We study a very general cost-sharing scheduling game.
An instance consists of $k$ jobs and $m$ machines and an arbitrary weighted bipartite graph denoting the jobs strategies.
An edge connecting a job and a machine specifies that the job may choose the machine;
edge weights correspond to processing times. Each machine has an activation cost that needs to be covered by the job assigned to it.
Jobs assigned to a particular machine share its cost proportionally to the load they generate.
Our game generalizes singleton cost-sharing games with weighted players.
We provide a complete analysis of the game with respect to equilibrium existence, computation, convergence and quality -- with respect to the maximal and the total cost.
We study both unilateral and coordinated deviations.
We show that the main factor in determining the stability of an instance and the quality of a stable assignment is the machines' activation-cost.
Games with unit-cost machines are generalized ordinal potential games, and every instance has an optimal solution which is also a pure Nash equilibrium (PNE).
On the other hand, with arbitrary-cost machines, a PNE is guaranteed to exist only for very limited instances, and the price of stability is linear in the number of players.
Also, the problem of deciding whether a given game instance has a PNE is NP-complete.
In our analysis of coordinated deviations, we characterize instances for which a strong equilibrium exists and can be calculated efficiently, and show tight bounds for the strong price of anarchy and the price of stability.