Fair Allocation of Indivisible Items With Externalities

One of the important yet insufficiently studied subjects in fair allocation is the externality effect among agents. For a resource allocation problem, externalities imply that a bundle allocated to an agent may affect the utilities of other agents. In this paper, we conduct a study of fair allocation of indivisible goods when the externalities are not negligible. We present a simple and natural model, namely network externalities, to capture the externalities. To evaluate fairness in the network externalities model, we generalize the idea behind the notion of maximin-share (MMS) to achieve a new criterion, namely, extended-maximin-share (EMMS). Next, we consider two problems concerning our model. First, we discuss the computational aspects of finding the value of EMMS for every agent. For this, we introduce a generalized form of partitioning problem that includes many famous partitioning problems such as maximin, minimax, and leximin partitioning problems. We show that a 1/2-approximation algorithm exists for this partitioning problem. Next, we investigate on finding approximately optimal EMMS allocations. That is, allocations that guarantee every agent a utility of at least a fraction of his extended-maximin-share. We show that under a natural assumption that the agents are α-self-reliant, an α/2-EMMS allocation always exists. The combination of this with the former result yields a polynomial-time α/4-EMMS allocation algorithm.