Achieving Envy-Freeness through Items Sale

We consider a fair division setting of allocating indivisible items to a set of agents. In order to cope with the well-known impossibility results related to the non-existence of envy-free allocations, we allow the option of selling some of the items so as to compensate envious agents with monetary rewards. In fact, this approach is not new in practice, as it is applied in some countries in inheritance or divorce cases. A drawback of this approach is that it may create a value loss, since the market value derived by selling an item can be less than the value perceived by the agents. Therefore, given the market values of all items, a natural goal is to identify which items to sell so as to arrive at an envy-free allocation, while at the same time maximizing the overall social welfare. Our work is focused on the algorithmic study of this problem, and we provide both positive and negative results on its approximability. When the agents have a commonly accepted value for each item, our results show a sharp separation between the cases of two or more agents. In particular, we establish a PTAS for two agents, and we complement this with a hardness result, that for three or more agents, the best approximation guarantee is provided by essentially selling all items. This hardness barrier, however, is relieved when the number of distinct item values is constant, as we provide an efficient algorithm for any number of agents. We also explore the generalization to heterogeneous valuations, where the hardness result continues to hold, and where we provide positive results for certain special cases.