This letter presents a planning system for autonomous driving among many pedestrians. A key ingredient of our approach is Pedestrian Optimal Reciprocal Collision Avoidance, a pedestrian motion prediction model that accounts for both a pedestrian's global navigation intention and local interactions with the vehicle and other pedestrians. Unfortunately, the autonomous vehicle does not know the pedestrians' intentions a priori and requires a planning algorithm that hedges against the uncertainty in pedestrian intentions. Our planning system combines a Partially Observable Markov Decision Process algorithm with the pedestrian motion model and runs in real time. Experiments show that it enables a robot scooter to drive safely, efficiently, and smoothly in a crowd with a density of nearly one person per square meter.