Envy-Free Makespan Approximation

We study envy-free mechanisms for assigning tasks to agents, where every task may take a different amount of time to perform by each agent, and the goal is to get all the tasks done as soon as possible (i.e., minimize the makespan). For indivisible tasks, we put forward an envy-free polynomial mechanism that approximates the minimal makespan to within a factor ofO(logm), wherem is the number of machines. This bound is almost tight, as we also show that no envy-free mechanism can achieve a better bound than Ω(logm/ log logm). This improves the recent result of Mu’alem [24] who introduced the model and gave an upper bound of (m+1)/2, and a lower bound of 2−1/m. For divisible tasks, we show that there always exists an envy-free poly-time mechanism with optimal makespan. Finally, we demonstrate how our mechanism for envy free makespan minimization can be interpreted as a market clearing problem. ∗AT&T Labs-Research, 180 Park Avenue, Florham Park, NJ. †School of Business Administration and Center for the Study of Rationality, The Hebrew University of Jerusalem. ‡The Blavatnik School of Computer Science, Tel Aviv University. §The Blavatnik School of Computer Science, Tel Aviv University. ¶The Blavatnik School of Computer Science, Tel Aviv University.