We investigate the problem of computation offloading in a mobile edge computing architecture, where multiple energy-constrained users compete to offload their computational tasks to multiple servers through a shared wireless medium. We propose a multi-agent deep reinforcement learning algorithm, where each server is equipped with an agent, observing the status of its associated users and selecting the best user for offloading at each step. We consider computation time (i.e., task completion time) and system lifetime as two key performance indicators, and we numerically demonstrate that our approach outperforms baseline algorithms in terms of the trade-off between computation time and system lifetime.