Parameters used in the test.

In many countries, public transport (PT) services are offered by multiple operators with different modes, such as trains, metros, and buses, which are interchangeable across a single journey. This paper presents a model with one origin (O)-destination(D) path operated by two operators, each of which is responsible for different parts of the OD journey. One operator competes with potential third-party transport companies by offering discount incentives. Such an abstract settings has not been discussed in the literature. We explore how prices, demand, profits, and social welfare change with discounts through a theoretical analysis and numerical simulations under five scenarios. The results indicate that in all the scenarios the operator offering a discount incentive can always attract more passengers and increase its profits. Moreover, reducing the service time of operators offering discounts contributes to an increase in social welfare. Notably, this paper deduces for the first time that the demand scenario aimed at maximizing social welfare is twice as high as that aimed at maximizing total profit. However, in the scenario of maximum social welfare, the profitability of operators becomes challenging.