Social optimum in public transport networks when users choose strategies: analysis and comparison with Wardrop equilibrium

The effective design and management of public transport systems are essential to ensuring the best service for users. The performance of a transport system will depend heavily on users behavior and decisions. Generally, users have individual objectives, such as seeking to minimize their own travel time. These decisions can affect the performance of the entire system, making it inefficient. A measure that allows to quantify the inefficiency of the system in the face of non-cooperative behavior is the price of anarchy. Our objective is to investigate the price of anarchy in the common-lines problem. In this approach, users choose which lines to use based on the best strategy for them. While Wardrop equilibrium has been studied for the common-lines problem, no contributions have been made toward the social optimum. In this work, we propose two optimization problems to obtain this optimum on simple networks, using strategy flow and line flow formulations. In addition, we prove that both optimization problems are equivalent, and we obtain a characterization of the social optimum flows. These contributions allow us to obtain results about the behavior of the price of anarchy in simple networks as demand increases.