Hierarchical Incentive Mechanism for Federated Learning: A Single Contract to Dual Contract Approach for Smart Industries

Federated learning (FL) has shown promise in smart industries as a means of training machine-learning models while preserving privacy. However, it contradicts FL’s low communication latency requirement to rely on the cloud to transmit information with data owners in model training tasks. Furthermore, data owners may not be willing to contribute their resources for free. To address this, we propose a single contract to dual contract approach to incentivize both model owners and workers to participate in FL-based machine learning tasks. Te single-contract incentivizes model owners to contribute their model parameters, and the dual contract in-centivizes workers to use their latest data to participate in the training task. Te latest data draw out the trade-of between data quantity and data update frequency. Performance evaluation shows that our dual contract satisfes diferent preferences for data quantity and update frequency, and validates that the proposed incentive mechanism is incentive compatible and fexible.


Introduction
In recent years, the rise of the Industrial Internet of Tings (IIoT) [1] and relevant intelligent technologies such as deep learning (DL) [2] algorithms have ushered in innovative changes and development opportunities for smart industries.Consequently, the deployment of the IIoT is becoming increasingly popular for various applications, e.g., smart grid [3], logistics [4], and healthcare [5].
IIoT integrates various technologies to digitally transform manufacturing and service operations.In 2020, Liu et al. [6] proposed a novel tracker based on response region reliability with edge computing, which achieved accurate and fast tracking with high reliability in IIoT applications.Ten, considering security concerns while bringing computing power to the edge of industrial systems, Houda et al. [7] designed a novel MEC-based framework to secure IIoT applications using FL.Feng et al. [8] proposed a trustworthy self-healing scheme based on blockchain and digital twin and implemented trustworthy self-healing in the edge AIenabled IIoT environment.Recently, IIoT has become a core component of smart applications, which can capture various valuable events and objects.Rahman et al. [9] proposed an AI-enabled IIoT to automate event management in smart cities. Farahani and Monsef [10] combined multiparty technologies, privacy-enhancing techniques, and AI to further facilitate the industrial data economy and innovation process.
DL algorithms require substantial amounts of data to outperform traditional methods in model training.Furthermore, restricted by regulations such as the General Data Protection Regulation (GDPR) [11], they are also reluctant to share data.Federated learning (FL) [12] as an emerging technology came into being in IIoT, where diferent model owners collaborate by sharing their gradients instead of raw data, thereby preserving privacy.FL is a technique that can train statistical models using data distributed over remote devices or siloed data centers while keeping data localized and preserving privacy.By leveraging the diversity and heterogeneity of data from diferent parties, FL can train a model that has more generalization ability and adaptability, thus improving the model's performance and reliability.Additionally, the hierarchical computing architecture [13] has been proposed to enhance data transmission efciency between the cloud and the terminal.In the hierarchical computing architecture, edge devices transmit data to the cloud through the nearest edge node, thereby reducing the number of communications for global aggregation and updates between terminal devices and remote cloud servers, energy consumption during communication, and latency.
Traditionally, FL research usually involves the model owner issuing model training tasks to workers and motivating them to participate through a single contract, i.e., the user contribution in the contract is related to only one independent variable.Workers only collect information when requested, and their contributions are only relevant to the data quantity.However, in many real-time tasks, service latency may be unbearable, requiring frequent data caching and updates to maintain data freshness.Tis trade-of between data quantity and data update frequency is underexplored.For example, disaster prediction in the mining industry relies on up-to-date information to prevent safety hazards [14].
In addition, when a model owner manages the trade-of between data quantity and data update frequency, it can create an incentive mismatch with the worker.For example, the model owner may prefer a large data quantity to ensure optimal model performance, but the collection expense such as energy consumption incurred in data collection and data storage in caching may be prohibitive for the worker.So, a uniform reward allocation may not result in optimal utility for both parties, as the model owner may lack information about the worker's unit cost of data collection.
Terefore, we propose an FL-based hierarchical incentive mechanism that utilizes a single contract to dual contract task-aware model.Our mechanism utilizes a single contract to incentivize model owners to collect gradients and a dual contract to motivate workers to perform both data collection and updates.We design the dual contract to fexibly adjust the diferent preferences of requesters regarding data quantity and update frequency.When data quantity is more important, the contract can be designed to motivate workers to collect more data.When data update frequency is more important, the contract can be designed to encourage more frequent updates.Trough the selfrevealing mechanism of contract theory, workers and model owners can maximize their utility by choosing the contract that best suits their needs.Te contributions of this paper can be summarized as follows: (i) We propose a new approach for federated learning that addresses the challenges of data quality in the context of edge computing.Specifcally, we use a single contract to dual contract structure that enables a three-way collaboration between model owners, workers, and requesters.Tis approach assesses data quality through two diferent aspects: data quantity and data update frequency.(ii) We utilize the self-revealing and weighting characteristics of the dual contract-theoretic incentive mechanism design to incentive worker.Specifcally, the model owners can adjust the proportion of data quantity and data update frequency in their dual contracts according to their preferences of singlecontract requesters.Te self-revealing mechanism of the contract allows workers to maximize their benefts when choosing a contract that suits them.So the approach can justly compensate workers for the costs associated with data collection and updating.(iii) We show that model owners can calibrate to suit the varying preferences for data quantity and data update frequency in the dual contract according to their willingness to participate in the single contract.Tis fexibility in calibration enables model owners to strike a balance between maximizing the quality of the data set and minimizing their own resource usage and workload, ultimately promoting a more efective and efcient federated learning process.
Te paper is structured as follows.Section 2 discusses the related works, Section 3 presents the system model and problem formulation, Section 4 formulates the contract design, Section 5 discusses the performance evaluation, and Section 6 concludes the paper.

Related Work
Te percentage of studies involving Industrial Internet of Tings (IIoT) applications has signifcantly increased in recent years, including in the areas of smart grids [15] and supply chains [16].Te former investigates the use of IIoT to improve task resilience in the event of accidents, while the latter explores the potential of IIoT to enhance task efciency.In particular, the study presented in [17] proposes a cloud-based solution for detecting patients in their homes.
Te above studies usually assume that machine learning models are trained in an ideal environment, which is not always the case.Tus, it is necessary to investigate decentralized learning schemes, such as federated learning (FL), for industrial applications.FL was designed to enable efcient machine learning among multiple computing nodes while ensuring data owner privacy and security.In the context of FL, a privacy-preserving Byzantine-robust federated learning (PBFL) scheme based on blockchain was proposed in [18], which achieves convergence and provides privacy protection on diferent datasets.Additionally, a compressed and privacy-preserving FL scheme in deep neural network (DNN) architecture was proposed in [19] to address the curse of dimensionality in FL.
FL has developed successful applications in various felds, including autonomous driving car [20] and smart home [21].In particular, the study in [22] enabled the FLbased diferential privacy algorithm to enhance the privacy level of the feature in smart healthcare.Infuenced by transmission efciency, the implementation of the data transmission architecture has changed from the initial central server to the edge server, which provides edge computing [23] services locally to meet the demands of real time, security, and privacy preservation.As such, the study in [24] proposes a hierarchical edge-cloud framework to 2 International Journal of Intelligent Systems reduce the system resource requirements and data transmission time, to satisfy efcient storage and rapid response.However, existing works have primarily focused on user privacy preservation or efcient problem-solving, rather than incentive mechanism design.With the popularization of FL tasks, designing an incentive mechanism to encourage data owners to participate in FL has become increasingly important.He et al. [25] propose a game theory-based incentive mechanism for collaborative security of FL in an energy blockchain environment, which can discourage nodes from taking malicious behaviors in iterative training of FL.Chen et al. [26] devise a multifactor reward function based on reputation, model accuracy, and reward rate, which ensures that data owners with a high reputation and high model accuracy will receive more rewards.Te study in [27] selected users with bids and contributions through a reverse auction mechanism.Reputation was integrated into incentives in studies such as [28].Te study in [29] used deep reinforcement learning to design a learning-based incentive mechanism.
Currently, the contract theory approach of FL has been well explored in the literature.Xu et al. [30] present a contract-based dynamically federated learning optimized personal deep learning scheme, which enabled edge devices to reach a consensus on the optimal weights of personalized models.Also, the study in [31] designed a multidimensional contract approach.Te study in [32] jointly considered data quality and computation efort.Privacy concerns have led to the emergence of contracts that incorporate privacy cost considerations, such as in the study presented in [33].Te study in [34] designed a two-period incentive mechanism that allowed for extension to multi-periods.Li et al. [35] develop a novel incentive-based federated learning framework, where a contract-based reputation mechanism and a Stackelberg-based interclient incentive mechanism are incorporated.Chen et al. [36] propose a contract-based edge-assisted federated learning model-sharing incentive mechanism, which maximize the EFL model consumers' proft and ensure the quality of training services.Feng et al. [37] incorporate local diferential privacy into contract theory-based private data trading to support personalized privacy preferences.However, most of these incentive mechanisms consider worker contribution based on the data quantity without focusing on data updates in FL.
According to the work of [38], the age of information (AoI) has been considered in FL tasks, and a hierarchical incentive mechanism framework has been proposed in [39] to improve the efciency between cloud and terminal.In this paper, we propose an FL-based single contract to dual contract hierarchical incentive mechanism that introduces a dual contract to solve the worker data multifaceted issue contribution problem.We model data quantity along with data update frequency to design incentives that take into account worker efort and appropriate proft.By adding the hierarchical framework, we aim to improve the efciency of the federated learning task.
Table 1 provides a comparison of the prominent features of this paper's incentive mechanism with other studies discussed in this paper.

System Model and Problem Formulation
3.1.System Model.Our system model consists of requester, model owner, and worker.Te requester publishes its model training task requirements to the model owner through the corresponding platform and associates with them by the contract.Te model owner completes the tasks by collecting relevant gradients from the worker who has signed contracts with them, and workers train the model on data in response to tasks.Te specifc details are given in Figure 1.
Our federated learning model is based on a three-tiered architecture of terminal-to-edge-to-cloud. When uploading parameters at the terminal, they are not directly transmitted to the cloud server.Instead, the parameters are frst subjected to edge aggregation on the edge server located at the network edge and close to the terminal device.Among these, the edge nodes in the blockchain will select the most reputable ones as the consensus committee to check the gradient parameters uploaded by the edge devices, and a leader is responsible for collecting qualifed gradient parameters.Ten, the edge-aggregated parameters are uploaded to the cloud server for global model aggregation and updating.When uploading global parameters to the cloud, they are not directly transmitted to the terminal.Instead, the global parameters are frst subjected to edge on the edge server located at the network edge and close to the terminal device.Subsequently, the global parameters are transmitted from the edge server to the terminal.Unlike the traditional terminal-to-cloud architecture, our architecture performs edge aggregation at the edge layer, reducing unnecessary updates and communications, thereby reducing the number of communications for global aggregation and updates between terminals and remote cloud servers, energy consumption during communication and latency.Tis improves the computational and communication efciency of the federated learning model.
We assume that the requester initiates a task that involves a set F � 1, . . ., f, . . ., F   of F model owners.As tasks are initiated, each model owner initiates a task that involves a set G � 1, . . ., g, . . ., G   of G workers in a synchronous task that spans a fxed duration T, with multiple instances of model training requests.In this scenario, the tasks issued by the model owner follow the Poisson process [38].Unlike in conventional FL studies where workers collect data after request arrival, we now consider the International Journal of Intelligent Systems possibility of data caching due to service delays.Workers periodically update the cached data and proceed to train the model on the gateway.In a privacy-preserving FL method adopting diferential privacy encryption, only the model parameters are sent to their corresponding model owners for aggregation.Te model owners generate a consensus committee and leader based on their reputation with the requester and proceed to verify the legal gradient and perform on-chain operations.Finally, the leader submits the aggregation gradient to the requester.

Problem Formulation.
In the FL network, the single contract of the incentive mechanism is comprised of , which represents M types of willingness to participate for model owners.Each model owner type φ m can be characterized by a probability mass function P(φ m ), where the types are indexed in nondecreasing order such that 0 , where the quality of gradient types is indexed in nondecreasing order such that 0 Te model owner type refects each model owner's level of willingness to participate and determines the quality of the collected gradient, whereby model owners who are more willing to participate collect higher-quality data.We defne the utility function of the requester for a model owner with type m as follows: where R m represents the reward paid by the requester to the model owner and σ 1 is the conversion parameter from gradient performance to profts.Te function represents the , where the data quantity types are indexed in a decreasing order Immediately after, there has of M data update cost type belonging to the unit data collection cost type c i j , where the data update cost types are indexed in a nondecreasing order 0 . Similarly, the data update frequency is denoted as , where the data update frequency types are indexed in a decreasing order η 1 where z T refers to the energy consumed for unit data transmission from the IIoT network to the gateway and z S refers to the energy consumed for unit data caching [40].
Te type m worker utility of data quantity denoted by ρ m can be expressed as follows: where q m and r m represent the amounts of data used to train the model and reward from the i model owner of worker m, respectively.Moreover, σ 2 and 9 represent the conversion parameter from data quantity performance to profts.Te diminishing returns of data quantity are represented by a concave function.
Te type m worker utility of data update frequency denoted by χ m can be expressed as follows: where s a and s b refer to the weighted preference for the data freshness and service delays, respectively.In our hierarchical incentive mechanism design, we take into account the contract formulated by the requester to obtain an FL-based model from the model owner.In order to gather gradients for the relevant data quality, the model owner has the fexibility to adjust z a and z b to accommodate varying preferences for data quantity and update frequency, as well as s a and s b to cater to varying preferences for data freshness and service delays.Tese adjustments are crucial to incentivize workers to participate in the task.

Contract-Theoretic Incentive Mechanism Design
In this section, the requester incentivizes the model owner to collect models with a single contract that takes into account data quality, while the model owner incentivizes workers to train the model with a dual contract that considers both data quantity and freshness.We begin by studying the single contract of a representative model owner and then studying the dual contract of a representative worker.Ten, we discuss the conditions for contract feasibility and relax the constraints to derive the optimal contract.

Single Contract.
For ease of notation, we study one of the model owners as a representative for now.Te model owner of type m utility maximize problem is denoted by U m as follows: max where V m refers to the quality of the gradient collected, R m refers to the contract rewards from the requester, and C represents the cost incurred per unit quality of the gradient collected.In the formulation of contract theory, each model owner has a bundle R m , V m   that maximizes its utility U m .From (1), the requester utility maximization function denoted by c 1 can be expressed as follows: where P(φ m ) refers to the proportion of model owner type m,  M m�1 P(φ m ) � 1, N refers to the number of model owners, and R m refers to the reward to each model owner of type m for its gradient collection eforts.For ease of reference, we refer the readers to Table 2 for commonly used notations.

International Journal of Intelligent Systems
To ensure the feasibility of the contract, if and only if satisfy the following constraints.
Defnition 1 (individual rationality (IR)).Each model owner participates in the task when they can get the positive utility, i.e., Defnition 2 (incentive compatibility (IC)).Te utility of each model owner can be maximized if and only if they choose the contract design for its type, i.e., To guarantee a feasible contract, we have to deal with M IR constraint and M(M − 1) IC constraint to reduce and relax conditions.

Lemma 3. For any feasible contract, we have
Proof.Using the IC constraint in Defnition 2, we frst prove if As such, we have Ten, we add these two inequalities By swapping left and right, we can obtain When φ m − φ z ≥ 0, it follows R m ≥ R z .Likewise, from the IC constraints, we have When R m ≥ R z ≥ 0, it follows φ m ≥ φ z .Lemma 3 is proven.

□
Lemma 3 implies that model owners with a higher willingness to participate φ to collect the higher quality of gradient V, and the more rewards R will be obtained.As such, the contract bundles are designed such that higher gradient quality contributed translate to higher rewards.As such, a feasible contract establishment must have the necessary condition for the following monotonicity conditions.

Theorem 4 (monotonicity). A feasible contract must satisfy the following conditions:
Next, we further relax the IR and IC constraints.Intuitively, the maximum utility model owner incurs the highest quality of data, i.e., the type M model owner.

Lemma 5 (reduce single-contract IR constraints). If the IR constraints of model owner type 1 are satisfed, the other IR constraints will also remain the same.
Proof.According to the IC constraints and conditions As such, if the IR constraint of model owner type 1 is satisfed, the type m, m ∈ 1, . . ., M { }, IR constraints are automatically satisfed.

□ Lemma 6 (reduce single-contract IC constraints). Te constraints can be reduced to local down incentive constraints (LDIC).
Proof.Consider three model owner types, where φ m−1 ≤ φ m ≤ φ m+1 .Te two LDICs, i.e., constraints between type m and type m − 1 model owners, are as follows: It can be obtained from Lemma 3 that when R m ≥ R z , it follows φ m ≥ φ z , and we can rewrite LDICs as follows: As such, we have As such, if the IC constraint applies to model owners of type m, it will also apply to model owners of type m − 1. Tis process can be extended down from type m − 1 to model owners of type 1, i.e., all LDICs remain the same, so we can rewrite that as follows: We can fnd that if the local upward incentive constraint (LUIC) holds, all UICs are also satisfed.From the monotonicity condition in Teorem 4, LDIC also implies a local upward incentive constraint (LUIC) as follows: As such, the IC constraints can be reduced to LDIC constraints, and it also guarantees that all UIC and DIC constraints hold.
With the constraints relaxed, we can derive a tractable set of sufcient conditions for the feasible contract.

□ Theorem 7. A feasible contract must meet the following sufcient conditions:
From the optimization contract established, we take V as the only infuencing factor to study gradient collection.As such, the optimal rewarding scheme can be summarized in the following theorem.

Theorem 8. For a known set of data quantity
the optimal reward is given by Proof.We use contradiction to validate this theorem.Tere exists a R Γ that yields greater proft for the model owner, meaning that Teorem 7 is incorrect, i.e., c 1 (R Γ ) ≥ c 1 (R * ).Tis implies there exists at least a t ∈ 1, 2, . . ., M { } that satisfes the inequality R Γ t ≤ R * t .According to Lemma 6, we have From Teorem 7, we also can get From ( 22) and ( 23), we can deduce that Tis violates the IR constraint.Terefore, there does not exist the rewards R Γ in the feasible contract that yields greater proft for the model owner.□ Substitute ( 10) into ( 6), the variable of each data quantity V m can be derived by separately optimizing each V * m as follows: V * m � arg max where Θ m � 1/φ m CV m − 1/φ m+1 CV m and Θ M � 0. Te derived solutions are feasible when they satisfy the monotonicity constraint.Otherwise, we adopt the "Bunching and Ironing" algorithm [39] to adjust the solutions iteratively (see Algorithm 1).

Dual Contract.
For ease of notation, we study one of the workers as a representative for now.Te worker of type m data collect and data update frequency utility maximize the problem denoted by u m and λ m as follows: max where r m represents the contract reward from data quantity and k m represents the contract reward from data update frequency.In the formulation of contract theory, each worker has a bundles r m , q m   and k m , η m   that, respectively, maximizes its utility u m and λ m .
From ( 3) and ( 4), the model owner utility is expressed as follows.
For the data quantity performance proft c 2 , For the data update frequency performance proft c 3 , International Journal of Intelligent Systems where n represents the number of the worker participating in the task and p(ω m ) denotes the proportion of m type worker.In the formula of contract theory, q m represents the data quantity provided by workers, η m represents the data update frequency provided by workers, and the quality of data is equal to the weighted preference of data quantity and data update frequency.
Overall, the utility of the worker is denoted by L m that can be expressed as follows: where z a and z b represent the weighted preference for data quantity and data update frequency, respectively.In addition, z a + z b � 1 and z a , z b ∈ [0, 1].Also, for the data quantity and data update frequency contract, each contract must satisfy the following constraints.
Defnition 9 (individual rationality (IR)).Each worker participates in the task if and only if its utility is not less than zero, i.e., Defnition 10 (incentive compatibility (IC)).Each worker of type m only chooses contracts designed for its type, not any other contract to maximize utility, i.e., and However, this means that we have to deal with 2 M IR constraints and 2 M(M − 1) IC constraints, both of which are nonconvex.As such, we continue to reduce and relax the conditions that guarantee the contract is feasible.

Lemma 11. For any feasible contract, we have if
Proof.Using the IC in Defnition 10, we have Ten, we add these two inequalities Tidying it up, we can get When c m ≤ c z , it follows q m ≥ q z .Likewise, we can prove if Q m ≤ Q z , it follows η m ≥ η z in this way.

□
Lemma 11 implies that workers with the lower unit data collection cost c collect the higher data quantities q and the more reward r.Similarly, workers with the lower cost of data update Q to update the higher data and more rewards k will be obtained.As such, the contract bundles are designed such that higher data quantities and data update frequency contribute translate to higher rewards.Also, a feasible contract has the necessary conditions for the following monotonicity conditions.

Theorem 12 (monotonicity). Te feasible dual contract must satisfy the following conditions:
and Next, we further relax the IR and IC constraints.Intuitively, the maximum utility worker is the worker that incurs the highest quality of data, i.e., the type M worker. (1) ALGORITHM 1: "Bunching and Ironing" adjusted algorithm. 8 International Journal of Intelligent Systems Lemma 13 (reduce dual contract IR constraints).If the IR constraints of worker type 1 are satisfed, the other IR constraints will also remain the same.
Proof.In the data collection stage, according to the IC constraints in Defnition 10 and conditions In the data update stage, the worker data update cost

and we have
As such, if the IR constraint of worker type 1 is satisfed, the other IR constraints are automatically satisfed.

□ Lemma 14 (reduce dual contract IC constraints). Te constraints can be reduced to local down incentive constraints (LDIC).
Proof.In the data collection stage, consider three worker types, where c m−1 ≥ c m ≥ c m+1 .Te two LDICs, i.e., constraints between owners of type m and type m − 1, are as follows: ( It can be obtained from Lemma 11 that when c m ≤ c z , it follows q m ≥ q z .As such, we can rewrite LDICs as follows: Tidy it up, we have As such, we have If the IC constraint in Defnition 10 applies to workers of type m, it will also apply to workers of type m − 1. Tis process can be extended down from type m − 1 to workers of type 1, i.e., all LDICs remain the same, and we can rewrite that as follows: We can fnd that if the local upward incentive constraint (LUIC) holds, then all UICs are also satisfed.From the condition in Teorem 12, if r m ≥ r m−1 , LDIC also implies a local upward incentive constraint (LUIC) as follows: (46) Likewise, we can prove the data update contract in this way.As such, we have shown that the IC constraints in Defnition 10 can be reduced to LDIC constraints since it also guarantees that all UIC and DIC constraints hold.□ With the constraints relaxed, we can derive a tractable set of sufcient conditions for the feasible contract.Theorem 15.A feasible contract must meet the following sufcient conditions: Teorem 15 implies that the weighted quality of data quantity and data update frequency is equal to the data quality required in the single contract, where we add a quality conversion parameter μ from data update frequency to data quantity.Te reward obtained by the weighted data quantity and data update frequency is smaller than the reward obtained in the single contract.Tereafter, we take c and η as the only infuencing factor to study data collection and data update, respectively.As such, the optimal rewarding scheme can be summarized in the following theorem.
Theorem 16.For a known set of data quantity q satisfying a feasible contract, the optimal reward is given by Similar, for a η satisfying Proof.We adopt the proof by contradiction to validate this theorem.We frst assume that there exists a r Γ that yields greater proft for the worker, meaning that Teorem 15 is incorrect, i.e., δ 1 (r Γ ) ≥ δ 1 (r * ).Tis implies there exists at least a x ∈ 1, 2, . . ., M { } that satisfes the inequality r Γ x ≤ r * x .According to Lemma 13 and Teorem 15, we have International Journal of Intelligent Systems From (50) and (51), we can deduce that r Γ 1 ≤ r * 1 � c 1 q 1 .Tis violates the IR constraint.Terefore, there does not exist the rewards c Γ in the feasible contract that yields greater proft for the model owner.Likewise, we can prove the (26) in this way.

□
Substitute (25) into (16), the variable of each data quantity q m can be derived by separately optimizing each q * m as follows: Similarly, substitute (49) into (28), the variable of each data update frequency η m can be derived by separately optimizing each η * m as follows: Te derived solutions are feasible if and only if they satisfy the monotonicity constraint.Otherwise, we can replace V with q or η using the Bunching and Ironing algorithm.

Performance Evaluation
In this section, we evaluate the feasibility of the single contract and the dual contract and the optimality of our designed dual contract-based incentive mechanism.Ten, we evaluate the data quantity and data update frequency changes under the diferent preferences.Also, we evaluate the utility of workers, the utility of model owners, and social welfare changes under the diferent worker types.
Te key simulation parameters are provided in Table 3, and we assume that there are 100 workers to research our incentive mechanism, with varying collection costs q m modeled by the normal distribution.We adopt the k-means clustering method to derive M � 5 clusters of workers and M � 8 clusters of model owners.We keep s a � s b � 0.5, i.e., both data freshness and service delays are of equal importance to the model owner.
For the single contract of the incentive mechanism, Figure 2 shows that both the quality of the gradient contributed and rewards for the model owners increase as the model owner's willingness to participate increases.Te monotonicity constraint in Teorem 4 is satisfed.Also, Figure 3 shows that all model owner utilities are positive and can maximize utility when they choose the contract design for its type.Te IR constraint in Defnition 1 and the IC constraint in Defnition 2 are satisfed.
Te dual contract of the incentive mechanism contains two superimposed contracts.For the frst stage contract, Figure 4 shows that both the data quantity contributed and rewards for the workers decrease as the cost incurred per unit quantity of the data collected increases.Te monotonicity constraint in Teorem 12 is satisfed.Also, Figure 5 shows that all worker utilities are positive and can maximize utility when they choose the contract design for its type.Te IR constraint in Defnition 9 and the IC constraint in Defnition 10 are satisfed.Te workers collect more data in the frst stage, i.e., the higher update costs in the second stage.As such, Figure 6 shows that both the data update frequency contributed and rewards for the workers increase as the cost of the data update decreases.Te monotonicity in Teorem 12 is satisfed.Also, Figure 7 shows that all worker utilities are positive and can maximize utility when they choose the contract design for its type.Te IR constraints in Defnition 9 and IC constraints in Defnition 10 are satisfed.

Performance Comparison.
To facilitate the analysis, we consider one task as representative of the continuous task, while the other tasks are similar.Terefore, our proposed incentive mechanism can be applied to any number of tasks in a continuous task with a fxed duration T.Moreover, we take the example of the requester who prefers data quantity with a set of values for σ 2 � 5.9z 2 a + 0.85z a + 20.88 and σ 3 � 6z 2 a − 11z a + 75. Figure 8 illustrates the variations in the model owner's utility as the data quantity preference z a changes.As expected, the model owner's utility increases as the data quantity weighing z a becomes higher, indicating a greater inclination toward the requester's preference.Furthermore, when z a > 0.5, the value of z a is equivalent to the model owner's willingness to participate φ in the single contract.A higher z a implies a greater willingness to participate.Tis fnding validates that the model owner can adjust the weights z a and z b to accommodate the requester's preference and own willingness to participate.
We present a comparison between the proposed incentive scheme and two other contract-based schemes: the contract-based social maximization scheme (CS) and the contract-based complete information scheme (CC) presented in [33].Te CC scenario assumes that the model owner has full knowledge of the cost types of each worker, 10 International Journal of Intelligent Systems while the CS scenario aims to maximize social welfare.We observe from Figure 8 that the utility of model owners cannot always be maximized under the single contract in CC.On the other hand, the single contract in CS yields similar results to the dual contract in terms of model owner utility, which is consistently lower than that of the proposed scheme.We attribute this diference in performance to the second stage of the proposed contract, where the average data update cost is smaller compared to that of the CS and CC scenarios.Tis leads to a better model owner utility under the proposed scheme compared to the single contract in CS and CC.Furthermore, the proposed contract also outperforms the single contract in CA, as the efect of the second stage contract improves its overall performance.Overall, our results highlight the importance of considering a multistage contract in incentivizing workers for collaborative data updates.

Managing the AoI-Service Latency Trade-Of.
In practice, a model owner may have diferent preferences for varying tasks.We vary the weights z a and z b within the range [0.1, 0.9] to study the changes in data quantity and data update frequency when the model owner preferences vary.

Simulation parameters Value
Gradient collect parameters: ϕ, C, and μ N(0.8, 0.01), 1.5, and 0.5 Data collect parameters: c and 9 N(0.7, 0.01) and 1 Data update parameters: α, β, ϑ, z T , and z S 1, 100, 1, 0.015, and 0.04 International Journal of Intelligent Systems Figure 9 depicts the changes in the number of data quantities as the preference towards z a varies.As expected, data quantities and the growth rate increase as z a increases.Figure 10 depicts the changes in data update frequency as the preference towards z a varies.As expected, the number of data update frequencies decreases, and the rate of decline increases as z a increases.

Impact of Worker
Types.Figures 11-13 depict the system performance concerning z a under a diferent number of worker types.When the number of z a increases, both the model owner and the workers obtain higher utilities.Because the collected quality of data improves, model owners can obtain more utility for training the model and gaining more rewards.Tus, social welfare is also improved.Also, we found when the number of worker types increases, the utility of the model owner decreases but the utilities of workers increase.Te reason is that when the number of worker types increases, it becomes more difcult for the model owner to mine the information of the worker type and design the corresponding contract.Terefore, the workers can extract more rewards from the model owner.International Journal of Intelligent Systems

Conclusion
Tis paper presents a hierarchical incentive mechanism for an FL-based system with caching where the models trained by the workers are all based on their latest data and investigate the trade-of between data quantity and data update frequency.Specifcally, we design a single contract to a dual contract based on the model owner's willingness to participate and the gradient quality that the worker provides.Our proposed mechanism uses contract theory to incentivize high-quality gradient updates from diferent types of workers.
As a future research direction, we can explore superior incentive mechanisms to improve FL efciency.Furthermore, by considering worker data quality in more aspects and practical conditions limitations, this may involve incorporating other metrics, such as data completeness or relevance or data correlation between diferent people into the incentive mechanism design.

Table 1 :
Te comparison of the prominent features of this paper's incentive mechanism with other studies.
Te dual contract of the incentive mechanism has L 1 � c i m : 1 ≤ m ≤ M   of M unit data collection cost types belonging to model owner i. Te worker type c i m can be characterized by a probability mass function p(c i m ), where the worker types are indexed in a nondecreasing order 0 are calibratable system model parameters that determine the service delays on model accuracy.σ 3 represents the conversion parameter from update frequency performance to profts.Te data freshness is ft by a monotonically increasing concave function showing incremental return decreases with the data update frequency.Te service delay is ft by a monotonically decreasing concave function showing the decline rate of return decreases with the data update frequency.
In addition, s a + s b � 1 and s a , s b ∈ [0, 1].α represents calibratable system model parameters that determine the data freshness on model accuracy.β and ϑ

Table 2 :
Table of commonly used notations.

Table 3 :
Table of key simulation parameters.