# A Learning Game-Based Approach to Task-Dependent Edge Resource Allocation
## 1 Introduction  
With the rapid advancement of hardware and software technologies, there is an increasing demand for computationally intensive real-time applications, such as facial recognition, virtual reality, and augmented reality [1, 2, 3]. These applications often require lower response times to meet user experience expectations. However, due to the limitations of device hardware and available resources, the devices cannot complete task computations in the required time [4, 5, 6]. Cloud computing is one of the candidate technologies to solve this problem, but the distance between the cloud and the devices is considerable. Transferring a large number of tasks to the cloud imposes a significant communication burden on the entire network. Edge computing (EC) deploys resources closer to devices, thereby enabling devices to offload tasks to a nearby edge server (ES) with less overhead, thus further reducing the time used for computational tasks [7]. Although edge servers (ESs) can provide the necessary resources for devices, the behavior of ESs and the user in practice is usually driven by interests [8, 9]. In order to offset operational costs, ESs charge a certain fee for processing tasks, while the user needs to pay for the costs incurred from using these resources, thus ensuring a satisfactory quality of experience (QoE). On the other hand, dependent tasks differ from tasks that can be arbitrarily divided (such as atomic tasks). The order of scheduling of its sub-tasks andthe amount of resources allocated to each sub-task are important factors that affect the execution time of the dependent task [10; 11]. Therefore, motivating ESs to engage in task computation and the effective allocation of resources and tasks are key issues in edge computing (EC). Game theory is a mathematical apparatus that studies the decision-making process of players in a competitive environment. It involves exploration of the interactions between participants, as well as their selection of strategies and expected outcomes [12; 13; 14]. Using the strengths of game theory, numerous studies in recent years have applied it to the research of edge resource allocation strategies. For example, [PERSON] et al. [15] employed a Stackelberg game for the dynamic allocation and pricing of communication and computational resources in networks, in which they proposed a joint optimal pricing and resource allocation algorithm based on game theory. [PERSON] et al. [16] investigated multiple resource allocation and pricing issues by modeling the problem as a Stackelberg game. [PERSON] et al. [17] introduced a game model for edge resource allocation with the aim of maximizing the utilization of computational resources. However, these studies all rest on a strong assumption that the game participants should share their information, such as the user needing to disclose resource preference parameters and the edge servers (ESs) required to reveal their cost parameters. Yet, game participants are rational, and they are typically reluctant to disclose such private information. This paper presents a two-stage resource allocation method designed for task-dependent scenarios. In the first stage, an incentive mechanism is proposed and modeled as a multivariate Stackelberg game. We then analyze the uniqueness of the Stackelberg equilibrium (SE) in an information sharing scenario, as well as design an iterative optimization algorithm that can approximate the SE solution. The research is then extended to non-information-sharing scenarios, where the game problem is formulated as a POMDP. A reinforcement learning algorithm based on a learning game is then proposed, which can learn from the participants' historical decisions while ensuring the privacy of game participants. The second stage aims to allocate the resources purchased by users under the incentive mechanism in a rational manner. A greedy-based deep reinforcement learning algorithm is designed to minimize task execution time.