We consider a task of scheduling with a common deadline on a single machine. Every player reports to a scheduler the length of his job and the scheduler needs to finish as many jobs as possible by the deadline. For this simple problem, there is a truthful mechanism that achieves maximum welfare in dominant strategies. The new aspect of our work is that in our setting players are uncertain about their own job lengths, and hence are incapable of providing truthful reports (in the strict sense of the word). For a probabilistic model for uncertainty our main results are as follows. 1) Even with relatively little uncertainty, no mechanism can guarantee a constant fraction of the maximum welfare. 2) To remedy this situation, we introduce a new measure of economic efficiency, based on a notion of a fair share of a player, and design mechanisms that are @math -fair. In addition to its intrinsic appeal, our notion of fairness implies good approximation of maximum welfare in several cases of interest. 3) In our mechanisms the machine is sometimes left idle even though there are jobs that want to use it. We show that this unfavorable aspect is unavoidable, unless one gives up other favorable aspects (e.g., give up @math -fairness). We also consider a qualitative approach to uncertainty as an alternative to the probabilistic quantitative model. In the qualitative approach we break away from solution concepts such as dominant strategies (they are no longer well defined), and instead suggest an axiomatic approach, which amounts to listing desirable properties for mechanisms. We provide a mechanism that satisfies these properties.
A relatively small part of the literature on scheduling in game theoretic settings deals with deadlines. Porter @cite_21 studies a problem of online scheduling of jobs on a single processor. Each job is characterized by a release time, a deadline, a processing time, and a value for successful completion by its deadline. Monetary transfers are used in order to lead agents to a relatively efficient behavior. On the other hand the work by Cres and Moulin @cite_7 deals with a scheduling domain, where deadlines (rather than job lengths) are private information, and compares two scheduling policies that do not involve money. As in all other work on scheduling that we are aware of, a player does not have uncertainty about his private parameters.
@cite_21 @cite_7
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For the problem of online real-time scheduling of jobs on a single processor, previous work presents matching upper and lower bounds on the competitive ratio that can be achieved by a deterministic algorithm. However, these results only apply to the non-strategic setting in which the jobs are released directly to the algorithm. Motivated by emerging areas such as grid computing, we instead consider this problem in an economic setting, in which each job is released to a separate, self-interested agent. The agent can then delay releasing the job to the algorithm, inflate its length, and declare an arbitrary value and deadline for the job, while the center determines not only the schedule, but the payment of each agent. For the resulting mechanism design problem (in which we also slightly strengthen an assumption from the non-strategic setting), we present a mechanism that addresses each incentive issue, while only increasing the competitive ratio by one. We then show a matching lower bound for deterministic mechanisms that never pay the agents. In a scheduling problem where agents can opt out, we show that the familiar random priority RP mechanism can be improved upon by another mechanism dubbed probabilistic serial PS. Both mechanisms are nonmanipulable in a strong sense, but the latter is Pareto superior to the former and serves a larger expected number of agents. The PS equilibrium outcome is easier to compute than the RP outcome; on the other hand, RP is easier to implement than PS. We show that the improvement of PS over RP is significant but small: at most a couple of percentage points in the relative welfare gain and the relative difference in quantity served. Both gains vanish when the number of agents is large; hence both mechanisms can be used as a proxy of each other.