As people pay more attention to environment protection, the number of electric vehicles (EVs) is gradually increasing. Energy trading management for EVs is becoming a challenge. However, existing research has not considered the problem of information sharing between energy traders and issues surrounding the protection of user privacy. Therefore, in this paper, we propose a vehicle-to-vehicle (V2V) and vehicle-to-grid (V2G) electricity trading architecture based on blockchain. All energy transactions of EVs can be recorded on the blockchain ledger to ensure privacy and smart contracts work as agents for pricing and optimal energy allocation. Furthermore, we introduce a two-way auction mechanism based on the Bayesian game and design a new price adjustment strategy. Finally, we propose a bidirectional auction mechanism based on the Bayesian game approach. We use extensive simulations to evaluate the performance of our proposed algorithm. Simulation results show that the social welfare and cost performance of our algorithm can be improved by up to 102.8 and 319%, respectively.