The conceptualization of fog computing in Fog–Cloud Network (FCN) comes as an addition of Mobile Cloud Computing (MCC) for mobile edge computing where the primary emphasize has been given towards energy savings and delay minimization in the recent years. However, it is observed that the task provisioning of computational-intensive task processing in FCN is challenging as the fog node is constrained with computational and storage resources when compared to cloud and hence may not be able to process all the tasks within the defined deadline. The task allocation problem in fog–cloud networks has been extensively studied in the past, yet most of the approaches are found to be computationally complex and incur higher execution time resulting significant resource costs. This paper addresses the problem of delay-sensitive task allocation in the FCN and proposes an idea of collaborative task scheduling and resource allocation schemes considering game theory and payoff computation. The experimental outcome shows that, unlike the existing systems, the proposed approach for task scheduling and resource allocation in FCN has significantly retained a balance between resource utilization and task processing time and also ensures optimal resource cost metrics for variable data traffic.