Privacy-preserving Information Sharing in Oligopoly Competitions
arXiv:2606.02348v1 [econ.TH] 1 Jun 2026
Yuxin Liu1 , M. Amin Rahimian1 1
Industrial Engineering, University of Pittsburgh [email protected], [email protected]
Information sharing among competing suppliers can improve decision-making under uncertainty, yet strategic concerns regarding rival exploitation often deter voluntary disclosure. We study information-sharing mechanisms in a Cournot oligopoly with uncertain demand, where a platform aggregates suppliers’ signals through privacy-preserving channels and may also possess an exogenous external signal. The central challenge is to balance strategic safety with informational utility: privacy noise reduces the exposure of individual signals, but also lowers the value of the shared information pool. We first characterize a baseline setting in which access to aggregated information is contingent on participation. In a two-firm market without an external signal, firms refuse to share regardless of the privacy level. In an n-firm market, sharing may arise even without privacy safeguards because non-participating firms lose access to the aggregated signal. Building on this baseline, we show that privacy protection alone is insufficient to incentivize disclosure; it must be combined with a sufficiently informative external signal. We further show that firms with more accurate private signals require stronger privacy protection. Overall, our results characterize the sharing-feasible region and highlight the complementarity between privacy design and the external information environment. Key words : information sharing, differential privacy, oligopoly competition, platform design, Bayesian games, strategic firms
1. Introduction Digital platforms such as Amazon increasingly serve as critical coordination hubs for modern supply chains. Suppliers operating on these platforms face substantial uncertainty arising from volatile demand, upstream disruptions, and cost shocks, and timely information exchange can significantly improve production planning, inventory management, and system-wide resilience (Ha et al. 2011, Guan et al. 2020, Zhang 2006). A large operations and supply chain literature has documented that limited information sharing amplifies demand variability and exacerbates inefficiencies such as inventory shortages and price instability, commonly referred to as the bullwhip effect (Lee et al. 1997, Chen et al. 2000). 1
Despite these benefits, information sharing among competing suppliers remains limited in practice, largely due to strategic and competitive considerations (Cachon and Fisher 2000, Aviv 2001, Li 2002). In competitive markets, disclosing private information alters rivals’ beliefs and actions, often intensifying strategic interaction and eroding individual incentives to share (GalOr 1985, Raith 1996). Suppliers competing in the same market are reluctant to disclose private information about demand forecasts, costs, or operational constraints, as such disclosures may be exploited by rivals to gain competitive advantage. These concerns are often reinforced by privacy regulations, contractual restrictions, and the need to protect proprietary business data (Acquisti et al. 2016, Goldfarb and Tucker 2019). As a result, competitive markets may exhibit inefficiently low levels of information exchange, especially when firms can benefit from others’ disclosures without making reciprocal contributions (Gal-Or 1985). Understanding how such strategic and privacy barriers arise, and how they can be mitigated through mechanism design, is therefore a first-order problem for platform operators and policymakers. This paper studies the design of information-sharing mechanisms among competing suppliers from a game-theoretic perspective. We focus on Cournot competition, a canonical model of quantity competition widely used in operations management and economics, and analyze suppliers’ incentives to participate in a platform-mediated information-sharing scheme. Our analysis highlights a fundamental trade-off: while aggregating information improves firms’ estimates of uncertain market fundamentals, it simultaneously intensifies strategic coupling and competition, which discourages voluntary participation. This tension has important implications for the role of privacy-preserving mechanisms in competitive environments. In particular, privacy protection by itself cannot create participation incentives: while it reduces strategic exposure, it simultaneously erodes the informational gains from aggregation. We examine how two institutional features—platform-owned external information and privacy-preserving mechanisms—reshape this trade-off. First, we study the role of external signals observed by the platform but not contributed by suppliers, such as aggregate demand indicators or market-level analytics. Access to such information creates informational value that does not originate from competitors’ reports, thereby increasing the benefit of participation without proportionally increasing strategic exposure. 2
Second, we analyze privacy-preserving mechanisms that add controlled noise to firms’ reports before aggregation. By attenuating the extent to which any single firm’s report affects what competitors learn, privacy protection reduces strategic coupling and softens competitive pressures. However, because privacy operates by reducing signal precision, it also diminishes the informational gains from aggregation. This dual effect implies that privacy alone cannot restore participation incentives, but when combined with sufficiently informative platform-owned signals, it can complement external information and stabilize voluntary sharing. Our results deliver several insights that build on the classic literature on information sharing in Cournot competition (Gal-Or 1985, Raith 1996), while emphasizing the role of participation-contingent access. In a two-supplier market, we show that information sharing cannot be sustained as a Nash equilibrium, even under privacy-preserving aggregation, unless it is accompanied by sufficiently informative external platform signals. This result is consistent with the standard strategic-substitutes logic: when information sharing intensifies competitive interaction, firms have incentives to limit disclosure. In our setting, privacy protection alone cannot overturn this logic, because reducing strategic exposure also reduces the precision of the aggregated information. Only when participation grants access to exogenous informational value, through platform-owned signals, can sharing be sustained in equilibrium. The multi-supplier case clarifies how this logic changes under a reciprocal-access arrangement. Following the quid-pro-quo information exchange structure in Kirby (1988), we consider a participation-contingent access rule under which firms receive the aggregated signal only if they contribute their own signals. Under this arrangement, full information sharing may arise even without privacy protection or external platform information, because a firm that unilaterally opts out also loses access to the pooled information generated by the other participants. We therefore interpret the no-privacy, no-external-signal result as a baseline characterization of reciprocal information exchange. For parameter regimes in which reciprocal access alone is insufficient to sustain sharing, we show that a suitable combination of platform-owned external information and calibrated privacy noise can restore voluntary participation. By clarifying how market size and institutional design jointly determine participation incentives, this paper contributes to the literature on information sharing under competition. The baseline analysis connects our model to reciprocal information exchange settings 3
in which access to pooled information is conditional on participation. Building on this baseline, our main contribution is to identify a structural complementarity between platformprovided external information and privacy design. Privacy protection alone cannot sustain sharing because it reduces informational precision together with strategic exposure. External signals, by contrast, create exogenous informational rents that can be combined with calibrated privacy noise to stabilize voluntary participation. This complementarity persists under heterogeneous signal precision, where asymmetric privacy protection can restore joint sharing even when firms differ in informational strength. Together, these insights highlight that privacy should be understood not merely as a compliance constraint, but as a strategic design instrument whose effectiveness depends critically on the surrounding information environment. The remainder of the paper is organized as follows. Section 2 reviews the related literature on information sharing under competition, platform intermediation and mechanism design, and privacy-preserving mechanisms. Section 3 introduces the model and benchmark environments, and formalizes the participation game and firms’ incentive structure in competitive and uncertain markets. Section 4 presents the main results and mechanism design insights, characterizing when information sharing can be sustained as an equilibrium and how platform-owned external information and privacy-preserving aggregation jointly expand the sharing-feasible region. Section 5 extends the analysis to environments with heterogeneous private signals and sequential participation, showing that the complementarity between external information and privacy design remains robust under informational asymmetries. Section 6 concludes with implications for platform design and directions for future research.
2. Literature Review Our work relates to three main strands of literature: information sharing under competition, platform intermediation and mechanism design, and privacy-preserving mechanism design. We discuss each in turn and clarify how our analysis builds on these literatures. 2.1. Information Sharing under Competition A foundational literature in economics examines whether competing firms have incentives to share private information (Gal-Or 1986, Sakai and Yamato 1989, Ha et al. 2017). Early 4
work shows that disclosure can intensify competition by aligning firms’ beliefs and increasing strategic responsiveness, thereby reducing profits (Clarke 1983, Vives 1984). Gal-Or (1985) provides a seminal analysis of strategic information disclosure in oligopoly, and Raith (1996) generalizes the analysis of information sharing in Cournot competition. A central modeling distinction in this literature concerns the access rule governing disclosed information. In some classical formulations, a firm may receive information disclosed by others even if it does not disclose its own information. Under such an access rule, firms can benefit from rivals’ disclosures while withholding their own signals, which weakens incentives for voluntary sharing. By contrast, Kirby (1988) studies trade associations as quid-pro-quo information exchange mechanisms, where access to shared information is tied to a firm’s own participation. Our baseline adopts this reciprocal-access structure: a firm receives the aggregated signal only if it contributes its own signal. This access rule is consequential for baseline participation incentives. When the number of firms is sufficiently large, full information sharing can arise as a Nash equilibrium even without privacy protection or external platform signals, because a unilateral nonparticipant loses access to the pooled information generated by the other firms. We therefore interpret this result as a baseline characterization of reciprocal information exchange, closely related to Kirby (1988). Building on this baseline, our main contribution is to study how privacy-preserving aggregation and platform-owned external information jointly affect the feasibility of voluntary information sharing. 2.2. Platform Intermediation and Mechanism Design A growing literature studies information sharing in platform-mediated markets, where platforms act as intermediaries that design information disclosure and sharing mechanisms (Li et al. 2021, Zha et al. 2023). Recent work examines how platform-controlled information affects competition, pricing, and supply-chain structure, including platform information provision to competing sellers (Avinadav et al. 2025), personalized pricing (Hu et al. 2025), and sales-format decisions (Gong et al. 2024). This stream of research highlights the role of platforms as information intermediaries that shape market outcomes through the design of disclosure policies. Among these studies, Liu et al. (2021) is most closely related to our work. They analyze a retail platform that owns superior demand information and optimally designs informationsharing strategies toward competing sellers. Their results show that the platform may have 5
incentives to share information, which they interpret as standing in contrast to the classical no-sharing results of Gal-Or (1985) and Raith (1996). However, the distinction between their setting and ours is fundamental. While Liu et al. (2021) study whether a centralized platform discloses information it exclusively controls, we study whether competing firms, each possessing private information, voluntarily participate in a common information-sharing mechanism. Thus, our model returns to the participation question at the center of the information-sharing literature, while embedding it in a platform-mediated environment. Relative to this platform-disclosure literature, our focus is on suppliers’ voluntary participation under reciprocal access. The reciprocal-access baseline clarifies when participation may be sustained without additional platform instruments. Our main analysis then studies how platform-owned external information and privacy-preserving aggregation expand the set of environments in which voluntary sharing is feasible. 2.3. Privacy-Preserving Mechanism Design Our paper is closely related to the growing literature on privacy-preserving mechanism design. As digital platforms increasingly rely on fine-grained information for prediction and decision-making, privacy concerns have become a central barrier to participation in information-sharing environments. While shared data improves forecasting accuracy, the risk of strategic exposure creates disincentives, especially in competitive settings. This trade-off between informational utility and privacy loss has been widely studied. Foundational works incorporate differential privacy into economic mechanism design without undermining truthfulness or efficiency (McSherry and Talwar 2007, Nissim et al. 2012, Pai and Roth 2013). Other studies examine how privacy constraints reshape incentives and data sharing in strategic environments (Kearns et al. 2014, Acquisti et al. 2016, Ezhei and Ladani 2017). More recently, work on optimal data acquisition analyzes how platforms interacting with privacy-aware agents design privacy guarantees and compensation to induce data provision (Chen et al. 2018, Cummings et al. 2023, Fallah et al. 2024, Acemoglu et al. 2023). Our model contributes by showing that privacy protection can affect participation incentives, but only in conjunction with the surrounding information environment. Privacy noise reduces strategic exposure, but it also lowers the precision of the aggregated signal. 6
As a result, privacy protection alone does not restore voluntary sharing in our baseline model. Instead, calibrated privacy noise can sustain participation when it is combined with sufficiently informative platform-owned external information. This result connects privacypreserving mechanism design with the classical problem of voluntary information sharing among competitors.
3. The Efficiency Dilemma in Competitive and Uncertain Markets We consider a set of n suppliers indexed by i = 1, . . . , n who compete in a Cournot market by choosing non-negative quantities qi ≥ 0. The market price depends on total output P Q = ni=1 qi and an unknown common parameter θ ∈ R that captures shared market uncertainty. This includes, for example, demand fluctuations or macroeconomic shocks that symmetrically affect all firms. Following the standard setup in the information-sharing literature (e.g., Gal-Or 1985), we adopt a linear inverse demand function of the form P (Q, θ) = a + θ − Q, where a > 0 is a deterministic intercept and θ is a zero-mean random variable representing the uncertain market condition. The linear specification ensures analytical tractability while capturing the essential economic trade-off between quantity and price. The additive term θ enters the intercept of the inverse demand function, modeling uncertainty in baseline market conditions—such as shifts in demand shocks—without altering marginal price sensitivity. This structure is particularly suitable for settings such as commodity markets, energy procurement, or digital platform competition, where the quantity sensitivity of demand is stable but the baseline market level fluctuates across periods. By placing uncertainty in the intercept rather than the slope, we isolate the informational role of signals in driving strategic quantity choices without conflating it with slope-driven volatility. We assume the common parameter θ ∈ R is drawn from a normal prior, θ ∼ N (0, u),
u > 0,
capturing market-wide uncertainty in demand. This is standard in the oligopoly information literature (e.g., Liu et al. 2021), enabling closed-form Bayesian inference. 7
Each supplier privately observes a noisy signal of θ, εi ∼ N (0, τi ),
si = θ + εi ,
where εi is independent across suppliers and independent of θ. Conditional on the true state, si | θ ∼ N (θ, τi ). For simplicity, we assume that all suppliers have symmetric information precision and set τi = τ for all i. This assumption preserves analytical tractability while allowing us to isolate the incentive effects of information sharing in a symmetric environment. In addition to the suppliers’ private signals, the platform has access to its own platform signal about the market state, εp ∼ N (0, τp ),
sp = θ + εp ,
where τp > 0 is fixed and exogenously given. This signal represents market intelligence collected independently by the platform through methods such as customer-side behavior tracking, demand forecasting, or external market research. It does not rely on supplier participation and is available to the platform regardless of the firms’ sharing decisions. If a subset J ⊆ {1, . . . , n} of suppliers participates, each firm i ∈ J reports its signal si to the platform. The platform applies a firm-specific privacy parameter mi and perturbs each report as vi ∼ N (0, mi ),
s̃i = si + vi ,
where the noise terms {vi }i∈J are mutually independent and independent of (θ, {si }i∈J , sp ). Adding independent Gaussian noise to each report provides a form of approximate differential privacy, ensuring that each supplier’s report remains statistically indistinguishable from nearby alternatives (Dwork and Roth 2014). Intuitively, larger mi values increase uncertainty about any single supplier’s private signal, limiting the ability of rivals or the platform to infer sensitive information. This protects suppliers from being strategically exposed through participation, while still enabling aggregate information to be extracted and shared. In this paper, we do not model privacy guarantees in formal privacy-budget terms (e.g., (ε, δ)-DP), but rather adopt the mechanism design perspective: noise levels mi are design parameters that trade off informativeness and strategic protection. 8
Then the platform aggregates their noisy reports {s̃j }j∈J together with its own signal sp to compute the posterior mean of the market state: µJ := E[θ | {s̃j }j∈J , sp ], which is then shared with all participating firms. Under the Gaussian assumptions, this posterior takes the closed-form: µJ = VJ
s̃j sp + τ + mj τp j∈J
X
! ,
VJ =
1 X 1 1 + + u j∈J τ + mj τp
!−1 .
Importantly, the platform does not disclose any raw supplier data—only the posterior mean. The platform signal sp increases the informativeness of this posterior without depending on any individual firm’s input, and thus plays a key role in mitigating the loss of signal accuracy caused by privacy noise. This shared posterior mean can also be viewed as a public signal, available exclusively to participants, which aggregates all available information in a privacy-preserving manner. In symmetric environments where all participating firms face identical privacy noise levels mj = m, the posterior depends only on the number of sharing firms k = |J|. We therefore write µk (with variance Vk ) to denote the posterior mean when k firms share their signals. In particular, µ1 corresponds to the one-firm case, µ2 to the two-firm case, and µn to the full n-firm sharing profile. This shorthand notation will be used throughout whenever symmetry applies. 3.1. Information Sharing Platform and Participation Decisions Each supplier chooses whether to join the platform. Let xi ∈ {0, 1} denote supplier i’s participation decision, where xi = 1 means joining the platform and xi = 0 means non-participation. If supplier i does not join the platform, its information set is IiN S = {si }. If firm i participates, IiS = {si , µJ }. 9
Then firms choose quantities according to Bayesian best responses: qiN S (si ) = arg max Eθ|si [qi (a + θ − Q)] , qi ≥0
qiS (si , µJ ) = arg max Eθ|si ,µJ [qi (a + θ − Q)] . qi ≥0
3.2. Equilibrium Structure and Benchmark Regimes The model defines a three-stage game: 1. Mechanism commitment stage (Platform): The platform commits to a privacypreserving aggregation mechanism by announcing the privacy parameters {mi }ni=1 . 2. Participation stage (Suppliers): Given the announced privacy guarantees, each supplier chooses xi ∈ {0, 1}, where xi = 1 denotes joining the platform (sharing) and xi = 0 denotes non-participation. 3. Quantity stage (Market competition): Suppliers choose quantities given the induced information structure. Our focus is on Nash equilibrium existence. Specifically, we study whether there exists a privacy-preserving mechanism {mi }ni=1 such that platform participation (information sharing) is a Nash equilibrium of the participation game. We define the participation incentive for supplier i as ∆i := E πiS (xi = 1, x−i = 1) − E πiN S (xi = 0, x−i = 1) , which captures the expected profit gain from joining the information-sharing platform when all other suppliers participate. Sharing is a Nash equilibrium if and only if ∆i ≥ 0
for all i ∈ {1, . . . , n}.
When the suppliers are symmetric, we write ∆ := ∆i , and the condition reduces to ∆ ≥ 0. 10
4. Privacy-Preserving Mechanism Design for Information Sharing 4.1. A Two-Firm Information-Sharing Model We begin with a stylized example involving two symmetric competing firms. The inverse demand function in the market is given by: P (Q, θ) = a + θ − Q, where Q = q1 + q2 denotes total output, and θ represents an uncertain market condition. Firm i’s profit is given by πi = qi · P (Q, θ) = qi (a + θ − q1 − q2 ). Each firm observes a noisy but unbiased signal about the underlying state θ. To isolate the role of privacy noise in discouraging participation, we first consider a benchmark in which the platform does not possess any external signal. This corresponds to taking the platform signal variance τp → ∞, so that any posterior released by the platform is based solely on supplier reports. Proposition 4.1 (No Sharing Equilibria for Two Firms Without External Signals) Consider two competing symmetric firms in the absence of external information provided by the platform (i.e., τp → ∞). There exists no privacy-preserving information aggregation mechanism that implements information sharing as a Nash equilibrium of the participation game. Proposition 4.1, proved in Section A.1, establishes a sharp negative result: with two competing symmetric firms, no privacy-preserving information aggregation mechanism can implement information sharing as a Nash equilibrium of the participation game. The intuition is structural. Any mechanism that aggregates firms’ reports and releases a common posterior necessarily strengthens strategic coupling between competitors, thereby intensifying competition. Introducing privacy noise can only reduce the precision of the released information; it does not alter the direction of the competitive externality induced by shared information. As a result, when the informational content of the platform is generated solely from firms’ own submissions, the incremental benefit from participation is insufficient to offset the strategic loss from increased competition, and participation cannot be sustained in equilibrium. 11
These findings are consistent with the classic insights of Gal-Or (1985). In that setting, when firms can choose the precision of information disclosure—effectively engaging in partial or noisy information sharing—symmetric competitors optimally choose not to share. Proposition 4.1 strengthens this insight by showing that even when noise is imposed exogenously by a platform through a privacy-preserving mechanism, information sharing remains unsustainable in symmetric duopoly. These results highlight a fundamental limitation of endogenous information aggregation and underscore the critical role of exogenous, platform-owned information in supporting equilibrium participation. This negative result naturally raises the question whether additional sources of information can restore participation incentives. In particular, if the platform also observes an external signal that is not contributed by the firms, then participation grants access to informational value that does not arise from disclosing one’s private signal to a rival. Such external information fundamentally changes the information structure and may expand the parameter region in which the net gain from participation is nonnegative. Our subsequent proposition shows that, under appropriate conditions, the presence of a platform-owned external signal can enlarge the feasible region satisfying ∆ ≥ 0, thereby enabling information sharing to be sustained in equilibrium (proof in Section A.2). Proposition 4.2 (Two-Firm Information Sharing Equilibrium with Platform Signal) Consider the symmetric two-firm Cournot model with common state θ ∼ N (0, u), private signals si | θ ∼ N (θ, τ ) for i = 1, 2, and a platform-provided signal sp | θ ∼ N (θ, τp ). For fixed accuracy parameters u, τ, τp > 0: 1. There exists a threshold τp∗ = τp∗ (u, τ ) ∈ (0, ∞) such that the Sharing profile is the unique symmetric Bayesian Nash equilibrium if and only if τp ≤ τp∗ . 2. The threshold τp∗ is uniquely characterized by the indifference condition: Var(µ2 ) = β0 (τp )2 Var(µ1 ), (1) 9 −1 where µ2 = E[θ | s1 , s2 , sp ] = V2 sτ1 + sτ2 + sτpp with V2 = u1 + τ2 + τ1p , and µ1 = E[θ | s1 ] = −1 V1 sτ1 with V1 = u1 + τ1 ; β0 (τp ) is the equilibrium response coefficient defined in (14). Proposition 4.2 highlights a simple trade-off. The indifference condition in (1) equates the information gain from sharing with the strategic gain from not sharing. The term Var(µ2 ) captures the benefit of improved demand estimation when both firms participate, while 12
β0 (τp )2 Var(µ1 ) reflects the advantage of staying out and avoiding the stronger competition that arises when both firms respond to more precise common information. The threshold τp∗ therefore represents the minimal accuracy of the platform signal required for the information gain from sharing to outweigh the strategic benefit of non-participation. When the platform signal is sufficiently precise, full sharing becomes the unique equilibrium. A natural follow-up question is whether a privacy-preserving mechanism can further induce information sharing when a platform signal exists but is insufficient on its own to sustain participation. Conceptually, privacy noise affects participation incentives through two opposing channels. On the one hand, adding noise reduces the precision of the released public signal, thereby weakening the informational value of joining the platform. On the other hand, by attenuating the extent to which a firm’s report is reflected in what its rival learns, privacy can mitigate strategic coupling and soften competitive pressures. The dominating effect depends on the underlying information environment. The following proposition (proved in Section A.3) characterizes how to design the noise level (m) of a privacy-preserving mechanism when the platform signal alone is not accurate enough to sustain participation. In contrast to Proposition 4.1, which establishes an impossibility result in the absence of external information, this result delivers a positive insight: when the platform provides a signal (or, more generally, any exogenous benefit), appropriately designed privacy protection can be used as a mechanism-design tool to induce firms to join the information-sharing platform. Proposition 4.3 (Privacy Noise Induces Voluntary Sharing for Two Firms) In the two-firm symmetric Cournot model, suppose that the platform signal has finite variance, i.e., τp < ∞, but is not precise enough to sustain full participation under the non-private mechanism, i.e., τp∗ (u, τ ) < τp < ∞. Then there exists a finite privacy noise level m∗ > 0 such that, under the privacy-preserving mechanism with m ≥ m∗ , each firm strictly prefers to participate when the other does. Thus, full participation becomes a unique symmetric equilibrium. Proposition 4.3 delivers a positive and economically intuitive message. Even when the platform’s external signal is too noisy to sustain voluntary participation under a nonprivate design (i.e., when τp > τp∗ (u, τ )), an appropriately chosen privacy noise level can 13
restore participation incentives. The key insight is that privacy noise reshapes the informational environment: by partially masking how a firm’s report affects what the rival learns, privacy attenuates strategic coupling and softens competitive pressures, while the platform’s external signal ensures that participation continues to deliver some exogenous informational value. As a result, participation incentives can become positive for sufficiently large privacy noise, and by continuity there exists an intermediate noise level m∗ > 0 that makes the firm exactly indifferent (with any slightly larger noise yielding strict incentives). While Proposition 4.3 is stated for the two-firm benchmark to highlight the mechanism most transparently, the same logic extends beyond duopoly. In the next section, we study the general n-firm Cournot model and characterize how the sharing-feasible region depends jointly on the number of firms n, the accuracy of the platform signal τp , and the privacy noise level m. This allows us to quantify how market competitiveness scales with n and how privacy can be used to stabilize sharing equilibria in larger industries. 4.2. An n-Firm Information-Sharing Model We extend the model to a setting with n ≥ 2 symmetric Cournot competitors. The platform contributes two key features: an platform signal about the market obtained independently, and a privacy mechanism that perturbs firm signals before aggregation. Participating firms receive a shared posterior mean of the market state, computed from the privatized firm signals and the platform’s external signal. This design ensures firms gain useful information even under strong privacy protection, while limiting strategic exposure to competitors. Before analyzing the full privacy-aware mechanism, it is useful to first understand firms’ incentives in a baseline setting without privacy protection or platform signals. In this simpler scenario, the platform merely aggregates raw signals submitted by firms. Unlike the two-firm case where non-participation dominates, the n-firm model exhibits richer strategic dynamics, and voluntary participation may arise even without platform-provided information. The next proposition characterizes this baseline equilibrium behavior. Proposition 4.4 (n-Firm Sharing Equilibrium without Privacy Protection) Consider the n-firm Cournot information-sharing model without privacy protection (m = 0). Let ∆ denote the unilateral participation gain of a representative firm when all other firms share. 14
Information sharing constitutes a Nash equilibrium if and only if ∆ ≥ 0, which under symmetry is equivalent to the condition 2 (u + τ ) (n − 1)u + nτ n ≥ 2 . (n + 1)2 (nu + τ ) (n2 − 1)u2 + 2n2 uτ + 2nτ 2
(2)
Proposition 4.4, proved in Section A.4, characterizes a baseline reciprocal-access environment without privacy protection or external platform signals. Following the quid-pro-quo information exchange structure in Kirby (1988), firms receive the aggregated signal only if they contribute their own signals. Under this participation-contingent access rule, information sharing may be sustained as a Nash equilibrium in an n-firm Cournot market even without privacy protection or platform-owned information. The mechanism behind Proposition 4.4 is the access-loss effect generated by reciprocal information exchange. Following the quid-pro-quo structure in Kirby (1988), a firm receives the aggregated signal only if it contributes its own signal. Thus, a unilateral nonparticipant not only avoids disclosing its private information, but also loses access to the pooled information generated by the other firms. When many firms participate, this pooled signal can be sufficiently precise, making the informational loss from opting out larger than the strategic benefit of withholding one’s signal. This is the force that can sustain full participation in the n-firm baseline, even without privacy protection or platform-owned information. This interpretation also clarifies the role of the platform in the baseline model. Without privacy protection and without an external platform signal, the platform does not create an additional source of information; it only implements the reciprocal-access aggregation rule. Therefore, the baseline result should be viewed as a characterization of participation under reciprocal access. Our subsequent analysis builds on this benchmark by asking when additional platform instruments—external information and privacy-preserving aggregation—are needed to sustain sharing. Proposition 4.4 provides an exact characterization of the participation condition. However, the inequality in (2) involves high-order terms and does not immediately reveal how market size and signal precision interact. To obtain a clearer understanding of its structure, we rewrite the condition in terms of the precision ratio r = τ /u. After multiplying both sides by the common denominator and simplifying, the participation condition ∆ ≥ 0 is equivalent to Φ(n, r) ≥ 0, 15
where Φ(n, r) is a cubic polynomial in r defined by Φ(n, r) = a0 (n) + a1 (n)r + a2 (n)r2 + a3 (n)r3 , with coefficients a0 (n) = n4 − 2n3 − 2n2 + 2n + 1,
a1 (n) = n4 − 2n3 − 3n2 − n + 1,
a2 (n) = −n4 + 2n3 − 3n2 − 2n,
a3 (n) = −(n − 1)n2 < 0 (n ≥ 2).
Thus, the existence of a sharing equilibrium is determined by the sign of this polynomial. Corollary 4.5 (Characterizing the Threshold for n-Firm Sharing Equilibrium) For each n ≥ 3, the cubic polynomial Φ(n, r) has a unique positive root, denoted by r0 (n). Information sharing constitutes a Nash equilibrium if, and only if, τ ≤ r0 (n). u Moreover, the threshold r0 (n) is increasing in n. Therefore, as the number of firms increases, the range of signal precision ratios that sustain voluntary sharing expands monotonically. Corollary 4.5, proved in Section A.5, shows that the cutoff value of relative signal uncertainty r = τ /u, defined as the ratio of individual signal uncertainty to common-state uncertainty, increases monotonically with the number of firms. In larger markets, information sharing can therefore be sustained even when individual firms’ private signals are relatively imprecise. The intuition is straightforward. When more firms participate in the platform, the shared statistic aggregates information from a larger pool of signals. As a result, a unilateral deviation becomes more costly: a deviating firm foregoes access to an increasingly informative aggregate signal while its competitors continue to condition their actions on it. This growing informational disadvantage strengthens participation incentives and relaxes the precision requirement on individual signals. Consequently, an increase in the number of firms expands the sharing-feasible region in terms of the relative signal noise r. Figure 1 visualizes the implicit threshold structure characterized in Corollary 4.5. The curve r0 (n) represents the unique positive root of Φ(n, r) = 0, which separates the parameter space into a sharing-feasible region (∆ ≥ 0) and a non-sharing region (∆ < 0). The 16
Feasible region: Δ ≥ 0 (u = 1)
2.0
τ/u
1.5
1.0
0.5
0.0 5
10
15
20
25
30
n Figure 1
Sharing region without privacy protection (m = 0). The figure illustrates the implicit threshold r0 (n) = τ /u as a function of the number of firms n. The shaded region corresponds to parameter values satisfying ∆ ≥ 0, i.e., where full information sharing constitutes a Nash equilibrium. The boundary √
curve is defined implicitly by Φ(n, τ /u) = 0. Its asymptote r∞ = 1+2 5 can be obtained by letting as n → ∞ in Equation (2) and solving for the equality.
monotone increase of r0 (n) in n confirms that as the number of firms grows, the equilibrium region supporting full information sharing expands. Economically, this reflects the fact that the strategic externality generated by others’ information sharing becomes increasingly beneficial relative to the competitive effect, making participation incentives stronger in larger markets. The analysis above shows that, in an n-firm market, full information sharing can arise endogenously even in the absence of platform signals or privacy protection. However, this sharing-feasible region is not exhaustive. For parameter values outside this region, unilateral deviations remain profitable and the sharing profile fails to be an equilibrium. This raises the question of whether additional institutional features can restore participation incentives in those non-sharing regimes. In line with the two-firm analysis, we first examine the role of an external platform-owned signal. When participation grants access to information that does not originate from competitors’ reports, the informational benefits of joining may increase without proportionally intensifying competitive pressures. 17
The following proposition formalizes this intuition by characterizing how the presence of a platform signal affects equilibrium participation in the n-firm model. Proposition 4.6 (n-Firm Information-Sharing Equilibrium with Platform Signal) Consider the n-firm Cournot model with fixed parameters n, u, τ, τp > 0. 1. There exists a threshold τp∗ = τp∗ (n, u, τ ) ∈ (0, ∞) such that the Sharing profile is the symmetric Bayesian Nash equilibrium if and only if τp ≤ τp∗ . 2. The threshold τp∗ is uniquely characterized by the indifference condition: Var(µn ) = β0 (τp )2 Var(µ1 ), 2 (n + 1) −1 P sp n si 1 n 1 + with V = + + , and µ1 = V1 sτ1 with V1 = where µn = Vn n i=1 τ τp u τ τp
(3) 1 1 −1 + . u τ
Proposition 4.6, proved in Section A.6, shows that the presence of a platform-owned external signal can restore equilibrium information sharing in parameter regions where endogenous sharing fails in the baseline n-firm model. When the platform signal is sufficiently precise (i.e., τp ≤ τp∗ ), participation grants firms access to informational value that does not arise from their competitors’ reports. This additional source of information increases the benefit of joining without proportionally intensifying strategic competition, thereby sustaining the sharing profile as a symmetric equilibrium. The characterization in (3) makes this trade-off explicit. The left-hand side captures the informational value generated by aggregating all firms’ information together with the platform signal, while the right-hand side reflects the strategic cost of participation, as measured by the sensitivity of a deviating firm’s optimal response to its private posterior. The threshold τp∗ therefore delineates the boundary between regimes in which external information alone suffices to induce voluntary sharing and regimes in which it does not. Figure 2 illustrates the equilibrium sharing regions characterized in Proposition 4.6. For each n, the shaded region in the (τ, τp )-plane corresponds to parameter values for which full information sharing constitutes a Bayesian Nash equilibrium. The dashed red vertical line marks the benchmark boundary in the absence of an external platform signal. Comparing panels (a) and (b) reveals two opposing effects of increasing the number of firms from n = 5 to n = 10. First, in the benchmark case without an external signal, the red dashed boundary shifts to the right as n increases. This implies that sharing can be sustained for a wider range 18
Sharing region, n = 5, u = 1. 10
Sharing region, n = 10, u = 1. 10
No external signal boundary
6
6 τp
8
τp
8
No external signal boundary
4
4
2
0
2
Share
0
2
4
6
8
0
10
Share
0
2
4
τ
(a) n = 5 Figure 2
6
8
10
τ
(b) n = 10
Sharing regions under an external platform signal. The shaded areas represent the parameter regions in the (τ, τp )-plane where the sharing profile constitutes a symmetric Bayesian Nash equilibrium. Panel (a) corresponds to n = 5, and panel (b) to n = 10 (with u = 1 and a = 1). The red dashed vertical line indicates the no-external-signal boundary, corresponding to the benchmark case without a platform signal. The equilibrium boundary is characterized by the indifference condition (3), which defines the threshold τp∗ (n, u, τ ).
of τ when the market is larger. The reason is that informational externalities are stronger in larger markets: each firm’s information contributes to a more substantial aggregate improvement in the precision of beliefs, increasing the collective value of sharing. However, once an external platform signal is introduced, the mechanism changes. The public signal reduces uncertainty commonly across firms, which strengthens strategic substitutability in quantities and intensifies competition. When n increases, this competitive effect grows faster than the incremental informational gain from an additional participant. As a result, the net private benefit from sharing declines with n, and the equilibrium sharing region becomes smaller when moving from n = 5 to n = 10. Importantly, Proposition 4.6 does not exhaust all non-sharing cases. When the platform signal is too noisy to sustain participation on its own, additional design instruments may be required. This observation motivates the introduction of privacy-preserving mechanisms in the next section, where we study how appropriately calibrated noise can further reshape firms’ incentives and expand the sharing-feasible region. 19
Proposition 4.7 (Privacy Noise Induces Voluntary Sharing among n Firms) Consider the n-firm symmetric Cournot model described above. Suppose the platform signal has finite variance, i.e., τp < ∞, but is not precise enough to sustain full participation under the non-private mechanism, i.e., τp∗ (n, u, τ ) < τp < ∞. Then there exists a finite privacy noise level m∗ > 0 such that, under the privacy-preserving mechanism with noise variance m ≥ m∗ , every firm strictly prefers to participate when all other firms participate. Consequently, the sharing profile is sustained as a symmetric equilibrium. Proposition 4.7, proved in Section A.7, shows that privacy protection restores voluntary information sharing only in the presence of external platform information. When the platform signal is too weak to sustain participation under a non-private design, calibrated privacy noise can reduce strategic coupling and make sharing incentive-compatible. However, without external information, privacy protection alone cannot sustain participation, as noise simultaneously reduces informational gains. This result highlights a complementarity between external signals and privacy design. Platform information provides the informational benefit needed for participation, while privacy noise limits the competitive cost of disclosure. Only when both instruments are present can the platform sustain information sharing in environments where each mechanism in isolation would fail. Figure 3 provides a geometric illustration of Proposition 4.7. The figure depicts the indifference surface in the (τ, τp , m) space defined by ∆n (τ, τp , m) = 0, along which firms are indifferent between participating in and abstaining from the information-sharing platform. For parameter values lying above the indifference surface, the privacy-preserving mechanism sufficiently mitigates firms’ privacy concerns, rendering information sharing profitable and thereby inducing participation. Several comparative statics emerge from the geometry of the indifference surface. First, holding τ fixed, an increase in τp —corresponding to a less precise external signal provided by the platform—raises the minimum level of privacy noise required to restore firms’ participation. Intuitively, when the platform’s signal is less informative, firms’ participation decisions rely more heavily on mitigating the competitive externality induced by information sharing, which in turn requires stronger privacy protection. As a result, sustaining voluntary participation under a noisier platform signal necessitates a larger privacy noise variance m. Similarly, holding τp fixed, a decrease in firms’ signal accuracy (i.e., an increase 20
n=3 n=5
Figure 3
Indifference surfaces under the privacy-preserving mechanism. The figure shows the indifference surfaces defined by ∆(τ, τp , m) = 0 in the (τ, τp , m) space, where firms are indifferent between participating in and abstaining from the information-sharing platform. The shaded regions on the m = 0 plane correspond to parameter values (τ, τp ) for which, in the absence of privacy protection, participation is unprofitable, i.e., ∆(τ, τp , 0) < 0. For parameter combinations located above the indifference surface, the privacy-preserving mechanism reduces effective information leakage sufficiently to restore profitability and induce participation. The surfaces are plotted for u = 1 and a = 1, with red corresponding to n = 3 and blue corresponding to n = 5.
in τ ) shifts the indifference surface upward, implying that stronger privacy protection is required to offset the intensified strategic interaction arising from noisier private information. The geometry also highlights the role of market size n. An increase in the number of firms intensifies strategic competition, which amplifies the competitive externality associated with information sharing. As a result, the non-sharing region expands as n increases. In particular, when n = 5 (blue surface), the set of parameter values (τ, τp ) for which firms would abstain from participation at m = 0 is strictly larger than under n = 3 (red surface). This implies that, in larger markets, firms require a more precise external signal (i.e., lower τp ) in order to sustain participation absent privacy protection. The comparison across n also reveals a nuanced interaction between market size and privacy noise. Although the required privacy noise level m is increasing in both τ and τp for any fixed n, the level of privacy noise needed to restore participation is not uniformly higher 21
when n is larger. When the external signal is sufficiently precise (low τp ), the stronger informational value of the platform mitigates competitive pressures, so that even under n = 5 a relatively smaller privacy noise level can restore participation. However, as τp rises and the platform signal becomes noisier, the intensified competition under larger n dominates, and substantially stronger privacy protection is required to sustain sharing. In sum, the figure illustrates that privacy does not operate as a uniformly beneficial intervention. Rather, privacy acts as a structural incentive mechanism whose effectiveness depends critically on the interaction among firms’ private signal precision, the accuracy of the platform’s external signal, and the magnitude of the injected privacy noise. When appropriately calibrated, privacy protection expands the region in which information sharing is incentive compatible and stabilizes information-sharing equilibria. In contrast, insufficient or poorly calibrated noise may fail to alleviate competitive externalities and can therefore undermine firms’ incentives to participate.
5. Information Sharing with Heterogeneous Signals and Under Sequential Entry 5.1. Information Sharing with Heterogeneous Private Signals We extend the baseline model to allow for heterogeneity in firms’ private information. Firm i observes a private signal εi ∼ N (0, τi ),
si = θ + εi ,
where signal noise variances (τi )ni=1 may differ across firms. As before, the platform observes an external signal εp ∼ N (0, τp ).
sp = θ + εp ,
Under the privacy-preserving mechanism, if firm i participates, its reported signal is perturbed as vi ∼ N (0, mi ),
s̃i = si + vi ,
where the platform may choose firm-specific privacy noise levels (mi )ni=1 . Given the privatized reports of participating firms and the platform signal, the platform releases the posterior mean of the common state to all participants. Proposition 5.1 (Privacy Restores Voluntary Sharing under Heterogeneous Signals) Consider the n-firm model with heterogeneous signal precisions (τi )ni=1 described above. 22
Suppose the platform signal has finite variance, i.e., τp < ∞, but is not precise enough to sustain full participation under the non-private mechanism. Then there exists a finite vector of privacy noise levels m∗ = (m∗1 , . . . , m∗n ) ∈ (0, ∞)n such that, under the privacypreserving mechanism with m ≥ m∗ , every firm strictly prefers to participate when all other firms participate. Consequently, the sharing profile is sustained as an equilibrium of the participation game. Proposition 5.1, proved in Section A.8, shows that the participation-restoring role of privacy noise is robust to heterogeneity in firms’ information quality. Even when firms possess private signals of different precisions, the platform can design a privacy-preserving mechanism—potentially with asymmetric noise levels across firms—that induces voluntary participation by all firms. The key economic insight is unchanged from the homogeneous benchmark. When privacy noise is sufficiently large, the information revealed through firms’ reports becomes negligible, and the platform’s released statistic converges to the public posterior based solely on the platform signal. Because the platform signal has finite precision, access to this public posterior strictly improves each firm’s information set relative to non-participation, under which a deviating firm relies only on its private signal while competitors condition on a richer information set. As a result, participation becomes a strictly dominant response to full participation by others, despite heterogeneity in private signal accuracy. To illustrate how heterogeneity in signal accuracy affects the required level of privacy noise, we present a numerical example in a two-firm setting. Figure 4 depicts the information-sharing regions in the (m1 , m2 ) plane under the privacy-preserving mechanism. The shaded regions indicate parameter values for which firm 1 and firm 2, respectively, are willing to participate in the platform, while their intersection corresponds to values of (m1 , m2 ) for which joint participation constitutes a Nash equilibrium. Across panels, we vary firm 1’s signal precision by changing τ1 ∈ {1, 3, 5}, while holding firm 2’s signal noise fixed at τ2 = 5. Other parameters are set to u = 1, τp = 20, and a = 1. The figure reveals a clear comparative-static pattern. As firm 1’s private signal becomes more precise (i.e., as τ1 decreases), the minimum privacy noise required to induce participation by firm 2 decreases, whereas the privacy noise required for firm 1’s own participation increases. Intuitively, a more informative signal from firm 1 raises the informational benefit 23
of participation for firm 2, reducing its need for privacy protection, while simultaneously increasing firm 1’s exposure to information leakage, thereby necessitating stronger privacy noise. In the extreme case τ1 = 1, firm 2 is willing to participate even when m2 = 0, provided that the privacy noise applied to firm 1’s report is not too large. In this regime, firm 2 benefits sufficiently from accessing firm 1’s highly informative signal, making participation attractive despite the absence of direct privacy protection. To complement the two-dimensional sharing regions in Figures 4 and 5 provide a onedimensional comparative-static perspective. As firm 1’s signal noise τ1 increases, its equilibrium privacy noise requirement m∗1 declines, reflecting the reduced privacy cost associated with a less informative signal. In contrast, the equilibrium noise required to induce firm 2’s participation, m∗2 , increases as the informational benefit from firm 1’s signal diminishes. 5.2. Heterogeneous Signals under Sequential Participation In this subsection, we study participation incentives in a heterogeneous information environment under sequential decision-making. We consider a setting in which all firms other than firm i have already joined the platform and share their signals using a common privacy noise level m. Firm i observes a private signal with noise variance τi that may differ from that of its competitors. Rather than solving for a joint equilibrium in participation and privacy choices, we analyze the participation decision of firm i conditional on the existing participation of its competitors. The platform is allowed to tailor the privacy noise level mi applied to firm i’s shared signal in order to induce its participation. This formulation captures a natural dynamic interpretation in which firms join the platform sequentially, and the platform designs firm-specific privacy protection to encourage additional participation. In the following proposition, by focusing on this best-response problem, we isolate how firm-level signal heterogeneity shapes participation incentives, abstracting from the equilibrium feedback effects of simultaneous participation decisions (proof in Section A.9). Proposition 5.2 (Privacy-Induced Participation under Sequential Entry) Consider the n-firm Cournot model under sequential participation. Suppose all firms other than firm i have already joined the platform and share their signals using a common 24
30
25
25
20
20
15
15
m2
m2
30
10
10
5
5
0
0 0
5
10
15
20
25
30
0
5
10
m1
15
20
25
30
m1
(a) τ1 = 5
(b) τ1 = 3 5
4
m2
3 Δ1 ≥ 0 Δ2 ≥ 0
2
Δ1 ≥ 0 and Δ2 ≥ 0
1
0 25
26
27
28
29
30
m1
(c) τ1 = 1 Figure 4
Information-sharing regions under heterogeneous private signals. Each panel depicts the regions in the (m1 , m2 ) plane where firm 1 and firm 2 are willing to participate in the privacy-preserving information-sharing platform. The intersection corresponds to parameter values for which joint participation constitutes a Nash equilibrium. Across panels, firm 1’s signal precision varies (τ1 ∈ {1, 3, 5}), while firm 2’s signal noise is fixed at τ2 = 5. Other parameters are u = 1, τp = 20, and a = 1.
privacy noise level m. Firm i observes a private signal with noise variance τi and, if participating, is subject to a firm-specific privacy noise mi . Assume the platform signal has finite variance τp < ∞ but is not sufficiently informative to induce firm i to participate without privacy protection, i.e., τp∗ (n, u, τi , τ ) < τp < ∞. Then there exists a finite threshold m̄i > 0 such that assigning mi ≥ m̄i induces firm i to prefer participation, taking other firms’ participation as given. 25
12
15
11
m2 *
m1 *
20
10
10 9
5
8
0 2
3
4
5
6
7
7
8
2
3
τ1
5
6
7
8
τ1
(a) Equilibrium m∗1 vs τ1 Figure 5
4
(b) Equilibrium m∗2 vs τ1
Impact of firm 1’s signal precision τ1 on equilibrium privacy noise levels. The left panel shows how the platform’s equilibrium privacy noise level for firm 1, m∗1 , varies with firm 1’s signal precision τ1 , while the right panel shows the corresponding cross-effect on the equilibrium noise level m∗2 assigned to firm 2. Fixed parameters are τ2 = 5, u = 1, τp = 20, and a = 1.
We complement the analytical results with a numerical illustration that characterizes how the firm-specific privacy noise threshold varies with signal heterogeneity. Throughout the experiment, we fix the prior variance at u = 1 and the demand intercept at a = 1. Firm i’s signal noise variance τi varies over the interval [2, 6], while all other firms share signals with a common noise variance τ = 5. The platform provides an external signal with noise variance τp = 10, which by itself is insufficient to induce participation in the absence of privacy protection. All other firms apply a common privacy noise level m = 10. For each value of τi and each market size n ∈ {3, 5, 10}, we numerically compute the minimal firm-specific privacy noise m∗i that makes firm i indifferent between participating and not participating, taking the participation of all other firms as given. The threshold m∗i is obtained by solving ∆i (τi , mi ) = 0, where ∆i denotes firm i’s unilateral participation gain. Figure 6 plots the resulting relationship between τi and m∗i . Two patterns emerge clearly. First, for any fixed market size n, the required privacy noise m∗i is decreasing in τi : firms with noisier private signals require less privacy protection to be willing to share. Second, for any given τi , the threshold m∗i is increasing in n, reflecting stronger competitive externalities in larger markets, which in turn necessitate stronger privacy protection to sustain individual participation. 26
mi *
120
100
n=3
80
n=5 60
n=7
40
20
2
Figure 6
3
4
5
6
τi
Firm-specific privacy noise threshold m∗i as a function of signal noise τi , for different numbers of competing firms n.
6. Conclusion This paper studies voluntary information sharing among competing firms on digital platforms. We show that sharing is governed by a fundamental trade-off: better inference about market uncertainty raises profits, but aligning beliefs can also intensify strategic coupling in quantity competition. This tension explains why socially valuable data exchange may fail to emerge in equilibrium. Our analysis delivers three structural insights. First, in small markets, voluntary information sharing cannot be sustained as a Nash equilibrium without sufficiently informative external platform signals. Privacy protection alone does not resolve this problem, because adding noise simultaneously reduces strategic exposure and weakens informational gains. Second, following the reciprocal-access structure in Kirby (1988), we consider a participation rule under which firms receive the aggregated signal only if they contribute their own signals. Under this arrangement, the multi-firm baseline differs from settings in which non-sharing firms can still receive others’ disclosed information. When many competitors participate, a unilateral deviation becomes costly because an opting-out firm loses access to the pooled information generated by the other firms. For sufficiently large n, this access loss can sustain full participation even in the absence of privacy protection or external platform signals. We interpret this result as a baseline characterization of reciprocal information exchange. Third, in regimes where reciprocal access alone does not sustain sharing, participation can be stabilized when calibrated privacy protection is combined with 27
platform-provided external information. External signals create informational value that does not depend on competitors’ disclosures, while privacy noise tempers how strongly a firm’s report affects rivals’ beliefs. The effectiveness of privacy protection therefore depends on the precision of the external signal, providing guidance for how the platform should calibrate its privacy mechanism given the level of exogenous information available. These findings have direct implications for platform design. Privacy protection should not be viewed merely as a compliance tool, but as a strategic instrument whose effectiveness depends on the surrounding information environment. Privacy alone is insufficient: without an external informational anchor, noise reduces both strategic exposure and aggregate informational value. Sustainable data-sharing platforms therefore require joint calibration of privacy protection and external information provision, taking into account market size and signal quality. More broadly, our results underscore the importance of integrating competition, information design, and privacy in the study of digital platforms, and suggest directions for future work on dynamic learning, networked markets, and endogenous platform investment in information.
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Appendix A: Proofs A.1. Proposition 4.1: No sharing equilibria for two firms without external signals We analyze the unilateral deviation incentive of firm 1. When firm 1 participates (and there is no platform-provided external signal, i.e., τp → ∞), the platform releases the posterior mean µ = E[θ | s̃1 , s̃2 ] ,
s̃i = si + vi ,
vi ∼ N (0, mi ).
Under symmetry, we restrict attention to m1 = m2 = m. In this case, the posterior mean µ is a linear function of the aggregate statistic 2
s̃ =
1X s̃i . 2 i=1
Since the mapping between µ and s̃ is linear and one-to-one (with coefficients depending only on parameters), conditioning on µ is equivalent to conditioning on s̃. Therefore, without loss of generality, we may write firm 1’s information set under participation as I1S = {s1 , s̃}.
Sharing Equilibrium. Because (θ, s1 , s2 , s̃) are jointly Gaussian and the profit function is quadratic, the Bayesian Nash equilibrium in the quantity stage is linear in the sufficient statistics. Hence, for i ∈ {1, 2}, qiS = c1 + d1 si + e1 s̃. Firm 1’s first-order condition under inverse demand p = a + θ − q1 − q2 is q1S =
1 a + E[θ | s1 , s̃] − E[q2S | s1 , s̃] . 2
Substituting q2S = c1 + d1 s2 + e1 s̃ yields 1 c1 + d1 s1 + e1 s̃ = a − E[c1 + d1 s2 + e1 s̃ | s1 , s̃] + E[θ | s1 , s̃] . 2 30
(4)
Let s Y = 1 . s̃ Then E[s2 | s1 , s̃] = Σs2 ,Y Σ−1 Y Y Y, with
E[θ | s1 , s̃] = Σθ,Y Σ−1 Y Y Y,
τ h u+τ u+ τi ΣY Y = τ τ 2 m , Σs2 ,Y = u u + 2 , Σθ,Y = u u . u+ u+ + 2 2 2 Substituting these conditional expectations into (4) and matching coefficients gives
a c1 = , 3
d1 =
2mu 6mu + τ 2 + 2τ u + 4mτ
,
e1 =
2τ u 3 (6mu + τ 2 + 2τ u + 4mτ )
.
Substituting the equilibrium strategies into the profit function yields firm 1’s ex-ante expected profit under participation: S 1 2u2 (18m2 (τ + u) + τ 2 (τ + 2u) + mτ (7τ + 12u)) 2 E π1 = . a + 2 9 (4mτ + τ 2 + 6mu + 2τ u)
(5)
Uniliteral Deviation. If firm 1 deviates and does not participate, its information set is I1N S = {s1 }.
Since there are only two firms and no external signal, information aggregation does not occur. Accordingly, firm 2 must also base its decision solely on its own private signal s2 . We look for a linear strategy q1N S = c0 + d0 s1 . The first-order condition is q1N S =
1 a + E[θ | s1 ] − E[q2 | s1 ] . 2
Gaussian updating implies E[s2 | s1 ] =
u s1 , u+τ
E[θ | s1 ] =
u s1 . u+τ
Matching coefficients yields a c0 = , 3
d0 =
u , 3u + 2τ
and hence q1N S =
a u + s1 . 3 3u + 2τ
Substituting into the profit function gives firm 1’s ex-ante expected profit under nonparticipation: a2 u2 (τ + u) E π1N S = + . 9 (2τ + 3u)2 31
(6)
Define the unilateral participation gain: ∆ = E π1S − E π1N S . Using (5) and (6), direct algebra yields ∆=
τ 2 u2 (−2m(2τ + 3u)(4τ + 9u) − τ (τ + 2u)(τ + 3u)) < 0, 2 9(2τ + 3u)2 (2u(3m + τ ) + τ (4m + τ ))
∀ m ≥ 0.
Therefore, participation is strictly dominated by non-participation when the other firm participates, for any privacy noise level m ≥ 0. Hence, full participation cannot be a Nash equilibrium under any privacy-preserving mechanism of the stated form. This proves the proposition. A.2. Proposition 4.2: Information-sharing equilibrium for two firms with platform signal We analyze the unilateral incentive of firm 1. For any participating set J ⊆ {1, 2}, the platform releases the posterior mean µJ := E[θ | {si }i∈J , sp ] . Under joint Gaussianity, mJ admits the precision-weighted form 1 X 1 |J | 1 −1 1 µJ = V J + + , si + sp , VJ := τ i∈J τp u τ τp where VJ = Var(θ | {si }i∈J , sp ) is the (deterministic) posterior variance. Moreover, since µJ = E[θ | {si }i∈J , sp ], the law of total variance yields Var(µJ ) = u − VJ . In particular, under full participation J = {1, 2}, Var(µ2 ) = u − V2 ,
V2 =
1 u
+
2 1 −1 + . τ τp
Firm 1’s private posterior mean is µ1 := E[θ | s1 ] =
u s1 , u+τ
Var(µ1 ) =
u2 . u+τ
(7)
Sharing Equilibrium. When both firms participate, the platform releases µ2 = E[θ | s1 , s2 , sp ]. Given µ2 , the posterior mean of θ is exactly µ2 ; hence each firm behaves as if demand intercept were a + µ2 . A symmetric best response implies qiS =
a + µ2 , 3
i ∈ {1, 2}.
As in the n-firm case, conditional expected profit at the best response equals the square of the chosen quantity: E[πi | µ2 ] = (qiJ )2 . Therefore the ex-ante expected profit is a2 Var(µ2 ) E[π1S ] = E[(q1S )2 ] = + . 9 9 32
(8)
Uniliteral Deviation. Suppose firm 1 does not participate, so its information set is I1N S = {s1 } and its sufficient statistic is m1 in (7). Firm 2 participates; the platform releases 1 1 1 1 1 −1 V2 := µ2 := E[θ | s2 , sp ] = V2 s2 + sp , + + . τ τp u τ τp
(9)
We look for a linear Bayesian Nash equilibrium of the form q1N S = α0 + β0 µ1 ,
q2S = α1 + β1 µ2 .
(10)
Key regressions. Because (µ1 , µ2 ) are jointly Gaussian with mean 0, the conditional expectations are linear: E[µ2 | µ1 ] = δ µ1 ,
E[µ1 | µ2 ] = γ µ2 ,
(11)
where δ=
Cov(µ2 , µ1 ) , Var(µ1 )
γ=
Cov(µ2 , µ1 ) . Var(µ2 )
A direct covariance calculation gives the closed form for δ. Let A(τp ) :=
1 1 + . τ τp
Then u2 V2 A(τp ), Cov(µ2 , µ1 ) = Cov E[θ | s2 , sp ], E[θ | s1 ] = u+τ and therefore, using Var(µ1 ) = u2 /(u + τ ), δ = V2 A(τp ). Also, using Var(µ2 ) = u − V2 from (A.2), 2
u Cov(µ2 , µ1 ) u+τ V2 A(τp ) γ= = . Var(µ2 ) u − V2
Solving for (α0 , β0 , α1 , β1 ). Firm 1’s best response to q2 is 1 1 q1 = a + E[θ | µ1 ] − E[q2 | µ1 ] = a + µ1 − E[q2P | µ1 ] , 2 2 and by (10) and (11), E[q2P | µ1 ] = α1 + β1 E[µ2 | µ1 ] = α1 + β1 δ µ1 . Equating coefficients in q1N S = α0 + β0 m1 yields α0 =
a − α1 , 2
β0 =
1 − β1 δ . 2
Firm 2’s best response is 1 1 NS q2 = a + E[θ | m2 ] − E[q1 | m2 ] = a + µ2 − E[q1 | µ2 ] , 2 2 33
(12)
where E[q1N S | µ2 ] = α0 + β0 E[µ1 | µ2 ] = α0 + β0 γ µ2 . Matching coefficients in q2S = α1 + β1 m2 gives α1 =
a − α0 , 2
β1 =
1 − β0 γ . 2
(13)
Solving (12)–(13) yields α0 = α1 = a/3 and the closed form β0 (τp ) =
1 − 21 δ = 2 − 12 δ γ
1 − 12 V2 A(τp ) u2 V A(τ ) . p 2 1 2 − 2 V2 A(τp ) · u+τu−V 2
(14)
Using the same “best-response value” identity, conditional expected profit equals (q1N S )2 ; hence E[π1N S ] = E[(q1N S )2 ] =
a2 u2 a2 + β0 (τp )2 Var(µ1 ) = + β0 (τp )2 . 9 9 u+τ
(15)
Define the unilateral gain from joining when the rival joins: ∆2 (τp ) := E[π1J ] − E[π1N S ]. Combining (8) and (15) yields ∆(τp ) =
u2 Var(µ2 ) − β0 (τp )2 , 9 u+τ
Var(µ2 ) = u −
1 u
+
2 1 −1 + , τ τp
(16)
with β0 (τp ) given by (14) and V2 defined in (9). The sharing profile is a symmetric equilibrium if and only if ∆2 (τp ) ≥ 0. Define τp∗ as the (unique) solution to ∆2 (τp ) = 0, which is exactly the indifference condition (1). This equation provides an explicit computable characterization of τp∗ (e.g., via a one-dimensional root search), completing the proof. To see why the Sharing profile is unique, note that when neither firm shares, each firm can still obtain the platform-provided signal sp , and hence by unilaterally deviating to share it can additionally induce the platform to incorporate its private signal into the public posterior. This strictly increases the firm’s informational advantage relative to the no-sharing outcome (where only sp is available), and therefore yields a strictly higher expected payoff. Consequently, the NoSharing profile cannot be a symmetric Bayesian Nash equilibrium whenever τp is sufficiently small, implying that Sharing is the only symmetric equilibrium in that regime. 34
A.3. Proposition 4.3: Privacy noise induces voluntary sharing for two firms We analyze the unilateral incentive of firm 1. When a firm participates, the platform observes a privatized report yi := si + ξi = θ + (εi + ξi ),
εi ∼ N (0, τ ), ξi ∼ N (0, m),
so that εi + ξi ∼ N (0, τ + m). If both firms participate, the platform releases the posterior mean t12 (m) := E[θ | y1 , y2 , sp ] . Under joint Gaussianity, 1 1 (y1 + y2 ) + sp , t12 (m) = V2 (m) τ +m τp
V2 (m) :=
1 2 1 + + u τ + m τp
−1 .
(17)
Moreover, by the law of total variance, Var t12 (m) = u − V2 (m). If firm 1 does not participate while firm 2 does, the platform releases −1 1 1 1 1 1 t2 (m) := E[θ | y2 , sp ] = V1 (m) y2 + sp , V1 (m) := + + . τ +m τp u τ + m τp
Sharing equilibrium. Suppose both firms participate. Firm i observes (si , t12 (m)) and we look for a symmetric linear equilibrium qiS (m) = c + d si + e t12 (m),
i = 1, 2.
Let t := t12 (m). Firm i’s first-order condition under inverse demand p = a + θ − q1 − q2 is qiS (m) =
1 a − E[qjS (m) | si , t] + E[θ | si , t] , 2
j ̸= i.
Using Gaussian projection, both conditional expectations are linear in (si , t). Matching coefficients yields a unique solution, with the constant term unchanged: a c= . 3 The remaining coefficients (d, e) are uniquely determined as continuous functions of (u, τ, τp , m). Conditional expected profit equals the square of the chosen quantity, hence E[π1J (m)] = E (q1J (m))2 = c2 + d2 Var(s1 ) + e2 Var(t) + 2de Cov(s1 , t), where Var(s1 ) = u + τ , Var(t) = u − V2 (m), and Cov(s1 , t) follows from (17). 35
Unilateral deviation. Suppose firm 1 does not participate and uses q1N S (m) = c0 + d0 s1 . Firm 2 participates and observes (s2 , t2 (m)), using q2S (m) = c1 + d1 s2 + e1 t2 (m). Firm 1’s best response is q1N S =
1 a + E[θ | s1 ] − E[q2S (m) | s1 ] . 2
A direct calculation yields E[t2 (m) | s1 ] =
Cov(t2 (m), s1 ) s1 , Var(s1 )
and matching coefficients implies a c0 = c1 = , 3 with (d0 , d1 , e1 ) uniquely determined as continuous functions of (u, τ, τp , m). As before,
a2 E[π1N S (m)] = E (q1N S (m))2 = + d20 (u + τ ). 9
Define ∆(τp , m) := E[π1S (m)] − E[π1N S (m)]. When m = 0, the privacy-preserving platform coincides with the non-private platform. Since τp > τp∗ (u, τ ), full participation is not an equilibrium without privacy, implying ∆(τp , 0) < 0. As m → ∞, the precision of privatized reports vanishes and m↗∞
t12 (m) −−−−→ E[θ | sp ],
m↗∞
t2 (m) −−−−→ E[θ | sp ],
in L2 . Thus, participation grants access to the informative public posterior E[θ | sp ], while nonparticipation leaves firm 1 conditioning only on s1 . Since τp < ∞ implies Var(E[θ | sp ]) > 0, this strictly raises firm 1’s optimized ex-ante payoff, yielding lim ∆(τp , m) > 0.
m→∞
Finally, ∆(τp , m) is continuous in m. By the intermediate value theorem, there exists m∗ > 0 such that ∆(τp , m∗ ) = 0, and any m > m∗ yields ∆(τp , m) > 0. For any m > m∗ we have ∆(τp , m) > 0, so sharing is a strict best response when the opponent shares. If both firms do not share, a unilateral deviation to Sharing adds non-degenerate information (for any finite m) to the public posterior based on sp , which strictly increases the deviator’s optimized ex-ante payoff. Hence No-Sharing cannot be an equilibrium. Therefore, for all m > m∗ , Sharing is the unique symmetric Bayesian Nash equilibrium. 36
A.4. Proposition 4.4: n-Firm sharing equilibrium without privacy protection Suppose first that all firms participate and privacy noise is symmetric, i.e., mj = m for all j. Each shared signal is perturbed as vj ∼ N (0, m).
s̃j = sj + vj , The platform releases the posterior mean
µ = E[θ | s̃1 , . . . , s̃n ] . Under symmetry, the posterior mean is a linear function of the aggregate statistic n
s̃ =
1X s̃j . n j=1
Because the mapping between µ and s̃ is linear and one-to-one (with coefficients depending only on model parameters), conditioning on µ is equivalent to conditioning on s̃. Hence, although the platform releases µ, the statistic s̃ is sufficient, and the informational content is identical. Therefore, without loss of generality, firm i’s information set under sharing can be written as IiS = {si , s̃}.
If firm i deviates while all other firms share, the platform aggregates only the remaining n − 1 perturbed signals: s̃−i =
1 X s̃j . n − 1 j̸=i
The platform releases µ−i = E[θ | {s̃j }j̸=i ] , which is again a linear one-to-one transformation of s̃−i . Thus, the deviating firm observes only its private signal, IiD = {si },
while each participating firm j ̸= i observes {sj , s̃−i }. 37
Sharing Equilibrium. Because (θ, s1 , . . . , sn , s̄) are jointly Gaussian and profits are quadratic, a Bayesian Nash equilibrium is linear in the sufficient statistics. We look for a symmetric linear strategy qiS = cS + dS si + eS s̄. Conditional on s̄, the private signal si only contains idiosyncratic noise about εi and provides no additional information about the common state θ. Hence in equilibrium the coefficient on si vanishes: dS = 0 and qiS = cS + eS s̄. Imposing the Cournot first-order condition qiS =
X 1 a + E[θ | si , s̄] − E[ qjS | si , s̄] , 2 j̸=i
and matching coefficients yields cS =
a , n+1
Substituting these strategies into πi = (a + θ −
eS =
nu . (n + 1)(nu + τ )
P
j qj )qi gives the ex-ante expected profit under full
sharing: E[πiS ] =
a2 nu + a2 τ + nu2 . n3 u + n2 τ + 2n2 u + 2nτ + nu + τ
Uniliteral Deviation. Now suppose firm i deviates and does not share while all n − 1 rivals share among themselves. We consider linear strategies of the form qiN S = c0 + d0 si , and for each participating rival j ̸= i, qjN S = c1 + d1 sj + e1 s̄−i . Conditional on s̄−i the private signal sj only reveals idiosyncratic noise and does not improve inference about θ; thus d1 = 0 and insiders use only s̄−i : qjN S = c1 + e1 s̄−i . Imposing the best-response (first-order) conditions for the deviator and for an insider and matching coefficients yields c0 = c1 =
a , n+1
d0 =
u (nτ + nu − u) 2n2 τ u + n2 u2 + 2nτ 2 − u2
,
e1 =
u (n − 1)(2τ + u) 2n2 τ u + n2 u2 + 2nτ 2 − u2
Substituting into the profit function gives the deviator’s ex-ante expected profit E[πideviate ]. 38
.
Define ∆ = E[πiS ] − E[πiN S ]. Direct algebra yields a closed-form expression for the unilateral participation gain: 2 ! (u + τ ) (n − 1)u + nτ n 2 ∆=u − 2 . (n + 1)2 (nu + τ ) (n2 − 1)u2 + 2n2 uτ + 2nτ 2 Since u > 0, the sign of ∆ is fully determined by the bracketed expression. Therefore, 2 (u + τ ) (n − 1)u + nτ n ∆ ≥ 0 ⇐⇒ ≥ 2 . (n + 1)2 (nu + τ ) (n2 − 1)u2 + 2n2 uτ + 2nτ 2 Hence, full information sharing constitutes a Nash equilibrium if and only if the above inequality holds. A.5. Corollary 4.5: Characterizing the threshold for n-Firm sharing equilibrium We begin with the following helper lemma. Lemma A.1 (Concavity and single-threshold structure) Let r = τ /u > 0, and let Φ(n, r) denote the polynomial obtained by clearing denominators in the condition ∆ ≥ 0, so that ∆≥0
⇐⇒
Φ(n, r) ≥ 0.
Then, for all n ≥ 3 and all r > 0, the function Φ(n, r) is strictly concave in r, i.e., ∂ 2 Φ(n, r) < 0. ∂r2 Moreover, for each fixed n ≥ 3, Φ(n, r) satisfies Φ(n, 0) > 0
and
lim Φ(n, r) = −∞.
r→∞
Consequently, for every n ≥ 3, the equation Φ(n, r) = 0 admits a unique positive solution r > 0, and Φ(n, r) ≥ 0
⇐⇒
0 < r ≤ r0 (n),
where r0 (n) denotes this unique positive root. Equivalently, ∆≥0
⇐⇒
τ ≤ r0 (n). u
By construction, the inequality ∆ ≥ 0 is equivalent to Φ(n, r) ≥ 0, where Φ(n, r) is a polynomial in r whose coefficients depend on n. Direct differentiation yields ∂ 2 Φ(n, r) = −2n4 + (4 − 6r)n3 + (6r − 6)n2 − 4n. ∂r2 39
We show that this expression is strictly negative for all n ≥ 3 and all r > 0. Rewrite it as ∂ 2 Φ(n, r) = − 2n4 − 4n + (4 − 6r)n3 + (6r − 6)n2 . 2 ∂r The first term is strictly negative. For the second term, factor out n2 : (4 − 6r)n3 + (6r − 6)n2 = n2 (4 − 6r)n + (6r − 6) . For n ≥ 3 and r > 0, we have (4 − 6r)n + (6r − 6) = 4n − 6 + r(6 − 6n) ≤ 4n − 6, since 6 − 6n ≤ 0 and r > 0. Therefore, (4 − 6r)n3 + (6r − 6)n2 ≤ 4n3 − 6n2 . It follows that ∂ 2 Φ(n, r) ≤ −2n4 + 4n3 − 6n2 − 4n = −2n n3 − 2n2 + 3n + 2 . 2 ∂r Let g(n) = n3 − 2n2 + 3n + 2. Then g ′ (n) = 3n2 − 4n + 3, whose discriminant is (−4)2 − 4 · 3 · 3 = −20 < 0, implying g ′ (n) > 0 for all n. Hence g(n) is strictly increasing and
g(3) = 27 − 18 + 9 + 2 = 20 > 0, so g(n) > 0 for all n ≥ 3. Since −2n < 0 for n ≥ 3, we conclude ∂ 2 Φ(n, r) <0 ∂r2
∀ n ≥ 3, r > 0.
Thus Φ(n, r) is strictly concave in r on (0, ∞). Evaluating at r = 0 yields Φ(n, 0) > 0 for all n ≥ 3. Moreover, since the leading term in r is cubic with negative coefficient, it follows that lim Φ(n, r) = −∞.
r→∞
Strict concavity together with Φ(n, 0) > 0 and limr→∞ Φ(n, r) = −∞ implies that for each fixed n ≥ 3, the equation Φ(n, r) = 0 admits a unique positive solution r > 0. Moreover, Φ(n, r) ≥ 0 ⇐⇒ 0 < r ≤ r0 (n), where r0 (n) denotes this unique positive root. Rewriting in terms of the original parameters yields τ ∆ ≥ 0 ⇐⇒ ≤ r0 (n). u 40
We can now give the proof of Corollary 4.5. The existence and uniqueness of a positive solution r > 0 to Φ(n, r) = 0 for each fixed n ≥ 3, as well as the threshold characterization Φ(n, r) ≥ 0 ⇐⇒ 0 < r ≤ r0 (n)
⇐⇒
∆ ≥ 0 ⇐⇒
τ ≤ r0 (n), u
follow directly from Lemma A.1. It remains to prove the monotonicity of r0 (n) in n. Since Φ is homogeneous of degree 3 in (τ, u), substituting r = τ /u and dividing by u3 > 0 yields an equivalent polynomial (with the same sign) of the form Φ(n, r) = a0 (n) + a1 (n)r + a2 (n)r2 + a3 (n)r3 , where
a0 (n) = (n − 1)(n + 1) (n − 2)n − 1 = n4 − 2n3 − 2n2 + 2n + 1, a1 (n) = n (n − 3)n(n + 1) − 1 + 1 = n4 − 2n3 − 3n2 − n + 1, a2 (n) = −n n (n − 2)n + 3 + 2 = −n4 + 2n3 − 3n2 − 2n, a3 (n) = −(n − 1)n2 < 0
(n ≥ 2).
By Lemma A.1, for each n ≥ 3 there is a unique r0 (n) > 0 such that Φ n, r0 (n) = 0 and Φ(n, r) > 0 for r ∈ (0, r0 (n)), while Φ(n, r) < 0 for r > r0 (n). √
Let φ := 1+2 5 be the golden ratio, so that φ2 = φ + 1 and φ3 = 2φ + 1. Evaluating Φ(n, r) at r = φ and using these identities gives Φ(n, φ) = a0 + a1 φ + a2 φ2 + a3 φ3 = a0 + a2 + a3 + a1 + a2 + 2a3 φ. A direct simplification yields a0 (n) + a2 (n) + a3 (n) = 1 − n3 − 4n2 < 0, and a1 (n) + a2 (n) + 2a3 (n) = 1 − 2n3 − 4n2 − 3n < 0, for every n ≥ 3. Since φ > 0, it follows that Φ(n, φ) < 0 for all n ≥ 3. Because Φ(n, 0) = a0 (n) > 0 (Lemma A.1) and Φ(n, ·) has a unique positive root, we conclude that 0 < r0 (n) < φ
for all n ≥ 3.
Fix n ≥ 3 and consider the point (n, r0 (n)) where Φ(n, r0 (n)) = 0. Since Φ(n, ·) is strictly concave and changes sign from positive to negative at r0 (n), the crossing is strict and hence ∂r Φ n, r0 (n) < 0. 41
(Indeed, if ∂r Φ(n, r0 (n)) ≥ 0, strict concavity would prevent Φ from becoming strictly negative for all r > r0 (n) without creating an additional zero, contradicting uniqueness.) We now show that ∂n Φ n, r0 (n) > 0
for all n ≥ 3.
A direct differentiation gives ∂n Φ(n, r) = a′0 (n) + a′1 (n)r + a′2 (n)r2 + a′3 (n)r3 . On the zero set {Φ(n, r) = 0} we may eliminate the cubic term by using r3 = −
a0 + a1 r + a2 r2 , a3
which yields the identity (valid at any (n, r) with Φ(n, r) = 0) a′ a′ a′ ∂n Φ(n, r) = a′0 − 3 a0 + a′1 − 3 a1 r + a′2 − 3 a2 r2 . a3 a3 a3 Define bk (n) := a′k (n) −
a′3 (n) ak (n), a3 (n)
k ∈ {0, 1, 2}.
Then, on Φ(n, r) = 0, ∂n Φ(n, r) = b0 (n) + b1 (n)r + b2 (n)r2 . Since a3 (n) = −(n − 1)n2 and a′3 (n) = −n(3n − 2), we have a′3 (n) 3n − 2 = >0 a3 (n) n(n − 1)
(n > 1).
A straightforward simplification using the explicit ak (n) yields b0 (n) = n3 − n2 + 3n − 1 − b1 (n) =
2 >0 n
(n ≥ 3),
n5 − 2n4 + 5n3 + 2n2 − 4n + 2 >0 n(n − 1)
(n ≥ 3),
and b2 (n) =
−n4 + 2n3 + n2 + 4n − 2 <0 n−1
Thus, for each fixed n ≥ 3, the function Qn (r) := b0 (n) + b1 (n)r + b2 (n)r2 is a strictly concave quadratic in r (since b2 (n) < 0). 42
(n ≥ 3).
At the root we have 0 < r0 (n) < φ. Therefore it suffices to show Qn (r) > 0 on the interval [0, φ]. Because Qn is concave, its minimum over the compact interval [0, φ] is attained at an endpoint, so min Qn (r) = min{Qn (0), Qn (φ)}.
r∈[0,φ]
We already have Qn (0) = b0 (n) > 0. It remains to verify Qn (φ) > 0. Using φ2 = φ + 1, Qn (φ) = b0 (n) + b1 (n)φ + b2 (n)φ2 = b0 (n) + b2 (n) + b1 (n) + b2 (n) φ. Substituting the explicit expressions for b0 , b1 , b2 above and simplifying yields √ √ √ √ (8 + 3 5)n3 + (3 + 3 5)n2 − (6 + 3 5)n + (3 + 5) Qn (φ) = . n(n − 1) The denominator is positive for n ≥ 3, and the numerator is a cubic polynomial with strictly positive √ leading coefficient 8 + 3 5 > 0 and positive n2 coefficient as well; in particular it is strictly positive for all n ≥ 3. Hence Qn (φ) > 0, and consequently Qn (r) > 0 for all r ∈ [0, φ]. Since r0 (n) ∈ (0, φ), we obtain ∂n Φ n, r0 (n) = Qn r0 (n) > 0
(n ≥ 3).
By the implicit function theorem, in a neighborhood of each n ≥ 3 the unique root r0 (n) is differentiable and satisfies
∂n Φ n, r0 (n) dr0 (n) . =− dn ∂r Φ n, r0 (n) Since ∂r Φ n, r0 (n) < 0 and ∂n Φ n, r0 (n) > 0. Therefore dr0 (n) >0 dn
for all n ≥ 3,
so r0 (n) is strictly increasing in n. Finally, since the equilibrium condition is τ /u ≤ r0 (n), the feasible region for full information sharing expands monotonically with n and is bounded by the implicit curve Φ(n, τ /u) = 0. A.6. Proposition 4.6: n-Firm information-sharing equilibrium with platform signal We analyze the unilateral incentive of firm 1. Fix any participating set J. By the platform rule in Section 3, the released statistic is the posterior mean µJ := E[θ | {si }i∈J , sp ] . Under joint Gaussianity, µJ is linear in the signals and can be written in the precision-weighted form µJ = V J
1 X τ i∈J
si +
1 sp , τp
VJ := 43
1 u
+
|J | 1 −1 + , τ τp
where VJ is the posterior variance Var(θ | {si }i∈J , sp ). Moreover, since µJ = E[θ | {si }i∈J , sp ], the law of total variance yields Var(µJ ) = Var E[θ | {si }i∈J , sp ] = Var(θ) − E Var(θ | {si }i∈J , sp ) = u − VJ , where the last equality uses Var(θ) = u and the fact that, in the Gaussian signal model, the posterior variance Var(θ | {si }i∈J , sp ) = VJ is deterministic (i.e., does not depend on the realized signals). In particular, under full participation J = [n], Var(µn ) = u − Vn ,
Vn =
1 u
+
n 1 −1 . + τ τp
Firm 1’s private posterior mean is µ1 := E[θ | s1 ] =
u s1 , u+τ
Var(µ1 ) =
u2 . u+τ
(18)
Sharing Equilibrium. When all firms participate, the platform releases m[n] = E[θ | s1 , . . . , sn , sp ]. Given m[n] , the posterior mean of θ is exactly m[n] ; hence each firm behaves as if demand intercept were a + m[n] . A symmetric best response implies qiS =
a + µn , n+1
i = 1, . . . , n.
Under linear demand and risk-neutrality, conditional expected profit at the best response equals the square of the chosen quantity: E[πi | m[n] ] = max E[(a + θ − qi
X
qj )qi | m[n] ] = (qiS )2 .
j
Therefore the ex-ante expected profit is E[π1S ] = E[(q1S )2 ] =
a2 Var(µn ) + . 2 (n + 1) (n + 1)2
(19)
Unilateral deviation Suppose firm 1 does not participate, so its information set is I1N S = {s1 } and its sufficient statistic is µ1 in (18). The remaining n − 1 firms participate; the platform releases µ−1 := E[θ | s2 , . . . , sn , sp ] = V−1
n 1 X
τ i=2
si +
1 sp , τp
V−1 :=
1 u
+
n−1 1 −1 + . τ τp
(20)
We look for a linear Bayesian Nash equilibrium of the form q1N S = α0 + β0 µ1 ,
qjS = α1 + β1 µ−1 , j = 2, . . . , n.
(Again µ−1 is sufficient for θ given participating information, so qjP depends only on m−1 .) 44
(21)
Key regressions. Because (m1 , m−1 ) are jointly Gaussian with mean 0, the conditional expectations are linear: E[µ−1 | µ1 ] = δ µ1 ,
E[µ1 | µ−1 ] = γ µ−1 ,
(22)
where δ=
Cov(µ−1 , µ1 ) , Var(µ1 )
γ=
Cov(µ−1 , µ1 ) . Var(µ−1 )
A direct covariance calculation using (18)–(20) gives a simple closed form for δ. Let A(τp ) :=
1 n−1 + . τ τp
Then u2 Cov(µ−1 , µ1 ) = Cov E[θ | s2 , . . . , sn , sp ], E[θ | s1 ] = V−1 A(τp ), u+τ and therefore, using Var(µ1 ) = u2 /(u + τ ), δ = V−1 A(τp ).
(23)
Also, using Var(µ−1 ) = u − V−1 from (20), 2
u Cov(µ−1 , µ1 ) u+τ V−1 A(τp ) = . γ= Var(µ−1 ) u − V−1
(24)
Solving for (α0 , β0 , α1 , β1 ). Firm 1’s best response to opponents’ total quantity Q−1 is q1 =
1 1 a + E[θ | µ1 ] − E[Q−1 | µ1 ] = a + µ1 − (n − 1)E[q2P | µ1 ] , 2 2
and by (21) and (22), E[q2P | µ1 ] = α1 + β1 E[µ−1 | µ1 ] = α1 + β1 δµ1 . Equating coefficients in q1N S = α0 + β0 µ1 yields α0 =
a − (n − 1)α1 , 2
β0 =
1 − (n − 1)β1 δ . 2
(25)
For any participating firm j ≥ 2, the best response condition is qj =
1 1 a + E[θ | µ−1 ] − E[q1N S + (n − 2)q2P | µ−1 ] = a + µ−1 − E[q1N S | µ−1 ] − (n − 2)q2P , 2 2
where E[q1N S | µ−1 ] = α0 + β0 E[µ1 | µ−1 ] = α0 + β0 γµ−1 . Matching coefficients in qjP = α1 + β1 µ−1 gives α1 =
a − α0 , n
β1 = 45
1 − β0 γ . n
(26)
Solving (25)–(26) yields α0 = α1 = a/(n + 1) and β0 (τp ) =
1 − n−1 δ n = n−1 2− n δγ
1 − n−1 V−1 A(τp ) n u2 V A(τ ) . p −1 2 − n−1 V A(τ ) · u+τu−V −1 p n −1
(27)
(Here δ, γ are given explicitly by (23)–(24).) Using the same best-response identity, conditional expected profit equals (q1N S )2 ; hence E[π1N S ] = E[(q1N S )2 ] =
2 a2 a2 2 2 u + β (τ ) Var(µ ) = + β (τ ) . 0 p 1 0 p (n + 1)2 (n + 1)2 u+τ
(28)
Define the unilateral gain from joining when others join: ∆(τp ) := E[π1S ] − E[π1N S ]. Combining (19) and (28) yields the fully explicit expression ∆(τp ) =
2 Var(µn ) 2 u − β (τ ) , 0 p (n + 1)2 u+τ
Var(µn ) = u −
1 u
+
n 1 −1 + , τ τp
with β0 (τp ) given by (27) and V−1 defined in (20). The sharing profile is a symmetric equilibrium if and only if ∆n (τp ) ≥ 0. Define τp∗ as the (unique) solution to ∆n (τp ) = 0, which is exactly the indifference condition 2 Var(µn ) 2 u = β (τ ) , 0 p (n + 1)2 u+τ
equivalently (3). This equation provides an explicit computable characterization of τp∗ (e.g., via a one-dimensional root search), completing the proof. A.7. Proposition 4.7: Privacy noise induces voluntary sharing among n firms We analyze the unilateral incentive of firm 1. When a set J ⊆ {1, . . . , n} participates, the platform observes privatized reports yi := si + ξi = θ + (εi + ξi ),
εi ∼ N (0, τ ), ξi ∼ N (0, m),
so that εi + ξi ∼ N (0, τ + m). The platform releases the posterior mean tJ (m) := E[θ | {yi }i∈J , sp ] . Under joint Gaussianity, tJ (m) is linear and admits the precision-weighted form ! −1 1 X 1 1 |J | 1 tJ (m) = VJ (m) . yi + sp , VJ (m) := + + τ + m i∈J τp u τ + m τp Moreover, since tJ (m) = E[θ | {yi }i∈J , sp ], the law of total variance yields Var tJ (m) = Var(θ) − E Var(θ | {yi }i∈J , sp ) = u − VJ (m), 46
(29)
where the last equality uses Var(θ) = u and the fact that in the Gaussian signal model the posterior variance Var(θ | {yi }i∈J , sp ) = VJ (m) is deterministic. We will use two special cases. Under full participation J = [n], ! −1 n 1 1 X 1 n 1 t[n] (m) = Vn (m) Vn (m) := . yi + sp , + + τ + m i=1 τp u τ + m τp Under unilateral non-participation by firm 1, the platform aggregates only firms 2, . . . , n: ! −1 n 1 1 X 1 n−1 1 Vn−1 (m) := . t−1 (m) := t[n]\{1} (m) = Vn−1 (m) yi + sp , + + τ + m i=2 τp u τ + m τp
Sharing equilibrium. Suppose all firms participate. Firm i observes (si , t[n] (m)) and we look for a symmetric linear equilibrium of the form qiS (m) = c + d si + e t[n] (m),
i = 1, . . . , n.
(30)
Let t := t[n] (m) and define X := (si , t)T . By Gaussian projection, for any j ̸= i and for θ, E[θ | si , t] = ΣθX Σ−1 XX X,
E[sj | si , t] = Σsj X Σ−1 XX X,
where the covariance objects are explicit. Writing t = α α=
Vn (m) , τ +m
β=
(31)
Pn
k=1 yk + βsp with
Vn (m) , τp
we have Var(si ) = u + τ,
Var(t) = u − Vn (m),
Cov(θ, t) = Var(t) = u − Vn (m),
and Cov(si , t) = Cov(sj , t) =: Cst , Cov(si , sj ) = u
Cst = α(nu + τ ) + βu
(j ̸= i),
(j ̸= i).
Cov(θ, si ) = u.
Hence
u+τ α(nu + τ ) + βu ΣXX = , α(nu + τ ) + βu u − Vn (m) Σsj X = u α(nu + τ ) + βu . ΣθX = u u − Vn (m) , Firm i’s first-order condition under inverse demand p = a + θ − qiS (m) =
1 a − E[Q−i | si , t] + E[θ | si , t] , 2 47
(32)
Pn
k=1 qk is
Q−i :=
X k̸=i
qkJ (m).
(33)
Using (30) and (31), E[Q−i | si , t] = (n − 1)c + (n − 1)d E[sj | si , t] + (n − 1)e t. Substituting into (33), and matching coefficients on (1, si , t) yields a linear system in (c, d, e) with unique solution.1 In particular, the constant term is unchanged: c=
a . n+1
(34)
The remaining coefficients can be written compactly in terms of the regression (projection) coefficients obtained from (32). Define E[sj | si , t] = λs si + λt t,
E[θ | si , t] = κs si + κt t,
where (λs , λt ) and (κs , κt ) are computed explicitly from (31)–(32). Matching coefficients on (si , t) yields the linear system 1 κs − (n − 1)dλs , 2 n−1 1 e. e = κt − (n − 1)dλt − 2 2
d=
(35)
i.e., equivalently, d=
1 κs − (n − 1)dλs , 2
e=
n−1 1 κt − (n − 1)dλt − e. 2 2
(36)
(These yield closed-form rational expressions in (n, u, τ, τp , m) upon substituting λs , λt , κs , κt from (32).) Under linear demand and risk-neutrality, conditional expected profit at the best response equals the square of the chosen quantity, hence E[πiS (m) | si , t] = (qiS (m))2 ,
⇒
E[π1S (m)] = E (q1S (m))2 .
Using (30) and E[s1 ] = E[t] = 0, E[π1S (m)] = c2 + d2 Var(s1 ) + e2 Var(t) + 2de Cov(s1 , t), where Var(s1 ) = u + τ , Var(t) = u − Vn (m), and Cov(s1 , t) = α(nu + τ ) + βu. 1
Uniqueness follows because ΣXX is positive definite for u > 0, τ > 0, τp ∈ (0, ∞), and m ≥ 0.
48
(37)
Unilateral deviation. Suppose firm 1 does not participate, so it observes only s1 and uses a linear strategy q1N S (m) = c0 + d0 s1 .
(38)
The remaining n − 1 firms participate. The platform releases t−1 (m) in (A.7). Each participating firm j ∈ {2, . . . , n} observes (sj , t−1 (m)) and uses a linear strategy qjS (m) = c1 + d1 sj + e1 t−1 (m).
(39)
Firm 1 (non-participation). Firm 1’s best response is 1 q1 = a + E[θ | s1 ] − E[Q−1 | s1 ] , 2
Q−1 :=
n X
qjS (m).
j=2
u u s1 and E[s2 | s1 ] = u+τ s1 . Moreover, since t−1 (m) is Gaussian and linear in We have E[θ | s1 ] = u+τ
(θ, ε2 , . . . , εn , ξ2 , . . . , ξn , η), E[t−1 (m) | s1 ] = Writing t−1 (m) = α−1
Cov(t−1 (m), s1 ) s1 . Var(s1 )
Pn
k=2 yk + β−1 sp with
α−1 =
Vn−1 (m) , τ +m
β−1 =
Vn−1 (m) , τp
a direct covariance calculation gives Cov(s1 , t−1 (m)) = α−1 (n − 1)u + β−1 u,
Var(s1 ) = u + τ.
Therefore, E[t−1 (m) | s1 ] =
α−1 (n − 1)u + β−1 u s1 . u+τ
(40)
Using (39) and linearity, E[Q−1 | s1 ] = (n − 1)c1 + (n − 1)d1
u s1 + (n − 1)e1 E[t−1 (m) | s1 ]. u+τ
Substituting into the best response and matching (1, s1 ) with (38) yields a − (n − 1)c1 1 u u α−1 (n − 1)u + β−1 u c0 = , d0 = − (n − 1)d1 − (n − 1)e1 . 2 2 u+τ u+τ u+τ
(41)
Participating firms j ≥ 2. Fix j ∈ {2, . . . , n} and let X := (sj , t−1 (m))T . Then Gaussian projection implies E[s1 | sj , t−1 (m)] = Σs1 X Σ−1 XX X,
E[θ | sj , t−1 (m)] = ΣθX Σ−1 XX X.
Moreover, for any k ∈ {2, . . . , n} \ {j }, E[sk | sj , t−1 (m)] = Σsk X Σ−1 XX X. 49
The covariance elements are explicit: Var t−1 (m) = u − Vn−1 (m),
Var(sj ) = u + τ,
Cov(sj , t−1 (m)) = α−1 ((n − 1)u + τ ) + β−1 u, Cov(s1 , sj ) = u,
Cov(θ, sj ) = u,
Cov(θ, t−1 (m)) = α−1 (n − 1)u + β−1 u, Cov(s1 , t−1 (m)) = α−1 (n − 1)u + β−1 u,
Cov(sk , sj ) = u,
Cov(sk , t−1 (m)) = Cov(sj , t−1 (m)).
Thus the 2 × 2 covariance matrix ΣXX of (sj , t−1 (m)) and the cross-covariance vectors Σs1 X ,ΣθX , Σsk X are known explicitly and can be inverted in closed form. Since s1 = θ + ε1 , where ε1 is independent of (sj , t−1 (m)) and has zero mean, it follows that E[s1 | sj , t−1 (m)] = E[θ | sj , t−1 (m)]. Therefore, Gaussian projection yields E[s1 | sj , t−1 (m)] = E[θ | sj , t−1 (m)] = ϕs sj + ϕt t−1 (m), and for any k ̸= j, k ≥ 2, E[sk | sj , t−1 (m)] = χs sj + χt t−1 (m), where −1 (ϕs , ϕt ) := Σs1 X Σ−1 XX = ΣθX ΣXX ,
(χs , χt ) := Σsk X Σ−1 XX .
The first-order condition for firm j is qj =
1 a + E[θ | sj , t−1 (m)] − E[q1N S (m) + 2
X
qkS (m) | sj , t−1 (m)] .
k∈{2,...,n}\{j}
Using the linear strategies q1N S (m) = c0 + d0 s1 ,
qkS (m) = c1 + d1 sk + e1 t−1 (m),
we obtain E[q1N S (m) | sj , t] = c0 + d0 (ϕs sj + ϕt t), #
" E
X
qkS (m) | sj , t = (n − 2)c1 + (n − 2)d1 (χs sj + χt t) + (n − 2)e1 t.
k̸=j,k≥2
Matching coefficients on (1, sj , t−1 (m)) with qjS (m) = c1 + d1 sj + e1 t−1 (m) yields the linear system 1 a − c0 − (n − 2)c1 , 2 1 ϕs − d0 ϕs − (n − 2)d1 χs , d1 = 2 1 e1 = ϕt − d0 ϕt − (n − 2)d1 χt − (n − 2)e1 . 2 50 c1 =
(42)
Combining (41) and the first equation in (42) gives c0 = c1 =
a , n+1
and the remaining coefficients (d0 , d1 , e1 ) are uniquely determined by the resulting 3 × 3 linear system in (41) and (42). Conditional expected profit at the best response equals the square of the chosen quantity, hence E[π1N S (m) | s1 ] = (q1N S (m))2 ,
E[π1N S (m)] = E (q1N S (m))2 = c20 + d20 Var(s1 ),
⇒
so E[π1N S (m)] =
a2 + d20 (u + τ ). (n + 1)2
(43)
Define the unilateral participation gain under privacy noise, ∆(τp , m) := E[π1S (m)] − E[π1N S (m)]. By construction, the privacy mechanism with m = 0 coincides with the non-private platform. Since τp > τp∗ (n, u, τ ) implies that full participation is not an equilibrium under the non-private platform, we have ∆n (τp , 0) < 0.
(44)
Next, consider the limit m → ∞. In (29), the precision of each privatized report is 1/(τ + m) → 0, so the platform statistic satisfies m↗∞
m↗∞
t[n] (m) −−−−→ E[θ | sp ],
t−1 (m) −−−−→ E[θ | sp ],
in L2 (and hence in distribution). Thus, as m → ∞, the sharing environment grants firm 1 access (through participation) to the informative public posterior E[θ | sp ], whereas under unilateral nonparticipation firm 1 remains restricted to s1 while its rivals condition on (sj , E[θ | sp ]). Since τp < ∞ implies that E[θ | sp ] is non-degenerate and payoff-relevant, this strictly increases firm 1’s optimized ex-ante payoff, implying lim ∆(τp , m) > 0.
m→∞
(45)
Finally, ∆n (τp , m) is continuous in m on [0, ∞): all equilibrium coefficients are obtained by solving linear systems whose entries are polynomials in the covariance elements of (θ, si , sp , {yi }), and these covariance elements depend continuously on m (indeed, rationally through Vn (m) and Vn−1 (m)). Therefore, by (44)–(45) and the intermediate value theorem, there exists m∗ > 0 such that ∆n (τp , m∗ ) = 0, and any m > m∗ yields ∆n (τp , m) > 0. This proves that under the privacypreserving platform, full participation is sustained as a symmetric equilibrium for a suitable privacy noise level. 51
A.8. Proposition 5.1: Privacy restores voluntary sharing under heterogeneous signals We prove the claim by analyzing each firm’s unilateral incentive and then taking a common high-noise limit. The argument follows the same structure as in the homogeneous case. Firm i observes a private signal si = θ + εi with εi ∼ N (0, τi ), and the platform observes an external signal sp = θ + εp with εp ∼ N (0, τp ). If a set J ⊆ {1, . . . , n} participates, the platform receives privatized reports yi := si + ξi = θ + (εi + ξi ),
ξi ∼ N (0, mi ),
so that εi + ξi ∼ N (0, τi + mi ). The platform releases the posterior mean tJ (m) := E[θ | {yi }i∈J , sp ] ,
m := (m1 , . . . , mn ).
Under joint Gaussianity, tJ (m) is linear and admits the precision-weighted form ! !−1 X 1 1 1 1 X 1 tJ (m) = VJ (m) VJ (m) := + . yi + sp , + τi + mi τp u i∈J τi + mi τp i∈J
(46)
Moreover, since tJ (m) = E[θ | {yi }i∈J , sp ], the law of total variance yields Var tJ (m) = u − VJ (m),
(47)
where the posterior variance VJ (m) is deterministic in the Gaussian model. Under full participation J = [n], t[n] (m) = Vn (m)
! 1 1 yi + sp , τi + mi τp i=1
n X
and under unilateral non-participation by firm i, the platform aggregates only [n] \ {i}: ! X 1 1 t−i (m) := t[n]\{i} (m) = V−i (m) yj + sp . τj + mj τp j̸=i
Sharing equilibrium. Fix m ≥ 0 and suppose all firms participate. Because primitives are jointly Gaussian and payoffs are quadratic under linear demand, there exists a (unique) Bayesian Nash equilibrium in which each firm i uses an affine strategy in its information (si , t[n] (m)): qiS (m) = ci (m) + di (m) si + ei (m) t[n] (m). The equilibrium coefficients solve a linear system obtained from the first-order conditions and Gaussian projection identities of the form E[θ | si , t[n] (m)] and E[sj | si , t[n] (m)]. All entries of this system are polynomials in the relevant covariance elements of (θ, s1 , . . . , sn , sp , y1 , . . . , yn ), and hence depend continuously on m through the terms (τi + mi )−1 and VJ (m) in (46). Therefore, the equilibrium coefficients and the ex-ante sharing payoff E[πiS (m)] vary continuously with m. 52
Unilateral deviation. Fix a firm i. Suppose firm i does not participate while all other firms do. Then firm i observes only si and uses an affine best response qiN S (m) = c̄i (m) + d¯i (m) si , whereas each participating firm j ̸= i observes (sj , t−i (m)) and uses an affine strategy qjS (m) = c̄j (m) + d¯j (m) sj + ēj (m) t−i (m). Then, these coefficients solve a linear system whose entries are covariance elements depending continuously on m. Hence the deviation payoff E[πiN S (m)] is continuous in m. Define firm i’s unilateral participation gain under privacy noise, ∆i (m) := E[πiS (m)] − E[πiN S (m)]. By the preceding discussion, each ∆i (m) is continuous in m on [0, ∞)n . By assumption, under the non-private mechanism (m = 0) the platform signal is not precise enough to sustain full participation; equivalently, the sharing profile fails to be incentive compatible, so there exists at least one firm i with ∆i (0) < 0. Next consider a common high-noise scaling path m(t) := t 1 with t ≥ 0. As t → ∞, each privatized report precision satisfies (τi + t)−1 → 0. From (46), t↗∞
t↗∞
t[n] (m(t)) −−−→ E[θ | sp ],
t−i (m(t)) −−−→ [θ | sp ],
in L2 (hence in distribution), because the contribution of the privatized reports vanishes and the posterior is driven only by (θ, sp ). Since τp < ∞, the public posterior E[θ | sp ] is non-degenerate and payoff-relevant. In the limit environment t → ∞, if firm i participates when all others participate, it observes (si , E[θ | sp ]); if it does not participate while others do, it observes only si while its rivals condition on (sj , E[θ | sp ]). Thus, participation strictly enlarges firm i’s information set by adding the informative signal E[θ | sp ]. In linear-quadratic Cournot, the value of (Blackwell) more informative signals is strictly positive whenever the additional signal is non-degenerate, so firm i’s optimized ex-ante profit is strictly higher under participation in this limit. Hence, lim ∆i (m(t)) > 0
t→∞
for each i ∈ {1, . . . , n}.
(48)
By continuity of ∆i (m(t)) in t, for each firm i there exists a finite threshold t∗i > 0 such that ∆i (m(t)) > 0 for all t ≥ t∗i . Let t∗ := max t∗i , i∈{1,...,n}
53
m∗ := t∗ 1.
Then for all m ≥ m∗ (componentwise), in particular for m = m∗ and any larger vector, we have ∆i (m) ≥ ∆i (m∗ ) > 0 for every firm i. Therefore, under the privacy-preserving mechanism with m ≥ m∗ , each firm strictly prefers to participate when all other firms participate. This establishes that sharing is sustained as an equilibrium under sufficiently large privacy noise, completing the proof. A.9. Proposition 5.2: Privacy-induced participation under sequential entry Fix firm i and suppose all firms j ̸= i have already joined the platform and share using a common privacy noise level m. Firm i has signal noise variance τi and, if it participates, the platform applies a firm-specific privacy noise level mi to its report. Throughout, inverse demand is Pn p = a + θ − k=1 qk , the prior is θ ∼ N (0, u), and the platform signal is sp = θ + η with η ∼ N (0, τp ). For each participating firm k, the platform observes the privatized report yk := sk + ξk = θ + (εk + ξk ),
εk ∼ N (0, τk ), ξk ∼ N (0, mk ),
with all shocks independent across firms and from θ. For firms j ̸= i, (τj , mj ) = (τ, m), while for firm i, (τi , mi ) = (τi , mi ). If firm i participates, the platform releases the posterior mean t[n] (mi ) := E[θ | {yj }j̸=i , yi , sp ] . Under joint Gaussianity, t[n] (mi ) is linear and admits the precision-weighted form ! 1 X 1 1 t[n] (mi ) = Vn (mi ) yj + yi + sp , τ + m j̸=i τi + mi τp −1 1 n−1 1 1 Vn (mi ) := + + + . u τ + m τi + mi τp
(49)
If firm i does not participate, the platform aggregates only firms j ̸= i and releases
t−i (m) := E[θ | {yj }j̸=i , sp ] = Vn−1 (m) Vn−1 (m) :=
1 n−1 1 + + u τ + m τp
! 1 1 X yj + sp , τ + m j̸=i τp
−1 .
Moreover, since each released statistic is a conditional expectation, the law of total variance yields Var t[n] (mi ) = u − Vn (mi ),
Var t−i (m) = u − Vn−1 (m),
and these are deterministic in the Gaussian model. 54
For later use, write t[n] (mi ) = αO αO = and write t−i (m) = α−i
P
j̸=i yj + αI yi + βsp , where
Vn (mi ) , τ +m
αI =
Vn (mi ) , τi + mi
β=
Vn (mi ) , τp
P
j̸=i yj + β−i sp , where
α−i =
Vn−1 (m) , τ +m
β−i =
Vn−1 (m) . τp
Assume firm i participates, so all firms observe a private signal and the released statistic t[n] (mi ). We look for a linear Bayesian equilibrium with two types of strategies: qiJ = cI + dI si + eI t[n] (mi ),
qjJ = cO + dO sj + eO t[n] (mi )
(j ̸= i).
Let t := t[n] (mi ). For firm i, define XI := (si , t)T . By Gaussian projection, for any j ̸= i and for θ, E[sj | si , t] = Σsj XI Σ−1 XI XI XI ,
E[θ | si , t] = ΣθXI Σ−1 XI XI XI .
The required covariance elements follow from sk = θ + εk and (49): Var(si ) = u + τi ,
Var(t) = u − Vn (mi ),
Cov(θ, t) = Var(t),
Cov(si , t) = αI (u + τi ) + αO (n − 1)u + βu =: Cit , Cov(sj , t) = αO (n − 1)u + τ + αI u + βu =: Cot (j ̸= i), Cov(si , sj ) = u,
Cov(θ, si ) = u.
Hence u + τi Cit ΣXI XI = , Cit u − Vn (mi )
ΣθXI = u u − Vn (mi ) ,
Σsj XI = u Cot .
Write the projection coefficients as E[θ | si , t] = κS si + κT t,
E[sj | si , t] = λS si + λT t,
−1 where (κS , κT ) = ΣθXI Σ−1 XI XI and (λS , λT ) = Σsj XI ΣXI XI .
A representative other firm j ̸= i conditions on XO := (sj , t)T , with covariance u+τ Cot ΣXO XO = , Cot u − Vn (mi ) and corresponding projection coefficients O E[θ | sj , t] = κO S sj + κT t,
E[si | sj , t] = ϕS sj + ϕT t, 55
E[sk | sj , t] = χS sj + χT t (k ̸= i, j),
−1 −1 −1 O where (κO S , κT ) = ΣθXO ΣXO XO , (ϕS , ϕT ) = Σsi XO ΣXO XO , and (χS , χT ) = Σsk XO ΣXO XO .
Firm best responses satisfy, for each k, qk =
1 a + E[θ | Ik ] − E[Q−k | Ik ] , 2
Q−k :=
X
qℓ ,
ℓ̸=k
where Ii = σ(si , t) and Ij = σ(sj , t) for j ̸= i. Substituting the linear strategies and the Gaussian projections above and matching coefficients on (1, si , t) and (1, sj , t) yields a linear system in the six unknown coefficients (cI , dI , eI , cO , dO , eO ), which has a unique solution because the relevant covariance matrices are positive definite for u > 0, τ > 0, τi > 0, τp ∈ (0, ∞), and m, mi ≥ 0. In particular, matching constant terms gives cI = cO =
a . n+1
The remaining coefficients are uniquely determined by coefficient matching on the signal and statistic components. Firm i’s ex-ante expected profit under participation is ΠJi (mi ) = E (qiJ )2 = c2I + d2I Var(si ) + e2I Var(t) + 2dI eI Cov(si , t), where Var(si ) = u + τi , Var(t) = u − Vn (mi ), and Cov(si , t) = Cit .
Non-participation outcome (firm i does not participate). Assume firm i does not participate, so it observes only si and uses a linear strategy qiN = c0 + d0 si . All other firms participate and observe (sj , t−i (m)), using qjJ = c1 + d1 sj + e1 t−i (m)
(j ̸= i).
u Let t− := t−i (m). For firm i, E[θ | si ] = u+τ si . Moreover, since t− is Gaussian and linear in i
(θ, {yj }j̸=i , sp ), E[t− | si ] = Using t− = α−i
Cov(t− , si ) si . Var(si )
P
j̸=i yj + β−i sp and Cov(si , yj ) = u for j ̸= i,
Cov(si , t− ) = α−i (n − 1)u + β−i u,
Var(si ) = u + τi ,
so E[t− | si ] =
α−i (n − 1)u + β−i u si . u + τi 56
Substituting these projections into firm i’s best response and matching coefficients on (1, si ) yields linear equations relating (c0 , d0 ) to (c1 , d1 , e1 ). For a representative participating firm j ̸= i, define X := (sj , t− )T . Gaussian projection gives E[si | sj , t− ] = ϕS sj + ϕT t− ,
E[sk | sj , t− ] = χS sj + χT t− (k ̸= i, j),
− − E[θ | sj , t− ] = κ− S sj + κ T t ,
with coefficients computed from the explicit covariance matrix of (sj , t− ). Plugging the linear strategies into firm j’s first-order condition and matching coefficients on (1, sj , t− ) yields a 3 × 3 linear system in (d0 , d1 , e1 ) (and c0 = c1 = a/(n + 1)). Firm i’s ex-ante expected profit under non-participation is N 2 ΠN = c20 + d20 Var(si ) = i = E (qi )
a2 + d20 (u + τi ), (n + 1)2
Define the unilateral participation gain ∆i (mi ) := ΠJi (mi ) − ΠN i . All equilibrium coefficients above are obtained by solving linear systems whose entries are polynomials in the covariance elements of (θ, sp , s1 , . . . , sn , y1 , . . . , yn ). These covariance elements depend (mi ) continuously on mi through Vn (mi ) and αI = Vτni +m , hence ΠJi (mi ) and ∆i (mi ) are continuous in i
mi on [0, ∞). By assumption τp∗ (n, u, τi , τ ) < τp < ∞, the platform signal is not sufficiently informative to induce firm i’s participation when no privacy noise is applied to firm i’s report. Equivalently, given the other firms’ participation, firm i strictly prefers not to participate at mi = 0, so ∆i (0) < 0. Consider mi → ∞. In (49), the precision on firm i’s privatized report satisfies 1/(τi + mi ) → 0, hence αI → 0 and Vn (mi ) → Vn−1 (m). Therefore t[n] (mi ) → t−i (m) in L2 . In the limit, participation by firm i no longer reveals any payoff-relevant information about its private signal to competitors, while it grants firm i access to the non-degenerate statistic t−i (m) constructed from existing participants and sp . Under non-participation, firm i does not observe any platform statistic and remains restricted to si . Since τp < ∞ implies that t−i (m) is informative about θ, access to t−i (m) strictly expands firm i’s information and hence strictly increases its optimized ex-ante profit, implying limmi →∞ ∆i (mi ) > 0. By continuity of ∆i (·) and the inequalities ∆i (0) < 0 and limmi →∞ ∆i (mi ) > 0, the intermediate value theorem implies that there exists m̄i > 0 such that ∆i (m̄i ) = 0. Thus, a finite firm-specific privacy noise level can render firm i (weakly) willing to participate given that all other firms participate. 57