The safety of vulnerable road users is a global concern for road safety. Among the most vulnerable groups on the road are pedestrians. Every road user becomes a pedestrian at some point in the transportation process, whether accessing a vehicle or transferring between vehicles. In daily transportation, designated pedestrian crossing points (where vehicles and pedestrians interact) pose significant safety risks for pedestrians. Previous studies have identified various factors that can impact pedestrian priority, but most of this research has focused on the pedestrian's viewpoint. Research has not analyzed the cognitive behavior of drivers and pedestrians. Our research aimed to explore how the cognitive abilities of pedestrians and drivers influence their behavior at designated pedestrian crossings. We carried out on-site measurement investigations at 4 sites in Budapest. We have recorded several data on the behavior and socio-demographic characteristics of pedestrians and the behavior of vehicles. We examined the data using both descriptive statistics and a predictive decision model. Our results show that pedestrians’ cognitive characteristics influenced the behavior of both pedestrians and drivers.