Edge computing has become a very popular service that enables mobile devices to run complex tasks with the help of network-based computing resources. However, edge clouds are often resource-constrained, which makes resource allocation a challenging issue. In addition, edge cloud servers must make allocation decisions with only limited information available, since the arrival of future client tasks might be impossible to predict, and the states and behavior of neighboring servers might be obscured. We focus on a distributed resource allocation method in which servers operate independently and do not communicate with each other, but interact with clients (tasks) to make allocation decisions. We follow a two-round bidding approach to assign tasks to edge cloud servers, and servers are allowed to preempt previous tasks to allocate more useful ones. We evaluate the performance of our system using realistic simulations and real-world trace data from a high-performance computing cluster. Results show that our heuristic improves system-wide performance by 20-25% over previous work when accounting for the time taken by each approach. In this way, an ideal trade-off between performance and speed is achieved.