Weighted Envy-free Allocation with Subsidy

We consider the problem of fair allocation of indivisible items with subsidies when agents have weighted entitlements. Specifically, we extend the envy-freeability studied in the unweighted case to the weighted envy-freeability and delve deeper into its properties. We first highlight various important differences from the unweighted case, e.g., the sufficient conditions that lead to envy-freeability in the unweighted case do not lead to weighted envy-freeability in the weighted case. We then present various results concerning weighted envy-freeability including general characterizations, algorithms for achieving and testing weighted envy-freeability, lower and upper bounds of the amount of subsidies for weighted envy-freeable allocations. Additionally, we design algorithms that ensure weighted envy-freeability while incorporating other fairness properties, such as weighted envy-freeness up to one item transfer.