Almost budget-balanced VCG mechanisms to assign multiple objects

Subsidy-free VCG mechanisms assign p identical objects to n agents. The efficiency loss is the largest ratio of budget surplus to efficient surplus, over all profiles of non-negative valuations. The smallest efficiency loss satisfies . If is bounded away from , converges to zero exponentially in n. Participation is voluntary in the optimal mechanism achieving if p=1, but not if p[greater-or-equal, slanted]2. Among voluntary mechanisms, the optimal efficiency loss is not significantly larger than if . But it does not converge to zero in n if .