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 Lˆ(n,p) satisfies Lˆ(n,p)⩽Lˆ(n,⌊n2⌋)⩽43n. If pn is bounded away from 12, Lˆ(n,p) converges to zero exponentially in n.Participation is voluntary in the optimal mechanism achieving Lˆ(n,p) if p=1p=1, but not if p⩾2p⩾2. Among voluntary mechanisms, the optimal efficiency loss is not significantly larger than Lˆ(n,p) if pn⩽12. But it does not converge to zero in n   if pn>12.