We consider a cost sharing system where users are selfish and act according to their own interest. There is a set of facilities and each facility provides services to a subset of the users. Each user is interested in purchasing a service, and will buy it from the facility offering it at the lowest cost. The notion of social welfare is defined to be the total cost of the facilities chosen by the users. A central authority can encourage the purchase of services by offering subsidies that reduce their price, in order to improve the social welfare. The subsidies are financed by taxes collected from the users. Specifically, we investigate a non-cooperative game, where users join the system, and act according to their best response. We model the system as an instance of a set cover game, where each element is interested in selecting a cover minimizing its payment. The subsidies are updated dynamically, following the selfish moves of the elements and the taxes collected due to their payments. Our objective is to design a dynamic subsidy mechanism that improves on the social welfare while collecting as taxes only a small fraction of the sum of the payments of the users. The performance of such a subsidy mechanism is thus defined by two different quality parameters: (ⅰ) the price of anarchy, defined as the ratio between the social welfare cost of the Nash equilibrium obtained and the cost of an optimal solution; and (ⅱ) the taxation ratio, defined as the fraction of payments collected as taxes from the users.
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