  Fair division of indivisible goods is a central challenge in artificial
intelligence. For many prominent fairness criteria including envy-freeness (EF)
or proportionality (PROP), no allocations satisfying these criteria might
exist. Two popular remedies to this problem are randomization or relaxation of
fairness concepts. A timely research direction is to combine the advantages of
both, commonly referred to as Best of Both Worlds (BoBW). We consider fair
division with entitlements, which allows to adjust notions of fairness to
heterogeneous priorities among agents. This is an important generalization to
standard fair division models and is not well-understood in terms of BoBW
results. Our main result is a lottery for additive valuations and different
entitlements that is ex-ante weighted envy-free (WEF), as well as ex-post
weighted proportional up to one good (WPROP1) and weighted transfer envy-free
up to one good (WEF(1,1)). It can be computed in strongly polynomial time. We
show that this result is tight - ex-ante WEF is incompatible with any stronger
ex-post WEF relaxation. In addition, we extend BoBW results on group fairness
to entitlements and explore generalizations of our results to instances with
more expressive valuation functions.
