Analysis of two-tier public service systems under a government subsidy policy

• Two-tier service system.
• Focus on competition and coordination.
• Formulate a game model.
• Obtain Nash equilibrium.
• Find properties of the competitive strategies.

Some public service systems such as healthcare systems consist of both free public service provider with a long wait time and paid private service provider with a short wait time. Such service systems are often called a two-tier service system. In general, more customers will choose the free service provider (SP). To reduce the congestion in the free system, the government may encourage customers to use the pay system by offering them a subsidy. This paper studies whether such a subsidy can reduce the free system’s waiting time and improves the social welfare. While the objective of the free system is to maximize its own total customers’ utility, the objective of the pay system is to maximize its profit. We develop a mixed duopoly game to analyze the Nash equilibrium for the competition between the free and toll systems. Two scenarios with unregulated and regulated prices are considered. When the pay system price is unregulated (the private SP can set prices freely), we find that if customers are less sensitive to the service delay, the subsidy policy can effectively reduce the expected waiting time for the free system and increase the customer utility surplus of the entire two-tier system. However, if customers are more sensitive to the service delay, the subsidy policy may have the opposite effect. When the pay system price is regulated (the price determined by government), the subsidy policy can effectively reduce the expected waiting time for the free system and improve the social welfare of the two-tier system. And there exists an optimal regulated price to maximize the social welfare of the entire public service system.