Two dimensional optimal mechanism design for a sequencing problem

We consider optimal mechanism design for a sequencing problem with $n$ jobs which require a compensation payment for waiting. The jobs' processing requirements as well as unit costs for waiting are private data. Given public priors for this private data, we seek to find an optimal mechanism, that is, a scheduling rule and incentive compatible payments that minimize the total expected payments to the jobs. Here, incentive compatible refers to a Bayes-Nash equilibrium. While the problem can be efficiently solved when jobs have single dimensional private data along the lines of a seminal paper by Myerson, we here address the problem with two dimensional private data. We show that the problem can be solved in polynomial time by linear programming techniques. Our implementation is randomized and truthful in expectation. The main steps are a compactification of an exponential size linear program, and a combinatorial algorithm to compute from feasible interim schedules a convex combination of at most n deterministic schedules. In addition, in computational experiments with random instances, we generate some more theoretical insights.