Fair by design: Multidimensional envy-free mechanisms

We address the common scenario where a group of agents wants to divide a set of items fairly, and at the same time seeks to optimize a global goal. Suppose that each item is a task and we want to find an allocation that minimizes the completion time of the last task in an envy-free manner, where no agent prefers anyone else's allocated task bundle over its own. This optimization goal is called makespan minimization, and the agents are often treated as machines. We give tight deterministic bounds for: (1) two unrelated machines; and (2) m≥2 related machines.