +(pI−cj)qIj <(pI − c)(qIi + qIj ), where the last step follows from r(c) > c and we used theaccounting deÞnition of proÞts. Combining, ∆(c, σ) > 0 ⇔ monopoly proÞtexceeds πIi . This holds if q

Ii + q

Ij exceeds the monopoly output of

12β(α − c).

Substituting for quantities from Lemma 1, this reduces to s∗ > c∗, which is valid.Hence, ∆(c, σ) > 0 for c ∈ [0, 2σ − α] when 2σ − α > 0.We now sort out the parameter cases for (g, c̄,α) that lead to 2c∗ − α ≷ 0.

Lemma 6. If g < 1/3 and c̄ > αr(g) ≡ α4[3− (1− 3g)h(g)], then 2c∗ − α > 0.

If g < 1/3 and c̄ < αr(g) or if g > 1/3, then 2c∗ − α < 0.

Proof: From Lemma 4 and (4.2), we see that c∗(g,α, c̄) rises in value from0 at c̄ = αk(g) to s∗ at c̄ = α

1+g. When g > 1/3, we have s∗ < α

2and, hence,

2c∗ − α < 0. When g < 1/3, we have s∗ > α2. Thus, c∗ crosses α

2as c̄ varies.

Simplifying with (4.2) then shows that c∗ T α2as c̄ T αr(g).¤

One further preliminary result will be useful.

Lemma 7. If g < 1/3 and c̄ > αm(g), then we have (3−5g)c∗ > 2gα. If g < 1/3and c̄ < αm(g) or if g > 1/3, then we have (3− 5g)c∗ < 2gα.

Proof: At c̄ = αk(g), we have c∗ = 0. At c̄ = α1+g, we have c∗ = s∗. Note

that (3 − 5g)s∗ T 2gα ⇐⇒ 3 − 10g + 3g2 T 0. This is a convex quadraticwith roots at 1/3 and 3. If g > 1/3, it is negative and, by c∗ < s∗, we have(3 − 5g)c∗ < (3 − 5g)s∗ < 2gα. If g < 1/3, the quadratic is positive and, with(3− 5g)s∗ > 2gα, we see that (3− 5g)c∗ crosses 2gα as c̄ varies. From (4.2), wehave (3− 5g)c∗ T 2gα⇐⇒ c̄ T αm(g).¤

42

g

c

1/3 1

α/2

α

0

αm(g)

αr(g)

αk(g)

α/(1+g)3α/4 I

II III

IV

V

Figure 7.1:

Direct calculations show that m(g) rises from m(0) = 0 to m(1/3) = 3/4 as gvaries and that r(g) rises from r(0) = 1/2 to r(1/3) = 3/4 as g varies. Further,m(g) < r(g) for g < 1/3. See Figure 7.1 for reference.We now have Lemma A5.

Lemma A5 Assume αk(g) < c̄ < α1+g. Suppose (g, c̄,α) satisfy g < 1/3 and

c̄ > αm(g). Then, for each σ ∈ [σ, c∗] there exists a unique type cσ suchthat ∆(cσ, σ) = 0. Further, we have i) ∆(c, σ) R 0 as c Q cσ, ii) cσ increaseswith σ, and iii) cσ is betweenMax{2σ−α, 0} and σ, with cσ = 0 for σ = σ.Suppose, instead, that g ≥ 1/3 or that g < 1/3 and c̄ < αm(g). Then∆(c, σ) < 0 for all c ∈ [0, σ] and any σ ∈ [0, c∗].

Proof: We apply the above results on ∆ to (g, c̄,α) in each of regions I, IIand III in Figure 7.1. Take (g, c̄,α) ∈ Region III. Then 2c∗ − α < 0 by Lemma6 and, hence, 2σ − α < 0 for any σ ∈ [0, c∗]. We know ∆(σ, σ) < 0 and that∆ is strictly convex in c ∈ [0, σ]. We are done if ∆(0, σ) < 0. From above,∆(0, σ) < 0 ⇔ σ < σ. But σ > c∗ ⇔ (3 − 5g)c∗ < 2gα and this holds in III byLemma 7. Hence, σ ≤ c∗ < σ and ∆(0, σ) < 0.Consider (g, c̄,α) ∈ Region II. Again, by Lemma 6, we have 2σ−α ≤ 2c∗−α <

0 for σ ∈ [0, c∗]. Also, ∆(σ,σ) < 0 and ∆ is strictly convex in c ∈ [0, σ]. FromLemma 7, we have σ < c∗ as 2gα < (3−5g)c∗ in II. Now, note that σ < σ implies

43

∆(0, σ) < 0 from above, so strict convexity implies ∆(c, σ) < 0 for all c ∈ [0, σ] inthis case. Take σ > σ so that ∆(0, σ) > 0, from above. By continuity, ∆ crosseszero at some cσ between 0 and σ, and convexity implies cσ is unique. Hence,∆(c, σ) ≷ 0 as c ≶ cσ. Since the partials satisfy ∆c(cσ, σ) < 0 < ∆σ(cσ,σ), we seethat cσ increases with σ.Let (g, c̄,α) ∈ Region I. Then 2c∗ − α > 0 by Lemma 6 and σ< c∗ by Lemma

7. Further, with g < 1/3 we have σ < α/2 < c∗. First, suppose, σ < α/2. Thenthe analysis follows the same logic as that for Region II. Next, suppose σ > α/2.Then, as 2σ − α > 0, we know from above that ∆(c, σ) > 0 for c ∈ [0, 2σ − α].Since ∆(σ, σ) < 0, continuity and strict convexity of ∆ for c ∈ [2σ − α, σ] implya unique cσ where ∆ crosses zero. Further, ∆(c, σ) ≷ 0 as c ≶ cσ. As above, weÞnd cσ increases with σ.¥Lemma 5 in the text follows directly as a special case of Lemma A5 where we

take σ = c∗ and let cL denote the crossing value cσ.D. Proof of Proposition 1The main task is to verify that equilibrium payoff to i for each c ∈ [0, c̄] at

the candidate disclosure and patenting choice exceeds the payoff for any feasibledeviation. We also verify that j is choosing optimally from {I,N} and thatquantities are optimal. Out-of-equilibrium supporting beliefs are speciÞed at theend of the proof. Note that (g, c̄,α) ∈ A implies that the three ranges are welldeÞned as 0 < cL < c∗ < c̄ holds.Consider the equilibrium payoff for type c ∈ [0, c̄] of i, denoted by U(c).

For c ≥ c∗, we have full disclosure with ϕP(c) = c. Then, Þrm j optimallychooses N since (c,ϕP(c)) lies above the EP line for c ≥ c∗ (see Table 7.1).Further from Lemma 2, j is active in the competition stage at (ϕP(c),P,N ) sincec ≥ c∗ > 2c̄− α. We then calculate U(c) = 1

9β(α− 2c+ c̄)2.

For c∗ > c > cL, we have partial disclosure at ϕP(c). Since (c,ϕP(c)) liesbelow the EP line, Þrm j optimally chooses I (see Table 7.1). Further, it is easyto show ϕP(c) <

1−g2(α+ c) and, hence, j is active at (ϕP(c),P, I) by Lemma 1.

Then, from (3.2), we calculate

U(c) = πi(c,ϕP(c),P ,I) =1

β(3− g)2·α− 3− g

2c +

1 + g

2c∗¸2+

g

β(3− g)·α− 1 + g

1− g c∗¸c.

Next, for cL ≥ c, we have partial disclosure at ϕS(c). By Lemma 3, j is active at(ϕS(c),S) provided c > max{2ϕS(c)−α, 0}; since c > 2ϕS(c)−α⇔ cL > 2c

∗−α,we see from Lemma 5 that j is active. We then calculate from Lemma 3, U(c) =πi(c,ϕS(c),S) = 1

9β

£α− 3

2c+ c∗ − cL

2

¤2.

44

For later reference, we note the following properties of U(c). First, it isstrictly decreasing and convex in each of the regions. This is obvious for c ≤ cLand for c > c∗. For cL < c ≤ c∗, differentiation shows that U 0(c) < 0 ⇔ghα− 1+g

1−gc∗i< [α− (1− g)c∗]. To see that this last inequality is valid, note

that we have (1+g)c∗ > c∗ > (1−g)2c∗ and this implies α−(1−g)c∗ > α− 1+g1−gc

∗.Since g < 1, we have U 0(c) < 0 for cL < c < c∗.Next, consider c = cL and c = c∗. By construction of cL (see Lemma 5), U

is continuous at c = cL. At c = c∗, however, there is a downward jump in U .This is because j is indifferent between I and N since (c∗,ϕP(c

∗)) lies on the EPline. The equilibrium speciÞes a choice of I by j in this case and, by Lemma A4,we have U(c∗) > limc↓c∗ U(c). Because of j�s indifference, we could also specify achoice of N by j; this would require that type cL choose (ϕP (cL),P) rather than(c∗,S). Either speciÞcation works equally well and has no impact on the structureof the equilibrium.We will need the following result to compare the equilibrium payoff, U(c), for

c ≤ c∗, to deviation payoffs.Lemma 8. Assume (g, c̄,α) ∈ A. Let c0 and s0 satisfy 0 < c0 ≤ cL and 1+g

2c∗ +

1−g2c0 ≤ s0 <

1−g2(α + c0). Suppose σ0 satisÞes πi(c0, σ0,S) = πi(c0, s0,P, I).

Then πi(c, σo − 12(c0 − c),S) R πi(c, s0 − 1−g

2(c0 − c),P, I) as c Q c0, for any

c ∈ [0, c00], where c00 > c0 is deÞned by e(c00) = s0 − 1−g2(c0 − c00).

Proof: Consider the line s = s0− 1−g2(c0−c). Note that c00 is deÞned by where

this line crosses the EP line. From Lemma 4 and Table 7.1, it is easy to verifythat a choice of (s,P) by c ∈ [0, c00], where s = s0 − 1−g

2(c0 − c) implies that j

chooses I and is active. Consequently j is active at (σ0,S) by c0 since i earnsless than the monopoly payoff at (s0,P); in turn, j is found to be active at (σ,S)choice by c, for σ = σ0 − 1

2(c0 − c).

DeÞne W (c) ≡ πi(c,σ0− 12(c0− c),S) and V (c) = πi(c, s0− 1−g

2(c0− c),P, I).

Let δ(c) ≡ W (c)−V (c).Then δ(c0) = 0, by construction, and differentiation showδ(c) is linear. We are done if we show δ(0) > 0. Evaluating the proÞt functionsat c = 0, we have δ(0) > 0 ⇔ (3 − g)σ0 > gα + 3s0 + gc0. The next step is toshow this inequality is valid.Substituting s = s0 − 1−g

2(c0 − c) into πi(c, s,P , I) from (3.2) in the text, we

have

V (c) =1

β(3− g)2·α− 3− g

2c+ s0 − 1− g

2c0

¸2+

gc

β(3− g)·α− 2

1− gs0 + c0¸

45

≡ V1(c)+V2(c). Clearly, V2(c0) > 0 holds as 1−g2 (α+ c0) > s0. Therefore, W (c0) >V1(c0). Simplifying the proÞt expressions then reveals that this is equivalent tothe inequality for W (0) > V (0).¤For Proposition 1 we will apply Lemma 8 with c0 = cL, s0 = ϕP(cL), and

σ0 = c∗. We now consider deviations for i types in each of the three ranges.

Case 1: c̄ ≥ c > c∗. Note that all types below c either disclose fully ordisclose no more than c∗. Thus, the only feasible deviation for c is to patent anddisclose �c = ϕP(�c) for some �c > c. Upon inferring type �c, the same argumentas above implies that j chooses N and produces (actively) at �qj = qj(�c,P ,N )as given by Lemma 2. The best response for i of type c is positive and followsqBRi = 1

2β(α− c− β�qj) and results in a payoff of β(qBRi )2. Simplifying then

yields the deviation payoff of u(�c, c) = 19β

¡α− 3

2c− 1

2�c+ c̄

¢2. Since this is strictly

decreasing in �c, we have U(c) > u(�c, c) and the equilibrium choice ϕP(c) = c isoptimal for c.Case 2: c∗ ≥ c > cL. There are three kinds of feasible deviations: i) to the

small innovation region at a �c where �c > c∗, ii) within the medium innovationregion to a �c where c < ϕP(�c) ≤ c∗, and iii) to the large innovation region at �cwhere c ≤ ϕS(�c) ≤ c∗. We take each of these sub-cases in turn.i) �c > c∗ : By the same argument as in case 1, we Þnd u(�c, c) = 1

9β

¡α− 3

2c− 1

2�c+ c̄

¢2.

This is decreasing in �c. Thus, to rule out a deviation to �c > c∗, it is sufficientto show U(c) ≥ lim�c↓c∗ u(�c, c). We apply Lemma A4. Noting that (c∗, c∗) lieson the EP line, set t = µ = c∗ in Lemma A4. Then, we see that U(c) = πIi (c)and lim�c↓c∗ u(�c, c) = πNi (c) as constructed in Lemma A4. As Lemma A4 assertsπIi (c) > π

Ni (c), we are done.

ii) c < ϕP(�c) ≤ c∗. Upon observing ϕP(�c), Þrm j infers type �c and, therefore,chooses I and the (positive) quantity �qj = qj(ϕP(�c),P , I) as in Lemma 1. Thebest response for i of type c is positive and follows qBRi = 1

2β[α− c− β(1 + g)�qj ] .

Calculating the payoff to i (as in (5.1) in the text) reveals that U(c) = u(�c, c); aswe see later, ϕP(c) necessarily involves weak incentive compatibility in (cL, c

∗).iii) c ≤ ϕS(�c) ≤ c∗. Upon observing ϕS(�c), Þrm j infers type �c and

chooses the (positive) quantity �qj = qj(ϕS(�c),S), as in Lemma 3. Calculat-ing the best response for i of type c and the resulting payoff yields u(�c, c) =19β

¡α− 3

2c+ c∗ − cL

2

¢2; thus, c is indifferent across the set of feasible �c deviations

into the large innovation region. Since c > cL, we apply Lemma 8 directly andU(c) > u(�c, c) holds.Case 3: cL ≥ c. The three kinds of feasible deviations are i) �c > c∗, ii)

46

c∗ ≥ �c > cL, and iii) c∗ ≥ ϕS(�c) ≥ c. We take each in turn.i) �c > c∗. As above, we Þnd that u(�c, c) = 1

9β

¡α− 3

2c− 1

2�c+ c̄

¢2is decreasing

in �c, so it is sufficient to show U(c) ≥ lim�c↓c∗ u(�c, c). From case 2, we knowU(c0) > lim�c↓c∗ u(�c, c0) for c0 ∈ (cL, c∗). Since U is continuous at cL, we haveU(cL) > lim�c↓c∗ u(�c, cL) and this implies 3c∗ ≥ cL + 2c̄. Comparing U(c) andlim�c↓c∗ u(�c, c), we see this last inequality implies U(c) ≥ lim�c↓c∗ u(�c, c) holds.ii) c∗ ≥ �c > cL. We calculate u(�c, c) exactly as in case 2 (ii). For c < cL, we

apply Lemma 8 directly and U(c) > u(�c, c) holds. At c = cL, we have equalityand the type cL is indifferent.iii) c∗ ≥ ϕS(�c) ≥ c. Calculate u(�c, c) exactly as in case 2 (iii). This yields

U(c) = u(�c, c); as we see later, ϕS(c) necessarily involves weak incentive compat-ibility in [0, cL].Finally, we must specify supporting beliefs for out-of-equilibrium (s,P) and

(s,S) choices. The simple linear extension of ϕP(c) to [0, cL] and the mean beliefµ = ϕ−1P (s) for s ∈ [ϕP(0),ϕP(c∗)], along with µ = c for s < ϕP(0) is sufficient tosupport the equilibrium. A linear extension of ϕs to (cL, c̄], however, will inducedeviations (by types near c∗). It suffices to take beliefs at (s,S) for s > c∗ to beµ = c̄ or µ = s. Intermediate inferences also work.¥E. Proof of Proposition 2To prove that the conditions are sufficient for a PBE, we simply apply the

proof of Proposition 1. The only change is that we replace cL and ϕS(cL) = c∗

with cσ and ϕS(cσ) = σ.To establish payoff dominance, simply compare the equilibrium with cL and

ϕS(cL) = c∗ from Proposition 1 to an equilibrium with cσ and σ for σ < σ < c∗.

Since cσ < cL, all types above cL earn the same payoff. Since πi(c, c∗ − 12(cL −

c),S) > πi(c,1+g2c∗ + 1−g

2c,P, I) for c ≤ cL, by Lemma 8, all types c ∈ [cσ, cL]

strictly prefer the equilibrium from Proposition 1.This leaves types c ∈ [0, cσ]. We must show πi(c, c∗− 1

2(cL− c),S) > πi(c,σ−

12(cσ − c),S). This reduces to c∗ − 1

2cL > σ − 1

2cσ. From Lemma 5, we know

cσ and σ satisfy πi(cσ, σ,S) = πi(cσ,1+g2c∗ + 1−g

2cσ,P, I). Combining with the

inequality from Lemma 8 in the previous paragraph evaluated at c = cσ, we haveπi(cσ, c

∗ − 12(cL − cσ),S) > πi(cσ, σ,S) and this implies c∗ − 1

2cL > σ − 1

2cσ.

We now turn to necessary conditions.For reference, (s,P) or (s,S) denotes a disclosure and patent choice by i. We

reserve ϕP(c) and ϕS(c) for candidate equilibrium choices. From Corollary 1, j isactive at any equilibrium choice by i. Finally, let r(c) ≡ 1+g

2c∗ + 1−g

2c for all c.

The Þrst step is to derive continuation payoffs in a PBE for any feasible patent

47

and disclosure choice by i. Suppose j observes (t,S) by i and holds mean beliefµ ∈ [0, t]. We apply Lemma A1 with cj = t and g ≡ 0 to Þnd the optimal quantityfor j; further, the quantity choice for i in Lemma A1 for each type c is a best-response to this choice by j. The continuation payoff for type c of i from (t,S)is

ui(c, t,S;µ) =(

19β

¡α− 3

2c− 1

2µ+ t

¢2, for t < α+µ

214β(α− c)2 , for t ≥ α+µ

2.

If (t,S) is observed in equilibrium, then µ = ϕ−1S (t). Otherwise, (t,S) is off theequilibrium path and the belief is only required to support the equilibrium andsatisfy Bayes� rule (µ ≤ t); by Lemma A1, the mean is the only payoff relevantproperty of j�s belief.Now, suppose j observes (t,P) and has mean belief µ ∈ [0, t]. PBE requires

that j makes an optimal choice between I and N at (t,P) for belief µ. For eachpossible t and µ, we apply Table 7.1 to Þnd the optimal j choice of I or N , andthen apply Lemma A1 as above to Þnd the quantities. The continuation payofffor type c of i is

ui(c, t,P ;µ) =

1

β(3−g)2£α− 3−g

2c− 1−g

2µ+ t

¤2+ gc

β(3−g)hα− 2

1−g t+ µi, µ ∈ KI at t

19β

£α− 3

2c− 1

2µ+ c̄

¤2,

(α−c)24β

,

µ ∈ KN at tµ ∈ KM at t

As before, if (t,P) occurs in equilibrium, then µ = ϕ−1P (t) must hold. Note that uiis single-valued except in the case where µ and t are on the EP line, t = e(µ) andµ ∈ KI ∩KN . In this case, each of N and I is optimal for j and the continuationpayoff for i can assume two possible values.The following result for ui is useful

Lemma 9. The continuation payoff ui is non-increasing in µ and strictly decreas-ing in µ whenever j is active in the production stage.

Proof : This is obvious for ui(c, t,S;µ). For ui(c, t,P;µ), the result followsdirectly from Lemma A2 when µ is in the interior of the KM , KI or KN intervals(see Table 7.1). Across the various boundary cases, we Þnd that ui is continuousand, hence, the result is established, except for the boundary between KI and

48

KN . Recall that this occurs when µ ∈ (max{0, 2c̄ − α}, c∗] and t = e(µ). FromLemma A4, we see that πIi (c) > π

Ni (c) are the two continuation payoff values for

ui. Hence, the jump in ui is downward since we cross from KI to KN as µ rises.¥We now examine the necessary structure of an equilibrium (separating PBE).

Clearly, under separation, a disclosure of c̄ leads j to infer i is type c̄ since, withno innovation, c̄ is the only feasible disclosure for type c̄. It is convenient to adoptthe convention that i chooses P in this case. Hence, ϕP(c̄) = c̄.Now consider types between c∗ and c̄. First, we show that only P is used in

equilibrium for c ∈ (c∗, c̄). Suppose, instead, that such a c type chooses (s,S)in equilibrium for some s ∈ [c, c̄]. From above, separation implies s < c̄. Inequilibrium, j infers type c for i upon observing (s,S) and, by Lemma 3, i mustearn the payoff of πi(c, s,S) = 1

9β(α− 2c+ s)2.

Consider a deviation by type c to (t,P) for some t ≥ c. By Lemma 9, the lowestpossible deviation payoff is ui(c, t,P; t), where j holds the pessimistic mean beliefof µ = t. Note that c∗ < c ≤ t implies j will chooseN in this case. Comparing, wehave ui(c, t,P ; t) > πi(c, s,S)⇔ (α− 3

2c− 1

2t+ c̄) > (α−2c+s)⇔ 1

2(c−t)+ c̄ > s.

Under a full-disclosure deviation of t = c, this reduces to c̄ > s and the deviation isstrictly proÞtable. Therefore, all types c ∈ (c∗, c̄) necessarily use P in equilibrium.It is straightforward to show that each c ∈ (c∗, c̄) chooses full disclosure under

a patent in equilibrium. Suppose, instead, that some type c ∈ (c∗, c̄) chooses(s,P) with s > c. This implies that (s,P) is a feasible deviation for any type�c ∈ (c, s]. Let (�s,P) be the equilibrium choice of such a �c type. Note that Þrm jnecessarily chooses N at each of (s,P) and (�s,P) since each of (c, s) and (�c, �s) liesabove the EP line. Then, �c prefers the deviation (s,P) to (�s,P)⇔ ui(�c, s,P; c) >πi(�c, �s,P,N )⇔ (α− 3

2�c− 1

2c+ c̄) > (α−2�c+ c̄)⇔ �c > c, which is valid. Therefore,

in equilibrium, we must have ϕP(c) = c for c ∈ (c∗, c̄).Now consider types c ≤ c∗. We claim that if (s,S) is chosen by any c ≤ c∗

in equilibrium, then s ≤ c∗. Suppose, instead, that some c ≤ c∗ chooses (s,S)with s > c∗. Since c ≤ c∗, it is feasible for type c to deviate to (�c,P) forany �c > c∗. In equilibrium, type c must prefer (s,S) to (�c,P) and this holds⇔ πi(c, s,S) ≥ ui(c, �c,P ; �c)⇔ (α−2c+ s) ≥ (α− 3

2c− 1

2�c+ c̄)⇔ s ≥ c̄− 1

2(�c− c).

As this must hold for any �c > c∗, we then have s ≥ c̄− 12(c∗ − c).

Consider the type �c = c̄− 12(c∗ − c). Clearly, we have c∗ < �c ≤ c̄ since c ≤ c∗.

Further, a deviation by type �c to (s,S) is feasible since s ≥ �c. Then, type �c strictlyprefers (s,S) to (�c,P) ⇔ ui(�c, s,S; c) > πi(�c, �c,P,N ) ⇔ (α − 3

2�c − 1

2c + s) >

(α−2�c+ c̄)⇔ s > c̄− 12(�c− c). This last inequality necessarily holds since �c > c∗.

Thus, we must have s ≤ c∗ for any equilibrium choice of (s,S).

49

The above argument implies that type c∗ chooses (c∗,P) in equilibrium. Thisis because the only possible equilibrium choice of the form (s,S) for type c∗ hass = c∗. The payoff πi(c∗, c∗,S), however, is strictly dominated by ui(c∗, �c,P , �c) for�c > c∗ and �c sufficiently close to c∗, which type c∗ can obtain by choosing (�c,P).Further, we have πi(c∗, c∗,P ,N ) > ui(c∗, �c,P ; �c) for any �c > c∗. Thus, separationimplies type c∗ must choose (c∗,P) in equilibrium and this holds whether j (whois indifferent) chooses I or N upon observing (c∗,P) and inferring type c∗ for i.Now consider types c ∈ (cL, c∗). We claim all such types use P in equilibrium.

Suppose, instead, some c ∈ (cL, c∗) chooses (s,S). From above, s ≤ c∗ holds andthe payoff to type c is πi(c, s,S). Consider a deviation by type c to (t,P) for somet ∈ (c, c∗). By Lemma 9, the deviation payoff is at least ui(c, t,P , t), where jholds the pessimistic belief of µ = t. Since µ = t < c∗, we see from Table 7.1 thatµ = t ∈ KI = [0, t] in this case and, hence, j necessarily chooses I.This deviation is strictly proÞtable for a choice of t sufficiently close to c∗. First,

πi(c, s,S) ≤ πi(c, c∗,S), by Lemma A2. Next, (from the proof of Lemma 5) 0 >∆(c) ≡ πi(c, c∗,S) − πi(c, r(c),P, I) for c ∈ (cL, c∗). Now, ui(c, t,P ; t) convergesto πi(c, r(c),P , I) as t increases to c∗. Combining, we then have πi(c, s,S) <ui(c, t,P ; t) for t < c∗ and t sufficiently close to c∗. Thus, the deviation is strictlyproÞtable and no type c ∈ (cL,c∗) uses S in equilibrium.We now show all types c ∈ (cL, c∗) choose partial disclosure and P. From

above, we know any such c chooses P and separation implies s < c∗ for anequilibrium choice (s,P) by c. We also see j chooses I at (s,P), by Table 7.1.Hence, the equilibrium payoff for c is πi(c, s,P, I).Consider a deviation by type c to (�s,P), where (�s,P) is the equilibrium choice

of a type �c for c < �c < c∗. In equilibrium, type c must prefer (s,P) to (�s,P)and this requires πi(c, s,P, I) ≥ ui(c, �s,P; �c). DeÞne the function n(c, x) =

1β(3−g)2

£α− 3−g

2c+ x

¤2+ gc

β(3−g)hα− 2

1−gxi. From the proof of Lemma A2, we see

n(c, x) is strictly increasing in x. Further, we have πi(c, s,P ,I) = n(c, s − 1−g2c)

and ui(c, �s,P ; �c) = n(c, �s − 1−g2�c). Hence, we must have s − 1−g

2c ≥ �s − 1−g

2�c in

equilibrium. As this holds for �c arbitrarily close to c∗, we must have s − 1−g2c ≥

c∗− 1−g2c∗ = 1+g

2c∗ since �c ≤ �s < c∗. Then, s ≥ 1+g

2c∗+ 1−g

2c ≡ r(c) > c, as c < c∗,

and we have partial disclosure.Now, partial disclosure implies that a deviation to (s,P) is feasible for types �c

such that c < �c ≤ s. Since type �c must prefer (�s,P) to (s,P), the same argumentas above implies �s− 1−g

2�c ≥ s− 1−g

2c. Hence, �s = s+ 1−g

2(�c− c).

We can now show ϕP(c) = r(c) for c ∈ (cL, c∗). Suppose some c0 ∈ (cL, c∗)

50

chooses (s0,P) in equilibrium with s0 > r(c0). DeÞne c1 = s0 and, noting that s0is feasible for c1, the above incentive compatibility argument implies that ϕP(c1) =s1 = s0 +

1−g2(c1 − c0). DeÞne c2 = s1 and continue on to construct a sequence

(cn) where cn = cn−1 +1−g2(cn−1 − cn−2) for n ≥ 2. Then (cn) is easily found to

converge to [2s0 − (1− g)c0]/(1 + g); since s0 > r(c0), this limit exceeds c∗. Butthen ϕ(cn) > cn > c∗ holds for large n, which is not possible. Hence, ϕP(c) = r(c)for c ∈ (cL, c∗).Now consider c ∈ [0, cL]. DeÞne Σ to be the set of types who choose S in

equilibrium. If Σ = φ, then all types choose P and the arguments above implythat ϕP(c) = r(c). Thus, suppose Σ 6= φ. We Þnd that an equilibrium choice(s,S) by c ∈ Σ must satisfy i) s ≤ c∗; ii) c < c̄ − 1

2(c∗ − c) < s; iii) if �c ∈ Σ

and �c < c, then s > �s ≥ s − 12(�c − c), with equality if c ≤ �s. Claim i) is from

above. Claim ii) follows from considering a deviation by c to the equilibriumchoice of types above c∗. Claim iii) follows from a deviation by �c to (s,S) and,when feasible, vice versa.DeÞne cσ ≡ supΣ and σ ≡ sup{s | s = ϕS(c) for some c ∈ Σ}. From above,

cσ ≤ cL and σ ≤ c∗ must hold. Also, by property iii) of Σ, for any sequence oftypes in Σ converging to cσ, we must have ϕS(c) converging monotonically to σ.From our earlier arguments, all c > cσ choose (r(c),P) in equilibrium.We claim that cσ and σ must satisfy πi(cσ, σ,S) = πi(cσ, r(cσ),P, I). To see

this, take c ∈ Σ such that cσ − ε < c ≤ cσ for ε > 0. By property (ii) of Σ,we have ϕS(c) > c. For ε small enough, (ϕS(c),S) is feasible for type �c wherecσ < �c < cσ+ε. With c < �c, it is clearly feasible for c to deviate to (ϕP(�c),P). Theclaim then follows from the incentive compatibility conditions as ε → 0. FromLemma A5, we see that this proÞt equality condition requires σ ≤ σ.The next step is to show that all c < cσ choose S and disclose ϕS(c) = σ −

12(cσ−c) in equilibrium. Suppose some c < cσ chooses (s,P) in equilibrium. Sincej must be active, the payoff to c is πi(c, s,P, I). By construction of cσ = supΣ,we can Þnd �c ∈ Σ such that c < �c ≤ cσ. Then (�s,S), where �s = ϕs(�c), is feasiblefor c since c < �c and incentive compatibility implies πi(c, s,P ,I) ≥ ui(c, �s,S; �c).Letting �c approach cσ (or taking �c = cσ if cσ ∈ Σ), we know that ui(c, �s,S; �c)converges to ui(c,σ,S; cσ). Thus, we must have πi(c, s,P , I) ≥ ui(c,σ,S; cσ).It is also feasible for c to deviate to (r(�c),P), the equilibrium choice of �c for

cσ < �c < c∗, and incentive compatibility implies s > r(c). We can now rule outthe possibility of such a c type in a neighborhood of cσ: suppose r−1(cσ) < c <cσ so that (s,P) is feasible for �c > cσ sufficiently close to cσ. Then incentivecompatibility implies s = r(c). With s = r(c), however, we can apply Lemma 8

51

(set c0 = cσ, s0 = r(cσ) and σ0 = σ) to see that ui(c, σ,S; cσ) = πi(c, σ − 12(cσ −

c),S) > πi(c, r(cσ)− 1−g2(cσ− c),P , I) = πi(c, r(c),P ,I), which violates incentive

compatibility. Thus, we must have c ∈ Σ for all c ∈ (r−1(cσ), cσ).Then, we show ϕS(c) = σ − 1

2(cσ − c) for all c in this lower neighborhood of

cσ by an analogous argument to the one we employed to show ϕP = r for typesbetween cL and c∗. Note that if cσ < r(0), then we have shown c ∈ Σ and ϕS(c) =σ− 1

2(cσ−c) for all c below cσ. If, instead, cσ ≥ r(0), we can employ a variation of

our argument to rule out P for all types below cσ. DeÞne η = sup{c | c < cσ andc /∈ Σ}.We Þnd that η and sη must satisfy πi(η, sη,P , I) = πi(η, σ− 1

2(cσ−η),S).

But Lemma 8 implies that incentive compatibility will be violated and some typeat or below η will strictly prefer a deviation to (ϕS(c),S) for η < c < cσ. Thus,all types below cσ choose S and disclose ϕS(c) = σ − 1

2(cσ − c).

Finally, note that type cσ can choose (σ,S) or (ϕP(cσ),P) in equilibriumwhen σ < c∗. For σ = c∗, however, the equilibrium connection with j indifferencebetween I and N at (c∗,P) by type c∗ of i comes into play. If j chooses I,then equilibrium requires that type cL choose (c∗,S); if j chooses N , then cLmust choose (ϕP(cL),P). Except for these minor open set considerations, theequilibrium structure is not impacted.¥

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