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