The potential market for modern self-driving cars is enormous, as they are developing remarkably rapidly. At the same time, however, accidents of pedestrian fatalities caused by autonomous driving have been recorded in the case of street crossing. To ensure traffic safety in self-driving environments and respond to vehicle-human interaction challenges such as jaywalking, we propose Level-$k$ Meta Reinforcement Learning (LK-MRL) algorithm. It takes into account the cognitive hierarchy of pedestrian responses and enables self-driving vehicles to adapt to various human behaviors. %which takes into account pedestrian responses while learning the optimal strategies. As a self-driving vehicle algorithm, the LK-MRL combines level-$k$ thinking into MAML to prepare for heterogeneous pedestrians and improve intersection safety based on the combination of meta-reinforcement learning and human cognitive hierarchy framework. We evaluate the algorithm in two cognitive confrontation hierarchy scenarios in an urban traffic simulator and illustrate its role in ensuring road safety by demonstrating its capability of conjectural and higher-level reasoning.