Overlapping Coalition Formation Game via Multi-Objective Optimization for Crowdsensing Task Allocation

Abstract

With the rapid development of sensor technology and mobile services, the service model of mobile crowd sensing (MCS) has emerged. In this model, user groups perceive data through carried mobile terminal devices, thereby completing large-scale and distributed tasks. Task allocation is an important link in MCS, but the interests of task publishers, users, and platforms often conflict. Therefore, to improve the performance of MCS task allocation, this study proposes a repeated overlapping coalition formation game MCS task allocation scheme based on multiple-objective particle swarm optimization (ROCG-MOPSO). The overlapping coalition formation (OCF) game model is used to describe the resource allocation relationship between users and tasks, and design two game strategies, allowing users to form overlapping coalitions for different sensing tasks. Multi-objective optimization, on the other hand, is a strategy that considers multiple interests simultaneously in optimization problems. Therefore, we use the multi-objective particle swarm optimization algorithm to adjust the parameters of the OCF to better balance the interests of task publishers, users, and platforms and thus obtain a more optimal task allocation scheme. To verify the effectiveness of ROCG-MOPSO, we conduct experiments on a dataset and compare the results with the schemes in the related literature. The experimental results show that our ROCG-MOPSO performs superiorly on key performance indicators such as average user revenue, platform revenue, task completion rate, and user average surplus resources.

1. Introduction

Due to the rapid advancements in wireless communication technologies and the Internet of Things, smart terminal devices such as mobile phones are increasingly used in daily life [1,2]. With an array of embedded sensors, including gyroscopes, light sensors, the global positioning system (GPS), and cameras, mobile phones can not only serve as communication devices but also sense their surroundings and collect data at any time [3]. This has led to the emergence of the mobile crowd sensing (MCS) service model. The MCS platform recruits users with mobile smart terminals and assigns them tasks. These users perform the tasks and then deliver the collected sensing data to the MCS platform. Compared to other sensing models, MCS can significantly reduce sensing and maintenance costs and leverage the “strength in numbers” feature to carry out tasks and collect data in a defined area with high accuracy [4,5]. As a result, the MCS service model has been increasingly utilized in various fields, including traffic detection [6,7], environmental monitoring [8,9], social networking [10,11], and healthcare [12,13].

In numerous studies on task allocation in MCS, users are typically assumed to be able to complete only one task [14]. However, with the rapid development of mobile sensing technology, the demands of MCS on users are gradually becoming more complex, often requiring them to participate in multiple tasks simultaneously [15]. In practical MCS scenarios, users are expected to effectively distribute their resources across multiple tasks to meet the resource demands of task execution. In such situations, research methods that solely focus on individual users and tasks are clearly insufficient to meet these complex needs. Therefore, this study aims to address this problem by proposing a novel task allocation scheme. This scheme is designed to effectively handle multi-user and multi-task scenarios in MCS, thereby enhancing the overall efficiency and performance of the system.

In the early research on MCS, smartphone users were generally treated as volunteers of the system [16]. However, due to the self-interested nature of users, the quality of task completion could not be guaranteed. Moreover, as the platform may acquire users’ geographical location data, this increases the risk to privacy security. These factors have led to users’ reluctance to participate in sensing activities without compensation, reflecting the necessity for incentives in the form of rewards to encourage user participation in task execution [17]. This means that there is an interest game relationship between the platform and users, that is, in order to maximize their own interests, both users and the platform will make a series of decisions and finally achieve an acceptable result [18]. However, this situation is not conducive to the long-term cooperation and development of the platform and its users.

To address the above issues, this paper introduces a crowd sensing task allocation scheme based on OCF games. In this model, each task can be assigned to multiple users, and participants of the same task form a coalition. Consequently, the coalitions of different tasks may overlap, forming an OCF game between users and tasks. The OCF theory permits users to participate in multiple coalitions, thus more accurately simulating the multi-task environment in reality. This adjustment aims to better reflect how users share resources and allocate tasks through cooperation and competition in complex multi-task environments, thereby more accurately describing the process where users allocate resources across multiple tasks and achieve more balanced benefits.

MCS task allocation schemes need to consider the interests of three parties: users seek to gain more returns for the resources they invest, the platform aims to regulate the price of resources to generate more revenue, and task publishers hope their tasks will be completed more effectively. If only the interests of one party are considered, neglecting those of others, it may lead to an imbalance in utility, thereby reducing the overall benefit of the MCS system. To address this issue, we employ a multi-objective optimization algorithm to adjust the parameters in the overlapping coalition game, aiming to obtain the optimal task allocation scheme under the premise of a balanced tripartite.

This paper proposes a ROCG-MOPSO scheme. We will outline the main contributions and contents of this research:

• We introduce the OCF game model to accurately depict the resource allocation relationship between users and tasks, and design two game strategies to optimize the coalition formation for sensing tasks. This method can utilize resources more efficiently and obtain superior task allocation schemes.
• We adopt the multiple-objective particle swarm optimization (MOPSO) algorithm to compute the optimal values of parameters in the overlapping coalition game, thereby harmonizing the objectives of task publishers, users, and the platform. This optimized model allows us to achieve superior task allocation schemes.
• To validate the effectiveness of the proposed method, we conduct experiments and compare the findings with schemes from related literature. The experimental results affirm that our scheme enhances the task completion rate and allows both users and the platform to reap more benefits.

The remainder of this paper is organized as follows. Section 2 reviews related research work and proposes the problems this paper aims to solve. Section 3 establishes the OCF game model for MCS task allocation, defines the related indexes and parameters of the model, and describes the task allocation process through two game strategies. Section 4 establishes the evaluation metrics for the task allocation results of the OCF game and the multi-objective optimization model. Section 5 introduces the MOPSO algorithm to balance the objectives of the three parties, and by determining the optimal parameters of the task allocation model, a superior task allocation scheme is obtained. Section 6 validates the effectiveness of the task allocation model proposed in this paper through simulation experiments. Section 7 summarizes the contents of this paper’s research and discusses future work.

2. Related Work

The key to MCS lies in assigning appropriate tasks to selected users under the constraints of various factors such as cost, energy consumption, and task completion time [19]. MCS task allocation can be divided into the worker selected tasks (WST) model and the server assigned tasks (SAT) model [20]. For the SAT model, [21] proposes a two-step strategy incentive mechanism in which the platform is responsible for distributing tasks and the user accepts them. The platform influences the length of time a user spends completing a task by controlling the total payment to the user. For the WST model, ref. [22] devises a distributed computing scheme where all participants of the same task receive the same payout and uses a pairwise decomposition approach to solve the platform revenue maximisation problem.

MCS task allocation can be achieved through game-theoretic approaches.

Study [23] introduces a Stackelberg game to construct the interaction between the server and users. While ensuring task completion, the method maximizes the utility of both the server and users. However, this scheme is difficult to apply in multi-task scenarios. Study [24] proposes a multi-strategy repeated game task allocation mechanism that provides high-quality services to users and ensures the authenticity of user data without manipulating their bids. Another game-theoretic approach introduced in [25] is the worker decision game based on congestion game theory, considering the competitive behavior among users. This effectively improves user satisfaction and platform benefits.

These game mechanisms, when addressing task allocation issues, often fail to effectively model the cooperation and resource allocation relationships among users. Therefore, many researchers have turned to coalition game theory as a tool to solve task allocation problems. However, most of the existing studies, when using coalition formation games, usually assume that a user can only join one coalition [26]. This is clearly significantly different from the situation in real-world MCS systems where users can participate in multiple sensing tasks. Therefore, OCF theory is more suitable for describing cooperation and distribution relationships under multi-task modes.

The theoretical framework of OCF has been extensively employed in task allocation studies across various application domains. Study [27] establishes a complex yet flexible overlapping coalition structure among vehicle fog nodes through OCF games. This effectively enhances computational efficiency and flexibility in task allocation, thereby maximizing the benefits users derive from vehicles. To obtain a better balance between task benefits and task costs during task allocation, ref. [28] proposes a sequential OCF game method, enabling drones to decide the number of resources allocated to each task. This makes for a better balance of task benefits and task costs during task allocation. Study [29] represents the task selection problem in smartphone collaborative sensing as an OCF game and proposes a distributed OCF algorithm aiming to maximize the profit of smartphone users. These studies explore the utilization of OCF theoretical and methodological approaches to optimize task allocation and demonstrate the broad application of OCF theory in task allocation, providing a robust theoretical foundation for further research.

Despite existing research effectively optimizing task allocation issues, most studies examine the problem from a singular perspective, thereby lacking a thorough consideration of the global viewpoint. These studies either focus solely on single-task scenarios or consider the interests of only one party—the platform, the users, or the task publishers—in multi-task scenarios. This approach, to some extent, restricts the comprehensive consideration and optimization of global benefits in MCS task allocation. There is an evident need for future research to adopt a more holistic approach, taking into account the interests of all parties involved in task allocation.

To address the aforementioned limitations, this study proposes the incorporation of the OCF to depict the interaction between users and platforms in multi-task scenarios, thus optimizing task allocation. Additionally, this study aims to consider the interests of task publishers, the platform, and users in a comprehensive manner. The research intends to introduce the MOPSO algorithm to select the optimal parameters of the OCF game. Through the implementation of this approach, it is anticipated that task allocation results can be further optimized. This, in turn, is expected to enhance overall benefits in situations where the interests of the three stakeholders—task publishers, the platform, and users—are balanced. This balanced strategy for task allocation not only optimizes individual outcomes but also significantly enhances the overall effectiveness and efficiency of the system, thereby maximizing general benefits.

3. MCS Task Allocation via OCF Game

3.1. Crowdsensing Task Allocation Architecture

The architecture of the crowdsensing system is shown in Figure 1. It includes three roles: task publisher, platform, and smart phone user.

Table 1. Parameters of the crowdsensing task allocation architecture.

Parameters	Meaning
$T=\{t_1,\dots ,t_i,\dots ,t_n\}$	Task set
$F=\{f_1,\dots ,f_i,\dots ,f_n\}$	Task minimum demand resource set
$G=\{g_1,\dots ,g_i,\dots ,g_n\}$	Task unit resource price set
$U=\{u_1^0,\dots ,u_i^0,\dots ,u_n^0\}$	Task minimum revenue set
$B=\{b_1,\dots ,b_j,\dots ,b_m\}$	Task completion result set
$P=\{p_1,\dots ,p_j,\dots ,p_m\}$	User set
$C=\{c_1,\dots ,c_j,\dots ,c_m\}$	User revenue penalty set
$A=\{a_1,\dots ,a_j,\dots ,a_m\}$	User resource set
$H=\{h_1,\dots ,h_j,\dots ,h_m\}$	User revenue set

lists the relevant symbols and their meanings for the crowdsensing task allocation model. Task publishers submit a set of tasks $T=\{t_1,\dots ,t_i,\dots ,t_n\}$ to the platform. The execution of the task requires the input of resources, the unit price set of which is $F=\{f_1,\dots ,f_i,\dots ,f_n\}$. The benefits set to be gained by completing each task is $U=\{u_1^0,\dots ,u_i^0,\dots ,u_n^0\}$. A set of smartphone users $P=\{p_1,\dots ,p_j,\dots ,p_m\}$ register on the platform, and it has a certain set of resources $A=\{a_1,\dots ,a_j,\dots ,a_m\}$. To simplify the model, the number of resources is an integer. Users submit the tasks they want to perform based on the task information published by the platform. Based on the task information submitted by the user and the task information they want to complete, the platform calculates the optimal task allocation scheme and assigns the task to the user. Users are able to earn a set of revenue $H=\{h_1,\dots ,h_j,\dots ,h_m\}$ from the platform after completing tasks, and the platform will also earn revenue K and return the results set of the task completion $B=\{b_1,\dots ,b_j,\dots ,b_m\}$ to the task publisher.

3.2. Build OCF Games Relationship between Users and Task

Task allocation involves the process of users dedicating their resources to perform a task. Therefore, the relationship between users and tasks constitutes an OCF game through resource allocation. Each user has a certain number of resources, and a task can be completed by multiple users. As shown in Figure 2, users $p_1$, $p_2$, and $p_3$ contributed their resources to accomplish task $t_1$, thereby forming a coalition. Similarly, users $p_3$, $p_4$, and $p_5$ also allocated their resources to carry out task $t_2$, resulting in the formation of another coalition. Notably, due to the participation of $p_3$, an overlap exists between these two coalitions. In the same vein, $p_5$ has become the overlapping part between the coalition in $(p_3,p_4,p_5)$ and the coalition in $(p_5,p_6,p_7)$. The OCF game relationship can be represented by the OCF game matrix of the Formula (1). Users can invest different numbers of resources in different tasks as desired, so they can participate in different coalitions. In multiple rounds of a game, users constantly adjust the input of resources in each task and finally achieve an optimized task resource allocation state, to obtain the optimized task allocation scheme. When the user completes the task assigned to them, the relevant information of the task is uploaded to the platform. Then, the platform sends the task completion to the publisher.

$$R^k=\left[\begin{array}{ccc}{r}_{11}^k& \dots & r_{1m}^k\\ ⋮& \ddots & ⋮\\ r_{n1}^k& \dots & r_{nm}^k\end{array}\right]$$

(1)

where k denotes the number of game rounds in the OCF game matrix. Row i of the matrix represents the task $t_i$, column j of the matrix represents the user $p_j$, and $r_{ij}^k(0⩽r_{ij}^k⩽r_j)$ denotes the resources allocated to the task by the user. $r_{ij}$ = 0 means that the user $p_j$ is not a member of the task $t_i$ game coalition in round k. Two game strategies are used in this paper, namely, the task’s attractiveness index game strategy for users and the unit resource price adjustment strategy. In each round of the game, users use these two game strategies to readjust the resources allocated to the task, optimising the resource allocation by changing the overlapping coalition game matrix. As the OCF game progresses through several rounds of resource allocation adjustments, the allocation gradually approaches an optimal level, resulting in optimal task allocation. This section may be divided by subheadings. It should provide a concise and precise description of the experimental results, their interpretation, and the experimental conclusions that can be drawn.

3.3. The Task Allocation Process of OCF Game

The OCF game matrix is updated by means of two game strategies: the attractiveness index and the adjustment of the price per unit of resources invested in the task. By repeatedly adjusting the resource allocation through several rounds of the game, the optimized resource allocation results are achieved, leading to optimal task allocation. This process of task allocation is represented in this paper as the repeated overlapping coalition formation game (ROCG), which converges to a stable overlapping coalition structure (OCS).

3.4. The Process of the ROCG

1

Initializtion

The initial value of the number of rounds in the OCF game is 1, and the maximum number is q. Initialize the attractiveness index matrix $E^1$, which is the attractiveness index matrix for the first round of the game. The rows of the matrix represent tasks, and the columns represent users. Set the initial $e_{ij}^1$ value to the same value, indicating that the tasks are equally attractive to the users.

$$E^1=\left[\begin{array}{ccc}{\mathit{e}}_{11}^1& \dots & {\mathit{e}}_{1m}^1\\ ⋮& \ddots & ⋮\\ {\mathit{e}}_{n1}^1& \dots & {\mathit{e}}_{nm}^1\end{array}\right]$$

(2)

2

Determine the Attractiveness Index

The attractiveness index matrix is updated in each round of the game using the attractiveness index adjustment strategy. The task attractiveness index is used to describe the extent to which users prefer to devote resources to which task. The resources required to complete a task and the price per resource of the task are two factors that influence the attractiveness of a task. On the one hand, users tend to allocate resources to tasks with a higher resource price per unit in order to gain more revenue. On the other hand, users also tend to invest resources in tasks that are easier to accomplish. The tasks require fewer resources left to complete, and users can avoid penalties for not completing them. Thus, the task attractiveness index $e_{ij}^k$ of the task $t_i$ to the user $p_j$ during the k round of the game is calculated in Formula (3).

$$e_{ij}^k=\left\{\begin{array}{ll}(1-\omega )\ast g_i+\omega \ast \sum _{\mathrm{i}=1}^{\mathrm{m}}{f}_{\mathrm{i}}\left(f_i-d_{ij}^k\right)& f_i-d_i^k>0\\ (1-\omega )\ast g_i& f_i-d_i^k\le 0\end{array}\right.$$

(3)

where $\omega \in (0,1)$ is the weight used to measure the extent to which users tend to devote resources to the task to be completed easily. $1-\omega$ is used to measure the extent to which users tend to devote resources to tasks with a higher price per resource. $d_{ij}^k$ is the number of resources already owned by the user $p_j$ at the time of assigning resources to the task $t_i$, and $d_i^k$ is the number of resources currently owned by the task $t_i$. Then,

$$d_{ij}^k=\sum _{\mathrm{p}=1}^{\mathrm{j}-1}{r}_{ip}^k,1\le i\le n$$

(4)

$$d_i^k=\sum _{p=1}^{\mathrm{m}}{r}_{ip}^k,1\le i\le n$$

(5)

When $f_i-d_i^k>0$, it means that in the current k round of the game, the task has a smaller number of resources than its minimum requirement resource. Thus, the task $t_i$ attractiveness index $e_{ij}^k$ includes both the resources required to complete the task and the price per resource. When $f_i-d_i^k\le 0$, it means that in the current k round of the game, the number of resources owned by the task $t_i$ is at or above its minimum required resources. Therefore, the task attractiveness index $e_{ij}^k$ no longer includes the factor of resources required for task completion. The attractiveness index forms the task attractiveness matrix $E^k$ in Formula (6).

$$E^{\mathrm{k}}=\left[\begin{array}{ccc}{e}_{11}^k& \dots & e_{1m}^k\\ ⋮& \ddots & ⋮\\ e_{n1}^k& \dots & e_{nm}^k\end{array}\right]$$

(6)

If $e_{ij}^{\mathit{k}}<\Gamma \ast {\sum }_{\mathrm{i}=1}^{\mathrm{n}}{e}_{ij}^{\mathit{k}}/n$, then $e_{ij}^{\mathit{k}}=0$. When the attraction of the task is too small compared to other tasks, the user will ignore the task and will not put resources into it, and then the attraction of the task can be regarded as 0. And $\Gamma <1$ is a factor used to measure coalition range rate.

3

Attractiveness index to adjust resource input strategy

In the course of the k round of the game, user $p_j$ determines the number of resources $r_{ij}^k$ to be invested in the task $t_i$ based on the task attractiveness index $e_{ij}^k$. It should be noted that if the resources invested by user $p_j$ in the previous round are withdrawn and this leads to the failure of task $t_i$, then the user cannot withdraw these resources and is unable to exit the coalition. Therefore, the number of resources $a_j$ that the user can allocate in this round is reduced ($a_j=a_j-r_{ij}^{k-1}$). $r_{ij}^k$ can be determined by Formula (7), which leads to the resource input matrix $R^k$ (also known as the coalition matrix).

$$r_{ij}^k=a_j\ast e_{ij}^{\mathit{k}}/\sum _{\mathrm{i}=1}^{\mathrm{n}}{e}_{ij}^{\mathit{k}}$$

(7)

In our model, the variations in resource contributions from user i serve as an indicator of the user’s membership status within the coalition. To elaborate, if $r_{ij}^{k-1}\ne 0$ and $r_{ij}^k=0$, we interpret this as user $p_j$ exiting the coalition. Conversely, if $r_{ij}^{k-1}=0$ and $r_{ij}^k\ne 0$, we interpret this as user $p_j$ joining the coalition. Participatory or withdrawal actions result in changes within the organizational coalition structure (OCS). Following each iteration, if no alterations in coalition membership are observed, the OCS is deemed stable. This state of stability serves as a pivotal indicator of convergence within our model, suggesting that the actions of coalition members have attained an equilibrium state, characterized by an absence of new user entries or existing user exits. This, in turn, suggests that a stable condition has been achieved within the OCS.

4

Unit resource price adjustment strategy

The aim of using the unit resource price adjustment strategy is to change the unit resource price of the task. Thus, the task attraction in the next round of the game can be changed, and finally the adjustment of resources can be further optimized to obtain a better coalition matrix. The unit resource price of the task is adjusted in the k round of the game using Formula (8), where $g_i^0$ is the initial unit resource price of the task $t_i$. Adjust the task unit resource price $g_i$ based on the value of $f_i-d_i^k$. When the task $t_i$ has fewer resources, i.e., $f_i>d_i^k$, increase $g_i$. And when the task $t_i$ has excess resources, i.e., $f_i<d_i^k$, decrease $g_i$.

$$g_i^{\mathrm{k}}=\left\{\begin{array}{c}\alpha \ast g_i^0\ast arctan\left(f_i/d_i^k\right)\ast \frac{4}{\pi }{f}_i>d_i^k\mathrm{and}{d}_i^k>0\\ 2\ast g_i^0{f}_i>d_i^k\mathrm{and}{d}_i^k=0\\ \beta \ast g_i^0\ast sin\left(\frac{f_i}{d_i^k}\right)f_i<d_i^k\end{array}\right.$$

(8)

where $\alpha$$\in (0,1)$ and $\beta$$\in (0,1)$ are the weights. $\alpha$ denotes the price weight when resources are lacking, and $\beta$ denotes the price weight when resources are in excess. The adjustment with $\alpha$ and $\beta$ is related to the demand for the task completion rate. When task completion rates are higher, $\alpha$ can be increased and $\beta$ reduced, making users more inclined to unfinished tasks. When a higher task completion rate is not required, $\alpha$ can be lowered and $\beta$ increased.

In this model, the trigonometric functions “arctan” and “sin” are utilized to simulate the relationship between unit resource price and task resource. In the k-th round, when the $d_i^k$ is less than the $f_i$, the task $t_i$ is incomplete, and the platform earns nothing from this task $t_i$. In this instance, the “arctan” function is used to reflect the increase in the $g_i^{\mathrm{k}}$. The characteristics of the “arctan” function result in a larger increase in $g_i^{\mathrm{k}}$ when the resource is less, and a gradual decrease in this increase as it approaches the $f_i$, reflecting the impact of resource scarcity on price. On the other hand, when the $d_i^k$ exceeds the $f_i$, its contribution to task quality decreases. The growth rate of task revenue is significantly lower than the growth rate of resources, leading to a decrease in $g_i^{\mathrm{k}}$. This situation is represented by the “sin” function, effectively depicting the process of resource over-investment, the diminishing contribution to task quality, and the subsequent decrease in $g_i^{\mathrm{k}}$. In summary, the “arctan” and “sin” functions in the model accurately portray how the platform adjusts the $g_i^{\mathrm{k}}$ based on $d_i^k$, thereby revealing the complex nonlinear relationship between $g_i^{\mathrm{k}}$ and $d_i^k$ in the k-th round.

5

Repeat the game process

If the number of games has not reached the set maximum value q or all members of the coalition will not quit the current coalition or join other new coalitions, the next game process continues.

6

The final result of the game

The final coalition matrix $R^q$ and the unit resource price matrix $G^q$ are obtained at the end of the game. And the results of the task allocation are available by $R^q$.

3.5. The ROCG Task Assignment Algorithm

The steps of the above-described ROCG task allocation process are shown in Algorithm 1.

Algorithm 1: ROCG algorithm

Input:  Task minimum demand resource set F; task initial unit resource price set G; task minimum utility set U; user resource set A; weight parameter $\omega$ Output:  The OCF game matrix $R^q$, and task unit resource price set $G^q$   1: Initialize $E^0$, $k=0$   2: while $k<q$ do   3:   for each $p_j\in P$ do   4:    for each $T_i\in T$ do   5:     if $f_i<d_i^{k-1}$ and $d_i^{k-1}-r_{ij}^{k-1}<f_i$ then   6:      $a_j=a_j-r_{ij}^{k-1}$   7:     end if   8:    end for   9:    for each $T_j\in T$ do 10:      Calculate the $r_{ij}^k$ according to Formula (7) and assign the resources to the task $t_i$ 11:    end for 12:   end for 13:   for each $T_j\in T$ do 14:     Calculate $e_{ij}^k$ according to Formula (3) 15:   end for 16:   Adjust the resource unit price according to Formula (8) 17:   $k=k+1$ 18: end while 19: return $R^q$,$G^q$

Lemma 1.

The ROCG algorithm converges to a stable coalition structure after a finite number of iterations.

Proof. 

Initially, we observe that the quantity of users, tasks, and resources are finite in each task allocation process. Furthermore, users allocate their resources in a discrete manner, which implies that the total number of possible OCS is also finite.

After each round of the game, we optimize the task allocation results through game strategy. Let the initial OCS be $C^0$, which forms after the initiation of user strategies. We represent the changes in OCS as $\{C^1,C^2,\dots ,C^k\}$.

According to Formula (7), a new coalition structure of task allocation results forms after each change. Additionally, a user may only exit a coalition if their withdrawal does not affect the task completion status. Therefore, when $C^k$ is the last coalition structure obtained through game strategy, we can consider $C^k$ as stable.

Assume $C^k$ is unstable; then, users would have the potential to exit the coalition or join others, and re-allocate their resources, which would alter the coalition structure $C^k$. This contradicts our previous observation that $C^k$ is the last coalition structure obtained through game strategy and that users cannot exit the coalition. Thus, we conclude that the ROCG algorithm will converge to a stable coalition structure OCS after a finite number of iterations.

Proven. □

4. The OCF Game via Multi-Objective Optimization

The OCF game model can provide a preferred task allocation scheme. However, the outcome of the game is impacted by the parameter $\omega$ in Formula (3). If we can choose better $\omega$ parameters on the basis of balancing the task publisher, platform, and user tripartite interests, we can obtain a better overlapping coalition game matrix $R^q$ to obtain better task assignment results. To find the optimal parameter $\omega$ of the OCF game, the MOPSO can be utilized. After $R^q$ and $G^q$ are obtained in the OCF game, they can be used to calculate the platform revenue, the average user revenue, and the task completion rate. These metrics are used to evaluate the effectiveness of task allocation in the OCF game and thus determine the degree of merit of the parameters. These metrics can be used as the fitness function of a multi-objective particle swarm to solve for the optimal parameters $\omega$.

4.1. Task Completion Rate

In the OCF game, the task completion rate is an important metric that measures the proportion of completed tasks to the total number of tasks posted on the platform. A higher completion rate increases the willingness of task publishers to post more tasks. A task is considered completed if the number of resources invested in it is greater than or equal to its minimum resource requirement. Conversely, if the amount invested is less than the minimum requirement, the task is not completed. When $f_i\le {\sum }_{p=1}^m{r}_{ip}$, this indicates that the task is complete. When $f_i>{\sum }_{p=1}^m{r}_{ip}$, this indicates that the task is not completed, then $b_i=1$. From this, the total number of completed tasks z can be obtained. The task completion rate is calculated from Formula (9).

$$L=\frac{z}{n}$$

(9)

4.2. User Average Revenue

4.2.1. Task Failure Punishment

In a game, when the choice of each member of the group favours individual interests over the interests of the group, it will eventually lead to the interests of the group members (including the chooser himself) being compromised, which is a social dilemma. The prisoner’s dilemma [30] is a classic example of such a social dilemma, which can decrease users’ willingness to engage in a given task. To motivate users to consider the group’s interests and perform the task better, a penalty should be imposed when a task is not completed. Additionally, when a task cannot be completed, the platform is then required to pay a fee without the benefit of the task. This can reduce the platform’s willingness to accept tasks for publication, in order to achieve a better allocation of resources and to ensure the benefits for all three parties. In this paper’s OCF game task allocation model, when a task is not eventually completed, the users who perform the task coalition are penalised by a reduction in their revenue. When the task $t_i$ is not completed, the failure penalty of user $p_j$ is calculated by Formula (10).

$$c_j=\zeta \ast r_{ij}\ast g_j\ast b_i$$

(10)

In order to attract users to participate in performing the task, users cannot lose money even if the task execution fails, so $\zeta$ is set to a proportion factor less than 1.

4.2.2. User Revenue

To compensate for the cost incurred by users in executing tasks, users require a certain return. In this study, instead of rewarding individual users, the platform rewards coalitions based on the number of resources they provide for the corresponding tasks. Subsequently, the coalition allocates rewards according to the resources furnished by the users to the coalition. The revenue $h_j^0$ for the user $p_j$ is is the total amount of resource revenue received in all task coaliation:

$$h_j^0=\sum _{i=1}^n{r}_{ij}^{}\ast g_j$$

(11)

Considering user in the failed task coalition, the user revenue needs to deduct the share of the revenue from the failure penalty, and the user’s revenue $h_j$ is:

$$h_j=h_j^0-c_j=\sum _{i=1}^n\left(1-\zeta \ast b_i\right)\ast r_{ij}\ast g_j$$

(12)

4.2.3. User Average Revenue

H represents the average revenue of all users, calculated by Formula (13). The higher average revenue can promote users to continue to participate in perceptual tasks.

$$H=\sum _{j=1}^m{h}_j/m$$

(13)

4.3. Platform Revenue

The revenue $u_i$ is used to assess the revenue that the platform receives from task $t_i$, which can be calculated by Formula (14). And $\theta <1$ is a factor used to measure resource conversion rates. When the resources $d_i$ invested by users in the task $t_i$ are less than the minimum required resources $f_i$, i.e., $f_i>d_i$, the task is considered incomplete and the platform revenue is 0. However, when $f_i<d_i$, the new resources invested are contributing less to the quality of the task $t_i$, so the rate of increase in task gain is significantly lower than the rate of increase in task resources.

$$u_i=u_i^0\left(1+\theta \left({\log }_{f_i}\max \left(f_i,d_i\right)-1\right)\right)$$

(14)

The platform revenue K is crucial to the sustainability of the platform, and it is calculated as the revenue obtained from the task publisher upon task completion minus the revenue given to all the task coalition in Formula (15). Thus, higher platform revenue is the key to keeping the platform running.

$$K=\sum _{i=1}^n{u}_i-\sum _{j=1}^m{h}_j=\sum _{i=1}^n{u}_i^0\left(1+\theta \left({\log }_{f_i}\max \left(f_i,d_i\right)-1\right)\right)-\sum _{i=1}^n\sum _{j=1}^m\left(1-\zeta \ast b_i\right)\ast r_{ij}\ast g_j$$

(15)

The application of the logarithmic function in Formulas (14) and (15) has a clear theoretical basis. The monotonically increasing characteristic of the logarithmic function delineates the relationship between the increase in $d_i$ and the elevation of $u_i$, and it also reflects the gradual deceleration of revenue growth as $d_i$ continues to expand. This is consistent with the relationship between $u_i$ and $d_i$ described by the model—while substantial $d_i$ can enhance $u_i$, the increment does not always exhibit linear growth. Furthermore, the logarithmic function effectively manifests the situation where the $u_i$ is zero when $d_i$ is below the $f_i$. These characteristics collectively validate the appropriateness of using the logarithmic function in this model.

4.4. Multi-Objective Optimization Model of OCF Game

Regarding Comprehensive Formulas (9), (13) and (15), the OCF game can be optimized for the three objectives by Formula (16) to make the selected parameter $\omega$ optimal, to obtain the optimal task allocation results.

$$f\left(\omega \right)=\left\{\begin{array}{c}maxL\hfill \\ maxH\hfill \\ maxK\hfill \end{array}\right.$$

(16)

5. MCS Task Allocation Algorithm for OCF Game Optimized by Multiple Objectives

Converting a multi-objective optimization problem into a single-objective optimisation solution by weighting the objectives will limit the accuracy of the solution, so a Pareto optimisation approach is taken to solve the multi-objective optimisation problem. The mathematical model of Formula (16) belongs to the NP-hard problem. Therefore, MOPSO is introduced in this paper to solve the optimal parameters of the OCF game model. To this end, the ROCG-MOPSO is designed. The ROCG-MOPSO algorithm is shown in Algorithm 2.

Algorithm 2: ROCG-MOPSO Algorithm

Input:  Task minimum demand resource set F; task initial unit resource price set G; task minimum utility set U; user resource set A; number of iterations of the population $smax$; and population size o Output:  Task allocation matrix for the $smax$ round iteration $R^{smax}$   1: $Y=\varnothing$   2: Random initialized population $\left\{x_1^0,\dots ,x_i^0,\dots ,x_o^0\right\}$   3: for $s=1$ to $smax$ do   4:   for $i=1$ to o do   5:    $\left(R_i^s,G_i^s\right)=ROCG\left(F,R_i^{s-1},G_i^{s-1},U,x_i^{s-1}\right)$   6:    Calculate ($L_i^s$,$H_i^s$,$K_i^s$) from Formulas (9), (13) and (15)   7:    Update the Pareto archive Y by sorting the non-dominated solutions according to ($L_i^s$,$H_i^s$,$K_i^s$)   8:   end for   9:   for $i=1$ to o do 10:     Obtain the ${\mathrm{pbest}}_i^s$ and ${\mathrm{gbest}}_i^s$ 11:     Update $X_i^{s+1}$ and $V_i^{s+1}$ according to Formulas (17) and (18) 12:   end for 13:   $s=s+1$ 14: end for 15: Randomly select an optimal solution corresponding to $R^{smax}$ in the Pareto archive Y 16: return $R^{smax}$

ROCG-MOPSO consists of three main steps. The number of MOPSO iterations is s and its initial value is 0. Each individual in $X_i^0$, the MOPSO population, is randomly initialized to the value of the weight parameter $\omega$ of the OCF game. The number of individuals in the population is o, and the initial velocity of the particles $V_i^0$ is 0. The Pareto archive Y is initialized to empty.

5.1. Update on the Pareto Archives

At the s-th iteration, the task allocation result $R_i^s$ and the unit resource price of the task $G_i^s$ are found separately for each individual of the population i using Formula (3). And the three-dimensional solution ($L_i^s$,$H_i^s$,$K_i^s$) of the fitness function is obtained by $R_i^s$, $G_i^s$ and Formula (16). If ($L_i^s$,$H_i^s$,$K_i^s$) belongs to the dominated solution in the Pareto profile Y, it is not added to Y. If it belongs to the non-dominated solution and the Y is not yet full, the solution is added to Y. If the Y is full, calculate the crowding degree of the solution in Y and the ($L_i^s$,$H_i^s$,$K_i^s$) [31], eliminate the one with the highest crowding degree, and add the remaining solution to Y.

5.2. Update Population Individuals

If the iteration number s is not larger than $smax$, go to Section 5.3.

The position and velocity of individuals in the MOPSO population are, respectively, $X_{\mathrm{i}}^s\in (0,1)$ and $V_{\mathrm{i}}^s\in (-0.5,0.5)$. So, the update formula in the $s+1$ generation is as follows:

$$\begin{array}{c}{V}_i^{s+1}=u\ast V_i^s+C_1\ast R_1\ast \left({\mathrm{pbest}}_i^s-X_i^s\right)+C_2\ast R_2\ast \left({\mathrm{gbest}}^s-X_i^s\right)\end{array}$$

(17)

$$\begin{array}{c}{X}_i^{s+1}=V_i^{s+1}+X_i^s\end{array}$$

(18)

where $u\in (0,1)$ is the inertial weight. $C_1,C_2\in (0,1)$ are the acceleration factors that regulate the degree of particles influenced, respectively, by global and own historical experience independently. $R_1$ and $R_2$ are the random number between the $[0,1]$ [32].

${\mathrm{pbest}}_i^{s-1}$ is the optimal position of the individual history in the $s-1$ round iteration of the population individual i, and the corresponding fitness function value is ($L_i^{s-1}$,$H_i^{s-1}$,$K_i^{s-1}$). The fitness value of the individual i in the s round is ($L_i^s$,$H_i^s$,$K_i^s$). If the fitness governs the fitness of $s-1$ rounds, the ${\mathrm{pbest}}_i^s$ value is $X_i^s$. Instead, ($L_i^s$,$H_i^s$,$K_i^s$) is dominated by ($L_i^{s-1}$,$H_i^{s-1}$,$K_i^{s-1}$), then ${\mathrm{pbest}}_i^s$ retains the value of ${\mathrm{pbest}}_i^{s-1}$. If ($L_i^s$,$H_i^s$,$K_i^s$) and ($L_i^{s-1}$,$H_i^{s-1}$,$K_i^{s-1}$) do not form a dominant relationship, one of them is randomly selected and the corresponding position is taken as the historical optimal position of individual i.

${\mathrm{gbest}}_i^{s-1}$ is the historically optimal position of the population in the s round, which needs to be reselected in each iteration. The specific method is to take the reciprocal of each solution of Y in the $s\text{-}th$ iteration as the probability, one of the solutions in Y by the roulette algorithm, and the corresponding $X_i^s$ of this solution is selected as ${\mathrm{gbest}}_i^s$.

Go to Section 5.1 after updating the particle position.

5.3. Output the Optimal Task Allocation Scheme

A solution ($L_i^{smax}$, $H_i^{smax}$, $K_i^{smax}$) is chosen at random from the Pareto file Y. The $R^{smax}$ corresponding to this solution is the optimal task allocation scheme optimized by the weights.

6. Experiment and Analysis

6.1. Experimental Setting

The experiments run on a machine with a Win10 operating system, a 2.30 GHz CPU, and 16 GB of RAM. The algorithm is written using MATLAB R2020b. The dataset for the experiments uses the dataset GPS Trajectories with transportation mode labels from literature [33] to simulate realistic schemes, and the ROCG-MOPSO are compared with schemes from the relevant literature.

The relevant parameter settings for the experiment is shown in

Table 2. Experimental scheme.

Scheme Abbreviation	Full Form
ROCG	Repeated overlapping coalition formation game
ROMCS	Repeated overlapping coalition game model for MCS
VTCF	Virtual terminal coalition formation game
MOCFF	Multi-responsibility–oriented coalition formation framework

.

• The ROMCS scheme [34]. It uses OCF game to adjust price strategies and dynamic learning methods to change the resource allocation of users for MCS task allocation.
• The VTCF scheme [35]. It utilizes the theory of game theory in OCF and conducts task allocation by transforming the problem of overlapping coalitions into a situation of non-overlapping coalition formation.
• The MOCFF scheme [36]. Agents form two types of coalitions, respectively, for communication and task execution, thereby facilitate the task allocation of multi-agent responsibilities.

The relevant scheme settings for the experiment are shown in

Table 3. Experimental parameters.

Parameters	Value
Number of tasks n	[10, 100]
Number of users m	[10, 300]
Task minimum required resources f	[10, 250]
Number of individual user resources a	[20, 50]

.

6.2. Experimental Analysis

The experiment evaluates the performance of ROCG-MOPSO through four evaluation metrics: the task completion rate (from Formula (9)), the user average revenue (from Formula (13)), the platform revenue (from Formula (15)), and the user average surplus resources (from Formula (19)). The user average surplus resources is the average value of the resources that the user devotes to the task over the minimum resources required for the task, with higher values indicating a lower level of reasonable degree of resource input by the user. It can also reflect the extent to which users are using their resources wisely.

$$V=\sum _{i=1}^n\left(d_i-f_i\right)\ast b_i/n$$

(19)

6.2.1. Parameter $\omega$ Selection

Table 4. Experiment results of four metrics.

	TC	PR	UAR	USR
ROCG-MOPSO	0.87	128,620.376	170.729	1.04
ROCG ($\omega =0$)	0.46	21,242.314	130.377	3.48
ROCG ($\omega =0.3$)	0.72	97,184.774	146.207	1.89
ROCG ($\omega =0.6$)	0.88	125,554.502	169.246	1.09
ROCG ($\omega =0.9$)	0.90	125,843.43	165.265	1.11

illustrates the task completion rate (TC), platform revenue (PR), user average revenue (UAR), and user average surplus resources (USR) metrics obtained by selecting different $\omega$ parameter values for the ROCG under conditions of 250 users and 50 tasks. The ROCG-MOPSO uses multi-objective optimization to select the optimal parameters. Additionally, several representative parameter values of ROCG, namely, 0, 0.3, 0.6, and 0.9, are also selected. The ROCG-MOPSO achieves superior values on the PR, UAR, and USR metrics compared to other ROCG parameter selections. Since the TC increases as the $\omega$ parameter value increases, to achieve optimal overall performance, TC cannot be considered in isolation. Instead, metrics such as PR, UAR, and USR should be considered in a comprehensive manner. Therefore, by selecting the optimal parameters, ROCG-MOPSO can balance various factors, ensuring optimal overall performance metrics at the expense of an appropriate decrease in the TC.

6.2.2. Effectiveness Evaluation on Four Metrics

1

Task completion rate

Figure 3a displays the impact of the number of users on the task completion rate. As the number of users increases, the number of participants in the task coalition also rises. This signifies an increase in the total resources that can be allocated to tasks, thereby correspondingly enhancing the task completion rate. However, as illustrated in Figure 3b, the task completion rate gradually diminishes with an increase in the number of tasks. This is because the increase in the number of missions leads to an increase in the number of coalitions. On the premise of a constant user quantity, the average number of participants per coalition decreases.

Regardless of whether the number of users or tasks increases, the task completion rate under the ROCG-MOPSO scheme is higher than other schemes. This is primarily attributed to this strategy’s ability to adjust resource inputs through two game strategies, enabling users to effectively participate in unfinished task coalitions instead of completed ones. Consequently, in the ROCG-MOPSO scheme, users have a greater probability of completing more tasks than those in other schemes.

2

Platform revenue

As depicted in Figure 4a, with the increase in the number of users, the platform revenue also shows an upward trend. This aligns with the results displayed in Figure 3a, where an increase in the number of users leads to an enhancement in the task completion rate, thereby resulting in higher platform profits. Figure 4b demonstrates the impact of the initial number of tasks on platform revenue. With the increase in the initial number of tasks, the platform’s profit also trends upwards. However, once the number of tasks reaches a certain value, the platform’s profit no longer increases and even begins to decrease. This is because when the number of tasks in the MCS system is too large, it is difficult for users to effectively allocate their resources to tasks.

It is noteworthy that the revenue performance of the platform under the ROCG-MOPSO scheme outperforms that of other schemes. Through this scheme, the platform is able to adjust the unit resource price of tasks in a reasonable manner, consequently modifying the attractiveness of tasks and influencing the resource allocation decisions of users. This approach allows for a more optimal resource allocation. As such, the ROCG-MOPSO scheme ensures that the platform not only increases profits but also enhances the task completion rate.

3

User average revenue

As Figure 5a depicts, with an increase in the number of users, the average user revenue shows a declining trend. The primary reason for this is that as the number of users grows, the total number of resources also increases proportionally, intensifying the competition among users. Consequently, the unit price of resources falls, leading to a decline in users’ average revenue. Figure 5b illustrates the impact of the rise in the number of tasks on average user revenue. As more tasks are posted by the platform, competition among users will lessen, and the range of task choices for users will expand. This, to a certain extent, aids users in avoiding the selection of tasks that are difficult to complete, thereby avoiding penalties. Therefore, as the number of tasks increases, the average revenue of users also rises.

In both scenarios, the average user revenue under the ROCG-MOPSO scheme outperforms that of other schemes. This superior performance can be primarily attributed to the fact that, under this scheme, users will not only choose more profitable tasks but also tend to choose easy tasks to invest in resources. Consequently, this ensures an efficient allocation of resources. Such an approach enables users to accrue profits whilst avoiding penalties associated with task failures. Therefore, the ROCG-MOPSO scheme not only incentivizes users to accomplish more tasks, it also ensures they achieve relatively higher revenues, thus stimulating their sustained active participation in tasks.

4

User average surplus resources

Figure 6a illustrates the variation in the average surplus resources of users as the number of users increases. The results indicate that when the number of users rises while the number of tasks remains constant, the average surplus resources of users increases accordingly. This outcome aligns with the conclusion in Figure 3a: the higher the number of users, the fewer the number of uncompleted tasks. This leads to a greater number of resources being redundantly allocated to already completed tasks, thereby increasing the average resource wastage per user. On the other hand, as shown in Figure 6b, with the increase in the number of tasks, the average surplus resources of users shows a gradually decreasing trend. This is because the increase in the number of tasks elevates the number of task coalitions that users can participate in, leading to more resources being effectively utilized in yet-to-be-completed tasks.

In both scenarios, the average surplus resources of users under the ROCG-MOPSO scheme is lower than other schemes. This is because under this scheme, users invest fewer of their resources into already completed tasks and more into unfinished tasks to gain higher earnings, thereby effectively reducing resource wastage. In summary, the ROCG-MOPSO scheme can achieve a higher resource utilization rate in MCS task allocation compared to other schemes.

7. Conclusions

This study presenst a multi-objective optimization-based MCS task allocation scheme using an OCF game. The scheme delineates the resource allocation relationship between users and tasks through an OCF game, and the allocation of resources is adjusted through two game strategies. After multiple rounds of games, we obtain the optimal resource allocation results, thereby determining the task allocation scheme. To balance the interests of task publishers, users, and the platform, and further optimize the task allocation scheme, we utilized a MOPSO algorithm to optimize the parameters of the OCF game in task allocation model. Comparative experiments conducted on datasets confirmed that the scheme proposed in this paper, when compared with the schemes in the related literature, can significantly improve the task completion rate and bring higher returns to both the platform and users. However, it is noteworthy that there are risks of user identity and data leakage during the task allocation process. Therefore, in future research, we contemplate designing a more secure MCS scheme. Encrypting user data and authenticating user identity can effectively eliminate the concerns of users about privacy leakage. This will aid in attracting more users to participate in task execution, thereby promoting the further development of MCS technology.