Mechanism design with uncertain inputs (to err is human, to forgive divine)

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 {\em fair share} of a player, and design mechanisms that are $\mathrm{\Omega }(1)$-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 $\mathrm{\Omega }(1)$-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.