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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 schedulerneeds to finish as many jobs as possible by the deadline. For this simpleproblem, there is a truthful mechanism that achieves maximum welfare indominant strategies. The new aspect of our work is that in our setting playersare uncertain about their own job lengths, and hence are incapable of providingtruthful reports (in the strict sense of the word). For a probabilistic modelfor uncertainty our main results are as follows.  1) Even with relatively little uncertainty, no mechanism can guarantee aconstant fraction of the maximum welfare.  2) To remedy this situation, we introduce a new measure of economicefficiency, based on a notion of a {\em fair share} of a player, and designmechanisms that are $\Omega(1)$-fair. In addition to its intrinsic appeal, ournotion of fairness implies good approximation of maximum welfare in severalcases of interest.  3) In our mechanisms the machine is sometimes left idle even though there arejobs that want to use it. We show that this unfavorable aspect is unavoidable,unless one gives up other favorable aspects (e.g., give up$\Omega(1)$-fairness).  We also consider a qualitative approach to uncertainty as an alternative tothe probabilistic quantitative model. In the qualitative approach we break awayfrom solution concepts such as dominant strategies (they are no longer welldefined), and instead suggest an axiomatic approach, which amounts to listingdesirable properties for mechanisms. We provide a mechanism that satisfiesthese properties.

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