In the new era of mixed transportation systems, where autonomous vehicles (AV) coexist with human-driven vehicles (HV) in service operations, urban transportation network flows will be consisted of “decentralized” decisions made by commuters who choose HV mode, and “centralized” decisions made by the AV mobility service operators that assign each traveler who choose AV mobility service with a route. This paper presents the first hybrid behavior equilibrium model that considers the combination of “elastic demand-travel service/mode selection-route allocation/selection” decision-making processes. This model fills the theoretical gap in the equilibrium of urban traffic networks under mixed traffic conditions. Two numerical examples were used to verify the effectiveness of the model, which revealed that the optimal supply mode for the AV service market is a joint effort by the government and private enterprises. More importantly, this market structure can simultaneously achieve optimal efficiency of the transportation system and perceived fairness for travelers, providing valuable insights into important management practice issues such as “what role should the government play in the AV service market.”