Mathematical Programming Approach to Multidimensional Mechanism Design for Single Machine Scheduling

Abstract

We consider the following optimal mechanism design problem in single machine scheduling. Given are n jobs which we regard as selfish agents. Jobs need to be processed non-preemptively on a single machine. Each job j has processing time pj and a disutility wj for waiting one unit of time. This defines the 2-dimensional type tj = (wj, pj) of a job j. The allocation rule f is a mapping of type profiles t = (tj)j=1,...,n to the set of all n! sched-ules. If in a given schedule σ, Sj(σ) denotes the starting time of job j, the jobs valuation for that schedule is −wjSj(σ). We assume that the mechanism designer needs to compensate jobs for waiting in the form of a payment pij, such that pij−wjSj ≥ 0. We consider this problem in a Bayes-Nash setting, that is, given are publicly known, discrete probability distributions describ-ing the jobs ’ possible types. Our goal is to find a (Bayes-Nash) incentive compatible mechanism that is optimal, which is a mechanism that minimizes the total expected payment j Epij. This Problem has been considered earlier in a paper by Heydenreich et al. [3], where mainly the special case of 1-dimensional type spaces has been analyzed (that is, public processing times pj and private wj). Also in that paper, an example has been proposed to show tha

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