Auctioning or assigning an object: some remarkable VCG mechanisms

We construct a variant of the Vickrey auction of a single object where the surplus is split in exogenously fixed shares between the seller and the buyers, up to a margin of error vanishingly exponentially as the number of buyers grows. When the object is the common property of the participants, we can similarly construct VCG mechanisms with a vanishingly small cash transfer to the residual claimant. For any integer q, 3 ≤ q ≤ n, we find the mechanism guaranteeing to each participant a fair share of the qth highest valuation, while minimizing the worst possible ratio of the cash transfer to the efficient surplus. We perform a parallel analysis when the object is undesirable. We compare the cash lost to the largest spread between individual valuations, and obtain the same trade-offs between fairness and the relative loss of surplus