A characterization of 2-player mechanisms for scheduling

TL;DR

This paper characterizes all truthful mechanisms for two-player scheduling with positive and negative values, showing they are limited to task partitioning and affine minimizers, with a maximum approximation ratio of 2.

Contribution

It provides a complete characterization of decisive truthful mechanisms for two-player scheduling, including their structure and approximation bounds, extending insights to multi-player scenarios.

Findings

Truthful mechanisms partition tasks into groups for independent allocation.

Maximum approximation ratio for two players is 2, with VCG being the unique optimal.

Threshold mechanisms are equivalent to additive mechanisms.

Abstract

We study the mechanism design problem of scheduling unrelated machines and we completely characterize the decisive truthful mechanisms for two players when the domain contains both positive and negative values. We show that the class of truthful mechanisms is very limited: A decisive truthful mechanism partitions the tasks into groups so that the tasks in each group are allocated independently of the other groups. Tasks in a group of size at least two are allocated by an affine minimizer and tasks in singleton groups by a task-independent mechanism. This characterization is about all truthful mechanisms, including those with unbounded approximation ratio. A direct consequence of this approach is that the approximation ratio of mechanisms for two players is 2, even for two tasks. In fact, it follows that for two players, VCG is the unique algorithm with optimal approximation 2. This characterization provides some support that any decisive truthful mechanism (for 3 or more players) partitions the tasks into groups some of which are allocated by affine minimizers, while the rest are allocated by a threshold mechanism (in which a task is allocated to a player when it is below a threshold value which depends only on the values of the other players). We also show here that the class of threshold mechanisms is identical to the class of additive mechanisms.