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.

公平分配中重要但尚未充分研究的主题之一是代理商之间的外部性效应。对于资源分配问题，外部性意味着分配给代理的份额可能会影响其他代理的效用。在本文中，我们对具有正外部性的不可分割商品的公平分配进行了研究。受网络扩散背景下的模型的启发，我们提出了一个简单自然的模型，即网络外部性，以捕获外部性。为了评估网络外部性模型中的公平性，我们对maximin-share（\（\ mathsf {MMS} \））概念背后的思想进行了概括，以实现一个新的标准，即扩展maximin-share（\（\ mathsf { EMMS} \））。接下来，我们考虑关于模型的两个问题。首先，我们讨论计算每个代理的\（\ mathsf {EMMS} \）值的计算方面。为此，我们引入了分区问题的一种广义形式，其中包括许多著名的分区问题，例如maximin，minimax和leximin。我们进一步表明存在1/2近似算法用于此分区问题。接下来，我们研究近似的\（\ mathsf {EMMS} \）分配，即保证每个代理使用至少其扩展极大值份额的效用的分配。我们证明，在自然假设代理是\（\ alpha \） -自力更生的情况下，\（\ alpha / 2 \） - \（\ mathsf {EMMS} \）分配始终存在。结合前一个结果，得出多项式时间\（\ alpha / 4 \） - \（\ mathsf {EMMS} \）分配算法。