Pigouvian tolls and welfare optimality with parallel servers and heterogeneous customers

Congestion externalities are a well-known phenomenon in transportation and communication networks, healthcare etc. Optimization by self-interested agents in such settings typically results in equilibria which are sub-optimal for social welfare. Pigouvian taxes or tolls, which impose a user charge equal to the negative externality caused by the marginal user to other users, are a mechanism for combating this problem. In this paper, we study a non-atomic congestion game in which heterogeneous agents choose amongst a finite set of heterogeneous servers. The delay at a server is an increasing function of its load. Agents differ in their sensitivity to delay. We show that, while selfish optimisation by agents is sub-optimal for social welfare, imposing admission charges at the servers equal to the Pigouvian tax causes the user equilibrium to maximize social welfare. In addition, we characterize the structure of welfare optimal and of equilibrium allocations.