A multi-deck cage has a deck floor with a ramp drive for fine positioning. The deck floor is configured as a ramp rotatable about a rotary axle and is movable up and down by means of actuators of the ramp drive. A sensor, which locates a mark arranged at a story sill, is arranged at a deck sill which is constructed as part of the deck floor. If the sensor has, at a stopping of the multi-deck cage at the story, travelled past the mark, the ramp drive is activated in a direction opposite to the travel direction of the multi-deck cage past the story stop until the sensor and the mark are disposed at the same level.