  In a decentralized system with $m$ machines, we study the selfish scheduling
problem where each user strategically chooses which machine to use. Each
machine incurs a cost, which is a function of the total load assigned to it,
and some cost-sharing mechanism distributes this cost among the machine's
users. The users choose a machine aiming to minimize their own share of the
cost, so the cost-sharing mechanism induces a game among them. We approach this
problem from the perspective of a designer who can select which cost-sharing
mechanism to use, aiming to minimize the price of anarchy (PoA) of the induced
games.
  Recent work introduced the class of \emph{resource-aware} cost-sharing
mechanisms, whose decisions can depend on the set of machines in the system,
but are oblivious to the total number of users. These mechanisms can guarantee
low PoA bounds for instances where the cost functions of the machines are all
convex or concave, but can suffer from very high PoA for cost functions that
deviate from these families.
  In this paper we show that if we enhance the class of resource-aware
mechanisms with some prior information regarding the users, then they can
achieve low PoA for a much more general family of cost functions. We first show
that, as long as the mechanism knows just two of the participating users, then
it can assign special roles to them and ensure a constant PoA. We then extend
this idea to settings where the mechanism has access to the probability with
which each user is present in the system. For all these instances, we provide a
mechanism that achieves an expected PoA that is logarithmic in the expected
number of users.
