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
,
, and
contributed their resources to accomplish task
, thereby forming a coalition. Similarly, users
,
, and
also allocated their resources to carry out task
, resulting in the formation of another coalition. Notably, due to the participation of
, an overlap exists between these two coalitions. In the same vein,
has become the overlapping part between the coalition in
and the coalition in
. 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.
where
k denotes the number of game rounds in the OCF game matrix. Row
i of the matrix represents the task
, column
j of the matrix represents the user
, and
denotes the resources allocated to the task by the user.
= 0 means that the user
is not a member of the task
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.
3.4. The Process of the ROCG
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
, 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
value to the same value, indicating that the tasks are equally attractive to the users.
- 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
of the task
to the user
during the
k round of the game is calculated in Formula (
3).
where
is the weight used to measure the extent to which users tend to devote resources to the task to be completed easily.
is used to measure the extent to which users tend to devote resources to tasks with a higher price per resource.
is the number of resources already owned by the user
at the time of assigning resources to the task
, and
is the number of resources currently owned by the task
. Then,
When
, 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
attractiveness index
includes both the resources required to complete the task and the price per resource. When
, it means that in the current
k round of the game, the number of resources owned by the task
is at or above its minimum required resources. Therefore, the task attractiveness index
no longer includes the factor of resources required for task completion. The attractiveness index forms the task attractiveness matrix
in Formula (
6).
If , then . 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 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
determines the number of resources
to be invested in the task
based on the task attractiveness index
. It should be noted that if the resources invested by user
in the previous round are withdrawn and this leads to the failure of task
, then the user cannot withdraw these resources and is unable to exit the coalition. Therefore, the number of resources
that the user can allocate in this round is reduced (
).
can be determined by Formula (
7), which leads to the resource input matrix
(also known as the coalition matrix).
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 and , we interpret this as user exiting the coalition. Conversely, if and , we interpret this as user 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
is the initial unit resource price of the task
. Adjust the task unit resource price
based on the value of
. When the task
has fewer resources, i.e.,
, increase
. And when the task
has excess resources, i.e.,
, decrease
.
where
and
are the weights.
denotes the price weight when resources are lacking, and
denotes the price weight when resources are in excess. The adjustment with
and
is related to the demand for the task completion rate. When task completion rates are higher,
can be increased and
reduced, making users more inclined to unfinished tasks. When a higher task completion rate is not required,
can be lowered and
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 is less than the , the task is incomplete, and the platform earns nothing from this task . In this instance, the “arctan” function is used to reflect the increase in the . The characteristics of the “arctan” function result in a larger increase in when the resource is less, and a gradual decrease in this increase as it approaches the , reflecting the impact of resource scarcity on price. On the other hand, when the exceeds the , 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 . 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 . In summary, the “arctan” and “sin” functions in the model accurately portray how the platform adjusts the based on , thereby revealing the complex nonlinear relationship between and in the k-th round.
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 and the unit resource price matrix are obtained at the end of the game. And the results of the task allocation are available by .