This paper considers a Mobile Edge Computing scenario with multiple mobile units (MUs), multiple access points (APs) and one cloudlet server. The MUs have to decide whether offloading their computation tasks to the cloudlet is energy wise beneficial. As there are multiple APs available to connect the MUs to the cloudlet and communication and computation resources have to be shared among all MUs, each MU also has to choose the AP for transmission that minimizes its offloading energy under the given fraction of the overall resources. The problem is formulated as a energy minimization problem with a maximum offloading time constraint. MUs not only need to consider the energy required for local computation or offloading, but simultaneously avoid an overlong processing time of offloaded computation. This joint offloading decision and resource allocation is divided into two subproblems in the proposed approach. The resource allocation problem is reformulated by using Lagrange multipliers and closed-forms for the calculation of the shared resources are found. These results can be integrated into the proposed game theoretic algorithm for the offloading decision problem. The algorithm is based on a potential game and therefore, can be proven to converge to a Nash equilibrium. Numerical results show a benefit of the proposed resource allocation strategy, a performance of the proposed game algorithm near the optimal solution and a fast algorithm execution time that can even be significantly improved by proposed sorting metrics. mobile edge computing, joint optimization, resource allocation strategy, game theory