  We study the classic problem of dividing a collection of indivisible
resources in a fair and efficient manner among a set of agents having varied
preferences. Pareto optimality is a standard notion of economic efficiency,
which states that it should be impossible to find an allocation that improves
some agent's utility without reducing any other's. On the other hand, a
fundamental notion of fairness in resource allocation settings is that of
envy-freeness, which renders an allocation to be fair if every agent (weakly)
prefers her own bundle over that of any other agent's bundle. Unfortunately, an
envy-free allocation may not exist if we wish to divide a collection of
indivisible items. Introducing randomness is a typical way of circumventing the
non-existence of solutions, and therefore, allocation lotteries, i.e.,
distributions over allocations have been explored while relaxing the notion of
fairness to ex-ante envy freeness.
  We consider a general fair division setting with $n$ agents and a family of
admissible $n$-partitions of an underlying set of items. Every agent is endowed
with partition-based utilities, which specify her cardinal utility for each
bundle of items in every admissible partition. In such fair division instances,
Cole and Tao (2021) have proved that an ex-ante envy-free and Pareto-optimal
allocation lottery is always guaranteed to exist. We strengthen their result
while examining the computational complexity of the above total problem and
establish its membership in the complexity class PPAD. Furthermore, for
instances with a constant number of agents, we develop a polynomial-time
algorithm to find an ex-ante envy-free and Pareto-optimal allocation lottery.
On the negative side, we prove that maximizing social welfare over ex-ante
envy-free and Pareto-optimal allocation lotteries is NP-hard.
