1 A Game-theoretic Approach Towards Collaborative Coded Computation Offloading Jer Shyuan Ng, Wei Yang Bryan Lim, Zehui Xiong, Dusit Niyato, Fellow, IEEE, Cyril Leung, Dong In Kim, Fellow, IEEE, Junshan Zhang, Fellow, IEEE, Qiang Yang, Fellow, IEEE Abstract—Coded distributed computing (CDC) has emerged as a promising approach because it enables computation tasks to be carried out in a distributed manner while mitigating straggler effects, which often account for the long overall completion times. Specifically, by using polynomial codes, computed results from only a subset of edge servers can be used to reconstruct the final result. However, incentive issues have not been studied systematically for the edge servers to complete the CDC tasks. In this paper, we propose a tractable two-level game-theoretic approach to incentivize the edge servers to complete the CDC tasks. Specifically, in the lower level, a hedonic coalition formation game is formulated where the edge servers share their resources within their coalitions. By forming coalitions, the edge servers have more Central Processing Unit (CPU) power to complete the computation tasks. In the upper level, given the CPU power of the coalitions of edge servers, an all-pay auction is designed to incentivize the edge servers to participate in the CDC tasks. In the all-pay auction, the bids of the edge servers are represented by the allocation of their CPU power to the CDC tasks. The all-pay auction is designed to maximize the utility of the cloud server by determining the allocation of rewards to the winners. Simulation results show that the edge servers are incentivized to allocate more CPU power when multiple rewards are offered, i.e., there are multiple winners, instead of rewarding only the edge server with the largest CPU power allocation. Besides, the utility of the cloud server is maximized when it offers multiple homogeneous rewards, instead of heterogeneous rewards. Index Terms—Coded distributed computing, straggler effects mitigation, hedonic game, all-pay auction, Bayesian Nash equilibrium F 1 INTRODUCTION OUPLED with reliable wireless communication tech- tation tasks collaboratively, the communication costs can be C nologies, IoT devices can serve as important sources of high due to the frequent exchange of intermediate results. sensor data for Artificial Intelligence (AI) technologies to be Secondly, the response times vary across the edge servers leveraged, towards the development of data-driven applica- due to several factors such as imbalanced work allocation, tions [1]. In particular, many machine learning models are contention of shared resources and network congestion [6], developed to monitor various large-scale physical phenom- [7]. Thirdly, the confidentiality of the data may be com- ena for smart city applications, such as prediction of road promised as eavesdroppers may monitor data transmission conditions [2], air quality monitoring [3] and tracking of over wireless channels. medical conditions [4]. Edge computing [5] has emerged as a Coded distributed computing (CDC) [8] has been pro- promising approach that extends cloud computing services posed as an efficient method for distributed computation to the edge of the networks. In particular, by leveraging tasks at the edge of the network. In particular, coding tech- on the computational capabilities, e.g., Central Processing niques are used to design computation strategies that divide Unit (CPU) power, of the edge servers, e.g., base stations the entire dataset and allocate subsets of data to the edge and edge devices, e.g., laptops and tablets, the cloud server servers for computations. In the distributed edge computing can offload its computation tasks to the edge servers and network, one of the main challenges is the straggler effects arXiv:2102.08667v1 [cs.GT] 17 Feb 2021 devices. where the task completion time is determined by the slowest However, there are several challenges pertaining to the edge server as the cloud server needs to wait for all edge distributed edge computing network that need to be ad- servers to return their results before it can reconstruct the dressed for efficient and scalable implementation. Firstly, final result. As a result, the latency of the distributed com- since several edge servers perform the distributed compu- putation tasks can be high [9], [10]. By using CDC schemes1, instead of having to wait for all edge servers to complete • JS. Ng and WYB. Lim are with Alibaba Group and Alibaba-NTU Joint their computation tasks, the cloud server only needs to wait Research Institute, Nanyang Technological University, Singapore. for a subset of edge servers to return their results. Hence, • Z. Xiong is with Pillar of Information Systems Technology and Design, CDC schemes can reduce computation latency by obviating Singapore University of Technology Design. • D. Niyato is with School of Computer Science and Engineering, Nanyang the need to wait for the slower edge servers. Technological University, Singapore. However, incentives are essential for the edge servers • C. Leung is with The University of British Columbia and Joint NTU-UBC to participate in or to complete their allocated CDC sub- Research Centre of Excellence in Active Living for the Elderly (LILY). tasks. To design an appropriate incentive mechanism, it is • DI. Kim is with Sungkyunkwan University, South Korea. • J. Zhang is with School of Electrical, Computer and Energy Engineering, 1. CDC schemes do not only mitigate straggler effects, but can also Arizona State University, USA. reduce communication costs and ensure security in the distributed edge • Q. Yang is with Hong Kong University of Science and Technology, Hong computing network. This paper focuses on CDC schemes that aim to Kong, China. mitigate straggler effects. 2 important to consider the unique characteristics of the CDC formation game, and (iii) auction design. framework. Specifically, even though the edge servers are each allocated a subset of the entire dataset for computa- 2.1 Coded Distributed Computing (CDC) tions, some of the edge servers’ computed results may not Given the emergence of big data which necessitates be used to reconstruct the final result, e.g., due to straggling. computation- and storage-intensive processing, large-scale These edge servers in turn do not receive any compensation. distributed systems have received significant attention from As a result, this may discourage the participation of certain both the research and industrial communities. A number edge servers. To address this challenge, we propose an all- of studies in the literature have focused on the minimiza- pay auction to model the competition between the different tion of communication load of the distributed computation edge servers and at the same time, improve the participation tasks. Network coding in the context of distributed cache of edge servers so as to elicit more CPU power for the CDC systems has been a promising approach to increase network tasks. throughput and improve performance by jointly optimizing In distributed edge computing networks, the edge data placement and delivery phases [11], [12]. servers may work together with various edge devices, by Recently, coding techniques have increasingly been used forming coalitions in order to complete their computation in distributed computing networks. One of the active re- tasks. To model the cooperation between the edge servers search areas is the minimization of the communication and devices, we propose a hedonic coalition formation game load in the data shuffling phase through coded multicast in which the edge devices decide which edge server to join transmission as this phase accounts for a large proportion of based on their utility-maximizing objectives. In analogy to the overall execution time [13]. There is a tradeoff between practical scenarios, the edge devices make decisions that computation load and communication load [14]. In order maximize their utilities without taking into consideration to reduce the number of communication rounds, which is the effect of their decisions on other edge servers or devices. significant for distributed iterative algorithms, [15] proposes The main aim of this work is to develop an incentive a computing technique that jointly codes the computation mechanism for enabling efficient completion of CDC tasks at multiple iterations by leveraging on the storage and for IoT applications. Our key contributions are summarized computation redundancy of the workers. The work in [16] as follows: considers the network topology of the distributed systems 1) We highlight the importance of incentives in CDC, in designing an efficient CDC scheme for practical imple- which is an issue ignored, but crucial toward eco- mentation. It relaxes the assumption that the physically- nomically sustainable distributed systems, by exist- separated servers are connected to a single error-free com- ing works. mon communication bus. 2) We propose a two-level game theoretic approach to Apart from the studies that focus on the minimization of incentivize the edge servers to contribute their CPU communication load in the distributed computing networks, power for the CDC tasks. coding techniques are also used to alleviate the stragglers’ 3) We formally show that the edge servers may im- delays that limit the performance as distributed computing prove their utilities by forming coalitions. We, there- systems are scaled up. This is achieved by reducing the fore, introduce a hedonic coalition formation game recovery