We consider the computation offloading problem in an edge computing system in which an operator jointly manages wireless and computing resources across devices that make their offloading decisions autonomously with the objective to minimize their own completion times. We develop a game theoretical model of the interaction between the devices and an operator that can implement one of two resource allocation policies, a cost minimizing or a time fair resource allocation policy. We express the optimal cost minimizing resource allocation policy in closed form and prove the existence of Stackelberg equilibria for both resource allocation policies. We propose two efficient decentralized algorithms that devices can use for computing equilibria of offloading decisions under the cost minimizing and the time fair resource allocation policies. We establish bounds on the price of anarchy of the games played by the devices and by doing so we show that the proposed algorithms have bounded approximation ratios. Our simulation results show that the cost minimizing resource allocation policy can achieve significantly lower completion times than the time fair allocation policy. At the same time, the convergence time of the proposed algorithms is approximately linear in the number of devices, and thus they could be effectively implemented for edge computing resource management. .