Ex ante equitable, ex post EQ1 randomized allocations always exist for two agents, but may not exist for three or more agents and their existence decision is NP-complete.
Designing random allocation mechanisms: Theory and applications
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Best of Both Worlds Guarantees for Equitable Allocations
Ex ante equitable, ex post EQ1 randomized allocations always exist for two agents, but may not exist for three or more agents and their existence decision is NP-complete.