Maximin share guarantee for goods with positive externalities

Abstract

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 the share allocated to an agent may affect the utilities of other agents. In this paper, we conduct a study of fair allocation of indivisible goods with positive externalities. Inspired by the models in the context of network diffusion, 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 ( $$\mathsf {MMS}$$ ) to achieve a new criterion, namely, extended-maximin-share ( $$\mathsf {EMMS}$$ ). Next, we consider two problems concerning our model. First, we discuss the computational aspects of finding the value of $$\mathsf {EMMS}$$ 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. We further show that a 1/2-approximation algorithm exists for this partitioning problem. Next, we investigate approximate $$\mathsf {EMMS}$$ allocations, i.e., allocations that guarantee each agent a utility of at least a fraction of his extended-maximin-share. We show that under a natural assumption that the agents are $$\alpha$$ -self-reliant, an $$\alpha /2$$ - $$\mathsf {EMMS}$$ allocation always exists. This, combined with the former result yields a polynomial-time $$\alpha /4$$ - $$\mathsf {EMMS}$$ allocation algorithm.