HOW A FIRM’S VALUE CAPTURE AFFECTS VALUE CREATION IN ITS ECOSYSTEM

企业的价值捕获如何影响其生态系统中的价值创造

KOSE JOHN New York University 高丝约翰 纽约大学

DAVID GADDIS ROSS University of Florida DAVID GADDIS ROSS 佛罗里达大学

Using a formal model, we develop a theory of how value capture by a firm with market or bargaining power dampens the incentives of other agents in its ecosystem, showing that the implications for value creation depend critically on the nature of the complementarities these agents generate. If complementarities are net positive, then dampening agents’ incentives can dramatically reduce the value created in the ecosystem, even to zero—that is, the prospect of a firm’s value capture may even prevent the firm itself from operating. Conversely, if complementarities are net negative, dampening agents’ incentives can increase value creation by discouraging agents from imposing these negative complementarities on each other. The disincentivizing effects of a firm’s value capture can be mitigated by enforceable contracts, cooperative governance, and self-organization by agents, but what is optimal for the agents or the firm may not be optimal for value creation in the ecosystem. 我们使用一个正式模型，构建了一个理论，探讨具有市场或议价能力的企业对价值的捕获如何削弱其生态系统中其他主体的激励，并表明价值创造的影响关键取决于这些主体产生的互补性的性质。如果互补性是净正向的，那么削弱主体的激励可能会大幅降低生态系统中创造的价值，甚至降至零——也就是说，企业捕获价值的前景甚至可能阻止企业自身运营。相反，如果互补性是净负向的，削弱主体的激励可以通过阻止主体相互施加这些负向互补性来增加价值创造。企业对价值的捕获所产生的抑制效应可以通过具有约束力的合同、合作治理以及主体的自我组织来缓解，但对主体或企业而言最优的方案可能并非生态系统中价值创造的最优方案。

Market and bargaining power allow a firm to capture value through the terms of trade. For economists, this has been a source of concern since The Wealth of Nations (Smith, 1776), given the inefficiency of “deadweight loss” that arises in the standard economic model of monopoly (Tirole, 1988). Strategy, by contrast, focuses on firm profit (Makadok, 2011). So, a key goal in classic strategy theory is developing market and bargaining power (Brandenburger & Stuart, 2007; Porter, 1979), and a rich body of literature has emerged on the antecedents of such power by (a) finding a “blue ocean” market niche, (b) building competitive strength," (c) “restraining rivalry,” or (d) acquiring, developing, and leveraging valuable, rare, inimitable, and nonsubstitutable resources (e.g., Adegbesan, 2009; Coen & Maritan, 2011; Ghemawat & 市场和议价能力使企业能够通过贸易条款获取价值。自《国富论》（Smith, 1776）以来，经济学家一直对此表示担忧，因为标准的垄断经济模型中会产生“无谓损失”的低效率问题（Tirole, 1988）。相比之下，战略研究关注企业利润（Makadok, 2011）。因此，经典战略理论的一个关键目标是构建市场和议价能力（Brandenburger & Stuart, 2007; Porter, 1979），并且通过以下途径已形成了大量关于此类能力前因的文献：（a）发现“蓝海”市场利基，（b）建立竞争优势，（c）“抑制竞争”，或（d）获取、开发和利用有价值、稀缺、难以模仿且不可替代的资源（例如，Adegbesan, 2009; Coen & Maritan, 2011; Ghemawat &

Thomas, 2008; Gimeno & Woo, 1999; Kim & Mauborgne, 2005; Makadok, 2010; Makadok & Ross, 2013; Schmidt & Keil, 2013). 托马斯，2008；吉梅诺和吴，1999；金和莫布格尼，2005；马卡多克，2010；马卡多克和罗斯，2013；施密特和凯尔，2013）。

Yet, strategy theorists have recently become aware that many dominant firms, like Walmart or Microsoft, owe their success to their ecosystems—that is, the suppliers, buyers, and complementors with which they cocreate value (Iansiti & Levien, 2004) and whose activity generates complementarities (i.e., spillovers or externalities) for the whole system (Jacobides, Cennamo, & Gawer, 2018). This codependence creates a tension for ecosystem governance. On the one hand, in accordance with classic strategy theory, it is critical to obtain “market power” and “bargaining power” (Hannah & Eisenhardt, 2018) by capturing an industry “bottleneck” (Jacobides, Knudsen, & Augier, 2006), and to be “tough” with partners (Iansiti & Levien, 2004) so as to capture as much value as possible. On the other hand, in order to ensure that the ecosystem creates as much value as possible under conditions of asset specificity (Williamson, 1985), it is equally critical to align the incentives of all ecosystem members by sharing value (Teece, 2018; Wareham, Fox, & Giner, 2014; Williamson & De Meyer, 2012). Thus, understanding how value capture by a lead firm with market and bargaining power affects the incentives of other ecosystem members to cocreate value and the total value the ecosystem creates is pivotal. 然而，战略理论家最近意识到，许多主导企业（如沃尔玛或微软）的成功归功于其生态系统——即与其共同创造价值的供应商、买家和互补者（Iansiti & Levien，2004），而这些主体的活动为整个系统产生互补性（即溢出效应或外部性）（Jacobides, Cennamo, & Gawer，2018）。这种相互依存关系给生态系统治理带来了张力。一方面，根据经典战略理论，通过捕捉行业“瓶颈”（Jacobides, Knudsen, & Augier，2006）获得“市场权力”和“议价能力”（Hannah & Eisenhardt，2018），并对合作伙伴保持“强硬”态度（Iansiti & Levien，2004）以尽可能多地攫取价值，这一点至关重要。另一方面，为确保生态系统在资产专用性条件下（Williamson，1985）创造最大价值，同样关键的是通过价值共享协调所有生态系统成员的激励机制（Teece，2018；Wareham, Fox, & Giner，2014；Williamson & De Meyer，2012）。因此，理解具有市场和议价能力的领先企业的价值攫取行为如何影响其他生态系统成员的共同创造价值激励，以及生态系统创造的总价值，是至关重要的。

However, despite recognizing these coopetitive tensions in ecosystems (e.g., if lead firms enter complementor niches [Gawer & Henderson, 2007; Zhu & Liu, 2018]), the liteatue has not ffee - orous formal analysis of this fundamental question. The closest work is a stream of formal models on optimal pricing, royalties, and intellectual property in platform ecosystems (Panico & Cennamo, 2020; Parker & van Alstyne, 2018; Parker, van Alstyne, & Jiang, 2017), which treated the lead firm as principal in a mechanism design problem, effectively fixing its market and bargaining power at a high level and managing incentives through enforceable, potentially detailed contracts. However, such precise contracting is not generally available in ecosystems, because members are hierarchically independent and may not transact with or even know each other (Jacobides et al., 2018; Thomas & Autio, 2020; Williamson & De Meyer, 2012). With the exception of Panico and Cennamo (2020), moreover, this formal work did not model members’ incentives to invest in value cocreation. In addition, although the ecosystem literature has emphasized positive complementarities such as knowledge and innovation spillovers (Fjeldstad, Snow, Miles, & Lettl, 2012; Rosenberg, 1982), negative complementarities from wasteful advertising (Grossman & Shapiro, 1984; Mankiw & Whinston, 1986), depletion of resources (Ostrom, 2009), or the generation of “bads” such as pollution (Weitzman, 1974) have been neglected. 然而，尽管人们认识到生态系统中的这种竞争性紧张关系（例如，主导企业进入互补者细分市场[Gawer & Henderson, 2007; Zhu & Liu, 2018]），但文献中尚未对这一基本问题进行自由、系统性的形式化分析。最接近的研究是关于平台生态系统中最优定价、特许权使用费和知识产权的一系列形式化模型（Panico & Cennamo, 2020; Parker & van Alstyne, 2018; Parker, van Alstyne, & Jiang, 2017），这些模型将主导企业视为机制设计问题中的委托人，实际上将其市场和议价能力固定在较高水平，并通过可执行的、可能非常详细的合同来管理激励。然而，在生态系统中，这种精确的契约通常是不可行的，因为成员是层级独立的，可能不会进行交易，甚至彼此不了解（Jacobides et al., 2018; Thomas & Autio, 2020; Williamson & De Meyer, 2012）。此外，除了Panico和Cennamo（2020）之外，这些形式化研究并未对成员在价值共创中的投资激励进行建模。另外，尽管生态系统文献强调了知识和创新溢出等积极互补性（Fjeldstad, Snow, Miles, & Lettl, 2012; Rosenberg, 1982），但诸如浪费性广告（Grossman & Shapiro, 1984; Mankiw & Whinston, 1986）、资源枯竭（Ostrom, 2009）或污染等“不良品”的产生等负面互补性却被忽视了（Weitzman, 1974）。

We address this gap in the literature with a formal analysis of how value capture by a lead firm with market and bargaining power affects value creation in its ecosystem. Specifically, we develop a biform model (Brandenburger & Stuart, 2007) of an ecosystem comprising a community of economic agents (firms and individuals) that undertake costly effort to cocreate value with a lead firm, whose market and bargaining power may vary, where complementarities may be either net positive or net negative, and where contracting may not be available, so that value creation and value capture are determined in unrestricted bargaining or using other coordination devices. We show that value capture by a lead firm with market and bargaining power dampens the incentives of the agents to exert costly effort, so they do less of it. If complementarities are net positive, the reduction in agent effort leads to a vicious circle. Each agent not only exerts less effort in the first instance but also anticipates that other agents will exert less effort too, prompting the agent to reduce effort even further. The result is a severe reduction in value creation, which may even suffocate the ecosystem out of existence. Conversely, if complementarities are net negative, value capture by a lead firm with market or bargaining power prompts agents to curtail their activity on the margin, leading to greater value creation overall. We believe we are the first to identify this potential benefit of market and bargaining power. 我们通过对具有市场和议价能力的主导企业如何影响其生态系统中价值创造的正式分析，来填补这一文献空白。具体而言，我们构建了一个双形态模型（Brandenburger & Stuart，2007），该模型中的生态系统由经济主体（企业和个人）组成，这些主体会为与主导企业共同创造价值而付出成本高昂的努力，而主导企业的市场和议价能力可能各不相同，互补性可能是净正或净负，且可能不存在契约安排，因此价值创造和价值捕获由不受限制的议价或其他协调机制决定。我们发现，具有市场和议价能力的主导企业对价值的捕获会削弱经济主体付出成本高昂努力的积极性，导致他们投入更少的努力。如果互补性为净正，那么经济主体努力的减少会形成恶性循环：每个主体不仅最初会减少努力，还会预期其他主体也会减少努力，从而促使该主体进一步减少努力。最终，价值创造会大幅减少，甚至可能导致生态系统消亡。相反，如果互补性为净负，具有市场或议价能力的主导企业对价值的捕获会促使经济主体在边际上缩减活动，从而总体上带来更大的价值创造。我们认为，我们是首个识别出市场和议价能力潜在益处的研究。

Then, in a modeling innovation that may be of independent interest to formal theorists in strategy, we show that a contract that inverts the standard order of the cooperative and noncooperative stages of the biform game may be a solution to the frictions associated with value capture by the lead firm: The lead firm and agents first bargain over each party’s ultimate value capture and then set the agents’ marginal value capture such that they noncooperatively exert the effort that maximizes value creation in the ecosystem. However, with positive complementarities the contract requires agents to post a bond, which may often not be feasible. 然后，在一项可能对策略领域的形式理论学者具有独立研究价值的建模创新中，我们表明，一种颠倒双形式博弈中合作阶段与非合作阶段标准顺序的契约，可能是解决领先企业价值攫取相关摩擦的方案：领先企业与代理方首先就各方最终的价值攫取进行谈判，然后设定代理方的边际价值攫取，以使他们非合作地付出能最大化生态系统价值创造的努力。然而，在存在正互补性的情况下，该契约要求代理方缴纳保证金，而这在很多时候可能不可行。

As an alternative to contracting, we consider the possibility that the lead firm is a cooperative in which value is distributed to agents in proportion to their cocreation of value with the lead firm (Cuevas & Fischer, 2006; Hansmann, 1996; McBride, 1986). This standard governance feature of cooperatives nullifies the disincentivizing effects of the lead firm’s value capture because agents anticipate both (a) that the lead firm captures some of the value they cocreate with it and (b) that the claim they have on the lead firm’s value capture is increasing in their contribution to value cocreation. These two effects offset each other exactly. To our knowledge, this motivation for cooperative governance has not been contemplated in previous research. As another alternative to contracting, agents may be able to selforganize on the effort that maximizes their own value capture. We find that agents always benefit from self-organization, but it unambiguously increases total value creation only if complementarities are positive. 作为合同的替代方案，我们考虑牵头企业是合作社的可能性，在这种合作社中，价值根据代理机构与牵头企业共同创造的价值比例分配给代理机构（Cuevas & Fischer, 2006; Hansmann, 1996; McBride, 1986）。合作社的这一标准治理特征消除了牵头企业攫取价值的消极激励效应，因为代理机构既预期（a）牵头企业会攫取他们与自身共同创造的部分价值，又预期（b）他们对牵头企业价值攫取的诉求会随着其对价值共同创造的贡献增加而增加。这两种效应恰好相互抵消。据我们所知，合作社治理的这种动机在以往研究中尚未被考虑。作为合同的另一种替代方案，代理机构可能能够在最大化自身价值攫取的努力上进行自我组织。我们发现，代理机构始终能从自我组织中获益，但只有当互补性为正时，自我组织才会明确地增加总价值创造。

Finally, an ecosystem is ultimately a collective enterprise (Jacobides et al., 2018), so we follow Chatain and Plaksenkova (2019) in focusing on total value created, a formalization of what Mahoney, McGahan, and Pitelis (2009) called “global sustainable value creation.” However, we also address the question posed by Jacobides et al. (2006: 1206): “Given scarce resources, does it make sense to keep the biggest part of a potentially shrinking pie, or a modest part of a growing pie?” Our answer is that the lead firm may capture more value with a bigger share of a smaller pie than a smaller share of the biggest pie. Thus, our analysis has implications for public policy. 最后，生态系统归根结底是一项集体事业（Jacobides等人，2018），因此我们遵循Chatain和Plaksenkova（2019）的研究，聚焦于创造的总价值，这是Mahoney、McGahan和Pitelis（2009）所谓的“全球可持续价值创造”的一种形式化表述。然而，我们也回应了Jacobides等人（2006：1206）提出的问题：“在资源稀缺的情况下，是保留一个可能正在缩小的‘大蛋糕’的最大部分更有意义，还是保留一个正在增长的‘大蛋糕’的较小部分更有意义？”我们的答案是，领先企业可能通过在一个较小的蛋糕中占据更大的份额，比在一个最大的蛋糕中占据较小的份额获得更多的价值。因此，我们的分析对公共政策具有启示意义。

THEORETICAL BACKGROUNDAND POSITIONING

理论背景与定位

Our theory focuses on a business ecosystem—that is, “an economic community supported by a foundation of interacting organizations and individuals” (Moore, 1996: 26), with a lead firm (also called keystone, architect, core, focal actor, leader, captain, orchestrator, or hub) and other, usually smaller, firms and individuals (agents) who cocreate value with the lead firm and each other (Adner, 2017; Jacobides et al., 2018; Jacobides, MacDuffie, & Tae, 2016).1 我们的理论聚焦于商业生态系统——即“由相互作用的组织和个人构成基础支撑的经济共同体”（Moore, 1996: 26），其中包含一家核心企业（也被称为关键企业、架构者、核心企业、焦点行为者、领导者、主导者、协调者或枢纽），以及其他通常规模较小的企业和个人（代理人），他们与核心企业及彼此共同创造价值（Adner, 2017；Jacobides et al., 2018；Jacobides, MacDuffie, & Tae, 2016）。1

Cocreation of value makes ecosystem governance fundamentally coopetitive (Brandenburger & Nalebuff, 1996; Hoffman, Lavie, Reuer, & Shipilov, 2018; Lado, Boyd, & Hanlon, 1997). On the one hand, in line with traditional strategy theory that emphasizes maximizing a firm’s market and bargaining power with trading partners (Porter, 1979), lead firms are enjoined to seize a “bottleneck” in the industry architecture (Dattée, Alexy, & Autio, 2018; Jacobides et al., 2006; Jacobides & Tae, 2015), and then be “tough” with business partners (Iansiti & Levien, 2004) in order to capture as much value as possible (Dedrick, Kraemer, & Linden, 2009; Hannah & Eisenhardt, 2018). 价值共创使生态系统治理从根本上具有竞争合作性（Brandenburger & Nalebuff，1996；Hoffman，Lavie，Reuer，& Shipilov，2018；Lado，Boyd，& Hanlon，1997）。一方面，与强调企业最大化与贸易伙伴的市场和议价能力的传统战略理论一致（Porter，1979），领先企业被要求抓住行业架构中的“瓶颈”（Dattée，Alexy，& Autio，2018；Jacobides et al.，2006；Jacobides & Tae，2015），然后对业务伙伴“强硬”（Iansiti & Levien，2004），以尽可能多地获取价值（Dedrick，Kraemer，& Linden，2009；Hannah & Eisenhardt，2018）。

On the other hand, ecosystem theory also holds that the “appropriability regime” must incentivize members to invest in the cocreation of value (Dhanarj & Parkhe, 2006; Helfat & Raubitschek, 2018; Nambisian & Sawhney, 2011) in the face of a classic problem of “asset specificity”: If a lead firm’s ecosystem partners make cospecialized investments in the ecosystem without safeguards, the lead firm may opportunistically use its market and bargaining power to expropriate the value these investments create (Williamson, 1985). 另一方面，生态系统理论还认为，“可占有性机制”必须激励成员在面对“资产专用性”这一经典问题时，投资于价值共创（Dhanarj & Parkhe，2006；Helfat & Raubitschek，2018；Nambisian & Sawhney，2011）。如果核心企业的生态系统合作伙伴在没有保障措施的情况下对生态系统进行专门化投资，核心企业可能会利用其市场和议价能力机会主义地侵占这些投资所创造的价值（Williamson，1985）。

Contracts are a potential solution to asset specificity, but neither detailed contracts nor hierarchical control are generally available in ecosystems because of the absence of direct transactional relationships or efficient intellectual property enforcement throughout the system (Jacobides et al., 2018; Teece, 2018; Williamson & De Meyer, 2012). Indeed, a prevalence of contracts or formal interorganizational relationships is said to make an economic community a supply chain or a network (e.g., Shipilov & Gawer, 2020; 合同是解决资产专用性的潜在方案，但由于整个系统中缺乏直接的交易关系或有效的知识产权保护，生态系统中通常既没有详细的合同，也没有层级控制（Jacobides等人，2018；Teece，2018；Williamson & De Meyer，2012）。事实上，合同或正式的组织间关系的普遍存在被认为会使经济共同体成为供应链或网络（例如，Shipilov & Gawer，2020；

Thomas & Autio, 2020). Moreover, although relational governance via social systems, trust, and norms may replace formal contracts (Jones, Hesterly, & Borgatti, 1997), evidence suggests that relational governance is more a complement than a substitute for formal contracts (Poppo & Zenger, 2002; Ryall & Sampson, 2009). It is thus important to understand value creation and value capture in ecosystems using noncontractual arrangements, such as unrestricted bargaining, the governance form of the lead firm, or agent self-organization, the last of which has been observed where actors value nonpecuniary rewards (e.g., open source software) (Lee & Cole, 2003; von Hippel & von Krogh, 2003). 托马斯和奥蒂奥，2020）。此外，尽管通过社会系统、信任和规范进行的关系治理可能会取代正式契约（琼斯、赫斯特利和博加蒂，1997），但有证据表明，关系治理更多的是正式契约的补充而非替代（波普和赞格，2002；赖尔和桑普森，2009）。因此，理解生态系统中使用非契约安排（如无限制议价、核心企业的治理形式或代理自组织）的价值创造和价值捕获非常重要，其中最后一种形式在参与者重视非货币奖励的情况下已被观察到（例如开源软件）（李和科尔，2003；冯·希普尔和冯·克罗格，2003）。

Another complication is that agents in an ecosystem generate complementarities for the system. The ecosystem literature has extensively discussed positive complementarities such as economies of scale in consumption or production (Thomas & Autio, 2020), cospecialized components (Jacobides et al., 2018), knowledge spillovers (Fjeldstad et al., 2012), and complementary innovations (Rosenberg, 1982). However, complementarities may also be negative. As shown in industrial organization economics, because firms enter markets, develop new products, and advertise with only their own profits in mind, they generate negative complementarities by “stealing business” from other firms (Grossman & Helpman, 1991; Grossman & Shapiro, 1984; Mankiw & Whinston, 1986) whose effort and expense to compete for customers or patents is wasted (Grossmann, 2007; Roy, 1997). Likewise, from the standpoint of choosing an appropriate tax or quota in the face of uncertainty, natural resource economics has studied negative complementarities such as resource depletion and congestion (Ostrom, 2009; Ryan, Holland, & Herrera, 2014), as well as the generation of public “bads” like chemical waste or air pollution (Weitzman, 1974). Granted, Schumpeterian “creative destruction” involves displacement of incumbents even if it is a net positive for society (Schumpeter, 1934), but the complementarities we contemplate here are net negative for other ecosystem players: Advertising may shift business from one firm to another without increasing value creation (Grossman & Shapiro, 1984), pollution interferes with businesses that depend on clean air and water, and undifferentiated entry may only serve to reduce ecosystem profits, for example, if firms compete à la Cournot (1838). 另一个复杂性是，生态系统中的主体会为系统产生互补性。生态系统文献广泛讨论了积极的互补性，例如消费或生产中的规模经济（Thomas & Autio, 2020）、共同专业化的组件（Jacobides et al., 2018）、知识溢出（Fjeldstad et al., 2012）以及互补性创新（Rosenberg, 1982）。然而，互补性也可能是负面的。正如产业组织经济学所表明的，由于企业进入市场、开发新产品并仅考虑自身利润进行广告宣传，它们会通过“抢走其他企业的业务”产生负面互补性（Grossman & Helpman, 1991；Grossman & Shapiro, 1984；Mankiw & Whinston, 1986），而这些企业为争夺客户或专利所付出的努力和成本是浪费的（Grossmann, 2007；Roy, 1997）。同样，从面对不确定性选择适当税收或配额的角度来看，自然资源经济学研究了负面互补性，如资源枯竭和拥堵（Ostrom, 2009；Ryan, Holland, & Herrera, 2014），以及化学废物或空气污染等公共“坏事”的产生（Weitzman, 1974）。诚然，熊彼特式的“创造性破坏”涉及对在位者的替代，即使这对社会是净积极的影响（Schumpeter, 1934），但我们这里所考虑的互补性对其他生态系统参与者是净负面的：广告可能会将业务从一家企业转移到另一家，而不会增加价值创造（Grossman & Shapiro, 1984）；污染会干扰依赖清洁空气和水的企业；以及无差异的进入可能只会降低生态系统的利润，例如，如果企业进行古诺式（1838）竞争。

There is thus a clear theoretical and practical need to understand how value capture by a lead firm with market and bargaining power affects the cocreation of value in its ecosystem, where contracting may be unavailable and complementarities may be positive or negative. However, the literature has yet to offer a rigorous formal analysis of this question, despite promising work on standards-setting (Ranganathan, Ghosh, & Rosenkopf, 2020; Simcoe, 2014), access (Miller & Toh, 2020; O’Mahony & Karp, 2020; Zhang, Li, & Tong, 2020), and information disclosure (Alexy, George, & Salter, 2013; Alexy, West, Klapper, & Reitzig, 2018), as well as the disincentivizing impact of lead firms entering the markets of complementors (Ceccagnoli, Forman, Huang, & Wu, 2012; Farrell & Katz, 2000; Gawer & Henderson, 2007; Pierce, 2009; Zhu & Liu, 2018). 因此，明确存在理解领先企业（凭借市场和议价能力）的价值捕获如何影响其生态系统中价值共创的理论和实践需求，因为在该生态系统中，契约可能不可用，互补性可能为正或负。然而，尽管在标准制定（Ranganathan、Ghosh & Rosenkopf，2020；Simcoe，2014）、准入（Miller & Toh，2020；O’Mahony & Karp，2020；Zhang、Li & Tong，2020）、信息披露（Alexy、George & Salter，2013；Alexy、West、Klapper & Reitzig，2018）以及领先企业进入互补者市场的抑制影响（Ceccagnoli、Forman、Huang & Wu，2012；Farrell & Katz，2000；Gawer & Henderson，2007；Pierce，2009；Zhu & Liu，2018）方面已有有前景的研究，但该问题尚未得到严格的形式化分析。

The work that comes closest to addressing our research question is an emergent stream of formal models of platform ecosystems. Parker and van Alstyne (2018) and Parker et al. (2017) developed a model in which external developers are able to generate more complementary innovations the greater their access to the platform and the longer they retain intellectual property protection for their innovations. The platform sponsor captures a fixed share of the value from developers’ innovations. Developer incentives are not modeled. These models yield important insights about the sponsor’s conflicting incentives: The sponsor increases value creation by making the platform more accessible and extending intellectual property protection, but increases its value capture by restricting access and bundling innovations into the platform more quickly. 最接近解决我们研究问题的研究是平台生态系统正式模型的新兴研究流。Parker和van Alstyne（2018）以及Parker等人（2017）开发了一个模型，其中外部开发者对平台的访问程度越高、对其创新的知识产权保护期限越长，就越能产生更多互补性创新。平台发起方从开发者的创新中获取固定比例的价值。模型未对开发者的激励进行建模。这些模型揭示了发起方相互冲突的激励机制的重要见解：发起方通过提高平台可访问性和延长知识产权保护来增加价值创造，但通过限制访问并更快地将创新捆绑到平台中以增加其价值捕获。

In the platform model of Panico and Cennamo (2020), end-user demand grows via the number of complementors and their innovativeness, the latter of which, importantly, is growing through complementors’ endogenous level of investment. This model offers a number of important insights on the platform sponsor’s optimal mix of pricing to end users and royalty rates to complementors, how complementors respond by choosing their prices to end users and their investment in innovativeness, and endogenous platform size given a fixed cost of complementor entry. 在Panico和Cennamo（2020）的平台模型中，终端用户需求通过互补者的数量及其创新性增长，而后者（创新性）至关重要地通过互补者的内生投资水平得以提升。该模型提供了关于平台发起方对终端用户定价与对互补者版税费率的最优组合、互补者如何通过选择终端用户价格和在创新性方面的投资做出回应，以及在互补者进入固定成本下内生平台规模的诸多重要见解。

We have learned much from this emergent stream of formal theory, but by focusing on optimal contracting and pricing this work did not address the implications of a lead firm’s market or bargaining power for value creation and value capture in the common situation where the lead firm’s inability to make binding, detailed, take-it-or-leave-it offers with a significant number of ecosystem partners leads to unrestricted bargaining. Neither did this work consider alternative coordination mechanisms, such as self-organization among the lead firm’s partners or changing its governance form. With the important exception of Panico and Cennamo (2020), the incentive of a lead firm’s partners to make costly investments in the cocreation of value has not been modeled. In addition, although negative complementarities have implicitly arisen in some of this work, they have not been a focus of attention or parameterization, being limited to a reduction in market share and pricing power for ecosystem players as their number increases, much as verbal theories of ecosystems have discussed limiting ecosystem access to avoid excess competition (e.g., Wareham et al., 2014). Thus, the literature has not focused on negative complementarities, especially with respect to the lead firm’s market and bargaining power. The purpose of this paper is to address these gaps in our understanding with a formal model that incorporates and foregrounds these missing elements. 我们从这一新兴的正式理论流中学到了很多，但通过专注于最优契约和定价，这项研究并未探讨在主导企业无法与大量生态系统合作伙伴达成具有约束力、详细且“要么接受要么放弃”的报价的常见情况下，主导企业的市场或议价能力对价值创造和价值捕获的影响，这种情况会导致无限制的议价。该研究也未考虑替代协调机制，例如主导企业合作伙伴之间的自组织或改变其治理形式。除了Panico和Cennamo（2020）的重要例外情况外，主导企业合作伙伴为共同创造价值进行高成本投资的动机尚未得到建模。此外，尽管一些研究中隐含出现了负互补性，但它们并未成为关注或参数化的焦点，仅限于随着生态系统参与者数量增加，其市场份额和定价权的降低，这与生态系统的口头理论中讨论的限制生态系统准入以避免过度竞争类似（例如Wareham等人，2014）。因此，文献中并未聚焦于负互补性，尤其是关于主导企业的市场和议价能力。本文的目的是通过一个正式模型来填补我们理解中的这些空白，该模型纳入并突出了这些缺失的要素。

Our paper is also related to other literatures. Our model is a biform game (Brandenburger & Stuart, 2007), which has cooperative and noncooperative stages. The noncooperative stage is formally similar to some macroeconomics models that study the potential for multiple equilibria (for a survey of this literature, see Matsuyama, 1995). Our paper differs from this research in several ways, including our focus on the value capture and governance of a lead firm. 我们的论文还与其他文献相关。我们的模型是一种双态博弈（Brandenburger & Stuart, 2007），它包含合作阶段和非合作阶段。非合作阶段在形式上与一些研究多重均衡可能性的宏观经济学模型相似（关于这一文献的综述，参见 Matsuyama, 1995）。我们的论文与这项研究在几个方面存在差异，包括我们对领先企业的价值捕获和治理的关注。

Agents’ failure to internalize complementarities is a form of agency behavior, which has been the subject of a rich formal body of literature on optimal contracting. Some work has studied vertical contracting (for a review, see Mookherjee, 2006; for an example in strategy, see Obloj & Zemsky, 2015). Other work has studied coordination and multitasking (e.g., Makadok & Coff, 2009; Puranam & Kretschmer, 2008; Ross, 2012, 2014). Especially relevant is the formal literature on value cocreation in the context of coinvestment (Jia, 2013), innovation (Fulghieri & Sevilir, 2011; Gambardella & Panico, 2014; Gambardella, Panico, & Valentini, 2015), alliances (Arslan, 2018; Panico, 2017), and relations with nonmarket actors (Chatain & Plaksenkova, 2019), as well as empirical work on how complementarities and bargaining power in dyadic and multilateral relationships determine who creates value with whom, and how much each party captures (Adegbesan & Higgins, 2011; Chatain & Mindruta, 2017; Mindruta, 2013; Mindruta, Moeen, & Agarwal, 2016). Unlike this work, our model includes several critical features for our research question, including the coordination problem involving a lead firm and agents in an ecosystem, how value capture by a lead firm with market or bargaining power affects this problem depending on the nature of complementarities, and noncontractual remedies in an ecosystem context. 代理人未能内化互补性是一种代理行为，这一主题在关于最优契约的丰富正式文献中已有研究。一些研究关注纵向契约（相关综述见Mookherjee, 2006；战略领域的例子见Obloj & Zemsky, 2015）。其他研究则探讨了协调与多任务处理（例如Makadok & Coff, 2009；Puranam & Kretschmer, 2008；Ross, 2012, 2014）。特别相关的是关于共同投资（Jia, 2013）、创新（Fulghieri & Sevilir, 2011；Gambardella & Panico, 2014；Gambardella, Panico, & Valentini, 2015）、联盟（Arslan, 2018；Panico, 2017）以及与非市场参与者关系（Chatain & Plaksenkova, 2019）背景下价值共创的正式文献，还有关于二元和多边关系中互补性与议价能力如何决定谁与谁创造价值以及各方捕获多少价值的实证研究（Adegbesan & Higgins, 2011；Chatain & Mindruta, 2017；Mindruta, 2013；Mindruta, Moeen, & Agarwal, 2016）。与这些研究不同，我们的模型针对研究问题纳入了几个关键特征，包括涉及生态系统中牵头企业与代理人的协调问题、拥有市场或议价能力的牵头企业的价值捕获如何根据互补性的性质影响这一问题，以及生态系统背景下的非契约性补救措施。

We respond to calls for research on alternatives to the shareholder-owned for-profit firm (Luo & Kaul, 我们回应了关于股东所有制营利性公司替代方案的研究呼吁（罗&考尔，

2019; Mahoney et al., 2009) by considering cooperatives, which have been neglected in the governance literature (Ménard, 2004) despite there being over 1,150 cooperatives worldwide with over $$ 100$ million in annual revenue (World Co-Operative Monitor, 2018)—including Crédit Agricole, a bank network with over $\mathtt { \ t 1 . 7 }$ trillion in assets, and Ace Hardware, a retailer with over 5,200 stores. Cooperatives, whose owners are their transaction partners (e.g., suppliers or buyers), may be formed because of communal values (Boone & Özcan, 2014; Rao & Greve, 2018), tax and regulatory benefits, or common preferences among a class of stakeholders (Hansmann, 1996), as well as to negotiate with large buyers (Sykuta & Cook, 2001) or to align worker incentives (Alchian & Demsetz, 1972). 2019; Mahoney et al., 2009) by considering cooperatives, which have been neglected in the governance literature (Ménard, 2004) despite there being over 1,150 cooperatives worldwide with over $$ 100$ million in annual revenue (World Co-Operative Monitor, 2018)—including Crédit Agricole, a bank network with over $\mathtt { \ t 1 . 7 }$ trillion in assets, and Ace Hardware, a retailer with over 5,200 stores. Cooperatives, whose owners are their transaction partners (e.g., suppliers or buyers), may be formed because of communal values (Boone & Özcan, 2014; Rao & Greve, 2018), tax and regulatory benefits, or common preferences among a class of stakeholders (Hansmann, 1996), as well as to negotiate with large buyers (Sykuta & Cook, 2001) or to align worker incentives (Alchian & Demsetz, 1972).

We do not provide a complete treatment of cooperatives, but zero in on a canonical governance feature of cooperatives that is relevant to some ecosystems, yet has received little analysis in the literature (Cuevas & Fischer, 2006; Hansmann, 1996; McBride, 1986): the allocation of residual cash flow rights in proportion to transaction volume (value cocreation). We show that this feature eliminates some effects of monopolistic pricing even without eliminating monopolistic pricing itself, offering a novel explanation for the cooperative form. 我们并未全面论述合作社，而是聚焦于合作社的一个典型治理特征——其与某些生态系统相关，但在文献中却鲜少得到分析（Cuevas & Fischer, 2006；Hansmann, 1996；McBride, 1986）：即按交易量（价值共创）分配剩余现金流权。我们发现，这一特征即便不消除垄断定价本身，也能消除其部分影响，从而为合作社形式提供了一种新颖的解释。

In classic strategy theory, bargaining power arises from market power: “Buyer bargaining power is greater when there are only a few large buyers than when there are many smaller ones. Similarly, a seller’s bargaining power is higher in a monopoly or oligopoly situation than in a highly competitive market” (Dedrick et al., 2009: 86). By contrast, we adopt the terminology of the value-based theory underlying our biform model: Market power arises from a firm facing less competition for its role in cocreating value, and bargaining power arises from “persuasive resources” or bargaining skill (Gans & Ryall, 2017). 在经典战略理论中，议价能力源于市场力量：“当只有少数几个大买家时，买方的议价能力比有许多小买家时更强。同样，在垄断或寡头垄断情况下，卖方的议价能力比在高度竞争的市场中更高”（Dedrick et al.，2009：86）。相比之下，我们采用基于价值理论的术语，该理论是我们的双形式模型的基础：市场力量源于企业在共同创造价值的角色中面临的竞争较少，而议价能力源于“有说服力的资源”或议价技巧（Gans & Ryall，2017）。

In transaction cost economics, integration is a solution to asset specificity (Coase, 1937; Williamson, 1985), especially where contracting hazards are high, such as across borders (Pitelis & Teece, 2010). Likewise, resource dependence theory focuses on how organizational survival relies on the resources of external parties (Pfeffer & Salancik, 1978). The agents and lead firm are mutually dependent, so the lead firm may employ “power use operations” against the agents, who would employ “operations to restructure dependencies,” including merging with the lead firm (Adegbesan & Higgins, 2011). Yet, firm integration is not fully dispositive in our setting. Detailed contracts and command-and-control in a single firm are not always feasible (Makadok & Coff, 2009) because outputs are hard to specify (Prendergast, 1999), and coordinating development is not necessarily more difficult between firms than inside a single firm (Gerwin, 2004). Thus, an ecosystem may exist inside a single firm—for example, as relates to production and service in a manufacturer or sales and service in an automobile dealership (Adner, 2017). 在交易成本经济学中，整合是解决资产专用性的一种方案（Coase, 1937；Williamson, 1985），尤其是在契约风险较高的情境下，比如跨国交易（Pitelis & Teece, 2010）。同样，资源依赖理论关注组织的生存如何依赖外部方的资源（Pfeffer & Salancik, 1978）。代理方与核心企业相互依赖，因此核心企业可能会对代理方采取“权力使用行动”，而代理方则会采取“重构依赖关系的行动”，包括与核心企业合并（Adegbesan & Higgins, 2011）。然而，在我们的研究情境中，企业整合并非完全具有决定性作用。单一企业中的详细契约和命令控制并不总是可行的（Makadok & Coff, 2009），因为产出难以明确界定（Prendergast, 1999），而且企业间协调开发的难度未必高于单一企业内部（Gerwin, 2004）。因此，单一企业内部也可能存在一个生态系统——例如，制造商的生产与服务、汽车经销商的销售与服务（Adner, 2017）。

MODEL

模型

Premise

前提

We follow Chatain and Plaksenkova (2019) by starting with a benchmark model, and then develop our theory by modifying this initial model. The motivation for this is that although our initial results might be amenable to verbal argument, our subsequent results are more distant from existing theory and in some cases more technically complex. The base model provides a foundation for this later analysis. As we discuss below, we also make simplifying assumptions that could be relaxed in future research. Formal proofs of propositions are provided in Appendix A. 我们遵循Chatain和Plaksenkova（2019）的做法，从一个基准模型开始，然后通过修改这个初始模型来发展我们的理论。这样做的动机是，虽然我们的初始结果可能可以用口头论证来解释，但我们后续的结果与现有理论的距离更远，在某些情况下技术上更复杂。基础模型为后续分析奠定了基础。正如我们下面讨论的，我们还做了一些简化假设，这些假设在未来的研究中可以放宽。命题的正式证明见附录A。

We model an ecosystem with a community of firms or individuals (“agents”) and a “lead firm” with which the agents cocreate value. Examples are an original equipment manufacturer (lead firm) and makers of components or complements, a branded food processor (lead firm) and farms, or, inside a single firm, a cost center (lead firm) and divisions that transact with it in an internal market. Formally, we model a biform game with a set $A$ of identical riskneutral atomistic agents of measure 1 and a profitmaximizing lead firm whose shareholders are a distinct zero-measure subset of $A$ . (We need at least two agents to model complementarities in their efforts; using atomistic agents instead of a finite $N \geq 2$ of them only economizes on notation because we do not have to normalize our equations by the number of agents.) The agents could be suppliers to or buyers from the lead firm, or may not transact directly with it. (These different transactional relationships are formally equivalent because biform games determine value capture through market and bargaining power rather than a specific price-setting mechanism—see below.) Neither the lead firm nor the agents can create value without the other. The lead firm and agents have sufficient “capacity” to cocreate value together, but no additional capacity to cocreate value with third parties. 我们构建了一个包含企业或个人（“代理”）社区以及一个与代理共同创造价值的“主导企业”的生态系统模型。例如，原始设备制造商（主导企业）及其零部件或互补品制造商，品牌食品加工商（主导企业）及其农场，或者在单一企业内部，一个成本中心（主导企业）及其在内部市场中与之交易的部门。形式上，我们构建了一个双态博弈模型，其中包含一个由测度为1的相同风险中性原子化代理组成的集合\( A \)，以及一个利润最大化的主导企业，其股东是\( A \)中一个不同的零测度子集。（我们至少需要两个代理来建模其努力中的互补性；使用原子化代理而非有限个\( N \geq 2 \)的代理仅能节省符号表示，因为我们无需按代理数量对方程进行归一化。）代理可以是主导企业的供应商或买方，也可能不直接与其交易。（这些不同的交易关系在形式上是等价的，因为双态博弈通过市场和议价能力而非特定的定价机制来决定价值捕获——见下文。）主导企业和代理缺一不可，无法单独创造价值。主导企业和代理具有共同创造价值的足够“能力”，但与第三方共同创造价值时则不具备额外能力。

In the first stage of our model, agents noncooperatively decide how much costly effort to exert in value cocreation with the lead firm, which could represent investments in the design or use of a product, or in production volume. The amount of value cocreated with each agent is growing in the effort exerted by that agent and general economic conditions (e.g., weather for an agricultural product). We also assume for now that complementarities are positive, so the amount of value cocreated with each agent is also growing in the efforts of other agents. Our model follows recent formal work in taking the nature of complementarities as exogenous (Panico & Cennamo, 2020; Parker & van Alstyne, 2018; Parker et al., 2017), and so may apply best to complementarities over which agents have less control, such as knowledge spillovers or, if negative (see below), congestion in or depletion of a common resource. Formally, in period $t = 0$ , each agent $_ i$ exerts effort $e _ { i } \geq 0$ at cost $e _ { i } ^ { 2 } / 2$ . In period $t = 1$ , the lead firm cocreates value with agent $_ i$ in an amount that is proportional to the agent’s effort and those of other agents: 在我们模型的第一阶段，代理方会非合作地决定在与牵头企业的价值共创中投入多少有成本的努力，这可能包括对产品设计或使用的投资，或是生产规模的投资。每个代理方共创的价值量随其自身投入的努力以及整体经济状况（例如农产品的天气条件）而增长。我们目前还假设互补性是正的，因此每个代理方共创的价值量也随其他代理方的努力而增长。我们的模型借鉴了近期的正式研究，将互补性的性质视为外生的（Panico & Cennamo, 2020；Parker & van Alstyne, 2018；Parker et al., 2017），因此可能最适用于代理方控制力较弱的互补性，例如知识溢出，或者如果是负互补性（见下文），则可能是公共资源的拥堵或耗竭。从形式上看，在时期 \( t = 0 \)，每个代理方 \( i \) 投入努力 \( e_i \geq 0 \)，成本为 \( e_i^2 / 2 \)。在时期 \( t = 1 \)，牵头企业与代理方 \( i \) 共创的价值量与该代理方及其它代理方的努力成正比：

e _ { i } \left( \theta + \delta \int _ { j \in A , j \neq i } e _ { j } d j \right) \times Y e_{i}\left( \theta + \delta \int_{j \in A, j \neq i} e_{j} dj \right) \times Y

$\theta$ is a state variable for economic conditions at $t = 1$ . Higher values of $\theta$ represent better conditions. $Y$ , a positive constant, represents a constant-returnsto-scale technology. The integral is the total effort exerted by other agents and thereby captures the complementarities among agents. The magnitude of the complementarities is parameterized by δ. We restrict δ to be less than $( 2 \dot { Y } ) ^ { - 1 }$ , so the level of agent effort is finite in all scenarios. (The need for this restriction is apparent from the optimal level of effort derived in Appendix A.) $\theta$ 是 $t = 1$ 时经济状况的状态变量。$\theta$ 的值越高，代表经济状况越好。$Y$ 是一个正常数，代表规模报酬不变的技术。积分项是其他主体的总努力程度，从而捕捉了主体之间的互补性。互补性的大小由参数 $\delta$ 来表示。我们将 $\delta$ 限制为小于 $(2 \dot{Y})^{-1}$，因此在所有情景中，主体的努力水平都是有限的。（这种限制的必要性从附录 A 中推导出的最优努力水平可以明显看出。）

After value is cocreated, it is shared among the parties in the cooperative stage using the core (see below), which represents unrestricted bargaining (Ross, 2018). If the core generates a range of values, we apportion value in this range according to the parties’ respective bargaining powers. The lead firm has bargaining power of $\alpha _ { L } > 0$ , and each agent $_ i$ has bargaining power of $\alpha _ { i } > 0$ , such that the agents collectively have $\textstyle \alpha _ { a } = \int _ { i \in A } \alpha _ { i } d i$ , where these parameters represent “persuasive resources” or bargaining skill (Gans & Ryall, 2017). Following Panico (2017), we translate these parameters into “contest functions”: in period $t = 2$ , if the core is a range of values, agents firm capures $\begin{array} { r } { \beta = \frac { \alpha _ { a } } { \alpha _ { a } + \alpha _ { I } } \in [ 0 , 1 ] } \end{array}$ $\begin{array} { r } { ( 1 - \beta ) = \frac { \alpha _ { L } } { \alpha _ { a } + \alpha _ { L } } . } \end{array}$ of this range, and the lead relative bargaining power, $\beta$ tends toward 0, whereas if the lead firm has low relative bargaining power, $\beta$ tends toward 1. 在价值共同创造之后，在合作阶段，各方将通过核心（见下文）共享该价值，这代表不受限制的议价（Ross, 2018）。如果核心产生一个价值范围，我们将根据各方各自的议价能力在该范围内分配价值。牵头企业的议价能力为 $\alpha_L > 0$，每个代理 $i$ 的议价能力为 $\alpha_i > 0$，使得代理们的总议价能力 $\alpha_a = \int_{i \in A} \alpha_i , di$，其中这些参数代表“说服资源”或议价技能（Gans & Ryall, 2017）。遵循 Panico (2017)，我们将这些参数转化为“竞赛函数”：在时期 $t=2$，如果核心是一个价值范围，代理企业捕获 $\beta = \frac{\alpha_a}{\alpha_a + \alpha_I} \in [0, 1]$ 和 $(1 - \beta) = \frac{\alpha_L}{\alpha_a + \alpha_L}$。在这种情况下，如果牵头企业的相对议价能力较低，$\beta$ 趋向于 0，而如果牵头企业的相对议价能力较低，$\beta$ 趋向于 1。

Lead Firm Bargaining Power and Positive Complementarities

牵头企业议价能力与正向互补性

Denote by $V$ the value created by the ecosystem (net of effort). Define $V ^ { M }$ as the maximum amount of value the ecosystem can create and $V ^ { u , \beta }$ as the Nash equilibrium amount, given $\beta$ Define $e _ { i } ^ { M }$ as the valuemaximizing level of effort per agent—that is, the effort associated with $V ^ { M }$ ddefine $e _ { i } ^ { u , \beta }$ as the equilibrium level of effort per agent associated with $V ^ { u , \beta }$ . 用 \( V \) 表示生态系统创造的价值（扣除努力成本后）。定义 \( V^M \) 为生态系统能创造的最大价值，\( V^{u,\beta} \) 为给定 \( \beta \) 时的纳什均衡价值量。定义 \( e_i^M \) 为每个主体的价值最大化努力水平——即与 \( V^M \) 相关的努力水平；定义 \( e_i^{u,\beta} \) 为与 \( V^{u,\beta} \) 相关的每个主体的均衡努力水平。

Proposition 1. With positive complementarities $( \delta > 0 )$ , equilibrium effort is less than the value-maximizing level—that is, $\dot { e } _ { i } ^ { u , \beta } < e _ { i } ^ { M } .$
命题1. 在存在正互补性（$\delta > 0$）的情况下，均衡努力水平低于价值最大化水平，即 $\dot { e } _ { i } ^ { u , \beta } < e _ { i } ^ { M }$。

is lower than the maximum—that is, $V ^ { u , \beta } < V ^ { M }$ . In addition, $e _ { i } ^ { M } - e _ { i } ^ { u , \beta }$ and $V ^ { M } - V ^ { u , \beta }$ decrease in $\beta$ and these differences increase in the magnitude of the positive complementarities (δ). 小于最大值——即，\( V^{u, \beta} < V^M \)。此外，\( e_i^M - e_i^{u, \beta} \) 和 \( V^M - V^{u, \beta} \) 随 \( \beta \) 减小，且这些差异随正互补性（δ）的大小增加。

Suppose that agents and the lead firm could coordinate to maximize net value created by the ecosystem. They would choose the level of effort to maximize the following: 假设代理方和牵头公司能够协调行动，以最大化生态系统创造的净价值。他们会选择努力程度来最大化以下目标：

\operatorname* { m a x } _ { e _ { i , \forall i \in A } } V = e _ { i } \left( \theta + \delta \prod _ { j \in A , j \neq i } e _ { j } d j \right) Y - \frac { e _ { i } ^ { 2 } } { 2 } $\operatorname*{\max}{e{i,\forall i \in A}} V = e_i \left( \theta + \delta \prod_{j \in A, j \neq i} e_j , dj \right) Y - \frac{e_i^2}{2}$

The term on the left is the value cocreated with the agents—that is, Equation (1)—and the term on the right is the cost of agent effort. Equation (2) is depicted as the solid line in Figure 1, which is maximized at a high level of effort, $e _ { i } ^ { M }$ . $\beta$ does not matter, because relative bargaining power only shifts value from agents to the lead firm without affecting the marginal returns to agent effort. 左侧的项是与代理方共同创造的价值——即公式（1）——右侧的项是代理方努力的成本。公式（2）在图1中以实线表示，在较高的努力水平 \( e_i^M \) 处达到最大值。\(\beta\) 无关紧要，因为相对议价能力只会将价值从代理方转移到牵头公司，而不会影响代理方努力的边际回报。

We now obtain the model equilibrium using backward induction. After agents exert effort and cocreate value with the lead firm, their respective shares of that value are bounded in the core by stability and feasibility (MacDonald & Ryall, 2004). Feasibility requires that the upper bound of each party’s value capture be its added value—that is, its contribution to the value created by the grand coalition, $\nu ( A )$ , which represents the entire ecosystem. This value corresponds to Equation (1) for each agent i. This value corresponds to Equation (1) for the lead firm, as well, because $A$ is measure one. Stability requires that the lower bound of each party’s value capture be what it could capture in any subgroup, including on its own. Since the value cocreated by each agent with the lead firm is additively separable in the sense of Makadok and Ross (2018), we can determine stability by considering the subgroups of the lead firm and each agent i. Since neither party can create value without the other, the lower bound on value capture for both parties is 0. (If the cocreation of value were not fully additively separable, we would replace the single agent $_ i$ with the additively separable subsets that partition $A$ ) Thus, the upper bound on both the lead firm’s and the agents’ value capture is the amount in Equation (1) and the lower bound is 0. In this range, value capture is determined by the relative bargaining powers of the agents, $\beta$ , and lead firm, $( 1 - \beta )$ . 我们现在通过逆向归纳法得到模型的均衡。在参与者付出努力并与核心企业共创价值后，他们各自在该价值中的份额受到核心稳定性和可行性的约束（MacDonald & Ryall，2004）。可行性要求各方价值捕获的上限是其附加价值，即其对大联盟创造的价值 $\nu(A)$ 的贡献，其中 $\nu(A)$ 代表整个生态系统。对于每个参与者 $i$，该价值对应式（1）。对于核心企业而言，该价值也对应式（1），因为 $A$ 是测度为1的集合。稳定性要求各方价值捕获的下限是其在任何子群（包括自身）中能捕获的价值。由于每个参与者与核心企业共创的价值在 Makadok 和 Ross（2018）的意义上是可加分离的，我们可以通过考虑核心企业和每个参与者 $i$ 的子群来确定稳定性。由于任何一方单独都无法创造价值，因此双方价值捕获的下限均为0。（如果价值共创不完全可加分离，我们将用划分 $A$ 的可加分离子集替换单个参与者 $i$）。因此，核心企业和参与者的价值捕获上限均为式（1）中的数值，下限为0。在此范围内，价值捕获由参与者的相对议价能力 $\beta$ 和核心企业的相对议价能力 $(1-\beta)$ 决定。

Thus, in the first stage, a focal agent $_ i$ who seeks to maximize their value capture, net of the cost of their effort $\left( u _ { i } ^ { \beta } \right)$ Thus, in the first stage, a focal agent $_ i$ who seeks to maximize their value capture, net of the cost of their effort $\left( u _ { i } ^ { \beta } \right)$

\begin{array} { r } { \operatorname* { m a x } _ { e _ { i } } u _ { i } ^ { \beta } = e _ { i } \left( \theta + \delta \displaystyle \int _ { j \in A , j \neq i } e _ { j } d j \right) Y \beta - \frac { e _ { i } ^ { 2 } } { 2 } } \end{array}

Intuitively, the agent invests costly effort to cocreate value with the lead firm in return for their share of that value, which increases with their bargaining power. Moreover, the agent does this noncooperatively with the other agents. The result is that two frictions prevent $V ^ { M }$ from being achieved in equilibrium, as we now explain. 直观上，代理人会投入成本高昂的努力与领先企业共同创造价值，以换取其在该价值中的份额，而这一份额会随着其议价能力的提升而增加。此外，代理人会与其他代理人进行非合作性的行动。接下来我们将解释，这一结果导致了两种摩擦，使得均衡状态下无法实现 \( V^M \)。

The first friction arises from an inability to coordinate on the value maximizing level of effort. Suppose that $\beta = 1$ , so agents capture all the cocreated value. Then, Equation (3) becomes: 第一个摩擦源于在价值最大化努力水平上无法协调。假设 $\beta = 1$，即参与者捕获所有共同创造的价值。此时，式 (3) 变为：

\begin{array} { r } { \operatorname* { m a x } _ { e _ { i } } u _ { i } ^ { 1 } = e _ { i } \left( \theta + \delta \underset { j \in A , j \neq i } { \uparrow } \right) Y - \frac { e _ { i } ^ { 2 } } { 2 } } \end{array}

Though similar in appearance, Equation (4) has one important difference from Equation (2). In Equation (4), the agent takes the efforts of other agents as given (in the integral), whereas in Equation (2), the efforts of all agents are considered simultaneously. If, in Equation (4), all agents besides our focal agent exert $e _ { i } ^ { M }$ , the focal agent will exert somewhat less, free-riding on the positive complementarities from their efforts. Referring to the line with long dashes labeled $e _ { i } ^ { M }$ in Figure 1, note that its maximum is to the left of and above the maximum associated with $V ^ { M }$ . In fact, our focal agent is not so fortunate, because, in equilibrium, our focal agent anticipates that other agents will exert the lower level of effort, $e _ { i } ^ { u , 1 } < e _ { i } ^ { M }$ private marginal return for their own effort. In consequence, the agent exerts $e _ { i } ^ { u , 1 }$ as well. Thus, agents underprovide effort in equilibrium. Referring to the line with long dashes labeled $e _ { i } ^ { u , 1 }$ in Figure 1, note that the equilibrium level of effort $e _ { i } ^ { u , \bar { 1 } }$ results in value created of $V ^ { u , 1 } < V ^ { M }$ . 尽管外观相似，但公式（4）与公式（2）存在一个重要差异。在公式（4）中，主体将其他主体的努力视为给定值（体现在积分中），而在公式（2）中，所有主体的努力是同时被考虑的。如果在公式（4）中，除我们的焦点主体外的所有主体都付出 \( e_{i}^{M} \)，那么焦点主体的付出会稍少一些，从而搭便车于他们努力带来的正互补性。参考图1中标注为 \( e_{i}^{M} \) 的长横线，注意其最大值位于与 \( V^{M} \) 相关的最大值的左侧且上方。事实上，我们的焦点主体并不那么幸运，因为在均衡状态下，焦点主体会预期其他主体会付出较低水平的努力 \( e_{i}^{u,1} < e_{i}^{M} \)（这是他们自身努力的私人边际回报）。因此，该主体也会付出 \( e_{i}^{u,1} \)。因此，主体在均衡状态下存在努力供给不足的问题。参考图1中标注为 \( e_{i}^{u,1} \) 的长横线，注意均衡努力水平 \( e_{i}^{u,\bar{1}} \) 产生的价值为 \( V^{u,1} < V^{M} \)。

The second friction arises if the lead firm has at least some bargaining power—that is, $\beta < 1$ . 第二个摩擦产生于领先企业至少拥有一定议价能力的情况下，即 $\beta < 1$。

Referring again to Equation (3), our focal agent only captures $\beta$ of the value they cocreate. Because the agent derives a lower marginal return from effort, they exert less of it. Referring to the lines with small dashes in Figure 1 (which impute $\begin{array} { r } { \beta = \frac { 1 } { 2 } ) } \end{array}$ , in equilibrium, value capture by the lead firm drives agent effort even further below the value-maximizing level $\left( e _ { i } ^ { M } > e _ { i } ^ { u , 1 } > e _ { i } ^ { u , \frac { 1 } { 2 } } \right)$ (see bottom left of Figure 1) and net value created even further below what is theoretically possible $\left( V ^ { M } > V ^ { u , 1 } > V ^ { u , \frac { 1 } { 2 } } \right)$ . 再次参考公式（3），我们的核心代理仅获取他们共同创造价值的$\beta$部分。由于代理从努力中获得的边际回报较低，他们的努力程度也较低。参考图1中带有小破折号的线条（这些线条估算出$\begin{array} { r } { \beta = \frac { 1 } { 2 } ) } \end{array}$，在均衡状态下，领先企业对价值的捕获会进一步将代理的努力程度推至低于价值最大化水平（$e_i^M > e_i^{u,1} > e_i^{u,\frac{1}{2}}$）（见图1左下角），并且净创造的价值也会进一步低于理论上的可能值（$V^M > V^{u,1} > V^{u,\frac{1}{2}}$）。

Figure 2 graphs net value created $\left( V ^ { M , N } , V ^ { u , 1 } , V ^ { u , \frac { 1 } { 2 } } \right)$ as a function of the magnitude of the complementarities (δ). For now, focus on the values of $\delta \geq 0$ If there are no complementarities $\left( \delta = 0 \right)$ , the friction associated with the lack of agent coordination is not relevant. Hence, if the lead firm has no market power $( \beta = 1 )$ , equilibrium value created attains the theoretic maximum $V ^ { u , 1 } = V ^ { M } )$ . As δ rises, so too does this friction, so the gap between $V ^ { u , 1 }$ and $V ^ { M }$ grows. 图2将创造的净值 $\left( V ^ { M , N } , V ^ { u , 1 } , V ^ { u , \frac { 1 } { 2 } } \right)$ 绘制为互补性大小（δ）的函数。目前，重点关注 $\delta \geq 0$ 的情况。如果不存在互补性（$\delta = 0$），则与代理缺乏协调相关的摩擦无关紧要。因此，如果领先企业没有市场力量（$\beta = 1$），均衡创造的价值达到理论最大值 $V ^ { u , 1 } = V ^ { M }$。随着δ的增加，这种摩擦也会增大，因此 $V ^ { u , 1 }$ 与 $V ^ { M }$ 之间的差距也会扩大。

In addition, for positive values of δ, the frictions associated with lead firm bargaining power and lack of agent coordination are mutually reinforcing, giving rise to a ratcheting effect. The intuition is that if the lead firm has bargaining power, an agent not only reduces their effort in response to value capture by the lead firm but also anticipates that other agents will reduce their efforts as well. The greater the complementarities, the more this reduction in effort by other agents reduces the focal agent’s private returns to effort, and so the more the focal agent reduces their own effort. Thus, as δ rises, the gap between $V ^ { u , \frac { 1 } { 2 } }$ and $V ^ { u , 1 }$ grows. 此外，对于δ的正值，与主导企业议价能力和代理方缺乏协调相关的摩擦会相互强化，从而产生棘轮效应。直觉上，如果主导企业拥有议价能力，代理方不仅会因主导企业的价值攫取而降低自身努力程度，还会预期其他代理方也会降低努力程度。互补性越强，其他代理方努力程度的降低对焦点代理方的私人努力回报的负面影响就越大，因此焦点代理方自身的努力程度也会降低得越多。因此，随着δ的上升，$V^{u,\frac{1}{2}}$与$V^{u,1}$之间的差距会扩大。

Lead Firm Bargaining Power and Negative Complementarities

牵头企业议价能力与负互补性

We now modify the model by making complementarities negative. 我们现在通过将互补性设为负来修改模型。

Proposition 2. With negative complementarities $\left( \delta < 0 \right)$ , for any given $\beta$ there exists a $\hat { \boldsymbol { \delta } } < \boldsymbol { 0 }$ , such that if $\delta = \hat { \delta }$ , then $V ^ { u , \beta } = V ^ { M }$ . $\hat { \delta }$ increases (becomes less negative) in $\beta$ . 命题2. 存在负互补性（$\delta < 0$）时，对于任意给定的$\beta$，存在$\hat{\boldsymbol{\delta}} < \boldsymbol{0}$，使得当$\delta = \hat{\delta}$时，$V^{u,\beta} = V^M$。$\hat{\delta}$随$\beta$增大（即负向程度降低）而增大。

Suppose, as we did above, that agents and the lead firm coordinate to maximize value created by the ecosystem and that complementarities are negative. In this case, they would choose a relatively low level of effort to account for the negative complementarities arising from agents’ efforts. See $e _ { i } ^ { M }$ and the resulting $V ^ { M }$ in Figure 3. 假设，正如我们上面所做的那样，代理方和牵头企业会协调行动以最大化生态系统创造的价值，并且互补性为负。在这种情况下，他们会选择相对较低的努力水平，以考虑到代理方努力带来的负互补性。参见图3中的\( e _ { i } ^ { M } \)以及由此产生的\( V ^ { M } \)。

As before, consider a focal agent and suppose that $\beta = 1$ , so that the agent captures the full value they As before, consider a focal agent and suppose that $\beta = 1$ , so that the agent captures the full value they

FIGURE 1 Value Functions with Positive Complementarities 图1 具有正互补性的价值函数

Notes: This figure illustrates how net value captured by a focal agent if $\beta = 1$ $\left( u _ { i } ^ { 1 } \right)$ and $\begin{array} { r } { \beta = \frac { 1 } { 2 } \left( u _ { i } ^ { \frac { 1 } { 2 } } \right) } \end{array}$ , as well as net value created by the ecosystem $( V )$ , change with effort when complementarities are positive $\left( \delta > 0 \right)$ . The labels on the net value captured lines indicate the effort undertaken by other agents. (Figures 1 and 3 are not drawn to the same scale.) 注：此图展示了当互补性为正（$\delta > 0$）时，焦点主体捕获的净值（当 $\beta = 1$ 时为 $\left( u_i^1 \right)$，当 $\beta = \frac{1}{2} \left( u_i^{\frac{1}{2}} \right)$ 时）以及生态系统创造的净值 $(V)$ 如何随努力程度变化。净值捕获线旁的标签表示其他主体的努力程度。（图1和图3未按相同比例绘制。）

cocreate. If other agents exert the value-maximizing effort $e _ { i } ^ { M }$ , then—like a tragedy of the commons—the focal agent will exert a higher level of effort because they do not consider the negative complementarities from their effort in their private marginal returns. 共创。如果其他代理付出了价值最大化的努力 \( e_i^M \)，那么——就像公地悲剧一样——焦点代理将付出更高水平的努力，因为他们在私人边际收益中没有考虑到自身努力带来的负互补性。

Referring to the line with long dashes labeled $e _ { i } ^ { M }$ in Figure 3, note that its maximum is to the right of and above the maximum associated with $V ^ { M }$ . Again, however, our focal agent is not so fortunate, because all agents behave in this selfish way by 参考图3中标注为 \( e _ { i } ^ { M } \) 的长横线所在行，需注意其最大值位于 \( V ^ { M } \) 对应的最大值的右侧和上方。然而，我们的焦点主体（focal agent）再次未能如此幸运，因为所有主体都以这种自私的方式行事，从而

FIGURE 2 Value Created as a Function of Complementarities
图2 互补性的函数关系所创造的价值

$V ^ { M }$ $\beta = 1$ $V ^ { u , 1 }$ and $\begin{array} { r } { \beta = \frac { 1 } { 2 } } \end{array}$ $\left( V ^ { u , \frac { 1 } { 2 } } \right)$ as a function of complementarities (8). $V ^ { M }$ $\beta = 1$ $V ^ { u , 1 }$ 和 $\begin{array} { r } { \beta = \frac { 1 } { 2 } } \end{array}$ $\left( V ^ { u , \frac { 1 } { 2 } } \right)$ 作为互补性的函数（8）。

FIGURE 3 Value Functions with Negative Complementarities 图3 具有负互补性的价值函数

$\beta = 1 \ \left( u _ { i } ^ { 1 } \right)$ $\begin{array} { r } { \beta = \frac { 1 } { 2 } \left( u _ { i } ^ { \frac { 1 } { 2 } } \right) } \end{array}$ tem $( V )$ , change with effort when complementarities are significantly negative $\left( \delta \tilde { < } \tilde { \delta } \right)$ . The labels on the net value captured lines indicate the effort undertaken by other agents. (Figures 1 and 3 are not drawn to the same scale.) $\beta = 1 \ \left( u _ { i } ^ { 1 } \right)$ $\begin{array} { r } { \beta = \frac { 1 } { 2 } \left( u _ { i } ^ { \frac { 1 } { 2 } } \right) } \end{array}$ tem $( V )$，当互补性显著为负时（$\delta \tilde { < } \tilde { \delta }$）会随努力程度变化。净值捕获线上的标签表示其他主体所付出的努力。（图1和图3未按相同比例绘制。）

overproviding effort. In equilibrium, each agent exerts $e _ { i } ^ { u , 1 } > e _ { i } ^ { M }$ , anticipating that other agents will do so too. Referring to the line with long dashes labeled $e _ { i } ^ { u , 1 }$ in Figure 3, the result is that the ecosystem only creates net value of $V ^ { u , 1 } < V ^ { M }$ . 过度提供努力。在均衡状态下，每个主体都付出 \( e_i^{u,1} > e_i^M \)，并预期其他主体也会如此。参考图3中标注为 \( e_i^{u,1} \) 的长横线所在行，结果是生态系统仅创造了净价值 \( V^{u,1} < V^M \)。

Now, suppose that the lead firm has sufficient bargaining power to capture $( 1 - \beta )$ of the value created. We found with that with positive complementarities, the friction from lead firm bargaining power reinforced the friction associated with lack of agent coordination. With negative complementarities, however, these two frictions offset each other. The reason for this is that value capture by the lead firm reduces an agent’s private returns to effort and thereby mitigates their tendency to overprovide it. If negative complementarities are precisely the right level $\left( \delta = \hat { \delta } \right)$ , the friction associated with lead firm bargaining power exactly offsets the friction associated with lack of agent coordination such that equilibrium value created attains the theoretic maximum. Refer to Figure 2, which shows that value created with lead firm bargaining power $\begin{array} { r } { \beta = \frac { 1 } { 2 } ) } \end{array}$ is larger than without $\displaystyle { \beta = 1 }$ if complementarities are less (farther from zero) than a $\tilde { \delta }$ , where $\hat { \delta } < \tilde { \delta } < 0$ , and that value creation attains the theoretic maximum if $\delta = \hat { \delta }$ . (We discuss $\tilde { \delta }$ further below.) $\hat { \boldsymbol \delta }$ moves closer to 0 as the lead firm’s bargaining power declines $\beta$ increases). 现在，假设主导企业有足够的议价能力来获取创造价值的 $(1 - \beta)$ 部分。我们发现，在存在正互补性的情况下，主导企业议价能力带来的摩擦会加剧因代理人间缺乏协调而产生的摩擦。然而，在存在负互补性的情况下，这两种摩擦会相互抵消。原因在于，主导企业对价值的捕获会降低代理人为努力付出的私人回报，从而减轻他们过度提供努力的倾向。如果负互补性恰好处于适当水平（$\delta = \hat{\delta}$），那么主导企业议价能力带来的摩擦会恰好抵消因代理人间缺乏协调而产生的摩擦，使得均衡创造的价值达到理论最大值。参考图2，该图显示，当互补性小于 $\tilde{\delta}$（且 $\hat{\delta} < \tilde{\delta} < 0$）时，具有主导企业议价能力（$\beta = \frac{1}{2}$）的价值创造量大于没有议价能力（$\beta = 1$）的情况，并且当 $\delta = \hat{\delta}$ 时，价值创造达到理论最大值。（我们将在下文进一步讨论 $\tilde{\delta}$。）$\hat{\delta}$ 随着主导企业议价能力的下降（$\beta$ 增大）而更接近0。

Refer also to the lines with small dashes in Figure 3, depicting a case where δ is negative enough for equilibrium value created to be greater with lead firm bargaining power $\left( V ^ { u , \frac { 1 } { 2 } } \right)$ than without $\left( V ^ { u , 1 } \right)$ 还可参考图3中带有小短横线的线条，该线条描述了一种情况：当δ足够负时，具有领先企业议价能力的均衡值（\( V^{u, \frac{1}{2}} \)）比不具有领先企业议价能力时（\( V^{u, 1} \)）更大。

but not quite negative enough to attain the theoretic maximum of $V ^ { M }$ .For this value of δ, the friction associated with lead firm bargaining power offsets the friction associated with lack of agent coordination by slightly too much, pushing $e _ { i } ^ { u , \frac { 1 } { 2 } }$ close to, but below, the optimal $e _ { i } ^ { M }$ . 但还不足以达到 \( V^M \) 的理论最大值。对于这个 \( \delta \) 值，领先企业议价能力相关的摩擦，以稍多的程度抵消了代理方缺乏协调所带来的摩擦，使得 \( e_i^{u, \frac{1}{2}} \) 接近但低于最优值 \( e_i^M \)。

Market Power and Competition

市场力量与竞争

In our analysis so far, the lead firm implicitly had market power because agents did not have another option for cocreating value. We now consider the possibility that the lead firm has a rival that is already in the market, or could enter it. The rival is an imperfect substitute for the lead firm in that, due to differences in technology, learning curves and other costs of entry, or physical distance, the rival can cocreate value with the agents in an amount equal to Equation (1) multiplied by $\gamma _ { L } \in [ 0 , 1 ]$ If $\gamma _ { L } = 1$ , then the rival is a perfect substitute and the rival and lead firm are in unbridled competition; as the lead firm becomes more differentiated from the rival or protected by entry barriers, $\gamma _ { L }$ tends toward 0. 在我们目前的分析中，牵头企业隐含地拥有市场力量，因为代理方没有其他选择来共同创造价值。我们现在考虑这样一种可能性：牵头企业有一个已经进入市场或可能进入市场的竞争对手。这个竞争对手是牵头企业的不完全替代品，原因在于技术差异、学习曲线和其他进入成本，或物理距离等因素，竞争对手与代理方共同创造的价值量等于公式（1）乘以 $\gamma_L \in [0, 1]$。如果 $\gamma_L = 1$，那么竞争对手就是完全替代品，竞争对手与牵头企业处于无限制竞争状态；随着牵头企业与竞争对手的差异化程度提高或受到进入壁垒的保护，$\gamma_L$ 会趋近于 0。

The rival reduces the lead firm’s market power, because the added value of the lead firm to the grand coalition is now only $( 1 - \gamma _ { L } ) \times \nu ( A )$ .Thus, the lead firm’s bargaining power applies to a narrower range of values, which are less favorable to it and more oabe e Slly $( 1 - \gamma _ { L } )$ , so the lead firm captures less value and the agents capture more in a shift that is growing in $\gamma _ { L }$ 竞争对手削弱了领先企业的市场影响力，因为领先企业对大联盟的附加价值现在仅为 $(1 - \gamma_L) \times \nu(A)$。因此，领先企业的议价能力所适用的价值范围变窄，这些价值对其更为不利，且更倾向于 $(1 - \gamma_L)$，所以领先企业获取的价值减少，而参与者获取的价值增加，这种转变随着 $\gamma_L$ 的增大而加剧。

All told, competition for the lead firm’s contribution to value creation reduces its market power and thus its share of value, just as lower bargaining power does. 总的来说，领先企业在价值创造方面的贡献竞争会削弱其市场力量，进而降低其价值份额，这与议价能力较低的情况类似。

Coordination Devices to Optimize Value Creation and Value Capture

优化价值创造与价值捕获的协调机制

（注：这里“Coordination Devices”译为“协调机制”更符合中文技术/商业语境，若需严格直译“协调装置”也可，但“机制”更通用。根据规则，仅翻译人类可读内容，标题属于人类可读，故翻译。）

The foregoing analysis raises the question of whether coordination devices could improve value creation. To explore this, we derive the optimal market and bargaining power to maximize value creation in the ecosystem, characterize the optimal contract as a default solution, analyze cooperative governance and agent self-organization as alternatives, and consider the lead firm’s preferences over its market and bargaining power. As in the transaction cost literature, we assume that governance form does not affect production costs, except through incentives (Pitelis, 2012: 1364). 上述分析提出了一个问题：协调机制是否能够提升价值创造。为了探究这一点，我们推导了最优的市场和议价能力，以最大化生态系统中的价值创造，将最优合同描述为默认解决方案，分析合作治理和代理自组织作为替代方案，并考虑核心企业对其市场和议价能力的偏好。与交易成本文献中的假设一致，我们认为治理形式不会影响生产成本，除非通过激励措施（Pitelis，2012：1364）。

Optimal lead firm value capture. For any given set of parameter values, it is possible to find an optimal combination of market and bargaining power $\left( { \hat { \boldsymbol { \beta } } } \right)$ such that, in equilibrium, the ecosystem creates the maximum possible value, $V ^ { M }$ . 最优牵头企业价值捕获。对于任何给定的参数值集合，都可以找到市场和议价能力的最优组合 $\left( { \hat { \boldsymbol { \beta } } } \right)$，使得在均衡状态下，生态系统创造出最大可能的价值 $V ^ { M }$。

Proposition 3. If $\beta = \hat { \beta } = \left( 1 - \delta Y \right) ^ { - 1 }$ , then equilibrium effort attains the value maximizing level—that is, $e _ { i } ^ { { \bar { u } } , \beta } = e _ { i } ^ { M }$ maximum—that is, Vu,β = VM . 命题3。若 $\beta = \hat{\beta} = \left(1 - \delta Y\right)^{-1}$，则均衡努力达到最大化水平，即 $e_{i}^{\bar{u},\beta} = e_{i}^{M}$（最大值），即 $V^{\bar{u},\beta} = V^{M}$。

$\hat { \boldsymbol { \beta } }$ has interesting properties. If complementarities are positive $\left( \delta > 0 \right)$ then $\hat { \boldsymbol { \beta } } > 1$ —that is, in order to induce the agents to exert the optimal level of effort, they must be given more than the marginal product of that effort in terms of subsequent value capture. However, in the absence of contracting (see below), the most a party can capture is $1 0 0 %$ of its added value. $\hat { \boldsymbol { \beta } }$ 具有有趣的性质。如果互补性为正（$\delta > 0$），则 $\hat { \boldsymbol { \beta } } > 1$——也就是说，为了促使代理人付出最优努力水平，必须给予他们超过该努力在后续价值捕获中的边际产品的报酬。然而，在缺乏契约的情况下（见下文），一方最多只能捕获其新增价值的 $100%$。

If complementarities are negative $\left( \delta < 0 \right)$ , then $\hat { \boldsymbol { \beta } }$ is between 0 and 1, so value capture by the lead firm through its market and bargaining power can, if at precisely the right level, permit the ecosystem to attain the maximum level of value creation, $V ^ { M }$ . As $\beta$ approaches $\hat { \boldsymbol { \beta } }$ from above or below, value created in the ecosystem increases monotonically toward the maximum. 如果互补性为负（$\delta < 0$），则 $\hat{\boldsymbol{\beta}}$ 介于 0 和 1 之间，因此领先企业通过其市场和议价能力所捕获的价值，若处于恰好合适的水平，可使生态系统实现最大价值创造水平 $V^M$。当 $\beta$ 从上方或下方趋近于 $\hat{\boldsymbol{\beta}}$ 时，生态系统创造的价值会单调递增至最大值。

Contracts. If it is possible to write an enforceable contract a priori, the ecosystem can attain its maximum possible value creation. To model this, we invert the order of the standard biform game such that bargaining over value capture comes first and agents noncooperatively choose their effort levels second. The bargaining stage contains an enforceable contract with two elements: (a) an optimal marginal apportionment to the agents from the value cocreated with the lead firm, $\overset { \vartriangle } { \hat { \boldsymbol { \beta } } }$ (see above), that induces them to exert the value-maximizing level of effort, $e _ { i } ^ { M }$ ; and (b) if, required, a lump-sum transfer to the value available for distribution such that, on net, the value each party captures reflects their respective bargaining powers, $\alpha _ { L } , \alpha _ { a }$ . 合同。如果能够事先签订一份可执行的合同，生态系统就能实现其最大可能的价值创造。为了对这一点进行建模，我们颠倒了标准双态博弈的顺序，使得对价值捕获的议价先于参与者非合作地选择其努力水平。议价阶段包含一份可执行的合同，其中有两个要素：(a) 从与牵头企业共同创造的价值中向参与者进行最优边际分配 $\overset { \vartriangle } { \hat { \boldsymbol { \beta } } }$（见上文），这会促使他们付出价值最大化的努力水平 $e _ { i } ^ { M }$；以及 (b) 如果需要，向可供分配的价值进行一笔总付转移，使得各方净捕获的价值反映其各自的议价能力 $\alpha _ { L } , \alpha _ { a }$。

This is easiest to understand with an example. Let $\begin{array} { r } { Y = \theta = 1 , \delta = \frac { 1 } { 4 } } \end{array}$ Suppose that bargaining power for the lead firm is $\alpha _ { L } = 1$ and for the agents is $\alpha _ { a } = 2$ ultimately, value should be shared one part to the lead firm, two parts to the agents. $\begin{array} { r } { \hat { \boldsymbol { \beta } } = \frac { 4 } { 3 } . } \end{array}$ This excess share of the cocreated value induces the agents to exert the value-maximizing level of effort for the ecosystem, $e _ { i } ^ { u , \beta } = e _ { i } ^ { M } = 2$ , resulting in a value of 3 being cocreated with the lead firm. The agents’ “share” of this is greater than $1 0 0 %$ , namely ${ \frac { 4 } { 3 } } \times 3 = 4$ To balance accounts, the agents would make a lump sum transfer of 2 to the pool of value for distribution, permitting the lead firm to receive 1 and the agents $- 2 + 4 = 2$ . Importantly, the lump sum transfer is owed by the agents regardless of how much effort they exert—akin to posting a bond—so it is a sunk cost and does not figure into their subsequent choice of effort. In a study of 52 joint development projects in technology, Ryall and Sampson (2009) did not report any contract provision like this, suggesting that implementing the optimal contract described here may be challenging if complementarities are strongly positive. This is easiest to understand with an example. Let $\begin{array} { r } { Y = \theta = 1 , \delta = \frac { 1 } { 4 } } \end{array}$ Suppose that bargaining power for the lead firm is $\alpha _ { L } = 1$ and for the agents is $\alpha _ { a } = 2$ ultimately, value should be shared one part to the lead firm, two parts to the agents. $\begin{array} { r } { \hat { \boldsymbol { \beta } } = \frac { 4 } { 3 } . } \end{array}$ This excess share of the cocreated value induces the agents to exert the value-maximizing level of effort for the ecosystem, $e _ { i } ^ { u , \beta } = e _ { i } ^ { M } = 2$ , resulting in a value of 3 being cocreated with the lead firm. The agents’ “share” of this is greater than $1 0 0 %$ , namely ${ \frac { 4 } { 3 } } \times 3 = 4$ To balance accounts, the agents would make a lump sum transfer of 2 to the pool of value for distribution, permitting the lead firm to receive 1 and the agents $- 2 + 4 = 2$ . Importantly, the lump sum transfer is owed by the agents regardless of how much effort they exert—akin to posting a bond—so it is a sunk cost and does not figure into their subsequent choice of effort. In a study of 52 joint development projects in technology, Ryall and Sampson (2009) did not report any contract provision like this, suggesting that implementing the optimal contract described here may be challenging if complementarities are strongly positive.

Cooperative governance. Cooperative governance may be an alternative where contracting is not feasible. Suppose that instead of being owned by thirdparty shareholders, the lead firm is owned by its agents as patron owners, yet continues to act to maximize value capture and distribute that value to agents in proportion to their contribution to value cocreation (transaction volume). This is typical of cooperatives (e.g., Ocean Spray, the branded fruit processor vis-à-vis its patron owner farmers). 合作治理。合作治理可能是一种在契约不可行时的替代方案。假设主导企业并非由第三方股东所有，而是由其代理人作为赞助所有者所有，但仍继续采取行动以最大化价值捕获，并按代理人对价值共创（交易量）的贡献比例向其分配该价值。这是合作社的典型特征（例如，Ocean Spray，即面向赞助所有者农民的品牌水果加工商）。

Proposition 4. If the lead firm is a cooperative, its value capture does not affect the gap between the level of effort that maximizes value created by the ecosystem and the equilibrium level of effort. Equivalently, its value capture does not affect the gap between the theoretic maximum and equilibrium value created. 命题4. 如果牵头企业是合作企业，其价值捕获不会影响生态系统创造的最大价值水平与均衡努力水平之间的差距。换句话说，其价值捕获不会影响理论最大值与均衡创造价值之间的差距。

The reason that value capture by a lead firm organized as a cooperative does not affect agent effort lies in agents’ private returns to effort. Consider a focal agent. On the one hand, the lead firm captures $\left( 1 - \beta \right)$ of the value cocreated with the agent, dissuading them from undertaking effort, as we saw before in Equation (3). On the other hand, the agent’s right to share in the value cocreated with other agents grows in proportion to the focal agent’s contribution to the creation of value. This contribution in turn grows with the agent’s effort, thereby incentivizing the agent to undertake more of it. These two effects offset each other exactly—that is, agents lose $( 1 - \beta )$ of the value they cocreate but recapture precisely this amount from their claims on the lead firm’s overall value capture. 核心企业以合作社形式组织时，其价值捕获不会影响代理方努力程度的原因在于代理方的努力私人回报。以某一核心代理方为例：一方面，核心企业捕获了与该代理方共同创造价值的$(1 - \beta)$部分，这正如前文式(3)所示，会削弱代理方的努力意愿；另一方面，代理方与其他代理方共同创造的价值中，其分享权的增长与该核心代理方对价值创造的贡献成正比。而这种贡献又随代理方的努力程度增加而提升，从而激励代理方投入更多努力。这两种效应恰好相互抵消——即代理方损失了其共同创造价值的$(1 - \beta)$部分，但通过对核心企业整体价值捕获的权益主张，又恰好重新获得了这部分损失。

Proposition 5. Equilibrium value created is higher if the lead firm is a cooperative than if it is shareholderowned, if and only if, for a given $\beta _ { z }$ ,complementarities are $( i )$ positive or (ii) negative and smaller in magnitude than $a \tilde { \delta }$ , where $\hat { \delta } < \widetilde { \delta } < 0$ δincreases (grows less negative) in $\beta$ . 命题5。当领先企业为合作制企业而非股东所有制企业时，创造的均衡价值更高，当且仅当，对于给定的 $\beta _ { z }$，互补性满足：(i) 为正，或者 (ii) 为负且其绝对值小于 $a \tilde { \delta }$，其中 $\hat { \delta } < \widetilde { \delta } < 0$，且 $\delta$ 随 $\beta$ 增加（负向程度减小）。

The cooperative form eliminates the effects of lead firm value capture, such that agents behave as if $\beta = 1$ , regardless of its actual value. Whether this results in the ecosystem creating more value depends on the nature of complementarities, δ. As depicted in Figure 2, for any given $\beta$ there is a δ where $\hat { \delta } < \tilde { \delta } < 0$ , such that if and only if $\delta > \tilde { \delta }$ ,equilibrium value created is greater if the lead firm is a cooperative and agents accordingly behave as if $\beta = 1$ than if the lead firm is shareholder owned and captures $( 1 - \beta )$ of the cocreated value. If, by contrast, complementarities are sufficiently negative—that is, if $\delta { \dot { < } } { \tilde { \delta } }$ , equilibrium value created is greater with a shareholder-owned lead firm than with a cooperative. δ moves closer to 0 as the lead firm’s relative market and bargaining power declines $\beta$ increases). 合作形式消除了核心企业价值攫取的影响，使得参与者的行为表现为 $\beta = 1$，无论其实际值如何。这是否会导致生态系统创造更多价值取决于互补性 $\delta$ 的性质。如图 2 所示，对于任何给定的 $\beta$，存在一个 $\delta$ 使得 $\hat{\delta} < \tilde{\delta} < 0$，当且仅当 $\delta > \tilde{\delta}$ 时，核心企业为合作型且参与者相应表现为 $\beta = 1$ 时创造的均衡价值，大于核心企业为股东所有并攫取 $(1 - \beta)$ 共同创造价值时的情况。相反，如果互补性足够负（即 $\delta < \tilde{\delta}$），则股东所有的核心企业创造的均衡价值大于合作型核心企业。随着核心企业相对市场和议价能力下降（$\beta$ 增加），$\delta$ 会更接近 0。

Agent self-organization. As another alternative to contracting, agents may be able to use social norms or hierarchical control to self-organize to solve their problem in Equation (3) more efficiently by coordinating on the level of effort that maximizes their value capture as a group given their relative market and bargaining power, $\beta$ (In this case, we can think of the lead firm as representing the buyer side of the market.) Let $e _ { i } ^ { S , \beta }$ be the level of effort that yields can self-organize to maximize it. Let $\bar { V } ^ { S , \beta }$ be the resulting value creation for the ecosystem. Agent self-organization. As another alternative to contracting, agents may be able to use social norms or hierarchical control to self-organize to solve their problem in Equation (3) more efficiently by coordinating on the level of effort that maximizes their value capture as a group given their relative market and bargaining power, $\beta$ (In this case, we can think of the lead firm as representing the buyer side of the market.) Let $e _ { i } ^ { S , \beta }$ be the level of effort that yields can self-organize to maximize it. Let $\bar { V } ^ { S , \beta }$ be the resulting value creation for the ecosystem.

Proposition 6. If the agents self-organize to exert the effort $e _ { i } ^ { S , \beta } = ( \theta Y \beta ) / ( 1 - 2 \delta Y \beta )$ that maximizes their value capture $u _ { i } ^ { S , \beta }$ tem does not obtain the maximum possible, i.e., $V ^ { M } > V ^ { S , \beta }$ , because agents underprovide effort—that is, $e _ { i } ^ { M } > e _ { i } ^ { S , \beta }$ If complementarities are positive $\left( \delta > 0 \right)$ , then equilibrium value created is higher with agent self-organization than without—that is, $V ^ { S , \beta } > V ^ { u , \beta }$ , because, in that case, equilibrium agent effort with self-organization is higher than without, i.e., $e _ { i } ^ { S , \beta } > e _ { i } ^ { u , \beta }$ 命题6。如果代理自我组织以施加努力 \( e_i^{S,\beta} = \frac{\theta Y \beta}{1 - 2 \delta Y \beta} \)，该努力最大化其价值捕获 \( u_i^{S,\beta} \)，但系统并未获得最大可能值，即 \( V^M > V^{S,\beta} \)，因为代理提供的努力不足——即 \( e_i^M > e_i^{S,\beta} \)。如果互补性为正（\( \delta > 0 \)），则在代理自我组织的情况下，均衡创造的价值比无自我组织时更高——即 \( V^{S,\beta} > V^{u,\beta} \)，因为在这种情况下，自我组织下的均衡代理努力高于无自我组织时，即 \( e_i^{S,\beta} > e_i^{u,\beta} \)。

Agents capture the most value if they self-organize by construction. That agents underprovide effort relative to the value-maximizing level follows because agents do not capture all the value they cocreate with the lead firm and so have less incentive to exert costly effort than if they captured all this value. Agent self-organization, by construction, eliminates the friction associated with their lack of coordination. This unambiguously increases value creation if complementarities are positive, because, in that case, the friction from lack of agent coordination reinforces the friction associated with value capture by the lead firm. Eliminating the friction from lack of agent coordination nudges equilibrium effort higher, toward the value-maximizing level. However, if complementarities are negative, the effect of eliminating the friction associated with lack of agent coordination is an ambiguous complex function of other parameters because, then, this friction offsets the friction associated with value capture by the lead firm, as we saw earlier. 如果代理能够通过构建实现自组织，它们将捕获最大价值。代理提供的努力相对于价值最大化水平不足，这是因为代理无法捕获它们与牵头企业共同创造的全部价值，因此与能够捕获全部价值的情况相比，它们缺乏付出高成本努力的动力。通过构建实现的代理自组织消除了因缺乏协调而产生的摩擦。如果互补性为正，这种情况无疑会增加价值创造，因为在这种情况下，缺乏代理协调所产生的摩擦会加剧牵头企业捕获价值所带来的摩擦。消除因缺乏代理协调而产生的摩擦会将均衡努力水平推向更高，接近价值最大化水平。然而，如果互补性为负，消除因缺乏代理协调而产生的摩擦的影响是其他参数的一个复杂函数，具有不确定性，因为如前所述，此时这种摩擦会抵消牵头企业捕获价值所带来的摩擦。

Lead firm preferences. Lead firms play a decisive role in how their ecosystems are governed. So, it is worth asking what the lead firm’s optimal value capture is and how it may differ from the valuemaximizing level. Let $\pi _ { L } ^ { \beta }$ be the lead firm’s equilibrium value capture (profits) and $\beta _ { L } ^ { M }$ be the $\beta$ that that induces the agents to maximize $\pi _ { L } ^ { \beta }$ . 核心企业偏好。核心企业在其生态系统的治理方式中发挥着决定性作用。因此，值得探讨核心企业的最优价值捕获是什么，以及它可能如何不同于价值最大化水平。设 $\pi _ { L } ^ { \beta }$ 为核心企业的均衡价值捕获（利润），$\beta _ { L } ^ { M }$ 为促使参与者最大化 $\pi _ { L } ^ { \beta }$ 的 $\beta$。

Proposition 7. Equilibrium value capture by the lead firm, $\pi _ { I , : } ^ { \beta }$ is maximized at $\beta _ { L } ^ { M }$ where $0 < \beta _ { L } ^ { M } =$ $\left( 2 - \delta Y \right) ^ { \ b = 1 } < \hat { \beta }$ . 命题7. 领先企业捕获的均衡价值 $\pi_{I,:}^{\beta}$ 在 $\beta_{L}^{M}$ 处达到最大，其中 $0 < \beta_{L}^{M} = \left( 2 - \delta Y \right)^{\beta=1} < \hat{\beta}$。

The lead firm needs the agents to capture some positive share of cocreated value to incentivize them to exert costly effort, but the optimal share from the lead firm’s perspective is below the level that results in the ecosystem creating the maximum possible value. In essence, the lead firm captures more value if it takes a bigger share of a smaller pie than a smaller share of the largest possible pie. Thus, the lead firm only favors changes to its market and bargaining power that move $\beta$ in the direction of both $\beta _ { L } ^ { M }$ and $\hat { \boldsymbol { \beta } }$ librium value created $\left( V ^ { u , \beta } \right)$ , and the value capture of the agents $\left( u _ { i } ^ { \beta } \right)$ and of the lead firm $\left( \pi _ { L } ^ { \beta } \right)$ , as a function of $\beta$ The figure depicts positive complementarities, so $\hat { \boldsymbol { \beta } } > 1$ , but $\beta _ { L } ^ { M }$ is well below 1. 牵头企业需要代理商获取一定比例的共创价值，以激励他们付出高成本的努力，但从牵头企业的角度来看，最优比例低于生态系统创造最大可能价值的水平。本质上，牵头企业如果从较小的蛋糕中获取更大比例，比从最大的蛋糕中获取较小比例能获得更多价值。因此，牵头企业只倾向于改变其市场和议价能力，使β向β_L^M和均衡创造价值（V^{u,β}）的方向移动，以及代理商的价值捕获（u_i^β）和牵头企业的价值捕获（π_L^β）作为β的函数。该图描绘了正互补性，因此$\hat{\beta} > 1$，但β_L^M远低于1。

FIGURE 4 Value Capture and Value Creation as Agents’ Share of Cocreated Value Varies
图4 随着参与者共创价值的份额变化，价值捕获与价值创造的关系

$\left( V ^ { u , \beta } \right)$ $\left( u _ { i } ^ { \beta } \right)$ , and value cap. tured by the lead firm $\left( \pi ^ { \beta } \right)$ vary with $\beta$ Complementarities are positive $\left( \delta > 0 \right)$ . $\left( V ^ { u , \beta } \right)$ $\left( u _ { i } ^ { \beta } \right)$，以及由牵头企业 $\left( \pi ^ { \beta } \right)$ 捕捉的价值随 $\beta$ 变化。互补性为正 $\left( \delta > 0 \right)$。

DISCRETE CHOICES

离散选择

So far, we have conceived of agent effort as continuous; however, in practice, many such effort choices have a discrete character, such as whether to join a new technology consortium or whether to expend a lump sum for a new plant. As a lead firm tries to get a new ecosystem to coalesce, it must facilitate value capture for other players. As our previous analysis showed, one friction in doing this is the disincentivizing effect of the lead firm’s value capture. With discrete choices, this friction might not only result in lower total value creation but even prevent the ecosystem from forming at all if enough parties opt out. We explore this possibility by modifying our model to consider discrete choices in agent effort. For brevity of exposition, we only analyze positive complementarities. 到目前为止，我们将代理的努力设想为连续的；然而，在实际中，许多此类努力选择具有离散特征，例如是否加入新的技术联盟，或是否为新工厂支出一笔总费用。当主导企业试图让新生态系统凝聚时，它必须促进其他参与者的价值捕获。正如我们之前的分析所示，在这一过程中的一个障碍是主导企业的价值捕获所产生的抑制作用。在存在离散选择的情况下，这种障碍不仅可能导致总价值创造降低，而且如果足够多的参与方选择退出，甚至可能完全阻止生态系统的形成。我们通过修改模型以考虑代理努力中的离散选择来探索这种可能性。为了简化阐述，我们仅分析正互补性。

We formally assume that at $t = 0$ each agent makes a discrete choice of whether to “opt in” to or “opt out” of the ecosystem. If the agent opts out, the agent captures value of 0 at $t = 1$ . If and only $i f$ the agent opts in, the agent exerts effort at cost $e$ and cocreates value with the lead firm in an amount of $\left( \theta + \Delta \right) Y$ at $t = 1$ . θ and $Y$ , as well as the split of value, $\beta$ , are as in the base model. $\Delta \in [ 0 , 1 ]$ is the endogenously determined proportion of agents who have opted in. 我们正式假设，在 \( t = 0 \) 时，每个参与者会做出一个离散选择，决定是“加入”还是“退出”该生态系统。如果参与者选择退出，那么在 \( t = 1 \) 时，该参与者的价值捕获量为 0。当且仅当参与者选择加入时，该参与者会以成本 \( e \) 付出努力，并与领先企业共同创造价值，其价值量为 \( (\theta + \Delta) Y \)（在 \( t = 1 \) 时）。\(\theta\) 和 \( Y \)，以及价值分配比例 \(\beta\)，均与基础模型一致。\(\Delta \in [0, 1]\) 是内生决定的已加入参与者的比例。

We note that an agent’s willingness to undertake costly effort will be influenced by their sense of θ—that is, future business conditions (e.g., the market for a technology consortium’s product). Optimistic agents will be more apt to exert effort (e.g., to join and invest in the consortium), whereas pessimistic agents will be more apt to demur. We incorporate these principles and avoid a multiplicity of equilibria by adopting the form of a global game as follows (Carlsson & van Damme, 1993). Define $\bar { \theta } _ { \beta } = e / ( Y \beta )$ as the threshold level of $\theta$ high enough that, conditional on $\beta$ , agents would opt in regardless of what other agents do. Define ${ \underline { { \theta } } } _ { \beta } = e / ( Y \beta ) - 1$ as the threshold level of $\theta$ low enough that, conditional on $\beta$ agents would opt out regardless of what other agents do. Define $\underline { { \theta } } ^ { M }$ as the optimal threshold low level of $\theta$ such that value creation, net of effort, is maximized if $a l I$ agents opt in if and only if $ { \boldsymbol { \theta } } \geq \underline { { \boldsymbol { \theta } } } ^ { M }$ . It is immediate that $\underline { { \theta } } ^ { M } = \overline { { \underline { { \theta } } } } _ { 1 }$ and that $\underline { { \theta } } _ { \beta }$ increases as $\beta$ declines (the lead firm captures more value). 我们注意到，代理人愿意付出高昂努力的意愿会受到其对θ的感知影响——即未来的商业环境（例如，技术联盟产品的市场）。乐观的代理人更倾向于付出努力（例如，加入并投资于该联盟），而悲观的代理人更倾向于拒绝。我们纳入这些原则，并通过采用全局博弈的形式来避免多重均衡（Carlsson & van Damme，1993）。定义$\bar{\theta}\beta = e/(Y\beta)$为θ的阈值水平，该阈值足够高，使得在给定β的条件下，无论其他代理人做什么，代理人都会选择加入。定义$\underline{\theta}\beta = e/(Y\beta) - 1$为θ的阈值水平，该阈值足够低，使得在给定β的条件下，无论其他代理人做什么，代理人都会选择退出。定义$\underline{\theta}^M$为θ的最优阈值低水平，使得如果且仅当$\theta \geq \underline{\theta}^M$时，所有代理人都选择加入，此时净努力的价值创造最大化。显然，$\underline{\theta}^M = \overline{\underline{\theta}}1$，并且$\underline{\theta}\beta$随着β的下降而增加（领先企业捕获更多价值）。

At $t = 0$ , agents do not know what $\theta$ will be at $t = 1$ but do receive a private signal $s$ that is uniformly distributed about the true future level of $\theta$ that is, $s$ is uniformly distributed on $\left[ \theta - \varepsilon , \theta + \varepsilon \right]$ . Agents also know that the unconditional distribution of $\theta$ is uniform on an interval $[ a , b ]$ , where $b > \bar { \theta } _ { \beta } + \varepsilon$ and $a < \underline { { \theta } } _ { \beta } - \varepsilon$ —that is, the unconditional distribution of $\theta$ extends at least slightly beyond the threshold levels of $\underline { { \theta } } _ { \beta }$ and $\bar { \theta } ^ { \beta }$ , for any given $\beta$ (This means that $\beta$ cannot be arbitrarily close to 0.) We have: 在 \( t = 0 \) 时，参与者不知道 \( t = 1 \) 时 \( \theta \) 的值，但会收到一个私有信号 \( s \)，该信号在真实未来水平 \( \theta \) 周围均匀分布，即 \( s \) 在区间 \( [\theta - \varepsilon, \theta + \varepsilon] \) 上均匀分布。参与者还知道 \( \theta \) 的无条件分布在区间 \( [a, b] \) 上均匀分布，其中 \( b > \bar{\theta}_\beta + \varepsilon \) 且 \( a < \underline{\theta}_\beta - \varepsilon \)——也就是说，对于任何给定的 \( \beta \)，\( \theta \) 的无条件分布至少会略微超出阈值水平 \( \underline{\theta}_\beta \) 和 \( \bar{\theta}_\beta \)（这意味着 \( \beta \) 不能任意接近 0）。我们有：

Proposition 8. The iterated elimination of strictly dominated strategies results in a unique threshold signal such that an agent opts in if and only if they receive a signal $s > s _ { \beta } ^ { u }$ For $\forall \varepsilon < s _ { \beta } ^ { u } - { \underline { { \theta } } } ^ { M }$ , $\dot { \exists } \theta _ { \beta } ^ { \dot { u } } > \underline { { \theta } } ^ { \dot { M } }$ such that $\forall \theta < \theta _ { \beta } ^ { u }$ , no agent opts in. $\theta _ { \beta } ^ { u }$ increases as $\beta$ declines. 命题8. 严格占优策略的迭代剔除会产生唯一的阈值信号，使得代理人仅在接收到信号 \( s > s_{\beta}^u \) 时选择参与。对于所有 \( \varepsilon < s_{\beta}^u - \underline{\theta}^M \)，存在 \( \theta_{\beta}^{\dot{u}} > \underline{\theta}^{\dot{M}} \)，使得对于所有 \( \theta < \theta_{\beta}^u \)，没有代理人选择参与。\( \theta_{\beta}^u \) 随着 \( \beta \) 的减小而增大。

This proposition can be understood as follows. Suppose that an agent receives a high signal $s _ { \beta , 0 } \mathop { = } \bar { \theta } _ { \beta } + \varepsilon$ . The agent opts in, because even if every other agent opts out, the focal agent’s private returns make opting in worthwhile. We can eliminate other actions of such an agent as “dominated strategies.” Now, consider an agent with a slightly lower signal $s _ { \beta , 1 }$ . The agent “knows” that all agents with a signal $\stackrel { \cdot } { s _ { \beta , 0 } } \geq \stackrel { \overline { { \theta } } } { \beta } + \stackrel { \cdot } { \varepsilon }$ will opt in, and the measure of such agents must be close to 1, since $s _ { \beta , 1 }$ is only slightly lower than $s _ { \beta , 0 }$ .Thus, our new focal agent opts in as well. We have eliminated dominated strategies a second time. We can repeat this exercise until we converge to a unique $s _ { \beta } ^ { u } > \underline { { \theta } } _ { \beta }$ . $( s _ { \beta } ^ { u }$ is given a precise mathematical definition in Appendix A.) 这个命题可以理解为：假设一个参与者收到了一个高信号 \( s_{\beta, 0} = \bar{\theta}_\beta + \varepsilon \)。该参与者选择参与，因为即使其他所有参与者都选择退出，该核心参与者的私人收益也使得参与是值得的。我们可以将此类参与者的其他行动视为“占优策略”。现在，考虑一个信号略低的参与者 \( s_{\beta, 1} \)。该参与者“知道”所有信号 \( \stackrel{\cdot}{s}_{\beta, 0} \geq \stackrel{\overline{\theta}}{\beta} + \stackrel{\cdot}{\varepsilon} \) 的参与者都会选择参与，而这类参与者的数量必须接近1，因为 \( s_{\beta, 1} \) 仅略低于 \( s_{\beta, 0} \)。因此，我们的新核心参与者也会选择参与。我们再次消除了占优策略。我们可以重复这个过程，直到收敛到一个唯一的 \( s_\beta^u > \underline{\theta}_\beta \)。（\( s_\beta^u \) 在附录A中有精确的数学定义。）

Likewise, if we start with a signal so low $\tilde { s } _ { \beta , 0 } \leq$ $\underline { { \theta } } _ { \beta } - \varepsilon$ that the focal agent opts out regardless of what other agents do, we arrive at the same $s _ { \beta } ^ { u }$ As long as the information environment is not too ““coarse”— that is, as long as $\varepsilon < s _ { \beta } ^ { u } - \underline { { \theta } } ^ { M }$ (and we henceforth assume that $\varepsilon$ is “small” in this sense)—there are states of the world above the optimal threshold $\underline { { \theta } } ^ { M }$ where no agent opts in, because even the most $s _ { \beta } ^ { u }$ b $\theta _ { \beta } ^ { u }$ the theshold develops, converges to $s _ { \beta } ^ { u }$ from below as $\varepsilon \to 0$ . 同样地，如果我们从一个如此低的信号 $\tilde{s}{\beta,0} \leq \underline{\theta}{\beta} - \varepsilon$ 开始，使得焦点代理无论其他代理做什么都会选择退出，我们会得到相同的 $s_{\beta}^u$。只要信息环境不是太“粗糙”——即只要 $\varepsilon < s_{\beta}^u - \underline{\theta}^M$（并且我们此后在这个意义上假设 $\varepsilon$ 是“小的”）——就存在高于最优阈值 $\underline{\theta}^M$ 的世界状态，在这些状态中没有代理选择加入，因为即使最 $s_{\beta}^u$ b $\theta_{\beta}^u$ 阈值的发展，当 $\varepsilon \to 0$ 时从下方收敛到 $s_{\beta}^u$。

Alternatively, consider the thought process of an agent who receives a given signal $s$ The agent does not know θ, but does know that $\theta \in [ s + \varepsilon , s - \varepsilon ]$ . Opting out yields the agent 0. Opting in yields the agent an expected value (net of effort), calculated by integrating out θ over $[ s + \varepsilon , s - \varepsilon ]$ . The lowest $s$ where this calculation yields a nonnegative outcome, given that other agents are engaged in a similar reasoning process, is $s _ { \beta } ^ { u }$ . Alternatively, consider the thought process of an agent who receives a given signal \( s \). The agent does not know \( \theta \), but does know that \( \theta \in [ s + \varepsilon, s - \varepsilon ] \). Opting out yields the agent 0. Opting in yields the agent an expected value (net of effort), calculated by integrating out \( \theta \) over \( [ s + \varepsilon, s - \varepsilon ] \). The lowest \( s \) where this calculation yields a nonnegative outcome, given that other agents are engaged in a similar reasoning process, is \( s_\beta^u \).

The implication is that there is again a friction associated with lack of agent coordination. Now, however, that friction manifests as a gap between the threshold level of the state variable above which it would be value maximizing for an ecosystem to arise $\left( \underline { { \theta } } ^ { M } \right)$ and the equilibrium threshold $\left( \theta _ { \beta } ^ { u } \right)$ .For $\theta \in \left( \underline { { \theta } } ^ { M } , \theta _ { \beta } ^ { u } \right)$ , it would be value maximizing for all agents to opt in, but all agents opt out, so no ecosystem develops at all. 这意味着再次出现了与代理缺乏协调相关的摩擦。然而，这种摩擦表现为状态变量的阈值水平之间的差距：生态系统出现时对其具有价值最大化意义的阈值（$\underline{\theta}^M$），以及均衡阈值（$\theta_\beta^u$）。对于 $\theta \in (\underline{\theta}^M, \theta_\beta^u)$，所有代理选择加入会使价值最大化，但所有代理都选择退出，因此根本不会形成生态系统。

As before, there is also friction associated with lead firm value capture that aggravates the friction associated with lack of agent coordination. If $\beta$ declines, the lead firm captures more of the value that would otherwise be captured by agents, reducing the incentive of agents to opt in. Agents who receive signals that are close to but higher than the previous $s _ { \beta } ^ { u }$ accordingly opt out. Agents with somewhat (more optimistic) higher signals anticipate that the first set of agents with slightly (more pessimistic) lower signals will opt out and will, because of the complementarities in effort $\left( \Delta \right)$ , opt out too. Thus, value capture by the lead firm has a ripple effect: The lower the $\beta$ , the higher the equilibrium threshold above which the ecosystem arises. As before, there is also friction associated with lead firm value capture that aggravates the friction associated with lack of agent coordination. If $\beta$ declines, the lead firm captures more of the value that would otherwise be captured by agents, reducing the incentive of agents to opt in. Agents who receive signals that are close to but higher than the previous $s _ { \beta } ^ { u }$ accordingly opt out. Agents with somewhat (more optimistic) higher signals anticipate that the first set of agents with slightly (more pessimistic) lower signals will opt out and will, because of the complementarities in effort $\left( \Delta \right)$ , opt out too. Thus, value capture by the lead firm has a ripple effect: The lower the $\beta$ , the higher the equilibrium threshold above which the ecosystem arises.

A notable feature of the discrete effort choice model is its dynamics, as we now discuss. Equilibrium value created, $V ^ { u , \beta }$ , as a function of $\theta$ is as follows: 离散努力选择模型的一个显著特征是其动态性，我们现在来讨论。由均衡值$V^{u,\beta}$（作为$\theta$的函数）的定义如下：

\begin{array} { c c c } { { 0 } } & { { \Delta = 0 } } & { { i f } } & { { \theta < \theta _ { \beta } ^ { u } = s _ { \beta } ^ { u } - \varepsilon } } \ { { \left( \left( \theta + \Delta \right) Y - e \right) \Delta } } & { { \Delta = \displaystyle \frac { \theta + \varepsilon - s _ { \beta } ^ { u } } { 2 \varepsilon } } } & { { i f } } & { { \theta _ { \beta } ^ { u } \leq \theta \leq s _ { \beta } ^ { u } + \varepsilon } } \ { { \left( \theta + 1 \right) Y - e } } & { { \Delta = 1 } } & { { i f } } & { { \theta > s _ { \beta } ^ { u } + \varepsilon } } \end{array}

If $\theta < \theta _ { \beta } ^ { u }$ , no agent receives a signal above $s _ { \beta } ^ { u }$ , so no agent opts in $( \Delta = 0 )$ and value created is 0. If $\theta { \stackrel { . } { > } } s _ { \beta } ^ { u } + \varepsilon$ . every agent receives a signal above $s _ { \beta } ^ { u }$ , so every agent opts in $( \Delta = 1 )$ and value created increases linearly in $\theta$ . In a small band around $s _ { \beta } ^ { u }$ , value created increases at a faster than linear rate in’θ, because the proportion of agents that opt in is also increasing. Thus, as depicted in Figure 5, the equilibrium has the character of a tipping point.” Value created “jumps” from nothing to a significant level across a potentially very small region of θ. The better the information environment— that is, the smaller $\varepsilon$ the more “sudden” the jup. 如果 $\theta < \theta_{\beta}^u$，则没有代理收到高于 $s_{\beta}^u$ 的信号，因此没有代理选择加入（$\Delta = 0$），创造的价值为 0。如果 $\theta \stackrel{.}{>} s_{\beta}^u + \varepsilon$，则每个代理都收到高于 $s_{\beta}^u$ 的信号，因此每个代理都选择加入（$\Delta = 1$），且创造的价值随 $\theta$ 线性增加。在 $s_{\beta}^u$ 附近的一个小范围内，创造的价值随 $\theta$ 的增长速率快于线性速率，因为选择加入的代理比例也在增加。因此，如图 5 所示，均衡具有临界点的特征。“创造的价值”会在 $\theta$ 的一个可能非常小的区域内从 0 跳跃到显著水平。信息环境越好——即 $\varepsilon$ 越小——跳跃就越“突然”。

These dynamics have an interesting implication. Suppose that the true level of the state variable is just slightly smaller than $\theta _ { \beta } ^ { u }$ The friction associated with lead firm market and bargaining power cause all agents to opt out, making it impossible for the lead firm to operate, too. Surprisingly, value capture by the lead firm with market and bargaining power would prevent an ecosystem from arising, yet no lead firm may exist to exercise that power, leaving little empirical evidence, like two ships that pass in the night without colliding. However, if the putative lead firm had less market or bargaining power, the ecosystem would arise and thrive. 这些动态具有一个有趣的含义。假设状态变量的真实水平仅略小于 $\theta _ { \beta } ^ { u }$，那么与领先企业市场地位和议价能力相关的摩擦会导致所有参与者退出，这也使得领先企业无法运营。令人惊讶的是，拥有市场和议价能力的领先企业进行价值捕获会阻碍生态系统的形成，但可能不存在能够行使这种权力的领先企业，从而留下很少的实证证据，就像两艘夜航船擦肩而过却未相撞。然而，如果假设的领先企业缺乏市场或议价能力，生态系统就会形成并蓬勃发展。

DISCUSSION

讨论

Value capture by a shareholder-owned lead firm with market or bargaining power severely aggravates the coordination problem that arises when agents do not internalize the positive complementarities from their activities, not only reducing value creation in the ecosystem but possibly suffocating it out of existence. Thus, the standard prescription that a lead firm should seize market and bargaining power in an ecosystem needs to be tempered by the realization that doing so can dissuade other agents from cocreating value to the point that it harms the lead firm. 当具有市场或议价能力的股东所有制牵头企业捕获价值时，会严重加剧因代理人未将其活动产生的正向互补性内部化而产生的协调问题，不仅会减少生态系统中的价值创造，还可能使其因窒息而不复存在。因此，关于牵头企业应在生态系统中攫取市场和议价能力的标准建议，需要因以下认识而有所缓和：这样做可能会劝阻其他代理人共同创造价值，以至于损害牵头企业自身。

FIGURE 5 Equilibrium Value Created as a Function of Business Conditions with Discrete Effort Choices
图5 离散努力选择下均衡值随商业条件变化的函数关系

Notes: This figure graphs net value created $\left( V ^ { u , \beta } \right)$ as a function of the state variable () for two different levels ofnoise in agents’ sigals. 注：该图绘制了净创造价值 $\left( V^{u,\beta} \right)$ 作为状态变量（）的函数，其中考虑了代理信号中两种不同的噪声水平。

Conversely, if agents generate negative complementarities the lead firm’s value capture dissuades agents from imposing these negative complementarities on each other, resulting in more total value creation if the negative complementarities are of sufficient magnitude. Thus, monopoly power, which is a source of inefficiency in the standard economic market model, may in fact be a socially useful tool for managing ecosystems and other communities with public “bads.” 相反，如果代理方产生负互补性，核心企业的价值捕获会阻止代理方相互施加这些负互补性，从而在负互补性达到足够程度时创造更多的总价值。因此，在标准经济市场模型中作为无效率来源的垄断权力，实际上可能是管理生态系统和其他存在公共“坏品”的社区的社会有用工具。

We modeled the optimal contract by inverting the standard order of the biform game, such that bargaining over value capture occurred first, a bespoke combination of noncooperative and cooperative game theory that has been rare in the formal strategy literature (Ross, 2018). Although the contract solves the coordination problem among the lead firm and agents, if complementarities are positive the agents would need to capture over $1 0 0 %$ of cocreated value, which may be hard to effectuate in practice. 我们通过颠倒双形式博弈的标准顺序来构建最优合同，使得价值捕获的谈判首先进行，这是一种非合作博弈论与合作博弈论的定制化结合，在正式战略文献中较为罕见（Ross, 2018）。尽管该合同解决了牵头企业与代理方之间的协调问题，但如果互补性为正，代理方将需要捕获超过100%的共创价值，这在实践中可能难以实现。

If a lead firm is a cooperative that accords financial claims in proportion to transaction volume, then its value capture does not dampen agent incentives, because the claim an agent has on the value the lead firm creates with other agents compensates for the lead firm’s share of the value cocreated with the focal agent. Thus, the cooperative form increases value creation in an ecosystem if and only if complementarities are positive or not too negative and the lead firm has market or bargaining power. This previously unexplored link between a lead firm’s value capture and a canonical governance feature of cooperatives provides a new motivation for their existence. 如果牵头企业是一家按交易量比例分配财务权益的合作社，那么其价值捕获不会削弱代理激励，因为代理对牵头企业与其他代理共同创造的价值所拥有的权益，足以弥补牵头企业与核心代理共同创造的价值中其所占的份额。因此，只有当互补性为正或不太过负面，且牵头企业拥有市场或议价能力时，合作社形式才会在生态系统中增加价值创造。此前未被探索的牵头企业价值捕获与合作社典型治理特征之间的这种联系，为合作社的存在提供了新的动机。

If agents can self-organize to coordinate on the level of effort that maximizes their value capture, equilibrium value creation in the ecosystem is unambiguously higher if complementarities are positive but may be lower if complementarities are negative. This may explain why we tend to observe self-organization more often in ecosystems for open source software and the development of common technical standards, where, at least in the early years of the ecosystem, innovation spillovers may make complementarities net positive. 如果代理能够自我组织以协调在最大化其价值捕获的努力水平上，那么如果互补性为正，生态系统中的均衡价值创造无疑会更高；但如果互补性为负，均衡价值创造可能会更低。这或许可以解释为什么我们往往在开源软件生态系统和通用技术标准开发中更常观察到自我组织——在这些生态系统中，至少在生态系统的早期阶段，创新溢出可能会使互补性净为正。

A lead firm prefers to capture a larger share of a smaller pie than a smaller share of the largest pie. Therefore, the interest of the lead firm in maximizing its value capture and the interest of the ecosystem as such in maximizing total value created may be misaligned. If we take a stakeholder view of the ecosystem (Klein, Mahoney, McGahan, & Pitelis, 2012), then management of a lead firm’s market and bargaining power, whether by nudging it toward the value-maximizing level or mitigating its baleful effects through other governance devices, is critical. 领先企业宁愿占据较小蛋糕的较大份额，也不愿占据最大蛋糕的较小份额。因此，领先企业在最大化自身价值捕获方面的利益，与整个生态系统在最大化创造的总价值方面的利益可能存在不一致。如果我们从生态系统的利益相关者视角出发（Klein, Mahoney, McGahan, & Pitelis, 2012），那么领先企业市场和议价能力的管理——无论是通过促使其达到价值最大化水平，还是通过其他治理手段减轻其有害影响——都至关重要。

In that regard, an ecosystem’s lead firm is an “enfranchised stakeholder” that may resist curtailment of its market and bargaining power, even if that would increase the health of its ecosystem (Klein, Mahoney, McGahan, & Pitelis, 2019). Yet, the difficulty of managing the lead firm’s private interests suggests that establishing social norms, institutions, and governance structures (Dyer & Singh, 1998) to do so could serve as a competitive advantage. In other words, although dynamic capabilities are typically based on valuable, rare, inimitable, and nonsubstitutable resources at the firm level (Katkalo, Pitelis, & Teece, 2010), the ecosystem is a “metaorganization” (Gulati et al., 2012) for which managing the lead firm’s market and bargaining power could be a “meta-competence” that would serve as a critical dynamic capability (Teece, 2007). Likewise, the literature is clear that contracting is hard in ecosystems, and our optimal contract illustrated this, since it required agents to post a (possibly large) bond. However, contracting skill can improve over time, both at individual firms and relationally (Argyres & Mayer, 2007; Mayer & Argyres, 2004). So, ecosystems that can contract effectively may have a competitive advantage. 在这方面，生态系统的核心企业是一个“被赋予权利的利益相关者”，它可能会抵制对其市场和议价能力的限制，即使这会增强生态系统的健康度（Klein, Mahoney, McGahan, & Pitelis, 2019）。然而，管理核心企业私人利益的难度表明，建立社会规范、制度和治理结构（Dyer & Singh, 1998）来实现这一点可能会成为一种竞争优势。换句话说，尽管动态能力通常基于企业层面有价值、稀缺、难以模仿且不可替代的资源（Katkalo, Pitelis, & Teece, 2010），但生态系统是一个“元组织”（Gulati et al., 2012），对其而言，管理核心企业的市场和议价能力可能是一种“元能力”，并将成为关键的动态能力（Teece, 2007）。同样，文献明确指出，生态系统中的契约制定很困难，而我们的最优契约也体现了这一点，因为它要求代理人缴纳（可能数额较大的）保证金。然而，契约制定能力可以随着时间推移而提升，无论是在单个企业层面还是关系层面（Argyres & Mayer, 2007; Mayer & Argyres, 2004）。因此，能够有效制定契约的生态系统可能具有竞争优势。

Clusters and agglomeration economies are groupings of firms whose proximity improves productivity by driving innovation, knowledge flows, new business formation, and the pooling of skilled labor (Marshall, 1920; Porter, 1990; Tallman, Jenkins, Henry, & Pinch, 2004) (positive complementarities), but which may suffer from “congestion” effects (negative complementarities), and in which interfirm cooperation and value creation and value capture (appropriation) are critical (Pitelis, 2012). Entrepreneurial ecosystems operate similarly, but through digital rather than spatial affordances, and their outputs are new ventures (Autio, Nambisian, Thomas, & Wright, 2018). These configurations resemble ecosystems as contemplated herein but without a lead firm as part of their definition (e.g., Pitelis, 2012: 1361). Even so, our analysis suggests that value capture by important players in these configurations (e.g., local capital providers or a union) may have the same disincentivizing effects on interfirm cooperation that we considered in our model, with the implications for value creation depending on the nature of the relevant complementarities. 集群和集聚经济是企业的集群，其邻近性通过推动创新、知识流动、新企业形成以及熟练劳动力的集中（Marshall，1920；Porter，1990；Tallman、Jenkins、Henry和Pinch，2004）（正互补性）来提高生产率，但可能会遭受“拥堵”效应（负互补性），并且在这种集群中，企业间合作以及价值创造和价值捕获（占有）至关重要（Pitelis，2012）。创业生态系统的运作方式类似，但通过数字而非空间的便利条件，其产出是新企业（Autio、Nambisian、Thomas和Wright，2018）。这些配置类似于本文所考虑的生态系统，但不将核心企业作为其定义的一部分（例如，Pitelis，2012：1361）。即便如此，我们的分析表明，这些配置中重要参与者（例如，地方资本提供者或工会）的价值捕获可能会对企业间合作产生与我们在模型中考虑的相同的抑制作用，而价值创造的影响取决于相关互补性的性质。

More generally, in ecosystems with positive complementarities, we should observe arrangements to mitigate the potentially suffocating effects of value capture by lead firms. These may take several forms, such as competition for the lead firm’s contribution to cocreated value in terms of actual rivals and lower barriers to entry. Not all markets can support multiple competing firms, however, because network effects among users, switching costs, economies of scale in production, or proprietary technology lead to monopoly. In these cases, we should expect ecosystems to strive to reduce the suffocating effects of the lead firm’s value capture through reducing its actual bargaining power, deciding on the division of value before costly investments are undertaken—as in our optimal contract—or the lead firm adopting the cooperative form. 更一般地说，在存在正向互补性的生态系统中，我们应该观察到一些安排，以减轻领先企业价值攫取可能带来的窒息性影响。这些安排可能有多种形式，例如实际竞争对手对领先企业为共创价值所做贡献的竞争，以及降低市场进入壁垒。然而，并非所有市场都能支持多家竞争企业，因为用户之间的网络效应、转换成本、生产规模经济或专有技术会导致垄断。在这些情况下，我们应该期望生态系统通过以下方式努力减轻领先企业价值攫取的窒息性影响：降低其实际议价能力，在进行高成本投资之前确定价值分配（如我们的最优合约中所述），或者领先企业采用合作形式。

Conversely, in ecosystems that generate negative complementarities—such as the depletion of a common resource (e.g., a fishing stock), or “bads” like pollution—we should observe lead firms with sufficient market or bargaining power to capture enough value to disincentivize economic activity down toward the level that maximizes the value created in the ecosystem. It follows that lead firms will tend to be shareholder owned and face less competition in such ecosystems. 相反，在产生负互补性的生态系统中（例如，共有资源的枯竭，如渔业资源，或污染等“不良品”），我们应该观察到主导企业拥有足够的市场或议价能力，以获取足够的价值，从而抑制经济活动降至生态系统中创造的价值最大化的水平。因此，主导企业往往会是股东所有的，并且在这样的生态系统中面临较少的竞争。

Consider as an example the market for cranberry products in North America. Cranberry farms generate positive complementarities in their production activities if they pool processing and marketing: the greater the production of each farm, the further the pooled operations move down an average cost curve, improve bargaining power with customers, and diversify risk. However, the resulting entity, such as the company Ocean Spray, has significant market and bargaining power vis-à-vis these farms, because of its size and the cost of transporting produce. Hence, Ocean Spray may be a cooperative so that its value capture does not dampen the incentive of farms to produce cranberries. As a contrary example, many small merchants use Amazon to compete for the same customers in their category, possibly generating negative complementarities as contemplated herein. Amazon has been accused of abusing its market and bargaining power vis-à-vis these merchants (Soper & Brody, 2019), but Amazon’s value capture may serve to reduce merchant efforts toward the value-maximizing level. 以北美蔓越莓产品市场为例。如果蔓越莓农场集中进行加工和营销，其生产活动会产生积极的互补性：每个农场的产量越大，集中运营就越能降低平均成本曲线，增强与客户的议价能力，并分散风险。然而，由此形成的实体（如Ocean Spray公司）由于规模和产品运输成本的原因，对这些农场具有显著的市场和议价能力。因此，Ocean Spray可能是一个合作社，以确保其价值获取不会削弱农场生产蔓越莓的积极性。作为相反的例子，许多小型商家利用亚马逊在其类别中争夺相同的客户，这可能会产生本文所考虑的负互补性。亚马逊被指控滥用其相对于这些商家的市场和议价能力（Soper & Brody，2019），但亚马逊的价值获取可能有助于降低商家向价值最大化水平努力的程度。

Our results have implications for public policy. First, the nature of complementarities in an economic community is an important, unappreciated contingency in evaluating whether monopoly power there is harmful to the public. Second, fostering the use of the cooperative form among economic actors may be a useful way to mitigate the effects of their market and bargaining power without resorting to more intrusive regulations. Third, some potentially valuable ecosystems may not exist because the firms that would lead them would have too much market and bargaining power, suggesting that antitrust policy should focus not only on actual monopolies but also on those that would likely form if a supporting ecosystem arose around them. 我们的研究结果对公共政策有启示。首先，在评估一个经济共同体中的互补性本质是否会对公众造成危害时，这种互补性是一个重要但未被充分认识的意外因素。其次，促进经济主体采用合作形式，可能是在不诉诸更具侵入性监管的情况下，减轻其市场和议价能力影响的有效途径。第三，一些潜在有价值的生态系统可能不存在，因为引领这些生态系统的企业会拥有过大的市场和议价能力，这表明反垄断政策不仅应关注实际存在的垄断，还应关注如果围绕这些企业形成支持性生态系统，它们可能形成的垄断。

Large firms have common resources that are used by profit centers, such as document production, human resources, central research and development, and advertising; as well as financial resources, such as tax capacity—or the firm’s tax bill that may be amortized using leases and investment in depreciable assets. In accessing these resources, profit centers generate both negative complementarities (e.g., by using up the resource) and positive complementarities (e.g., internal technology spillovers). Using transfer pricing or managerial contracts to align managers’ incentives with respect to these complementarities is often infeasible, requiring the use of internal markets. In these cases, our theory suggests that if complementarities are negative, allowing the managers of the common resource to price monopolistically may result in the greatest value creation; conversely, if complementarities are positive, the pricing power of the managers of the common resource should be mitigated though external sourcing or cooperative governance, whereby revenue generated by the common resource is redistributed to the profit centers in proportion to their use of it. 大型企业拥有被利润中心使用的共同资源，例如文档制作、人力资源、中央研发和广告；以及财务资源，如纳税能力——即企业的税单，可通过租赁和对折旧资产的投资进行摊销。在获取这些资源时，利润中心会产生负互补性（例如，耗尽资源）和正互补性（例如，内部技术溢出）。使用转让定价或管理合同来使管理者的激励与这些互补性保持一致通常不可行，因此需要使用内部市场。在这种情况下，我们的理论表明，如果互补性是负的，允许共同资源的管理者进行垄断定价可能会创造最大价值；相反，如果互补性是正的，则应通过外部采购或合作治理来减轻共同资源管理者的定价权，将共同资源产生的收入按利润中心对其的使用比例重新分配给它们。

Our analysis has limitations that could serve as fruitful topics for future research. Our model was abstract because we focused on first-order questions of lead firm market and bargaining power, but some ecosystems have important application-specific features. One example is platforms associated with firms like Uber, Visa and Mastercard, or Google, where network effects play a key role. Likewise, De Beers was historically the lead firm for the diamond industry. One of its activities was ensuring that mines, cutters, wholesalers, and jewelers acted to maintain prices, which is distinct from the standard way in which lead firms cocreate value in other ecosystems. We need to understand how these factors interact with lead firm value capture and complementarities. 我们的分析存在局限性，这些局限性可能成为未来研究的有益主题。我们的模型较为抽象，因为我们专注于主导企业的市场和议价能力这一一阶问题，但一些生态系统具有重要的特定应用功能。一个例子是与优步、维萨、万事达卡或谷歌等公司相关的平台，在这些平台中，网络效应发挥着关键作用。同样，戴比尔斯（De Beers）在历史上曾是钻石行业的主导企业。其活动之一是确保矿场、切割商、批发商和珠宝商采取行动维持价格，这与主导企业在其他生态系统中共同创造价值的标准方式不同。我们需要了解这些因素如何与主导企业的价值获取和互补性相互作用。

Our model was static. In a dynamic setting, a lead firm might not only consider how its market and bargaining power influenced its profits in the short-tomedium term but also how maintaining a large market share at the expense of short-term profits might foreclose entry. Likewise, value capture by the lead firm may motivate agents to replace the lead firm, merge with other agents to improve bargaining power, make their product less cospecialized, or seek to be acquired by the lead firm. We did not model opportunistic behavior, for instance, where the lead firm induces investments by complementors with a view to entering their markets later. Similarly, we assumed that complementarities were exogenous, except for agent effort, but an agent might foster negative complementarities in response to a lead firm’s market or bargaining power, perhaps driving a competitor out of business, to justify its membership of the ecosystem, or a lead firm might foster negative complementarities to encourage an agent to accept an acquisition offer. 我们的模型是静态的。在动态环境中，核心企业不仅可能考虑其市场和议价能力如何影响其短期至中期的利润，还可能考虑以牺牲短期利润为代价维持较大市场份额可能如何阻止新进入者。同样，核心企业的价值攫取可能会促使代理方替换核心企业、与其他代理方合并以提高议价能力、降低其产品的互补性，或寻求被核心企业收购。例如，我们没有对机会主义行为进行建模，即核心企业诱导互补方进行投资，以便日后进入其市场。同样，我们假设互补性是外生的，除非代理方的努力，但代理方可能会对核心企业的市场或议价能力做出反应，培育负向互补性，例如将竞争对手排挤出市场以证明其在生态系统中的成员资格，或者核心企业可能培育负向互补性以鼓励代理方接受收购提议。

We considered the possibility that the lead firm faces competition for its role in value creation, but we did not consider agents joining multiple ecosystems or moving among them, nor competition between ecosystems. We derived the optimal level of effort to maximize agents’ value capture but did not model how agents could coordinate on this effort level; and we derived the optimal level of market and bargaining power for the lead firm to maximize its value capture but did not model how the lead firm could influence its market or bargaining power. 我们考虑了核心企业在价值创造过程中面临竞争的可能性，但并未考虑代理方加入多个生态系统或在不同生态系统间转移的情况，也未考虑生态系统之间的竞争。我们推导了最大化代理方价值获取的最优努力水平，但未建模代理方如何在这一努力水平上进行协调；并且我们推导了核心企业为最大化自身价值获取所需的最优市场及议价能力水平，但未建模核心企业如何影响其市场或议价能力。

In bargaining over value, biform game formalism treats all agents in the ecosystem the same, be they suppliers to, buyers from, or not a direct counterparty of the lead firm. This makes our results more general, but an agent’s place in industry architecture may affect how bargaining over value occurs. We maintained the standard assumption that production costs do not vary with governance arrangements, but that may not be true in practice. There is a need for more research in these areas. We hope that our work serves as a starting point for these efforts. 在价值谈判中，双态博弈形式主义将生态系统中的所有主体一视同仁，无论他们是领先企业的供应商、买方，还是非直接交易对手。这使得我们的研究结果更具普遍性，但主体在行业架构中的位置可能会影响价值谈判的发生方式。我们维持了生产成本不随治理安排变化的标准假设，但这在实践中可能并不成立。这些领域需要更多的研究。我们希望我们的工作能为这些努力提供一个起点。

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Kose John (kjohn@stern.nyu.edu) is the Charles William Gerstenberg Professor of Banking and Finance at New York University, Stern School of Business. He holds a PhD from the University of Florida. His recent research interests include FinTech, Blockchain economics, corporate bankruptcy, dividend policy, corporate governance, top-management compensation, social entrepreneurship, institutions and innovation. Kose John (kjohn@stern.nyu.edu) 是纽约大学斯特恩商学院的查尔斯·威廉·格斯坦伯格银行与金融教授。他拥有佛罗里达大学的博士学位。他近期的研究兴趣包括金融科技、区块链经济学、企业破产、股息政策、公司治理、高管薪酬、社会创业、机构与创新。

David Gaddis Ross (david.ross@warrington.ufl.edu) is an associate professor at the University of Florida’s Warrington College of Business Administration. He received his PhD from New York University’s Stern School of Business. His research examines the governance of firms, with a focus on gender in management and the formal foundations of strategy. David Gaddis Ross (david.ross@warrington.ufl.edu) 是佛罗里达大学沃灵顿商学院的副教授。他拥有纽约大学斯特恩商学院的博士学位。他的研究考察公司治理，重点关注管理中的性别问题以及战略的正式基础。

X X

APPENDIX A

附录 A

PROOF OF PROPOSITIONS

命题证明

Proof of Proposition 1

命题1的证明

Since the integral in the parentheses in Equation (2) integrates to $e _ { i }$ , the solution is readily obtained by differentiation: 由于式(2)中括号内的积分结果为\( e_i \)，因此可通过微分轻松得到解：

\begin{array} { l } { \displaystyle { \frac { d V } { d e _ { i } } = \left( \theta + 2 \delta e _ { i } \right) Y - e _ { i } = 0 } } \ { \displaystyle { e _ { i } ^ { M } = \frac { \theta Y } { 1 - 2 \delta Y } } } \end{array}

The integral in the parentheses in Equation (3) integrates to $e _ { j }$ , because each agent takes the efforts of other agents as given. Differentiating and applying symmetry such that $e _ { j } = e _ { i }$ yield: 式(3)中括号内的积分结果为\( e_j \)，因为每个主体将其他主体的努力视为给定值。对其求导并应用对称性（即\( e_j = e_i \)）后得到：

\begin{array} { l } { \displaystyle \frac { d u _ { i } ^ { \beta } } { d e _ { i } } = \left( \theta + \delta e _ { j } \right) Y \beta - e _ { i } = 0 } \ { \displaystyle e _ { i } ^ { u , \beta } = \frac { \theta Y \beta } { 1 - \delta Y \beta } } \end{array}

Claim. This is the only equilibrium. That it is an equilibrium is clear from that fact that it satisfies the first-order conditions for every agent. That there can be no asymmetric equilibrium follows from the fact that each agent’s objective function is (i) quadratic in its own effort with a negative square term and (ii) a positive linear function of the efforts of other agents. Therefore, in any candidate asymmetric equilibrium in which $e _ { i } > e _ { j }$ , the marginal returns to effort for agent $j$ exceed those of agent $_ i$ Thus, the first-order conditions cannot be simultaneously zero for both agents, meaning at least one of them would have a profitable deviation. 主张。这是唯一的均衡。它是一个均衡是显而易见的，因为它满足了每个参与者的一阶条件。不存在非对称均衡的原因在于，每个参与者的目标函数是（i）其自身努力的二次函数，且带有负平方项，以及（ii）其他参与者努力的正线性函数。因此，在任何候选的非对称均衡中，若\( e_i > e_j \)，则参与者\( j \)的努力边际回报超过参与者\( i \)的边际回报。因此，两个参与者的一阶条件不可能同时为零，这意味着至少其中一个参与者会有有利可图的偏离。

That $e _ { i } ^ { M } > e _ { i } ^ { u , \beta }$ follows from inspection. That e decreases in $\beta$ follows from the bounds on δ and: That \( e_{i}^{M} > e_{i}^{u, \beta} \) follows from inspection. That \( e \) decreases in \( \beta \) follows from the bounds on \( \delta \) and:

\frac { d } { d \beta } e _ { i } ^ { u , \beta } = \frac { d } { d \beta } \frac { \theta Y \beta } { 1 - \delta Y \beta } = \frac { \theta Y } { \left( 1 - \delta Y \beta \right) ^ { 2 } } > 0 \(\frac{d}{d\beta} e_{i}^{u, \beta} = \frac{d}{d\beta} \frac{\theta Y \beta}{1 - \delta Y \beta} = \frac{\theta Y}{\left(1 - \delta Y \beta\right)^2} > 0\)

Substitution yields 替代产生

V ^ { M } = \frac { Y ^ { 2 } \theta ^ { 2 } } { 2 \bigl ( 1 - 2 \delta Y \bigr ) } > V ^ { u , \beta } = \frac { Y ^ { 2 } \theta ^ { 2 } \beta \bigl ( 2 - \beta \bigr ) } { 2 \bigl ( 1 - \delta Y \beta \bigr ) ^ { 2 } } \( V^{\{ M \}} = \frac{Y^2 \theta^2}{2 \bigl(1 - 2 \delta Y \bigr)} > V^{\{ u, \beta \}} = \frac{Y^2 \theta^2 \beta \bigl(2 - \beta \bigr)}{2 \bigl(1 - \delta Y \beta \bigr)^2} \)

That $V ^ { M } - V ^ { u , \beta }$ decreases in $\beta$ follows from the bounds on δ and: That \( V^M - V^{u, \beta} \) decreases in \( \beta \) follows from the bounds on \( \delta \) and:

\frac { d } { d \beta } { V } ^ { u , \beta } = \frac { d } { d \beta } \frac { Y ^ { 2 } \theta ^ { 2 } \beta ( 2 - \beta ) } { 2 { \left( 1 - \delta Y \beta \right) } ^ { 2 } } = - \frac { Y ^ { 2 } \theta ^ { 2 } { \left( { \left( \delta Y - 1 \right) \beta + 1 } \right) } } { { \left( \delta Y \beta - 1 \right) } ^ { 3 } } > 0 \frac{d}{d\beta} V^{u,\beta} = \frac{d}{d\beta} \frac{Y^2 \theta^2 \beta (2 - \beta)}{2 \left(1 - \delta Y \beta\right)^2} = -\frac{Y^2 \theta^2 \left( \left( \delta Y - 1 \right) \beta + 1 \right)}{\left( \delta Y \beta - 1 \right)^3} > 0

Proof of Proposition 2

命题2的证明

The difference between $e _ { i } ^ { M }$ and $e _ { i } ^ { u , \beta }$ is a linear function of δ with equality at $\hat { \delta } = - ( 1 - \beta ) / Y \beta < 0$ It is obvious from inspection that δ increases (becomes less negative) as $\beta$ increases toward its upper bound of 1. $e_{i}^{M}$ 与 $e_{i}^{u,\beta}$ 之间的差异是 $\delta$ 的线性函数，在 $\hat{\delta} = -(1 - \beta)/Y\beta < 0$ 时等式成立。显然，当 $\beta$ 向其上限 1 增加时，$\delta$ 会增大（变得不那么负）。

Proof of Proposition 3

命题3的证明

Solving for $\hat { \boldsymbol { \beta } }$ by equating $e _ { i } ^ { M }$ and $e _ { i } ^ { u , \beta }$ ensures that it is a global maximum. That $V ^ { u , \beta }$ is continuous and has no other local maxima in $\beta$ is obvious from the linearity of the numerator in Equation (A5). But $V ^ { u , \beta }$ can have both concave and convex regions, as can be seen from the fact that $\beta$ is also in the denominator in Equation (A5). 通过令 \( e_i^M \) 和 \( e_i^{u, \beta} \) 相等来求解 \( \hat{\boldsymbol{\beta}} \)，可确保其为全局最大值。从式 (A5) 中分子的线性性可以明显看出，\( V^{u, \beta} \) 是连续的，并且在 \( \beta \) 中没有其他局部最大值。但是，由于 \( \beta \) 在式 (A5) 的分母中，\( V^{u, \beta} \) 可能同时包含凹区域和凸区域，这一点可以从该事实中看出。

Proof of Proposition 4

命题4的证明

Denote by $\pi _ { i }$ the portion of the lead firm’s profits to which agent $_ i$ is entitled by virtue of the lead firm’s cooperative form. We have: 记 $\pi_i$ 为代理 $i$ 根据牵头企业的合作形式有权获得的牵头企业利润份额。我们有：

\pi _ { i } = \frac { e _ { i } \left( \theta + \delta \displaystyle \int _ { j \in A , j \neq i } e _ { j } d j \right) } { \displaystyle \int _ { j \in A } e _ { j } \left( \theta + \delta \displaystyle \int _ { k \in A , k \neq j } e _ { k } d k \right) d j } π_i = \frac{e_i \left( \theta + \delta \int_{j \in A, j \neq i} e_j dj \right)}{\int_{j \in A} e_j \left( \theta + \delta \int_{k \in A, k \neq j} e_k dk \right) dj}

\times Y ( 1 - \beta ) \int _ { j \in A } e _ { j } \left( \theta + \delta \int _ { k \in A , k \neq j } e _ { k } d k \right) d j \times Y ( 1 - \beta ) \int _ { j \in A } e _ { j } \left( \theta + \delta \int _ { k \in A , k \neq j } e _ { k } d k \right) d j

The term to the left of the multiplication symbol is the agent’s share of total patronage (value cocreation) and thus of the lead firm’s value capture. The term to the right of the multiplication symbol is the lead firm’s share of value created. We then have by canceling terms: 乘法符号左侧的项是代理方在总惠顾（价值共创）中的份额，因此也是牵头企业价值捕获的份额。乘法符号右侧的项是牵头企业创造的价值份额。然后我们通过消去项得到：

\pi _ { i } = e _ { i } \left( \theta + \delta \int _ { j \in A , j \neq i } e _ { j } d j \right) Y ( 1 - \beta ) $\pi_{i} = e_{i} \left( \theta + \delta \int_{j \in A, j \neq i} e_{j} dj \right) Y(1 - \beta)$

Hence, the function that each agent $i$ maximizes when the lead firm is a cooperative is 因此，当主导企业是合作型企业时，每个代理人 \( i \) 最大化的函数是

\begin{array} { c } { { u _ { i } ^ { \beta < 1 } = e _ { i } \left( \theta + \delta \displaystyle \int _ { j \in A , j \neq i } e _ { j } d j \right) Y \beta + \pi _ { i } - \frac { e _ { i } ^ { 2 } } { 2 } } } \ { { { } } } \ { { = e _ { i } \left( \theta + \delta \displaystyle \int _ { j \in A , j \neq i } e _ { j } d j \right) Y - \frac { e _ { i } ^ { 2 } } { 2 } = u _ { i } ^ { \beta = 1 } } } \end{array}

Proof of Proposition 5

命题5的证明

If the lead firm is owned by third party shareholders, then, because $V$ is quadratic in effort, $V ^ { u , \beta = 1 } > V ^ { u , \beta < 1 }$ if and only if 如果牵头公司由第三方股东持有，那么，由于 \( V \) 是努力程度的二次函数，当且仅当 \( V^{u, \beta = 1} > V^{u, \beta < 1} \) 时

\begin{array} { l } { { \displaystyle e _ { i } ^ { u , \beta = 1 } - e _ { i } ^ { M } < e _ { i } ^ { M } - e _ { i } ^ { u , \beta < 1 } } } \ { { \displaystyle \frac { \theta Y } { 1 - \delta Y } - \frac { \theta Y } { 1 - 2 \delta Y } < \frac { \theta Y } { 1 - 2 \delta Y } - \frac { \theta Y \beta } { 1 - \delta Y \beta } } } \end{array}

The bounds of this inequality are quadratic in δ and have only one negative solution, namely $\tilde { \delta }$ where $\hat { \delta } < - \left( \sqrt { \beta ( 2 - \beta ) } - \beta \right) / 2 Y \beta = \tilde { \delta } < 0$ We then have: 这个不等式的边界关于δ是二次的，且仅有一个负解，即$\tilde{\delta}$，其中$\hat{\delta} < - \left( \sqrt{\beta (2 - \beta)} - \beta \right) / (2 Y \beta) = \tilde{\delta} < 0$。我们接下来有：

\frac { d } { d \beta } \widetilde { \delta } = \frac { d } { d \beta } \frac { - \left( \sqrt { \beta ( 2 - \beta ) } - \beta \right) } { 2 Y \beta } = \frac { 1 } { 2 Y \beta \sqrt { \beta ( 2 - \beta ) } } > 0 \frac { d } { d \beta } \widetilde { \delta } = \frac { d } { d \beta } \frac { - \left( \sqrt { \beta ( 2 - \beta ) } - \beta \right) } { 2 Y \beta } = \frac { 1 } { 2 Y \beta \sqrt { \beta ( 2 - \beta ) } } > 0

Proof of Proposition 6

命题6的证明

If agents self-organize to exert the optimal effort given $\beta$ , we solve Equation (3) as we did Equation (2) in Proposition 1, i.e., the integral integrates to $e _ { i }$ . We then obtain: 如果代理在给定 $\beta$ 的情况下自发组织以付出最优努力，我们将像在命题 1 中求解方程 (2) 那样求解方程 (3)，即积分结果为 $e_i$。然后我们得到：

\begin{array} { l } { { e _ { i } ^ { S , \beta } = \displaystyle \frac { \theta Y \beta } { 1 - 2 \delta Y \beta } } } \ { { e _ { i } ^ { M } - e _ { i } ^ { S , \beta } = \displaystyle \frac { \theta Y ( 1 - \beta ) } { ( 1 - 2 \delta Y ) ( 1 - 2 \delta Y \beta ) } > 0 } } \ { { e _ { i } ^ { S , \beta } - e _ { i } ^ { u , \beta } = \displaystyle \frac { \delta \theta Y ^ { 2 } \beta ^ { 2 } } { ( 1 - \delta Y \beta ) ( 1 - 2 \delta Y \beta ) } > 0 \Longleftrightarrow \delta > 0 } } \end{array}

Proof of Proposition 7

命题7的证明

Substituting the equilibrium value of $e _ { i } ^ { u , \beta }$ into Equation (1) and multiplying by the lead firm’s share of value created yields: 将 \( e_{i}^{u, \beta} \) 的平衡值代入式 (1)，并乘以领先企业创造的价值份额，得到：

\pi _ { L } ^ { u , \beta } = \frac { Y ^ { 2 } \theta ^ { 2 } \beta ( 1 - \beta ) } { \left( 1 - \delta Y \beta \right) ^ { 2 } } $\pi_{L}^{u, \beta} = \frac{Y^{2} \theta^{2} \beta (1 - \beta)}{\left(1 - \delta Y \beta\right)^{2}}$

Differentiating and setting to zero yields: 求导并令其等于零，得到：

\begin{array} { l } { \displaystyle \frac { d } { d \beta } \pi _ { L } ^ { u , \beta } = - \frac { Y ^ { 2 } \theta ^ { 2 } \left( \left( \delta Y - 2 \right) \beta + 1 \right) } { \left( \delta Y \beta - 1 \right) ^ { 3 } } = 0 } \ { \displaystyle \beta _ { L } ^ { M } = \left( 2 - \delta Y \right) ^ { - 1 } < \hat { \beta } } \end{array}

Proof of Proposition 8

命题8的证明

Given a signal $s$ , the expected value to an agent from opting in if no other agent opts in is: 给定一个信号 \( s \)，如果没有其他代理选择加入，代理选择加入的预期价值为：

\int _ { s - \varepsilon } ^ { s + \varepsilon } \theta Y \beta d \theta - e = s Y \beta - e \int _ { s - \varepsilon } ^ { s + \varepsilon } \theta Y \beta d \theta - e = s Y \beta - e

By assumption, there is a signal $s _ { \beta , 0 } \mathop { = } \bar { \theta } _ { \beta } + \varepsilon$ such that the foregoing expectation is nonnegative. Thus, every agent who receives a signal of $s _ { \beta , 0 }$ or higher opts in. Consider an agent who receives a signal $s < s _ { \beta , 0 }$ . This implies $\theta \in [ s + \varepsilon , s - \varepsilon ]$ .Given the assumption that only those agents with a signal above $s _ { \beta , 0 }$ opt in, the expected value to the focal agent from opting in is: 假设存在一个信号 \( s_{\beta, 0} = \bar{\theta}_\beta + \varepsilon \)，使得上述期望非负。因此，每个收到 \( s_{\beta, 0} \) 或更高信号的参与者都会选择加入。考虑一个收到信号 \( s < s_{\beta, 0} \) 的参与者。这意味着 \( \theta \in [s + \varepsilon, s - \varepsilon] \)。鉴于只有信号高于 \( s_{\beta, 0} \) 的参与者会选择加入，焦点参与者选择加入的期望值为：

\begin{array} { l } { \displaystyle \frac { 1 } { 2 \varepsilon } \int _ { s - \varepsilon } ^ { s + \varepsilon } \left( \theta + \Delta \right) Y \beta d \theta - e } \ { \displaystyle = \frac { 1 } { 2 \varepsilon } \int _ { s - \varepsilon } ^ { s + \varepsilon } \left( \theta + \operatorname* { m a x } \left( \frac { \theta + \varepsilon - s _ { \beta , 0 } } { 2 \varepsilon } , 0 \right) \right) Y \beta d \theta - e } \end{array}

It is clear that for $s$ sufficiently close to (but less than) $s _ { \beta , 0 }$ , Equation (A15) is positive. Define $s _ { \beta , 1 }$ as the lowest $s$ for which Equation (A15) is non-negative. We can then construct a decreasing sequence ${ s _ { \beta , n } }$ , where $s _ { \beta , n }$ is substituted for $s _ { \beta , 0 }$ above and $s _ { \beta , n + 1 }$ is calculated as we calculated $s _ { \beta , 1 }$ . Since the sequence is bounded below by $\underline { { \theta } } _ { \beta }$ and the support of $\theta$ is continuous, the sequence converges to an $s _ { \beta } ^ { u }$ such that: 显然，对于充分接近（但小于）\( s_{\beta,0} \) 的 \( s \)，式 (A15) 为正。定义 \( s_{\beta,1} \) 为使式 (A15) 非负的最小 \( s \)。然后我们可以构造一个递减序列 \( \{ s_{\beta,n} \} \)，其中 \( s_{\beta,n} \) 替换上述的 \( s_{\beta,0} \)，并按计算 \( s_{\beta,1} \) 的方式计算 \( s_{\beta,n+1} \)。由于该序列以 \( \underline{\theta}_\beta \) 为下界，且 \( \theta \) 的支撑集是连续的，因此该序列收敛到某个 \( s_\beta^u \)，满足：

\begin{array} { l } { { \displaystyle \frac { 1 } { 2 \varepsilon } \int _ { s _ { \beta } ^ { u } - \varepsilon } ^ { s _ { \beta } ^ { u } + \varepsilon } \left( \theta + \frac { \theta + \varepsilon - s _ { \beta } ^ { u } } { 2 \varepsilon } \right) Y \beta d \theta - e } } \ { { \displaystyle = \left( s _ { \beta } ^ { u } + \frac { 1 } { 2 } \right) Y \beta - e = 0 } } \end{array}

It is clear that $s _ { \beta } ^ { u } > \underline { { \theta } } ^ { M }$ and that $\theta _ { \beta } ^ { u } \longrightarrow s _ { \beta } ^ { u }$ from below as $\varepsilon \to 0$ . The uniqueness of $s _ { \beta } ^ { u }$ is apparent from the fact that the foregoing expression only has one solution. That $\theta _ { \beta } ^ { u } , s _ { \beta } ^ { u }$ both incease as $\beta$ declines follows from inspection of Equation (A16). 显然，\( s_{\beta}^{u} > \underline{\theta}^{M} \)，并且当\( \varepsilon \to 0 \)时，\( \theta_{\beta}^{u} \)从下方趋近于\( s_{\beta}^{u} \)。\( s_{\beta}^{u} \)的唯一性从上述表达式仅有一个解这一事实中显而易见。通过检查式(A16)可以发现，\( \theta_{\beta}^{u} \)和\( s_{\beta}^{u} \)均随\( \beta \)的减小而增大。

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