  How can a system designer exploit system-level knowledge to derive incentives
to optimally influence social behavior? The literature on network routing
contains many results studying the application of monetary tolls to influence
behavior and improve the efficiency of self-interested network traffic routing.
These results typically fall into two categories: (1) optimal tolls which
incentivize socially-optimal behavior for a known realization of the network
and population, or (2) robust tolls which provably reduce congestion given
uncertainty regarding networks and user types, but may fail to optimize routing
in general. This paper advances the study of robust influencing, mechanisms
asking how a system designer can optimally exploit additional information
regarding the network structure and user price sensitivities to design pricing
mechanisms which influence behavior. We design optimal scaled marginal-cost
pricing mechanisms for a class of parallel-network routing games and derive the
tight performance guarantees when the network structure and/or the average user
price-sensitivity is known. Our results demonstrate that from the standpoint of
the system operator, in general it is more important to know the structure of
the network than it is to know distributional information regarding the user
population.
