Abstract
Strategy-proof, budget-balanced, and envy-free rank mechanisms assign q identical objects to n agents. The efficiency loss is the largest ratio of surplus loss to efficient surplus, over all profiles of non-zero valuations. The smallest efficiency loss $$\frac{n-q}{n^{2}-n}$$ is uniquely achieved by the following simple allocation rule: assign one object to each of the $$q-1$$ agents with the highest valuations, a large probability to the agent with the qth highest valuation, and the remaining probability to the agent with the $$(q+1)$$ th highest valuation.
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