Public and private roads co-exist in a practical transport network system, and this in general is known as a public–private mixed road network. Two stakeholders, that is, the government and a private firm, with inconsistent objectives exist during the operation of the public–private mixed network. This study considers competition between the government and the private firm while determining toll pricing scheme and credits charging scheme in the public–private mixed road network under the elastic origin–destination (O–D) demand case. The private firm maximizes its profit via pricing tolls on its controlled roads (private roads for short) while the government maximizes its total social gains via charging credits on its controlled roads (public roads for short). It is formulated as an equilibrium problem with equilibrium constraints (EPEC). To measure the inefficiency of EPEC, total social welfare is used by optimizing the sum of their objective functions as a system optimum (SO) model. Subsequently, a diagonalization method is used to solve the models. Numerical experiments are conducted on in two typical networks. The experiment with the small network demonstrates the feasibility of the proposed EPEC model. The Sioux Falls network example illustrates the competition between the government and the private firm when operating the public–private mixed network via determining the charging credits scheme and pricing tolls scheme. Moreover, EPEC is less inefficient because the EPEC lowers the sum of the objective function values of the government and private firms compared with that of SO where the sum of their objective functions is maximized.
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