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Cooperate to Compete: Strategic Data Generation and Incentivization Framework for Coopetitive Cross-Silo Federated Learning

Unknown · 2026 · arxiv_cs

arXiv CS · Papers · License: Open Access · 2026

cloud distributedcomputing parallelcomputing

distributed computing, parallel computing, cloud

arXiv:2604.14886v1 [cs.AI] 16 Apr 2026

Thanh Linh Nguyen, Graduate Student Member, IEEE, Nguyen Van Huynh, Member, IEEE, and Quoc-Viet Pham, Senior Member, IEEE

Abstract—In data-sensitive domains such as healthcare and finance, cross-silo federated learning (CFL) allows organizations to collaboratively train AI models without sharing raw data. However, many practical CFL deployments are inherently coopetitive, in which organizations cooperate during model training while competing in downstream markets. In such settings, training contributions, including data volume, quality, and diversity, can improve the global model yet inadvertently strengthen rivals, eroding one’s competitive advantage. This dilemma is amplified by non-independent and non-identically distributed data, which leads to asymmetric learning gains and undermines sustained participation. While existing competition-aware CFL and incentive-design approaches reward organizations based on marginal training contributions, they fail to account for the implicit costs of strengthening competitors. In this paper, we introduce CoCoGen+, a coopetition-compatible data generation and incentivization framework that jointly models statistical data heterogeneity and inter-organizational competition while endogenizing GenAI-based synthetic data generation as a strategic decision in CFL. Specifically, CoCoGen+ formulates each training round as a weighted potential game, where organizations strategically decide how much synthetic data to generate by balancing learning performance gains against computational costs and utility losses caused by competition. We then provide a tractable equilibrium characterization and derive implementable generation strategies to maximize system-wide social welfare. To promote long-term collaboration, we integrate a payoff redistribution-based incentive mechanism to compensate organizations for their training contributions and competition-caused utility degradation. Experiments on increasingly challenging learning tasks using Fashion-MNIST, CIFAR-10, and CIFAR100 validate the feasibility of CoCoGen+. The results show how heterogeneity, competition intensity, and incentives shape organizational strategies and social welfare, while CoCoGen+ outperforms baseline methods in overall efficiency. Index Terms—Coopetition, Cooperation, Competition, Federated Learning, Game Theory, Generative AI.

I. I NTRODUCTION Modern artificial intelligence (AI) services and products increasingly depend on access to large, diverse, and highquality data [2]. However, in highly regulated domains, such as healthcare and finance, relevant data is often scarce and siloed across multiple organizations (e.g., hospitals or banks) Thanh Linh Nguyen and Quoc-Viet Pham (corresponding author) are with the School of Computer Science and Statistics, Trinity College Dublin, Dublin 2, D02 PN40, Ireland (e-mail: {tnguyen3, viet.pham}@tcd.ie). Nguyen Van Huynh is with the School of Computer Science and Informatics, University of Liverpool, Liverpool, L69 3DR, UK (e-mail: [email protected]). Part of this article was presented at IEEE GLOBECOM 2025 [1].

Non-IID data Global model distribution

Competition

Org 1 Cooperation in training global model

Central CFL server Business Market

Org 2

Customers

Org 3

Fig. 1: Coopetitive cross-silo federated learning setting with heterogeneous organizations. Organizations cooperate by exchanging model updates to build a shared global model, while competing in downstream markets for market share and revenue. Each organization holds non-IID local data.

and protected by strict privacy regulations, such as the General Data Protection Regulation (GDPR) [3]. These constraints are more pronounced in the era of generative AI (GenAI) [4], [5], where training robust AI models requires massive and heterogeneous datasets, while centralized data collection is both costly and becoming harder to sustain due to legal and platform restrictions on large-scale data acquisition [6]. In light of these constraints, cross-silo federated learning (CFL) has emerged as a promising paradigm, enabling organizations to collaboratively train a global model while keeping private data local, exchanging only model updates through a central server, preserving data privacy and regulatory compliance [7], [8]. Despite the potential, CFL faces significant challenges in real deployments. In particular, organizations are rational, benefit-driven, and often reluctant to fully contribute their proprietary data, computing, and communication resources, primarily due to the phenomenon of coopetition, wherein organizations act as both collaborators in model training (i.e., cooperation) and competitors in leveraging the shared trained global model in downstream markets (i.e., competition) [9], [10]. This dual role fosters self-interested behavior, as organizations may fear exposing valuable information that may

strengthen their competitors in the business market, alongside concerns about potential privacy leakage and the operational burden of iterative local training and communication. This challenge is compounded by the existence of non-IID data across organizations, which can degrade global model generalization and cause uneven benefits across participating organizations [8]. Under non-IID, some organizations may gain little from cooperation relative to training alone, further increasing dropout risks, thereby weakening the sustainability of the CFL system. For example, Fig. 1 illustrates a scenario involving three geographically distributed private and forprofit healthcare organizations, each possessing heterogeneous datasets (e.g., retinal images from different patient populations and age groups). These organizations seek to cooperatively develop an AI-based automated diabetic retinopathy detection system to address the limitations of their individual datasets, such as insufficient data volume and diversity. However, as they serve overlapping customer bases and try to offer the most accurate and reliable AI solutions, they are also direct competitors. Consequently, concerns over exposing proprietary data, losing competitive advantage, and incurring a significant operational overhead present significant barriers to data sharing and global model development. Therefore, from a systemwide perspective, there is an urgent need to develop solutions that quantify how coopetition and non-IID affect sustainable and efficient cooperation, also known as maximizing social welfare [11], under a competitive nature among organizations in CFL. Social welfare can be defined as the sum of the utilities of all participating organizations [12]. Technically, maximizing social welfare ensures that the collective utility of all organizations is optimized, making training cooperation both effective and sustainable for all organizations involved. Recent studies have shown that GenAI-based data augmentation can mitigate non-IID by generating synthetic data to reduce inter-organizational distribution gaps [13]. In competitive CFL, where data sharing is restricted and distributional mismatch across organizations is pronounced, GenAI can provide a scalable method to close these data distribution gaps without direct data exchange. However, GenAI-based synthetic data generation is computationally intensive and costly, particularly when performed on the organization side [5], [14], [15]. This introduces a consequential strategic decision: "how much synthetic data to generate on the local organization". Organizations need to balance the utility gains from improved learning performance against the costs of cooperative data generation and local training, as well as the competitive losses of enabling competitors to benefit from a more accurate shared global model. Despite its significance, existing CFL literature and incentive mechanisms have not yet provided a unified framework that endogenizes strategic GenAI-based data generation. Moreover, the coupled effects of data heterogeneity and inter-organizational competition on utilities and system-wide welfare remain underexplored. Motivated by these challenges and discussions, we aim to address the following research questions (RQs) in this work: #RQ1: How does the interplay between non-IID and economic inter-competition influence organizational strategic behaviors, incentive mechanism design, and the overall social welfare in

a CFL system? #RQ2: Given coupled effects incurred by non-IID and interorganizational competition, how can we develop a GenAIbased data generation strategy for each organization to maximize system-wide social welfare? #RQ3: How can we design a redistribution-based incentive mechanism that internalizes competitive externalities, promotes efficient data generation, and improves social welfare, while remaining budget-balanced under non-IID CFL? To answer these RQs, we propose a CoopetitionCompatible Data Generation and Incentivization Framework for CFL, termed CoCoGen+. CoCoGen+ jointly models interorganizational competition and non-IID through gains based on learning performance (i.e., training loss) and payoff redistribution incentives. It treats GenAI-based synthetic data generation as a strategic training decision, allowing organizations to optimize their strategies under competition and nonIID data conditions. We show that the resulting per-round interaction can be formulated as a weighted potential game [16], which enables systemic equilibrium characterization of organizations’ synthetic data generation strategies. Building on this formulation, we derive implementable Nash equilibrium solutions via Karush–Kuhn–Tucker (KKT) conditions and a provably convergent fixed-point iteration. Moreover, we incorporate a payoff-redistribution-based incentive mechanism that not only compensates organizations according to their marginal contributions to global model performance, but also redistributes payoffs from lower-contribution organizations to those of higher ones, thereby promoting stable participation under practical constraints (e.g., individual rationality and budget balance). The main contributions of this paper are as follows: ▶ We develop a unified framework for CFL that jointly captures data heterogeneity and inter-organizational competition, which are two tightly coupled factors that determine participation incentives and system-wide social welfare. ▶ We model GenAI-based synthetic data generation as a strategic decision for mitigating non-IID effects, enabling organizations to determine the amount of generated data under competitive, cost, and heterogeneity constraints. ▶ We design a payoff redistribution-based incentive mechanism to mitigate utility losses caused by competition. This addresses the dual challenge of only rewarding marginal training contributions and competitive externalities (i.e., strengthening market rivals), thereby improving fairness and supporting sustainable collaboration. ▶ We formulate the per-training-round strategic interaction as a weighted potential game. This enables equilibrium characterization under coupled organizational utilities. ▶ We provide empirical insights into the joint effects of coopetition and data heterogeneity across different learning task complexities. Particularly, our experiments show that stronger competition and milder data heterogeneity can stimulate synthetic data generation and improve social welfare. They also reveal that the effect of payoff redistribution strength is learning task-dependent and nonmonotonic, implying that effective incentive design must be co-tuned with task complexity and data heterogeneity.

TABLE I: Comparison between prior research & CoCoGen+. Work [9] [17] [18]–[20] [11], [21], [22] [23] [24] [25] [26] [27] [12], [28] CoCoGen+

F1 ✗ ✔ ✗ ✔ ✔ ✗ ✔ ✔ ✗ ✔ ✔

F2 ✗ ✗ ✗ ✗ ✗ ✔ ✔ ✔ ✗ ✗ ✔

F3 ✔ ✔ ✔ ✗ ✗ ✗ ✗ ✗ ✔ ✗ ✔

F4 ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✔

F5 ✔ ✔ ✔ ✔ ✔ ✗ ✗ ✗ ✔ ✗ ✔

F6 ✗ ✔ ✗ ✔ ✗ ✗ ✗ ✔ ✔ ✔ ✔

F7 ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✔ ✔ ✔

F1: Coopetition-aware incentive; F2: Strategic GenAI-based data generation; F3: Coopetition; F4: Joint non-IID and coopetition modeling; F5: Cross-silo settings; F6: Social welfare; F7: Payoff redistribution protocol.

A key distinction between CoCoGen+ and our earlier framework in [1] is the incorporation of a payoff-redistributionbased incentive mechanism, together with a systematic analysis of key factors shaping organizations’ synthetic-data generation strategies and social welfare across different learning task complexities. These additions offer significant benefits, including incentive fairness, enhanced robustness to free riders, enhanced overall social welfare under competition and data heterogeneity, and deeper insights into the feasibility of the framework under real-world learning tasks. Table I summarizes the relevant literature across multiple contribution dimensions, which are further discussed in Section II. II. R ELATED W ORK A. Competition-aware Cross-silo Federated Learning Understanding inter-organizational competition in CFL has recently emerged as a critical research direction. This competitive dynamic significantly influences organizations’ incentives to participate, the resources they contribute, and the value they can exploit from a shared global model, as well as from other organizations. One line of research focuses on market-outcome modeling, where learning performance is linked to downstream economic outcomes (e.g., revenue, customer retention, or market share). For instance, Huang et al. [9] studied a duopoly setting and showed that competition, data contribution, and privacy costs jointly affect global model performance and organizational profits. Specifically, by formulating interactions among organizations and customers as a three-stage Stackelberg game, they showed that high privacy costs coupled with strong competition reduce profits and deteriorate model performance. Extending this direction, Huang et al. [17] analyzed an oligopoly competition model and proposed a model-differentiation mechanism that allows organizations to further personalize the shared global model for downstream services. Their mechanism can improve learning model accuracy, organizational revenues, and social welfare. Similarly, Wu et al. [18] developed a decision support framework among market competitors that models market-share dynamics based

on customer loyalty, switching behaviors, and potential market growth. By introducing the concepts of market stability and friendliness, they established tight lower bounds on necessary performance gains, ensuring that model improvements are distributed such that no organization’s market share decreases beyond a negotiated threshold. A second line of work addresses competition-aware collaboration formation. For example, Tan et al. [19] leveraged graphtheoretic constructs and balance theory to quantify competitive relationships and proposed a collaborator-selection algorithm that avoids conflicts of interest while improving social welfare. Chen et al. [20] further proposed an optimal collaboration formation strategy, specifically designed to address the challenges of self-interest and business competition among organizations. The framework introduces two core principles of ensuring an organization only benefits from the system if they contribute to it, thereby eliminating free-rider concerns, where organizations will benefit from others’ contributions without truthfully contributing to the training process, coalition formation, and preventing any contribution to competitors or their supporters to avoid conflicts of interest. While these studies and others in the literature establish that competition affects the CFL dynamics (e.g., social welfare, revenues, or choosing collaborators), they do not explicitly or theoretically quantify how non-IID drives learning-performance disparities that feed back into economic incentives, nor analyze the combined effects of non-IID and competition on organizational decision-making and social welfare [9], [17]– [20]. Furthermore, these works typically assume voluntary participation, overlooking the practical necessity of compensating organizations for their contributions, such as data, computation, and communication resources. This limitation is particularly critical in CFL, where all organizations receive the same final global model, while simultaneously competing in downstream tasks. In such settings, the lack of contributionaligned incentive designs may discourage organizations from allocating sufficient resources to the training process due to free-rider concerns, especially under competitive pressure [9], [17], [18]. B. Incentive mechanisms in Cross-silo Federated Learning Game-theoretic incentive design has become a prevalent and effective tool for coordinating and sustaining CFL among rational and self-interested organizations. Existing studies mainly focus on mitigating the free-rider problem and aligning private utilities with system-wide objectives (e.g., fairness, social welfare) under practical constraints such as individual rationality and budget balance. For example, Tang et al. [11] proposed a non-cooperative game formulation to derive the organization’s optimal processing capacity decision during training, explicitly targeting free-riding and public-good aspects of CFL, while maximizing the social welfare. They further integrated the incentive mechanism into a lightweight and transaction-efficient blockchainbased system to ensure transparency and auditability, eliminating dependence on a central party for incentive coordination in [21]. From a privacy perspective, Mao et al. [22]

developed a multi-stage game-theoretic incentive mechanism to analyze the behavior of privacy-aware organizations and their impact on CFL model performance, particularly in terms of convergence rate. The mechanism incentivizes organizations to optimally select noise levels for local model updates while enabling the central server to allocate corresponding rewards. It further ensures truthful reporting and aims to achieve social welfare maximization. Liu et al. [23] explored a different methodology for incentive mechanism designs by utilizing a supermodular game for characterizing strategic interactions and deriving data size strategies among organizations. The use of strategic complementarities captures the collaborative yet self-interested nature of CFL, where an organization’s incentive to contribute increases with the contributions of others. This structure ensures the existence and convergence of a pure-strategy Nash equilibrium, supporting a stable and sustained collaboration framework. The equilibrium outcomes are used to design a fairness-aware aggregation weighting mechanism, which addresses performance bias arising from data heterogeneity. However, existing incentive mechanisms largely focus on cooperation-centric CFL settings and do not explicitly model business competition among organizations. As a result, they do not capture the competitive externality that an organization may suffer when its contributions improve competitors’ downstream services, an effect that can materially change resource-allocation decisions (e.g., the amount of GenAIgenerated synthetic data), participation stability, global model performance, and social welfare even when standard incentive constraints are satisfied. C. GenAI-based Federated Learning Non-IID data remains a central challenge in CFL, often causing optimization instability and deviation from the global optimum [8]. It also provides the key link between the two previous strands of work. Particularly, in competitive CFL, organizations need incentives to invest in GenAI-based synthetic data generation as a costly yet effective means of reconstructing underrepresented local distributions, reducing distribution gaps, and improving generalization. For instance, Li et al. [24] proposed FIMI, a resource-aware data generation framework that determines synthetic data amounts jointly with bandwidth and power allocation to meet performance targets while minimizing energy consumption to fill the missing categories in local datasets for cross-device FL. Zhang et al. [25] considered offloading data generation from energy-constrained devices to generative AI service providers. The system models three entities, including the buyer, devices, and providers, who act as sellers. To incentivize participation and ensure efficient resource allocation, a reverse auction–based mechanism is employed to coordinate transactions among entities. This integration of GenAI and economic incentives leads to improved convergence speed and higher model accuracy. From a contract-theoretic viewpoint, Huang et al. [26] proposed a data quality–aware incentive mechanism that jointly considers data volume, data quality (including local and GenAI-generated data), and unit computation cost. The

framework derives optimal contracts to minimize the cloud server’s cost under information asymmetry settings, where only the distributions of client data sizes and computation costs are known. This proposed solution achieves higher accuracy, better social welfare, and lower server costs. However, these GenAI-based approaches are primarily tailored to cross-device settings with resource-constrained clients and lack applicability to cross-silo FL, where organizations are strategic competitors, face different cost structures (e.g., for pricing or generating data), and must strategically decide whether and how much data to generate, while anticipating the spillover, also known as competitive externality [29], to competitors in business markets. Consequently, GenAI-based synthetic data generation is not only a technical consideration but also a strategic and economic one, which remains insufficiently addressed in existing work. III. S YSTEM MODEL AND PROBLEM FORMULATION A. Overview Leveraging the concepts of coopetition [4] and external economies [29] from economics, we define CFL coopetition. Definition 1 (CFL Coopetition). CFL coopetition is a strategic regime in which organizations simultaneously cooperate to train a shared global model and compete in downstream markets for economic returns. Its defining feature is that an organization’s contribution generates positive collective learning gains but also produces negative competitive externalities1 by unintentionally improving rival organizations’ market position. As illustrated in Fig. 2, we consider a practical CFL architecture with multiple coopetitive organizations and a trustworthy central server, which can be either rented or jointly established for coordination. These organizations, whose utilities depend on the model performance improvements across all participants [18], form a consortium for model training. We denote by N =Δ {1, . . . , 𝑛, . . . , 𝑁 } the set of 𝑁 organizations, where each organization 𝑛 ∈ N has a processing capability 𝑓𝑛 , and a dataset consisting of original local data and/or GenAI-generated synthetic data. Organizations collaboratively participate in training a global model over a set of training rounds T =Δ {1, ..., 𝑡, ...,𝑇 } to improve performance on downstream tasks used for their services or products. Let gen D𝑛loc and D𝑛 denote the original local data and synthetic data of organization 𝑛, respectively, with corresponding sizes gen gen 𝑑 𝑛loc =Δ |D𝑛loc | and 𝑑 𝑛 =Δ |D𝑛 |. In traditional CFL settings, organization 𝑛 trains its local model solely on dataset D𝑛loc . In contrast, in our work, each organization 𝑛 adopts a mixed local gen gen dataset of D𝑛mix = D𝑛loc ∪ D𝑛 and 𝑑 𝑛mix =Δ |D𝑛mix | = 𝑑 𝑛loc + 𝑑 𝑛 for its local model update, where we apply the equal-generated gen data allocation strategy to each class. Let 𝒅 gen =Δ {𝑑 𝑛 : ∀𝑛 ∈ N } denote the GenAI-augmented data size strategy profile of gen all organizations in each round 𝑡, and let 𝒅 −𝑛 be the data size strategy profile of all organizations excluding organization 𝑛 gen gen (i.e., 𝒅 −𝑛 =Δ {𝑑 𝑛′ : ∀𝑛′ , 𝑛 ∈ N , 𝑛′ ≠ 𝑛}). 1 When an organization has a sheer volume of local/synthetic data, it disproportionately shifts inter-organizational performance gaps and contribution weights in the global model. Under competition, rivals may benefit from such contributions without compensating the contributing organization.

Payoff

Uplink

Heterogeneous datasets

Downlink

Local models

6 Incentivization 0

Market (business) competition

Global model

Central server

Market shares

we use the power-law function to characterize 𝑛’s local model performance (i.e., training loss) on the training data combination of private data and synthetic data, as expressed as follows: gen

Generated data

𝜖 𝑛 ≈ 𝛼(𝑑 𝑛loc + 𝑑 𝑛 ) −𝛽 − 𝛿,

2 4

Data label

(After)

Data label

....

Org N (Before)

Data label

Amount

(Before)

Amount

(After)

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(Before)

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Org 2

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3 Org 1

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1 Local GenAI-based data generation process

Fig. 2: CoCoGen+ architecture with workflow.

B. GenAI-based Cross-silo Federated Learning Workflow CFL organizations collaboratively train the global model under the coordination of a central server using their private data. Each organization owns a local model architecture with the same dimension as the global model architecture [30]. The per-round CFL training process is as follows (see Fig. 2). 0) Global model initialization & Synthetic data generation strategy profile allocation. We denote 𝒘 𝑡 as a set of global model parameters in the training round 𝑡. Necessary information about the training process, such as the number of global model parameters and the learning network architecture, is public to all organizations within the system. Besides, based on gathered information (e.g., high-level information about data distribution, computational capabilities), which is approved by organizations for learning tasks, the central server can compute data generation strategies (as elaborated in Sec. IV) and send this guidance to organizations. 1) Local GenAI-based data generation. After receiving guidance from the central server, each organization 𝑛 puts efforts into augmenting its local dataset by generating gen 𝑑 𝑛 (e.g., using a variational transposed convolutional neural network [15]), considering their local data, utilities, and competition intensity with other organizations. 2) Global model downloading. In round 𝑡, each organization 𝑛 downloads the global model parameter set 𝒘 𝑡 −1 . 3) Local model updating. Then, each organization 𝑛 selects gen local data size 𝑑 𝑛mix including 𝑑 𝑛loc and 𝑑 𝑛 , and utilizes 𝑡 −1 local compute resources 𝑓𝑛 to train 𝒘 over one local update, performing stochastic gradient descent to obtain a local model update 𝑤 𝑡𝑛 . The local empirical loss of organization 𝑛 in the training round 𝑡 is defined by: 1 ∑︁𝑑𝑛mix 𝐿 𝑡𝑛 (𝒘 𝑡 −1 ) = mix 𝑙 (𝒘 𝑡 −1 ; 𝑥 𝑖 ), (1) 𝑖=1 𝑑𝑛 where 𝑥𝑖 is a data sample in the mixed dataset D𝑛mix and 𝑙 (𝒘 𝑡 −1 ; 𝑥 𝑖 ) is the classification loss function (e.g., crossentropy) of training 𝒘 𝑡 −1 on 𝑥𝑖 . Based on [24], [31]–[33],

(2)

where 𝛼 > 0, 𝛽 > 0, and 𝛿 ≥ 0 are experimentally determined hyperparameters2 (conducted in Sec. V), with 𝛼 representing a task-dependent scaling constant, 𝛽 controlling the steepness of the learning curve, and 𝛿 is a task-dependent offset. This function shows that local model performance, thereby global model performance, depends on both local and synthetic data, underscoring the importance of optimizing the volume of synthetic data to achieve higher learning performance. 4) Local model uploading. The organization 𝑛 uploads its model update 𝑤 𝑡𝑛 to the server. 5) Local model aggregation. Finally, the server aggregates local model updates using an aggregation algorithm (e.g., FedAvg [30]) to get a new global model 𝒘 𝑡 . The global loss function is computed by: ∑︁ 𝑑 mix 𝑛 𝑡 𝑡 (3) 𝐿 𝑡 (𝒘 𝑡 ) = Í 𝑁 mix 𝐿 𝑛 (𝒘 𝑛 ). 𝑑 𝑛∈ N 𝑖=1 𝑖 6) Payoff computation. The central server computes the payoff redistribution (formally defined in Def. 3) based on the uploaded local model performance and the intensity of competition among organizations. It also determines the compensation provided to organizations for their computational costs and data (e.g., through access to the trained global model). 7) Reward distribution. The central server distributes rewards to compensate organizations in proportion to their contributions to global model improvements through local data and computational resources. It also enforces the payoff transfers by collecting payments from lowercontributing organizations and allocating them to highercontributing organizations. C. Computational Cost Let 𝜂 𝑛 and 𝜇 𝑛 be the computation workload required for the organization 𝑛 to train and generate a single data sample, respectively. The energy consumption in one training round of organization 𝑛 can be profiled by [34]: cmp

𝐸𝑛

gen

= 𝜅 𝑛 (𝜂 𝑛 𝑑 𝑛mix + 𝜇 𝑛 𝑑 𝑛 ) 𝑓𝑛2 ,

(4)

where 𝜅 𝑛 is the effective capacitance coefficient of organization 𝑛’s computational chipset. The total cost for computation tasks of 𝑛 in each training round can be represented by: cmp

cmp

𝐶𝑛 = 𝐶𝑛 𝐸 𝑛 ,

(5)

cmp where 𝐶𝑛 represents the computational operating cost per

energy unit. Besides, each organization pays the central server the same fee 𝐶0 for calculating the data generation strategy 2 We can fit the performance curves using two ways such as extrapolation or approximation depending on the learning tasks’ complexity [32].

profile and aggregating the global model. The size of model updates for each organization is fixed [30], and thereby the communication cost is neglected in this work [35], [36]. D. Organization Utility given Coopetition and Heterogeneity This subsection formalizes how each organization’s utility is shaped by the interplay of cooperative learning gains, competitive losses, computational costs, and payoff redistribution. We decompose the utility into four components and define each in turn. a) Cooperation gain: Organization 𝑛 contributes a mixed dataset of size 𝑑 𝑛mix to achieve a high-accuracy global model (i.e., minimizing training loss). Following [11], we quantify the contribution of each organization as a function of the trained global model performance (i.e., training model error). We denote 𝜖 as the global model performance after one training round. The smaller the trained global performance, the better the fit of the global model to the training data. According to [34], [37], we establish the relationship between the global model performance 𝜖 under all organizations’ data contribution strategy profile 𝒅 gen and the local model performance 𝜖 𝑛 (see Eq. (2)), which is profiled as follows: ! 1 Í 𝑛∈ N 𝜖 𝑛 − 1 𝑁 gen , (6) 𝜖 ( 𝒅 ) = exp 𝜚

This loss captures the key coopetitive tension. Particularly, contributing more efforts improves the global model but simultaneously strengthens competitors’ model performance or downstream services, reducing the organization 𝑛’s market positions and advantage. c) Payoff redistribution: To sustain cooperation among competing organizations and guarantee fairness, CoCoGen+ compensates higher-contributing organizations (e.g., higher volumes of synthetic data) at the expense of lower-contributing ones [27], [38]. The shared global model functions as a public good [11], without redistribution, free-riding incentives can erode participation. We therefore introduce a bilateral payoff transfer mechanism. Definition 3 (Payoff Redistribution). Organization 𝑛 receives a payoff transfer from each competing organization 𝑛′ , proportional to its marginal contribution to global model performance improvement and scaled by the competitive intensity between them: gen gen gen gen 𝑝 𝑛,𝑛′ = 𝜉𝛾𝑛,𝑛′ 𝜖 𝑛 (𝑑 𝑛 , 𝒅 −𝑛 ) − 𝜖 𝑛′ (𝑑 𝑛′ , 𝒅 −𝑛′ ) , (10) where 𝜉 ≥ 0 is the payoff compensation rate for each contribution gap unit. Therefore, the total payoff of 𝑛 gained from its competitors 𝑛′ , can be calculated by: gen

gen

𝑃𝑛 (𝑑 𝑛 , 𝒅 −𝑛 ) =

∑︁

(11)

𝑝 𝑛,𝑛′ .

𝑛′ ∈ N

where 𝜚 > 0 is a fine-tuning parameter based on learning tasks. The revenue of organization 𝑛, obtained from its contribution gen 𝑑 𝑛 to improve the global performance, is given by: gen

gen

𝑟 𝑛 (𝑑 𝑛 , 𝒅 −𝑛 ) = 𝜓 𝑛 [𝜖 0 − 𝜖 ( 𝒅 gen )] ,

(7)

where 𝜓 𝑛 ≥ 0 is the organization 𝑛’s revenue per global model’s performance unit or its valuation on global model precision [9], [11], and 𝜖0 denotes the performance of the initial, untrained global model, evaluated on the test set. Intuitively, a higher 𝜓 𝑛 means higher profitability that the organization 𝑛 gains from the global model-related services they can create. b) Competition loss: While cooperation improves the global model, it also creates a negative competitive externality, where rival organizations benefit from each other’s contributions without bearing the associated costs. To quantify this externality, we introduce the notion of competitive intensity. Definition 2 (Competitive Intensity). CFL competitive intensity 𝛾𝑛,𝑛′ ∈ [0, 1], for 𝑛 ≠ 𝑛′ , quantifies the degree of market rivalry between organizations 𝑛 and 𝑛′ . It captures the degree of strategic overlap in their global-model-derived products and the congruence of their target customer segments. Using this metric, the revenue that a competing organization 𝑛′ , extracts from organization 𝑛’s contribution is calculated by: gen gen gen gen 𝑟 𝑛′ = 𝜙 𝑛′ 𝛾𝑛,𝑛′ 𝜖 𝑛 (𝑑 𝑛 , 𝒅 −𝑛 ) − 𝜖 𝑛′ (𝑑 𝑛′ , 𝒅 −𝑛′ ) , (8) where 𝜙 𝑛′ ≥ 0 is the organization 𝑛′ ’s revenue per unit of contribution gap. The total competition loss that organization 𝑛 incurs aggregated over all rival organizations is given by: ∑︁ gen gen 𝑅𝑛 (𝑑 𝑛 , 𝒅 −𝑛 ) = 𝑟 𝑛′ . (9) ′ 𝑛 ∈N

d) Discussion: The redistribution constraints 𝜉 ≤ 𝜙 𝑛′ : The stability of the redistribution mechanism requires that the compensation rate 𝜉 not exceed the per-unit revenue 𝜙 𝑛′ that any organization derives from its rival’s contribution. Remark 1 (𝜉 ≤ 𝜙 𝑛′ ). We impose 𝜉 ≤ 𝜙 𝑛′ 3 to ensure that no organization 𝑛′ is required to compensate a rival 𝑛 at a rate exceeding the benefit it derives from that rival’s efforts in CoCoGen+. This constraint preserves the economic rationality of the cooperation and is critical for participation stability under competition. It also accommodates regulated domains (e.g., healthcare, finance) where cooperation may be mandated despite potential economic losses. Example 1. In the mergers and acquisitions sector, in Facebook’s acquisition of WhatsApp, the $3B restricted stock unit allocation (𝜉) was structured to vest over four years, capping payouts in proportion to the projected synergy revenue of 𝜙 𝑛′ , expected from WhatsApp’s integration. This ensures that the compensation for continued cooperation did not exceed the marginal utility Facebook anticipated gaining, aligning with our CFL assumption 𝜉 ≤ 𝜙 𝑛′ [39]. Example 2. In 5G standard-essential patent licensing. Competing firms such as Ericsson, Nokia, and Qualcomm license their patented innovations, which are essential to the standard’s functionality, to all manufacturers of end-user devices. Within this framework, the aggregate economic benefit generated by the universally adopted standard, monetized through licensing fees collected from all manufacturers, represents 𝜙 𝑛′ . This collected revenue is then distributed to patent-holding 3 We

gen

use 𝜉 and 𝜙𝑛′ to represent 𝑃𝑛 (𝑑𝑛 , 𝒅 −𝑛 ) and 𝑅𝑛 (𝑑𝑛 , 𝒅 −𝑛 ), respectively, for the ease of presentation of the assumption because two equations are affected by these two parameters.

members via royalty payments (𝜉). The financial viability and sustainability of the entire framework are predicated on the core assumption that 𝜉 ≤ 𝜙 𝑛′ , as the distributed royalties are by definition a fraction of the collected licensing fees [40]. Remark 2 (𝜉 > 𝜙 𝑛′ ). When the compensation rate (𝜉) exceeds the benefit (𝜙 𝑛′ ), an organization must pay its competitor more than it gains, violating CoCoGen+’s economic rationale. The dominant response in such a scenario is exit, blocking, or renegotiation. Example 3. In 2014, the Spanish government passed a new copyright law, mandating non-waivable payments (𝜉) from news aggregators (e.g., Google News) to publishers for posting links or excerpts of news articles. However, Google News service’s local benefit (𝜙 𝑛′ ) could not cover the compulsory fees, leading to their shutdown in Spain [41]. E. Problem Formulation Having defined four components of organizational utility, gen gen including cooperation gain 𝑟 𝑛 (𝑑 𝑛 , 𝒅 −𝑛 ), computational cost gen gen 𝐶𝑛 + 𝐶0 , competition loss 𝑅𝑛 (𝑑 𝑛 , 𝒅 −𝑛 ), and payoff redistrigen gen bution 𝑃𝑛 (𝑑 𝑛 , 𝒅 −𝑛 ), we now combine them into a per-round utility function. The utility of organization 𝑛 when choosing gen a synthetic data generation strategy 𝑑 𝑛 , is as follows: gen

gen

𝑈𝑛 (𝑑 𝑛 , 𝒅 −𝑛 ) =𝑟 𝑛 (𝑑 𝑛 , 𝒅 −𝑛 ) + 𝑃𝑛 (𝑑 𝑛 , 𝒅 −𝑛 ) − 𝑅𝑛 (𝑑 𝑛 , 𝒅 −𝑛 ) − 𝐶𝑛 − 𝐶0 .

(12)

Social welfare is defined as the sum of organizational utilities, specifically, ∑︁ gen gen 𝑈𝑛 (𝑑 𝑛 , 𝒅 −𝑛 ). (13) 𝑛∈ N

CoCoGen+ seeks a synthetic data generation strategy profile

that maximizes social welfare while ensuring that individual participation remains rational and the redistribution mechanism is self-financing. These requirements are formalized in the following two properties. Constraint 1 (Individual Rationality-IR). Each organization 𝑛 participates in the CFL consortium only if its utility is nonnegative, specifically, gen

gen

𝑈𝑛 (𝑑 𝑛 , 𝒅 −𝑛 ) ≥ 0, ∀𝑛 ∈ N .

Algorithm 1 Data generation using fixed-point iteration cmp

gen

Require: 𝑑 𝑛loc , 𝛼, 𝛽, 𝛿, 𝜚, 𝜅 𝑛 , 𝐶𝑛 , 𝑓𝑛 , 𝜂 𝑛 , 𝜇 𝑛 , 𝑧 𝑛 , 𝑑min , 𝑑max , tolerance 𝜖 tol , maximum iterations 𝐾max , A1, A2, A3 ∗,gen Ensure: Data generation strategies 𝒅 ∗,gen = {𝑑 𝑛 : ∀𝑛 ∈ N } gen gen 0 1: Initialize 𝑘 = 0, 𝒅 , Calculate 𝐹 ( 𝒅 ), and Set 𝐹 −1 ( 𝒅 gen ) = 𝐹 0 ( 𝒅 gen ) ▷ Fixed-point iteration: 2: while |𝐹 𝑘 ( 𝒅 gen ) − 𝐹 𝑘−1 ( 𝒅 gen )| > 𝜖 tol and 𝑘 ≤ 𝐾max do 3: 𝑘 ← 𝑘 +1 4: for each 𝑛 = 1 to 𝑁 do ▷ Case 1 (Theorem 2): 5: if 𝐴3 𝑁𝛼𝛽𝜚 exp 𝐴1𝜚−1 < −𝐴2 then gen gen 6: 𝑑 𝑛 ← 𝑑min ▷ Case 2 (Theorem 2): 7: else if 𝐴3 𝑁𝛼𝛽𝜚 exp 𝐴1𝜚−1 > −𝐴2 then gen gen 8: 𝑑 𝑛 ← 𝑑max ▷ Case 3 (Theorem 2): 9: else 1 − 𝛽+1 gen 𝑁𝜚 10: 𝑑 𝑛 + 𝑑 𝑛loc ← − 𝐴2𝛼𝛽 exp − 𝐴1𝜚−1 11: end if 12: end for 13: Update 𝒅 gen and Calculate 𝐹 𝑘 ( 𝒅 gen ) 14: end while

▷ Nearest-neighbor rounding: 15: for each 𝑛 = 1 to 𝑁 do gen gen 16: if 𝐹 (ceil(𝑑 𝑛 )) < 𝐹 (floor(𝑑 𝑛 )) then gen gen 17: 𝑑 𝑛 = ceil(𝑑 𝑛 ) 18: else gen gen 19: 𝑑 𝑛 = floor(𝑑 𝑛 ) 20: end if gen gen gen 21: Clip 𝑑 𝑛 to the interval [𝑑min , 𝑑max ] 22: Update 𝒅 gen 23: end for 24: return 𝒅 ∗,gen • Strategies: each organization 𝑛 decides how much data gen 𝑑 𝑛 it should generate in the training round 𝑡. • Objectives: each organization 𝑛 aims to maximize its

utility 𝑈𝑛 expressed in (12). ∗,gen

Definition 4 (Nash Equilibrium of G). A strategy profile 𝑑 𝑛 is a pure strategy Nash Equilibrium (NE) point of the G if and only if no organization can improve its utility by deviating ∗,gen ∗,gen gen ∗,gen unilaterally [42], i.e., 𝑈𝑛 (𝑑 𝑛 , 𝒅 −𝒏 ) ≥ 𝑈𝑛 (𝑑 𝑛 , 𝒅 −𝒏 ).

(14)

Constraint 2 (Budget Balance). The payoff redistribution mechanism is self-financing, requiring no external subsidy [38]. Formally, aggregate payoff transfers sum to zero, i.e., ∑︁ gen gen (15) 𝑃𝑛 (𝑑 𝑛 , 𝒅 −𝑛 ) = 0. 𝑛∈ N

IV. A P OTENTIAL G AME - BASED PROPOSED MECHANISM In this section, we demonstrate that the interaction between coopetitive organizations is a weighted potential game. We then use it to derive GenAI-based data generation strategies. A. Game Formulation Game G: (Stage Game in the training round 𝑡) • Players: competitive, heterogeneous, self-interested, rational, and strategic organizations 𝑛 ∈ N .

B. Nash Equilibrium Analysis Directly finding the NE of G is challenging because deriving the closed-form expression of the fixed point of organizations’ best response mapping is complicated. To analyze G, we show that G is a weighted potential game [43]. Then, we derive the NE of G by solving the minimization problem over the corresponding weighted ordinary potential function. Theorem 1. G is a weighted potential game with the potential function 𝐹 ( 𝒅 gen ), which is calculated by ∑︁ 𝜅 𝑛 𝐶 cmp (𝜂 𝑛 + 𝜇 𝑛 )𝑑 gen 𝑓 2 𝑛 𝑛 𝑛 , (16) 𝐹 ( 𝒅 gen ) = 𝜖 ( 𝒅 gen ) − 𝑧 𝑛 𝑛∈ N Í where 𝑧 𝑛 = 𝑛′ ∈ N 𝛾𝑛,𝑛′ (𝜉 − 𝜙 𝑛′ ) − 𝜓 𝑛 < 0. Proof. The proof can be referred to Appendix A.

□

Since G is a weighted potential game and 𝑧 𝑛 < 0, its NE corresponds to the (global or local) optimal solution to the following minimization problem, i.e., , min gen

𝐹 ( 𝒅 gen )

s.t.

𝑑min ≤ 𝑑 𝑛 ≤ 𝑑max , 𝑑 𝑛 ∈ Z+ ,

(17b)

(14), (15).

(17c)

𝒅

gen

(17a) gen

gen

(17) is a non-convex optimization problem. To overcome this gen non-convexity challenge, we relax the value of 𝑑 𝑛 (in number gen gen of data samples) to the interval [𝑑min , 𝑑max ]. The problem (17) can therefore be recast as follows: min gen

𝐹 ( 𝒅 gen )

s.t.

𝑑min ≤ 𝑑 𝑛 ≤ 𝑑max , 𝑑 𝑛 ∈ R+ ,

(18b)

(14), (15).

(18c)

𝒅

gen

(18a) gen

gen

By solving (18), we show NE of G. ∗,gen Theorem 2. G possesses a NE of 𝒅 ∗,gen = {𝑑 𝑛 : ∀𝑛 ∈ N } under different scenarios, where we define the following variables: ∑︁   ∗,gen Δ 1   𝐴 = (19a) 𝛼(𝑑 𝑛loc + 𝑑 𝑛 ) −𝛽 − 𝛿 1  𝑛∈ N   𝑁 cmp 𝐴2 =Δ 𝜅 𝑛 𝐶𝑛 (𝜂 𝑛 + 𝜇 𝑛 ) 𝑓𝑛2 /𝑧 𝑛 (19b)      𝐴3 =Δ (𝑑 𝑛loc + 𝑑 𝑛∗,gen ) −𝛽−1 (19c)  ∗,gen gen • Scenario 1: 𝑑 𝑛 = 𝑑min iff 𝐴3 𝑁𝛼𝛽𝜚 exp 𝐴1𝜚−1 < −𝐴2 , ∗,gen gen • Scenario 2: 𝑑 𝑛 = 𝑑max iff 𝐴3 𝑁𝛼𝛽𝜚 exp 𝐴1𝜚−1 > −𝐴2 , gen ∗,gen gen < 𝑑max and we have • Scenario 3: 𝑑 min < 𝑑 𝑛 1 − 𝛽+1 𝐴1 − 1 𝐴2 𝑁 𝜚 ∗,gen exp − 𝑑𝑛 = − − 𝑑 𝑛loc , (20) 𝛼𝛽 𝜚

TABLE II: Parameter settings. Param

Value

Param

Value

𝜅𝑛 𝜚 𝑓𝑛 N 𝜇𝑛 gen 𝑑min 𝜂𝑛

10−28 [51] 6 𝑈 (1, 5) GHz 10 3 × 106 0 samples 3 × 106

𝜙𝑛 𝑑𝑛loc 𝜉 𝛾𝑛,𝑛′ 𝛼𝐷 gen 𝑑max 𝜓𝑛

𝑈 (2 × 102 , 4 × 102 ) 𝑈 (10, 3000) samples {10, 50, 90, ..., 250} {𝑈 (0, 0.2) , 𝑈 (0, 1) , 𝑈 (0.8, 1) } {0.1, 0.5, 0.9} 6000 samples 𝑈 (4 × 102 , 6 × 102 )

between training data volume and local learning error, and by examining how competition and data heterogeneity influence synthetic data generation strategies and the resulting social welfare (#RQ1 #RQ2). Second, we analyze how the payoff redistribution-based incentives affect the social welfare and individual utilities, verifying that IR is sustained and that redistribution provides fair compensation to coopetitive organizations (#RQ1, #RQ3). Third, we demonstrate the superior social welfare of CoCoGen+ compared with other baselines across varying competitive intensities, heterogeneity levels, and learning task complexities (#RQ1, #RQ2, and #RQ3). A. Experiment Setup

We consider a CFL system with 𝑁 = 10 organizations, consistent with the typically small number of participants in CFL [8]. To represent different levels of interorganizational competition, we set the competitive intensity to 𝛾¯ = {0.0956, 0.4782, 0.8956}, where each value is computed as the mean of competitive intensity across all organizations with individual intensities drawn from 𝑈 (0, 0.2),𝑈 (0, 1), and 𝑈 (0.8, 1), respectively. These three cases correspond to low, moderate, and high competition. Datasets. Our experiments are conducted on three datasets, including Fashion-MNIST [44], CIFAR-10 [45], and CIFAR𝑁 with the following update rule for {𝑢 𝑛(𝑘 ) } 𝑛=1 using the fixed 100 [45], with different learning difficulty levels being simple, medium, and complex, respectively [46]. To model data hetpoint iteration: erogeneity among organizations, we sample label proportions gen ∀𝑛 ∈ {1, ..., 𝑁 }, 𝑢 𝑛 =Δ 𝑑 𝑛 + 𝑑 𝑛loc , 𝑝 ∼ Dir(𝛼𝐷 ), where 𝛼𝐷 = {0.1, 0.5, 0.9} is the Dirichlet dis1 h i tribution parameter to control the degree of label imbalance. −   𝛽+1 1 Í  𝐴 𝑁𝜚 𝛼(𝑢 𝑛(𝑘 ) ) −𝛽 − 𝛿 − 1 ª A smaller 𝛼𝐷 means a higher heterogeneity level. 𝑛∈ N © 𝑁  2 ® exp − 𝑢 𝑛(𝑘+1) = − , Implementation. We conduct experiments over three ®  𝛼𝛽  𝜚   datasets with Flower framework [47] on 4 NVIDIA L40 « ¬  GPUs with 48GB memory. For Fashion-MNIST and CIFAR∗,gen where 𝑢 𝑛(𝑘 ) converges to (𝑑 𝑛 + 𝑑 𝑛loc ) as 𝑘 → ∞. We can 10, we use MobileNetV3-small [48] and MobileNetV2 [49], ∗,gen then get the value of 𝑑 𝑛 . respectively. For CIFAR-100, we use ResNet-34 [50]. The Proof. The proof, including the fixed point iteration conver- batch size is 16, and the learning rate is 0.01. Unless otherwise gence proof, can be referred to Appendix B. □ stated, key parameter settings are shown in Table II. The procedure for computing approximate solutions 𝒅 ∗,gen Baselines. Existing work offers limited applicable baselines is presented in Algorithm 1. The solution is verified against for coopetitive-aware data generation among organizations in constraints in (14) and (15). The algorithm has a time com- heterogeneous CFL settings. We therefore adopt the following plexity of O (𝑁 (𝑁 + 𝐾)), where 𝐾 is the number of iterations approaches as comparative baselines. until convergence. • Vanilla CFL (VCFL). Traditional CFL without GenAIbased data generation approach and without considering gen V. E XPERIMENTS competition among organizations (i.e., 𝑑 𝑛 = 0, 𝛾𝑛,𝑛′ = ′ ′ This section empirically evaluates CoCoGen+ with three 0, ∀𝑛, 𝑛 ∈ N , 𝑛 ≠ 𝑛). objectives. First, we assess the feasibility of the proposed mod• Without Competition among Organizations (WCO). eling framework and algorithmic design by verifying whether In WCO, there is no competition among organizations gen the adopted scaling law accurately captures the relationship (i.e., 𝑑 𝑛 ≠ 0, 𝛾𝑛,𝑛′ = 0, ∀𝑛, 𝑛′ ∈ N , 𝑛′ ≠ 𝑛).

L o c a l le a r n in g e r r o r (

2 .0

)

2 .4 )

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----------------------------------------

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(a) Fashion-MNIST with MobilenetV3-small.

0 .0

) ) )

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)

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)

S a m p le S a m p le S a m p le

L o c a l le a r n in g e r r o r (

2 .8

0 .8

1 0 0 0 0

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N u m b e r o f tr a in in g d a ta s a m p le s (

(b) CIFAR-10 with MobileNetV2.

5 0 0 0 0 )

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1 0 0 0 0

2 0 0 0 0

3 0 0 0 0

4 0 0 0 0

N u m b e r o f tr a in in g d a ta s a m p le s (

5 0 0 0 0 )

Fig. 3: 𝜖 𝑛 with respect to the number of local training data using different datasets and deep neural network architectures.

• Without Data Generation (WDG). In WDG, GenAI76

6. 0

9. 0

2. 0

5. 0

(a) Fashion-MNIST with MobileNetV3-small. 81

B. CoCoGen+ Feasibility

based data generation mechanisms are not applied, but payoff redistribution-based incentive solution is still used gen (i.e., 𝑑 𝑛 = 0, 𝛾𝑛,𝑛′ ≠ 0, ∀𝑛, 𝑛′ ∈ N , 𝑛′ ≠ 𝑛). • Random Data Generation (RaDG). In this RaDG scheme, organizations randomly generate the amount of GenAI-based augmented data for local training. • Maximum Data Generation (MaDG). In this MaDG scheme, organizations exhaust their local resources to generate the maximum amount of GenAI-based auggen gen mented data for local training (i.e., 𝑑 𝑛 = 𝑑max , ∀𝑛 ∈ N ).

2. 0

3. 0

4. 0

5. 0

(b) CIFAR-10 with MobileNetV2. 79

4. 0

0. 0

a) Scaling laws for CFL: We demonstrate that the scaling law function (cf. Eq. (2)) effectively captures the interplay of data heterogeneity, data quantity, comprising both local and newly generated data, and the local learning performance (i.e., loss) across different datasets and model architectures in CFL. As shown in Fig. 3, for each heterogeneity level 𝛼𝐷 , we empirically obtain a corresponding set of hyperparameters {𝛼, 𝛽, 𝛿} that characterize the local learning curve on three datasets. As the volume of GenAI-generated data increases while the number of data samples per organization remains fixed, the local learning error decreases significantly across different 𝛼𝐷 values. This reduction saturates once the generated data reaches a sufficient volume, at which point the loss stabilizes. Notably, our hyperparameterized scaling law yields learning curves that closely match the empirical trajectories, supporting the validity and feasibility of the proposed model. These results demonstrate that GenAI-based data augmentation not only improves local model performance but also enables predictable and controllable improvements in downstream tasks in federated settings. Consequently, our framework provides a principled basis for deciding when and to what extent to scale synthetic data generation under different heterogeneous settings. b) Impacts of average competition intensity (𝛾) ¯ and data heterogeneity (𝛼𝐷 ) levels: In Fig. 4, we examine the coupled effects of data heterogeneity (𝛼𝐷 ) and average competitive intensity (𝛾) ¯ on 1) the average amount of GenAI-generated data among organizations and 2) the resulting social welfare, across

Fig. 4: Impact of 𝛾¯ and 𝛼𝐷 on the organization’s data generation strategy and the social welfare of CoCoGen+. three datasets with a fixed value4 of 𝜉 = 90. Under highly heterogeneous data conditions (e.g., 𝛼𝐷 = 0.1), increasing 𝛾¯ exacerbates the demand for substantial data augmentation volumes among organizations. This behavior is driven by the proposed payoff redistribution mechanism in Eq. (10), wherein competitive intensity functions as a scaling multiplier 4 Similar trends are observed on other values of 𝜉 .

2 2 5 0

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(a ) F a s h io n -M N IS T w ith M o b ile N e tV 3 -s m a ll.

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(c ) C IF A R -1 0 0 w ith R e s N e t-3 4 .

Fig. 5: Impacts of payoff redistribution rate 𝜉 on data contributions across organizations and social welfare over three datasets with 𝛼𝐷 = 0.1 and 𝛾¯ = 0.8956 in CoCoGen+.

for rewards. As 𝛾¯ intensifies, the marginal utility gained from outperforming competitive organizations increases, effectively incentivizing a tournament dynamic [52] where organizations try to maximize their payoff defined in Eq. (11) through larger contributions. Also, organizations are compelled to scale their synthetic data volumes to offset the heightened competition loss (see Eq. (9)) arising from the strong competitive organizations that have more local data and to satisfy individual rationality constraints defined in Constraint 1. While such data generation increases organizations’ computational burden, the integrated incentive-based payoff redistribution compensates organizations in proportion to their contributions (e.g., larger local datasets and/or higher amounts of synthetic data). This, in turn, raises individual utilities and leads to a net increase in social welfare across all three datasets. On the other hand, under conditions of fixed competitive intensity (e.g., 𝛾¯ = 0.8956) but increasing heterogeneity, organizations are required and motivated to generate more synthetic data through Theorem 2 to close the statistical data distribution gaps due to heterogeneous local data to promote global model convergence. Particularly, harder learning tasks demand larger augmentation budgets. For example, on CIFAR-100, the maximum number of generated data samples reaches 2470, compared to 1220 on Fashion-MNIST and 1360 on CIFAR-10. However, organizations may incur substantial losses from cooperation, including the exposure of proprietary data features that contribute to their business advantage to other competitive organizations. Besides, the costs in training and generating data become significant under the highest heterogeneity level (i.e., 𝛼𝐷 = 0.1). As a result, it can further diminish the incentives for cooperation and also lower overall social welfare. c) Sensitivity to the payoff compensation rate 𝜉: Fig. 5 evaluates the sensitivity of CoCoGen+ to the payoffredistribution coefficient 𝜉 under strong heterogeneity (𝛼𝐷 = 0.1) and high competition intensity (𝛾¯ = 0.8956). Since 𝜉 linearly scales the bilateral transfer 𝑝 𝑛,𝑛′ and thus 𝑃𝑛 (i.e., Eqs. (10)-(11)), it alters each organization’s best response for data generation strategy. Although the redistribution-based incentive mechanism is budget-balanced (see Def. 3 and Constraint 2) and thus cancels in aggregate, it still affects social

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Fig. 7: Efficiency of CoCoGen+ compared to baseline methods with (top) varying 𝛼𝐷 and 𝛾 = 0.4782 and (bottom) varying 𝛾 and 𝛼𝐷 = 0.5 for three datasets FMNIST, CIFAR-10, and CIFAR100 from left to right.

can dominate, reducing social welfare. As 𝜉 increases toward 𝜉 ∗ , redistribution partially internalizes competitive externalities such as losses from exposing valuable data features, curbing excessive generation, and improving net welfare. Beyond 𝜉 ∗ (see Remark 2), 𝑃𝑛 becomes large and can approach or exceed 𝑅𝑛 , so contributing 𝑑 𝑛mix of organization 𝑛 yields limited net benefit to competitors because they need to redistribute huge payoff. Consequently, organizations strategically reduce contributions, degrading global performance and social welfare. Fig. 6 further illustrates the utilities of organizations together with their local and synthetic data volumes across values of payoff compensation rate 𝜉 = {10, 50, 90, 130, 170, 210, 250} for the most challenging learning task, CIFAR-100. All organizations satisfy the IR constraint (cf. Constraint 1). As 𝜉 increases, organizations with limited local data reduce the synthetic data generation volume but gain improved utility because the proposed redistribution mechanism compensates them for the competitive loss associated with strengthening rivals through larger data contribution and model update sharing. In contrast, organizations with larger local datasets, which contribute less synthetic data, experience a decline in utility because they are required to transfer part of payoffs under the redistribution mechanism. C. CoCoGen+ Efficiency Fig. 7 evaluates the effectiveness of CoCoGen+ in terms of social welfare against baseline methods on Fashion-MNIST,

CIFAR-10, and CIFAR-100 under varying levels of competitive intensity 𝛾¯ and heterogeneity level 𝛼𝐷 with 𝜉 = 90. Organizations with less volume of data tend to generate more synthetic data and vice versa. Thanks to the payoff redistribution-based incentive mechanism, they can receive compensation for losses from competition along with the gains from the shared global model and compensation for computational costs. a) Impact of data heterogeneity (𝛼𝐷 ): With a fixed value5 of moderate competitive intensity 𝛾¯ = 0.4782 (see Tab. II and V-A), Figs. 7(a)-(c) show that social welfare improves across all methods as 𝛼𝐷 increases, representing lower heterogeneity on three datasets. Especially, CoCoGen+ outperforms all baselines with the performance gap widening significantly as data distributions become more heterogeneous (smaller 𝛼𝐷 ). This underscores the effectiveness of CoCoGen+ in optimizing the volume of synthetic data generation based on local resources and competition intensity among organizations. Conversely, RaDG and MaDG yield the lowest welfare, indicating that randomized and aggressive data generation, respectively, incur high costs without proportional payoff. Consistent with learning task complexity, welfare is highest for Fashion-MNIST, followed by CIFAR-10, and finally CIFAR100.

5 Similar trends are observed on other values of 𝛾¯ = {0.0956, 0.8956}.

b) Impact of competitive intensity (𝛾): ¯ Under a fixed heterogeneity level (𝛼𝐷 = 0.5) in Figs. 7(d)-(f), CoCoGen+ achieves the highest social welfare as competition level intensifies, because its embedded payoff redistribution mechanism compensates organizations with higher contributions (e.g., those generating more data), thereby mitigating coopetitive losses and potentially reinforcing long-term participation incentives under stronger market rivalry. By comparison, social welfare on VCFL and WCO remain unchanged since they do not account for the competition and largely provide only proportional returns gained from the collective global model. WDG degrades with 𝛾, ¯ suggesting that insufficient or nonadaptive data generation fails to counteract heterogeneityamplified competition effects, while RaDG and MaDG further reduce welfare due to inefficient data generation that increases costs and erodes business advantage under intense competition. D. Discussion and Key Findings The experimental results yield three principal findings that directly address the RQs posed in Section I. a) Coupled effects of data heterogeneity (𝛼𝐷 ) and competition (𝛾) (#RQ1): Non-IID severity and inter-organizational competition intensity levels jointly shape equilibrium strategies and social welfare outcomes. Under stronger competition, organizations are pushed toward a tournament-like behavior, where outperforming rivals yields higher marginal payoffs, thus increasing synthetic-data demand, thereby boosting the social welfare, especially under strong heterogeneity where class imbalance among organizations is severe. However, social welfare remains bounded by the statistical nature of local data, as the heterogeneity level increases (i.e., smaller 𝛼𝐷 ), all methods benefit less from additional data, while costs and competitive losses become more pronounced, lowering welfare. In this regime, CoCoGen+ remains robust because payoff redistribution-based incentive internalizes part of the competitive externality and sustains incentives alignment across 𝛼𝐷 and 𝛾 settings. b) Strategic data generation via potential game formulation (#RQ2): Formulating GenAI-based data generation as a weighted potential game converts the equilibrium design problem into a tractable potential-minimization problem. The resulting pure-strategy Nash equilibrium generation levels, computed via Algorithm 1, are implementable under practical constraints (i.e., IR, budget balance) and consistently outperform all non-strategic baselines across the parameter space. c) Budget-balanced payoff redistribution (#RQ3): The 𝜉scaled redistribution incentive mechanism effectively internalizes competitive externalities by rebalancing payoffs toward higher-contributing organizations (i.e., generating more data). Crucially, however, the optimal redistribution strength is taskdependent, where complex learning tasks benefit from moderate redistribution (e.g., 𝜉 ∗ = 90 for CIFAR-100), whereas excessive redistribution uniformly suppresses contributions and degrades welfare. This non-monotonicity indicates that incentive design in coopetitive CFL must be jointly calibrated with the complexity of the downstream task and the severity of data heterogeneity, rather than specified as a single universal parameter.

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A PPENDIX A. Proof of Theorem 1 Proof. We adopt the forward method [16], where we consider the case where an organization 𝑛 can deviate from strategy gen ′,gen {𝑑 𝑛 } to {𝑑 𝑛 }, while other organizations’ strategy set gen {𝒅 −𝒏 } keeps unchanged. We have the following equality (21), shown at the bottom of the page. According to formulas (16) and (21), we obtain 1 gen gen ′,gen gen 𝐹 ( 𝒅 gen ) − 𝐹 ( 𝒅 ′,gen ) = 𝑈𝑛 (𝑑 𝑛 , 𝒅 −𝒏 ) − 𝑈𝑛 (𝑑 𝑛 , 𝒅 −𝒏 ) . 𝑧𝑛 (22) Eq. (21) shows that any change of the 𝐹 ( 𝒅 gen ) is equal to the 1 𝑧𝑛 -scaling change of the organization utility function caused by any unilateral deviation of the organization, which indicates that the G is a weighted potential game. This completes the proof. □ B. Proof of Theorem 2 We first prove that the optimization problem (18) is a strictly convex optimization problem by verifying that the Hessian matrix of 𝐹 ( 𝒅 gen ) is positive definite (i.e., ▽2 H (𝐹 ( 𝒅 gen )) ≻ 0). Specifically, we present the first and second partial derivatives

of the objective function of problem (17) as Eqs. (23)–(25), shown at the bottom of this page. Let 𝒁 =

𝛼𝛽 𝛼𝛽 , , ..., gen gen 𝑁 𝜚 (𝑑1loc +𝑑1 ) 𝛽+1 𝑁 𝜚 (𝑑2loc +𝑑2 ) 𝛽+1 𝑇

𝛼𝛽 gen 𝛽+1 𝑁 𝜚 (𝑑 loc 𝑁 +𝑑 𝑁 )

, we have the expression of Hessian matrix as Eq. (26), shown at the bottom of the next page.

gen

(28) with respect to 𝑑 𝑛

𝛼𝛽 𝜕L gen = − gen 𝜕𝑑 𝑛 𝑁 𝜚(𝑑 𝑛loc + 𝑑 𝑛 ) 𝛽+1 ! Í gen −𝛽 1 loc −𝛿 −1 𝑛∈ N 𝛼(𝑑 𝑛 + 𝑑 𝑛 ) 𝑁 × exp 𝜚 cmp

− Given any vector 𝜻 = (𝜁1 , 𝜁2 , ..., 𝜁 𝑁 ) 𝑇 and 𝜻 ≠ 0, we have Eq. (27), shown at the bottom of the next page. Therefore, the Hessian matrix of 𝐹 ( 𝒅 gen ) is positive definite, thereby 𝐹 ( 𝒅 gen ) is strictly convex.

𝑛∈ N

(28) where 𝜆 𝑛 ≥ 0, 𝜐 𝑛 ≥ 0, ∀𝑛 ∈ N are the Lagrangian multiplier for the corresponding constraint. The partial derivative of function

gen

′,gen

gen

𝜅 𝑛 𝐶𝑛 (𝜂 𝑛 + 𝜇 𝑛 ) 𝑓𝑛2 − 𝜆𝑛 − 𝜐𝑛 . 𝑧𝑛 ∗gen

The optimal solution 𝑑 𝑛 equation: −

The problem (18) is convex and satisfies Slater’s condition; hence strong duality holds and the KKT conditions are necessary and sufficient for optimality. Moreover, since 𝐹 ( 𝒅 gen ) is strictly convex on the feasible set, the optimal solution is unique. The corresponding Lagrangian of the optimization problem (18) is as follows ! gen −𝛽 1 Í loc −𝛿 −1 𝑛∈ N 𝛼(𝑑 𝑛 + 𝑑 𝑛 ) 𝑁 gen L ( 𝒅 , 𝝀, 𝝊) = exp 𝜚 ∑︁ 𝜅 𝑛 𝐶 cmp (𝜂 𝑛 + 𝜇 𝑛 )𝑑 gen 𝑓 2 𝑛 𝑛 𝑛 − 𝑧 𝑛 𝑛∈ N ∑︁ ∑︁ gen gen gen gen − 𝜆 𝑛 (𝑑 𝑛 − 𝑑min ) − 𝜐 𝑛 (𝑑max − 𝑑 𝑛 ),

is as follows:

(29)

should satisfy the following

𝛼𝛽 ∗gen 𝑁 𝜚(𝑑 𝑛loc + 𝑑 𝑛 ) 𝛽+1 ! ∗,gen −𝛽 1 Í loc ) −𝛿 −1 𝑛∈ N 𝛼(𝑑 𝑛 + 𝑑 𝑛 𝑁

× exp

𝜚 −

cmp 𝜅 𝑛 𝐶𝑛 (𝜂 𝑛 + 𝜇 𝑛 ) 𝑓𝑛2

− 𝜆 𝑛 − 𝜐 𝑛 = 0.

(30)

𝑧𝑛 Moreover, the following conditions can be obtained:  Eq. (30)     gen gen gen gen  𝑑 − 𝑑 ≥ 0, 𝑑 − 𝑑 𝑛 ≥ 0,    𝑛 gen min gen max  KKT conditions 𝜆 𝑛 (𝑑 𝑛 − 𝑑min ) = 0,   gen gen   𝜐 𝑛 (𝑑max − 𝑑 𝑛 ) = 0,    𝜆 𝑛 ≥ 0, 𝜐 𝑛 ≥ 0, ∀𝑛 ∈ N . 

(31a) (31b) (31c) (31d) (31e)

By analyzing the KKT conditions, three possible solutions to the optimization problem (18) can be presented as follows: ∗,gen

If 𝜆 𝑛 > 0, the optimal solution 𝑑 𝑛

gen

= 𝑑min , and based on

gen

𝑈𝑛 (𝑑 𝑛 , 𝒅 −𝒏 ) − 𝑈𝑛 (𝑑 𝑛 , 𝒅 −𝒏 ) ∑︁ gen gen gen gen cmp gen 𝛾𝑛,𝑛′ (𝜉 − 𝜙 𝑛′ ) 𝜖 𝑛 (𝑑 𝑛 , 𝒅 −𝑛 ) − 𝜖 𝑛′ (𝑑 𝑛′ , 𝒅 −𝑛′ ) − 𝜅 𝑛 𝐶𝑛 = 𝜓 𝑛 [𝜖 0 − 𝜖 ( 𝒅 gen )] + (𝜂 𝑛 + 𝜇 𝑛 )𝑑 𝑛 + 𝜂 𝑛 𝑑 𝑛loc 𝑓𝑛2 𝑛′ ∈ N

− 𝜓𝑛 𝜖0 − 𝜖 ( 𝒅

= 𝜓𝑛 𝜖 ( 𝒅

′ ,gen

) −

) − 𝜖 (𝒅

∑︁

′,gen gen gen gen cmp ′gen 𝛾𝑛,𝑛′ (𝜉 − 𝜙 𝑛′ ) 𝜖 𝑛 (𝑑 𝑛 , 𝒅 −𝑛 ) − 𝜖 𝑛′ (𝑑 𝑛′ , 𝒅 −𝑛′ ) + 𝜅 𝑛 𝐶𝑛 (𝜂 𝑛 + 𝜇 𝑛 )𝑑 𝑛 + 𝜂 𝑛 𝑑 𝑛loc 𝑓𝑛2

𝑛′ ∈ N gen

∑︁

) +

gen gen ′,gen gen cmp ′gen gen 𝛾𝑛,𝑛′ (𝜉 − 𝜙 𝑛′ ) 𝜖 𝑛 (𝑑 𝑛 , 𝒅 −𝑛 ) − 𝜖 𝑛 (𝑑 𝑛 , 𝒅 −𝑛 ) + 𝜅 𝑛 𝐶𝑛 (𝜂 𝑛 + 𝜇 𝑛 ) (𝑑 𝑛 − 𝑑 𝑛 ) 𝑓𝑛2

𝑛′ ∈ N

= −𝜓 𝑛 𝜖 ( 𝒅

gen

) − 𝜖 (𝒅

′ ,gen

∑︁ cmp ′,gen gen ) + 𝜖 ( 𝒅 gen ) − 𝜖 ( 𝒅 ′ ,gen ) 𝛾𝑛,𝑛′ (𝜉 − 𝜙 𝑛′ ) + 𝜅 𝑛 𝐶𝑛 (𝜂 𝑛 + 𝜇 𝑛 ) (𝑑 𝑛 − 𝑑 𝑛 ) 𝑓𝑛2

′

= 𝜖 ( 𝒅 gen ) − 𝜖 ( 𝒅 ′ ,gen )

∑︁

𝑛 ∈ N cmp ′,gen gen 𝛾𝑛,𝑛′ (𝜉 − 𝜙 𝑛′ ) − 𝜓 𝑛 +𝜅 𝑛 𝐶𝑛 (𝜂 𝑛 + 𝜇 𝑛 ) (𝑑 𝑛 − 𝑑 𝑛 ) 𝑓𝑛2 .

(21)

𝑛′ ∈ N

}

𝑧𝑛

! gen cmp 𝛼(𝑑 𝑛loc + 𝑑 𝑛 ) −𝛽 − 𝛿 − 1 𝜅 𝑛 𝐶𝑛 (𝜂 𝑛 + 𝜇 𝑛 ) 𝑓𝑛2 − , 𝜚 𝑧𝑛 ! ! gen −𝛽 1 Í loc −𝛿 −1 𝜕 2 𝐹 ( 𝒅 gen ) 𝛼2 𝛽2 𝛼𝛽(𝛽 + 1) 𝑛∈ N 𝛼(𝑑 𝑛 + 𝑑 𝑛 ) 𝑁 + exp , gen gen = gen gen 𝜚 𝜕𝑑 𝑛 𝜕𝑑 𝑛 𝑁 2 𝜚2 (𝑑 𝑛loc + 𝑑 𝑛 ) 2(𝛽+1) 𝑁 𝜚(𝑑 𝑛loc + 𝑑 𝑛 ) 𝛽+2 ! gen −𝛽 1 Í loc −𝛿 −1 𝜕 2 𝐹 ( 𝒅 gen ) 𝛼2 𝛽2 𝑛∈ N 𝛼(𝑑 𝑛 + 𝑑 𝑛 ) 𝑁 exp . gen gen = gen gen 𝜚 𝜕𝑑 𝑛 𝜕𝑑 𝑛′ 𝑁 2 𝜚2 (𝑑 𝑛loc + 𝑑 𝑛 ) 𝛽+1 (𝑑 𝑛loc′ + 𝑑 𝑛′ ) 𝛽+1 𝜕𝐹 ( 𝒅 gen ) 𝛼𝛽 =− exp gen gen loc 𝜕𝑑 𝑛 𝑁 𝜚(𝑑 𝑛 + 𝑑 𝑛 ) 𝛽+1

1 Í 𝑛∈ N 𝑁

(23) (24) (25)

(31d), it means 𝜐 𝑛 = 0. Therefore, from (30), the following condition can be obtained: 𝛼𝛽 𝜆𝑛 = − ∗,gen loc 𝑁 𝜚(𝑑 𝑛 + 𝑑 𝑛 ) 𝛽+1 ! Í ∗,gen −𝛽 1 loc ) −𝛿 −1 𝑛∈ N 𝛼(𝑑 𝑛 + 𝑑 𝑛 𝑁 × exp 𝜚 cmp

−

𝜅 𝑛 𝐶𝑛 (𝜂 𝑛 + 𝜇 𝑛 ) 𝑓𝑛2 > 0. 𝑧𝑛

(32)

∗,gen gen If 𝜐 𝑛 > 0, the optimal solution 𝑑 𝑛 = 𝑑max , and based on

(31c), it means 𝜆 𝑛 = 0. Therefore, from (30), the following condition can be obtained: 𝛼𝛽 𝜐𝑛 = ∗,gen loc 𝑁 𝜚(𝑑 𝑛 + 𝑑 𝑛 ) 𝛽+1 ! Í ∗,gen −𝛽 1 loc ) −𝛿 −1 𝑛∈ N 𝛼(𝑑 𝑛 + 𝑑 𝑛 𝑁 × exp 𝜚 𝜅 𝑛 𝐶𝑛 (𝜂 𝑛 + 𝜇 𝑛 ) 𝑓𝑛2 > 0. 𝑧𝑛

(33) ∗,gen

gen

If 𝜆 𝑛 = 0 and 𝜐 𝑛 = 0, the feasible solution 𝑑min < 𝑑 𝑛 < 𝑑max ∗,gen based on (31c) and (31d). And, 𝑑 𝑛 should satisfy −

𝛼𝛽 ∗,gen 𝑁 𝜚(𝑑 𝑛loc + 𝑑 𝑛 ) 𝛽+1 ! ∗,gen −𝛽 1 Í loc ) −𝛿 −1 𝑛∈ N 𝛼(𝑑 𝑛 + 𝑑 𝑛 𝑁

× exp −

𝜚 −

cmp 𝜅 𝑛 𝐶𝑛 (𝜂 𝑛 + 𝜇 𝑛 ) 𝑓𝑛2

For the ease of presentation, we define the following variables:  1 ∑︁  gen   𝛼(𝑑 𝑛loc + 𝑑 𝑛 ) −𝛽 − 𝛿 , 𝐴1 =  𝑛∈ N   𝑁 cmp 𝐴2 = 𝜅 𝑛 𝐶𝑛 (𝜂 𝑛 + 𝜇 𝑛 ) 𝑓𝑛2 /𝑧 𝑛 ,      𝐴3 = (𝑑 𝑛loc + 𝑑 𝑛gen ) −𝛽−1 . 

(37a) (37b) (37c)

∗,gen

Because Eq. (36) of 𝑑 𝑛 is transcendental, we propose an 𝑁 update rule for {𝑢 𝑛(𝑘 ) } 𝑛=1 using FPI as follows: gen

cmp

Based on Eq. (35), we have the following equation: " cmp 𝜅 𝑛 𝐶𝑛 (𝜂 𝑛 + 𝜇 𝑛 ) 𝑓𝑛2 𝑁 𝜚 gen loc 𝑑𝑛 + 𝑑𝑛 = − 𝛼𝛽𝑧 𝑛 1 ! # − 𝛽+1 gen −𝛽 1 Í loc −𝛿 −1 𝑛∈ N 𝛼(𝑑 𝑛 + 𝑑 𝑛 ) 𝑁 . × exp 𝜚 (36)

(34)

𝑧𝑛 Then, we obtain the following necessary and sufficient condi∗,gen tion for 𝑑 𝑛 : " cmp 𝜅 𝑛 𝐶𝑛 (𝜂 𝑛 + 𝜇 𝑛 ) 𝑓𝑛2 𝑁 𝜚 ∗,gen 𝑑𝑛 = − 𝛼𝛽𝑧 𝑛 1 ! # − 𝛽+1 Í ∗,gen −𝛽 1 loc ) −𝛿 −1 𝑛∈ N 𝛼(𝑑 𝑛 + 𝑑 𝑛 𝑁 × exp − 𝜚 − 𝑑 𝑛loc .

∀𝑛 ∈ {1, ..., 𝑁 }, 𝑢 𝑛 =Δ 𝑑 𝑛 + 𝑑 𝑛loc , 1 h i   − 𝛽+1 (𝑘 ) −𝛽 1 Í  𝐴 𝑁𝜚  𝛼(𝑢 ) − 𝛿 − 1 𝑛 𝑛∈ N © ª 𝑁  2 ® 𝑢 𝑛(𝑘+1) = − exp − ®  𝛼𝛽 𝜚   « ¬  (𝑘 ) = M(𝑢 𝑛 ).

Based on the definition of the standard interference function in [53], we have the following proposition to guarantee the convergence of our proposed fixed point iteration. 𝑁 Proposition 1. {M(𝑢 𝑛 )} 𝑛=1 converges to an unique fixed point ∗ 𝑢𝑛. 𝑁 Proof. We prove that {M(𝑢 𝑛 )} 𝑛=1 satisfies the following properties: • Positivity: M(𝑢 𝑛 ) > 0, • Strictly decreasing: if 𝑢 𝑛 > 𝑢 𝑛′ then M(𝑢 𝑛 ) < M(𝑢 𝑛′ ) 6 , • Scalability: ∀𝜑 > 1, 𝜑M(𝑢 𝑛 ) > M(𝜑𝑢 𝑛 ). gen

With 𝑢 𝑛 = 𝑑 𝑛 + 𝑑 𝑛loc > 0, we have M(𝑢 𝑛 ) > 0, so positivity holds.

(35) 6 The more total local data 𝑢 yields better global model performance (i.e., 𝑛 lower loss), so M(𝑢𝑛 ) must be strictly decreasing.

▽ H (𝐹 ( 𝒅

gen

! gen 𝛼(𝑑 𝑛loc + 𝑑 𝑛 ) −𝛽 − 𝛿 − 1 )) = exp 𝒁𝒁𝑇 𝜚 ! ! gen −𝛽 1 Í loc −𝛿 −1 𝑛∈ N 𝛼(𝑑 𝑛 + 𝑑 𝑛 ) gen 𝛽+2 𝑁 2 2 loc + 𝑑𝑖𝑎𝑔 𝑁 𝜚 (𝑑 𝑛 + 𝑑 𝑛 ) exp . 𝜚 1 Í 𝑛∈ N 𝑁

! !2 loc + 𝑑 gen ) −𝛽 − 𝛿 − 1 ∑︁ 𝛼(𝑑 𝛼𝛽𝜁 𝑛 𝑛 𝑛 𝜻 𝑇 H (𝐹 ( 𝒅 gen ))𝜻 = exp gen 𝜚 𝑁 𝜚(𝑑 𝑛loc + 𝑑 𝑛 ) 𝛽+1 𝑛∈ N ! gen −𝛽 1 Í loc − 𝛿 − 1 𝛼𝛽(𝛽 + 1) ∑︁ 𝜁 𝑛2 𝑛∈ N 𝛼(𝑑 𝑛 + 𝑑 𝑛 ) 𝑁 + exp > 0. gen 𝜚 𝑁𝜚 (𝑑 𝑛loc + 𝑑 𝑛 ) 𝛽+2 𝑛∈ N 1 Í 𝑛∈ N 𝑁

(26)

(27)

Regarding monotonicity and scalability, we take the first derivative of the M(𝑢 𝑛 ): " 𝜕M(𝑢 𝑛 ) 𝐴2 𝐴3 𝐴2 𝑁 𝜚 − = 𝜕𝑢 𝑛 𝛽+1 𝛼𝛽 h i Í # 1 −𝛽 𝑁1 𝑛∈ N 𝛼(𝑢 𝑛 − 𝛿 − 1 − 𝛽+1 −1 × exp − 𝜚 𝐴1 − 1 < 0, (38) × exp − 𝜚 showing that M(𝑢 𝑛 ) is a strictly decreasing function. Therefore, we have 1) if 𝑢 𝑛 > 𝑢 𝑛′ then M(𝑢 𝑛 ) < M(𝑢 𝑛′ ), and 2) ∀𝜑 > 1, 𝜑M(𝑢 𝑛 ) > M(𝑢 𝑛 ) > M(𝜑𝑢 𝑛 ). This completes the proof. □

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