I Introduction
Following the advancements in the Internet of Things (IoT) and edge computing paradigm, traditional Vehicular AdHoc Networks (VANETs) that focus mainly on VehicletoVehicle (V2V) and VehicletoInfrastructure (V2I) communications [li2007routing, hartenstein2008tutorial] are gradually evolving into the Internet of Vehicles (IoV) paradigm [xu2017internet, wan2016mobile].
The IoV is an open and integrated network system which leverages on the enhanced sensing, communication, and computation capabilities of its component data sources, e.g., vehicular sensors, IoT devices, and Roadside Units (RSUs) [yang2014overview], to build datadriven applications for Intelligent Transport Systems, e.g., for traffic prediction [wang2018internet], traffic management [kumar2018ant], route planning [florian2014privacy], and other smart city applications [ang2018deployment]. Coupled with the rise of Deep Learning, the wealth of data and enhanced computation capabilities of IoV components enable effective Artificial Intelligence (AI) based models to be built.
Beyond ground data sources, aerial platforms are increasingly important today given that modern day traffic networks have grown in complexity. In particular, Unmanned Aerial Vehicles (UAVs) are commonly used today to provide data collection and computation offloading support in the IoV paradigm. The UAVs feature the benefits of high mobility, flexible deployment, cost effectiveness [zhou2014efficient], and can also provide a more comprehensive coverage as compared to ground users. UAVs can be deployed, e.g., to capture images of car parks for the management and analysis of parking occupancy [zhou2017car], to capture images of roads and highways for traffic monitoring applications [elloumi2018monitoring, coifman2006roadway, ke2018real], and also to aggregate data from stationary vehicles and roadside units that in turn collect data of other passing vehicles periodically [binol2018time]. Apart from data collection, the UAVs have also been used to provide computation offloading support for resource constrained IoV components [zhang2018energy, bekkouche2018uavs].
As such, studies proposing the Internet of Drones (IoD) and DronesasaService (DaaS) [gharibi2016internet, koubaa2018dronetrack, koubaa2017service] have gained traction recently. Moreover, the DaaS industry is a rapidly growing one [walia2019global] that comprises independent drone owners which provide ondemand data collection and model training for businesses and city planners.
Naturally, to build a better inference model, the independently owned UAV companies can collaborate by sharing their data collected from various sources, e.g., carparks, RSUs, and highways, for collaborative model training. However, in recent years, the regulations governing data privacy, e.g., General Data Protection Regulation (GDPR) are increasingly stringent. As such, this can potentially prevent the sharing of data across DaaS providers. To this end, we propose the adoption of a Federated Learning (FL) based [mcmahan2016communication] approach to enable privacypreserving collaborative ML across a federation of independent DaaS providers.
In our system model (Fig. 1), a client, hereinafter model owner, is interested in collecting data from a region for model training, e.g., for traffic prediction. Given the energy constraints of UAVs [zeng2017energy], the region is further divided into smaller subregions. The model owner then announces an FL task, e.g., the capturing of realtime traffic flow over highways or the collection of data from RSUs for model training [ke2018real]. Then, only the DaaS providers, hereinafter UAVs, that are able to complete the task within the stipulated time and energy constraints respond to the model owner. Thereafter, the model owner assigns an optimal UAV to each subregion. After the UAV collects the sensing data, model training takes place on each UAV separately, following which only the updated model parameters are transmitted to the model owner for global aggregation.
Our proposed approach has three advantages. Firstly, the resource constrained IoV components are aided by the UAV deployment for completion of time sensitive sensing and model training tasks. Secondly, it preserves the privacy of the UAVcollected data through eliminating the need of data sharing across UAVs. Thirdly, it is communication efficient. The reason is that traditional methods of data sharing will require the raw data to be uploaded to an aggregating cloud server. With FL, only the model parameters need to be transmitted by the UAVs.
However, there exists an incentive mismatch between the model owner and the UAVs. On one hand, the model owners aim to maximize their profits by selecting the optimal UAVs which can complete the stipulated task at the lowest cost, e.g., in terms of sensing, transmission, and computation costs. On the other hand, the UAVs can take advantage of the information asymmetry and misreport their types so as to seek higher compensation. To that end, we leverage on the selfrevealing properties of contract theory [bolton2005contract] as an incentive mechanism design to appropriately reward the UAVs based on their actual types. In particular, given the complexity of the sensing and collaborative learning task, we consider a multidimensional contract to account for the multidimensional sources of heterogeneity in terms of UAV sensing, learning, and transmission capabilities.
After deriving optimal contracts to which the UAVs respond to, the possibility that multiple UAVs prefer a particular subregion still remains. To that end, we leverage on the GaleShapley (GS) [dubins1981machiavelli] matchingbased algorithm to assign the optimal UAVs to each subregion.
The contribution of this paper is as follows:

We propose an FL based sensing and collaborative learning scheme in which UAVs collect the data and participate in privacypreserving collaborative model training for applications in the IoV paradigm towards the development of an Intelligent Transport System.

In consideration of the incentive mismatches and information asymmetry between the UAVs and model owner, we propose a multidimensional contractmatching based incentive mechanism design that aims to leverage on the selfrevealing properties of an optimal contract, such that the most optimal UAV can be matched to a subregion.

Our incentive mechanism design considers a general UAV sensing, computation, and transmission model, and thus can be extended to specific FL based applications in the IoV paradigm.
The organization of this paper is as follows. Section II reviews the related works, Section III introduces the system model and problem formulation, Section IV discusses the multidimensional contract formulation, Section V considers a matchingbased UAVsubregion assignment, Section VI presents the performance evaluation of our proposed incentive mechanism design, and Section VII concludes.
Ii Related Work
In recent years, given the rising popularity of UAVs, there is an increasing number of UAVrelated studies in the literature. One group of studies focus on the fundamental issues related to the challenges of UAV deployment, e.g., trajectory optimization [zeng2017energy, oleynikova2016continuous, zhang2018cellular], communication constraints [abdulla2014optimal, alemayehu2017efficient], as well as the efficient assignment and deployment of UAVs [boccardo2015uav, brust2015networked]. Another group of studies propose specific applications of UAVs, e.g., as flying base stations [alzenad20173], with mobile cloudlets for computation offloading [jeong2017mobile], and for search and rescue missions [scherer2015autonomous]. In particular, the UAVs are also increasingly considered for providing sensing services, i.e., data collection, [zhou2018mobile] and for the development of IoV related applications, e.g., for traffic prediction [elloumi2018monitoring], localization of ground vehicles [liu2018uav], and to facilitate vehicular communications [zhuang2019sdn].
The market of UAVs as service providers, e.g., in ondemand data collection, is a rapidly growing one [walia2019global]. Given the heterogeneity in UAV types, e.g., in energy constraints and computation capabilities, the incentive mechanism design for UAV systems is an important issue. The study in [ma2019strategic] adopts a game theoretic approach to analyze the offloading decisions of UAVs acting as flying cloudlets for IoT devices. In contrast, the study in [zhang2018predictive] proposes the contracttheoretic approach to incentivize UAV base stations to contribute higher transmit power for enhanced coverage over wireless networks. In consideration of the limited availability of mobile charging stations for UAVs, the study in [shin2019auction] proposes an auctionbased approach to efficiently assign the UAVs to specific charging time slots so as to reduce congestion.
However, given the nascent field of FL, there are relatively few works that propose FL based collaborative learning schemes involving UAVs. To the best of our knowledge, the study of [zeng2020federated] is the first to propose the implementation of FL for joint power allocation and scheduling of UAV swarms. With the increasingly stringent regulations related to data privacy, the adoption of FL can facilitate collaborative learning for the development of effective AI models, without the exchange of potentially sensitive raw data. As such, there is an urgent need to consider the incentive mechanism design for FL in UAV networks.
To that end, we can take reference from the growing literature related to incentive mechanism design for FL. For example, the study in [kang2019incentive] adopts a contracttheoretic approach to motivate workers to contribute more computation resource for efficient FL. On the other hand, the study in [feng2019joint] formulates the Stackelberg game to analyze the inefficiency in model update transfer. As an extension, the study in [zhan2020learning]
uses a Stackelberg game formulation together with Deep Reinforcement Learning to design a learningbased incentive mechanism for FL. For a comprehensive survey in this area, we refer the readers to
[lim2019federated].Apart from the traditional considerations of incentive design in FL, the UAV systems involve other sources of heterogeneity in UAV types, e.g., traversal costs. As such, the multidimensional sources of heterogeneity in UAVs have inspired us to adopt the multidimensional contract theoretic approach [wang2019multi] in our incentive mechanism design. Moreover, in contrast to traditional works in contract theoretic mechanism design, our system model only involves the matching of a single, optimal UAV type to each subregion. This necessitates the use of the matchingbased algorithm such as the GS algorithm. The use of matching for UAVs to subregions have also been studied in [zhou2018mobile]. However, [zhou2018mobile] does not have any mechanism in place to ensure truthful reporting, while we leverage on the selfrevealing properties of contract theory to that end. While the study of contractmatching has also been explored for resource allocation in vehicular fog computing [zhou2019computation], the contract considered is singledimensional with simpler considerations.
In summary, our study considers the adoption of FL to facilitate privacy preserving sensing and collaborative learning in the UAV services market, and proposes a multidimensional contractmatching design that aims to match the most optimal UAV to each sensing subregion, while accounting for the multiple sources of heterogeneity in UAV types.
Iii System Model and Problem Formulation
We consider a network in which a model owner aims to collect data from stipulated nodes, e.g., from RSUs or images of segments in the highway, in a target sensing region to fulfill a timesensitive task. One UAV is selected by the task publisher to cover each of the subregions. Given information asymmetry and the multiple sources of heterogeneity in UAV cost types, the model owner leverages on the selfrevealing properties of a multidimensional contract theoretic approach to choose one UAV suited to cover each of the subregion. After data collection, the UAV returns to their respective UAV bases for Federated Learning (FL) based model training.
Following [zhou2018mobile], the target sensing region can be modeled as a graph and divided into smaller graphs, i.e., subregions whose set is denoted , e.g., through the multilevel graph partition algorithm [karypis1995multilevel]. The set of nodes in subregion is denoted with the node in subregion located at . The Euclidean distance between two nodes and located within subregion , is expressed as where , i.e., all nodes are interaccessible.
A set of unmanned aerial vehicles (UAVs) are located at bases situated around the target sensing region. Without loss of generality, we assume that each base owns a single UAV and . Moreover, our model can be easily extended to scenarios in which a UAV swarm is required for sensing in each subregion. Denote as the set of bases where refers to the base of UAV located at . The Euclidean distance between the base of UAV and subregion is expressed as , where and denotes the centre of the subregion.
There are two stages in our system model as follows:

MultiDimensional Contract Design: The UAV types, e.g., sensing, traversal, and transmission costs, are private information not known to the model owner. As such, the model owner designs a multidimensional contract to leverage on the selfrevealing mechanism so as to select the optimal UAV to cover each subregion. In particular, the model owner can maximize its profits by choosing the lowest cost UAV among all feasible UAVs that can complete the task within the time constraint.

UAVSubregion Assignment: Each UAV reports its type and ranks the subregions based on its preferences of coverage to the FL model owner. Then, a stable UAVsubregion matching is derived using the GaleShapley (GS) algorithm. Note that each UAV’s preference can vary across different subregions. As an illustration, we consider a representative UAV with base and two subregions and where . In this case, UAV has to traverse a longer distance to reach subregion . As such, it is able to cover a smaller proportion of subregion relative to due to energy and time constraints.
In the following, we consider the sensing, computation, and data transmission model of a representative UAV.
Iiia UAV Sensing Model
Notation  Description 

Subregion  
UAV  
Base of UAV  
Total sensing distance  
Total traversal distance  
Total duration taken for traversal and sensing  
Total energy taken for traversal and sensing  
Marginal cost of node coverage for sensing  
Traversal cost  
Local computation duration  
Total energy taken for computation  
Marginal cost of node coverage for computation  
Total duration for transmission  
Energy taken for transmission  
UAV utility  
Contractual rewards  
Unit cost of energy for the UAV  
Model owner profit  
Contract set and individual contract  
Compensation for sensing and computation costs  
Compensation for traversal and transmission costs  
Marginal cost of node coverage  
UAV auxiliary type 
We consider a representative UAV tasked by the model owner to cover a proportion of nodes in the subregion . Denote the node coverage assignment of UAV in subregion to be where represents that the UAV has to fly through the segment between nodes and , and implies otherwise.
The total distance traveled for sensing by UAV under assignment is as follows:
(1) 
Denote where indicates cardinality, i.e., refers to the proportion of node coverage by UAV in subregion where .
Apart from traveling between the nodes, the UAV has to travel to and from its base. Denote the total distance traveled by the UAV as . Hereinafter, we refer to as the sensing distance, whereas refers to the traversal distance.
Following the works of [zeng2019energy, zhang2018predictive], each UAV travels with an average velocity and expends a fixed propulsion power throughout the task for tractability, where and refers to the required power to balance the parasitic drag caused by skin friction and required power to balance the drag force of air redirection respectively^{1}^{1}1In practice, the propulsion power is in turn a function of other factors, e.g., reference area of the UAV and wing aspect ratio and weight. For simplicity, we consider that accounts for these factors.. Note that the propulsion power consumed by the UAV when it changes its direction is negligible [zeng2017energy]. The total duration taken for traversal and sensing is denoted , whereas the total energy consumed to cover the traversal and sensing distance is as follows:
(2) 
where is the distance traveled by the UAV if it covers all nodes, i.e., , and for notation simplicity. Note that represents the sensing cost, i.e., marginal cost of node coverage for sensing in the subregion, whereas refers to the traversal cost, i.e., the energy cost of traveling to and from the base. A higher can imply that the UAV requires greater propulsion power to complete the task, e.g., due to its larger weight or wingaspect ratio, whereas a higher implies either a greater propulsion power to move, or a greater traversal cost, i.e., the subregion is farther away from the base. While the value of varies across subregions due to the varying , i.e., the marginal cost of node coverage varies according to the sensing area of the subregion, the ordering of the UAV types based on the sensing costs is retained. On the other hand, the order of UAVs by traversal costs varies across subregions, based on the distance between the UAV base and each of the subregions.
IiiB UAV Computation Model
After the UAV covers its assigned set of nodes following assignment , it returns to the base for an FL based model training over global iterations where to minimize the global loss . Following [konevcny2016federated], each training iteration consists of three steps namely: (i) Local Computation, i.e., the UAV trains the received global model locally using the sensing data, (ii) Wireless Transmission, i.e., the UAV transmits the model parameter update to the model owner, and (iii) Global Model Parameter Update, i.e., all parameter updates derived from the subregions are aggregated to derive an updated global model , where , which is then transmitted back to the UAVs for the training iteration.
In general, a series of local model training is performed by the UAV to minimize an Lipschitz and
strongly convex local loss function
up to the target accuracy defined by the model owner to derive the parameter update. Note that a larger value of implies greater deviation from the optimal value. Moreover, , i.e., the local solution does not have to be trained to optimality, e.g., to reduce local computation duration especially for time sensitive tasks. In particular, following the formulation in [yang2019energy]:(3) 
The FL training is completed after global iterations where and . The total local computation duration is as follows:
(4) 
whereas the energy consumption of UAV for computation is as follows:
(5) 
is the effective switched capacitance that depends on the chip architecture [mao2016dynamic], is the cycles per bit for computing one sample data of UAV , is the unit of data samples collected by UAV , refers to the lower bound on number of local iterations required to achieve local accuracy [yang2019energy] where , and refers to the computation capacity of the UAV , measured by CPU cycles per second. For ease of notation, we denote , i.e., a higher implies greater energy cost for computation per additional node coverage. Similar to , the value of varies across subregion due to the different units of data samples available for computation. However, the ordering of the UAV types based on computation costs is retained.
IiiC UAV Transmission Model
After local computation, the wireless transmission takes place from the selected UAVs to the model owner. For simplicity, we denote the achievable rate of the UAV to be a product of its transmit power and a scaling factor which covers other considerations, e.g., bandwidth allocation and channel gain.
The total time taken by the UAV to upload parameter update of size is as follows: . Note that the model upload size is constant regardless of the number of global iterations or quantity of data collected, given the fixed dimensions of the model update. The transmission energy consumption, denoted , is as follows:
(6) 
IiiD UAV and Model Owner Utility Modeling
The utility function of a representative UAV covering subregion can be expressed as follows:
where refers to the contractual rewards and refers to the unit cost of energy.
Following [liu2018edge, khan2019federated, zhan2020learning], the FL model accuracy is a concave function of the aggregate data collected across subregions by the selected UAVs. In particular, the inference accuracy of the model is improved when more nodes are covered, i.e., a model trained using data across a more comprehensive coverage of classes may be built. Without loss of generality, we consider the aggregate model performance to be an average of node coverage across all regions, analogous to the Federated Averaging algorithm:
(8) 
where is the system parameter. The total profit obtained from all UAVs is thus as follows:
(9) 
where refers to the conversion parameter from model performance to profits, and the contractual reward expense for each selected UAV is denoted as . In the next section, we devise the optimal contract which satisfies the Individual Rationality and Incentive Compatibility constraints.
Iv MultiDimensional Contract Design
In this section, we first consider a multidimensional contract formulation. To solve the multidimensional contract, we sort the UAV types according to an auxiliary variable which reflects the marginal cost of node coverage. Then, we relax the constraints for contract feasibility and include a fixed compensation component for traversal and transmission costs so as to solve for the optimal contract.
Iva Contract Condition Analysis
Given that the sensing cost , traversal cost , computation cost , and transmission cost are all private information that are not precisely known by the model owner, we consider the multidimensional contract theoretic incentive mechanism design to leverage on its selfrevealing properties.
The UAVs can be classified into different types to characterize their heterogeneity. In particular, the UAVs can be categorized into a set
of traversal cost types, set of sensing cost types, set of computation cost types, and set of Q transmission cost types.Without loss of generality, we also assume that the user types are indexed in nondecreasing orders in all four dimensions: , , , and . For ease of notation, we represent a UAV of traversal cost type , sensing cost type , computation cost type , and transmission cost type to be that of type.
To enforce the UAVs to truthfully reveal their private information, we adopt a twostep procedure for the contract design:

MultiDimensional Contract Design: We convert the multidimensional problem into a singledimensional contract formulation following the approach in [wang2019multi]. In particular, we sort the UAVs by an auxiliary, onedimensional type in the ascending order based on the marginal cost of node coverage, i.e., sensing and computation cost types. Then, we solve for the optimal contract for each subregion denoted where to derive the optimal node coveragecontract reward bundle .

Traversal Cost Compensation: In contrast to existing works on multidimensional contracts, the UAVs also incur the additional traversal cost and transmission cost components, both of which are not coupled with the marginal cost of node coverage. In other words, these costs have to be incurred regardless of the number of nodes a UAV decides to cover in the subregion. For each contractual reward, we add in a fixed compensation to derive the final contract bundle .
We first discuss the multidimensional contract formulation as follows. A contract is feasible only if the Individual Rationality (IR) and Incentive Compatibility (IC) constraints hold simultaneously.
Definition 1.
Individual Rationality (IR): Each type UAV achieves nonnegative utility if it chooses the contract item designed for its type, i.e., contract item .
(10) 
Definition 2.
Incentive Compatibility (IC): Each type UAV achieves the maximum utility if it chooses the contract item designed for its type, i.e., contract item . As such, it has no incentive to choose contracts designed for other types.
(11) 
The multidimensional contract formulation is as follows:
(12) 
However, the optimization problem in (IVA) involves , i.e., IR constraints and , i.e., IC constraints, all of which are nonconvex. Therefore, we first convert the contract into a singledimensional formulation in the next section.
IvB Conversion Into A SingleDimensional Contract
In order to account for the marginal cost of node coverage, we consider a revised utility of the UAV type that excludes the traversal and transmission costs as follows:
(13) 
where we denote for ease of notation, and refers to the contractual reward arising from the multidimensional contract design. To focus on a representative contract, we drop the superscripts for now. Given that the ranking of marginal cost types does not change across subregion, note that our contract design is a general one applicable to all subregions.
We derive the marginal cost of node coverage for the type() UAV as follows:
(14) 
Intuitively, since the coverage of an additional node results in the additional expenses of sensing and computation costs. A larger value of implies a larger marginal cost of node coverage, due to the greater sensing and computation costs incurred for a particular UAV type.
We can now sort the UAVs according to their marginal cost of node coverage in a nondecreasing order as follows:
(15) 
where denotes the auxiliary type user. Given the sorting order, the UAV types are in an ascending order based on their marginal cost of node coverage:
(16) 
Note that for ease of notation, we use type or type interchangeably to represent the auxiliary type user. In addition, we refer to and interchangeably to represent the new ordering subsequently. Similarly, to represent the marginal cost of node coverage, we use and . In the next section, we derive the necessary and sufficient conditions for the contract design.
IvC Conditions For Contract Feasibility
We derive the necessary conditions to guarantee contract feasibility based on the IR and IC constraints as follows.
Lemma 1.
For any feasible contract , we have if and only if , .
Proof.
We first prove the sufficiency, i.e., if . From the IC constraint of type UAV we have:
(17) 
which implies:
(18) 
Given that , we can deduce .
Next, we prove the necessity, i.e., . Similarly, we consider the IC constraint of the type UAV:
(19) 
Given , we deduce , which follows that . The proof is now completed. ∎
Lemma 2.
Monotonicity: For any feasible contract , if , it follows that .
Proof.
We adopt the proof by contradiction to validate the monotonicity condition. We first assume that there exists such that , i.e., the lemma is incorrect.
We consider the IC constraints for the type and UAV:
Then, we add the constraints together and rearrange the terms to obtain:
(20) 
As such, Lemmas and give us the necessary conditions of the feasible contract in the following theorem.
Theorem 1.
A feasible contract must meet the following conditions:
(22) 
Next, we further relax the IR and IC constraints. Due to the independence of on the contract item , i.e., , , the UAV type does not change with the node coverage and contract rewards. In addition, the ordering of the type by marginal costs does not change with the subregion . As such, we are able to deduce the minimum utility UAV , i.e., the UAV type that incurs the highest marginal cost of node coverage, as follows:
(23) 
Intuitively , i.e., the UAV characterized by is the UAV which incurs the highest marginal cost of node coverage, and hence it is the minimum utility UAV.
Lemma 3.
If the IR constraint of the minimum utility UAV type is satisfied, the other IR constraints will also hold.
Proof.
From the IC constraint and the sorting order , we have the following relation:
As such, as long as the IR constraint of the UAV type is satisfied, it follows that the IR constraints of the other UAVs will also hold. ∎
Lemma 4.
For a feasible contract, if and then .
Note that the relation , implies the Pairwise Incentive Compatibility (PIC), which is fulfilled under the following condition:
Proof.
Suppose we have three UAV types , , and where . The Local Upward Incentive Constraint (LUIC), i.e., IC constraint between the and UAV is as follows:
(24) 
In addition, we consider:
(25) 
Given the order of marginal cost of node coverage in (16), it follows that (25) is positive. As such, we have:
(26) 
Adding the LUIC inequalities presented in (33) together with that of (26), we have:
(27) 
By considering the Local Downward Incentive Constraint (LDIC), i.e., IC constraint between the and UAV, as well as adopting the approach in (25), we are able to derive that:
(28) 
As such, given that the LUIC and LDIC hold, we have proven that the PIC of the contracts hold, i.e., . ∎
With Lemma 3, we are able to reduce IR constraints into a single constraint, i.e., as long as the minimum utility type UAV has a nonnegative utility, it follows that the other IR constraints will hold. Moreover, with Lemma 4, we are able to reduce constraints into constraints, i.e., as long as the PIC constraint of the type and type UAV holds, it follows that the IC constraints between the type and all other UAV types will hold.
With this, we are able to derive a tractable set of sufficient conditions for the feasible contract in Theorem 2 as follows. The first condition refers to the reduced IR condition corresponding to Lemma (3), whereas the second condition refers to the PIC condition between the type and type UAV corresponding to Lemma (4).
Theorem 2.
A feasible contract must meet the following sufficient conditions:


.
IvD Contract Optimality
To solve for the optimal contract rewards , we first establish the dependence of optimal contract rewards on route coverage . Thereafter, we solve the problem in (IVA) with only. Specifically, we obtain the optimal rewards given a set of feasible node coverages from each UAV which satisfies the monotonicity constraint .
In addition, the multidimensional contract formulation that we have thus far only considers the selfrevelation for two types, i.e., sensing and computation costs. To account for traversal and transmission cost types, we add an additional fixed compensation into the contract rewards. The traversal cost can be derived from the historical information of the UAV, and can be calibrated based on the response that the model owner receives. In the following theorem, we prove that the addition of a fixed rewards compensation does not violate the IC constraints, i.e., the selfrevealing properties of the contract is still preserved, whereas it is inconsequential even if the IR constraint is violated, given that has already been designed to sufficiently compensate marginal costs, and only one optimal UAV is required to serve each subregion. The optimal rewarding scheme is summarized as follows.
Theorem 3.
For a known set of node coverage satisfying in a feasible contract, the optimal reward is given by:
(29) 
Proof.
There are two parts to the proof. Firstly, we prove that the reward design for the twodimensional contract is optimal. Therefore, we adopt the proof by contradiction. We first assume there exists some that yields greater profit for the model owner, meaning that the theorem is incorrect, i.e., . For simplicity, we need to consider only the rewards portion of the model owner’s profit function in this proof, i.e., . This implies there exists at least a that satisfies the inequality .
According to the PIC constraint of Lemma (4), we have:
(30) 
In contrast from Theorem 3, we have:
(31) 
From (30) and (31), we can deduce that . Continuing the process up to , we , which violates the IR constraint. As such, there does not exist the rewards in the feasible contract that yields greater profit for the model owner. Intuitively, the model owner chooses the lowest reward that satisfies the IR and IC constraints for profit maximization.
Secondly, we show that adding a fixed traversal cost reward does not violate the IC constraint. Within a subregion, when we consider the complete utility function of the auxiliary UAV with type , ,
(32) 
Intuitively, the traversal and transmission cost are structurally separate from the marginal costs, i.e., sensing and computation cost types of the UAV within a subregion . As such, the fixed reward terms cancel out and the selfrevealing properties of the contract is preserved.
Note that the IR constraint may no longer hold for some where
(33) 
However, this is inconsequential given that unlike the conventional contract theoretic formulations, we only require a type of UAV to serve a subregion. Moreover, is already designed such that the IR constraints hold to compensate marginal costs sufficiently. For certain subregions, can be calibrated upwards if no UAV responds to the model owner.
∎
Unlike conventional contract theoretic formulations, the model owner only requires a single contract per subregion for the optimal UAV. From (9), we can deduce that the optimal type UAV to serve each subregion is . Intuitively, for each region, the model owner leverages on the selfrevealing properties of the multidimensional contract formulation to obtain an optimal UAV with the lowest marginal cost of node coverage for profit maximization. In other words, this is the UAV that can cover the largest proportion of the subregion at the lowest cost, among all feasible UAVs that can complete the task. We can substitute the optimal rewards into the profit function of the model owner and rewrite the profit maximization problem as follows:
s.t.  
(35) 
where:
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