Automated vehicles (AVs) promise many benefits for future mobility. One of them is a reduction of the required total vehicle fleet size, especially if AVs are used predominantly as shared vehicles. This paper presents research on this potential reduction for the greater Zurich, Switzerland, region. Fleets of shared AVs serving a predefined demand were simulated with a simulation framework introduced in the paper. Scenarios combining levels of demand for AVs with levels of supply (i.e., AV fleet sizes) were created. An important contribution of this study is the use of travel demand at highly detailed spatial and temporal resolutions that goes beyond the simplifications used in previous studies on the topic. This detailed travel demand provides a more solid basis for the ongoing discussion about the future fleet size. It was found that for a given fleet performance target (here, the target was for 95% of all transport requests to be served within 5 min), the relationship between served demand and required fleet size was nonlinear and the ratio increased as demand increased. A scale effect was detected. This effect has the important implication that for different levels of demand the fleet is used more or less efficiently. This study also found that if waiting times of up to 10 min were accepted, a reduction of up to 90% of the total vehicle fleet could be possible even without active fleet management, like vehicle redistribution. Such effects require, however, that a large enough share of the car demand be served by AVs.