Cities worldwide struggle with overloaded transportation systems and their externalities, such as traffic congestion and emissions. The emerging technology of autonomous transportation systems bears a high potential to alleviate these issues. At the same time, this technology might also introduce negative effects, in particular by disproportionately cannibalizing public transportation. A careful analysis of this trade-off requires modeling both modes of transportation within a unified framework. In this paper, we propose such a framework, which allows us to study the interplay among mobility service providers, public transport authorities, and customers, and in particular to analyze the effect of autonomous ride-hailing services on the demand for public transportation. This framework combines a graph-theoretic network model for the transportation system with a game-theoretic model whereby mobility service providers are profit-maximizers and customers select individually-optimal transportation options. We apply our modeling approach to data for the city of Berlin, Germany and present sensitivity analyses to study factors that mobility service providers or municipalities can act upon to strategically steer the overall system. We show that depending on market conditions and policy restrictions, autonomous ride-hailing systems may complement or cannibalize a public transportation system, and discuss the main factors behind such different outcomes as well as strategic design options available to policymakers.