Abstract | We present three new coordination mechanisms for schedul-
ing n sel¯sh jobs on m unrelated machines. A coordination
mechanism aims to mitigate the impact of sel¯shness of jobs
on the e±ciency of schedules by de¯ning a local schedul-
ing policy on each machine. The scheduling policies induce
a game among the jobs and each job prefers to be sched-
uled on a machine so that its completion time is minimum
given the assignments of the other jobs. We consider the
maximum completion time among all jobs as the measure
of the e±ciency of schedules. The approximation ratio of
a coordination mechanism quanti¯es the e±ciency of pure
Nash equilibria (price of anarchy) of the induced game. Our
mechanisms are deterministic, local, and preemptive in the
sense that the scheduling policy does not necessarily process
the jobs in an uninterrupted way and may introduce some
idle time. Our ¯rst coordination mechanism has approxima-
tion ratio O(logm) and always guarantees that the induced
game has pure Nash equilibria to which the system con-
verges in at most n rounds. This result improves a recent
bound of O(log2 m) due to Azar, Jain, and Mirrokni and,
similarly to their mechanism, our mechanism uses a global
ordering of the jobs according to their distinct IDs. Next
we study the intriguing scenario where jobs are anonymous,
i.e., they have no IDs. In this case, coordination mechanisms
can only distinguish between jobs that have diffeerent load
characteristics. Our second mechanism handles anonymous
jobs and has approximation ratio O
¡ logm
log logm
¢
although the
game induced is not a potential game and, hence, the exis-
tence of pure Nash equilibria is not guaranteed by potential
function arguments. However, it provides evidence that the
known lower bounds for non-preemptive coordination mech-
anisms could be beaten using preemptive scheduling poli-
cies. Our third coordination mechanism also handles anony-
mous jobs and has a nice \cost-revealing" potential func-
tion. Besides in proving the existence of equilibria, we use
this potential function in order to upper-bound the price of stability of the induced game by O(logm), the price of an-
archy by O(log2 m), and the convergence time to O(log2 m)-
approximate assignments by a polynomial number of best-
response moves. Our third coordination mechanism is the
¯rst that handles anonymous jobs and simultaneously guar-
antees that the induced game is a potential game and has
bounded price of anarchy. |