Efficient, Strategy-proof and almost budget-balanced assignment

Call a Vickrey-Clarke-Groves (VCG) mechanism to assign p identi-cal objects among n agents, feasible if cash transfers yield no deficit. The efficiency loss of such a mechanism is the worst (largest) ratio of the budget surplus to the efficient surplus, over all profiles of non neg-ative valuations. The optimal (smallest) efficiency loss eL(n, p) satisfies eL(n, p) ≤ eL(n, {n