Braess Paradox of traffic networks with mixed equilibrium behaviors

• We extend the analysis of Braess Paradox to consider the mixed equilibrium behavior.
• The paradox occurrence conditions under the mixed equilibrium are investigated.
• The occurrence conditions of the Braess Paradox are generalized in a grid network.

Under the user equilibrium (UE) behavior assumption, the Braess Paradox (BP) and its variations have been well investigated. However, users do not always follow the UE behavior. In reality, there are likely quiet a few non-collaborative Cournot–Nash (CN) players coexisting with UE players in the common traffic network. Users in a CN player are completely collaborative to minimize their total travel cost and users subordinating to different players are perfectly competitive. Considering both UE and CN players in the congested network, it remains unclear that under what conditions the BP will occur. In this paper, the BP occurrence conditions under the UE–CN mixed equilibrium are firstly investigated using the classical Braess network with linear link cost function. Then, the BP conditions are studied to the ordinary grid network with nonlinear link cost function. It is shown that the BP occurrence in the conventional Braess network depends upon the link travel time function parameters and the demand level of users controlled by UE/CN players, and the BP occurs in the grid network only for certain demand combinations of users under one UE player and two CN players.