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.
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Notes
Note that these exchanges are only to prove the lemma, and not in the algorithm.
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Seddighin, M., Saleh, H. & Ghodsi, M. Maximin share guarantee for goods with positive externalities. Soc Choice Welf 56, 291–324 (2021). https://doi.org/10.1007/s00355-020-01278-8
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DOI: https://doi.org/10.1007/s00355-020-01278-8