IInnovation and Strategic Network Formation
Krishna Dasaratha ∗ February 11, 2020
Abstract
We study a model of innovation with a large number of firms that create new technologiesby combining several discrete ideas. These ideas can be acquired by private investment orvia social learning. Firms face a choice between secrecy, which protects existing intellectualproperty, and openness, which facilitates learning from others. Their decisions determineinteraction rates between firms, and these interaction rates enter our model as link proba-bilities in a learning network. Higher interaction rates impose both positive and negativeexternalities on other firms, as there is more learning but also more competition. We showthat the equilibrium learning network is at a critical threshold between sparse and densenetworks. At equilibrium, the positive externality from interaction dominates: the innova-tion rate and even average firm profits would be dramatically higher if the network weredenser. So there are large returns to increasing interaction rates above the critical threshold.Nevertheless, several natural types of interventions fail to move the equilibrium away fromcriticality. One policy solution is to introduce informational intermediaries, such as publicinnovators who do not have incentives to be secretive. These intermediaries can facilitate ahigh-innovation equilibrium by transmitting ideas from one private firm to another. ∗ Harvard University. Email: [email protected]. I am deeply grateful to Drew Fudenberg,Benjamin Golub, Matthew Rabin, and Tomasz Strzalecki for their guidance and support. I would also liketo thank Daron Acemoglu, Mohammad Akbarpour, Yochai Benkler, Laura Doval, Edward Glaeser, KevinHe, Nir Hak, Scott Kominers, Shengwu Li, Jonathan Libgober, Michael Rose, Stefanie Stantcheva, OmerTamuz, and Alexander Wolitzky for valuable comments and discussion. a r X i v : . [ ec on . T H ] F e b Introduction
A growing body of empirical research suggests that interactions between inventors are animportant part of innovation. New technologies are often produced by combining individualinsights with learning from peers, which confers large benefits on firms and inventors engagedin such learning. When highly-connected clusters of firms emerge in a location, as in thetechnology industry in Silicon Valley, inventors in these areas are much more productive.But frequent collaboration and learning are not assured even when inventors in a givenindustry co-locate (Saxenian, 1996 gives a well-known example). Rather, interaction patternsdepend on firms’ decisions, such as how much to encourage their employees to interactwith employees from other firms. These interactions let a firm learn from other companiesand inventors. There are also downsides for the firm, as employees may share valuableinformation. A more secretive approach allows firms to prevent potential competition byprotecting intellectual property. In making these types of decisions, firms and inventorsmust choose between openness and secrecy.We develop a theory of firms’ endogenous decisions about how much to interact withother firms, and the consequences for information flows, the rate of innovation, and relatedpolicy decisions. An important feature is that firms combine different ideas to create newtechnologies (see also Acemoglu and Azar, 2019 and Chen and Elliott, 2019 for relatedcombinatorial production functions). Strategic complementaries play a crucial role in manynetwork games (e.g., Ballester, Calv´o-Armengol, and Zenou, 2006 and Bramoull´e, Kranton,and D’amours, 2014), and in our setting complementarities arise endogenously from thisprocess of combining ideas.We use a framework inspired by recombinant growth (Weitzman, 1998) to explicitly modelthe creation of new technologies. Technologies are modeled as finite sets of distinct ideas.Ideas can be acquired in two ways: (1) via private investment and (2) via social learning.Firms generate profits by combining ideas to produce new technologies, but the profits from The benefits to interactions between inventors and movement of inventors have been quantified empiri-cally by Akcigit, Caicedo, Miguelez, Stantcheva, and Sterzi (2018), Kerr (2008), Samila and Sorenson (2011),among others. See Bessen and Nuvolari (2016) for historical examples and Chesbrough (2003) for examples in thetechnology industry. learning network . Wetherefore consider link formation decisions with uncertainty, while the leading approach inthe literature on network-formation games focuses on deterministic models (Jackson andWolinsky, 1996 and Bala and Goyal, 2000). Since we take actions to be continuous choicesthat translate to interaction rates, optimal behavior satisfies first-order conditions ratherthan a high-dimensional system of combinatorial inequalities.A key feature of our model is that ideas can spread several steps through this network:when one firm learns from another, the information transferred can include ideas learnedfrom a third firm. We refer to this as indirect learning . Under indirect learning, firms’incentives depend on the global structure of the network. Each firm would like to learnmany ideas, since then the firm could combine these ideas to produce a large number of newtechnologies, and much of this learning can be indirect.This analysis leads to our second contribution, which is to characterize equilibrium andquantify the associated externalities. Learning outcomes depend dramatically on whetherthe learning network is sparsely connected or densely connected. If firms’ interaction ratesare below a critical threshold, the learning network consists of many small clusters of firmswho learn few ideas. Above the threshold, the learning network has a giant component asymptotically: a large group of firms who learn a large number of ideas and can incorporatethese ideas into many new technologies. We analyze an individual firm’s best responses ineach of these two domains, i.e., when other firms form a sparse or dense network. By contrast, existing work on strategic formation of random networks largely focuses on direct connec-tions (Currarini, Jackson, and Pin, 2009).
2n our baseline model, we show that the equilibrium interaction rates are at the criticalthreshold between sparse and dense networks. Firms would deviate to interact more if thenetwork were likely to be sparse, and deviate to interact less if the network were likely tobe dense. Intuitively, in sparse networks firms would increase interaction rates to fill centralpositions in the network, known in sociology as ‘structural holes’, which enable the firm tocombine ideas learned via different interactions. As others interact more, these structuralholes disappear, and indeed firms tend to learn the same ideas repeatedly from differentinteractions. So the incentives to be more open are weaker relative to the incentives to besecretive.Since equilibrium is at the critical threshold, a giant component would emerge if all firmsshifted interaction rates slightly above equilibrium levels. Firms learn relatively few ideas atthe threshold, but could learn many more with a bit more interaction. Although more learn-ing leads to more competition, we show there are still unboundedly large profitability andwelfare gains (as the number of firms grows large) to interventions that increase interactionrates above equilibrium levels. To rephrase in terms of the underlying economic forces, thebenefits from more learning outweigh the lost profits from additional competition, even ifonly producers’ interests are considered. A consequence is that increasing interaction ratesis a first-order concern in designing policy. By contrast, policies targeting decisions aboutprivate investment rather than interaction, such as subsidies to R&D, have minimal effectat equilibrium—but can be valuable if paired with interventions to increase openness.We discuss one type of policy change that can induce more productive interaction pat-terns, which is to introduce public innovators who do not have incentives to be secretive. Forexample, governments could fund academic researchers who are especially willing to interactwith other researchers, including in industry. The key is that public innovators can serve asinformational intermediaries, transmitting ideas between private firms. They play a valuablerole even after considering the equilibrium response of the profit-maximizing firms, who mayadjust to be more secretive.We next explore which features of the baseline model are needed to obtain a critical The concept of structural holes, introduced by Burt (1992), refers to network positions allowing agentsto combine information from different connections or spread information between groups. how these ideas are learned, e.g., via manyinteractions or a few interactions. To capture these complementarities, we prove a key lemmarelating a firm’s equilibrium action to the extent to which technologies combine ideas fromdistinct interactions. An additional challenge is that because the network formation pro-cess is endogenous, vanishing-probability events and lower-order terms in link probabilitiescould affect payoffs. Our analysis therefore requires careful treatment of the graph branchingprocess governing the number of ideas learned from each interaction.
This paper relates to research in network theory, especially network formation, and to modelsof innovation.At a methodological level, we develop a theory of strategic network formation with prob-abilistic links. A large literature since Jackson and Wolinsky (1996) and Bala and Goyal(2000) considers endogenous network formation assuming that agents can choose their linksexactly. Because equilibrium is then characterized by a large system of inequalities, thesemodels illustrate key externalities in special cases but remain largely or entirely intractable inmany others. By instead considering agents facing uncertainty, we obtain a smooth model oflink formation that can be solved via basic optimization techniques combined with analysesof random graphs. Under this random-network approach to network formation, incentives to form linksdepend on the ‘phase transitions’ between sparse and dense networks. Economic modelsinvolving phase transitions have been recently explored in the context of diffusion processesby Campbell (2013), Akbarpour, Malladi, and Saberi (2018), and Sadler (2020), who let The pairwise stability solution concept from Jackson and Wolinsky (1996) and variants have been appliedto network formation in many settings, including innovation (K¨onig, Battiston, Napoletano, and Schweitzer,2011, K¨onig, Battiston, Napoletano, and Schweitzer, 2012). An alternate approach to smooth network formation is to consider weighted networks, so that eachlink has an intensity (Baumann, 2017 and Griffith, 2019). By analyzing random networks, we can studycontinuous link-formation decisions without requiring network weights. Golub and Livne (2010) also study network formation with phase transitions, and allow payoffs to dependon distance one and two connections. An important feature of our model is that firms’ decisions depend onthe global network structure rather than only local connections. We first describe our model formally and provide an example. We will then discuss interpre-tations of the model and its assumptions. The section concludes by introducing our solutionconcept. Indeed, by assuming a continuum of firms, macroeconomic models of imitation often implicitly restrictto subcritical interaction patterns. .1 Basic Setup There are n > , . . . , n . Each firm i can potentially discover a distinct idea , alsodenoted by i . We let I ⊂ { , . . . , n } be the set of ideas that are discovered.Each firm i chooses a probability p i ∈ [0 ,
1) of discovering this idea and pays investmentcost c ( p i ). We will assume that c is continuously differentiable, increasing, and convex with c (0) = 0 and lim p → − c ( p ) = ∞ . The realizations of discoveries are independent.A technology t = { i , . . . , i k } consists of k ideas i , . . . , i k ∈ I , where k > Each idea i ∈ t must be discoveredby the corresponding firm to be included in a technology. There are therefore (cid:0) nk (cid:1) potentialtechnologies, and a firm i can produce more than one technology.Each firm i chooses a level of openness q i ∈ [0 , q i and q j , the interactionrate between i and j is ι ( q i , q j ) = q i q j . The timing of the model is simultaneous: firms choose actions p i and q i and then alllearning occurs. We denote the vectors of actions by ( p , q ). When actions are symmetric,we will refer to p i by p and q i by q .Given actions p and q , we denote the set of ideas that firm i learns from others by I i ( p , q ) ⊂ I . This is a random set depending on realizations of learning and discoveries. Wenow describe how learning occurs.With probability ι ( q i , q j ) , firm i learns directly from firm j . In this case, firm i learnsidea j if j ∈ I . If firm i learns directly from firm j , then with probability δ ∈ [0 , i also learns indirectly through firm j . In this case, firm i also learns all ideas in I j ( p , q ).All realizations of direct and indirect learning are independent, and in particular, firm i canlearn from firm j without j learning from i .When δ = 0 there is only direct learning , while when δ > indirect learning canalso occur. When δ > , . . . , n and a link from node j to node i if firm i learns indirectlythrough firm j . The model will extended in Appendix D to allow firms to potentially discover multiple ideas. In the baseline model, the parameter k is the same for all firms.
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Figure 1: Network with four firms and k = 3. Black circles are firms that discover ideaswhile white circles do not discover ideas. Dashed lines indicate direct learning and solid linesindicate indirect learning. The only technology produced is t = { , , } and firm 3 receivesmonopoly profit.A firm i receives payoff 1 from each proprietary technology t . A technology t isproprietary for firm i if (1) i ∈ t and (2) i is the unique firm such that j ∈ { i } ∪ I i ( p , q ) forall j ∈ t . In words, the technology contains firm i ’s idea and firm i is the unique firm thatknows all ideas in the technology.If t is not a proprietary technology for firm i , then firm i receives payoff 0 from thetechnology t . In Section 5, we will consider more general payoff structures in which (1)payoffs are not additive across technologies and (2) firms instead receive payoff f ( m ) if m > To illustrate the mechanics of the model, we describe a simple example with n = 4 firms andcomplexity k = 3. Suppose that firms choose ( p , q ) and realizations are such that (1) ideasare discovered by firms in I = { , , } and (2) firm 1 learns indirectly through firm 2 anddirectly from firm 3, firm 3 learns indirectly through firm 1, and firm 3 learns directly fromfirm 4.The network and ideas are shown in Figure 1. Black circles correspond to firms withideas i ∈ I , i.e., firms that discover ideas, while white circles correspond to firms with ideas i / ∈ I , i.e., firms that do not discover ideas. Solid arrows denote indirect learning links, whiledashed arrows indicate only direct learning occurred.Since k = 3, the unique technology t consisting of ideas in I is t = { , , } . The8
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Figure 2: Network with four firms and k = 3. Black circles are firms that discover ideaswhile white circles do not discover ideas. Dashed lines indicate direct learning and solid linesindicate indirect learning. Firm 1 now learns indirectly from firm 3, unlike in Figure 1. Theonly technology produced is t = { , , } , and there are no profits because firms 1 and 3 bothproduce t .realizations of the sets I i ( p , q ) of ideas learned from others are: I ( p , q ) = { } , I ( p , q ) = ∅ , I ( p , q ) = { , } , I ( p , q ) = ∅ . Because firm 3 is the unique firm such that t ⊂ I i ( p , q ) ∪ { i } and we have 3 ∈ t , firm 3produces the technology t and receives monopoly profit of 1 for that technology. There areno profits from any other technologies.Suppose instead that firm 1 also learns indirectly through firm 3, as shown in Figure 2.Then we have I ( p , q ) = { , } , I ( p , q ) = ∅ , I ( p , q ) = { , } , I ( p , q ) = ∅ . The only potential technology remains t = { , , } . We now have t ⊂ I i ( p , q ) ∪ { i } for bothfirm 1 and firm 3, so both receive the competitive profit of zero for that technology. Thereare also no profits from other technologies. Before expressing expected payoffs of firms and defining equilibrium, we discuss interpreta-tion and assumptions in the model.
Actions:
Firm actions are choices ( p i , q i ). The first component p i corresponds to a level9f investment in R&D. A small probability of a discovery is cheap, while probabilities closeto one are very expensive.The second component q i corresponds to a level of openness or secrecy in interactionswith other firms. As one example, consider a technology company’s decision about whetherto locate in an area with many other technology firms. Locating near other firms will lead tomore casual interactions between the employees of the firm making the choice and employeesof other firms (e.g., at bars and restaurants), and information can be shared in either directionin these interactions. If there are enough relevant firms in the area, employees could revealinformation about their firm or information about other firms from previous discussions. Inaddition to a firm’s choice of location, the action q i could include decisions such as whetherto send employees to conferences and how much disclose ongoing R&D to employees.An important feature of the model is that increasing q i increases the probability that firm i learns from other firms but also increases the probability that other firms learn from i . The baseline model assumes that learning probabilities are symmetric: firm i learns fromfirm j with the same probability that firm j learns from firm i . In Section 4, we allow firmsto have heterogeneous propensities to learn across firms and find this symmetric structuredoes not drive results.The downside to interaction for a firm i is the increased probability of outgoing links,not an exogenous link-formation cost. Because the costs of links are an endogenous featureof the model, our equilibrium characterization does not depend on functional forms of costs,as it would with exogenous link costs separate from the innovation process. In Appendix F, we describe a related model in which firms instead face a tradeoff betweenlearning and private investment. In this extension, the probability that firm i learns fromfirm j depends only on firm i ’s action, and the downside to interaction comes from a budgetconstraint on the interaction rate and private investment. The techniques we use extendeasily, and we find equilibrium depends on the rate at which firms can substitute betweeninteraction and private investment. Storper and Venables (2004) discuss the importance of face-to-face interactions. Stein (2008) gives a microfoundation for bilateral communication in the context of innovation. Acemoglu, Makhdoumi, Malekian, and Ozdaglar (2017) consider a similar link cost in a setting wherethe benefits depend only on direct connections and the network-formation game is deterministic. Outgoinglinks are undesirable in their model because of a primitive preference for privacy. ormal and Informal Interactions: The model is meant to primarily describe infor-mal interactions between employees or firms, rather than more formal arrangements suchas licensing agreements or joint R&D ventures. As such, our results are most applicable toindustries where formal property rights are imperfectly enforced (Section 6 discusses the in-terplay between informal interactions and more formal property rights). Because informationtransmitted via informal interactions can often spread several steps, an analysis consideringglobal network structure is particularly relevant.In Appendix D, we compare the payoffs to firms with different numbers of private ideas.This analysis can also be interpreted as measuring the value of formal contracting arrange-ments allowing multiple firms to share ideas frictionlessly. We find that as the number offirms grows large, the benefits to such an arrangement are a vanishing fraction of a firm’sexpected profits.
Interaction Rate:
The multiplicative interaction rate ι ( q i , q j ) = q i q j has the featurethat firm i ’s probability of learning from another firm and that firm’s probability of learningfrom i are both proportional to q i . Thus, this is the (unique up to rescaling) interaction ratethat arises from a random matching process in which all agents choose a search intensity andthe probability of learning in each direction is proportional to that intensity. See Cabrales,Calv´o-Armengol, and Zenou (2011) for a microfoundation for a closely related deterministicmodel.Two key properties of the interaction rate are:(1) ι ( q, q (cid:48) ) = ι ( q (cid:48) , q ) for all q and q (cid:48) (2) ι ( q,
0) = 0 for all q .Property (1) says that the interaction rate is symmetric, as discussed above. Property (2)says that firms can choose not to interact with others. Much of the analysis, including ourexistence and characterization results for symmetric equilibria, generalizes to any strictlyincreasing and continuously differentiable interaction rate ι : [0 , × [0 , → ∞ satisfyingthese properties. Learning Network:
A useful assumption is that if firm i learns indirectly through firm j , then firm i learns all ideas known to j . This ensures that there is a well-defined learning11etwork, and this network is a central object in our analysis. If indirect learning were notperfectly correlated across ideas, there would be a separate learning network for each idea. Firm Profits:
The positive payoffs from producing technologies correspond to monopolypayoffs, which we normalize to 1. Formally, all technologies give the same monopoly profitsand that these profits are deterministic. It would be equivalent to take monopoly profitsto be randomly drawn from any distribution with finite mean, as long as firms have noinformation about the realizations a priori. For example, only a small constant fraction oftechnologies could actually be profitable enough to produce.If multiple firms know all ideas contained in t , then there is a competitive market andfirms receive zero profits. This baseline payoff structure, which we generalize in Section 5.2,corresponds to Bertrand competition.Our setup requires that monopolist firms must have privately developed one of the ideasin a technology to produce that technology, but competitors need not. To start a newmarket, some expertise and/or confidence in the quality of the relevant idea is needed. Oncea market exists, however, entrants do not require this expertise, perhaps because relevantdetails can be obtained from the competitor’s technology. Given actions ( p , q ), we define the proprietary technologies P T i ( p , q ) for i to be the setof technologies t such that i ∈ t , firm i learns all other ideas j ∈ t , and no other firm knowsall ideas in t . Note that this set is a random object depending on link realizations. Thenthe expected payoff to firm i is U i ( p , q ) = E [ | P T i ( p , q ) | ] − c ( p i ) . To further illustrate payoffs, we write the cardinality of
P T i ( p , q ) explicitly when δ = 1.Recall that I i ( p , q ) is the set of ideas learned by firm i given actions ( p , q ). Like P T i ( p , q ),this is also a random object. 12hen δ = 1, the expected payoffs to firm i are: U i ( p , q ) = p i · E (cid:20)(cid:18) | I i ( p , q ) | k − (cid:19)(cid:21) · (cid:89) j (cid:54) = i (1 − ι ( q i , q j )) − c ( p i ) . A technology t that i profits from consists of i ’s private idea, which is developed withprobability p i , and a choice of ( k −
1) other ideas known to i . The firm j faces competitionif and only if some firm learns all of i ’s ideas, and the probability that this does not occur is (cid:81) j (cid:54) = i (1 − ι ( q i , q j )). Finally, the private investment cost is c ( p i ).In general, a firm can face competition for a technology t in two ways. First, a firm j can learn all of firm i ’s ideas via indirect learning. Second, a firm j can learn i ’s private ideadirectly and then the other ideas in the technology t from links with firms other than i . Theprobability of the second possibility is more difficult to express in closed form, and in generaldepends on the technology t . We will show that when there is not too much interaction,then most competition comes via the first channel.Payoffs in this model depend on the number of ideas known to a firm in a particularcombinatorial manner. We will allow payoffs to be a more general function of the number ofideas learned in Section 5.1. We now define our solution concept:
Definition 1. An equilibrium ( p ∗ , q ∗ ) is a pure-strategy Nash equilibrium. An equilibrium( p ∗ , q ∗ ) is an investment equilibrium if p ∗ i > i .Because all choices p i and q i are probabilities of discoveries or interactions, we restrict topure strategies.If p i = 0 for all i , then any q will give an equilibrium: if no other firms are investing,there is no reason to invest and so payoffs are zero. It is easy to see these trivial equilibriaalways exist, and we will focus on investment equilibria.For some of our results, it will also be useful to make the stronger assumption thatprivate investment is non-vanishing asymptotically. We consider a sequence of equilibria as13he number of firms n → ∞ . Definition 2.
A sequence of equilibria ( p ∗ , q ∗ ) has non-vanishing investment iflim inf n min i p ∗ i > . Depending on c ( · ), there may be equilibria at which all firms choose very low levels ofprivate investment because others are investing very little. The definition excludes thesepartial coordination failures as well. In this section, we characterize equilibrium in our model. We first briefly describe investmentequilibria under direct learning ( δ = 0). The remainder of this section will characterizeinvestment equilibria under indirect learning ( δ > We summarize results with δ = 0 here, and give a full analysis in Appendix C. In this case,ideas can spread at most one step.There exists a symmetric investment equilibrium for n large, and at any sequence ofsymmetric investment equilibria the interaction rate is ι ( q ∗ , q ∗ ) ≈ (cid:18) k − n (cid:19) k . Since the interaction rate is of order n − k , the probability that a generic firm knows all theideas in a given technology is of order n . It follows that the probability that there existscompetition on a given technology is constant.14or n large, each firm learns from a large number of other firms with high probability. Wewill see that interaction rates are much lower in the indirect-learning case. With only directlearning much more interaction is needed to generate a substantial risk of competition, sothe interaction rate must be higher for potential competition to meaningfully deter openness.A key feature of the direct learning environment is that interaction between firms j and j (cid:48) imposes only a negative externality on a third firm i by increasing potential competition.Therefore, decreasing openness would increase average profits. Formally, at the symmetricequilibrium ( p ∗ , q ∗ ) we have: lim n →∞ ∂U i ( p ∗ , q ) ∂q ( q ∗ ) < . Once indirect learning is introduced, interaction between firms j and j (cid:48) also imposes a positiveexternality by facilitating indirect learning by firm i . We will compare the magnitudes ofthese positive and negative externalities. Our main focus is the indirect learning case ( δ >
0) in which ideas can spread multiple steps.Asymptotically, firms’ incentives will depend on the global structure of the indirect-learningnetwork. To better understand this dependence, we let the number of firms n → ∞ andbegin with outcomes under a sequence of symmetric actions.We say that an event occurs a.a.s. (asymptotically almost surely) if the probability ofthis event converges to 1 as n → ∞ . To simplify notation, we often omit the index n (e.g.,from the actions ( p i , q i ).) Definition 3.
A sequence of symmetric actions with openness q is: • Subcritical if lim sup n ι ( q, q ) δn < • Critical if lim n ι ( q, q ) δn = 1 • Supercritical if lim inf n ι ( q, q ) δn > i in the indirect-learning network is ι ( q, q ) δ ( n − ,
15o the three cases distinguish networks where each firm learns indirectly less than once,approximately once, and more than once in expectation. In the subcritical case, it followsthat the expected number of firms that learn a given idea is a finite constant. In thesupercritical case, there is a positive probability that a given idea is learned by a largenumber of firms (i.e., a number growing linearly in n ).This intuition is formalized by results from the theory of random directed graphs (Karp,1990 and (cid:32)Luczak, 1990). Adapting their results to this setting, we have the following result.A component of a directed network is a strongly-connected component, i.e., a maximal setof nodes such that there is a path from any node in the set to any other. Lemma 1 (Theorem 1 of (cid:32)Luczak (1990)) . Suppose q is symmetric.(i) If the indirect-learning network is subcritical, then a.a.s. every component has size O (log n ) .(ii) If the indirect-learning network is supercritical, then a.a.s. there is a unique compo-nent of size at least (cid:101) αn for a constant (cid:101) α ∈ (0 , depending on lim n ι ( q, q ) δn , and all othercomponents have size O (log n ) . These asymptotic results each imply that large finite graphs have the component struc-tures described with high probability. It follows from the lemma that in a subcritical sequenceof equilibria, all firms learn at most O (log n ) ideas a.a.s. In a supercritical sequence of equi-libria, there is a positive fraction of firms learning a constant fraction of all ideas a.a.s. At acritical equilibrium, the number of ideas learned lies between the subcritical and supercriticalcases.To discuss asymmetric strategies and later heterogeneity in firms, we now generalize thenotion of criticality to arbitrary strategies. Consider the matrix ( ι ( q i , q j ) δ ) ij . The entry ( i, j )is equal to the probability that firm i learns indirectly from firm j . Let λ be the spectralradius of this matrix, i.e., the largest eigenvalue. Definition 4.
A sequence of actions with openness q is: • Subcritical if lim sup n λ < • Critical if lim n λ = 1 16 Supercritical if lim inf n λ > Theorem 1.
For n sufficiently large, there exists a symmetric investment equilibrium. Anysequence of investment equilibria is critical. At a sequence of symmetric investment equilibrium, the theorem implies that ι ( q ∗ , q ∗ ) → δn , and in particular symmetric investment equilibria are asymptotically unique.While it is easy to see there must exist a symmetric equilibrium, a priori there need not bean equilibrium with non-zero interaction and investment. In fact, we show that there existsa sequence of symmetric equilibria with non-vanishing investment. Once we have analyzedfirms’ best responses in each region, the existence result follows by fixed point techniques.We are able to drop the assumption of symmetric strategies, which is standard in set-tings involving random networks (e.g., Currarini, Jackson, and Pin, 2009, Golub and Livne,2010, and Sadler, 2020), and show any equilibrium is at the critical threshold. Asymmetricequilibria could feature firms with ι ( q ∗ i , q ∗ i ) above and below δn .The proof of Theorem 1 builds on existing mathematical results on large random graphs,and generalizes them to allow complementarities between ideas and endogenous link prob-abilities. The first obstacle to applying existing results is that the combinatorial structureof technologies generates complementarities between ideas, so payoffs and incentives do notsimply depend on the expected number of ideas learned. A second issue is that link proba-bilities are endogenous, so lower-order terms in link probabilities and vanishing-probabilityevents can matter asymptotically. We now discuss the key ideas in the proof, including howwe address these challenges. 17 roof Intuition. We describe the basic idea of the proof in the case δ = 1, and the generalargument is similar. We also begin by discussing symmetric strategies.We will use the first-order condition for q i and the assumption of symmetry to characterizeequilibrium behavior. When δ = 1, whether a firm i faces competition depends only onwhether another firm j has learned from i . Since profits are zero when a firm j has learnedfrom i , we can take firm i ’s first-order condition in q i conditioning on the event that no firmhas yet learned from i . We can also condition on firm i discovering its private idea sinceprofits are zero otherwise.The first-order condition says that at any best response, the cost to firm i of allowinga firm j to learn from i is equal to the benefit from learning from an additional firm j .Applying this to a firm i at equilibrium, we obtain: E (cid:20)(cid:18) | I i ( p ∗ , q ∗ ) | k − (cid:19)(cid:21) = 1 q ∗ ( n − · E (cid:34) ∂ (cid:0) | I i ( p ∗ ,q i ,q ∗− i ) | k − (cid:1) ∂q i ( q ∗ ) (cid:35) (1)Recall that I i ( p ∗ , q ∗ ) is the set of ideas learned by firm i , which is a random variable. Theleft-hand side is the cost of an outgoing link, which erases the monopoly payoffs from anytechnologies produced by i . In particular, these costs are increasing in the number of ideas | I i ( p ∗ , q i , q ∗− i ) | learned by i . The right-hand side is the marginal benefit from increasing theprobability of learning from each other firm by n − , which is equal to the marginal benefitfrom an additional outgoing link.A key feature of equation (1) is that the left-hand side and right-hand side both dependon the distribution of the number of ideas learned from a given link. We will exploit thissymmetry between costs and benefits to solve for q ∗ n . We are able to do so because of theendogenous downside to outgoing links, which depends on the number of ideas that firm i learns.We use the first-order condition in equation (1) to obtain an expression for q ∗ n in termsof the number of incoming links used to learn the ideas in an average proprietary technology.Consider a technology t such that i produces t and gets monopoly profits. This technologyis a combination of ideas learned from different links. For example, if k = 4, an exampletechnology could consist of i ’s private idea, two ideas learned indirectly from firm j , and18ne idea learned directly from firm j (cid:48)(cid:48) . In this example, the technology would combine ideasfrom three different links.More generally behavior will depend on the number of links utilized in learning the ideasin a technology t . We refer to this number of links as τ ( t ), so that τ ( t ) = 3 in the examplein the previous paragraph. The key tool, which we state in the subcritical region, is: Lemma 2.
Along any sequence of symmetric investment equilibria with lim sup δι ( q ∗ , q ∗ ) n < , δι ( q ∗ , q ∗ ) n ∼ E t ∈ P T i ( p ∗ , q ∗ ) [ τ ( t )] for all i . Lemma 2 says that the expected number of other firms from whom i learns is equal tothe expected value of τ ( t ) for a random proprietary technology t . We give a brief intuitionfor the lemma. If τ ( t ) is higher, then there are stronger complementarities between links,because produced technologies combine ideas from more links. In this case, if a firm has afew existing links, an additional link will be more valuable than an existing link due to thesecomplementarities. Since additional links are relatively more valuable, firms are willing tointeract more.Since τ ( t ) is always at least one, Lemma 2 implies that lim n ι ( q ∗ , q ∗ ) n ≥ , so therecannot be a subcritical equilibrium.In the supercritical region, almost all proprietary technologies t ∈ P T i ( p ∗ , q ∗ ) are createdby combining a private idea with ( k −
1) ideas learned from observing the giant component.In particular, payoffs are determined up to lower order terms by whether firm i has a linkthat provides a connection to the giant component. Given such a link, additional links addlittle value. Thus there are not complementarities between links; indeed, links are strategicsubstitutes due to the potential redundancies.But because firms have more to lose from an outgoing link in the supercritical region,complementarities between links are needed to sustain high interaction rates. Since thesecomplementarities are not present, there is not a supercritical equilibrium either. We check This would not be the case if firms could produce technologies of any complexity. Then payoffs grow atan exponential rather than polynomial rate in the number of ideas learned, so additional ideas can be veryvaluable, as in the growth model of Acemoglu and Azar (2019). n →∞ P [ | I i ( p , q ) | = y ] that firm i learns y ideas decays exponen-tially in y . The proof bounds | I i ( p , q ) | above with the number of nodes in a multi-typePoisson branching process and then analyzes this branching process.We prove the existence result and characterization of symmetric equilibria for any strictlyincreasing and continuously differentiable interaction rate ι : [0 , × [0 , → [0 ,
1] satisfying:(1) ι ( q, q (cid:48) ) = ι ( q (cid:48) , q ) for all q and q (cid:48) (2) ι ( q,
0) = 0 for all q .We rely on the multiplicative functional form ι ( q i , q j ) = q i q j to extend the characterizationfrom symmetric equilibria to arbitrary equililbria.Theorem 1 makes a sharp prediction about equilibrium. This depends on the specificationof payoffs, which are linear in the number of monopoly technologies produced by a firm. Weconsider how our equilibrium characterization extends to more general payoffs in Section 5. We next discuss welfare consequences of Theorem 1. At any critical sequence of equilibria,the number of ideas | I i ( p ∗ , q ∗ ) | learned by each firm is o ( n ) asymptotically almost surely.Since each firm can produce at most (cid:0) | I i ( p ∗ , q ∗ ) | k − (cid:1) proprietary technologies, U i ( p ∗ , q ∗ ) = o ( n k − ) . Suppose instead that all firms choose ( p, q ) where p is non-vanishing and lim n ι ( q, q ) δn ∈ (1 , ∞ ). Then if α ∈ (0 ,
1) is fraction of ideas that are learned by all firms in the giant20omponent asymptotically almost surely, U i ( p , q ) = ( p ) k α (cid:18) αnk − (cid:19) (1 − δ · ι ( q, q ) − (1 − δ ) · ι ( q, q ) · α ) n − − c ( p ) + o ( n k − ) . (2)Asymptotically almost surely, a firm learns all ideas learned by the giant component withprobability α . In this case, the firm learns pαn + o ( n ) ideas and therefore can produceapproximately (cid:0) αnk − (cid:1) potential technologies if their component ideas are discovered. As weshow in the proof of Theorem 1, the probability of facing competition on a given technologyis approximately (1 − δ · ι ( q, q ) − (1 − δ ) · ι ( q, q ) · α ) n − . Since ι ( q, q ) δn converges, this probability converges to a positive constant. So the growthrate of U i ( p , q ) is of order n k − .Thus, average payoffs are much higher in the supercritical region than the critical regionwhen the number of firms is large. Theorem 1 showed that nevertheless individual incentivesto be secretive lead to a critical equilibrium.We could also consider social welfare by including the surplus obtained by consumersfrom monopoly products and competitive products. It follows easily that the socially optimaloutcome will be in the supercritical range, like the outcome maximizing average firm profits. We can interpret these findings in terms of informational externalities. Interactions im-pose two externalities on a third-party firm i : (1) there is a positive externality as theseinteractions can lead to more indirect learning by i , and (2) there is a negative externality,because these interactions can lead to potential competition with i . In the subcritical andcritical range, the positive externality dominates. This is because most competition comesfrom firms that learn indirectly through i and thus the negative externality has little impact.Theorem 1 and equation (2) have several policy implications, which we state informallyand then as formal results: More formally, suppose that consumer surplus is W c from each product with a competitive market and W m from each product with a monopoly, where W c ≥ W m ≥
0. We can then define social welfare to bethe sum of total producer payoffs and consumer surplus. The expected number of competitive productsand the expected number of total products produced are both increasing in q when firms choose symmetricstrategies. Therefore, the findings in the section that increasing q will increase average profits extend tosocial welfare. At or near equilibrium outcomes, there are large gains to policies (e.g., non-enforcementof non-compete clauses, establishing innovation clusters) that encourage or requiremore interaction between firms and thus shift outcomes to the supercritical region • Policies to increase private investment (e.g., subsidies for R&D) will not shift outcomesto the supercritical region, and thus have much smaller benefits at equilibrium • But once outcomes are in the supercritical region, policies to increase private invest-ment will have large benefits.We can formalize the first bullet point:
Corollary 1.
For any sequence of equilibria ( p ∗ , q ∗ ) with non-vanishing investment and any (cid:15) > , lim n →∞ U i ( p ∗ , (1 + (cid:15) ) q ∗ ) U i ( p ∗ , q ∗ ) = ∞ . The first corollary says that at equilibrium increasing all q i by any multiplicative factorhas a very large effect on payoffs asymptotically. The proof shows that such an increase willchange payoffs from o ( n k − ) to a polynomial of order n k − .Corollary 1 relates to Saxenian (1996)’s study of the Route 128 and Silicon Valley technol-ogy industries, which found that Silicon Valley had much more open firms and grew faster. Inthe terminology of our model, Route 128’s secrecy corresponds to equilibrium behavior. Butinstitutional features of Silicon Valley (including non-enforcement of non-compete clausesand common ownership of firms by venture capital firms) may have constrained firms’ ac-tions to prevent high levels of secrecy (subcritical or critical choices of q i ). Such constraintswould imply much higher payoffs at equilibrium.We also give the second and third bullet points in a corollary: Corollary 2.
For any sequence of equilibria ( p ∗ , q ∗ ) with non-vanishing investment and any (cid:15) > , lim n →∞ ∂U i ( p ∗ + x , (1+ (cid:15) ) q ∗ ) ∂x (0) ∂U i ( p ∗ + x , q ∗ ) ∂x (0) = ∞ . ubcritical Critical SupercriticalBest Response q i High Intermediate Low
Average Payoffs
Constant Intermediate Polynomial
Increasing q Large Benefit Large Benefit Ambiguous
Increasing p Small Benefit Intermediate Large BenefitTable 1: Best responses and policy implications when firms choose symmetric strategies( p , q ) in the subcritical, critical, and supercritical regions.If all firms are choosing private investment level p ∗ , increasing private investment slightlyhas a much larger effect in the supercritical region than at equilibrium. Thus a standardresult about underinvestment in R&D due to spillovers holds in our model, but this ineffi-ciency only first order once interaction rates are high enough. The proof shows that the effectof increasing p is o ( n k − ) at equilibrium but polynomial of order n k − in this supercriticalregion. Corollary 1 showed there are large gains to increasing interaction rates above equilibriumlevels. A natural question is whether these gains be realized via policy interventions otherthan directly restricting firms’ strategy spaces.We now show that introducing public innovators who are not concerned with secrecyleads to learning and innovation at the same rate as in the supercritical region. In particular,there exists a giant component of the learning network containing these public innovators.Public innovators could correspond to academics, government researchers, open-source soft-ware developers, or other researchers with incentives or motivations other than profitingfrom producing and selling technologies.A public innovator i pays investment cost c ( p i ) and receives a payoff of one for eachtechnology t such that: (1) i ∈ t and (2) j ∈ { i } ∪ I i ( p , q ) for all j ∈ t . We will rely onthe fact that for public innovators there is no downside to interactions, but not on the exactincentive structure.All firms have the same incentives as in the baseline model, and public innovators andfirms interact as in the baseline model. We now call an equilibrium symmetric if all public23nnovators choose the same action and the same holds for all private firms. Proposition 1.
Suppose a non-vanishing share of agents are public innovators. Then thereexists a sequence of symmetric equilibria with non-vanishing investment, and at any sequenceof equilibria with non-vanishing investment lim inf n U i ( p ∗ , q ∗ ) (cid:0) n − k − (cid:1) > for all firms i . The proposition says that at equilibrium, firms’ payoffs are at least a constant fraction ofthe maximum achievable profits (cid:0) n − k − (cid:1) . Thus payoffs are of order n k − , as in the supercriticalregion without public innovators. This holds for any positive share of public innovators, andindeed could be extended to a slowly vanishing share of public innovators.Proposition 1 assumes that firms cannot differentially interact with public innovatorsand private firms. In Appendix E, we show the same result holds when interactions can bedirected toward public innovators or private firms.Public innovators are valuable primarily as informational intermediaries rather than fortheir private ideas. Because public innovators do not face costs to interaction, they willchoose q i = 1 at equilibrium. Therefore, public innovators can learn many ideas via interac-tions and transmit these ideas to other public innovators or to private firms (e.g, academicslearning ideas from conferences and collaborations and then consulting for private industry).Conversely, the proposition would remain unchanged if all public innovators instead choose p i = 0 and q i = 1.Empirical research on collaboration between academia and industry supports the valueof academic researchers as informational intermediaries between firms. Azoulay, Graff Zivin,and Sampat (2012) study movement of star academics, and find that moves increase patent-to-patent and patent-to-article citations locally. Moreover, Jong and Slavova (2014) findthat firms that disclose high-quality R&D through publications with academics are moreinnovative, suggesting information flows exhibit symmetry properties within interactions.24 Asymmetric Learning Probabilities
The baseline model assumes that information flows are symmetric across pairs of firms. Inpractice, firms may have hetereogeneous probabilities of learning from others, even given afixed interaction rate.Suppose instead that firm i directly learns from firm j with probability β i ι ( q i , q j ) , where the propensity to learn β i ∈ [ β,
1) for some β ∈ (0 , λ be the spectral radius of the matrix ( β i ι ( q i , q j ) δ ) ij . As before entry ( i, j ) is equal to theprobability that firm i learns indirectly from firm j . Let λ be the spectral radius of thismatrix. Definition 5.
A sequence of actions with openness q is: • Subcritical if lim sup n λ < • Critical if lim n λ = 1 • Supercritical if lim inf n λ > Theorem 2.
Suppose firms have propensities to learn β . There exists an investment equi-librium for n large, and any sequence of investment equilibria is critical. Equilibria remain critical even when the directed link probabilities are asymmetric acrosspairs. The characterization result extends immediately to the case in which β i are chosenendogenously at a cost (cid:101) c i ( β i ), which can vary across firms. In this case, firms can nowcontrol the likelihood of learning along two dimensions. First, higher interaction rates allowa firm to learn more from from others at the expense of a higher probability of its ideas This choice can be made simultaneously with or prior to the choice of q i . q i relates the value of the ideas already known to firm i with the value ofincreasing the interaction rate. Fixing q i , a higher β i increases both sides of this first-ordercondition because firms with higher propensities to learn have already learned more ideasbut also will learn more from an additional interaction.At potential equilibria in the subcritical region, these two forces cancel out and a firm’soptimal choice of openness q ∗ i is approximately independent of that firm’s propensity to learn β i . The proof of Theorem 1 used the symmetry between the value of existing links and anadditional link to solve for q ∗ n . Even if each of these links only realizes with probability β i ,the symmetry persists and so q ∗ n is unchanged.The two opposing effects would not entirely cancel in the supercritical region, becausethere are potential redundancies between multiple links to the giant component. Theseredundancies matter more for firms with higher β i . Nevertheless, we can bound the averageinteraction rate when all firms choose q i to respond optimally to the giant component size. In Sections 2 and 3, we studied equilibrium when expected payoffs were U i ( p , q ) = E [ | P T i ( p , q ) | ] − c ( p i ) . Firms’ utility functions had two properties:1. Payoffs are linear in the number of proprietary technologies, and2. Payoffs do not depend on technologies for which the firm faces competition.The first assumption determines the benefits from incoming links, while the second deter-mines the costs of outgoing links. 26e now relax each of these assumptions. We find that equilibrium remains critical whenthe returns to producing more technologies are increasing. More generally, we show thatequilibrium is critical in a setting where profits are a convex function of the number of ideaslearned. Changing the profit structure in competitive markets, however, leads to supercriticalor subcritical equilibria. So outcomes depend on the specification of the costs of outgoinglinks, but are less sensitive to the specification of the gains from learning.
The baseline model assumed that a firm’s profits are linear in the number of proprietarytechnologies. In practice, there may be increasing or decreasing returns to producing moretechnologies. Suppose that the payoffs to firm i are instead | P T i ( p , q ) | ρ − c ( p i ) , where ρ > ρ = 1. When ρ >
1, there are increasing returns tocontrolling more monopolies. When ρ <
1, there are decreasing returns to controlling moremonopolies. Note that these increasing or decreasing returns to scale are not determined bythe innovative process, but rather by production costs or other market conditions.
Proposition 2.
There exists ρ ≤ such that for any ρ ≥ ρ , any sequence of symmetricinvestment equilibria is critical. When k > , we have ρ < . The proposition shows that the prediction of critical equilibria is not knife-edge withrespect to ρ . In particular, increasing returns to scale cannot move interactions above thecritical threshold. As long as k (cid:54) = 2, slightly decreasing returns to scale will not moveinteractions below the critical threshold either.Consider a firm i that does not face competition. We show that under the conditions ofthe proposition, the firm’s profits are convex in | I i ( p , q ) | . As a result, learning additionalideas is more appealing relative to protecting existing ideas, so openness will not decreasebelow the critical region. Checking concavity is delicate when ρ <
1, because in this case27rm profits are the composition of the binomial coefficient (cid:0) | I i ( p , q ) | k − (cid:1) , which is convex, andthe polynomial, | P T i ( p , q ) | ρ , which is concave.We also show that for any ρ , openness will not increase enough to push equilibrium intothe supercritical region either. At a potential supercritical sequence of symmetric investmentequilibria, profits are driven by the event that firm i learns from the giant component andproduces ( p ∗ ) k (cid:0) αnk − (cid:1) proprietary technologies, where α is the share of ideas learned by thegiant component.Firm i chooses q i to maximize the probability of this event. Asymptotically the optimal q i is independent of the payoffs from this event since these payoffs are very large, and thereforethe optimal q i is independent of ρ . Given this, the calculation is the same as in the case ρ = 1 (Theorem 1), where there is no supercritical sequence of investment equilibria.More generally, the proof shows that our criticality result relies on two features of thepayoff function. First, payoffs for a firm i that does not face competition are convex in thenumber of ideas | I i ( p , q ) | learned by i . Second, payoffs grow at a polynomial rate in thenumber of ideas learned by i .We can state this formally when δ = 1, so that when firm i learns from j it will learn allideas known to firm i . In this case, we let the profits for firm i be φ ( | I j ( p , q | ) when firm i discovers its private idea ( i ∈ I ) and no firm learns from i , and 0 otherwise. We will assumethat φ ( · ) is strictly increasing and continuously differentiable. Proposition 3.
Suppose δ = 1 and payoffs when no firm learns from i and i ∈ I are equalto φ ( | I i ( p , q | ) , where φ ( x ) is convex and φ ( x j ) φ ( x (cid:48) j ) → along any sequence of ( x j , x (cid:48) j ) such that x j x (cid:48) j → . Then any sequence of symmetric equilibria with non-vanishing investment is critical. The assumption that φ ( x j ) φ ( x (cid:48) j ) → x j , x (cid:48) j ) such that x j x (cid:48) j → φ ( · ). In particular, this assumption holds if φ ( x ) = Cx d + O ( x d − )for any C > d ≥
1. If payoffs instead grow at an exponential rate in thenumber of ideas, then a supercritical equilibrium is possible because an additional idea maybe very valuable (see Acemoglu and Azar, 2019 for a related effect).28 special case is that firms can produce technologies of multiple complexities k , perhapswith different payoffs for technologies of different complexities. The growth condition willhold as long as the allowed complexities are bounded (independent of n ). We found in Theorem 1 that equilibrium lies on the critical threshold. This result is robustto different payoffs structures for monopolist firms. We now show that Theorem 1 doesdepend on the structure of competition, and show that altering payoffs from competitivemarkets can lead to supercritical or subcritical outcomes.To generalize the payoff from technologies, we will now assume that firm i receives payoffs f ( m ) from a technology t such that i ∈ t , firm i learns all other ideas in t , and m otherfirms learn all ideas in t , where f ( · ) is weakly decreasing. We maintain the normalization f (0) = 1.A simple case is f ( m ) = a < m >
0. The analysis in previous sections corre-sponded to the case a = 0.We can also allow f ( m ) <
0, which could correspond to a fixed cost of production thatmust be paid before competition is known. We assume that firms make a single decisionabout whether to produce the technologies that they learn. Proposition 4. (i) If < f (1) < and f ( m ) ≥ for all m , then there exists a symmetricinvestment equilibrium for n large and any sequence of symmetric investment equilibria issupercritical.(ii) If f ( m ) < for all m > , then any sequence of symmetric investment equilibria issubcritical. Part (i) says that if the potential downside to enabling competitors is not as large, thenfirms will be more willing to interact. This pushes the equilibrium from the critical thresholdinto the supercritical region. Cournot competition, for example, would correspond to f ( m )satisfying the conditions of part (i) of the proposition. If firms can condition their production decision on the flow of ideas, the analysis becomes more compli-cated. i ’s proprietary technologies only include ideas learned from one otherfirm. Lemma 3.
For any critical sequence of symmetric actions with p > , lim n →∞ E t ∈ P T i ( p , q ) [ τ ( t )] = 1 . We use a pair of coupling arguments to show that most profits come from rare events inwhich a single link (indirectly) lets a firm learn many ideas. Comparing critical equilibria tosubcritical equilibria near the critical threshold, we find that the expected number of ideaslearned by a firm grows large. Then by comparing critical equilibria to supercritical equilibrianear the threshold, we verify that the probability of learning a large number of ideas is small.To complete the proof of the lemma, we show that few technologies are produced by two ormore of these rare events occurring simultaneously.
In the baseline model, technologies could only be protected via secrecy. We now considerthe possibility that a positive fraction of firms receive patents on their ideas. As motivationfor this setup, suppose that firms discover different types of ideas and patent law determineswhich types are patentable. For example, Bessen and Hunt (2007) discuss the boundaries ofpatent law in the software industry and how those boundaries have changed over time.30ore precisely, a fraction b ∈ (0 , patent on their private ideas.In this case, other firms cannot use this private idea, either as monopolists or competitors.Formally, a firm i receives payoff 1 from each technology t such that (1) idea i ∈ t ; (2) firm i knows all j ∈ t and no other j ∈ t receive patents; and (3) either i receives a patent or i isthe unique firm that knows all j ∈ t . Else the firm receives payoff 0 from the technology t .To focus on how patents relate to informal interactions, we analyze model patents in avery simple way. In particular, the model could be extended to allow for imperfect patentrights and/or licensing of patents.A firm now chooses either a level of openness q i (0), which is the action without a patent,or q i (1), which is the action with a patent. We will refer to the choices at symmetricequilibria as q ∗ (0) and q ∗ (1).For the first part of the following result ( δ = 0), we will also assume that each firm i pays cost (cid:15) > (cid:15) has little effect on the equilibrium. Proposition 5.
Suppose a fraction b ∈ (0 , of firms receive patents. If δ = 0 , then with k = 2 and any link cost (cid:15) > , there does not exist an investment equilibrium for n large. If δ > , then ι ( q ∗ (0) , q ∗ (0)) is o (1 /n ) along sequence of any symmetric investment equilibria. With only direct learning ( δ = 0), the proposition says that positive investment cannotbe sustained at equilibrium for n large. This is because of an adverse-selection effect thatdiscourages social interactions.Because firms receiving patents have no need for secrecy, firms with patents choose veryhigh interaction rates q i (1). Thus, most interactions are with firms with patents. On theother hand, firms with patents are undesirable to interact with because their ideas cannotbe used by others. Because of this adverse selection in the matching process, firms withoutpatents will have much lower expected profits than in the model without patents. When k = 2 and there is an arbitrarily small cost to links, this has the effect of shutting down all We do not allow firms to discriminate in their interactions based on others’ patents. Formally, for all n sufficiently large, all equilibria have p ∗ = 0. When there is no private investment,firms are indifferent to all choices of interaction rates. k > q ∗ (1) = 1. Firms withoutpatents now have some interactions with firms without patents, who can transmit ideas fromother firms without patents. Interactions between pairs of firms without patents, however,are rare.Proposition 5 relates to several strands of literature on patent rights. A theoretical andempirical literature considers firms’ choices between formal and informal intellectual propertyprotections, particularly patents versus secrecy (e.g., Anton and Yao, 2004, Kultti, Takalo,and Toikka, 2006, and the survey Hall, Helmers, Rogers, and Sena, 2014). We focus noton the choice between formal and intellectual property rights but on the interplay betweenthe two. The proposition finds that in markets with some patent rights, firms must sacrificemore learning to achieve a given level of secrecy.A second contrast is to theoretical findings on patents and follow-up innovation (e.g.,Scotchmer, 1991, Scotchmer and Green, 1990, Bessen and Maskin, 2009). This literatureinvestigates when granting patent rights for an idea decreases follow-up innovations involvingthat idea. In our random-interactions setting, patent rights can not only decrease follow-upinnovations involving patent ideas but also decrease follow-up innovations involving other unprotected ideas. 32 Patent Share b N u m be r o f M onopo l y P r odu c t s k=2k=3k=5k=10 Figure 3: Average profits from monopoly products for n large as a function of the patentshare b , when δ = 1 and k = 2 , , , and 10.We can use Theorem 1 and Proposition 5 to ask when patent rights improve welfare andwhat the optimal fraction b of patentable ideas would be. In the direct-learning case, theproposition gives conditions under which patents are harmful.In the indirect-learning case, average firm profits and social welfare are higher withinterior patent rights b ∈ (0 ,
1) than no patent rights because firms with patents are valuableas intermediaries. Under indirect learning, we can ask what value of b maximizes averagepayoffs. Any positive b provides the benefits of information intermediaries, and so there isa tradeoff between the higher private profits obtained by firms with patents and the socialbenefits provided by firms without patents, whose ideas can be used by others. The optimalvalue of b can be interior asymptotically for low k , but for high k the optimal value of b converges to zero as n grows large.This is easiest to see when δ = 1. In this case, we can compute that e − bq ∗ (0) n ≈ . So theaverage firm profits are approximately( p ∗ ) k − (cid:18) (1 − b ) nk − (cid:19) ( b + 14 · (1 − b )) . We graph these profits for n large and different values of k in Figure 3. The value of b ∗ as n → ∞ when k = 2 and converges to as n → ∞ when k = 3. For k ≥ , the optimal share of patents b ∗ → n → ∞ . We have studied strategic network formation in large random graphs in the context of aneconomic application to innovation and social learning. The model is particularly suitedto analysis of informal interactions, e.g., between employees of firms, which cannot be fullygoverned by formal contracts. We find that in these settings, if there are many firms and ideascan travel multiple steps, the global structure of the learning network has stark consequencesfor incentives and payoffs. In particular, expected payoffs and welfare are much higher whenthere is enough interaction to support a giant component.While we have focused on a network-formation game with a tradeoff between secrecyand learning, we have developed more broadly applicable tools for questions in networkeconomics, particularly those with complementarities between connections. In Appendix F,we show that our analysis extends easily to a related model where the key tradeoff is betweenprivate investment and interaction. Outside of network formation, the same techniques canalso be applied to optimizing diffusion processes, e.g., determining the optimal number ofseeds for a new product or technology. This includes settings with complementarities acrossadopters, such as diffusion of a new social media app.
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A Proof of Theorem 1
We will first prove Theorem 1. We begin by describing the analysis of symmetric equilibria,and show that there exists a sequence of symmetric investment equilibria and any sequenceof symmetric investment equilibria is critical. We then extend the characterization to showthat any sequence of investment equilibria, which need not be symmetric, is critical.In all appendices, we say that f ( n ) ∼ g ( n ) if f ( n ) /g ( n ) → n → ∞ . A.1 Symmetric Investment Equilibria
After passing to a subsequence if necessary, we can assume that any sequence of equilibriais either subcritical, critical, or supercritical.We begin by describing the structure of this section.38. We first show that there does not exist a subcritical sequence of symmetric investmentequilibria. The proof assumes such a sequence exists for the sake of contradiction andcharacterizes equilibrium learning and behavior via three lemmas. The first lemmastates that the probability of a firm learning a large number of ideas decays exponen-tially. To prove it we compare the number of ideas learned to the number of nodes ina Poisson branching process. The second lemma gives a first-order condition for thechoice of q i . The first and second lemmas are used to prove the third, which shows that δι ( q ∗ , q ∗ ) n is approximately equal to an expectation E [ τ ( t )], where τ ( t ) is the numberof links needed to learn the ideas in the technology t . Because τ ( t ) ≥ t , thisimplies that lim inf n δι ( q ∗ , q ∗ ) n is at least one. But this contradicts the assumptionthat the sequence of equilibria is subcritical.2. We then show that there does not exist a supercritical sequence of symmetric invest-ment equilibria. We show that given a supercritical sequence of symmetric actions,the payoffs for firm i can be computed (to first order) based on whether firm i learnsthe ideas learned from the giant component and which firms learn from i . This isbecause the number of ideas that firm i learns from outside the giant component isvery likely to be small, so such ideas have a small effect on payoffs. Using this fact, wecan compute the highest-order term in the firm i ’s payoffs explicitly. We show that atany supercritical sequence of symmetric investment equilibria each firm i would preferto deviate to a lower choice of q i for n large.3. We finally show that there exists a symmetric investment equilibrium for n large.The argument uses Kakutani’s fixed-point theorem, which we show applies using thepreceding analysis. Subcritical Case : Our first lemma shows that at a potential sequence of subcritical in-vestment equilibria, the probability of learning a large number of ideas decays exponentially.We prove the first two lemmas without restricting to symmetric strategies. Let q =max i q i be the maximum choice of openness given action q . Lemma A1.
Along any sequence of investment equilibria with δι ( q, q ) n < , there exists > and y such that the probability that P [ | I i ( p ∗ , q ∗ ) | = y ] ≤ e − Cy for all y ≥ y and all i and n .Proof. Let ID i ( q ∗ ) be the set of firms j such that there is a path from i to j in the indirect-learning network. Then I i ( p ∗ , q ∗ ) is the set of ideas such that idea j is discovered ( j ∈ I )and j ∈ ID i ( q ∗ ) or some firm in ID i ( q ∗ ) learns directly from j . The probability each firmin ID i ( q ∗ ) learns directly from j is at most ι ( q, q ), so | I i ( p ∗ , q ∗ ) | is first-order stochasticallydominated by the sum of ID i ( q ∗ ) random variables distributed as Binom ( n, ι ( q, q )).We claim that | ID i ( q ∗ ) | is first-order stochastically dominated by the number of nodesin the Poisson branching process with parameter ι ( q, q ) δ . To prove this, it is sufficient toshow that a random variable with distribution P oisson ( ι ( q, q ) δn ) first-order stochasticallydominates a random variable with distribution Binom ( n, ι ( q, q ) δ ), as ID i ( q ∗ ) is the setof nodes in a branching process with the distribution of offspring first-order stochasticallydominated by Binom ( n, ι ( q, q ) δ ). By Theorem 1(f) of Klenke and Mattner (2010), this holdsif (1 − δι ( q, q )) n ≤ e − ι ( q,q ) δn . Letting C (cid:48) = ι ( q, q ) δn , we observe that (1 − C (cid:48) n ) n is increasing in n and converges to e − C (cid:48) , sothe inequality holds.Now, a standard result shows that are finitely many nodes in the Poisson branchingprocess, and there are y nodes in the Poisson branching process with probability e − C (cid:48) y ( C (cid:48) y ) y − y !(Theorem 11.4.2 of Alon and Spencer, 2004). Using Stirling’s approximation, we can ap-proximate this probability as 1 √ π y − ( C (cid:48) ) − ( C (cid:48) e − C (cid:48) ) y .
40n particular, this probability decays exponentially because C (cid:48) e − C (cid:48) < C (cid:48) (cid:54) = 1.Since | I i ( p ∗ , q ∗ ) | is first-order stochastically dominated by the sum of ID i ( q ∗ ) randomvariables distributed as Binom ( n, ι ( q, q )), by the central limit theorem, the probability that | I i ( p ∗ , q ∗ ) | = y also decays exponentially in y .Our second lemma expresses the first-order condition for q i at a subcritical sequence ofinvestment equilibria. Lemma A2.
Along any sequence of investment equilibria with δι ( q, q ) n < , δ (cid:88) j (cid:54) = i ∂ι ( q i , q ∗ j ) ∂q i ( q ∗ i ) · E (cid:20)(cid:18) | I i ( p ∗ , q ∗ ) | k − (cid:19)(cid:21) ∼ E (cid:34) ∂ (cid:0) | I i ( p ∗ , ( q i ,q ∗− i ) | k − (cid:1) ∂q i ( q ∗ i ) (cid:35) for each i .Proof. Suppose other players choose actions p − i and q − i .We claim that due to the assumption that δι ( q, q ) n <
1, competition that is not based onlearning all of firm i ’s ideas indirectly is lower order. More formally, let T i ( p , q ) be the setof technologies t such that i ∈ t and firm i learns all other ideas j ∈ t . The claim is thatif there does not exist a link from firm i to another firm j in the indirect-learning network,the conditional probability E t ∈ T i ( p ∗ , q ∗ ) [ t ∈ P T i ( p ∗ , q ∗ ) ]that t ∈ P T i ( p ∗ , q ∗ ) for a technology t ∈ T i ( p ∗ , q ∗ ) chosen at random converges to one.Here, each technology t with i ∈ t is chosen with probability proportional to the probabilitythat firm i knows all ideas in t , i.e., t ∈ T i ( p ∗ , q ∗ ).Suppose t ∈ T i ( p ∗ , q ∗ ) and no firm learns indirectly from i . Choose some j ∈ t distinctfrom i . By Lemma A1, the probability that a given firm j (cid:48) learns y ≥ y ideas decaysexponentially in y at a rate independent of n . By independence, the probability that firm j (cid:48) learns ideas i and j is at most o ( n ). Therefore, the probability that any firm j (cid:48) learns ideas i and j is at most o (1). The technology t ∈ P T i ( p , q ) if there is no such j (cid:48) for any j ∈ t distinct from i , so this proves the claim. Recall that
P T i ( q i , q − i ) ⊂ T i ( p , q ) is the subset of technologies t such that no other firm learns all ideasin t . i choosing p i ∈ [0 ,
1) and q i ≥ p i E (cid:20)(cid:18) | I i ( p ∗ , ( q i , q ∗− i )) | k − (cid:19)(cid:21) (cid:89) j (cid:54) = i (1 − δ · ι ( q i , q ∗ j )) − c ( p i ) + o (1) . We first note that the optimal q i does not depend on p i , but instead is chosen to maximize E (cid:20)(cid:18) | I i ( p ∗ , ( q i , q ∗− i )) | k − (cid:19)(cid:21) (cid:89) j (cid:54) = i (1 − δ · ι ( q i , q ∗ j )) + o (1) . The first-order condition gives δ E (cid:20)(cid:18) | I i ( p ∗ , q ∗ ) | k − (cid:19)(cid:21) (cid:88) j (cid:54) = i ∂ι ( q i , q ∗ j ) ∂q i ( q ∗ i )(1 − δ · ι ( q i , q ∗ j )) − ∼ ∂ E (cid:34) (cid:0) | I i ( p , ( q i ,q ∗− i )) | k − (cid:1) ∂q i ( q ∗ i ) (cid:35) . Finally, we have (cid:88) j (cid:54) = i ∂ι ( q i , q ∗ j ) ∂q i ( q ∗ i )(1 − δ · ι ( q i , q ∗ j )) − → (cid:88) j (cid:54) = i ∂ι ( q i , q ∗ j ) ∂q i ( q ∗ i )because lim sup δι ( q, q ) n < t ∈ P T i ( p ∗ , q ∗ ) , let τ ( t ) be the smallest number of (direct or indirect) links suchthat firm i would still know all technologies j ∈ t with only τ ( t ) of its links.We next prove Lemma 2, which states that along any sequence of symmetric investmentequilibria with lim sup δι ( q ∗ , q ∗ ) n < δι ( q ∗ , q ∗ ) n ∼ E t ∈ P T i ( p ∗ , q ∗ ) [ τ ( t )]for all i .As above, each technology t ∈ P T i ( p ∗ , q ∗ ) under a given realization of all random vari-ables is chosen with probability proportional to the probability of that realization. Lemma 2.
Along any sequence of symmetric investment equilibria with lim sup δι ( q ∗ , q ∗ ) n < Note that I i ( p , q ) does not depend on p i . , δι ( q ∗ , q ∗ ) n ∼ E t ∈ P T i ( p ∗ , q ∗ ) [ τ ( t )] for all i .Proof of Lemma 2. We will apply Lemma A2, which gives δ · ∂ι ( q i , q ∗ ) ∂q i ( q ∗ ) · E (cid:20)(cid:18) | I i ( p ∗ , q ∗ ) | k − (cid:19)(cid:21) ∼ n ∂ E (cid:104)(cid:0) | I i ( p ∗ , ( q i ,q ∗− i )) | k − (cid:1)(cid:105) ∂q i ( q ∗ )at a symmetric equilibrium.Let Γ be the set of weakly increasing tuples γ = ( γ , . . . , γ l ( γ ) ) of integers such that (cid:80) l ( γ ) j =1 γ j = k −
1. We write l ( γ ) for the length of the tuple γ .Let X j be i.i.d. random variables with distribution given by the number of ideas that firm i would learn from a firm j (cid:48) conditional on learning directly from j (cid:48) . That is, X j is distributedas the sum of a Bernoulli random variable with success probability p ∗ (corresponding todirect learning) and a random variable distributed as | I j (cid:48) ( p ∗ , q ∗ ) | with probability δ andzero otherwise.Then we claim that E (cid:20)(cid:18) | I i ( p ∗ , q ∗ ) | k − (cid:19)(cid:21) = (cid:88) γ ∈ Γ (cid:18) n − l ( γ ) (cid:19) ι ( q ∗ , q ∗ ) l ( γ ) E l ( γ ) (cid:89) j =1 (cid:18) X j γ j (cid:19) + o (1) . (3)The right-hand side counts the expected number of choices of k − j , allowing for the same idea to be chosen multiple timesvia distinct neighbors. To show the claim, we must argue that the contribution from choicesof k − o (1).We first bound the probability that there exists a firm j such that there are two paths from i to j with distinct first edges. Given j (cid:48) , j (cid:48)(cid:48) ∈ N i , we want to bound above the probabilitythere is a j in the intersection I j (cid:48) ( p ∗ , q ∗ ) ∩ I j (cid:48)(cid:48) ( p ∗ , q ∗ ) . The expected number of firms witha path to j (cid:48) in the indirect learning network is bounded above by − ι ( q ∗ ,q ∗ ) nδ , and the sameholds for j (cid:48)(cid:48) . Therefore, the expected number of firms that each learns from is boundedabove by ι ( q ∗ ,q ∗ ) n − ι ( q ∗ ,q ∗ ) nδ . By independence, the probability of a non-empty intersection is thus43t most n ( (1+ ι ( q ∗ ,q ∗ )) − ι ( q ∗ ,q ∗ ) nδ ) . Because ι ( q ∗ , q ∗ ) δn is bounded away from one above and δ is finite,this implies that the probability there is a firm j observed via any such j (cid:48) and j (cid:48)(cid:48) is boundedabove by O ( n ).By Lemma A1, the probability that | I i ( p ∗ , q ∗ ) | = y decays exponentially in y . Since (cid:0) | I i ( p ∗ , q ∗ ) | k − (cid:1) is polynomial in | I i ( p ∗ , q ∗ ) | , it follows that the contribution to the expectation E (cid:104)(cid:0) | I i ( p ∗ , q ∗ ) | k − (cid:1)(cid:105) from any O ( n )-probability event is o (1). This completes the proof of the claimin equation (3).We can express the right-hand side of Lemma A2 similarly. Recall the right-hand sidecounts the number of additional sets of k − i if i added a direct link to an additional random agent. We then have:1 n E (cid:34) ∂ (cid:0) I i ( p ∗ , ( q i ,q ∗− i )) k − (cid:1) ∂q i ( q ∗ ) (cid:35) = (cid:88) γ ∈ Γ (cid:18) n − l ( γ ) − (cid:19) ι ( q ∗ , q ∗ ) l ( γ ) − E l ( γ ) (cid:89) j =1 (cid:18) X j γ j (cid:19) + o (1) . (4)The same argument shows that the contribution from choices of k − o (1).Substituting equations (3) and (4) into Lemma A2, we find that ι ( q ∗ , q ∗ ) δ ∼ E (cid:34) (cid:0) n − l ( γ ) − (cid:1)(cid:0) n − l ( γ ) (cid:1) (cid:35) , where the expectation is taken over all ( k − I i ( p ∗ , q ∗ ), and foreach such set, l ( γ ) is the number of direct links on a path to at least one idea in the set.Thus, lim n →∞ ι ( q ∗ , q ∗ ) δn = lim n →∞ E [ l ( γ )] = lim n →∞ E t ∈ P T i ( p ∗ , q ∗ ) [ τ ( t )] . We must have τ ( t ) ≥ t . So by Lemma 2, we have lim inf n δι ( q ∗ , q ∗ ) n ≥ Supercritical Case : Suppose there exists a supercritical sequence of symmetric invest-ment equilibria. We have lim inf ι ( q ∗ , q ∗ ) δn > ι ( q ∗ , q ∗ ) n exists or is infinite.Theorem 1 of Karp (1990) shows that a.a.s. the number of firms that all firms in thegiant component learn from is αn + o ( n ) for a constant α increasing in lim ι ( q ∗ , q ∗ ) n andthat the number of agents outside the giant component observed by any agent is o ( n ).If firm i chooses q i , the probability of i learning all ideas known to the giant componentis (1 − (1 − δι ( q i , q ∗ )) α ( n − o ( n ) ). Conditional on this event, firm i learns p ∗ ( αn + o ( n )) ideas.Therefore, we have: E (cid:20)(cid:18) | I i ( p ∗ , ( q i , q ∗− i )) | k − (cid:19)(cid:21) = ( p ∗ ) k − (1 − (1 − δι ( q i , q ∗ )) αn + o ( n ) )(( α ( n − k − + o ( n k − )) . In particular, to solve for firm i ’s choice of q i to first order, we need only consider technolo-gies consisting of i ’s private idea and ( k −
1) ideas learned by the giant component. Theprobability that such a technology faces competition is(1 − δ · ι ( q i , q ∗ ) − (1 − δ ) · ι ( q i , q ∗ ) · α ) n − + o (1) . The term δ · ι ( q i , q ∗ ) corresponds to the possibility of a firm j indirectly learning all of firm i ’s ideas. The term (1 − δ ) · ι ( q i , q ∗ ) · α corresponds to the possibility of a firm j directlylearning firm i ’s idea (but not indirectly learning from i ) and indirectly learning the ideaslearned by the giant component.Thus, we are looking for q i maximizing: E (cid:20)(cid:18) | I i ( p ∗ , ( q i , q ∗− i )) | k − (cid:19)(cid:21) (1 − δ · ι ( q i , q ∗ ) − (1 − δ ) · ι ( q i , q ∗ ) · α ) n − . (5)We want to find q i maximizing expression (5) asymptotically. We claim the derivative ofexpression (5) in q i is equal to zero at q ∗ only if ι ( q ∗ , q ∗ ) δn ≤
1. A fortiori, we can insteadshow this for E (cid:20)(cid:18) | I i ( p ∗ , ( q i , q ∗− i )) | k − (cid:19)(cid:21) (1 − δ · ι ( q i , q ∗ )) n − . This is because for n large, the derivative of this expression in q i is positive if the derivativeof expression (5) is positive. 45he first-order condition for the latter expression at q i = q ∗ implies that( p ∗ ) k n ( αn ) k − (1 − δι ( q ∗ , q ∗ )) n − ( δα (1 − δι ( q ∗ , q ∗ ))(1 − δι ( q ∗ , q ∗ )) α ( n − − − (1 − (1 − δι ( q ∗ , q ∗ )) α ( n − )is o ( n k − ). Solving for ι ( q ∗ , q ∗ ) such that this holds, we obtainlim n (1 − δι ( q ∗ , q ∗ )) α ( n − − = 11 + δα . Thus, lim n e − δαι ( q ∗ ,q ∗ ) n = 11 + δα . (6)The left-hand side is the asymptotic probability that a firm does not indirectly learn from afirm which learns all ideas known to all firms in the giant component, and this is 1 − α . So1 − α = 11 + δα , or equivalently α ( αδ + 1 − δ ) = 0 . Since δ ≤
1, our sequence of solutions to equation (6)must have α = 0, and therefore cannot be supercritical.We have now shown that any sequence of symmetric investment equilibria is critical. Wenext prove that there exists a symmetric investment equilibrium for n large. Existence : We first show that given private investment p >
0, there exists a level ofopenness q that is a best response when all other firms choose ( p, q ). We then show thatwhen all firms choose openness q , there exists an optimal p for all firms.To check existence of a symmetric investment equilibrium, let BR ( q ) be the set of bestresponses q when all other firms choose q j = q and p j = p >
0. Note that the set BR ( q ) doesnot depend on the value of p >
0. Because payoffs are continuous in q , the correspondence BR ( q ) has closed graph.We have shown that when ι ( q, q ) < δn , any element q (cid:48) of BR ( q ) has δι ( q, q (cid:48) ) n approx-imately equal to the expectation of τ ( t ) over proprietary technologies. Because τ ( t ) ≥ We stated this result above at equilibrium, but only used that the firm was choosing a best response. t , when ι ( q, q ) < δn we have ι ( BR ( q ) , q ) ⊂ [ 12 δn , kδn ]for n sufficiently large. Suppose that ι ( q, q ) ≥ δn . Then the firm can achieve a positive payoff by choosing q (cid:48) such that (cid:80) j (cid:54) = i ι ( q (cid:48) , q ) = 1. On the other hand expected payoffs to firm i vanish along anysequence of q (cid:48) →
0. Therefore 0 is not in the closure of BR ( q ) whenever ι ( q, q ) ≥ δn .By compactness we can choose (cid:15) ( n ) < such that ι ( BR ( q ) , q ) ≥ (cid:15) ( n ) δn when δn ≤ ι ( q, q ) . We therefore have ι ( BR ( q ) , q ) ⊂ [ (cid:15) ( n ) δn ,
1] when ι ( q, q ) ∈ [ (cid:15) ( n ) δn , n sufficiently large. So by Kakutani’s fixed-point theorem, for n sufficiently large thereexists a fixed point of BR ( q ) at which ι ( q, q ) ∈ [ (cid:15) ( n ) δn , q ∗ of BR ( q ). Fix any firm i and let the potential proprietarytechnologies P P T i ( q ) be the set of technologies t such that firm i will receive monopolyprofits for t if all ideas in the technology t are discovered. This is a random object dependingon the realizations of interactions but not on the realizations of private investment, and P P T i ( q ) ∩ I = P T i ( p , q ).We have shown that q ∗ is critical, so E [ | P P T i ( q ) | ] → ∞ .A symmetric equilibrium corresponds to p ∗ satisfying p ∗ = argmax p p ( p ∗ ) k − E [ | P P T i ( q ) | ] − c ( p ) . Indeed, by the lemma we could take any open interval containing [ δn , k − δn ] instead of [ δn , kδn ]. p ∗ satisfying c (cid:48) ( p ∗ ) = ( p ∗ ) k − E [ | P P T i ( q ) | ] . Because c ( · ) is continuously differentiable and convex with c (cid:48) (0) ≥ c ( p ) → ∞ as p → E [ | P P T i ( q ) | ] → ∞ , there exists a solution for n sufficiently large. So there exists asymmetric investment equilibrium for n sufficiently large. A.2 Arbitrary Investment Equilibria
To complete the proof of Theorem 1, it remains to extend our characterization from sym-metric equilibria to arbitrary equilibria. It is again sufficient to show that we cannot have asupercritical sequence of investment equilibria or a subcritical sequence of investment equi-libria, and we treat each case separately.We first consider the supercritical case, and show that there exists a giant component withthe same relevant properties as in our analysis of symmetric equilibria. We then consider thesubcritical, which follows the same basic outline as in our analysis of symmetric equilibria.The following lemma, which bounds agents’ actions uniformly, will be useful for thesubcritical and supercritical cases. Given q , we let q = max i q i and q = min i q i be themaximum and minimum choices of openness. Lemma A3.
Consider any sequence of investment equilibria ( p ∗ , q ∗ ) . There exists a con-stant C such that ι ( q, q ) n ≤ C for all n .Proof of Lemma A3. Suppose not. Relabelling firms, we can assume that q ∗ = q for each n .Passing to a subsequence if necessary, we can take q ∗ √ n → ∞ as n → ∞ .We first assume for the sake of contradiction that the expected number of times firm 1learns directly is unbounded. Passing to a subsequence we can assume that the expectednumber of times firm 1 learns directly converges to infinity, i.e., (cid:80) j (cid:54) =1 ι ( q ∗ , q ∗ j ) → ∞ .We now fix n large and consider the payoffs to firm 1 after deviating to choose q .By the Chernoff bound, the probability that no firm learns indirectly from firm i decaysexponentially in q . 48e claim that (cid:0) | I i ( p ∗ , ( q ,q ∗− )) | k − (cid:1) grows at most at a polynomial rate in q . Let X be a randomvariable equal to the number of ideas learned by learning from a random firm j , each chosenwith probability proportional to q j . For q i such that (cid:80) j (cid:54) =1 ι ( q , q ∗ j ) is an integer m , therandom variable (cid:0) | I i ( p ∗ , ( q ,q ∗− i )) | k − (cid:1) is first-order stochastically dominated by (cid:0) (cid:80) mj =1 X j k − (cid:1) , where X , . . . , X m are i.i.d. random variables distributed as X . This in turn within a constantmultiple of (cid:16)(cid:80) mj =1 X j (cid:17) k − . By Rosenthal’s inequality (Rosenthal, 1970), the growth rate ofthe expectation of this sum of moments in m is at most polynomial in m .Therefore, the payoffs to firm 1 conditional on no firm learning indirectly from 1 are atmost polynomial in q . The probability of this event decays exponentially, so for n sufficientlylarge, firm 1 could profitably deviate to a smaller choice of q . This gives a contradiction.The remaining case is that q ∗ √ n → ∞ but (cid:80) j (cid:54) =1 ι ( q , q ∗ j ) is bounded. Then theremust exist a sequence of i such that q ∗ i /q ∗ →
0, and for some such sequence of i we have (cid:80) j (cid:54) = i ι ( q ∗ i , q ∗ j ) →
0. This gives a contradiction since then for n large, firm i could profitablydeviate to choose q i = ( (cid:80) j (cid:54) = i q ∗ j ) − . This complete the proof of the lemma.
Supercritical Case : Because the sequence of actions is supercritical, we can assumethat the matrix ( ι ( q ∗ i , q ∗ j ) δ ) ij has spectral radius at least λ > n sufficiently large.We first claim that there exists α > n , there is a component ofthe learning network containing at least αn firms a.a.s. It is sufficient to show this afterdecreasing q i for some i , and therefore also λ . By Lemma A3, we have q ∗ i ≤ C/ √ n for each i . We can therefore assume without loss of generality that there are at most K choices of q i for each n . Here the number of distinct actions K can depend on the initial upper bound λ .We denote the number of firms choosing q i by n ( q i ).By Theorem 1 of Bloznelis, G¨otze, and Jaworski (2012), the largest component has atleast αn + o ( n ) nodes a.a.s., where α is the extinction probability of the multi-type branchingprocess with types corresponding to choices of q i and the number of successors of type q i (cid:48) of a node of type q i distributed as a Poisson random variable with mean δι ( q i , q i (cid:48) ) n ( q i (cid:48) ). ByTheorem 2 of Section V.3 of Athreya and Ney (1972), this extinction probability α > n large since λ > λ >
1. This proves the claim, and we now return to studying the originalactions q .Because there is a component of the learning network containing at least αn firms with49robability at least (cid:15) , we can also choose C such that q is at most C √ n for all n . To see this,note that the payoffs to choosing q i = √ n are of order n k − . On the other hand, the payoffsto choosing q i = C √ n are bounded above by 2(1 − e − CC ) n k − . So for C sufficiently small,the expected payoffs to choosing q i = C √ n conditional on any realization of all links betweenfirms other than i are less than the expected payoffs to choosing q i = √ n at equilibrium.As n grows large, for all i and j distinct the probability that j ∈ t for a uniformly chosen t ∈ P T i ( q ∗ ) approaches zero. We will show that for any sequence of best responses q i for firm i , the expected number of interactions (cid:80) j (cid:54) = i ι ( q ∗ i , q ∗ j ) has a unique limit which is independentof i .To show this, we next claim that there is at most one component of linear size a.a.s. Todo so, we will use equation (5) of Bloznelis, G¨otze, and Jaworski (2012). In the notation ofBloznelis, G¨otze, and Jaworski (2012), the type space S will be S = [ C, C ], and the kernel κ ( s, s (cid:48) ) = ss (cid:48) . We will identify the type of an agent i with q ∗ i √ n .The space of distributions ∆( S ) over types is compact. Fix such a distribution. Rela-belling so that q ∗ i are increasing in i , we can generate a random network for each n by takingthe action q ∗ i of agent i to be s/ √ n , where s is the ( i/n ) th quantile of the distribution. As n → ∞ , by equation (5) of Bloznelis, G¨otze, and Jaworski (2012), the largest component ofthe learning network learns αn + o ( n ) ideas for some α ∈ [0 , o ( n ) ideas. Because the space of distributions ∆( S ) is compact, this convergenceof component sizes is uniform. So passing to a convergent subsequence if necessary, we canassume that there is a unique giant component learning αn + o ( n ) ideas a.a.s., where α > q i are then equal to ( p ∗ ) k (cid:0) αnk − (cid:1) times the probability that firm i learns all ideas known to the giant component and no firm j learns i ’s idea and all ideasknown to the giant component, plus a term of order o ( n k − ). Formally, if G is the set offirms that learn all ideas known to the giant component, the action q i is chosen to maximize: (cid:18) αnk − (cid:19) (cid:32) − (cid:89) j ∈ G (1 − δι ( q i , q ∗ j )) (cid:33) (cid:89) j ∈ G (1 − ι ( q i , q ∗ j )) (cid:89) j / ∈ G (1 − δι ( q i , q ∗ j )) + o ( n k − ) . δ (cid:88) j ∈ G q ∗ j ∼ (cid:32) − (cid:89) j ∈ G (1 − ι ( q i , q ∗ j )) (cid:33) ( (cid:88) j ∈ G q ∗ j + δ (cid:88) j / ∈ G q ∗ j ) . Since the right-hand side is increasing in q i , the solution has a unique limit lim n (cid:80) j (cid:54) = i ι ( q ∗ i , q ∗ j ) . This limit does not depend on i . Passing to a subsequence if necessary, we can assume thelimit lim n →∞ (cid:80) j ι ( q i , q ∗ j ) exists and is independent of i . Moreover, this limit must be greaterthan δ for equilibrium to be supercritical.But then the same calculation as in the symmetric case shows that the best response q i for all firms is at most δ √ n asymptotically, which gives a contradiction. Subcritical Case : The largest component of the learning network has at most o ( n )nodes a.a.s. We will derive an asymmetric version of the characterization in Lemma 2.By Lemma A3, we can choose C such that q is at most C √ n for all n . We now proceed toderive a characterization of equilibrium as in Lemma 2. We will then use this characterizationto show the result.We first claim, as in the proof of Lemma 2, that the contribution to E (cid:20)(cid:18) | I i ( p ∗ , q ∗ ) | k − (cid:19)(cid:21) from ideas that are learned via multiple direct connections is lower order.The key to the claim is the following lemma, which generalizes Lemma A1 from thesymmetric case: Lemma A4.
Consider a subcritical sequence of actions such that ι ( q i , q i ) n is bounded aboveuniformly. Then lim n →∞ P [ | I i ( p , q ) | = y ] decreases at an exponential rate in y .Proof. Because the sequence of actions is subcritical, we can assume that the matrix ( ι ( q i , q j ) δ ) ij has spectral radius at most λ < n sufficiently large. Increasing q i for some i andtherefore also λ , we can assume without loss of generality that there are at most K choices51f q i for each n . Here the number of distinct actions K can depend on the initial upperbound λ . We denote the number of firms choosing q i by n ( q i ).We will bound the number of firms j with a path from j to i in the indirect learningnetwork above by the number of nodes in a multi-type branching process with the numberof successors distributed as Poisson random variables. The types will correspond to the (atmost K ) choices of q i .For each firm i , the number of firms choosing q j that firm i learns from indirectly is abinomial random variable with success probability δι ( q i , q j ) and at most n ( q j ) trials. Weshowed in the proof of Lemma A1 that such a random variable is first-order stochasticallydominated by a Poisson random variable with parameter δι ( q i , q j ) n ( q j ).Therefore, the number of firms that firms j with a path from j to i in the indirectlearning network is first-order stochastically dominated by the number of nodes in the multi-type branching process such that the number of successors of a node of type q i of each type q j is distributed as a a Poisson random variable with mean δι ( q i , q j ) n ( q j ). Call this numberof nodes y (cid:48) .We want to show that y (cid:48) < ∞ with probability one and the probability that y (cid:48) = y decaysexponentially in y . The (at most K × K ) matrix δι ( q i , q j ) n ( q j ) has spectral radius at most λ because ( ι ( q i , q j ) δ ) ij does. Therefore, by Theorem 2 of Section V.3 of Athreya and Ney(1972), the probability that y (cid:48) = ∞ is zero.Let Z j be the number of nodes in the j th generation of the branching process. By Theorem1 of Section V.3 of Athreya and Ney (1972), the probability that Z j > λ i . So the probability that Z T > T .Dropping nodes with zero probability of interaction if necessary, we can assume that all q i >
0. By the Perron-Frobenius theorem, there exists an eigenvector of ( δι ( q i , q j )) ij withpositive real entries and eigenvalue equal to the spectral radius of this matrix. We call thiseigenvector v .We claim that the probability that there are more than T v i nodes of some type q i inone of the generations 1 , . . . , T decays exponentially in T . There is one node in generationzero. Suppose that there are at most T v i nodes of each type q i in generation j . Then byour construction of v , the number of nodes of each type q i in generation j + 1 is Poisson52ith mean at most T v i λ . A Poisson random variable of mean T v i λ is the sum of T Poissonrandom variables of mean v i λ . So by the central limit theorem, the probability that thereare at least T v i such nodes decays exponentially in T , independent of j . This implies theclaim.Therefore, the probability that there are more than T v i nodes decays exponentially in T . We have completed the proof that y (cid:48) < ∞ with probability one and the probability that y (cid:48) = y decays exponentially in y . Finally, | I i ( p , q ) | is first-order stochastically dominated bythe sum of y (cid:48) Bernoulli random variables with n trials and success probability max i ι ( q i , q i ).The statement of the lemma now follows by the central limit theorem.We can now complete the proof of the claim that the contribution to E (cid:20)(cid:18) | I i ( p ∗ , q ∗ ) | k − (cid:19)(cid:21) from ideas that are learned via multiple direct connections is lower order. This is becausethe probability that there are links from j to i in the indirect learning network for l firms j decays exponentially in l . On the other hand, the probability of learning the ideas in acomponent via multiple direct connections converges to zero because each component is o ( n )a.a.s. By Lemma A4, the contribution from the this vanishing probability event vanishesasymptotically. This gives the claim.We now let X j be i.i.d. random variables with distribution given by the number ofideas that firm i would learn from a firm j conditional on learning directly from j . Thatis, X j is distributed as the sum of a Bernoulli random variable with success probability p ∗ j (corresponding to direct learning) and a random variable with the distribution of | I j ( p ∗ , q ∗ ) | with probability δ and equal to zero otherwise. E (cid:20)(cid:18) | I i ( p ∗ , q ∗ ) | k − (cid:19)(cid:21) = (cid:88) γ ∈ Γ (cid:88) j ,...,j l ( γ ) (cid:54) = i l ( γ ) (cid:89) r =1 q ∗ i q ∗ j r E l ( γ ) (cid:89) r =1 (cid:18) X j r γ r (cid:19) + o ( E (cid:20)(cid:18) | I i ( p ∗ , q ∗ ) | k − (cid:19)(cid:21) ) . The second summation is over choices of l ( γ ) of distinct firms other than i .53 n E (cid:34) ∂ (cid:0) I i ( p ∗ , ( q i ,q ∗− i )) k − (cid:1) ∂q i ( q ∗ ) (cid:35) = 1 n (cid:88) γ ∈ Γ (cid:88) j ,...,j l ( γ ) (cid:54) = i l ( γ ) (cid:88) r =1 q ∗ j r (cid:89) r (cid:48) (cid:54) = r q ∗ i q ∗ j r E l ( γ ) (cid:89) r =1 (cid:18) X j r γ r (cid:19) + o ( E (cid:20)(cid:18) | I i ( p ∗ , q ∗ ) | k − (cid:19)(cid:21) . By Lemma A2, we have δ ( (cid:88) j (cid:54) = i q ∗ j ) (cid:88) γ ∈ Γ (cid:88) j ,...,j l ( γ ) (cid:54) = i l ( γ ) (cid:89) r =1 q ∗ i q ∗ j r E l ( γ ) (cid:89) r =1 (cid:18) X j r γ r (cid:19) ∼ (cid:88) γ ∈ Γ (cid:88) j ,...,j l ( γ ) (cid:54) = i l ( γ ) (cid:88) r =1 q ∗ j r (cid:89) r (cid:48) (cid:54) = r q ∗ i q ∗ j r E l ( γ ) (cid:89) r =1 (cid:18) X j r γ r (cid:19) . Rearranging, (cid:88) γ ∈ Γ (cid:88) j ,...,j l ( γ ) (cid:54) = i l ( γ ) (cid:89) r =1 q ∗ i q ∗ j r E l ( γ ) (cid:89) r =1 (cid:18) X j r γ r (cid:19) ( δq ∗ i ( (cid:88) j (cid:54) = i q ∗ j ) − l ( γ )) ∼ . In particular, we have δq ∗ i ( (cid:88) j (cid:54) = i q ∗ j ) ∼ E t ∼ G i ( p ∗ , q ∗ ) [ τ ( t )] (7)where the expectation is taken with respect to the appropriate distribution G i ( p ∗ , q ∗ ) overtechnologies.As in the symmetric case above, this implies that lim inf n →∞ δq ∗ i ( (cid:80) j (cid:54) = i q ∗ j ) ≥ i . So the limit inferior of the row sums of ( δι ( q ∗ i , q ∗ j )) i,j is at least one. Thus the spectralradius of this matrix also satisfies lim sup n λ ≥ , which contradicts our assumption that thesequence of equilibria is subcritical.We conclude that any sequence of investment equilibria must be critical, which provesTheorem 1. 54 nline AppendixB Remaining Proofs Proof of Corollary 1.
Consider a sequence of equilibria ( p ∗ , q ∗ ) with non-vanishing invest-ment. The corollary states that for any (cid:15) > n →∞ U i ( p ∗ , (1 + (cid:15) ) q ∗ ) U i ( p ∗ , q ∗ ) = ∞ . By Theorem 1, each firm learns o ( n ) ideas at the equilibrium ( p ∗ , q ∗ ). Therefore, theexpected utility U i ( p ∗ , q ∗ ) is o ( n k − ).The spectral radius λ → λ (cid:48) be the spectral radius under actions ( p ∗ , (1 + (cid:15) ) q ∗ ) for each n .Since λ is the spectral radius of the matrix ( δq i q j ) ij and λ (cid:48) is the spectral radius of thematrix ( δ (1 + (cid:15) ) q i q j ) ij , we have λ (cid:48) = (1 + (cid:15) ) λ . In particular, λ (cid:48) > (cid:15) for n sufficientlylarge.Therefore, as shown in the proof of Theorem 1, when actions are ( p ∗ , (1 + (cid:15) ) q ∗ ), there isa giant component of firms learning at least αn ideas for some α >
0. The payoffs to firm i from the event that i learns all ideas known to the giant component and no other firm learnsfrom i grows at rate proportional to n k − .Due to Lemma A3, there is a non-vanishing probability that no firm learns from i . Since E [ | I i ( p ∗ , q ∗ ) | ] → ∞ by Theorem 1 and the assumption of non-vanishing investment, wecannot have q ∗ j √ n → j could profitably deviateby increasing q ∗ j to √ n . So under actions ( p ∗ , (1 + (cid:15) ) q ∗ ), there is a non-vanishing probabilityof firm i learning all ideas learned by the giant component. Thus, there is a non-vanishingprobability that firm i learns all ideas known to the giant component and no other firmlearns from i when actions are ( p ∗ , (1 + (cid:15) ) q ∗ ).So given any sequence of equilibria ( p ∗ , q ∗ ) with non-vanishing investment, expectedprofits U i ( p ∗ , (1 + (cid:15) ) q ∗ ) after increasing openness must grow at rate proportional to n k − .The result follows from comparing the two growth rates.55 roof of Corollary 2. Consider a sequence of equilibria ( p ∗ , q ∗ ) with non-vanishing invest-ment. The corollary states that for any (cid:15) > lim n →∞ ∂U i ( p ∗ + x , (1+ (cid:15) ) q ∗ ) ∂x (0) ∂U i ( p ∗ + x , q ∗ ) ∂x (0) = ∞ . Recall the potential proprietary technologies
P P T i ( q ) are the set of technologies t suchthat firm i will receive monopoly profits for t if all ideas in the technology t are discov-ered. This is a random object depending on the realizations of interactions but not on therealizations of private investment, and P P T i ( q ) ∩ I = P T i ( p , q ).Given actions ( p , q ) U i ( p , q ) = E [ (cid:88) t ∈ P P T i ( q ) (cid:89) j ∈ t p j ] − c ( p i ) . Along a sequence of equilibria with non-vanishing investment, we must have p ∗ → E [ | P P T i ( q ∗ ) | ] → ∞ by Theorem 1. Therefore, ∂U i ( p ∗ + x , q ∗ ) ∂x (0) ∼ k E [ | P P T i ( q ∗ ) | ] − c (cid:48) ( p ∗ i ) . By the first-order condition for p i , E [ | P P T i ( q ∗ ) | ] ∼ c (cid:48) ( p ∗ i ) . Combining these approximate equalities, it follows that ∂U i ( p ∗ + x , q ∗ ) ∂x (0) ∼ ( k − E [ | P P T i ( q ∗ ) | ]The same arguments as in the proof of Corollary 1 show that E [ | P P T i ( q ∗ ) | ] is o ( n k − )while E [ | P P T i ((1 + (cid:15) ) q ∗ ) | ] is a polynomial of order n k − .56ence we also have E [ | P P T i ((1 + (cid:15) ) q ∗ ) | ] > E [ | P P T i ( q ∗ ) | ] ∼ c (cid:48) ( p ∗ )for n large, so ∂U i ( p + x , (1 + (cid:15) ) q ∗ ) ∂x (0) > ( k − E [ | P P T i ((1 + (cid:15) ) q ∗ ) | ]for n large.The corollary follows from the growth rates of E [ | P P T i ( q ∗ ) | ] and E [ | P P T i ((1+ (cid:15) ) q ∗ ) | ]. Proof of Proposition 1.
Let b ( n ) be the share of public innovators for each n .We first show that lim inf n U i ( p ∗ , q ∗ ) (cid:0) n − k − (cid:1) > i at any sequence of equilibria with non-vanishing investment.It is weakly dominant and strictly preferred at any investment equilibrium for all publicinnovators to choose q i = 1. Therefore, all public innovators are in the same componentof the learning network. Private investment p i by public innovators is non-vanishing, soasymptotically almost surely all firms in this component learn at least αn ideas for some α > q and q be the maximum and minimum levels of openness q i chosen at equilibriumby private firms, respectively. Because the probability that no firm learns indirectly from i vanishes exponentially in ι ( q ∗ i , n while payoffs are O ( n k − ), the quantity ι ( q, n must bebounded at equilibrium.A consequence is that ι ( q, q ) n →
0. Therefore, the expected number of links to a firm i from other firms vanishes while the expected number of links to i from public innovators isnon-vanishing. So a.a.s., a given firm i ’s links are all with public innovators.Since learning indirectly from a public innovator implies learning at least αn ideas, itfollows that ι ( q, n does not vanish asymptotically at equilibrium. Therefore, the expectedpayoff U i ( p ∗ , q ∗ ) has order n k − for each firm i . This proves the characterization of equilibriawith non-vanishing investment. 57t remains to show there exists a sequence of symmetric equilibria with non-vanishinginvestment. Recall that we now call an equilibrium symmetric if all public innovators choosethe same action and the same holds for all private firms.Suppose that all firms other than i choose ( p, q ) with p ≥ and δqn ≤ C for some C > p i = p (cid:48) ≥ and q i = 1. If q i is the best responsefor i , then lim n q i n exists and is independent of ( p , q ) (given the restrictions in the previoussentence). This is because the probability of interactions between i and other firms vanishesasymptotically, while the best response does not depend on the number of ideas learned bythe unique giant component.Therefore, we can choose (cid:15) > C > q ∈ [ (cid:15)δn , Cδn ] , then for n large so isany best response q i for firm i . We claim that for n large, given p , there exists q that is abest response to ( p , q ) . This follows from Kakutani’s fixed point theorem as in the proof ofTheorem 1. We call this choice of openness q ( p ).Given such ( p , q ( p )), each firm has a non-vanishing probability of learning a linear num-ber of ideas. Therefore, E [ | I i ( p , q ( p )) | ] → ∞ . So for n large, any best response p i for eachpublic innovator and each firm i has p i ≥ . By Kakutani’s fixed point theorem, there existsymmetric actions ( p , q ( p )) such that p i ≥ is also a best response for each i . Thus thereexists a sequence of symmetric equilibria with non-vanishing investment. Proof of Theorem 2.
We first characterize equilibria by showing we cannot have a supercrit-ical and then subcritical sequence of investment equilibria. We then show there exists aninvestment equilibrium for n large. Supercritical Case : Suppose there is a supercritical sequence of investment equilibriapropensities to learn β i for each i .Passing to a subsequence if necessary, we can assume that all firms in the giant componentlearn (cid:101) αn + o ( n ) ideas for some (cid:101) α and that the number of firms learning all ideas learned bythe giant component is αn + o ( n ) for some α . The argument is the same as in the proof ofTheorem 1.For each i , let α i be the probability that firm i learns all ideas learned by all firms in the Because link probabilities are no longer symmetric within pairs, we do not assume that α = (cid:101) α . β = (cid:80) j β j q ∗ j (cid:80) j q ∗ j . As n → ∞ , this converges to the derivative of the number of firms that learn from firm i in q i divided by (cid:80) j (cid:54) = i q ∗ j .The first-order condition for firm i then implies: δα i β ≤ (1 − α i ) β i α + o (1) . (8)To see this, suppose we increase q i / (cid:80) j (cid:54) = i q ∗ j infinitessimally. We can condition on the eventthat no firm has learned indirectly from i . The left-hand side is the probability that firm i has learned indirectly from the giant component times the probability that a firm learnsindirectly from i after this increase. The right-hand side is the probability that firm i hasnot learned indirectly from the giant component times the probability that firm i learnsindirectly from the giant component after this increase.We have α i = 1 − e − αβ i δ (cid:80) j (cid:54) = i ι ( q ∗ i ,q ∗ j ) . Substituting into equation (8),(1 − e − αβ i δ (cid:80) j (cid:54) = i ι ( q ∗ i ,q ∗ j ) ) β ≤ e − αβ i δ (cid:80) j (cid:54) = i ι ( q ∗ i ,q ∗ j ) β i α + o (1) . Therefore, δ (cid:88) j (cid:54) = i ι ( q ∗ i , q ∗ j ) ≤ log(1 + αβ i ∗ /β ) αβ i + o (1) . (9)The expected number of firms from which firm i learns indirectly is β i δ (cid:80) j (cid:54) = i ι ( q ∗ i , q ∗ j ).By equation (9), this probability is bounded above log(1+ αβ i ∗ /β ) α + o (1). By the standardelementary inequality log(1 + x ) < x , this is bounded above by β i /β + o (1).The expected number of firms that learn indirectly from i is (cid:88) j (cid:54) = i β j ι ( q ∗ i , q ∗ j ) = ββ i (cid:88) j (cid:54) = i β i ι ( q ∗ i , q ∗ j ) . The right-hand side is the product of ββ i and the expected number of firms from which firm59 learns indirectly, and therefore is at most ββ i · ( β i /β + o (1)) = 1 + o (1) . Since firm i is arbitrary, this implies that each column sum of the matrix ( δι ( q ∗ i , q ∗ j )) is atmost 1 + o (1). Therefore, the spectral radius of this matrix is at most 1 + o (1). But thiscontradicts our assumption that the sequence of equilibria is supercritical. Subcritical Case : Suppose there is a subcritical sequence of investment equilibria.Introducing choices of secrecy, Lemma A2 states that δ ( (cid:88) j (cid:54) = i q ∗ j β j ) E (cid:20)(cid:18) | I i ( p ∗ , q ∗ ) | k − (cid:19)(cid:21) = 1 n E (cid:34) ∂ (cid:0) I i ( p ∗ , ( q i ,q ∗− i )) k − (cid:1) ∂q i ( q ∗ ) (cid:35) + o ( E (cid:20)(cid:18) | I i ( p ∗ , q ∗ ) | k − (cid:19)(cid:21) ) . The proof of the lemma remains the same.We now let X j be i.i.d. random variables with distribution given by the number ofideas that firm i would learn from a firm j conditional on learning directly from j . Thatis, X j is distributed as the sum of a Bernoulli random variable with success probability p ∗ j (corresponding to direct learning) and a random variable with the distribution of | I j ( p ∗ , q ∗ ) | with probability δ and equal to zero otherwise.By the same arguments as in the proof of Theorem 1, we have: E (cid:20)(cid:18) | I i ( p ∗ , q ∗ ) | k − (cid:19)(cid:21) = (cid:88) γ ∈ Γ (cid:88) j ,...,j l ( γ ) (cid:54) = i l ( γ ) (cid:89) r =1 β i q ∗ i q ∗ j r E l ( γ ) (cid:89) r =1 (cid:18) X j r γ r (cid:19) + o ( E (cid:20)(cid:18) | I i ( p ∗ , q ∗ ) | k − (cid:19)(cid:21) ) . and also1 n E (cid:34) ∂ (cid:0) I i ( p ∗ , ( q i ,q ∗− i )) k − (cid:1) ∂q i ( q ∗ ) (cid:35) = 1 n (cid:88) γ ∈ Γ (cid:88) j ,...,j l ( γ ) (cid:54) = i l ( γ ) (cid:88) r =1 β i q ∗ j r (cid:89) r (cid:48) (cid:54) = r β i q ∗ i q ∗ j r E l ( γ ) (cid:89) r =1 (cid:18) X j r γ r (cid:19) + o ( E (cid:20)(cid:18) | I i ( p ∗ , q ∗ ) | k − (cid:19)(cid:21) . Therefore, δ ( (cid:88) j (cid:54) = i q ∗ j β j ) (cid:88) γ ∈ Γ (cid:88) j ,...,j l ( γ ) (cid:54) = i l ( γ ) (cid:89) r =1 q ∗ i q ∗ j r E l ( γ ) (cid:89) r =1 (cid:18) X j r γ r (cid:19) ∼ (cid:88) γ ∈ Γ (cid:88) j ,...,j l ( γ ) (cid:54) = i l ( γ ) (cid:88) r =1 q ∗ j r (cid:89) r (cid:48) (cid:54) = r q ∗ i q ∗ j r E l ( γ ) (cid:89) r =1 (cid:18) X j r γ r (cid:19) . β i terms cancel. Rearranging, (cid:88) γ ∈ Γ (cid:88) j ,...,j l ( γ ) (cid:54) = i l ( γ ) (cid:89) r =1 q ∗ i q ∗ j r E l ( γ ) (cid:89) r =1 (cid:18) X j r γ r (cid:19) ( δq ∗ i ( (cid:88) j (cid:54) = i q ∗ j β j ) − l ( γ )) ∼ . In particular, we have δq ∗ i ( (cid:88) j (cid:54) = i q ∗ j β j ) ∼ E t ∼ G i ( p ∗ , q ∗ ) [ τ ( t )]where the expectation is taken with respect to the appropriate distribution G i ( p ∗ , q ∗ ) overtechnologies. Because τ ( t ) ≥ t , the limit superior of the expected number of firmsthat learn from i is at least one.So each column sum of the matrix ( δι ( q ∗ i , q ∗ j )) is at least 1 + o (1). Therefore, the spectralradius of this matrix is at least 1 + o (1). This contradicts our assumption that the sequenceof equilibria is subcritical. Existence : We will show there exists an investment equilibrium for n large. To do so,we first fix a sequence of p with lim inf n min i p i > i . Given such a p , we considerthe set of best responses BR ( q − i ) for firm i when other firms choose actions ( p − i , q − i ). Notethat unlike in the proof of Theorem 1, since the equilibrium is no longer symmetric, the bestresponse q i can depend on others’ levels of private investment.First suppose that a sequence of opponents’ actions ( p − i , q − i ) is subcritical. Our analysisabove showed that for n large, the best response BR ( p − i , q − i ) has q i ( (cid:88) j (cid:54) = i q j β j ) ∈ [ 12 δn , kδn ] , where the upper bound follows from the fact that τ ( t ) ≤ k − p − i , q − i ), the matrix of linkprobabilities for firms other than i has spectral radius λ > . Then firm i can achieve apositive payoff by choosing q i such that β i (cid:80) j (cid:54) = i ι ( q i , q j ) = 1. On the other hand expectedpayoffs to firm i vanish or are negative along any sequence of best-responses such that β i q i →
0. Therefore 0 is not in the closure of BR i ( q − i ) whenever the matrix of link probabilities for We extend our definition of criticality to the restriction of the random network to agents other than i . i has spectral radius λ > .Because 0 is not in the closure of BR i ( q − i ) for any q − i , by compactness we can choose (cid:15) ( n ) such that BR i ( q − i ) ≥ (cid:15) ( n ) √ n when λ > . We therefore have BR i ( q − i ) ∈ [ (cid:15) ( n ) √ n ,
1] when q j ∈ [ (cid:15) ( n ) √ n ,
1] for all j (cid:54) = i for n sufficiently large. So by Kakutani’s fixed-point theorem, for n sufficiently large thereexists a fixed point of q (cid:55)→ ( BR i ( q − i )) i . We will call this fixed point q ( p )) to indicate thedependence on p . It remains to show that there exists p such that p i is a best responseunder actions ( p , q ( p )) for all i .Suppose that p i ≥ for all i and all n . Our analysis of the subcritical and supercriticalregions above extends immediately to this fixed point, as we did not rely on p being chosenoptimally. Therefore, the sequence of outcomes q ( p ) must be critical. In particular, expectedpayoffs at ( p , q ( p )) converge to ∞ for all such sequences of p .The best response p i maximizes p i E t ∈ P P T i ( q ( p )) [ (cid:89) j ∈ tj (cid:54) = i p j ] − c ( p i )and therefore satisfies c (cid:48) ( p i ) = E t ∈ P P T i ( q ( p )) [ (cid:89) j ∈ tj (cid:54) = i p j ] . (10)Beacuse c ( p i ) is strictly increasing and strictly convex with c (cid:48) (0) ≥ c ( p ) → ∞ as p →
1, there exists a solution.Since p j ≥ for all j , for n large the optimal p i ≥ as well since the expected number ofpotential proprietary technologies converges to infinity by equation (10). So by Kakutani’sfixed point theorem, for n large there exists p ∈ [ , n such that p i is optimal under actions( p , q ( p )). This is a symmetric investment equilibrium.62 roof of Proposition 2. We first show there is no sequence of supercritical symmetric invest-ment equilibria for any ρ >
0. To do so, we consider firm i ’s choice of q i in the supercriticalregion. As in the proof of Theorem 1, E (cid:20)(cid:18) | I i ( p ∗ , ( q i , q ∗− i )) | k − (cid:19) ρ (cid:21) = ( p ∗ ) k (1 − (1 − δι ( q i , q ∗ )) α ( n − o ( n ) )( α ( n − ( k − ρ + o ( n ( k − ρ . In particular, to solve for firm i ’s choice of q i to first order, we need only consider tech-nologies consisting of i ’s private idea and ( k −
1) ideas learned by the giant component. Theprobability that such a technology faces competition is(1 − δ · ι ( q i , q ∗ ) − (1 − δ ) · ι ( q i , q ∗ ) · α ) n − + o (1) . The term δ · ι ( q i , q ∗ ) corresponds to the possibility of a firm j indirectly learning all of firm i ’s ideas. The term (1 − δ ) · ι ( q i , q ∗ ) · α corresponds to the possibility of a firm j directlylearning firm i ’s idea (but not indirectly learning from i ) and indirectly learning the ideaslearned by the giant component.Thus, we are looking for q i maximizing: E (cid:20)(cid:18) | I i ( p ∗ , ( q i , q ∗− i )) | k − (cid:19) ρ (cid:21) (1 − δ · ι ( q i , q ∗ ) − (1 − δ ) · ι ( q i , q ∗ ) · α ) n − . This expression is equal to: (cid:0) ( p ∗ ) k (1 − (1 − δι ( q i , q ∗ )) α ( n − o ( n ) )(( α ( n − k − + o ( n k − )) (cid:1) ρ (1 − δ · ι ( q i , q ∗ ) − (1 − δ ) · ι ( q i , q ∗ ) · α ) n − . Therefore, asymptotically the optimal q i will be a maximizer of:( p ∗ ) ρ ( k − (1 − δι ( q i , q ∗ )) α ( n − o ( n ) )(( α ( n − k − + o ( n k − )) ρ (1 − δ · ι ( q i , q ∗ ) − (1 − δ ) · ι ( q i , q ∗ ) · α ) n − . The terms containing q i do not depend on ρ to first order. Therefore, the optimizationproblem is the same as in Theorem 1, and the same argument shows there is no supercriticalsequence of symmetric investment equilibria.It remains to define ρ suitably and show there is no subcritical sequence of symmetric63nvestment equilibria when for ρ ≥ ρ . We will choose ρ to satisfy the conditions in thefollowing lemma: Lemma B5. If k = 2 and ρ ≥ , then (cid:0) yk − (cid:1) ρ is convex in y > . If k > , there exists ρ < such that (cid:0) yk − (cid:1) ρ is convex in y > for all ρ ≥ ρ. Proof.
We can assume y ≥ k −
1. We have (cid:0) yk − (cid:1) = (cid:81) k − j =0 ( y − j )( k − , and the right-hand side isdefined for all real y >
0. We will determine the sign of: d dy (cid:32)(cid:32) (cid:81) k − j =0 ( y − j )( k − (cid:33) ρ (cid:33) . This expression has the same sign as ddy ρ (cid:32) k − (cid:89) j =0 ( y − j ) (cid:33) ρ − k − (cid:88) i =0 (cid:89) j (cid:54) = i ( y − j ) . This derivative is equal to ρ ( ρ − (cid:32) k − (cid:89) j =0 ( y − j ) (cid:33) ρ − (cid:32) k − (cid:88) i =0 (cid:89) j (cid:54) = i ( y − j ) (cid:33) + ρ (cid:32) k − (cid:89) j =0 ( y − j ) (cid:33) ρ − k − (cid:88) i =0 (cid:88) i (cid:48) (cid:54) = i (cid:89) j (cid:54) = i,i (cid:48) ( y − j ) . (11)If ρ ≥
1, both the first and second term are non-negative for y ≥ k −
1, so expression (11)is non-negative as well.Suppose k >
2. Expression (11) has the same sign as( ρ − (cid:32) k − (cid:88) i =0 (cid:89) j (cid:54) = i ( y − j ) (cid:33) + (cid:32) k − (cid:89) j =0 ( y − j ) (cid:33) k − (cid:88) i =0 (cid:88) i (cid:48) (cid:54) = i (cid:89) j (cid:54) = i,i (cid:48) ( y − j ) . (12)s The first term may be negative if ρ <
1, while the second term is positive for y ≥ k − k − y . Therefore, we can choose y and ρ < ρ > ρ and y > y .We want the expression to be positive for k − ≤ y ≤ y . There are finitely many values,and for each expression (12) is positive when ρ is sufficiently close to one or at least one.Therefore, increasing ρ if needed, we find that expression (12) is positive for ρ > ρ and64 ≥ k −
1. This proves the lemma.Let ρ ≥ ρ , where ρ = 1 when k = 2 and ρ is chosen as in Lemma B5 for k > n δ · ι ( q ∗ , q ∗ ) n <
1. Passing to a subsequence if necessary, we can assume that δ · ι ( q ∗ , q ∗ ) n converges.We claim that for n sufficiently large δ ∂ι ( q i , q ∗ ) ∂q i ( q ∗ ) · E (cid:20)(cid:18) | I i ( p ∗ , q ∗ ) | k − (cid:19) ρ (cid:21) < n − E (cid:34) ∂ (cid:0) | I i ( p ∗ , ( q i ,q ∗ ) | k − (cid:1) ρ ∂q i ( q ∗ ) (cid:35) (13)for all i . Both sides of the inequality converge because δ · ι ( q ∗ , q ∗ ) n converges to a limit lessthan one.Let X be a random variable equal to | I i ( p , q ) | with probability δ and 0 with probability1 − δ . Then the left-hand side of equation (13) is equal to E (cid:20)(cid:18) X k − (cid:19) ρ (cid:21) asymptotically.Let X be the random variable with distribution equal to the change in | I i ( p , q ) | if firm i learned from an additional firm j chosen uniformly at random. Then the right-hand sideof equation (13) is equal to E (cid:20)(cid:18) | I i ( p , q ) | + X k − (cid:19) ρ − (cid:18) | I i ( p , q ) | k − (cid:19) ρ (cid:21) asymptotically.In this case, with probability 1 − δ , the firm i only learns directly from firm j . Withprobability δ , firm i learns indirectly through firm j , and then learns | I j ( p , q ) | − | I i ( p , q ) ∩ I j ( p , q ) | additional ideas.The expected cardinality | I i ( p , q ) ∩ I j ( p , q ) | is o (1), by the same independence argument65iven in the proof of Lemma A2. Therefore, we can ignore the intersection term in computingthe limit of the right-hand side of equation (13). Let (cid:101) X be the random variable withdistribution equal to the number of ideas firm i learns from firm j , including any ideas firm i already knows, i.e., X without this intersection term.Then (cid:101) X first-order stochastically dominates X , and is one higher with non-vanishingprobability. By Lemma B5, this implies E (cid:20)(cid:18) X k − (cid:19) ρ (cid:21) < E (cid:34)(cid:18) | I i ( p , q ) | + (cid:101) X k − (cid:19) ρ − (cid:18) | I i ( p , q ) | k − (cid:19) ρ (cid:35) for n large. It follows that the same inequality holds for n large with X replacing (cid:101) X , whichproves the claim.So along any sequence of symmetric investment equilibria with lim sup δι ( q ∗ , q ∗ ) n < n sufficiently large δι ( q ∗ , q ∗ ) n > E t ∈ P T i ( p ∗ , q ∗ ) [ τ ( t )]for all i . The proof is the same as the proof of Lemma 2, with the approximate equality fromLemma A2 replaced by the inequality from equation (13).In particular, δι ( q ∗ , q ∗ ) n > n large, which contradicts the assumption of subcriti-cality. So any sequence of symmetric investment equilibria is critical. Proof of Proposition 3.
The proof follows the same basic outline as the proof of Proposi-tion 2, with the function (cid:0) | I i ( p , q ) | k − (cid:1) ρ replaced by φ ( | I i ( p , q ) | ).We first show there is no sequence of supercritical symmetric equilibria with lim inf n p ∗ /n >
0. To do so, we consider firm i ’s choice of q i in the supercritical region. As in the proof ofTheorem 1, the payoffs to firm i are: E [ U i ( p ∗ , ( q i , q − i ))] = (1 − (1 − ι ( q i , q ∗ )) α ( n − o ( n ) )(1 − ι ( q i , q ∗ )) n − p ∗ φ (cid:0) p ∗ ( α ( n −
1) + o ( n k − )) (cid:1) − c ( p ∗ )when the giant component has size αn + o ( n ) . We will bound φ ( p ∗ ( α ( n −
1) + y )) , where y is o ( n ). This expression is less than or equal66o φ ( p ∗ α ( n − p ∗ yφ (cid:48) ( p ∗ ( α ( n −
1) + y )) . By the assumption of non-vanishing investment, we have lim n p ∗ >
0. By our assumptionthat φ ( x j ) φ ( x (cid:48) j ) → x j /x (cid:48) j →
1, we can conclude φ ( p ∗ ( α ( n −
1) + y )) = φ ( p ∗ α ( n − o ( φ ( p ∗ α ( n − . Therefore, q i is chosen to maximize:(1 − (1 − ι ( q i , q ∗ )) α ( n − o ( n ) )(1 − ι ( q i , q ∗ )) n − + o (1) . The maximization is the same as in Theorem 1 with δ = 1, and the same calculation showsthere is no supercritical sequence of symmetric investment equilibria.The proof that there is no the subcritical sequence of symmetric equilibria with lim inf n p ∗ /n > (cid:0) | I i ( p , q | k − (cid:1) ρ replaced by φ ( | I i ( p , q | ). We no longer needto prove Lemma B5, as we assume that φ ( · ) is convex. Proof of Proposition 4.
Proof of (i): We will use Lemma 3, which we now prove, to show wecannot have a critical sequence of symmetric investment equilibria.
Lemma 3.
For any critical sequence of symmetric actions with p > , lim n →∞ E t ∈ P T i ( p , q ) [ τ ( t )] = 1 . Proof of Lemma 3.
We can assume without loss of generality that p is bounded away fromzero, because E t ∈ P T i ( p , q ) [ τ ( t )] does not depend on the value of p as long as p is non-zero.Let (cid:15) >
0. The probability that firm i learns from d firms decays exponentially in d . ByRosenthal’s inequality (Rosenthal, 1970), the payoffs to learning from d firms grow at mostat a polynomial rate in d . Thus we can choose d such that the contribution to E t ∈ P T i ( p , q ) [ τ ( t )]from the event that firm i learns from more than d other firms is at most (cid:15) for n large.Since (cid:15) is arbitrary, we can restrict our analysis to the event that firm i learns from atmost d other firms. 67e claim that as n → ∞ , we have E [ | I i ( p , q ( λ )) | ] = ∞ . Let λ > q ( λ ) is defined by ι ( q ( λ ) , q ( λ )) = − λδn . For any q < q (cid:48) , the random variable | I i ( p , q ) | is first-order stochastically dominated by | I i ( p , q (cid:48) ) | . So it is sufficient to show thatlim λ → lim n →∞ E [ | I i ( p , q ) | ] → ∞ . We can bound | I i ( p , q ) | below by the expected number of firms j with a path from j to i in the indirect learning network. By Theorem 11.6.1 of Alon and Spencer (2004), the limitof this quantity as n → ∞ is equal to the number of nodes in a Poisson branching processwith parameter 1 − λ . As λ →
0, this number of nodes converges to infinity. This provesthe claim.The proof of Lemma 3 will also use the following lemma, which states that learning alarge number of ideas at a critical sequence of equilibria is rare for n large: Lemma B6.
Let ω ( n ) → ∞ . Then P [ | I i ( p , q ) | > ω ( n )] → as n → ∞ .Proof. Let (cid:15) >
0. We want to prove that P [ | I i ( p , q ) | > ω ( n )] < (cid:15) for n large.Let q ( λ ) be the solution to ι ( q ( λ ) , q ( λ )) = λδn . Once again, for any q < q (cid:48) , the randomvariable | I i ( p , q ) | is first-order stochastically dominated by | I i ( p , q (cid:48) ) | . So it is sufficient toshow there exists λ > P [ | I i ( p , q ( λ )) | > ω ( n )] < (cid:15) for n large.We showed in the proof of Lemma 2 that the number of descendants of i in the indirectlearning network is first-order stochastically dominated by the number of nodes in the Poissonbranching process with parameter 1 + λ . The probability that this Poisson branching processincludes infinitely many nodes converges to 0 as λ → λ such that this probability is at most (cid:15)/ λ , we can also choose y such that the probability that the Poisson branching processhas y nodes for any y ≤ y < ∞ is at most (cid:15)/
4. Because | I i ( p , q ) | is first-order stochasticallydominated by the sum of y Bernoulli random variables with success probability p and y binomial random variables distributed as Binom ( p (1+ λ ) δn , n − , we can choose y (cid:48) such thatthe probability that y (cid:48) ≤ | I i ( p , q ) | < ∞ is at most (cid:15)/ P [ | I i ( p , q ) | > y (cid:48) ] < (cid:15) for n large, which implies P [ | I i ( p , q ) | > ω ( n )] < (cid:15) for n largesince ω ( n ) → ∞ We now return to the proof of Lemma 3. Choose ω ( n ) → ∞ such that ω ( n ) E [ | I i ( p , q ) | ] → . We have assumed that i learns from at most d other firms. We can order these firms from1 to d . For each of these firms j , let the additional ideas AI j be the set of ideas that firm i learns from firm j and has not learned from any previous firm 1 , . . . , j − n → ∞ , a vanishing share of proprietary technologies include ideasthat firm i learns only from firms j with AI j ≤ ω ( n ) . The number of such ideas is boundedabove by ω ( n ) d . So the number of proprietary technologies including at least one such ideais bounded above by E (cid:20)(cid:18) | I i ( p , q ) | + dω ( n ) k − (cid:19) · ( dω ( n )) (cid:21) = dω ( n ) E (cid:20)(cid:18) | I i ( p , q ) | + dω ( n ) k − (cid:19)(cid:21) , (14)while the total number of proprietary technologies is on the same order as E (cid:20)(cid:18) | I i ( p , q ) | k − (cid:19)(cid:21) ≥ E (cid:20)(cid:18) | I i ( p , q ) | k − (cid:19)(cid:21) E [ | I i ( p , q ) | ] k − . (15)Since ω ( n ) E [ | I i ( p , q ) | ] → , the quotient of expression (14) divided by expression (15) vanishes as n → ∞ .Let (cid:15) >
0. For n sufficiently large, Lemma B6 implies that the probability that AI j >ω ( n ) is at most (cid:15) . We will show that the contribution to E t ∈ P T i ( p , q ) [ τ ( t )] from the event that69 I j > ω ( n ) for more than one j can be taken to be small.We condition on the event that AI j > ω ( n ) for at least one j . Then the probability that AI j > ω ( n ) for at least one other j is bounded above by d(cid:15) , while the expected number ofproprietary technologies increases by at most a constant multiplicative factor (depending on d ) in this case. Since (cid:15) can be taken to be arbitrarily small, it follows that the contributionto E t ∈ P T i ( p ∗ , q ∗ ) [ τ ( t )] from the event that AI j > ω ( n ) for more than one j vanishes as n → ∞ .The remaining technologies in P T i ( p , q ) consist of firm i ’s private idea and k − j . We thus have τ ( t ) = 1 for each of the remaining technologies t ∈ P T i ( p , q ) . This shows that E t ∈ P T i ( p , q ) [ τ ( t )] →
1, which proves the lemma.We will show that if lim inf n p > n ι ( q, q ) nδ ≤ , then given symmetricactions ( p , q ), ∂U i ( p , q ) ∂q i ( q ) > n sufficiently large. In words, given symmetric actions in the subcritical or criticalregion, increasing q i would increase payoffs. We can assume without loss of generality that p is bounded away from zero, because the sign of this derivative is independent of p .Let D i ( q ) be the set of firms j such that there is a path from i to j in the indirect-learningnetwork. We claim that when lim sup n ι ( q, q ) δn ≤
1, a.a.s. a random technology t ∈ T i ( p , q )is known by | D i ( q ) | other firms.We can write ι ( q, q ) δn = 1 + (cid:15) , where lim sup n (cid:15) ≤
0. Let (cid:15) = max( (cid:15), n − / log n ) . By the‘No Middle Ground’ claim from p. 210-211 of Alon and Spencer (2004), the probability thata given node in an undirected random network with link probability (cid:15)n is contained in acomponent of cardinality at least (cid:15)n at most n − k − for n large.A standard correspondence states that the size of the component containing a givennode in an undirected random graph first-order stochastically dominates the number ofnodes reachable by a path from that node in a directed random graph with the same linkprobability (see for example (cid:32)Luczak, 1990). So the probability that a given idea is learnedindirectly by more than (cid:15)n firms is at most n − k − for n large. Thus, we can choose aconstant such that the probability that any idea is learned indirectly by more than (cid:15)n firmsis at most n − k for n large. Since there are (cid:0) nk (cid:1) potential technologies, we can condition on70he event that no idea is learned indirectly by more than (cid:15)n firms.Now, choose t ∈ T i ( p , q ) and let j ∈ T i ( p , q ). The number of firms that learn idea j from each firm that indirectly learns j is bounded above by a Poisson random variable withparameter 1 /δ (by the same argument as in the proof of Lemma A2). So we can assumethat at most C(cid:15)n firms learn the idea for some constant
C >
0. For a firm j (cid:48) / ∈ D i ( q ) toknow all ideas in t , that firm must learn j and directly learn i . Each of the C(cid:15)n firms thatlearn j will directly learn i with probability at most (cid:15)δn , so the probability that any of thesefirms directly learns i vanishes asymptotically. This proves the claim.Thus, the payoff to firm i is U i ( p , q ) = E [ f ( | D i ( q ) | ) | T i ( p , q ) | ] − c ( p i ) + o ( E [ | P T i ( p , q )]) . Thus, ∂U i ( p , q ) ∂q i ( q ) = E (cid:20) ∂f ( | D i ( q ) | ) | ∂q i ( q ) | T i ( p , q ) | (cid:21) + E (cid:20) ∂ | T i ( p , q ) | ∂q i ( q ) f ( | D i ( q ) | ) (cid:21) + o ( E [ | P T i ( p , q )]) . We claim this is equal to E (cid:20) ∂f ( | D i ( q ) | ) | ∂q i ( q ) (cid:21) E [ | T i ( p , q ) | ]+ E (cid:20) ∂ | T i ( p , q ) | ∂q i ( q ) (cid:21) E [ f ( | D i ( q ) | )]+ o ( E [ | P T i ( p , q )]) . (16)The relevant random variables are independent conditional on the event that I i ( p , q ) ∩ D i ( q ) = ∅ and this intersection remains empty after adding an additional incoming or outgoing link.Because lim sup n ι ( q, q ) nδ ≤ , this occurs asymptotically almost surely. We must show thecontributions to T i ( p , q ) from the vanishing probability event that this intersection is non-empty vanish as n → ∞ . This follows from the bounds on the probability of this event in the‘No Middle Ground’ claim from p. 210-211 of Alon and Spencer (2004), and the argumentis the same as above. 71ecause f is non-negative with f (0) = 1 and f (1) >
0, we can choose (cid:15) > − qn · E (cid:20) ∂f ( | D i ( q ) | ) | ∂q i ( q ) (cid:21) < E [ f ( | D i ( q ) | )] − (cid:15) for all n . Here, the left-hand side is equal to the decrease in f ( | D i ( q ) | when an additionalfirm learns from firm i , which is at most f ( | D i ( q ) | and will be smaller with non-vanishingprobability.At a subcritical sequence of actions, we havelim sup n ι ( q, q ) δn ≤ lim inf n E t ∈ P T i ( p , q ) [ τ ( t )] . By the same argument used to prove Lemma 2, this implies that E [ | T i ( p , q ) | ] < qn · E (cid:20) ∂ | T i ( p , q ) | ∂q i ( q ) (cid:21) for all n sufficiently large. At a critical sequence of actions, Lemma 3 shows thatlim n →∞ E t ∈ P T i ( p , q ) [ τ ( t )] = 1 . So at a critical sequence of actions, it follows from Lemma 3 and the definition of τ ( t ) that E [ | T i ( p , q ) | ] ∼ qn · E (cid:20) ∂ | T i ( p , q ) | ∂q i ( q ) (cid:21) . In either case, substituting into expression (16) we obtain: ∂U i ( p , q ) ∂q i ( q ) > n sufficiently large. This proves the claim, so anysequence of symmetric investment equilibria is supercritical.It remains to show there exists a symmetric investment equilibrium. When all other firmschoose p j = p > q j = q , the optimal choice of q does not depend on p . Let BR ( q ) bethe optimal level of openness when other firms choose q .72irst, suppose that ι ( q, q ) = (cid:15)δn . Then there exists a giant component of αn + o ( n ) firms,where α is increasing in (cid:15) and converges to zero as (cid:15) → (cid:15) sufficiently small and n large, BR ( q ) > q. The expected profits to anarbitrary firm i choosing ( p, q i ) when all other firms choose ( p, q ) are:( p ) k (cid:18) αnk − (cid:19) (1 − (1 − δι ( q i , q )) αn ) f ( E t ∈ T i ( p , ( q i ,q ∗− i )) [ m ]) + o ( n k − ) , where m is the number of firms other than i who learn idea i and all ideas known to all firmsthat learn from the giant component.Suppose ι ( q i , q ) ≤ (cid:15)δn . We will argue that increasing q i would increase firm i ’s payoffs,and we can condition on the event that no firm who knows all ideas learned by the giant com-ponent has learned from i . By our assumptions on f ( · ), the change in f ( E t ∈ T i ( p , ( q i ,q ∗− i )) [ m ])from an additional firm j learning indirectly from i are bounded above by C <
1. If firm i has learned all ideas learned by the giant component, then for (cid:15) small the expected decreasein payoffs from a firm j learning from i is:1 q ∂f ( E t ∈ T i ( p , ( q i ,q ∗− i )) [ m ]) ∂q i ( p ) k (cid:18) αnk − (cid:19) + o ( n k − ) ≤ C · δ · ( p ) k (cid:18) αnk − (cid:19) + o ( n k − ) . As (cid:15) grows small, the expected increase in payoffs to learning from an additional firm j approaches: δ ( p ) k (cid:18) αnk − (cid:19) + o ( n k − ) . Since C <
1, firm i would deviate to increase q i . So any best response q i must be greaterthan q , which proves our claim.The function BR ( q ) is continuous and BR (1) ≤
1, so for n large we must have BR ( q ) = q for some q with ι ( q, q ) ≥ δn . There exists an optimal choice of p given q by the same argumentas in the proof of Theorem 1. So there exists an investment equilibrium for n large.Proof of (ii): Suppose there exists a critical or supercritical sequence of symmetric in-vestment equilibria. We claim that ∂U i ( p ∗ , ( q i , q ∗− i )) ∂q i ( q ∗ ) < , p isbounded away from zero, because the sign of this derivative is independent of p .First suppose there exists a critical sequence of investment equilibria. Lemma 3 showsthat lim n →∞ E t ∈ P T i ( p ∗ , q ∗ ) [ τ ( t )] = 1 . As a consequence, E (cid:20)(cid:18) | I i ( p ∗ , q ∗ ) | k − (cid:19)(cid:21) ∼ qn · ∂ E (cid:104)(cid:0) | I i ( p ∗ , ( q i ,q ∗− i )) | k − (cid:1)(cid:105) ∂q i ( q ∗ ) . Therefore, ∂ E (cid:2) | P T i ( p ∗ , ( q i , q ∗− i )) | (cid:3) ∂q i ( q ∗ ) ∼ . On the other hand, ∂ E (cid:2) | T i ( p ∗ , ( q i , q ∗− i )) | (cid:3) ∂q i ( q ∗ ) > n large, and increasing q i weakly increases the number of firms f ( m ) who know eachtechnology. Since firm i receives negative profits from each t ∈ T i ( p ∗ , ( q i , q ∗− i )) that is notproprietary, firm i ’s profits are decreasing in q i at q i = q ∗ .We next suppose there exists a supercritical sequence of equilibria. Let m be the numberof firms other than i who learn idea i and all ideas known to all firms that learn from thegiant component. The expected profits from choosing q i are:( p ∗ ) k (cid:18) αnk − (cid:19) (1 − (1 − δι ( q i , q ∗ )) αn ) f ( E t ∈ T i ( p ∗ , ( q i ,q ∗− i )) [ m ]) + o ( n k − ) , where α is the share of ideas learned by all firms in the giant component.Conditional on the event that m >
0, increasing q i will weakly decrease expected payoffsbecause increasing m and increasing | I i ( p ∗ , ( q i , q ∗− i )) | both weakly decrease payoffs. More-over, this increase is strict, because the probability of learning the ideas in the giant compo-nent is strictly higher under higher q i .Now consider the event that m = 0. We showed in the proof of Theorem 1 that when f ( m ) = 0 for all m >
0, the increase in expected payoffs from an additional incoming link is74ess than the decrease in expected payoffs from an additional outgoing link. Since decreasing f ( m ) for m > q i = q ∗ , increasing q i will weakly decrease expected payoffs conditional on the event m = 0.Therefore, when q i = q ∗ , increasing q i will weakly decrease expected payoffs uncondition-ally, which gives our contradiction. We have checked the critical and subcritical cases, sothis completes the proof of Proposition 4. Proof of Proposition 5. (i): Suppose there exists a sequence of investment equilibrium with n → ∞ .We first consider the first-order condition for q i (1). The expected payoff to firm i with apatent when k = 2 and link costs are (cid:15) is: p i p ∗ q ∗ (0) q i (1)(1 − b )( n − − (cid:15) ( q i (1)( q ∗ (0)(1 − b ) + q ∗ (1) b )( n − − c ( p i ) . The first term now does not depend on whether other firms learn the ideas involved in thetechnologies that firm i can produce. The second term is the expected link cost.The expected payoff is linear in q i (1) , so the coefficient of q i (1) must be non-negative atany investment equilibrium. Therefore:( p ∗ ) q ∗ (0)(1 − b ) ≥ (cid:15) ( q ∗ (0)(1 − b ) + q ∗ (1) b ) . If equality holds, then firms with patents are indifferent to all choices of interaction rates.But then firms without patents would not choose positive interaction rates, which they mustat any investment equilibrium. So the inequality is strict.Because payoffs are strictly increasing in q i (1) on [0 , q ∗ (1) = 1. Thus theinequality ( p ∗ ) q ∗ (0)(1 − b ) > (cid:15) ( q ∗ (0)(1 − b ) + q ∗ (1) b )implies that lim inf n q ∗ (0) is positive. But if all interaction rates are bounded below byconstants, the probability that a firm j without a patent receives monopoly profits from agiven technology t decays exponentially. Since firm j ’s link costs are linear in n , firm j ’s75xpected payoff is negative. This cannot occur at equilibrium, so we have a contradiction.(ii): It is weakly dominant for all firms to choose q ∗ (1) = 1, and is strictly optimal atany investment equilibrium. So for any δ >
0, almost surely all firms with patents are inthe same component of the indirect-learning network. All firms in this component learn αn ideas for some α > q ∗ (0) n → ∞ and consider firm i without a patent. The probability that nofirm with a patent learns indirectly from firm i decays exponentially in q ∗ (0) n , and so profitsmust be o ( n k − ). But firm i could receive higher profits by deviating to choose q i (0) = δn ,as this would give profits of order n k − . So it must be the case that q ∗ (0) is O (1 /n ), andthus ι ( q ∗ (0) , q ∗ (0)) = ( q ∗ (0)) = O (1 /n )as desired. C Direct Learning
We now analyze the model from Section 2 in the case δ = 0. Then firms can only learndirectly from other firms, and not indirectly. Proposition C1.
When δ = 0 , there exists a symmetric investment equilibrium for n large,and at any sequence of symmetric investment equilibria lim n ι ( q ∗ , q ∗ ) n k = ( k − k . Interaction rates are much higher than in the indirect-learning case, because withoutindirect learning much more interaction is needed for competition to be a substantial force.The expected number of ideas learned by each firm is now O ( n k − k ), which is still asymptoti-cally lower than in the supercritical case with indirect learning (where the expected numberof ideas learned is linear in n ). Proof.
For n large, the expected number of potential technologies that firm i produces and76hich include firm i ’s private idea: | T i ( p ∗ , ( q i , q ∗− i )) | ∼ ( p ∗ ) k ( k − ι ( q i , q ∗ )( n − k − The probability that no other firm produces any such technology is:(1 − ι ( q i , q ∗ ) ι ( q ∗ , q ∗ ) k − ) n − + o (1) . So q i is chosen to maximize the number of potential proprietary technologies for i : | P T i ( p ∗ , ( q i , q ∗− i )) | ∼ ( p ∗ ) k ( k − ι ( q i , q ∗ )( n − k − (1 − ι ( q i , q ∗ ) ι ( q ∗ , q ∗ ) k − ) n − . (17)The first-order condition for q i is:( k − (cid:18) ∂ι ( q, q ∗ ) ∂q ( q i ) (cid:19) (1 − ι ( q i , q ∗ ) ι ( q ∗ , q ∗ ) k − ) ∼ ( n − ι ( q i , q ∗ ) (cid:18) ∂ι ( q, q ∗ ) ∂q ( q i ) (cid:19) ι ( q ∗ , q ∗ ) k − . We claim that we must have ι ( q ∗ , q ∗ ) → q i such that the interaction rate ι ( q i , q ∗ ) is proportional to n . Thus, the first-ordercondition implies: lim n ( n − ι ( q i , q ∗ ) ι ( q ∗ , q ∗ ) k − = k − . At equilibrium, this implies ι ( q ∗ , q ∗ ) ∼ ( k − n − k as desired.Interactions between firms j and j (cid:48) now only impose a negative externality on a thirdfirm i . The negative externality appears because these interactions can facilitate competition.There is no longer a benefit to firm i , because learning between firms j and j (cid:48) cannot facilitateindirect learning by firm i .A consequence is that increasing all firms’ openness would decrease average profits:77 orollary C1. When δ = 0 , at any symmetric investment equilibrium ( p ∗ , q ∗ ) , lim n →∞ ∂U i ( p ∗ , q ) ∂q ( q ∗ ) < . Proof.
Firm i ’s optimization problem over q i given symmetric strategies ( p ∗− i , q − i ) by oppo-nents is equivalent to choosing an interaction rate ι ( q i , q − i ) with all other firms, given theirinteraction rates ι ( q − i , q − i ) with each other.By the envelope theorem, the derivative of U i ( p ∗ , q ) as we vary ι ( q i , q − i ) fixing ι ( q − i , q − i )is zero at q = q ∗ . We can see from equation (17) that the derivative of U i ( p ∗ , q ) as we vary ι ( q − i , q − i ) fixing ι ( q i , q − i ) is negative. Therefore, decreasing q symmetrically at equilibriumreduces the payoffs U i ( p ∗ , q ).The corollary shows that decreasing interaction rates will increase average profits. Iffirms respond to the new interaction rates by adjusting private investment, the effect onthe innovation rate will be more ambiguous: there will be an increase in R&D but a givendiscovery will be less likely to spread. In Acemoglu, Makhdoumi, Malekian, and Ozdaglar(2017), benefits from interactions also depend only on direct links, so a similar argumentshows that the network is denser than the social optimum.We can observe from the proof that adding a constant link cost (cid:15) > (cid:15) . We next discuss direct learning with patent rights when k > C.1 Patents
Proposition 5 considers granting patent rights to some types of ideas when δ = 0 and k = 2. We found that adverse selection interactions prevents the emergence of an investmentequilibrium.We now extend the analysis to k >
2. There is now an investment equilibrium, but thesame adverse selection effect implies that firms with patents choose much lower levels ofopenness than without patents.Suppose that a fraction b ∈ (0 ,
1) of firms receive patents, as in Proposition 5. Wemaintain the assumption that δ = 0. 78 roposition C2. Suppose a fraction b ∈ (0 , of firms receive patents and δ = 0 . For k > , at any sequence of symmetric invsetment equilibria lim n q ∗ (0) n /k = (cid:18) k − b (cid:19) /k . Proof.
It is weakly dominant for patent rights choose q ∗ (1) = 1, and this action is thebest response at an investment equilibrium. We claim that at any sequence of symmetricinvestment equilibria, q ∗ (0) n → ∞ . The probability that all ideas in a given technology t including i are known to another firm is(1 − q ∗ (0) k ) bn + o (1) . This converges to zero whenever q ∗ (0) n /k →
0, so if q ∗ (0) n were bounded along any subse-quence then firm i could profitably deviate by increasing q i when n is large.Therefore, by the law of large numbers, for a firm i without patents choosing q i (0) whenother firms choose equilibrium actions p ∗− i and q ∗− i , the number of ideas learned from firmswithout patents is | I i (( p i , p ∗− i ) , ( q i , q ∗− i )) | ∼ p ∗ (0) q i q ∗ (0)(1 − b ) n. So the expected number of proprietary technologies for a firm without patents choosing p i and q i (0) is: E [ | P T i (( p i , p ∗− i ) , ( q i , q ∗− i )) | ] ∼ p i (0)( p ∗ (0)) k − (1 − q i (0) q ∗ (0) k − ) bn (cid:18) q i q ∗ (0)(1 − b ) nk − (cid:19) . Note that we use our explicit formula ι ( q i , q j ) = q i q j for the interaction rate here.We can approximate the binomial coefficient with its highest-order term, so taking thefirst-order condition and cancelling terms gives: bnq ∗ (0) k − ( q i q ∗ (0)(1 − b ) n ) ∼ ( k − − b ) nq ∗ (0)(1 − q i (0) q ∗ (0) k − ) . q i = q ∗ (0) and solving, q ∗ (0) ∼ (cid:18) b · k − n (cid:19) /k as claimed above.As a result, the interaction rate between firms without patents is ι ( q ∗ (0) , q ∗ (0)) ∼ (cid:18) b · k − n (cid:19) /k , which is lower order than the interaction rate in Proposition C1. The payoffs to firmswithout patents are therefore of lower order than in Proposition C1, where no patent rightsare granted.The interaction rate between a firm with a patent and a firm without a patent is ι ( q ∗ (0) , q ∗ (1)) ∼ (cid:18) b · k − n (cid:19) /k , which is the same order as the interaction rate in Proposition C1. The payoffs to firms withpatents are therefore of the same order as in Proposition C1, where no patent rights aregranted. D Firm Size
The baseline model assumed that each firm can discover a single idea. In this section, weconsider firms that can instead discover 1 < σ < k private ideas. A firm with size σ > σ > σ small firms. The assumption that σ < k is not essential, but rules out investment equilibria with no interaction: allfirms choose q i = 0 but private investment p i > Decisions about private investment may be different in these two cases, but this will not affect ouranalysis. { , . . . , n } but now allow multipleideas for each firm. Each of the σ ideas corresponding to firm i is discovered independentlywith probability p i . A firm learning i directly from j will learn all private ideas discoveredby firm j . The analysis from Sections 3 and 5, including Theorem 1 and Propositions 2 and4, extends easily to any firm size σ < k .We will now compare the payoffs of firms of two different sizes σ and σ (cid:48) . Suppose thereare fixed positive shares of firms of each size. We can think of the exercise as measuring thevalue of increasing firm size. Proposition D3.
If firm i can discover σ ideas and firm i (cid:48) can discover σ (cid:48) ideas, then atany sequence of investment equilibria: lim n →∞ U i ( p ∗ , q ∗ ) U i (cid:48) ( p ∗ , q ∗ ) = σσ (cid:48) . The proposition says that when payoffs are such that equilibrium is critical or super-critical, then small and large firms obtain the same payoffs per idea asymptotically. Animplication is that merging two separate firms would increase their profits by very little for n large.We can give intuition in the case σ = 1 and σ (cid:48) = 2. Two separate firms of size one caneach potentially produce technologies by combining their private idea with n − n − n − (cid:0) yk − (cid:1) is much larger than (cid:0) yk − (cid:1) for y large, the additionaltechnologies have a small impact on profits in large markets.When equilibrium is subcritical, as under the payoff structure of Proposition 4(ii), wehave lim n →∞ U i ( p ∗ , q ∗ ) U i (cid:48) ( p ∗ , q ∗ ) > σσ (cid:48) . In this case, because firm profits are bounded asymptotically, technologies using multipleprivate ideas will generate a non-vanishing share of a firm’s profits.81n important assumption in Proposition D3 is that σ and σ (cid:48) do not depend on n , so thatfirms are still small relative to the overall market. A firm that can discover a non-vanishingfraction of all ideas can obtain much higher payoffs per idea than small firms, as such a firmwill obtain payoffs of order n k − even without interacting with other firms. Proof of Proposition D3.
We can show that equilibrium is critical or supercritical by a modi-fication of the argument used to prove Theorem 1 and Proposition 4, which we now describe.A version of Lemma A2 still applies at any subcritical equilibrium. The statement andproof must be modified, as in the proof of the subcritical asymmetric case of Theorem 1,to accommodate heterogeneity in firms. Because σ < k , we must have τ ( t ) ≥ t .Because δq ∗ n is equal to the expectation of τ ( t ) with respect to a suitable distribution overtechnologies t , we cannot have a subcritical equilibrium.Therefore, we have lim n →∞ U i ( p ∗ , q ∗ ) = ∞ for firms i of either size.So for any integer y > (cid:15) >
0, we have E (cid:2) | P T i ( p ∗ , q ∗ ) | | I i ( p ∗ , q ∗ ) | >y (cid:3) ≥ (1 − (cid:15) ) E [ | P T i ( p ∗ , q ∗ ) | ]for n sufficiently large, where is the indicator function. That is, almost all of the profitsof firm i are generated in the event that firm i learns at least y ideas.Because lim y →∞ (cid:0) yk − l (cid:1)(cid:0) yk − (cid:1) = 0for all l >
1, in expectation at least a share 1 − (cid:15) of technologies in P T i ( p ∗ , q ∗ ) include onlyone private idea developed by firm i (of either size).Suppose σ < σ (cid:48) and fix firms i of size σ and i (cid:48) of size σ (cid:48) . The preceding facts imply thatby choosing ( p i , q i ) = ( p ∗ i (cid:48) , q ∗ i (cid:48) ), for any (cid:15) > i can guarantee E (cid:2) | P T i (( p i , p ∗− i ) , ( q i , q ∗− i ) | (cid:3) ≥ (1 − (cid:15) ) E [ | P T i (cid:48) ( p ∗ , q ∗ ) | ]for n sufficiently large. Here the first two factors of (1 − (cid:15) ) correspond to the share of propri-etary technologies including only one private idea developed by firm i (cid:48) , while we introducethe third because at least a share 1 − (cid:15) of proprietary technologies for firm i (cid:48) do not include82dea i . This implies the result.This section introduced heterogeneity in firm size. Our results can similarly accommodateheterogeneity in other parameters, such as the complexity k of products produced by a firmor the private investment cost c ( · ). E Public Innovators and Directed Interaction
We now show that the result of Proposition 1 continues to apply if firms can direct theirinteractions toward private firms or public innovators.As in Section 3.4, public innovator i pays investment cost c ( p i ) and receives a payoff ofone for each technology t such that: (1) i ∈ t and (2) j ∈ { i } ∪ I i ( p , q ) for all j ∈ t .All firms have the same incentives as in the baseline model. Public innovators and firmscan now choose two interaction rates q i and q i , where q i is the interaction rate with publicinnovators and q i is the interaction rate with private firms.We show payoffs again grow at the same rate as in the supercritical region, up to aconstant factor: Proposition E4.
Suppose a non-vanishing share of agents are public innovators. Thenthere exists a sequence of symmetric equilibria with non-vanishing investment, and at anysequence of equilibria with non-vanishing investment lim inf n U i ( p ∗ , q ∗ ) (cid:0) n − k − (cid:1) > for all firms i .Proof. Let b ( n ) be the share of public innovators for each n .We first show that lim inf n U i ( p ∗ , q ∗ ) (cid:0) n − k − (cid:1) > i at any sequence of equilibria with non-vanishing investment.It is weakly dominant and strictly preferred at any investment equilibrium for publicinnovators to choose q ∗ i = q ∗ i = 1 . Therefore, all public innovators are in the same component83f the learning network. Private investment p i by public innovators is non-vanishing, soasymptotically almost surely all firms in this component learn at least αn ideas for some α > O ( n k − ) by choosing q i = n and q i = 0.This is because then the probability that firm i learns indirectly from a public innovator andno firm j learns from i is non-vanishing, and the payoffs from this event are O ( n k − ). Thisshows the desired bound on U i ( p ∗ , q ∗ ) ( n − k − ) .It remains to show there exists a sequence of symmetric equilibria with non-vanishinginvestment. Suppose that all public innovators choose p ≥ and q i = q i = 1 and all firmsother than i choose ( p , q , q ) with p ≥ , δq n ≤ q = 0. If q i is the best responsefor i , then lim n q i n exists and is independent of p and ( p , q , q ) given the constraints inthe previous sentence. This is because the probability of interactions between i and otherfirms vanishes asymptotically, while the best response does not depend on the number ofideas learned by the unique giant component.Therefore, we can choose (cid:15) > q ∈ [ (cid:15)δn , , then so is an arbitrary firm i ’sbest response q i . Because all other private firms choose q j = 0, firm i is indifferent to allchoices of q i and in particular q i = 0 is a best response. We claim that for n large, given p , there exists q such that each each q i is a best response to ( p , p , q , q ) . This followsfrom Kakutani’s fixed point theorem as in the proof of Theorem 1. We call this choices ofopenness q ( p , p ).Given such ( p , p , q ( p , p )), each firm has a non-vanishing probability of learning alinear number of ideas. Therefore, E [ | I i ( p , q ) | ] → ∞ . So any best response p i for each publicinnovator has p i ≥ , and the same holds for each firm. By Kakutani’s fixed point theorem,there exists ( p , p , q ( p , p )) such that p ≥ and p ≥ are also best responses. Thusthere exists a sequence of equilibria with non-vanishing investment.84 Investment/Interaction Tradeoffs
Our main model focuses on a tradeoff between learning and secrecy. The same techniquesalso let us characterize a related model in which firms instead face tradeoffs between learningand investment, and must decide how to allocate resources between these two tasks. Inparticular, we now model the probability that firm i learns from firm j as depending onlyon firm i ’s action rather than depending symmetrically on firm i and firm j ’s actions.A firm i continues to choose actions p i and q i , now subject to the budget constraint that p i + λq i n = 1for some λ >
0. The constant λ determines the cost of an additional expected interaction interms of probability of discovering a private idea.Firm i learns directly from each firm j with probability q i . As in the main model, inthis case firm j learns indirectly through firm j with probability δ ∈ [0 , c ( p i ) to private investment. So U i ( p , q ) = E [ | P T i ( p , q ) | ] , where P T i ( p , q ) is the set of technologies for which firm i receives monopoly profits.The equilibrium characterization depends on the rate at which the firm can substitutebetween interaction and private investment: Theorem F1.
Suppose δ > . Any sequence of symmetric equilibria with positive payoffsis: (i) Subcritical if λ < δ ;(ii) Critical if λ = δ ; and(iii) Supercritical if λ > δ . In particular, this learning rate no longer depends on q j .
85s the opportunity cost of interaction decreases, the equilibrium level q ∗ increases. Thekey intuition is that, as in the baseline model, the marginal downside to additional interactionis proportional to the current profits. In the baseline model that downside comes frominteraction potentially facilitating competition, while now the downside comes from a lowerprobability of discovering a private idea.Exploiting the similar structure, we can derive a variant of Lemma 2: λq ∗ n ∼ p ∗ E t ∈ P T i ( p ∗ , q ∗ ) [ τ ( t )] . At the critical threshold, we have δq ∗ n → E t ∈ P T i ( p ∗ , q ∗ ) [ τ ( t )] →
1. The theorem follows from these facts and the budget constraint.One could also show as in the proof of Theorem 1 that there exists an equilibrium withpositive payoffs for n large. We omit the proof here. Proof.
We will show an analogue of Lemma A2 in this context.We claim that along any sequence of symmetric equilibria with positive payoffs and δqn < n , λ E (cid:20)(cid:18) | I i ( p ∗ , q ∗ ) | k − (cid:19)(cid:21) ∼ p ∗ E (cid:34) ∂ (cid:0) | I i ( p ∗ , ( q i ,q ∗− i ) | k − (cid:1) ∂q i ( q ∗ ) (cid:35) (18)for each i .We can argue as in the proof of Lemma A2 that since δqn <
1, competition from indirectlearning is lower order. Therefore, we can condition on the event that no firm j has learnedindirectly from firm i , which is independent of | I i ( p ∗ , q ∗ ) | .The left-hand side of equation (18) is λ times the benefit from a marginal increase inprivate investment p i . The right-hand side of equation (18) is the benefit from a marginalincrease in q i . By the budget constraint p i + λq i n = 1 , these are equal at any interiorequilibrium. Payoffs are zero at equilibria with p i = 0 or p i = 1 for any i .It follows as in the proof of Lemma 2 that along any sequence of symmetric equilibria86ith positive payoffs and δq ∗ n < n , λq ∗ n ∼ p ∗ E t ∈ P T i ( p ∗ , q ∗ ) [ τ ( t )] . At any symmetric subcritical equilibrium we must have E t ∈ P T i ( p ∗ , q ∗ ) [ τ ( t )] > . So we must have λq ∗ n < p ∗ for n large. Combining this inequality with the definition ofsubcritical equilibria, we conclude that λ < δ .Next, suppose δq ∗ n >
1. Then if α is the number of ideas learned by firms in the giantcomponent, the payoff to choosing q i is proportional to:(1 − λq i n )(1 − e − δq i αn )(1 − λq ∗ n ) k − (cid:18) αnk − (cid:19) + o ( n k − ) . Indeed, this is the payoff if no firm learns i ’s idea and all other ideas learned by the giantcomponent, and the probability this occurs is independent of the choice of q i .Taking the first-order condition, we have λ (1 − e − δq ∗ αn ) ∼ δα (1 − λq ∗ n ) e − δq ∗ αn or equivalently λδ ∼ α (1 − λq ∗ n ) · e − δq ∗ αn − e − δq ∗ αn . Because α ∼ − e − δq ∗ αn , this implies λδ ∼ (1 − λδ δq ∗ n )(1 − α ) . To have a solution with α >
0, and therefore to have a sequence of symmetric supercriticalequilibria, requires λ > δ .Finally, consider a symmetric sequence of investment equilibria. Lemma 3 continues to87pply, so we must have E t ∈ P T i ( p ∗ , q ∗ ) [ τ ( t )] →
1. As a consequence, E [ (cid:18) | I i (( p i , p ∗− , ( q i , q ∗− i )) | k − (cid:19) ] ∼ ( p ∗ ) k − q i n (cid:18) E (cid:20)(cid:18) Xk − (cid:19)(cid:21) + o (cid:18) E (cid:20)(cid:18) Xk − (cid:19)(cid:21)(cid:19)(cid:19) , where X is a random variable distributed as the number of ideas known to an arbitrary firm j . Thus, p i E [ (cid:0) | I i (( p i ,p ∗− , ( q i ,q ∗− i )) | k − (cid:1) ] is maximized by ( p ∗ , q ∗ ), subject to the budget constraint p ∗ + λq ∗ n ∼
1, when p ∗ ∼ . So we have δq ∗ n ∼ λ = δ ..