An important trend in mobility is the consumption of mobility as-a-service heralding in the age of freefloating vehicle sharing (FFVS) systems. In many markets such fleets compete. We investigate how realtime competitor information can create value for operators in this context. We focus on the vehicle supply decision which is a large operational concern. We show empirically that local market shares directly depend on the share of available vehicles in a location, which underlines the value potential of competitor awareness. We leverage this insight by proposing a novel decision support system for optimal management of FFVS systems under competition. We proceed in two phases, (1) a predictive phase and (2) a prescriptive phase. In phase (1), we compile a spatio-temporal dataset based on Car2Go and DriveNow transactions in Berlin, which we supplement with temporal, geographical and weather data. We partition the city into hexagonal tiles and observe vehicle supply per tile at the start of each period. We train machine learning models to predict vehicle inflows and vehicle outflows during the next period to derive total supply and demand. We find that inflows and outflows can be predicted with high accuracy using similar models. We test different temporal and spatial resolutions and find that spatial resolution incurs larger performance penalties. In phase (2), we formulate a myopic mixed integer non-linear programming model with a margin-maximizing objective function. The model trades off additional market share gains against the cost of re-locating vehicles, which enables operators to assign vehicles optimally across the service network. Our numerical studies on the case of Car2Go and DriveNow demonstrate that this competitor-aware model is capable of profitably improving market share by up to 1.4% or 3.4% for human-based and autonomous relocation respectively in a prefect foresight scenario and by up to 0.8% and 1.8% respectively when using predicted values.