Fair by design: Multidimensional envy-free mechanisms

- Fair by design attempts solving optimization problems in an envy-free manner.
- We provide tight bounds on envy-free mechanisms for makespan minimization.
- Multidimensional envy-freedom is a far less restrictive constraint than truthfulness.

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.A natural question to ask is whether envy-free pricing techniques can improve the current known bounds for truthful mechanisms for the task-scheduling problem studied in the seminal paper of Nisan and Ronen (2001). We find that for two unrelated machines, envy-free in-expectation is a far weaker constraint (i.e. less restrictive) than truthful in-expectation.