Summary: Optimal Mechanisms for Single Machine Scheduling
July 16, 2008
We study the design of optimal mechanisms in a setting where job-agents compete for being processed
by a service provider that can handle one job at a time. Each job has a processing time and incurs a waiting
cost. Jobs need to be compensated for waiting. We consider two models, one where only the waiting
costs of jobs are private information (1-d), and another where both waiting costs and processing times
are private (2-d). Discrete probability distributions represent the public common belief about private
information. In this setting, an optimal mechanism minimizes the total expected expenses to compensate
all jobs, while it has to be Bayes-Nash incentive compatible. We derive closed formulae for the optimal
mechanism in the 1-d case and show that it is efficient for symmetric jobs. For non-symmetric jobs,
we show that efficient mechanisms perform arbitrarily bad. For the 2-d case, we prove that the optimal
mechanism in general does not even satisfy IIA, the `independent of irrelevant alternatives' condition.
Hence any attempt along the lines of the classical auction setting is doomed to fail. In the 2-d case, we
also show that the optimal mechanism is not even efficient for symmetric agents.