Fair Division in a Variable Setting

We study the classic problem of fairly dividing a set of indivisible items among a set of agents and consider the popular fairness notion of envy-freeness up to one item (EF1). While in reality, the set of agents and items may vary, previous works have studied static settings, where no change can occur in the system. We initiate and develop a formal model to understand fair division under a variable input setting: here, there is an EF1 allocation that is disrupted due to the loss/deletion of some item(s), or the arrival of new agent(s). The objective is to regain EF1 by performing a sequence of valid transfers of items between agents - no transfer creates any new EF1-envy in the system. We call this the EF1-Restoration problem.In this work, we develop efficient algorithms for the EF1-Restoration problem when the agents have identical monotone valuations and the items are either all goods or all chores. Both of these algorithms achieve an optimal number of transfers (at most $km/n$, where $m$, $n$, and $k$ denote the number of items, agents, and EF1-unhappy agents respectively) for identical additive valuations.Next, we consider a valuation class with graphical structure, introduced by Christodoulou et al. (EC 2023), where each item is valued by at most two agents, and can be seen as an edge between these two agents in a graph. Here, we consider EF1 orientations on multigraphs - allocations where each item is allocated to an agent who values it. When the valuations are also additive and binary, we present an optimal algorithm for the EF1-Restoration problem. We also consider pairwise-homogeneous graphical valuations (all items between a pair of agents are valued the same), and develop an optimal algorithm when the graph is a multipath.Finally, for monotone binary valuations, we show that the problem of deciding whether EF1-Restoration is possible is PSPACE-complete.