A lottery over two-agent allocations can be simultaneously ex-ante envy-free, ex-post EFX, and ex-post Pareto optimal, but this becomes impossible for three agents with four goods.
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Best-of-Both-Worlds Fairness and Pareto Optimality
A lottery over two-agent allocations can be simultaneously ex-ante envy-free, ex-post EFX, and ex-post Pareto optimal, but this becomes impossible for three agents with four goods.