EF1 for Mixed Manna with Unequal Entitlements

We study fair division of indivisible mixed manna when agents have unequal entitlements, with weighted envy-freeness up to one item (WEF1) as our primary notion of fairness. We identify several shortcomings of existing techniques to achieve WEF1. Hence, we relax WEF1 to weighted envy-freeness up to 1 transfer (WEF1T), and give a polynomial-time algorithm for achieving it. We also generalize Fisher markets to the mixed manna setting, and use them to get a polynomial-time algorithm for two agents that outputs a WEF1 allocation.