Simple mechanisms achieve arbitrary approximation to optimal revenue for bounded-support distributions with private budgets but cannot guarantee any positive fraction of optimal revenue for unbounded supports or certain bounded distributions, even under independence.
Optimal deterministic mechanisms for an additive buyer
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Evaluating the Performance of Approximation Mechanisms under Budget Constraints
Simple mechanisms achieve arbitrary approximation to optimal revenue for bounded-support distributions with private budgets but cannot guarantee any positive fraction of optimal revenue for unbounded supports or certain bounded distributions, even under independence.