On social envy-freeness in multi-unit markets

We consider a market setting in which buyers are individuals of a population, whose relationships are represented by an underlying social graph. Given buyers’ valuations for the items being sold, an outcome consists of a pricing of the objects and an allocation of bundles to the buyers. An outcome is social envy-free if no buyer strictly prefers the bundles of her neighbors in the social graph. We focus on the revenue maximization problem in multi-unit markets, in which there are multiple copies of the same item being sold, and each buyer is assigned a subset of identical items. We consider four different cases that arise when considering two different buyers valuations, i.e., single-minded or general, and by adopting two different forms of pricing, that is item- or bundle-pricing. For all the above cases we show the hardness of the revenue maximization problem and give corresponding approximation results. All our approximation bounds are optimal or nearly optimal. Moreover, under the assumption of social graphs of bounded treewidth, we provide an optimal allocation algorithm for general valuations with item-pricing. Finally, we determine optimal bounds on the corresponding price of envy-freeness, that is on the worst-case ratio between the maximum revenue that can be achieved without envy-freeness constraints, and the one obtainable in case of social relationships. Some of our results close hardness open questions or improve already known ones in the literature concerning the standard setting without sociality.