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.
Assigning Resources to Budget-Constrained Agents.Review of Economic Studies, 80(1):73–107, February 2013
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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.