Privacy-Aware Sensing-Quality-Based Budget Feasible Incentive Mechanism for Crowdsourcing Fingerprint Collection

Mobile crowdsourcing (MCS) has shown great potential in received signal strength (RSS) fingerprint collection, in which an incentive mechanism plays a critical role to motivate users’ participation. However, how to quantify the quality of the gathered fingerprint data is still not addressed well in the design of incentive mechanism for MCS-based fingerprint collection. In this paper, a sensing quality metric is proposed to characterize the joint impact of users’ privacy protection and the spatial coverage of the submitted data. Given a limited budget, a basic incentive mechanism is devised to recruit appropriate users to maximize sensing quality. Considering that the cost of each user is regarded as private information and users may be attempted to misreport their costs to increase the revenue. Hence, an auction-based incentive mechanism is proposed to achieve the truthfulness of users’ costs, which is truthful, individually rational, computationally efficient and budget feasible. Simulation results show that our proposed schemes outperform the baseline schemes and the experiment with real-world data is carried out to evaluate the performance of our proposed basic incentive mechanism.


I. INTRODUCTION
Mobile crowdsourcing (MCS) has emerged as a promising solution to leverage the power of mobile users (MUs) for massive data collection and processing [1]. A typical MCSbased system is mainly comprised of a platform residing in the cloud and a crowd of MUs, where MUs are recruited to participate in the required task and upload the sensing data back to the platform via the existing wireless network. Generally, it requires MUs' efforts and consumes diverse resources (e.g., data usage, time, battery) to participate in MCS activities. To incentivize MUs' participation, many incentive mechanisms have been developed with diverse optimization goals, e.g., reducing the platform's cost, maximizing social welfare [2].
The associate editor coordinating the review of this manuscript and approving it for publication was Wei Zhang. Over the past decade, many research efforts have been dedicated to tailoring MCS to acquire received signal strength (RSS) fingerprint data from the inertial sensors of users' smartphones with the aim to relieve the burden of RSS fingerprint collection [3], [4]. However, the fingerprint data gathered through MCS lacks quality guarantee, which, in turn, has a significant effect on users recruitment [5]. This necessitates the need for measuring the quality of sensing data (namely ''sensing quality'') and designing a quality-based incentive mechanism for crowdsourced fingerprint collection. While the state-of-the-art studies [6], [7] have been conducted to characterize the quality of fingerprint data gathered in the MCS manner, they are not fully suitable in practical scenarios. The success of the work [6] strongly depends on experiments of users' sensed locations, which may lead to unpredictable factors. The work in [7] is under the assumption that the prior knowledge of the indoor space is perfectly known, including the indoor map and the physical locations VOLUME 8, 2020 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ of reference points. Therefore, due to the lack of a perfect benchmark, it is quite challenging to evaluate sensing quality for MCS-based RSS fingerprint collection. Moreover, it is worth noting that none of the works mentioned above considers users' privacy protection and deals with the impact of the privacy issue on sensing quality measurement. Nowadays, users' privacy issue has been one of the key concerns for MCS-based systems [8], since individuals' sensitive information can be inferred from sensing data [9]. Hence, users will suffer privacy breach when such sensitive data is disclosed to the unauthorized third party. Given this challenge, a great number of techniques from different aspects of privacy have been developed to protect users' privacy [10] that can be broadly categorized into four types: encryption [11], anonymity [12], differential privacy [13] and game theory [14]. Among those, local differential privacy (LDP) has been widely adopted by many technology companies such as Microsoft, Google, Apple, where users are permitted to perturb personal data by adding random noise to achieve privacy protection [15]. However, there is a conflict between the incentives of privacy preservation by users and data aggregation accuracy maximization by the platform [16], which increases the difficulty in the sensing quality evaluation for crowdsourced fingerprint collection.
It is still an open research topic to quantify sensing quality for MCS-based fingerprint collection without the prior knowledge of ground truth and users' historical data. Moreover, the consideration of privacy issue increases the difficulty for the quality measurement and the design of a quality-based incentive mechanism. In this study, a sensing quality metric is first proposed to capture the joint impact of privacy protection and the spatial coverage of users' submitted data. Subsequently, the proposed sensing quality metric is incorporated into the design of incentive mechanism for users selection under a budget constraint. The key contributions of this paper are summarized as follows: • A sensing quality metric is proposed to characterize the joint impact of the privacy protection for users' sensory data and spatial coverage of the area covered by the submitted sensing data.
• The basic incentive mechanism is proposed to recruit users with an assumption that individual user truthfully reports its cost. To maximize sensing quality, a greedy algorithm is proposed for users selection constrained within a budget limit.
• An auction-based incentive mechanism is proposed to address the scenario where users misreport their costs, and theoretically analyzed to prove its truthfulness, individual rationality, computational efficiency, and budget feasibility. The reminder of this article is organized as follows. Section II analyses the related works and Section III introduces the system model. The sensing quality maximization problem is described in Section IV and the basic mechanism is presented in Section V. Section VI presents the auctionbased mechanism with theoretical analysis. Section VII and Section VIII present the performance evaluation results. Section IX makes a conclusion of this paper.

II. RELATED WORK
In recent years, many incentive mechanisms have been developed for mobile crowdsourcing (MCS) to attract enough number of users to participate in sensing tasks.
Game-theoretic incentive mechanisms have received increasing popularity, since they are capable to deal with users' strategic behaviors [17]- [19]. Yang et al. [17] proposed a platform-centric model to encourage users' participation within which the relationship between the platform and users is modeled as Stackelberg game. Pang et al. [18] proposed a Bayesian game based incentive mechanism for spatial crowdsourcing. Nie et al. [19] proposed a Bayesian game-theoretic incentive mechanism for social crowdsensing service, where the information of social network effects is incomplete.
Considering that selfish individual users seek to maximize their own gain by misreporting their truthful costs. To tackle this issue, auction theory has been an invaluable tool for MCS-based systems to discover users' truthful costs and effectively control the payment for the selected users. In the literature, many researchers have studied the auction-based incentive mechanisms for MCS-based applications [20]- [24]. For example, Zhao et al. [20] investigated an online auctionbased incentive mechanism for mobile crowdsourced sensing and Ji and Wang [21] proposed an auction-based incentive mechanism to study the correlated tasks allocation problem. The authors in [22] and [23] considered the user-centric model where each mobile user could ask for the reserve price, and further proposed a revenue maximizing reverse auction mechanism. In [24], authors proposed an incentive mechanism to maximize the social welfare including the workercentric task selection phase and the platform-centric worker selection phase. However, none of these studies considers the issue of estimating users' data quality.
In this study, we first propose a metric of sensing quality to characterize the joint impact of users' privacy protection and spatial coverage. Then, we incorporate this metric into the design of the basic incentive mechanism to maximize sensing quality, and the auction-based incentive mechanism to achieve the truthfulness of users' costs, respectively.

III. SYSTEM OVERVIEW
The system model considered in our work is described in more detail in this section. For convenience, Table 1 lists the frequently used notations.   number of MUs to contribute to sensing data for RSS fingerprint collection. 2 The system architecture is depicted in Fig. 1, where the interactions between MUs and MCP are as follows: • Step 1: The MCP announces the task to MUs and similar to [25], the upper bound of is also broadcasted.
• Step 2: The interested MUs perform the task and send the data perturbed by adding random noise with the claimed cost (sensing cost, privacy cost) to MCP.

• Step 3 & 4:
The MCP selects a subset S of MUs (winners) to maximize the sensing quality and determines the payment for winners to compensate their costs, while the total cost does not exceed the budget B.

B. USER COST
In this paper, the cost of each user comes from two aspects: (1) Sensing Cost: When participating in a MCS activity, it incurs nontrivial sensing cost to a user in terms of the effort to collect data, the charge to transmit data, the computing power of devices to process data, etc. Referring to [26], sensing cost for individual MU is modeled as a function of its trajectory distance d. Based on [27], a linear function c s d is adopted to formulate sensing cost, in which c s scales the value of unit movement cost for each MU.
(2) Privacy Cost: Individuals' sensitive information can be derived from MUs' sensed data, which leads to users' privacy loss (also termed as ''privacy cost'') if such data is leaked. To quantify MUs' privacy, the celebrated notion of local differential privacy [13] is adopted.
Definition 1 (Local Differential Privacy): An algorithm is -local differentially private ( -LDP), if and only if for any input κ and κ , the following inequation can hold: where K ∈ Range( ) and Range( ) is the set of all possible outputs of the algorithm . Pr is the probability that maps the input κ or κ to K. ∈ (0, 1) denotes the privacy protection level (or privacy budget) achieved by F, where data is strongly protected when approaches to 0. According to [25], for any user, its privacy cost is positively correlated with . Assuming that there is a positive relation between privacy cost and d, since more sensitive information is provided as d increases. In this paper, a linear model c p d is adopted to quantify privacy cost, where c p reflects how much a MU cares about its privacy.
In summary, MU i's cost is denoted as follows 3 :

IV. PROBLEM FORMULATION
This section defines a sensing quality metric, and presents the budget-constrained sensing quality maximization problem.

A. SENSING QUALITY FORMULATION
This subsection studies the joint impact of privacy protection and spatial coverage of the target area to characterize sensing quality for crowdsourced RSS fingerprint collection.
(i) The Impact of Privacy Protection: Basically, MUs' raw sensory data is perturbed with random noise to achieve -LDP. However, this perturbation influences MCP's data aggregation accuracy. To quantify the accuracy of the aggregated result, the notation (α, β)-accuracy is introduced.
Definition 2 (α, β)-Accuracy: An algorithm R with a perturbation mechanism is (α, β)-accurate where α, β ∈ (0, 1), if the following inequality can be held: where R is adopted by the platform to aggregate the uploaded data and the algorithm F is adopted to achieve -LDP. D is the corresponding dataset and β is a predetermined constant selected by the platform. α denotes the difference between the aggregation result before and after perturbation. Finally, by combining Definition 1 and 2, we can obtain the relationship between and α: as is small, large noise is locally added to MUs' raw data, which causes larger α and furthermore reduces the aggregation accuracy. Based on [28], the function 4 q(α) is exploited to characterize the impact of privacy protection on sensing quality, which decreases in α.
(ii) The Impact of Spatial Coverage: Generally, a larger number of fingerprints measured from diverse places contribute to enhancing the system performance. Meanwhile, the coverage scale of areas visited by participants is affected by the randomness of MUs' movement. To describe the places where users complete different activities (e.g., walking, turning), points of interest (PoIs) L = {l 1 , l 2 , . . . , l n } are introduced that can be extracted from users' indoor trajectories. Successively, Shannon Entropy [29] is used to capture the popularity of different PoIs, which denotes that how many unique users have visited a specific PoI as well as how many times each user have visited a PoI.
where λ i j denotes the number of visits of user u i ∈ to PoI l j and j = u i ∈ λ i j . The entropy for a PoI is regarded as the probability that the PoI is visited by users.
Definition 3 (Spatial Coverage): Given a collection of PoIs L and an associated weight set W = {ω 1 , ω 2 , . . . , ω n }, the spatial coverage of the target area is characterized by (5): where different weights in W are assigned to different PoIs to capture the information heterogeneity since different valuable information is provided by different PoIs, in which all elements in W sum up to 1. It is assumed that a higher weight is assigned to the PoI with lower entropy, since it has a smaller chance of being visited by users.
(iii) Sensing Quality Metric: Referring to [17], the power property of log function is used to characterize sensing quality with considering privacy protection and spatial coverage.
where the logarithm function is applied to reflect the MCP's diminishing returns on participating MUs while the total cost does not violate the predefined budget B.

B. PROBLEM STATEMENT
The MCP aims to select a subset S MUs to maximize the sensing quality given a limited budget B, in which the budget B could be intuitively represented by the total reward that all the selected users obtain.

V. BASIC INCENTIVE MECHANISM DESIGN
This section presents a basic incentive mechanism within the budget limitation to recruit users to maximize sensing quality, which is called BudgeTed Sensing Quality Maximization (BTSQM) problem. Similar to [30], this mechanism is designed under an assumption that MUs do not misreport their costs, which means that the costs claimed by MUs are treated as their real costs.

A. PROBLEM REDUCTION
To address this problem, we first claim that the BTSQM problem could be transformed into the budgeted maximum coverage problem [31]. Definition 4 (Budgeted Maximum Coverage Problem [31]): A collection of sets S with associated costs is defined over a domain of weighted elements X . The objective is to find a subset of S ⊆ S such that the total cost of all elements in S is at most of the budget B and the total weight of all elements covered by the subset S is maximize.
Similar to [30], given a PoI set L, X u i = {λ i j } n j=1 is used to denote the PoIs visited by u i . Then, the collection of all MUs' cover set . . , c m } is acquired. Consequently, the transform relation between our BTSQM problem and the budgeted maximum coverage problem is as follows: X → S, L → X with the corresponding target set S → S . As a result, our BTSQM problem is proved to be NP-hard, since the budgeted maximum coverage problem has been proved to be NP-hard [31]. To solve the BTSQM problem, the objective function is proved to be non-decreasing submodular such that the submodular function [32] could be exploited.
Definition 5 (Submodular Function): Leth be a finite set, a function f on the subsets ofh is submodular if Lemma 1: Given an arbitrary subset S 1 ⊆ and a PoI set L, the function Q S 1 (L) is non-decreasing submodular.
The proof can be found in Appendix A.

B. ALGORITHM DESIGN
Based on [31], a greedy algorithm is proposed to obtain the approximate solution that is illustrated in Algorithm 1.
• First, the collection H of of the cardinality less than the fixed integer τ > 0 is obtained, where the total cost of each set in H does not exceed B. Among them, the set with the largest sensing quality is selected as the candidate solution (line 1).
• Second, all subsets of of the cardinality τ are enumerated to acquire a feasible solution, whereas the total cost of each subset is at most of B (line 2). The greedy algorithm is used to complement all the subsets. At each iteration, MU u i is added to the candidate subset when meeting the following requirements: (1) the ratio (Q H ∪u i − Q H )/c(u i ) is maximized; (2) the total cost of MU u i and the current candidate set does not violate B. Then, the set with the highest sensing quality is chosen as the targeted solution (lines 3-14).
• Finally, the approximation solution of BTSQM problem is achieved (lines 15-16).  In summary, the complexity is O(m τ +1 log(m)), where τ is a integer and not less than 3.
The proof can be found in Appendix B.

VI. AUCTION-BASED INCENTIVE MECHANISM
In the previous section, a basic mechanism is proposed to maximize sensing quality under the assumption that each MU truthfully reports its cost. However, the cost is private information from MUs' perspective that is not public to MCP. To achieve the truthfulness of MUs' costs, a sensing quality-aware auction-based mechanism is presented in this section.

A. PROBLEM FORMULATION
As depicted in Fig. 2, the trade-offs between MCP and MUs are modeled as a reverse auction, where MCP acts as the auctioneer that purchases indoor trajectories from MUs who act as bidders. The utility of each winner is defined as (10): where p i denotes the payment to winner i Based on MUs' contributed sensing data, MCP is able to provide different kinds of location-based services to the public. Under this circumstance, the valuation of the platform is characterized as follows: where µ > 0 is the coefficient that transforms the sensing quality metric into the platform's valuation function. This auction-based mechanism aims to maximize the MCP's valuation under a constrained budget.
Meanwhile, this mechanism also aims to satisfy the following desirable properties: • Truthfulness: No MU can improve its utility by submitting a bid different from its true cost, no matter what others submit.
• Individual Rationality: The utility of individual winner is non-negative.
• Computational Efficiency: The result of this auctionbased mechanism can be obtained in polynomial time.

B. AUCTION BASED ALGORITHM DESIGN
To meet the above properties, the designed mechanism mainly depends on Myerson's well-known characterization [33] and budget feasibility [34]. Based on Lemma 1, given µ > 0, it is easily found that the platform's valuation function is submodular and monotonically non-decreasing. To select winners, the marginal contribution per bid of user i ∈ \S is defined as ζ i = ν i (S) b i with respect to S, where ν i (S) = ν(S ∪ {i}) − ν(S). In this study, Algorithm 2 contains the following phases: Winner Selection: First, the winner set is initialized as empty and MUs are sorted in the non-increasing order of ζ , in which the candidate user is selected with the largest marginal contribution per bid (line 2). Second, the difference between the candidate user's ζ and the value per budget of S is acquired. If the difference is positive, this user is added to S and the next iteration is continued until the difference is negative, Otherwise, the candidate user is discarded and this phase terminates (lines 3-6). VOLUME 8, 2020 Algorithm 2 Budgeted Auction Mechanism Input: PoI set L with associated weight set W , potential user set with associated cost set , and a limited budget B Output: A set of winning mobile users S ⊆ and payment profile P 1: // phase 1: winner selection 2: S ← ∅, i ← arg max j∈ S ← S ∪ {i}; 5: i ← arg max j∈ \S ν j (S) b j ; 6: end while 7: // phase 2: critical payment determination 8: for each i ∈ do; 9: P ← ∅; 10: end for 11: for each i ∈ S do;

12:
−i ← \{i}, ← ∅; 13: repeat 14: end if 21: end for 22: return S, P Payment Determination: First, the payment set P is initialized as empty for all users (lines 8-10). Second, the winner set is recomputed over −i ← \{i} and MUs in −i are sorted based on ζ to obtain the first j users denoted as j , where the j th user is denoted as i j in this sorting. Third, the largest bid that winner i can submit is achieved where MU i is selected not the user at j-th position in this sorting. This process is repeated until the position after the last winner over −i . Finally, for all candidate payments, the maximal payment is selected as the final payment to winners (lines 11-20).

C. MECHANISM ANALYSIS
In this subsection, our designed mechanism is proved to satisfy the predefined properties.
Theorem 1: The auction-based mechanism is truthful. The proof can be found in Appendix C. Theorem 2: The proposed auction-based mechanism is individually rational.
Proof: According to (10), if user i is not selected as a winner, its payoff is zero, otherwise its claimed cost should be lower than or equal to the critical value. Moreover, this auction-based mechanism is proved to be truthful, user i could report its true cost. Hence, winner i's utility p i − c i is nonnegative. Assume that there are two arbitrary user sets S ⊂ G ⊆ and η 0 = arg max η∈G\S where b(S) = g∈S b g . By utilizing (14), we can obtain the following lemma. Lemma 2: For the winner r ∈ S, its critical payment p r has a upper bound ν r (S)B ν(S) . The proof can be found in Appendix D. Theorem 4: The auction-based mechanism is budget feasible.
Proof: By using Lemma 3, we have Hence, the sum of the payment to all winners does not exceed the budget.

VII. NUMERICAL SIMULATIONS
This section presents the simulation results of our proposed mechanisms. The basic incentive mechanism is made comparison with the following schemes: (i) Random selection algorithm: in each round, users are randomly added to the candidate user set with satisfying the budget constraint; (ii) Maximum sensing quality (MSQ) algorithm: at each iteration, the user who maximizes sensing quality is selected, where the budget constraint is satisfied. However, in the basic mechanism, a user who maximizes the ratio between the increased sensing quality and the cost will be added to the candidate user set. The auction-based mechanism is compared against the following two benchmark schemes: Greedy-SM and Random-SM [35].

A. SIMULATION SETTINGS
Assuming that MUs' unit sensing cost c s and unit privacy cost c p are randomly distributed within the range (0, 1)$/m and the trajectory distance d for each user is normalized into a scale of 0-10m. The weighs for different PoIs are randomly distributed over (0, 1) and the value of the platform's valuation parameter µ is fixed as 1000$. Different types of q(α) (0 < α < 1) are adopted to study the impact of privacy protection on sensing quality, in which is randomly distributed within the range (0, 1). Meanwhile, we assume that a PoI is visited by the same user at most twice. MUs' number m is varied from 200 to 1000 with the increment of 200 and the budget B is within the range from 0 to 300$. Fig. 3, the impact of the budget on sensing quality is investigated. We observe that as the budget  increases, the proposed sensing quality metric satisfies the law of diminishing returns. Meanwhile, our proposed algorithm can achieve a higher sensing quality as compared to the random selection algorithm and MSQ algorithm. Due to the simulation setting, when the budget is greater than a threshold, sensing quality of our proposed algorithm is same to that of the random selection algorithm.

As shown in
In Fig. 4, we fix MUs' number m = 200. It is notable that the platform's valuation increases with the increasing budget and satisfies the diminishing returns rule. As shown in Algorithm 2, at the phase of winner selection, the user with the largest marginal contribution per bid is selected, hence our proposed mechanism can achieve a higher valuation in comparison with the baseline schemes. Fig. 5 shows the impact of m on the valuation of the platform, in which we fix B = 100$. It is found that the valuation indeed demonstrates diminishing returns as m increases. At the same time, our proposed auction-based mechanism outperforms the baseline schemes since the user with a larger marginal contribution per bid is chosen.
As illustrated in Fig. 6, our proposed auction-based mechanism is budget feasible by investigating users' total payment given different amount of budget. We observe that the total payment increases with the increasing users since more users are selected to perform the task. In Fig. 7, we note that the valuation is impacted by different types of q(α). Based on Definition 2, the larger α leads to lower accuracy achieved by the aggregation algorithm. Hence, the valuation of the platform's decreases with the increasing value of α.

VIII. EXPERIMENT RESULTS
In this section, the performance of the basic incentive mechanism is evaluated on UJIIndoorLoc database [36]. This database contains wireless local area network (WLAN) RSS measurements of three buildings of Universitat Jaume I with four or more floors, which is often utilized to evaluate indoor localization solutions that depend on RSS fingerprints.

A. EXPERIMENT SETTINGS
In this experiment, the location data of the first floor of the first building is adopted to validate the proposed mechanism, where the physical coordinate or location is comprised by the longitude and the latitude in the dataset. 5 Considering that the indoor scenario is roughly divided into subareas based on the signal coverage of access points (APs) deployed in the building. Hence, APs can be utilized to denote the mentioned PoIs in Section III. Here, the assumption is that if the RSS reading of an AP in a POI can be detected, it means that its corresponding PoI is covered by users. As mentioned in the previous sections, the mechanism performance is greatly influenced by privacy protection achieved by adding noise to raw data. Moreover, the study [13] has shown that the Laplace mechanism can be used to achieve -local differential privacy. Therefore, the noise drawn from a Laplace distribution with zero mean is added to the training data to investigate the impact of privacy protection, where the standard deviation σ is approximately proportional to 1/ . Moreover, k-nearest neighbors (KNN) [37] is adopted to evaluate the performance, where our proposed basic mechanism is compared with the random selection algorithm. Meanwhile, the total cost of each candidate user is distributed over (1, 10)$ and the number of candidate users is fixed as m = 20 and m = 30, respectively. Moreover, the average indoor localization error of 50 testing data is adopted as the metric to evaluate the system performance.

B. RESULTS
In the KNN solution, k neighbors of the target are selected based on Euclidean distance or other distances to estimate the target location. Fig. 8 investigates the impact of k on the localization error. It is noted that when k = 3, both the random selection algorithm and our proposed basic incentive mechanism can achieve a lower localization error compared to other values of k. Meanwhile, the localization error of our proposed incentive mechanism is smaller than that of the random selection algorithm. Fig. 9 studies the effect of the limited budget on different mechanisms under different user number, respectively, where k = 3. As shown in Fig. 9, on one hand, the localization error becomes lower with the increasing budget; on the other hand, the localization error decreases with the increasing user number. Compared to the random selection algorithm, the proposed basic incentive mechanism can achieve a lower localization error. Moreover, this experiment result is found in line with the numerical simulation result. Fig. 10 demonstrates the localization error under different σ , in which σ denotes the standard deviation of the Laplace distribution. The studies [13], [38] have shown that with larger σ , larger noise has to be added to users' data such  that higher privacy protection level can be achieved which increases the localization error.

IX. CONCLUSION AND FUTURE WORK
In this study, a novel metric of sensing quality for MCSbased RSS fingerprint collection was proposed to characterize the joint impact of privacy protection and spatial coverage. In the basic incentive mechanism, a greedy-based algorithm was proposed to obtain an approximation solution for the sensing quality maximization problem under a constrained budget. Aiming to achieve the truthfulness of users' private costs, an auction-based incentive mechanism was proposed and theoretically proved to meet the predefined properties. Simulation results verified that our proposed mechanisms outperformed the baseline schemes. Moreover, an experiment based on real-world data was carried out to evaluate the proposed basic incentive mechanism.
In the future, the P2P or D2D networking paradigm [39] would be used as a cost and energy effective way to provide the spatial coverage for crowdsourcing fingerprint collection.
According to Definition 4, to prove that the sensing quality function is submodular, we need to demonstrate that By leveraging MUs' cover sets, |X S 1 ∪{u} | − |X S 1 | denotes the amount of covered PoIs in MU u's cover set X u and not in the set X S 1 given a MU u ∈ \ S 2 . It is found that X S 1 ∪u − X u ≥ X S 2 ∪u − X u . Hence, it suffices to show that the inequality (15) can hold. Moreover, the marginal sensing quality from adding an additional user is non-negative which demonstrates that the objective function satisfies the law of diminishing returns.
To summarize, we have proved that the objective function is non-decreasing submodular.

APPENDIX B PROOF FOR APPROXIMATION RATIO OF ALGORITHM 1
Proof: The study [31] has shown that: (1) the candidate solution with the largest weight is selected as the approximation solution, whose associated cost does not exceed the limited budget; (2) the value of coverage quality achieved by the approximation solution is at least 1 − 1/e times greater than the optimal value of the coverage quality achieved by arbitrary subset. In this context, we have reduced our problem to the classical budgeted maximum coverage problem as well as captured the relation between these two problems. Therefore, for τ ≥ 3, the approximation factor achieved by our designed algorithm is greater than (1 − 1/e).

APPENDIX C PROOF FOR THEOREM 1
Proof: To prove the truthfulness of our proposed auction-based mechanism, it requires to show that our proposed mechanism meets Theorem 1. On one hand, base on marginal contribution per bid, user i who submits a lower bid will be sorted in the same or a prior position; on the other hand, the platform's valuation function is monotone. Hence, the monotonicity can be maintained.
Next, we show that the payment for each winner is the critical value. In the payment determination phase, users are essentially sorted according to their marginal contribution per bid. For user i, if b i > p i , it will be pushed backward after the position of the last user i j ∈ −i . Since its marginal contribution per bid is lower than that N th user, it will not be selected after the (N + 1) th iteration. If b i < p i , it is selected as a winner according to the winner selection rule.
In summary, our designed auction mechanism is truthful.

APPENDIX D PROOF FOR LEMMA 2
Proof: First, consider that ∅ = S ⊂, . . . , ⊂ S k ⊆ , due to the submodularity of the platform's valuation function, we have ν(S) b(S) ≥ ν(S k ) B . Then, for contradiction, we assume that user r ∈ S k reports a bid b r > ν r (S)B ν(S k ) and it still can be selected as a winner. Before user r is added to the winner set, we have r = arg max i∈ \S .
We make an assumption that S ∪ {r} S k ∪ S. If not we have S ∪ {r} = S k ∪ S and then Therefore, we have b r ≤ ν r (S)B ν(S k ) and get a contradiction. Then, we set M = S k \S and incorporate r and Lemma 2 into S k ∪ S and S ∪ {r}, respectively. For g 0 ∈ M\{r}, according to Lemma 2, we have Meanwhile, we have obtained that b r > ν r (S)B ν(S k ) and thus we have ν r (S) b r < ν(S k ) B . Combined with (17), we can get According to the inequality (17), for j ∈ [k], we have Besides, it is proved that winning user truthfully reports its bid. Therefore, we could = B/2. Based on these analytics, we can have As a result, we have ν(S k ) < 2ν(S ∪ {r}). Besides, we know that ν r (S)