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We here address the problem of fairly allocating indivisible goods or choresto $n$ agents with weights that define their entitlement to the set ofindivisible resources. Stemming from well-studied fairness concepts such asenvy-freeness up to one good (EF1) and envy-freeness up to any good (EFX) foragents with equal entitlements, we present, in this study, the first set ofimpossibility results alongside algorithmic guarantees for fairness amongagents with unequal entitlements.  Within this paper, we expand the concept of envy-freeness up to any good orchore to the weighted context (WEFX and XWEF respectively), demonstrating thatthese allocations are not guaranteed to exist for two or three agents. Despitethese negative results, we develop a WEFX procedure for two agents with integerweights, and furthermore, we devise an approximate WEFX procedure for twoagents with normalized weights. We further present a polynomial-time algorithmthat guarantees a weighted envy-free allocation up to one chore (1WEF) for anynumber of agents with additive cost functions. Our work underscores theheightened complexity of the weighted fair division problem when compared toits unweighted counterpart.

Almost Envy-Free Allocations of Indivisible Goods or Chores with Entitlements