Two Dimensional Optimal Mechanism Design for a Sequencing Problem

Abstract

We propose an optimal mechanism for a sequencing problem where the jobs' processing times and waiting costs are private. Given public priors for jobs' private data, we seek to find 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, 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, answering an open problem in [13]. 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 decompose feasible interim schedules. In addition, in computational experiments with random instances, we generate some more insights.

Efficient implementation of Carathéodory’s theorem for the single machine scheduling polytope

“…In such settings, some of the data, such as job processing times, are private to the jobs. One approach for computing BayesNash optimal mechanisms that recently has received attention is to use linear programming relaxations for the so-called reduced form of the mechanism [1,7]. Such relaxations yield so-called interim solutions, which are (interior) points of a certain polytope.…”

“…Specifically, for the single machine scheduling problem where jobs have private data, the last step to implement a mechanism requires the solution of the decomposition problem of the single machine scheduling polytope. We refer to [7] for a detailed discussion.…”

“…An O(n 9 ) algorithm follows directly from work by Fonlupt and Skoda [4] on the intersection of a line with an arbitrary polymatroid and using the GLS method. However, a closer look reveals that an O(n 3 log n) implementation is possible for the scheduling polytope [7]. Still, this result is unsatisfactory in the following sense.…”

Mechanisms for scheduling games with selfish players

“…This requires a decomposition of a point in the scheduling polytope into a convex combination of vertices, which can be done efficiently too. The results from Chapter 3 are published as [35,36].…”

“…Our algorithm not only generalizes this to the scheduling polytope, but also adds a completely new, geometric interpretation. The different parts from Chapter 5 are published as [35,36] and [37,38].…”